Chow - Open Channel Hydraulics

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MeGRA W-HItr, CIVIL ENGLNEERING SERIES HARMER E. DAVIS, Consulting Editor

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OPEN-CHANNEL )

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BABBI'IT '. Engineering.in Public Health BENJAMIN' Statically Indeterminate St~uctures Cnow . Open,-cha,nnel Hyqraulics DAVIS, TROXELL,'AND WrsKoCIL . Tl1e Testing and Inspection of ' Engineering Materials DUNl'iAM . Foundations of Structures DUNHAM' The Theory and Practice of Reinforced Concrete DUNHAM AND YOUNG.' Contracts, Specifications, and Law for Engineers GAYLORD AND GAYLORD' Structural Design HALLERT 'Photogrammetry HENNES AND EKSE . Fundamentals of Transportation Engineering KRYNINE AND JUDD' Principles of Engineering Geology and Geot.echnics LINSLEY AND FRANZINI . Elements of Hydraulic Engineering

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VEN TE CHOW, Ph.D.

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Proiessot of Hydtaulic En(finl!erin(f University of Illinois

INTERNATIONAL STUDENT EDITION

LmsLIDY, KOHLER, AND I'A ULHUB ' Applied Hydrology LINSLEY, KOHLER, AND PAULHUS' Hydrology f9r Engineers LU:8DER . Aerial Photographic Interpretation MA'l'SON, SMITH, AND HURD' Traffic Engineering MEAD, MEAD, AND AKERMAN' Contracts, Specifications, and Engineering Relations NORRIS, HANSEN, HOLLEY, BIGGS, NAMYET, AND 1fINAMI . :Structural Design for Dyiramic Loads PEURIFOY' Construction Planning, Equipment, and Methods' PEURIFOY' gstimating Constructi()u Costs • TROXELL AND DAVIS' Composition and Properties of Concrete TSCHEBOTARIOFF . Soil Mechanics, Foundations, and Earth Structures

HYDR~AULICS

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URQUHART, O'ROURl<::t, AND WINTER' Design Concrete Structures WANG AND ECKEr. . Elementary Theory of Structures

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OPEN-CHANNEL HYDRAULICS

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Exclusive rights by Kogakusha Co., Ltd., for manufacture and export from _. Japan. This book cannot be re-exported from the cou\ltry to which it is consigiled by Kogakllsha Co., Ltd., or by McGraw-Hill Boak Company, Inc., or any .of its subsidiaries.

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Copyright © 1-959 by the McGraw-Hill Book Company, Inc. All rights reserved. This bo"k, or parts thereof, may not be reprodlJced in any for~ without pClmissioll of the publishers. Libra,,' of COI1g1·es.r Catalog Card Number 58-13860

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In recent years water-resources projects and hydraulic engineering works' have been developing rapidly throughout the world. The knowledge of .open-chp,nnel hy~raulics, which is essential to the de~ign of many hydraulic structures, has thus advanced by leaps and bounds: To the students and engineers in the field of hydraulic engineering, such valuable new knowledge should be .made available in suitable book form. It is therefore not that some 'new books have already appeared. Howi:lver, most of them a.re present-ed in limited scope and 1111 2.re written in foreign In the· English langu!l.ge, the t\>,ro well-known books, respectively' by Bakhmeteff and by 1VoodwI1l'd and Posey, were published nearly two decades ago. 2 This book gives broad coverage.of recent developments; it should meet the present Ilead. It is designed as a. textbook fo\, both undergrachui.te and graduate stuuents .and also as a compendium for practicing <:;,ll'=>lU"'" Emphasis is given to the qualities of "teachability" and" practicability," and t1tt,empts were made in preaenting the material to bridge the gap which is generally to exist between the theory and the practice. In order to achieve these objectives, the use of .advanced mathematics is deliberately avoided as much as possible, and the exp~a.nation of hydraulic 1 Such. as: Etienne Orausse, "Hydrauliquc des can~ux decouvel·ts en regime permanent" (" Hydraulios of Open Channeld with Steady Flow"), Editions Eyrolles, Paris, 1951; R. Silber, "Etude at trace des ecouleroents permanents en canaUl( et rivieres" ("Study and Sket
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PREFACE

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theories is greatly simplified as far as practicable, ~Illustrativ.e example~ . are to show the application of the theories, and practical problems are provided Jor exercises. Furthermore, short historical accounts are given in footnotes in order to stimulate the reader's interest, and ample references are supplied for his independent .studies. Some references, hO'\'8ver, may not be readily available the reader, but they are liste:d for academicaud historical intercst. In essence, the book is the outgrowth of the author's 20 year,,' experience as a student,teacher, engineer, rese.iucher, and consultant in' the field of hydra\llic engineeTing, The manuscript of the book was drafted for the first time in the academic year of 1951-1952 for use in teaching the students of civil, agricultural, and mechanical engineering and of .. theoretical' ahd applied mechanics at the University of Illinois. Since then several revisions have been made. In .the beginning, the material was prepared solely for graduate students. Owing to the general demand for a book on the design of hydraulic structures for undergraduate studies, the manuscript was expanded to include more fundaments;! principles and design procedures. At the sa.me time, most of the advanced mathematics and theories were either omitted or replaced by more practic.'l..l approaches using mathematical operations of a level not higher than calculus.. Fro!!. 1951 to 1955, the author made several special visits to many major engineering agenCies and firms in the United Stat,es tQ discuss problems with their engineers. As a result, a Yl'.st fund of information on hydraulic design pnictices was collected and incorporated into the manuscript. Thereafter, the author also visited many hydraulic institutions and laboratories in other countries and exchanged knowledge with their staff members..In 1956 he visited England, France, Belgium, the Netherlands,.Germany, Italy,
to

l'REFACE

ix

In Part I on basic principles, the type of flow in open channels is cla~i-· fied according to the vliriation in the parrunetersof flow with respf.ct to space and time. For simplicity, the depth of flow is used as the flow parameter in the classification. The state of flow is classified according to the range of the invariants of flow with r~spect to viscosity and gravity. The flow invariants used are the Reynolds number and the Froude number. Since the effect of surface tension of water is insignificant l.n most engineering problems, the Weber nU!l1b~r as it flow invariant is not introduced. In fact, the state of fiow c~n be further classified for its stability in u.ccordu.nce with the Vedernikov number or other suitahlf?l:rlt!:!rla. However, ::mch a criterion h.as not been weli established in engil1eering praatice, and therefore it is taken up only briefly later in Chap,8. FOUl' coefficients for velocity and pressure distributions are introduced, particular, the f.mel'gy coefficient is presented throughout the book. This coefficient is usuHlly ignored ill most books on hydraulics. In practical applications, the effect of the energy coefficient on computations and hence on is quite significant f:Ll1d therefore should not be overlooked, e,reHl though the value of the coefficient miJ,y not always be determined accurately. The energy and momE,ntum principles constitute the basis of interpretation for most hydraulic phenomena. A thorough treatment of the two prfnclples is in Chap. 3. SilIce the book is intended for the use of practicing engineel's, the treatment of a problem i~ in m013t cases based on a ollEl- or two-dimensional flow: . In Part II 011 \ll1ifol'm flow, eeveral uniform-flow formulas are. introduced. ,Despite many new proposals for a formula having a theore~ical background, the Manning formula still holds its indisputable top position in the field of practical a;pplicatioI)s. This formula. is therefore used extensively in the book. In certain specific problems, however, the Che2iY formula is used occasionally. ' . The design for uniform flo\v covers nonerodible, erodible, and grassed channels. The erodible channels in general may be classified under three types: chapl1els which scour but do ilOt silt, channels which silt but do not ~cour, and channels which ;icour and silt simultlllleously. In channels of tlie second and third types, it is nec€6sary for the walter to carry sediments, As will be stated later, the sediment transportation il'; consid~red as a subject in the domain of river hydraulics, Therefore, only the channels: of the first type, which :carry relatively clear wa~er in stable condition, are treated in this bodk ,. Iii Part III ongl'adually va-ried flow, se,,-eral methods for the computation of flow profi1.~s are discussed. .A new method of direct integration is introduced which requires the use of a, varied-flow function table first

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developed by Professor Boris A~ Bakhmeteff in 1912. 1. The table given in Appendix D of th IX. book ]s an ~xten8ion of the table to n~arly: three times its original size. This extended table \1nd a table for negative slopes were prepared during 1952 to 1954 by the author for teaching purposes at the : j Universjty of Illinois,2 For the computation of fiow profiles in circular , e) conduits, a varied-ft.ow function table is also· provided in Appendix E. : The method of singular point is a powerful tool for the analysis of flow , • profiles. Since this method requires the use of advancedmatheml1tics} it is described only briefly in Chap. 9 for the purpose of st.imulating £ur. the I' interest in the theoretical study of flow problems. . alit In Part IV on rapidly varied fiow, the treatmellt of the problems is 1 I largely supported by experimental data, because this t.ype of flow is so ~:, complicated that a mere theoretical analysis. in most cases will not yield sufficient information fO!' the purpose of practical design. The use of the ~!i fiow-n!;i metllod and the method of characteristics is mentioned but no . details are given, becallse t.he former is so popular that it, can be found .. in most hydraulics books; while the latter requires t.he knowledge .0£ i I advanced Tiiathematics beyond the scope of this wbrk. In Pnrt V on llnsteady:fiow, the keatment is general but practical. It \ Iii I·1 should be recognized that this type of f10wia a highly specialized subj ect. 3 The knowledge of advanced mathematics would be required if a compre. hensive treatment were given. i It should be noted that the subject matter of this book dwells mainly t/) on the flow of water in channels where water contains little foreign material.· Consequently, problems related to sediment trAnsportation and I)) air entnl.inment are not fully discussed. In recent years, sedirrient tr!lnsi portatim in channels has become abroad subject that is generally COvered -'~ a1 in the study of river hydraulics, which is often treated iI).depc:mdently.1 •

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1 Boris A. Bakh~ete.ff,"O Neravnomernom Dvizhenii Zhidkoati v Otkrytorn Rusle" {"Varip.d Flow in Open Channels"),·St. Petersburg, Russia, 1912. , Ven Te Chow, Integratmg the eC\uation of gra~ually varied /lOw, paper no. 838, Proc!;ecHngs, A.men·caT' Society of Civil ETl{/in~er3, vol. 81, pp. 1-32, November} 1055. Closing disc~ssioil by the author in J01trnal of HydrauHcs Division,vol. 83, no. HY1, paper no. 1177, pp. 9·-22, February, 195;'. : ' • Spedal references are: J: J. Stoke". "W';'ter Waves," vol. IV of "Pilre and Applied Mathenlatics," Interscience Publishers, New Yorle, 1957; V. A. ArkhangelskiI, "Ra~chety Neustanovivsh~gosia pvizheniill. v Otkrytykh Vo40tokakh" ("Calcula.tion of Unsteady Flow in Op£,n Channels"), Academy of ScieRces, U.S.S.R., 1947; and ·8. A. Khristianovich, "N eustanovivsheiesia d vizhenie v kana.lakh i rekakh" ("Upstea.dy Motion ill Channels and.Rivers"), in "Nekotoryie!Vopl'osy Mekhaniki Sp19~hnol Sredy" (" Several Que~tions on the Mechanics of Oontinuous Media "), Aca~emy of Sciences, U.S.S.R., 1938, pp. l3-154. .. • ~pecial references on the subj'eet of river hytlrauiics are:· Serge Lelia.vsky, "An Introduction to Flilvial Hydrauli~s," Constable and Co., Ltd.; London, 1955; and T.Blench, "Regime Dehaviour of yanllis and Rivers," Butterwor;th & Co. (Publishers) Ltd" London, 1957. .

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Similarly, the transient flow in chanI).els subject to the iilfiuence of the tides is a special topic in the rapidly developed fields of tiditl hydraulics and cOD,stalel1gineering and is therefore b9yond the scope of this book. In a science which has reached so advanced a state of development, a large portion of the work is necessarily one of coordination of' existillg cOlltributions. Throughout the text} the auth6r hr,s attempted to make specinc acknowledgment regarding the source of m:lter.ial employed} and any failure to do so is o.n unintention:al oversight. . In the preparation of this \:lOok, engineers and a,dministrators in many engineering agencies hl1ve ~!1thusi<:1st.ically furnished information and extend~d cooperation. The author is especially indebted to those in the U.S. BUl'cau of Reclamation, U.S. Geological Survey, U.S. Soil Conser,ration Ser~ice. U.S. Agricultural Research Service, U.S. Army Engineer. . WiLterways E~periment Stat.ion, Offices .of the Chief Engineer and District EnD'ineers of the U.S. Army Corps of Engineers, U.S. Weather Bureau} U,S~Buref~u of Public Roads; and the Tennessee Vilrey Authority . Also, mn.ny friends and colleagues have kindly. supplied information and generously offered suggestions. In particular, the author wishes· . to thank Dr. Hunter Rouse, Professor of Fluid Mechanics and Director of Iowa Institute of Hydraulic Reseal'ch, State Unive·rsity of Iowa; Dr. Arthur '1'. Ippen, Professor of Hydraulics and Director of Hydrodynamics Laboratory, Massachusett.s Institute of Technology; Dr. Giulio De Marchi, Professor of I:1ydraulics and Director of Hydraulic Laboratory, Institute of Hydraulics and Hydraulic Construction, Polytechnic Institute of Milan,Italy; Dr. Roman R. Chugayv, Professor and Head of Hydraulic Con:;truction, Scientific Research Institute of Hydraulic Engineering, Polytechnic Jl1sti;tute of Leningrad, U,S.S.R.; Monsieur Pierre Danel, President of SOGREAH (Societe Grcnobloise d'Etudes et d'Applications Hydrauliques), France, and President of the Interun,tional AssociatiO!l of Hydraulic Research; Dr. Charles Jaeger, Special L.ectul'er at the Imperial College of Science and Technology, Univemityof Londoll, and COllsulting Enginee!' of. The English Electric Company, Ltd., England; Professor L. J. Tison, Director of Hydraulic Institute, University of Ghent, Belgium; Dr. Tojil'O' Ishihara, Professor of Hydraulics and Dean of Facult.y of Engineering, 'Kyoto University~ Japan; and Dr. Otto Xirschmer, Yrofessor 'of Hydraulics and Hydralllic Structures, Technical Jnstitute of Darmstadt, G e l ' m a n y . . , Special acknowledgments are due. Dr. Nathan M. Newmarlt, Professor and Head; ofthe Department of Civil Engineering} University; of Illinois} for his encouragement and unfailingisupport of this project; D~. James M. RobertsoiJ., Professor of Theoretica~ and Applied Mechanics, ;University of Illinoi.':\, for his review of and c~mmellts on Chapter 8 onitheoretical concepts;: nnd Dr. Steponas Kolu~aila, Professor of Civil Engineering,

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PREFACE'

University of Notre Dame, for h.is readi~g of the entire manuscript and his valuable suggestions, Dr. Kolupaila' also helped in interpreting and collecting information from the hydraulic literature written ill Russian, Polish, Lithuanian, and several. other languages which are unfamiliar to theauthol', . The author also wishes to express his warm gr[l.titucie t,o those whQ have constantly shown a keen intei'est ill .his work, as this iuterest lent a strong impetus toward the completion. of this volume.

CONTENTS

Ye.nTe Chow Prefacs

. PART I,

Chapter 1.

BASIC PRINCIPLES

Open-channel Flow !llld Its Classiffcations

1-1. D~scriptiori . 1-2. Types 'Of Flow . 1-3. State Qf FlQW 1·4. Regimes Qf Flow . Chapter 2.

2-6. 2·7. 2·8, 2-9. 2·10.

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Open Channels and Their·Properties

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Energy and Momentum Principles.

3-1. Energy in Open-channel Flow. 3·2.. Specific Energy. 3-3. Criterion for a CriticnJ State of Flow 3-4. Inte~pretation of Local'Phenomena 3-5. Energy in Nonpri~atic Chl!.nnels 3-6. Momentum in Open-charinel Flow 3-7, Specific Force. . ,. 3-8. Momentum Priu¢iple Applied to Nonprisffiatic, Channels. Chaptet 4.

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Channel Geometr)~ . Geometric Elements of Cha.nnel Section, Velocity Distribution in a. Channel Section. Wide Open Channel Measurement of Velocity , Velodty-dbtribution Coeflicienu, . DeterminaHon of Velocity-distribution Coefficients PrllS
Chapter 3.

3 3

2-1. Kinds of Open bl~annel

2-2, 2-3, 2-4, 2-5.

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Critical Flow: Its CQl~nputation and Appllcations .

4-1. Critical Flow 4-2. The8ection Factor for Crit.ical-fiow Computa.tion . for Criticru-fiow Computation _ 4-3. The HydrILulic' 4-4. Computation of Flow. 4-5. Control of Flow. 4-6; Fl9W Measurement.

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32 39 39 41 42

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CO!-T'l'ENTS

PART 11. Chapter 5.

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Development ()f Uniform Flow and Its Founulas

Computation of Uniform Flow .

The Conveyance of a Channel Section The Section Factor [or Uniform-fio;; Computation . . The Hydraulic EXjlQnent for Uniform,.flow COmputation. . Flow Chua.cteristics in Jl. Closed Conduit with Of/en-channel Flow Flow in a Channel Section' with Composite Roughness Determination of the Normal Depth and Velocity. Determination of the Normal a.nd Critice.l Slopes 6-8. Problems of Uuitorm-fiow Computa.tion. 6-9. Computation of Flood Discharge. 6-10. Uniform Surface Flow . Chapter 7.

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89 89 91

93 94 98 10l

101 108 114

D<"sign of Channels for Uniform Flow

128 ,12.8

131 134

136 140

142 144

146 148 157

A. NONERODIBLE CHANNELS

157 157

7-1. The Nonel"Ddlble Channel.

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. 7-2. Nonerodible r-,·laterilll and Lining.

158 158

7-3. The Minirmlm Permissible Velocity

7-4. Channel Slopes . . 7-5. FreebofJ.l"d 7-6. The Best Hydraulic Section 7-7. Determination of Section Dimensions

B,

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162

ERODIBLE CHANNELS WHICH SeOUR BU'l'

7-9. The Maximum Pel'missible Velocitv . 7-10. Method of Permissible Velocity . 7-11. The Tra.ctive· Force. 7-12. Tractive-force Riltio 7-13. Permissible '1;'raotive Force 7-14. Method of Tractive Force. 7-15. The Stable Hydra.ulic Section.

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7-8. MethOds of Approach .

C. GRA8SEDCHANNELS 7-16. The Gra.ssed. Channel . 7-17. The Retardance Coefficient 7-18. The Permissible Velocity

Chaptllr 8. Theoretical Concepts of Boundary Layer, Surface Roughness,. .-192 Velocity Distribution, and Instability of Uniform Flow.

167 168 . 170

172 175

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184

192 194

8-1. The Boundary Layer . 8-2. Concept of Surface Roughness 8-3. Computation of Boundary Layer. S.4. Velocity Distribution in Turbulent Flow 8-S. Theoretica.l Uniform-flow: Equations. . 8-6. TheOl·etica.l Interpretation of Manning's Roughneas Coefficient 8-7. ::Vlethods for Determining MalUling's Rougbneas Coefficient 8-8. Instability of U:uiform Flow . . PART III.

128

6-1. 6-2., 6-3. 6-4. 6-5. 6-6. 6-7.

184 184

7-19. Selection of Gr8.53. . . 7~20. Procedure of Design

UNIFOR.M FLOW

5-1. Qualifications for Uniform Flow : 5--2. Establishment of Uniform Flow . 5--3. Expressing the Velocity of a Uniform Flow. 5-4. The Chezy Formula 5-5. Determination of Ch,s'Oy's Resistanoe Factor 5-6. The Manning Formula. 5-7. Determination of l\-1anning's Pw:mghneas Coefficient 5-8. Factors Affecting Manning's Roughness Coefficient 5-9. The Table of :VIa.nnillg's Roughness Coefficient 5-1.0. Illustro.tions of Chimnels with Various Roughnesee3 , Chapter 6.

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CONTENTS

198 200

202 205

206 210

GRADUALLY VARIED FLOW.."

Chapter 9. . Theory and Analysis. 9-1. Basic Assumptio!l.9 . 9-2. Dynamic Equation of Graduallv Varied Flow 9-3. Che.ra.cteristiDs of Flow Profiles' 9-4. Cla.ssification of Flow Profiles. 9-5. Analysis of Flow Profile 9-6. Metnod of Singular Point. 9-7. The Transitio,nal Depth

Chapter 10.

Methods of Computa.tion

10·1. 10-2. 10-3. 10-4, 10-5.

The GrD.phical-int!lgration Method Method of Direct Integration. The Direct Step Method The Standard St.ep Method Computation of iii Family of Flow Profiles 10-6. Th~ Standard Stap Method for Natural Channels. 10,7. The SLage-fall-discharge Method ior Natural Channell! 10-8. The E.m. Method for Natural Channels.

Cb,apter 11.

Practica.l Problems .

11·1. Delivery of a Canal for Subcritical Flow 11-2. Delivery of a Cana.! for Supercrit.ical Flow 11-3. Prublems Related to Cana.! Design . . 11-4. Computation of Flow Profile in Nonprismlltic Channels 11-5. Design of Transitions . 11-6. Transitions between Canal and Flume or Tunnel 11-7. Transitions between Cana.l and Inverted Siphon 11-8. Backwater Effect of a Da.m 11-9. Flow P!l.SSing Isla.nds . . 11-10. River Confluence Chapter 12.

Spatia.!ly Vflried Flow

12-1. Basic Principles and' Assumptions 12-2. Dynamic Equa.tion for Spa.tia.lly Varied Flow

217

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217 218 222 227 232 237 242

249 249 252 262

265 268 274 280 284 297 297 302 303 306 307 310 .317 319 320

321 327 327 329

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12-3. 12-4. 12-5. 12-6.

Analysis of Flow Pro5Ie . . Method of Numerical Integration The Isoclinal flilethod . . Spati~.lly Varied Surface flow PART IV.

Chapter 13.

RAPIDLY VARIED FLOW

IntroducUon.

13-1. Characteristics of the Flow 13.2. Approa.ch to the Problem . Chapter 14.

Flow over Spillways

14-1. The Sharp-crested Weir 14-2. Aeration of the Nappe . 14-3. Crest Shape of Overflow Spillways 14-4. Discharge of the Overflow Spillway '14-5. Rating of Overflow Spillways. ' . ' . . 14-6. 'Upper Nappe Profile of Flow over Spillways 14-7. Effect of Pi~rs in Gated Spillways' .14-8. Pressure on Overflow Spillways . . 14-9. Drum Gates. . 14-10. Flow at the Toe of Ov.erllow Spillways 14-11. The Ski-jump Spillway 14-12. Submerged OverfloW' Spillways Chapter iii.

Hydra.ulic Jump and Its Use as EneIgy DissipatoI

15-1. The Hydra.ulic Jump . 15-2. Jump in Horizontal Rectangular Channels 15-3. Types of Jump . 15-4. BllSic Cha.racteristics of the Jump 15-5. Length' of Jump. 15-6. The Surface Profile .. 15-7. Location of Jump .' 15;8. Jump as Energy Dissipator 15-9. Control of Jump by Sills . 15-10. Control of Jump by Abrupt Drop 15-11. Stilling Basins of GeneraU"ed Design 15-12. The SAF stilling Basin 15-13. USBR Stilling Ba.3in II 15-14. USBR Stilling Basin IV 15-15. The St.l'aight Drop Spillway 15-16. Jump in Sloping Channels. 15-17. The Oblique Jump. Chapter 16. 16-L 16.2. 16-3; 16-4. 16-5; 16-6: 16.7.

333 341 346 347

Flow in Channels of Nonlinear Alignment

Nature of the Flow . Spiral Flow . Energy Loss. Super elevation Cross Waves. . . . . : . . . . . Design Considerations for Subcritical Flow. Design Considerations for S~percritical Flow

357 3.57 '357

360 360 362 363

365 368

Chapter 17.

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CONTENTS

CONTENTS

xvi

Flow through Nonprismatic Channel Sections

461 '--:,....

17-1. 17-2. 17-3. 17-4. 1t-5.

17-6. 17-7. 17-8. 17-9. 17-10. 17-11. 17..12. 17-13. 17-14.

Sudden Transitions,. Subcritica.l Flmv through Sudden Transitions Contractions in Supercritical Flo''''' Expansions in Supercritical Flow. Constrictions Sl!bcritical Flow through Constrictions Backwater Effect due to Constriction Flow through Culverts. Obotructions. ro'loW' between Bridge Piers Flow through Pile Trestles Flow through Trash Racks Underflow Gates Channel Junctions

461 464 468 470 475 476 490 493 499 501 506 506 507 512

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370 374 380 382 384 385 393 3113 395 395 396 398 399 399 404 408 412 414 415 417 422 423 425 429 439 439 439 441 444 448 455 456

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PART V.

UNSTEADY FLOW l~.

Chapter 18. 18-1. 18-2. 18-3. 18-4. 18-5. 18-6. 18-7. 18-8.

Gradually Varied Unsteady Flow

525

Continuity of Unsteady Fiow Dynamic Equation 'for Unsteady Flow,. Monoclinal Rising Wave . Dynamic Equa.tion for Uniformly Progressive Flow Wave Profile of UnIformly Progressive Flow Wave Prupagation . Solution of the UnsLeady-flow Equations Spatially Varied Unsteady Surface Flolv

Chapter 19.

Rapidly Varied UnsteadY Flow

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l_

554

19-1. Uniformly Progressive Flow 19-2. The Moving Hydraulic Jump, 19-3. Positive Surges . 19-4. Negative Surges. 19,5. Surge in Power Canals. 19-6 .. Surge in Naviga~ion Canals 19-7. Surge through Channel Transitio!IS 19-8. Surge at Channel.Junctions 19-9. Pulsating Flow. Chapter 20.

,~

525 626 528 631 533 537 640 543

..

Flood Routing

-.'

554 557 559 566 568 572 575 578 580

C '-...,'

l. I.. i, __

586 (

20-1. 20-2. 20-3. 20-4. 20-5. 20-6.

Ii

,.

Routing of Flood Method of Characteristics. Method of Diffusion Analogy. Principle of Hydrologic Routing Methods of Hydrologic Routing A Simple HydrologIC Method Routing

of

"~._~ • • ' _ _ _ _ V~·

y.'.----.-... ---

586 587 601 604 607 609

'-

\ ~,

i ,--

\

I

CHAPTER

,1

OPEN~C:HANNEL FLOW AND ITS CLASSIFICATIONS

1-1. Description.

The flow of water in a conduit may be either openThe two kinds of flow are similar in many ways but differ in ohe important respect. Open-channel flow must have a free surface, whereas pipe flow has none, since the water must fill the whole conduit. A free surface is subject to atmospheric pressure. _:PiR~ , flow, being connnedin a .closed "£Q~1duit, x~rts irect atl!lo,§ heri pressure but hydraulic pressure only. ., , "--,, ,,The t\VO kinds of flow are compared in Fig. 1-1. Shown on the left side is pipe flow. Two piezometer tubes are installed on the pipe at' sections 1 and 2. ,The water levels in the tubes are maintained by the pressure in the pipe at elevations represented by ~he so-cag~a.lYJdr,guliq g.rade line. The pressure exerted by the water in each section of the . pipe is indicated in the corresponding tube by the height y of the watet c;olumn above the center line of the pipe. The tIJt.al energy in the flow of the section with reference to a datum line is the sum of the elevation z of the pipe-center line, the piezoine'tric height y, and the velocity head V2/2g, where V is the mean velocity of flow. l The energy is represented in the figure by what is called the energy grade line or simply the energy line. The loss of energy that results when w:ater flows from seqtion 1 to sectiori 2 is .represented by hi. A similar diagram for open-channel flow is shown on the right side of Fig. 1-1. For simplicity, it is assumed . that the flow is parallel and has a uniform velocity distribu,tion and that the slope of the channel is, small. lP,..i,his c!1;,§e, the 'Y~~!t~ §~rj~ce is ,the .hldraulic m£l~dine, and ~~~.1t£.Qf..1~j~OIUjUg",,~ piezometric [email protected] '., . . ~ Despite the similarity between the two kinds of flow, it is much more cl.ifficult to solve problems of flow in open channels than.in pressure pipes. ~low conditions in open c~annels are complicated Py the ',fact that the channel flotu or 'Pip~ flow.

---'-->

~

!

It is here assumed that the velocity is uniformly distrib\.ited across the conduit othenvise a correc,tion ,,",ould have to be made, such Ilf is desaribed in Art. 2-7 fbr open c h a n r r e l s . ! , : 2 If the flow were curvilinear or if the slope of the channel w~re large, the piezometric height would be appreciably different from the depth of flow (l
-"1"'-

3

: 1

s~ction;

;~

I" .,

I

,,--.. :

.

'-....

" .~

xviii

CONTENTS

Geometric Elements for Circular Channel Sections

625

Appendix B. Geometric Elements for. Trapezoidal, Triangular, and Parabolic Cb-annel Sections

629

Appendix C.

640

Appendix A.

, Appendix D.

NQIDographic Solution of the Manning Formula Table of the Varied-flow FuncH,ms

.

F(u.,N)

Appendix E.

r " I-UN dlt

= Jo

and

F(u,N)_s,

=

r"

dtl )0 l+'uN

Table of the Varied-flow Functions for Circular Sections

,,

641

!

'~

·PART I

BASIC PRINCIPLES

657

Name Index.

663

. Subiect Index

669

(

I

)

r

,- .

)

4

""

BASIC PRINCIPLES

OPEN-CHANNEL FLOW AND ITS, cL'ASSlPICATI0NS

position of the free surfa.ce is likely to ch~nge with respect to time .and space and also by Lhe fact that the depth of flow, the discharge, and the slopes of the channel bottom and of the free surface are interdependent. Reliable experimental data. on flow in open channels are usually difficult to obtain. Furthermore, the physical condition of open channels varies -'much more widely than that of pipes. In pipes the cross section of flow is fixed, since it is completely defined by the geometry of the .conduit. ; _I The cross section of a pipe is generally round, but that of an open channel may be of any the circular to the irregular forms of natural streams.' In pipes, the interior Surface ordinarily ranges in roughness

Steady Flow and Unsteady Fww: Time as the Criterion, Flow in an open channel is said to be steady if the depth of flow does not change or if ' it can be assumed to be constant during the time interval under consideration. The flow is unsteady if the depth changes with time. In most open~channel problems it is necessary to studyftow behavior only under sready conditions., .If, however, the change in flow condition with respect . to time is of maior concern) the flow should be .treated as unsteady. In floods and surges, for instance, which are typical examples of unsteady now, the stage of flow changes instantaneously as the Wf1ves PMS by, and the time element becomes vitally important in the design of control structures. For any flow, the discharge Q at a channel section is expressed by

Q

'VA

(I-I)

where V is the mean velocity and A is the flow cross-sectional area normal to the direction of the flow, since the mean velocity is defined as the discharge divided by the cross-sedional area. In most problems of steady flow the discharge is constant throughout the reach of the channel under cOllsideration;in other words, the flow is continuous. rrhus, using (1-1), (1-2) Pipe flow

Open-chann,1 flow

FIG. 1-1. Comparison between pipe flow and open-channel flow.

from that of new smooth brass or wooden-stave pipes, on the one hand, t,o.that of old corroded iron or steel pipes, on the other. In open channels the surface varies from tha.t of the polished metal used in testing flumes to that of rongh inegular river bed'l. :.vloreover, the roughness in an open channel varies with the position of the free surface. Therefore, the selection of fdction coeffic~el1tS is attended by greater uncertainty for ~ open channels than for pipel$. In general, the treatment of open-channel flow is 30mew~at morl:l~Ip.Eirical than that of pipe flow. The empirical ) method is the best available at present and, if cautiously applied, can yield results of practical value. The flow in a. closed conduit is not necpssarily pipe flow. It must be " classified as open-channel flow if it has a free surface. The storm sewer, . .1 for example, which is a closed conduit, is generally designEld for openchannel flow because the flow in the sewer is ekpected to maintain a free surface most of .the .time. '. 1-2. Types at Flow. Open-channel fiow call be classified into many '\ types and descri:be'd in various ways, The following classification is made according to the change in flow depth with respect to time and space.

,

...

where' the subscripts designate different cllannel sections. .This is the '/::::,.. continuity equation for a Qontinuous steady flow. ' -------~----.-,.-~-"'- .. -.-,..-'-.-.--.~-- . --~~~----Equation (1-2) is obviously invalid, however, where the discharge of a steady flow is nonuniform along the channel, that is, where- water runs in or ont along th.&..Q~...QLfloW'. This type of flo,!!L. kn2wn as_~/Ltiql.l'!l ~!ied .!!!...!ii8c
.P. • ....

I,

6

I

i .~"

BASIC PRINCIPLES

is a practically impossible condition. The term «uniform flow" is , , therefore, used hereafter to refer only to steady uniform flow. Flow is varied if the depth of flow changes along the length of the channel. Varied flow may be either steady or unsteady. ~nce unsteady 2l~fI.~~e, the t~,rm II unsteady flow" is used hereafter to designate unsteady varied flow exclusively:-'-. ._ ... Varied. flow roay be furt~; da.ssified as either rapidly or gradually varied. The flow is rapidly; varied if the depth changes abruptly over a comparatively short distance; otherwise, it is gradually varied. A rapidly varied flow is also known as a local phenomenon,' examples are the hydraulic jump and the hydraulic drop.

OPEN-CHANNEL FLOW AND ITS CLASSIFICATIONS

I

For clarity, the classification of open-channel flow is summarized as follows:

f'

I

I I I

'{'

I r

I Chong'e of depth from time time

to

Uniform 'flow - Flow in a laboratory' channel

G.V.F.

RV.F.

Sluice

Confraction below the sluice

----:---~-

-

~ -'.

,

//

G.v. F. - Flood wove

R.V.F. - Bore

lJnsteqdy flow •

FlG. 1-2. Various types of open-channel Bow. RV.F. ~mpidly varied flow.

G.V.F. = gradually varied flow;

t ~'.

The flow is turbulent if the viscous forces are 'weak relative to the inertial fDrces. In turbulent flDW, the water particles move in irregular paths which are ,neither smooth nor fixed but which in the aggregate still represent the .forward motion of the entire stream. Between the laminar and turbulent ,states there is a mixed, or transitional, state. The effect .of viscosity relative to inertia can be represented by the Reynolds number, defined as

Varied flow

'

A. Steady flow L Uniform flow .2. Varied flow c. Gradually varied flow b. Rapidly varied flow B. Unsteady flow L Unsteady uniform flow (rare) 2. Unsteady now (i.e., unsteady varied flow) a.Gradually varied unsteady flow b. Rapidly varied unsteady flow Various types of flow are sketched in Fig, 1-2. For iIlu.strative purposes, these diagrams, as well as other similar sketches of open channels in this book, have been draw.l to a greatly exaggerated vertical scale, since ordinary channels have small bottom slopes. 1-3. State of Flow. The state 01' behavior of open-channel flow is governed basically by the effects of viscosity and gravity relative to the in.ertial forces of the flow. The surface tension of water may affect the behavior of-flow under certain circumstances, but it does not piay a significant role in most open-channel problems encountered in engineering. , Effect of Viscosity, Depeilding on the effect of viscosity relative to inertia, the flow may be laminar, turbulent, or transitional. The flow is laminar if t.he viscous forces are so strong relative to 'the inertial forces that viscosity plays a signiftcant part in determining flow behavior. i~ laminaLfl.mY...Jh~~arti.91esa,ppeal' to IAove in deE!lite smooth paths, or streamlines, and infinitesimally thin layers of. fi»id se~11l To slia:e ove~\:n~Q~!.lt l~ye.r~: . - . - . - - - . - - . - - - - . - . - - -

Unsteady unifarm flow - Race

RV.F.

7

I

1 i I I

II

i

j.

R == VL JJ

(1-3)

where V is the velocity of flow in fps; L is a characteristic length in ft, here consider'ed equal ,to the hydraulic radius R of a conduit; and 11 (nu) is the kinematic viscosity of water in fV/sec.

The kinematic viscosity

8

\..J I

,

"-

.

"BASIC PRINCIPLES

'

OPEN-CHANNEL FLOW AND ITS CLASSIFICATIONS

ill ft2/sec is equal ~o the dynamic viscosity, f.I (mu) in shlg/ft-sec divided by the mass densl(;Y p (rho) in slug/ft,·. For water at 68"F (2000) 5 f.I = 2.09 X 10- and p = 1.937; hence, 11 1.08 X 10-$. " An open~chan?,el flow is laminar if the ReYl10lds number R is small and tUl'bt~lent if. R IS ,large. Numerous experinlents have shown that the flow 10 n. pIpe . c~an~es, from laminar to tul'l)ul~l1t in the range of R betwee~ the cntlCal val.ue~&Q.Q. ~n~ a valuethfl,t inay be as high as 50,000. . I~ th~se exp~nment.s the dIameter of the pipe was taken as the chal'act~rIStlC. le~gth m defining the Reynolds number. When the hYdra~11C radlUS 15 taken as the characteristic length, the corresponding' range IS from ,500 to 11,50q, ,. since th.e~!.fl:,l!l~~~_ of '!..,Eipe is fo~r times its hydraulic mdlUs. ' --'--.•-.-.--~--'-''''------

I I

r I i

,

I

I

,,I i

;t

I

(1-4)

,where 14, ~ the fl'1::Lional los; i.n ft. for flow in the pipe, i is the friction fact~~. L L~ the l~ugth of the pIpe III ft, do is the diameter of the pipe in V IS the velocIty of flow in fps, and g is the acceleration due to gravity In ftfsec~. '

:t,

hJlL, the above equation

1. .."

(1-5)

I

. Th f-R[IiJrelati~l1S~pJ.~smo~t~J;_~~~ can ~~~e,!2I'.!lssed bv th, e Blas:ius equatton ' . '. - ----"-,---- , .. .---'- ., 0.223

?

=

RO.~'

(1-6)

t

.1 t8 a result ~f Darc(s. stud.: [21: ~n flo~ in pipes, his name is commonly associated wlthithat ~ WelS~o.ch [31 deslgna,tJng~hlS. equation which Weisbach firstformulated..

in

r'l

From Eq;S. (1-3)• and (1-5) it can be shown that i

In

Act~IlYI dAubulSson [4] presenteji, prIOr to Darcy, a formula. that can be reducEid to the form of Eq. (1-4). i . , t tn. this equation, ~he hydraulic :radius is used as the characteristic lengtll in defin.. -.. mg the Reynolds .number. If the; diameter of pipe were used as ~he characteristic length, the numerlllal constant of ~he numerator in this equationfwould be 0.316.

(1-7)

basis for comparing flow conditions in open channel.s. It may be noted that corresponding equations forflO'wi'n open channels have been derived byJf~~~l~L§>lld ~.epear to be very_similaLtQ_ihrnQe-fiow t:lJl'\l.!:l:jJ.QTI!3~ giYQ!L:;tj;lov:e. It must be remember.ed, however, that, owing to the free "mlrface and to the ~~oUhe }u~draulic ~us. discharge... and slope, the f-R relationship inopen-channel flow does not follow exactlyJ;htl. , sinlpleConcepts thathohlfor pipe flow. Some specific features of the '.f-R relationshiP open-chann.erfloware described below. ,t Experimental data available, for the .determinationof the f-R relationship in open-channel flow can be found in various publications on hydmu-' which plots the relationship for flow iIi lics. 1 Figure is based on data developed at the University of HUn an University of Minnesota [20]. In this plot the following features may be noted: l.The 'plot shows clearly how the state of flow changes from laminar to turbulent. as the Reynolds number, increases. The discontinuity of the plot and the spread of data characterize the transitional regIon, as they do in ,the Stanton for fio'w in pipes. The transitional range, however, so as "pipe flow. The iower critical Reynolds number depends to some extentoa channel shalle. The value :ia~0rQD1 Q.OQJ;9~-(i60;oemgg~i;~rally l~g~;-th-;:;-th-; value 'for pipe flow. For practical purpoSes,thetransitioUal range of:R Iol:-opei1='channel -~.~ flow mfty be a.~sumed to bc..§QQJ.Q..~..QQP. ' It should be noted, however, that the uppe.r value is arbitrary, since there is no definite upper limit ' for all flow conditions. 2. The: data in the laminar regiQu can be defined by a general equation 'I( f F -R • (1-8)

,

;. It snould be noted that there is actually no definite upper limit..

,!

+ 0.4 '

---

:'hls equation may a.lso be applied to uniform ahd nearly uniform flows channels. ' , - - -_________

~/-m...0pen

f

2 log (R ...;'1)

The resulting Prandil-:von

I),

and transitional stat.es ot open-.channel flow c~m be expressed by. a.~lagram thatshows a relation between the Reynolds number and the fnctlOn factor of the Darcy-Weisbach formula. Such 9~ dlUgra~, g~I1el'ally knoWn us the BtanloTL4:f.ggn;,m Ill, has beeg.
Since da = 48 and the ~ne~gy gradie.u,t S may be rewritten for the friction factor

Nikuradse 8 ~

-'--~Eciuations (1-6) and (1-7) will be used i'n the follo\ving discussion as

~Th~-i~mi~~:, turbul~ntJ

,

v..'ith the data obtained b d1'man equ~..!4sm Is 1

9

~

, I ,

I

I Y I

I

K = 8gR 2S ! .V'

(1-9)

Se~ [l(~1 to [23].' ! ' : . The da.;ta. for the rectangular channel; were furnished through the cdurtesy of Professor W. :/;1. lAnsford ann processedJoi the present purpose by the author. 1

2

11 10

BASIC PRINCrPLES

Since V and R have speeific ~'alues for any given chantiel shape, K is a purely numerical factor dependent only on channel shape. For laminar . flow in smooth cha.nllE)ls, the value of J( call be determined theoretically [20} .. The pJot in Fig. 1-3 indica.tes that Kia approximately 24. for tht> reetanguJar channels and 14 fol' the triangular channel under consideration.

e

R

o

F: '.

(C

$

',{I td

a.o AI

R ., 4,01;:11

a til

•

a ..

e

R • 0.0

~

to

..

R • 3,0*

(I

R

..

4,

~4 tK

tloll

A ... 1.:2 C!A

eta -i.lli:vtRS;'TY n,= fLl!NOlS DATA G

0••

fI£CTAlfGlA.AR c.HAflNa~,1.5Ff \\'10£. wlTK GL,us WALlS &. POI.JSH(O 9Fl4SS PLATE

0.2

!~

lhjrro~,

e . TRtANCiULi\J'i CHNfh£i..'(P VE~TEX MOLt. wr:'H SMOO'lM

.,..

•

RECrANGtJt.AI\ C:Io!IAIfN!:l..1.7 1M WfDt~ 'Wt!'H SJ,t0Q'l',.. SlJFl~

FACE; o

or

Sl'Flt;CTtiRAL

srUL

nU4NCAA...A.R CHA'HN£l.,l'd" TO

fsd''V~Rrt)t ,t~I.ES,WlrH

0.08

f

0.1

om

f

O.Oi

0.06

().04 Q

k

100.0rt9fM

.. k.

0.02.831"

O.qz 0.02

\

,

1

O.OIl-----4----+-....:..--·t---~·'r-~

0.01

0.000 0.008

O.OO41--+-+++4-+-~-+-'f+-+-'-JHh-+-'-I--+-l-+I--!·-++H

i

4

.J ~

•

$1'0'

I

FIG. 1-3. The f-R relationship for flow in smooth channels.

3. The data in the turbulent region correspond closely to the Bllisius:j=lrandtl-vo1n Karman curve. This ~ndicates that the law for turbulent fiowin smaoth pipes may be approximately representative of a.Usmooth channels. The plot also shows that the shape of the channel does not have an important influence 011 friction in turbulent flow, as'it does in ," laminar fl o:W. The dataJor laminar flow obtained at the University of Minnesota [20J and the da~ for turbulent flow cdllected individually by Kirschmer l

.' . h channelS. . Bo.zin's channeis; No.4, FIG. 1-4. The f-R. ~elatlonshlp for Row In rOU~ed wood; No. 14, 'unpolished wo~d gravel embedded In aement; No. Il,. unpo m lon 10 mm high, and·lO mm In roughened by tra.nsverse wooden stripS 27. sp.~in'" of mm; No. 24, cem, ent .. 7 Ne 14. except WIt .1 a te· : spacmg; No.1, same B.S. ' d K' her's cha.nnel: smooth concre . lining; and No. 26, u~p
;n

R ;

"--'

10

\

\ I,

50

.Q

"

. . . (23] are shown in the diagram for fio~ in (15,16}, Elsner [221, and Kozelnny f the' data channel roughnes.~ is (Fig. 1-4). 13Qme 0 ' • f I _ .r . • h' . e8.SUre of the roughness fll1rttcles or;m k whic IS a SIze m . f t ' s'. Y , . represen d' . illustrates the followmg ea ur.e ing the channel surface. The lsgram d fi d b Eq (1-8) In this • • . . th d til. can be e ne Y . . . L In the lammat reg~on e ~ly h' h than it is for smooth channels region, the value of K l.S genera 19 e r . .

\, I

'-.---

..

~

~-"

t

IJ. )-J (J.

r

L

r

.",

)

(

'ru

f'J I' )--

")

Ir}

~".

i

."-"'

12

~.-.-.---

-_._._-,_

t j) -1. [

and ranges bfltween 60 and 33, indicating the pronounced influence of the channell'oughnesson the friction factor. . 2. In the turbulent region the channel shape has a pronounced effect on the friction factor. It is believed that, when the degree of roughness i~ const.ant the fdction. factor decreases roughly in the order of rectangular, t:'iangular, trapezoidal, and circular channels. At the suggestion of Prandtl, Kirschmer [15,lDj explained that the effect of channel shape may be due to the development of secondary flow, which is ~pparently more pronounced in rect(1ngular channels than in,say, triangular channels: The secondary flow is the movement of water particles on /l, cross section normal to the longitudinal direction of the channel. A high secondary flowrinvolves high energy loss andthns r.,ccounts for high chaI!!lE11.J:{:!§.i..frul.nce. . . . . 3. In the turbl,1lent region most plots appear parallel to the Prandtlvon Karman curve, This curve serves as an approximate limiting position toward which a plot moves as the over-all resistance becomes less, Accoi'dingtQ a concept advanced by Morris [241 (Art. 8-2), the rise of the plots above the smooth-conduit curve may be explained as a resul~ of additional energy loss generated by the roughlless elements. When the Reynolds number is very high, some plots become essentially horizont,al, reaching a stllte of SO-Galled complete turbulrnce. At this state the value of fis independent of Reynolds number and dep~nds solely on roughness, hydraulic' radius, .and channel shape . . 4. The' plot of Varwick's data [16J for a give~ roughness,- hydraulic radius, and channel sha~ star~ off from ll. curve parallel to thLPran..Q.tj~ Voill'{arman curve, then rises as the Reypolds n~mber incre~§, and final1y becomes horizo'ntal as a state of complete turbulence is reached. The rise of the plot is a peculiar phenomenon which demands explanation, I and, since this finding has not been verified by other data, mor.e experimental studies .seem necessary to substantiate it. ~-'-~It should be noted that the ab'Jve descriptions are limited to lowvelocity, 01' subcritical, flow (which will be defined later in this.-ar.ti.cl.e)_ and to Row onwhich 8.ll:.rface tension dOEls not have a §1IDlificant influence. .. _._._----;---,. .. - .... -In most open channels laminar flow occurs very rarely. ,Th~!!l;~t tfu;.1 the surface of a streilr!!:J!:eE!~!g_~
-

---

According to thelloncept of Morris [241. thIS phenomenon probably represents a tJ'ansition of thefiow to a.nother type of flow having higher energy loss. As the Reynolds nu~ber increases, the fio'IV may be cha.nging from quasi-smooth flow to wakeinterference flow, and then tomolatecl·rou.ghness flow (~t. 8-2). 1

"

BASIC PRINCIPLES

-

OPEN-CHANNEL FLOW. AND ITS Ci..AftSIFICATIONS

13

As the flow in most channels is turbulent, a model empl.oyed to simulate a prototype channel should be designed so that the Reynolds number of flow of the model channel is in the turbulent range. E.ffect oj Gravity. The effect of gravity upon the .state of flow is represented bt aratio...91 ine~al forces to gravity forc~ This l'atio is,given by the FrQuile numbe1'z' defined» ',.

v

(1-10)

where, V is the mean velocity of fiow in fps, g is the acceleration of gra Yity in ft/sec z) and Lis a.characteristic length in ft. In open-channel fiow the characteristic leng~h is made equal to the lmdraulic deptAJ2., which is defined as the cross-sectional area of the water normal to the direction of. flow in the channel divided by the width of .the fr¢e surface. For rectangular channels this is equal to the depth of the flow' section, When F is equa.l to unity, Eq. (1-10) gives V

=

.../gD

(1-11)

and the flow is said to be in a critical sta.te. If F is less thaJ1 unity. or V < V(jJ5, the flow is s'libcriticaZ. In this state the role played. by gravity forces is mOfe pronounced; so the flow has a low velocity and is often described as tranquil a.nd streaming. If F is greater than unity, or V > the flow is 8upercriticaZ. In this state the inertial forces become domL'lant; so the flow has a high velocity and is usually described as rapid, shooting, and torrentiaL IE the mechanics of water wave~Jhe.critic& velocity v'gD is ideirtified as the celerity of the small gl'avitywaves that~ccu;fns-hallo;v wnterm channels as a result of any momentary change in the local depth of the . _water (Art. 18-6). Such a change may .Q~opecLbjullsJ;urha.ll.c~~u;l.!: obstacl~in...thJL!lb.!IJlp.el that cause a di§placement of waj;.!'lr.J.!.b.
v'{jD;

>IL

n-2

I O~ber dimensionless ratios used for the. sa.me purpoae.include (1) the lcinenc-flow factor}. VI/uL ... FI, first tlsed by Rehbock [251 and then by Ba.khmetefi' f26Ji (2) the Bouuinesq number B "'" V / v'2UR, first used by Engel [27J; 8.nd (3) the kinelicity or velocitY-head ratio 11; = V'/2gL, proposed by Stevens [28] alld Posey [29J respecti vaLr. . •

-I

itJ

'J 14

.

r

BASIC PRINCIPLES .

designed for this effect ;that is, the Froude llumhflr of the flow in the model' channel must be made "qual to tha,t of the flow in the prototype channeL 1-4. Regimes of Flow. A combined effect of viscosity and gravity may produce anyone of four 'regimes of flo7JJ in an open channel, namely, (1) 8ubcritical-larninar, when F is less than unity and R is in the 19,ininal' range; (2) 8upel'cl'itical-laminar, when F is greater than unity and R is in the laminar range; (3) ::mpercritical-turbulBnt, when F ia greater than unity

OPEN-CHANNEL FLOW AND ITS CLASSIFICATIONS

15

I I

1 \........,;;

~

1

1

I

1

,_I

I

'1

(

1

'-J

~\

I

j

,-,'

i

I

~J

f,

)

i I

C:--'\

I

\..-

\

!

~.~

Velocily, Ips

FIG.

1~5.

Depth-velocity relationships for four regimes of open':channel flow.

,_-

(Afler

Roberlson and Rouse [3D].)

and R is in the turbulent range'; and (4) subcritical-turbulent, when F is less than unity and R is in the tui:bulentrange. The depth-velocity relationships for the four flow regimes in a wide open channel can be shown' by a logarithmic plot (Fig. 1-5) [30]. The heavy line for F = 1 and the shaded band for the laminar-turbulent transi.tional range inters,ect on . th~ graph and divide the whole area into four portions, each pf which repres~nts a flow regime. The first two regimes, sub critical-laminar and ,supercritical-Iaminar, are not commonly encountered in applied openchannel hydraulics, since the flow is generally turbulent in the channels considered in engineering problems. However, these regimes occur : frequeI).tly where then~ is very thin depth-this is known as sheet flow. and they become significant in such problems as the testmg of hydraulic , models, the study of overland flow, and erOSlOn cOlltrol for such flQlY:• Photographs of the four regimes of flow are shown in Fig. 1-6. In each

'~

I

L Fw. 1-6. Photographs showing four flow regimes in a laboratory cll!1nnel. of H. Rouse.)

i

.

.

..

(Courtesy

photograph the direction of flow is from left to right. All flows are uniform except those on the right side of the middle and bottom views. The top view represents uniform subcritical-laminar flow, . The flow is su.b:ritical, since the Froude number was I',djusted to slightly below the cntical value; and the streak of undiffused dye indicates that it is laminar. Th~ middle ~i~w shows a uniform supercritical~laminar fl'ow changing to v~r~ed subcntical-turbulent. The bottom view shows a uniform superc:'1.tlCal-turbulent flow changing to varied subcritical-turbulent. In both cases, the diffusion of dye is the evidence. of turbulence.

{_.

,

16

I

(

\.

~l

,

BAstC ' PRINCIPLES

It is'believed th~t gravity action may have a definitive effect upon the flow resistance in cliurmels at the tut'bulent-flow range~ The experi,mental data studied by Jegorow [311 and Iwagaki [32J for smooth rec'tangular channels. and by Hom-:ma [33J for rough :channels have shown that, ~n the supol'critical-turhulent regil;l1e of flo~, the friflj;ion fact
( I

where A 1 and A. are the cross-sectional areas of the flow at sections' 1 and 2, respectively, a.nd lJ.y'is the drop in water surface between the sections. 1-2. Verify Eq. (1-10). 1~3. Verify by computation the depth-velocity' relationships shown in Fig. 1-5 £0: the four flow regimes in a wide rectangular open channel. The temperature, o,f the water is tnken as 68°F. . 1-4. A model channel is used to simulate a prototype channel 100 ft wi~e, carrying ;t disclmrge of 500 ds aL. a depth of 4 ft. The model is designed fOI' gravity effect, Itnd a: tUl'bu[ent-fiow condition is asaured. Determinet.he minimum size of the model and the scale ratio, BSSuming the upper limit of the \'r1lnsitionru-flow region to be R = 2,000. The sCII.le ratio is the ratio 'of the linear dimension of the model tot,ha.t of the prototype. . REFERENCES

'\

I

1. T. E. Stanton and J. R. Pannell: Similarity of, motion in relation to surfa.ce ftiction of fluids, PMlosophical1'r(l?!sactions, Roya! S()ciety of LaMon, vol. 214A, pp; 199224, 11114. : . ' : 2. H. Darcy: Sur des rechercbes experimentales relatives au mouvement des eaux dans les tuya~ (Experimental researches on the floW' of water in pipes), CiJmptes rendus des 8~an;ces del' Aoodemie des Sciences, vol. 38; pp. 1109-112~, June 2tlj 1854. 3. Julius Weishach: "Lefu-bueh der Ingenieur-! und Maschinemnechanik" (,ITextbook of Engineering Mechanics"), Brunswidk, Germa.ny, 1845. ' j. 4. J. F. d'Aubuisson de Voisins: "Tl"aite d'hydrl:\.ulique" ("Treatise on Hydraulics"), Levrant, Pari", '2d ed., 1840; trll.nslated int(l) English by Joseph Bennett, :Little, Brown &: COllj,pany, Boston, 1852, pp. 202-211. '

I

I ! !I ,I

OPEN-CHANNEL FLOW. AND JTS CLASSIFICATIONS

17

5. H. Blasius: Das Ahnlichkeitsgesetz bei Reibungsvorgiingenin Flilasigkeiten (The In.w oLsimilitude for: frictions in fl\1ids), Forschungsheft des VereitUl deutscher .' " Il1gimieure, No. 11:11, Berlin, Illi3. 6. Theodor von Ruman:. Mechauische Ahnlichkeit und Turbulenz (Mechanical similitude and turbulence), Proceedings afthe 3d In/ema/irma! ConfITBsl!liar Applied M ecka1!.ics, Stockhol1n, vol. I, pp. 85"-93, 1930. 7. L. Prandtl: The mechanics of viscous fluids, in W. F. Durand (editor-in-chiei): "Aerodynamic Theory./' Springer-Verlag,Berliu, 1935, vol. III, div. a,.p. 142. 8. J. Nikuradse: Gesetzmlissigkeiten der t,Ul'bulenten Stromung in glatten Rohren (Laws of turbulent flow· in smooth pipes), Forsck'ungsheft des .Vereins deutllcher Ingenieur~, No.'3M, Berlin, 1932. 9. Garbis H. Reulegan: Lews of turbulent flow in open channels, paper RP1151, Jou;'!1.al of Research, U.S. Na.t.iollaL Bureau of Sl.c.ndards, vol. 21, pp. 707-741, Dllcember, 1938. 10. J. Allen: Streamline and turbulent, flow in open channels, The London, EdinbuTgh and Dublin Philosophical Magazine and Journal of SciencfI, ser. 7; vol. 17, pp. 1081-1112, June, 1934. . , 11. H. Ba.zin: Recherches experimentrues, sur l'ecoulement de l'eau dans les canaux decouveris (Ex.perimental researches on the flO\v of wa.ter in open cha.nnels),' M emClires pT~senUs par d'!IIBrS savnll!~ d l' Academie des Sciences, Paris, voL 19, 1865. 12. Studies of river bed materials and their mQvement, with special referance to the lower Mississippi River, U.S. Waterways Experiment Sta.tion, Technical ' Paper 17, January, 193.5. 13. 8. P. Raju: Versuche uberden Stromungswidersta.nd gekrummter o:ffener Ka.n!!.le (Study of flow resistance in curved open ehannels), Miu"ilunge:TL des hydro.uli8chen In~tiluts der lel:h'rr.ischrn Hochschule Muncher£, no. 6, pp. 45-60, Munich, 1933. English translation by Clarence E. Bardsley: Resistance to flow in curved open channels, Proceedings, American Society of Civ'ii Engineers, vol. 63, pt. 2, p. 49 after p. 1834, Novembet:, 11)37. 14. Lorenz G. Strllub: Studies of the transition-region between laminar and turbulent flow in open channels, Transactions, American GeQphY~icat Union, vol. 20J pt. iv, pp. 649-663, 1939. 15. Otto Kirachtner: Reibungsverluste in Rohren und Kanalen (Frictional losses in pipEllj l1nd channels); Die Wass~rwirtschaft, Stuttgart, voL 39, no. 7, pp. 137-142, April; no. 8, pp. 168-174, May, 1949. 16. Ott.o Kirschmer: Pertes de charge dansles conduites forcees et lea eaI!o.ux decouVllJ'ts (En:ergy losses in pressure conduits and open channels), RllIIue q6n~rale de l'hydrauUllue, vol. 15, no. 51, pp.1l5-138, May-June, 1949. 17. Yuichi Iwaga.ki: em Lllmina.r to turbulent Row in a wide open channel, by W. hi. Owen, Tran8actions, .Ameriron Society of Civil .Engineers, vol. 119, pp. 1:165-1166, 1954.

•

.

18. Horace William King: :"Handbook of HydmIlIics," revised by Ernest F. Brater, McGrll.w-Hill Book Co,npany, Inc., New York, 4th ed., 1954, p. 7-35. 19. F. Bettes: Non.uniforln flow in cli.annels, Ciuil Ertgineering and P,.blicW01'ks Review, London,· vol. 52, no. 609, pp. 323-324, M~rch; no. 610, pp~ 434-4,36, April, 1957. , . 20. Lorenz G. Straub, Edvtard Silberman, and Herbert O. Nelson: Open-channel flow at. small Reynolds nUfnbers, Transacfions, American Society of Civil Engineers, vol. 123, pp. 685-706, ,1958. ! .21. Walll1ce M. Lansford ~nd James M. Robertson: DiB~lIssion·of Open-channel flow at small ReynCllds nUl)lbers, by.Lorenz G. Stra.ub, Edward Silberman, and H"r-

.j 18

BASIC PRINCIPLES :

b!lrt C. Nels01l, Trqnsaction8, American So.ciety o.J Civil Engineers, vol.. 123, pp. 707-712, 1 9 5 8 . . . 22. FtlLnz Eisner:- Offene Gerinne (Open channeL), sec. 4 of vol. IV, "Hydro- und Aerodynamik," in \V. Wien and F. Hanns (editors-in-chief): "Handbuch der Experimentalphysik," Akademische Veda.gsgesellschaIt mbH, Leipzig, 1932, p.298. 23. Josef l(ozeny: "Hydraulik" ("Hydraulics"), Springer-Verlag, Vienn9., 1953, p.574. . 24. Henry M. Mcirris, Jr.; Flow in rough condl,lits, Transactions, American Society oj Civil Engi1J,ee1's, vo.l.120, pp. 373-398, 1955. Discussio.ns on pp. 3.99-410. 25. Th.R.ehbock: Zur Frage des Briickenstaues (On the prDblem Df bridge constrictions), Zenlralblail der B(11!verwaZtun!l, BerUn, vol. 39, ~o. 37, pp. 197-200, 1919. 26. Boris A. Bakhmeteff: "Hydraulics of Open Channels,"McGraw-Hill Book Com. pany, Inc., New York, 1932, p. 64. . 27. F .. V. A. E. Engel: Non-uniform flow of water: Problems a.nd phenomena. in open channels with side cont.ractions, The Engineer, vol. 155, pp. 392-394, 429-430, 456-457, 1933. . ... . 28. 1. C. Stevens: Discussion Dn The hydraulic jump in sloping channels, by C. E. Kindsva.ter, TramactiO'nll, American Society of Civil Engineers, vol. 109, pp: 11251135, 1944. 29. C. J. Posey: DiscussiDn on The hydraulic jump in sloping channels, by C. E. Kindsvater, TransactioTLI/, American Society oj Civil Engineers, vol. 109, pp. 11351138, 1944. 30. J. M. Robertson and lIunter Rouse: On the four regimes· of open-channel flow, Civil E'lI.{Ji11eering, vol. 11, no.. 3, pp. 169-171, !vIa.rch, 1941. 31. S. A. JegorDw: Turbulente 'Oberwellenstromung(Schiesscnl in offenen Gerinnell mit gla.tten Wanden (Turbulent 5up'
I·I

\..:.·,1

I

\

i

i, CHAPTER

2

OPEN CHANNELS AND THEIR PROPERTIES

2-1. Kinds of Open Channel. An open channel is a conduit in which water flows with a iree surface. Classified according to its origin a channel may be either natural 01' artificial. ( Natural channels include all watercoUl:~ that exist naturally on the <......1 \ earth, varying in size from tiny hillside rivulets, through brooks, stl·eams,. smaH and large rivers, to tidal estuaries. Underground streams ca~ water with a fr~e surface are also considereclrulturul open chann~. The hydraulic properties of nll,tural channels are generally very irregular. In some cases empirical assumptions reasonably consistent with actual obs:ervl;ttiol1s and experience may be made such that the conditions of flow. in these channels become amenable ~o the analytical treatment of theoretical hydraulics. A comprehensive study of the behavior of flow in natural channels requires knowledge of other fields, such as hydrology, geomorphology, :sediment transportation, f1tc, . It constitutes, in' fact, a subject of its own, known as river lLJlP,raulics, Artificial channels are those constmcr,ed or developed by human effort: navigation channels, power canals, irrigation canals f),nd flumes, drainage ditches, trough spillways, floodways, log chutes, roadside gutters, etc., as well as model channels that are built in the laboratory for testing pm·poses. The hydraulic properties of such channels can be either controlled to the extent desired or designed to meet given requirements. The application ofhydra.ulic theories to artificial channels will, therefore, produce results . fairly close to actual conditions and, hence, are reasonably accurate fOl' ,......... practical design purposes. . Under various circumstances in engineering practic\'l the.artificial open channel is 'given different names, such as "canal," :'\ftume," "chute," "drop," "culvert," "open-flow tunnel," etc. These names, however, are used rather loosely and can be.defined only in a very genernl way. The canal is usually a long and mild-sloped channel built in, the ground, which may be unlined C!r lined with stone masonry, concrete, cement, wood, or bituminous materials. The flume is a channel of wood, metal, concrete, or masonry, usually supported on or above the surface of the ground to carry water across a depression. The chute is a channel having steep 19

20

(

BA.SIC PRINCIPLES

slopes. The drop is similar to a. chute, but the change in elevation is effected in a shott distance. The culvert flowing partly full is a covered channel of comparfl,tively short length installed to drain water through highway and railroad embankments. The open-flow t'unnel is a comparatively long covered channel used to ca.rry water through a hill or any obstruction on the ground. 2-2. Channel Geometry. A chann\'ll built witlulllY.a:t+ing cross sectiOll ti..'1dc~bottom slope is ~~lled a 'B.risma(ikma1Jnel. Otherwise, the channel is nonprismaticj an example is a trough spillway having variable width and curved alignment. Unl\lss specifically indicated, the channels dGScl:ibed in this book are prismatic. The term channel Bection used in this book refers to the cross section of a channel taken normal to the direction of the flow. A l'ertical channel s8ctio-n, howeve".is the vertical section passing through the lowest or bottom point of·the channel section. For hvl'izontal channels, therefore, the channel section is always a vertical channel section. Natural channel sections are in general very irregular, usually varying from an approximate parabola to an approximate tmpezoid. For streams subject to frequent fioods, the channel may consist of a main channel section carrying normal discharges and one or more side channel sections for accommodating overflows. Artificial channels are USUally designed with sections of regular geometric shapes. Tablc 2-1 lists seven geometric shapes that are in common use. The trapezoid is the commonest shape for channels with unlined earth banks, for it provides side slopes fol' stability. The rectangle and triangle are special cases of the trapezoid. Since the rectangle has vertical sides, it is commonly used for channels built of stable materials, such as lined masonry, rocks, metal, or timber. The triangular section is used only for small ditches, roadside gutters, and laboratory works. The cirde is Hle popular section for sewers and culverts of small and medium sizes. The parabola l is used as an approximation of sections of small and medium-size natural channels. The round-cQrnered rectangle is a modification of the rectEmgle. The round-bottom triangle L':! an . approximation of the parabola; it is a form usually created by excavation with .shovels. Closed geometric sections other than the circle are frequently used in sewerage, particularly for sewers enough for a man to enter. These sections are given various names according to their form; they may be

I

I

!

I I i i

I

r 1

i

I !

.i!:

+

I;'.>

+

~

1 The side slope z; 1 of a pa.rl!.boli~ section at the intersection of the sides with the free surface can be computed easily'by the simple formula z = Russian e!lgineers [11 also use semielliptica! and para.bolic of higher order: y azP with 11 .;, 3 Qr 4. The constant CL is computed fl'om the side slope assumed· at the' free surface ..

21

22

BASIC PRINCIPLES

egg-shaped, ovoid, semielliptical, U-shaped, catenary, horseshoe, baskethandle, 'etc. The complete rectangle and square are also common for large sewers. Dimensions and'properties of sewer sections may be found in textbooks on sewerage.! A special geometric section kilOwn as hydrostatic catenary or lintMrw [4,5J is the shape of the cross section of a trough, formed of flexible sheets assumed to be 'Iveightless, filled with water up to the top of the seCtion, . and firmly supported at the upper edges of the sides but with no effects of fixation~ The hydrostatic catenary has been used for the design of the sectiolls of some elevated irrigation flumes. These flumes are constmcted of metal plates so thih that their weight is negligible, and are :firmly attached to beams at the upper 2-3, Geometric Elements of Channel Section.' Geometl"l:c elements are pro'perties of a channel section tha.t can be defined entirely by the geometry of the section B.nd the depth of flow. These elemerits are very importa.nt and are used extensively in flow computations. For simple regular channel 'sections. the geometric element.s can be expressed mathematically ill terms of the depth of flow and other dimensions. of the sectioll. For complicated sections and sections of natural streams, however, ,110 simple formula can be 'written to express these elements, but curves representing the relation -between these elements and the depth of flow can be prepared for use in hydraulic computations. The definitioris of several geometric elements of basic importance are given below. Other geoIlwtric elements used in this book will be defined where they first appear. The depth of flow y is the vertical distance of the lowest point of a /' 'chanilel section from the free surface. This term is often used inter-' changeably with the depth offlow section d. Stridly sp~aking, the depth of flow section is the depth of flow normal to the direction of flow, or the height of the channel section containing the water. For a channel with a longitudinal slop~ angle e, it can be 8een that the depth of flow is equal to the depth of flow section divided by cos e. In the case of steep channels, therefore, the two tei'ms should be used discriminately. The stage is the elevation or vertical distance of the free smfa-ce above 'a' datum. If the lowest point of the channel section is chosen as the datum, the stage is identical with the depth of flow. The top width T is the width of channel section at the free surface. / The wafer al'ea A is the cro~s-sectional area of the fi~w norlD,al to the / direction of flow. The wetted perimeter P is the length of the line of intersection of the channel wetted surface' with a cross-sectional plane normal to the direction bf flow. ' I Many typica.! seWer sections arEi described in [2J nnd [3].

OPEN CHANNELS AND THEIR PROPERTIES

23,\

The hydr.aulic radiu,$ R is the ratio of the water area to its wetted perimet.er, or. A R (2-1)

'1

The hydr~ulic de1Jth D is the ratio of ~le wa.ter area to the top width~ or D

=

T

(2-2)

The slJction factor for critical-flow contputtllion Z is the product of the . water area and the square root of the hydraulic depth, or

Z=A

(2-3)

.j

. The section factor for uniform-flow compulaUon A]z% is the product of the water area and the two-thirds power of the hydraulic radius. .

\

"

I

.!

.

FIG. 2-1. Geometric elements of a. circular section.,

I

~lements

Tabie 2-1 furnishes 11 list of formulas for six basic geometric of' 1 seVf\11 . commonly' used channel sections, For a circular section, the curves in Fig. 2-1 represent the ratios of the geometric elements of the I s~ction to' the corresponding elements when t11e section is fl~wing full.] These curves are prepared from It table. given in Appendix,A. For cer- '. tain trapezoidal, triangular, and parabolic sections commonly found in practical uses, the diag.rams given in Appendix B provide a convenient Ii means of determining the geometric elements. . '

I

'{ [

·1

,\

\ ' \

i

--

24 )

), ,-

"

~',.#

, Example 2-1. Compute the'hydraulic radius, hydraulic dep~h, and section factor.?: Of the~ra.pezoidal chUlluel section in Fig. 2-2. The depth of flow is' 6 ft. ,

'

~~' , '

'

Q,?!/' ~ '

\

~~ .~' C~b=20':, .1' \ '~//l7~~0~

.

....&..

T:44"

P=46,8 -

25

OPEN CHANNELS ,AND 'TIIEIR PROPERTIES

BASIC PIIINCIPLES

,

1.5

t~.5'

,

.

'.

I

Tropezuidal chonnel

Triangular channel

,

FIG. 2-2. A channel crosS'section. $oluti{)ll. By formulas given in Table 2-1, the following are computed: P = 20 +' 2 X 6 ";5 = 46.8 ft; A = 0,.5(20 + 4'!L0 6 = 192.0 ft>; R = Hl2/46.8 = 4.10 ft; D = 19%4 = 4.37 ft; and Z = 192 -./4.37 = 401 ft,·,:

Sholtow ditch

.

NarroW' (@c!anglJlar 5~ction

a

),

\'I

2-4. Velocity Distribution in Channel Section. Owing to the pres~ ence of a free surface and to the friction along the Ghannel wall, the velocities in a channel are not uniformly distributed in the channel section. The measured maximum velocity in ordinary channels usually appears to occur below the free .sUlface at a distance of 0.05 to 0.25 of the depth;

Pipe Natural irregular channel

FIG. 2-4. Typical curvea of equal velocity in various channel sections.

FIG. 2-3. Velocity distrib,ution in'·a. rectangula; 'channeL

I r

r I, .

the closer to the banks, the deeper is the maximum. Figure 2-3 illustrates *e general pattern of veJocity distributio;l over \;al'ious vertical and horizontal sections of a rdctangular channel sectio~ and the curves of equal velocity in the cross section. The general patterns for velocity distribution in several channel sections of other shapes are illustrated in Ji]ig. 2-4. ; , I The velocity distributfoJ in a channel section depends also on other f('l.clors, such as the unusu~l shape of the section, tlle roughness of the

channel~
26

OPEN CHAN)lELS AND THEllR PROPERTIES

BASIC PRINClPLES

The pattern will include. one spira}.on each side of the center line , where . th€ water level is the highest. In practical considerations, it is quite safe to ignore the :;piral motion in straight prismatic channels. Spiral flow in curved channels, however, is an important phenomenon to be considered in design add wHl be discussed later (Art. 16-2).

(cleonlcur Jines

'or

equal componenl lv'l

(o)Con!our lines 01 equal ve c for (v)

Co I Con lou, lines of equal componenHv,1

(d) ConI our lines 01 . equal component (vyl

(e) Dire clio n Jines and

Jr

ma9niludes of the IOlerol currenlslvor '

FIG. 2-6. Distribution of the velocity components, fa.cing downstream a.t the midsection of a. straight flume. Voloeitiestl.re In em/sec (= 0.0328 fps); y/b = 1.0; R "" 73,500; and Q = 701iters/s.ee (= 2.47 cfs). (Afler .A. ShlLkry [6J.)

2-5. Wide Open Channel. Observations in very wide open channels have shown that the velocity disttibution in the central region of the section is essentially the same as it ;would be in rectangular channel of infinite width. In other words, under this condition, the sides of the !iliJ!imel h~i practically DO influence on the velocit.y distribution in the C~gi~l, and. the flow i~_Y;~~.Eehtral region c~n therefore be regarded as two-dimensional in hydraulic ana:1yses. Careful experiments indicate, further, that this central region '~Xi~t8 in rectangular channels only when

a

~

I

27

the width. h,grea~.E 5 to 10 times. the depth <:>f flow ...4~IJe~on the .2_'?~ion of l:lu):fac~ roughness. Thus, a wide Open channel can safely be defined as a rectangular channel whose width is greater than 10 tiri1es the dept.h of flow. For either experimental or anaJytical purposes, the flow in the cen tral region of a wide open channel may be' considered to be the same as the flow in !1 rectangular channel of infinite width. 2-6. Measurement of Velocity. According to the stream-gaging procedure of the U~S. Geological Survey,'lthe channei crosssecticH1 is-ciivid~d -~rti6a1 strips by a number of successive vert~cals, and mean velocities in verticals are determined by measuring t.he velocity at 0.6 of the depth J!-l_~Gh vertical, or, where morc reliable results are required, by taking the average of the velocities at 0.2 and 0.8 of the depth. When the stream i.s covered with ice, the mean velocity is no longer close t.Q 0.6 of the water dept.h, l;mUI!e average at 0.2 and 0.8 of the water depth still gives reliable resul.1§. The average of the ~ean velocities in any two adjacent verticals multiplied by the are:1- between the verticals gives the discharge through this vertical strip of the cross section.: ThG sum of discharges through all strips is t.he total discharge. The mean velocit.y of the who.Ie section is, therefore, eq ua! to the total discharge divided by the whole· area. It should be noted that the above methods are simple and approximate. For precise measurements more elaborate methods must be used, which are beyond the scope of this book. 2-7. Velocity-distribution Coefficients. As a result of nonuniform 9.istributiol1 of velocities over a channel section/the velocity hea.d of an open-channel flow is generally greater than the value computed according to the expression V 2 /2g, where V is the ri;lea.n velocity. When the,energy principJe is used in computation, the true velocity head may be expressed as a V 2/2g, where a is known as the energy coefficient or Co-riolis coe.fficient, in honor of G .. Coriolis [12J who first proposed it. Experimental data indicate that. the value of a varies from about 1.03 to 1.36 for fairly straight pri!:imR.tic channels. The value is generally higher for small channels and lower for large streams of considerable depth. The nOllUniform di.stribution of '/elocities also affects the co.rnputation of momen~umin open-channel flow. From the principle of mechanics, the "momentum of the fluid passing through a channel section per unit time is expressed· by {JwQVj(J, where f3 is known as·the mom.entum coefficient or Boussinesq coe:ffi.cient, after J. Boussinesq [13] who fii'st proposed it; w.is the unit weight of water; Q is the discharge; and V i.'3 the mean velocity. It is ,generally found that the value of (3 for fairly straight prismatic cha.nnels varies approximately from 1.01 to 1.12. The two velocity-distribution coefficients are always slightly larger than the limiting value of unity, a.t which the velocity distribution is 1

For details see

!9J

to [11 J.

.

_,. t:../

_

)

'I

!

) .

· ,

\

28

,

BASIC PRINCIPLES

OPEN CHANNELS AND THEIR PROPERTIES

strictly uniform across the channel section. For channels of regular cross section and fairly straight alignment, the effect of nonuniform velocity distrihution on the computed velocity head and momentum is small, especially in comparison with other uncertainties involved in ~he computation. Therefore, the coefficients are often assumed to be unity. In channels of complex cross section, the coefficients for energy and momentum Can easily be great as 1.6 and 1.2, respectively, and can vary quite rapidly from section to section in case of il'I'egular alignment. Upstream' from weirs, in the vicinity of obstructions, or near pronounced irregularities in alignment, values of a greater than 2.0 have been observed. 1 , Precise studies or analyses of flow in such channels will require measurement of the'actual velocity and accurate determination of the coefficients. In regard to the effect of channel slope, the coefficients are usually higher , in steep channels than in flat channels. For practical purposes, Kolupaila [16J proposed the values shown below for the velocity-distribution coefficients. Actual values of the coefficients for a number of channels may be found in {l7) and [18J.

as

I

"I

I!

Vo.lue of", '

Value of {j

Channels Min

Av

'Max

lVIin

Av

Ma.x '-"0-

Regular channels, flumes, spillway:;: .... , N o.tural streams and torrents ..... . . Rivers.under ice cover., ..... ....... River vlllleys" ove,flooded ..... ...... ~

r

I,

I r I

~

1.10 1.15 1.20 1.50

1.15 1. 30 1.50 1.75

1.20 1.50 2,00 2,00

1.03 1.05 L05 1.10 1.07 '1.17 L 17 1.25

1.07 1.17 1.33 1.33

2-8. Determination of Velocity-distribution Coefficients. Let LlA be an elementary area in the whole water area A, and w the unit weight of water; then the weight of water pasSing LlA per unit ,time with a velocity v is 'wv LlA. The kinetic energy of water passing AA per unit time is wv 3 AA/2g. This is equivalent to the product of the weight wu AA and the velocity head v2/2g. The total kinetic energy for the whole ,WB.ter area is equal to 2:wv ll t1Aj2g. Now, taking the whole area as A, the mean velocity as V, and the 1 A va.lue of", = 2.08 was computed by Lindquist [141 uslng da.~a from Wl}it' measurements made by Ernest W. Schoder and Kenneth B. Turner. In the case;of clolled conduits, much larger values of ,. have been obS,erved [15]. A value of '" = a.87, observed at the outlet section of a. dro.ft tube in the Rublevo power pmnt, is probably the largest known value obtained from actual measurements; the re",l value there must have been still lar!!~r-,-10,2% more, if the effect of a. IS" curvattire of the streamlines is taken into account. The largest known value from laboratory meO:suremen1;$ is believed to be '", = 7.4, which was derived by V. S. Kv:ia.tkovskii in 1940 in the VIGM (All-Union Institute for Hydraulic Machinery, U.S.S.R.) for the spiral flow under !l. model turbine wheeL

29

2

corrected velocity head for the whole area as a V /2g, the total kinetic energy 'is ,aw A/2g. Equating this quantity with 2:wv 3 AA/2g and reducing, J1]3 dA L;'~! LlA (2-4) V3A ,'"'"

va

The momentum of water passing AA, per unit time is the product of the mass W
, Jy 2 dA {J

=

V2 A

2:v 2 LlA "". V2 A

(2-5)

O'Brien nnci Johnson [19] used a graphical solution of the above formulas sa follows: From the measured velocity-distribution curves, the area ivithin each curve of equal velocity is planimetered. Taking the velocity indicated by each equal-velocity l!'!!rve as v, a curve of 1]3 against the corresponding planimetered area is constructed. . It is ,evident that the area beneath this v3 curve is the integral 2:v 3 AA, which can be obtained by planimetering again. Similai:ly, integrals 1:v 2 Ail. and 2:v LlA can also be obtained. The integral 2:v AA divided by A gives V. With these quantities determined, the above equations can be solved for the coef- . ficients a and p. For approximate values, the energy and momentum coefficients can be 'compl.lteaDytne following formulas: 1 Ct

'p

= 1 + 3.% 1

+

f2

2t 3

(2-6)· (2-7)

where Ii = v.u/V 1, VM being the maximum velocity and V being the mean velocity; . Computation of the velocity-distribution coefficients for. irregular natural channels will be discussed later (Art. 6-5). In most pl'nctical problems dealing with regular cha~nels it is not necessary to consider the varia.tion of velocity throughout the cross section, since use of the average· veiocity will give the accuracy required. The expressions V2/2g and wQ V j g a~e used extensively in this book with the understand:ing either that these items have been cOlTected for the effect of the non~ uniform velocity dis~ributionj Of that a value of unity is asiumed. 2 I These formula.s a.re obLained by I1ssuming a. logtLrithmic distribution Qf velocity (Art. 8-5, Prob. '8-9). Assuming a lincar velocity distribution, Rehbock [201 obtained c< = 1 f" I1nd.{j = 1 "/3. . • For discussions on this subject, the reader may look into. 121J and [22]. However, he should use' judgment in reading these references because they contain erroneous

+

+

OPEN CHAN,lIjELS AND THEIR PROFER'I'IES

BASIC PRINCIPLES

30

2-9. Pressu're Distribution in a Channel Section. The pressure at any point on the crO,';fi section of the flow in a channel of small slop: can be , measured by the height of the water column in a piezometer tub~ Instayed at the point (Fig. 2-7). Ignormg mInor disturbances due to turbulence, etc., it is apparent that this water column should rise from the point of measurement up to the hydraulic grade line or the water surB face. Therefore, the pressure at any point on the section is directly propor(al tional to the depth of the point below the free ~ui'face 9,nd equal to the hydrostatic pressure corresponding to this depth. In other words, the distribution of pressute over the cross section of the channel is the 'same as the distribution of hydrostatic pressure j that is;the distribution is linear and can be represented by a straight line AB (Fig. 2-7a). This is known ,as the, hydrostatic latO of preSS1lrB distribution . ./ Strictly speaking, the application of the hydrostatic law to the pressure distribution in the ,crQ§s se~tiQn of a flQwing , 8 s' channel is valid only if th&.!o\v filaments (c 1 have no accele~~n coinponertts in the FIG. 2-7. Pressure distribution in straight and curved channels of ·plane·gf cross sectioE' This type of flow small or horizontal slope at the ~oretically known as parallel flow, l section under considera.tion. h = that is, such th'at the streamlines ha VI! piez~metric head; h. = hydrosta.tic head; and c "" pressure-bead cor, neither substantial curvature nor diverrection for curvature. (a) Par~ gence. ~~uently, there are n.o agallel flow; (b) convex flow; (c) preciablc acceleration component.§..JlQ!:= concave flow. mal to the direction of flo'\' that \VO~ disturb the hydrostatic-pressure d~J;.iQ.u in thfLcros~ctien of-flt--

)

31

In actual problems uniform flow i:s practically parallel flow. Gradually varied flow may also be l'egarded .as parallel flow, since the cha.nge in depth of flow is so mild that the streamlines have neither appreciable curvature hoI' divorgence; that is, the curvature and divergellce are so small that the effect of the acceleration components in the cross-sectional plane is negligible. For practical p1.rposes, therefore, the hydrosta.tic law . of pressure dislribution is applicable to gra,dually varied flow as well as to 1miform flow. If the curvature of stl'eamlines is substantial, the flow is theoretically ~-> known as curvilinear flow. The effect of the curvature is to produce appreciable ELGceleration components or cell trifugal forces normal to the direction of flow. Thus, the pressure distribution over the section deviates from the h}'drostatic if ourvilinear flow occurs rn the vertical plane:' Such curviiinear flow may be either convex or concave (If;lg~nd c). In both cases the .nonlinear pressure distribution is represented by AB' instead of tlte straight distribution AB that would occur if the flow wel'e parallel. It is assumed that ail streamlines are horizontal at the section under consideration. In concave flow the centrifugal forces are pointing dDW:lward to reinforce the gravity action; so the resulting pressure is greater than the otherwise hydrostatic pressure of a parallel flow. In convex flow the centrifugal forces are acting upward against the gravity action; consequently, the resulting pressure is less than the otherwise hydrostatic pressure of a parallel flow. Similarly, when divergence of streamlines is great enough to developappl'eciable acceleration components normal to the flow; thehydrostati'c pressure distribution will be disturbed accQrdingly. Let the deviation from an otherwise hydrostatic pressure h. in a curvilinear flow be designated by c (Fig. 2-7b and c). Then the true pressure or the piezometric height h = h; + c. ' If the channel has a curved longitudinal profile, the approximate centrifugal pressure may be computed, by Newton's law of acceleration, ~~,~he E~~~~~ oL~L@iWt d and a croSQ..§ection. Qf 1 sq ft, that is, wd/g, and the centrifugal ac~ v2/r; or ' _----7

~.-~---

, ~l"tlow.

",/'

statements. Some authors have proposed the use of the mOluentUnl coeffic~en.tto , r'epJace the energy coefficient even in computations based on the energy PrJ?ClpJ~, " t ec't V'Nhether the energy coefficient or the momentum . , coeffiCient . . o1 ,IS d TI 1IS IS no carr . to be used depends on whether the energy or the moment?m prw::lple -:,e . The two coefficients are derived independently from baslca.lly different principles (Art. 3-6). Neither of them is wrong and neither ca.n be replaced by t~e other; both should be used in th~ correct sense. ' " ' I Specific qualifica.tions for parallel flow were clea:rly stated for the first tlme by Belanger [23]. '

=

wd~~ g

r

)

(2-8)

where w is the unit weight of water, g is the gravitational acceleration, v is the velocity of flow, and'r is the radius of curvature. The pl'esstirehead correction is, therefore, d v2 (2-9) c=-gr

IS Ill:r

;

\

.

p

~

.I

)

,

For computing the value ofc at the channel bottom, r is the radius of cu::',rature of the bottom, d is the depth of flow, and for practical purposes

J,

(

.

. i

32

"

BASIC PRINCIPLES

OPEN CHANNELS AND 'l'HEIR PROPERTIES

v m~y be ass~med equal to the average velocity of the flow. Apparently, c is positive for concave. flow, negative for convex flow, and. zero for parallel flow. ,. In parallel flow the pressure is hydrostatic, and the pressme head may be represe11ted by the depth of flow y. For simplicity, the pressure head of a curvilinear fiow may be represented by City, where a'is a correction ~oefficiimt for the curvature effect. The l:orrecti.on~oeffici~nt is referred to as apressure-dislribtttion coejfiy'ient.. Since this coefficient is applied to a pl'esswe head, it may be specifically SLa~'/m1~:!t!~~!1fficienl. It can be shown that the pressure coefficient is expressed by a'

/., i

I\ I

r.

~y}o

A

1

hI! dA

+

1

r

A

cv dA

sure head at any vertical depth is equal to this iiepth multiplied by a correction fact·or cos 2 e. . Apparently, if the angle (J is small, this factor will not differ apprecill.bly .fl'Om unity. In fact, the correction tends' to decrease the pressure head by an amount less than 1 % until e is nearly , 6" i a slope of about 1 in 10. Since the slope of ordinary channels is far less than 1 in 10; the correction foi: slope effect can usually be safely _ _-:i"" ignored .. However, when the cha~l_slope is large and its eff5l9t becoroe~ appreciable, thecorrection should be made if Mcumte comput~ttion is -'-~-.----.

-'

Pressure di.stribution

FrG. 2-8. Pressure tlistribution in parallel How in cha.nnels of lurge slope.

desired. A channel qf this type, say, with a slope gre~ter than 1 in 10,1.'.3 hereafter called a channel of large slope. Unless specifically mentioned, all chMlnels descrlbeanereafte-r are ~onsidered to be channels of small slope, where the slope effect is negligibie. If a channel of large slope ,has a. 10ngitudin!11 vertical profil£,; of appreciable. curvatul'e~ the pressure head should be cOl;rected fo]' the effect of the curvature of streamlines (Fig. 2-9). In simple notation, the pressure head may be expressed as g'y cos 2 Jt., """~~~~S== In channels of large slope the usu . andhighel' thl1n the critical velobity. When this velocity reaches a certain riHl.gni, "tude, the flowing water ,vill entrain nil', produ'cing a swell in its volume:i .and Ml increase in depth.l For this rel1son the pressUl'C computed uy Eq. {2-11) or (2-12) 4,~n shown in several gases to b.e higher than th~

h = Y cos 2, e h = d 90S e

'

.

'on. ~ertical section A'C

where d = ~ cos e, the depth measured perpendicularly from the water surface. It should be noted from geometry (Fig. 9-1) that Eq. (2-11) does not apply strictly to varied flow) piwticularly when 0 is v~ry large, whereas Eg. (2-12) still applies. Eq~lition ·(2-11) states that ,the presor--,

-

(2-10)

}o '

where Q is the total discharge andy is the depth of flow. It can easily be seen that a' is greater than 1.0 for concave flow, less than 1.0 for convex. flow, and equal to LO for parallel flow. . For complicated curved profiles, the total pressure distribution can be determined approximately by the fjow~net method or DlOre exactly by model testing. , In ra..mr!l.t.Y!1ried flow theghange in depth of fl.oJYJs so rapid and abl'Upt .:that th
or

1

=~

33

JAil' becomes entraine~.in ",,,,ter generally...!!! v"e19\l~ of !>~t go Ips and higher.[ Besides velocity, however. other factors such'as entrance condition, channel rough., ness, distance .traveled, cha.nnel cross section. volume discharge, etc •• all have some; bearing on a.ir entrainment. ' . ;

,

M. Hasurrh has measured the distributibn of pressure along the slopihg faces of we~ [241. The data. obtained from these ekperimenta have verified Eqs'. :(2 .. 11) and (2-12) very sa~Jsf!l.Ctorily (25]. . 1

or

i

I

I

35

BASIC PRINCIPLES

OPEN CHANNELS AND 'rHEIR PROPERTIES

actual measured pressure obtained 'by model testing. If the average density of the air-water mix~ure is known, it shouM be' used to, replace the density of pnrc water in the computation when air entrainment is expected. The actual density of the mixture varies from the bottom to: the surfafaTor the floV\L For practical purpo;es, however, the density may be assumed constant; this assumption of I!!llform air distributi_o_I?:_i~_

struct curves showing the relationships between the depth y and the section elementS A; R, D, and Z; and (b) determine, from the curves the geometric elements for y = 4.

34

Distance f,om a reference point neClr left bank, ft Left bank: -5

Stage, ft

5.6 4.6 4.0

-4 -2

1.9 0.8 0.2 0.3 0.2

0 1

2

3 5

'1

Distance fro~ a referenc(J point near .left bank, ft 7 9 11 13 15 17

Stage, ft -0.1 -0.1 -0.4 -0.1 0.7

19 Right bank: 20

3.2 4.1

j

\\

2.6

j \

2-6. The hydrostaLil" cll.tenary may be plott.ed for [Lny given depth l/ and slDpe angle O. at its ends by the following two approximate equations: :1:)=

fk [(1 -

%k' - 1)1.g,k4).p

+ (%k' + %2k4) sin 21/>

Concave flow

FIG. 2-9. Pressure distribution in curvilinear flow in channels of large slape.

the cross section wiII simplify cQmputation, with, the errors on the safe side.

.[

PROBLEMS 2-1. Verify the formulas far geometrio elements of the seven channel sections given in Tahle 2-1. $-2. Verify the cUrY'es shown in Fig. 2-1. 2-3. Construct curves siinilar tQ those shown in Fig. 2-1, for a square channel section. 2-4. Construct cu'rY'es similar to those shown in Fig. 2-1 for an equilateral triangle with one side as the channel bottom. 9.-5. From the data g(ven below on the cross section 1 of a natural stream con·

. I (2-13)

I

l

(2-14)

YI = if cos 10

Convex, flow

~ %56k~ sin 4.j.]

where :1'1 and YI, respectively, are the ordinate and abscissa measured from t,h.e midpoint of the free surface; k = sin (110/2); '" = sin- 1 f [sin (1/>/2)l!kl; and 11 is the slope angle at the point (XI,l!I), varying from 0 at the bottom of the curve to 9, at the ends. The above equations will define' the. cross section when the flow is at its full dept.h. The slope angle at the ends of a hydrostatic catenary of best hydraulic efficiency is found mathema.tically to be II, = 35'37'7". (a) Plot this section with' a depth y = 10 ft, and (bl determine the values of A, R, D, and Z at the full depth . 2-7. Estimate the Ylllues of momentum coefficient (j for., the- given values of energy c(lefficient ex = 1.00, 1.50, and 2.op. , 2-8. Compute the energy and mo~entum coefficients of the cross st'ction shown in Fig. 2-3 (a) by Eqs. (2-4) and (2-5), and (b) by Eq~; (2-6) and (2-7). The cross section and the curves of equal velocity can be transferred to a piece of drawing paper and enlarged for deSired ll.ccuracy. 2-9. In designing side walls steep chutes and overflow spillways, prove that the overturning moment due to the pressure of the flowing water is equal to Yswy' cos' 9, wherew.is the unit weight of water, y is the vertical depth of the flowing water, and 9 is t,he slope angle of the channel. 2 ..10. Prove Eq. (2-10). 2-11. A high-head overflow spillway (Fig. 2-10) has a 60-ft-radius flip bucket u.t its downstream end. The bucket is not submerged, but acts to change the direction of the flow from the slope of the lipillway face to the horizontal and to discharge the flov1 into the air' between vertical training walls so ft apart. , At: a discharge of 55,100 ds, ;the water surface at the vertical section OB is at El. 8.52. 'Verify t.he curve that represents the computed hydraulic ,pressure acting on the training wall at section DB. The computatiQn is bailed an Eq,' (2-9) and on'the following assumptions: (1) the velqcity is uniformly di~tl'ibuted across the section; (2) the vo.lu,e used for r, fQr pressur~ values near the wall base, is 'equal to the radius of the bucket but, for other pre;isure values, is equal to the radius of the concentric flow lines; and (3) the flow is entto.ined with air, and the density ,of the air-water mixtureca~ be estimated by the

(,

"

, i

of

/a)

1 It is common practioe to show the cross section of a stream in a direction looking , downstream and to prepare the lQngitudinal profile qf a channel so that the wate~ flows from left to right, ;unless this arrangement would bit to show the feature to b~ illustrated by the cross'section and profile. This practice is generally fqllowed bt most 'engineering offices. However, for geographical reasons or in order to depict clearly the location and profile of a stream, the profile may be shown with water ftow~ ing from right to left and the cross section ma.y be shown looking upstream. This happens in ma.ny drawings pre'pared by the TennesseEj Valley Authority, because the Tennessee River and most of its tributaries flow from:east tQ west, and so are shown with the direction of flow from right to left on a, conventional map.

1

f

)

j )

.1

,.

! 36

BASIC PRINCIPLES

OPEN CHANNELS. AND THEIR PRO'PERT1ES

Douma. formula,' that is, 'U -

10

~0.2V: . gR

- 1

(2-15)

where u is the percentage of entrained .air by voiume, V is the velocity of flow, and

R Is the hydraulic radIus. . 2-12. Compute the wall pressure on the section OA (Fig. 2-10) of the spillway described in Prob. 2-11. that at section DB.

It is assumed that the depth of tlow section is the same!l.S

/

\«

$PilIW1!Y

Iraining wall,

eo II

cpO!1

;:;

C .2 ::l

.)

:OJ

t;j

i. 2

~ I

/, i

4

,a Un;l pressare, II 01 woler

\I

FIG, 2-10. Side-wall pressures on the flip bucket of a spillwa.y. 2-13. Compute the wall pressure on the section OA (Fig. 2-10) of the spillway descrIbed in Prob. 2-11 if the bucket is submerged with a tailwater level at EL 75.0. It is !l.SSulned that the pressure resultbg from the centritugal force or the submerged jet need not be considered beca.use the submergence will reault in a severe reduction in velocity.

REFERENCES 1. S. F. Averillnov: 0 gidravlicheskom raschete rusel krivolineinoI formy poperech,

2.

3. .

4.

nogo secheniia (Hydraulic design of channels with curvilinear form oithe crosS section), lzvestiia Akademii Nauk S.S.S.R., Otdelenie ~'ekhnic"eskfk;h Nauk" Moscow, no. 1, pp. 54-58, 1956. Leonard Metcalf and H. P. Eddy: "American Sewerage Pra.ctice," McGraw-Hm Book Company, 1M., New York, 3d ed., ,1935, vo!. 1. Harold E. Babbitt: "Sewerage and Sewage Treatment," John Wiley &: Sons, Inc., New York, 7th ed., 1952, pp. 60-:.66. H. M. Gibb: Curves for solving the hydrostatic oatenary, Engineering News, vol. 73, no .. 14, pp. 668-670, Apr, 8, 1915.

I This iormull!. [26J is based on da.ta obtained from actual· conorete and wooden chutes, involving errOnl of ±10%. '

37

5. George Higgins: "Water Channels," Crosby, Lockwood &: Son Ltd., London, 1927, pp.15-36. . . 6. Ahmed Shukry: Flow around bends in an open flume, Transactions, AmericilTl Society of Civil Engineers, vol. 115, pp. 751-779, 1950.· ' 7. A. II. Gibson: "Hydraulics and Its Applications,'" Constable &: Co., Ltd., London, 4th ed., 1934, p .. 332. , 8.J. R. Freeman: "Hydr·a.ulic Laboratory Practice," Amedcan Society of Mecha.nical , Engineers, New York, 1929, p. 70: ' 9. Don M. Corbett and ot.hers:8trealn-ga.ging procedure, U.S. Geologicnl SlI1vey, Water Supply Paper 888, 1943. 10. N. C. Grover and A. W. Harri'ngtoo.: "S.ream FlOW," John Wiley &; 80·ns, Inc.) New York, Hl43. 11. Standards for methods and records of· hydrologi~ measurements, United Natio7ls Economic Comm.isslcn for Asia: and the Fa:r Ei.I$~, Flood Control Series, No.6, Ba.ngkok, 1954, pp. 26-30. , 12. G. CorioUs: Sur.l'etablissemellt de Ill. formule qui donne la figure des remons, et .sIU· 12. ilorrection tiu'on doH y int,roduire POllr tenir compte des diffel'ences de vitesse dans les diVers points d'une marne section d'un COUl'ant (On the ba.ckwater-curve equation a.tid the corrections to be introduced to !lccount for the difference of the velocitie$ at different points on the same cross section), Ivnmoire No. 268, ..,l,n'nalca du punts et chaw;sees, vol. 11, ser. 1, pp. 314-335, 1836. 13. J. Boussinesq: Esg's'i sur la theorie des eaux courantes (On the theory of flowing waters), M~moire& ]fr/;sentes par diven savants ri l'Academie des Sciences, Paris, 1877. . . 14. Erik G. W. Lindquist: Discussion un Precise. weir measurements, by Ernesf W. Schader andT(ennethB. Turner, 1"7'(tllaac:l.ions, American Society of Civil Engineers, vol. 93, pp. 1163-1176, 1929. 15. N. M. Shcha.pov: H Gidrometriia Gidrotelchnicheskikh SoorllllheniI i Gicir,omashin" (" Hydrometry of Hydrv.lllic Structures and MacJ:Jnery ") I Gosenel'goizciat, . . Moscow, 1957, p. 88. 16. Stcponas Kolupaila: Methods of determin!l.tion of the kinetic energy facto!', The Port Engineer; Calcutta, India., vol. 5, no. I, pp. 12-18, Januo.ry, 1956. 17. M. P. O'Brien and G: H. Hickox: "Applied Fluid Mechallics," McGraw-Hill Book Company, Inc., New York, 1st ed., 1937, p'.272. ' 18, Horace William·King; i'Handbook of Hydraulics," 4th ed., l'evised by Ernest F. Brater, McGraw-Hill Book Company, Inc., New York, 1954, p. '7-12. 19. Morrough P. O'Brien and Joe W. Johnson: Velocity-head correction for hydrau1ia flow, Engineering News-Record, vol. 113, 0.0.7, pp. 214-216, Aug. 16, 1934. . 20. Th. P..ehbQck': Die Bestimmung der I,age der Energielinie bei ftiessenden Gewfulsern mit HilIe des GeschwindigkeitshOhen-Ausgleichwertes (The determina.tion of the position of the energy line in flowing water with the o.id of velocity-head a.djustment), Der Bau.ingenieuT, Berlin, vol.. 3, no. 15, pp. 453-455,·· Aug. 15, 11122. 21. Boris A. Bak&meteff: CorioIis and the energy principle in hydraulics, in "Theodore von !Urman Anniversary Volume," California. Institute of Teohnology, Pasadena, 1941, pp. 59-65. 22. W. S. Eisenlohr: Coefficient's for velocity distribution in open-Channel flow, Tra.nsac:I.ior/.$, American. Socie4/ of Civil Enginee7's, voL 11:0, pp. 633-644, 1945. Discussions, pp. 645-668. 23. J. B. Bela.nger: "Essai sur la solution numeriqne de ql,lelques problemes relatifs au mou.-ement permanent des eaux courantes" ("Essa.y on tIle Numerica.l Solution of Some Problems Relative to Steady Flow of Wa.ter"), Carilian-Goeury, Paris, 1828, pp. 10-24.

\

,

38

BASIC PRINCIPI..ES

24. It Ehrenberger: Versuche Iiber die Verteilung der Drucke an Wehrriicken infolge des I1bsturzcnden '.Vassers (Experiments on the distribution 'of pressuresa\ong the f~~e of w(d ..;; resulting from the impact of the fa.lling water), Die W IMJserwirtschaft, Vienna, vol. 22, no. 5, pp. 65-72, 1929. 25. 'H&rald Lauffer: Druck, Energie und Fliesszustand in Gerinnen mit grossem Gefiille (Pressure, energy, and flow type in channels with high gradients), Wasserkrafl, und Wasserwirtschaft, Munich, vol. 30, no. 7, pp. 78--82, 1935. 26. J. H. Douma: Discussion on Open channel flow at high velocities, by L. Standish Hall, in Entra.inment of atr in flowing water: a symposium,T1'ansactions, American Society of Civil Engineers, vol. 108, pp. 1462-1473, 1943.

LeI!.'foI.l-

.2.

.

Pit' 67/\1c: ,Il.1 ,f;.-N

h}jc/rct,;.d.i :/'t'--10 f? .

d

,

.ftu;o 4,4ff luitf&

Yl

tU. ,lillJ-' foe-t ~

.

3

1

ENERGY AND MOMENTUM PRINCIPLES

eN

c: c ,'" Y . 5 . W/ibn

CHAPTER

(,

3-1. Energy in Open-channel Flow. It is known in elementary hydraulics that the total energy in foot-pounds per ponild of water in any streamline passing through 9, channel section may be expressed as the total head in feet of water, which is equal to the sum of the elevation

<

7 tUi, rv:5 (id..

t

Y-'.c,tt/).Je1'

@ ,

..k.": In a CCt·n~l.pdjW 4t- /~1{."fp"U.1/f'A(iCL

_ P ,~,> J: .. ::>..;t;.1".!/f"''' V"'r1.tl.-7

I

HI]~II tt'l"1

.

c. "e J' .,t:;

'I

VW' _..:!.~,~" .it.-.:.~(j

~'i._ .:di.l:£nnd~~L4;'1~ -t&.4i-:5 p,ttt'a . e>J rret ,/·1/0 fall . ..vV ,he. ellA-of /:J~"m ;$ dtre,·r/6..,;tLul ·'n'YII!'. .s~t:rt, Vr';,t.. cd cbf #.. is /l. J'Vi4 v..u ¢ ,pver.( e @'we <5 (I!

~y.& o,C;...l."< ~

"'"'t

iJ d /,1 (f?)e.

'!, 6 rtf..~'( • .§

/u..-",,' fYr..f t! Ci.~"Vl/ r1 f~J .

/

/J

/

'.

IS

r1.!1 /!.~Ci//. bkrl~)~""'p1)

t·'1' .

'7....

('/.

d

Saelioil 0

~

I

I

I

"

.) • (

ftvtAP. f

t'k,~e l',., ctylv) '

Jt#rry.: (IU9A .~) . ~it..d-(c4blh ( .bq:...... e It:;rY:J

< .5q,-'Ui4'V"t

d.c;ot/..,

I

1.__ ~:lum _1_1. 1.~_ __ •

FrG. 3-1. Energy in grad'.lo1ly varied 'open-channel Row.

above a datum, the pressure head, and the velocity head. For example, with respect to the datum plane, the totltl head H at a section 0 conta.ining point A on a streamline of flow in a channel of large slope (Fig. 3-1) may be written H

!/YiM..ltU... (l!':71tdVt) ~-;P. T>/~~cf

I

CttYD.iJ,.u.:; r;~yo;o (}v'Iv: vI<

1

= ZA +

d A, cos

8+ V2;

2

(3-1)

where ZA is the elevation of pain t A above the datum plane, dA is the depth of point A below the, water surface measured along the channel section, 8 is the slope angle of the channel bottom, and V.{2j2g is the velocity hea~ of the flow in the streamline passing through A. In general, every streamline passing through a channel section will have . 39

_. r"',

• i

,

40

BASIC PRINCIPLES

ENERGY AND MOMENTUM PRINCIPLES

~ different velocity head, owing to· the nonuniform vel~ity distribution ~ctual fl~w, .Onl~ in un ideal pal'a:lel fl~w of uniform velocity distributl.on can .he velocIty head be truly IdentlCal for aU points on the cross sectlOn. In the ~ase of gradually varied flow, however,· it may be assumed, for pr~ctlCal purposes, that the velocity heads for all points on the ch~nnel sectIOn are equa.l , and the el~ergy coefficient may be used ·to correct for the over-all effect of the nonuniform velocity distribution. Thus, the total energy at the channel section is

-3-2. Specific Energy, Specific energyl in a channel section is defined as the energJi per pound of water at any section of· a channel measured with respect to the channel bottom. Thus, according to Eq. (3:-2) with Z = 0, the specific energy becomes .

U1

H

=

z + rl cos () .

For channels of small slope, 0 = O. nel section is

S. = d cos or, for 8: channel of small slope and a

2

.'

+ c ;P2g .-

(3-3)

Consid~r !lOW a prismatic channel of large slope (Fig. 3-1). The line r~pl'eSelltlltgJhL~_~~va~ion of the tota.Lhea..cLO-f.Jlow is the lill.&gy_line. ;rhe slope of the hne IS known as the energy g'radient, denoted by Sf.

The slope oLthe water surface is de;noted by Sw and the slope of the channel bottom l by So = sin () . In uniform £10'" - . () . ", S J-- S w-- S o-Sln. According to the principle of conservation· of energy, the total energy head at the upstream section 1 should be eql.lal to the· total energy head at t~e downstream seetion 2 plus the loss of energy hf between the two sectIons; or ...

)

I,

Zl

+d

l

V;2

+ al ~2·g .

cos /1

.

=

Z2

V"

+ d z COs /1 + az ---.l: 2g

.L

,

. h

'f

(3-4)

This equation applies to parallel or gradualJy varied flow. For a channel of small slope, it becomes .

.

ZI

V; + YI .+ al -2 = g

Z2

. + Y2 + a2 -V~ +h 2g .f

Either of these tW? equations is known as the energy equation. = .a2 = 1 and hr = 0, Eq. (3-5) becomes

(3-5)

Wh~n

al

. +

V I2

. . Yl+ -2 g

Zl

.·.

=

Z2

+y~

V 22

+ -2g

const

(3-())

This is the well-known BernoHl,li energy equation.2 1 The slo~e is generally defined as tan O. For the present purpose, however it is defined as sm O. ' 2 .. It ~s .believed th~t this equil-tioni is ascribed to the Swiss mathematician Daniel Berno\llh only lllf~l'ence, to gjve recognition to his pioneer achievement in hy.dro1yn~mICs, m ,partlcular the in~roductio'n of the concept of "~ead." Actuall}', tlu~ e~uatlOn was first formulated by; Leonhard Euler a.nd later popularized by Julius WelSbl\ch [1 J. .. . .

, \ 'JI

?y

+

V2

a 2g

,.

)

(3-7)

1,

=

E =y+-

(3-2)

Thus, the tots'! energy at the chan..

H = z +d

8

V!

V + a --.:. 2g

41

(3-8)

. 2g

which illdic!l.tes that the specific energy is equal to the sum of the depth of water and the velocity head. For simplicity, the following discussion' will be based on Eq. (3-8) for a channel of small slope. Since V = Q/ A, Eq. (3-8) may be written E = y + Q2/2gAz. It can.be seen that, for a given channel section and discharge Q, the specific energy in a channel section is a function of the depth of flow only. When the depth of flow is plotted against the specific energy for a given channel section and discharge, a specific-energy CUTve (Fig. 3-2) is obtained. This curve has ,.two limbs AGand BG. 'rh~ .lifl.lb AC.~J)proache;3the ,horiz.ontall\-xis asymptoticallx toward the righ.t. The limb Be approaches the line OD as. it extends l.lpward and to the right. Line OD is a line that passes through the origin and !ul:,s _ananglfi Qf lr~eJlP.ati.QJUiQl!j11 to '!5~!-­ For a channel of large slope, the angle of inclination of the line OD will be different from 45°. (Why?) At any point P on this curve, the ordinate represehts the depth, and the: ,abscissa represents the specific energy, which is equal to the snm:6f the pressure head y a.nd the'velocity head V 2 /2g. . The curve shows that, for a given specific energy, there are tW9 possible depths,. for instance, the low stage YI and the high stage y~. The low stage is· called the alternate depth· of the high stage, and vice vel'sa. At point G; the specific energy is a n~inimum. It will be proved Inter that this condition of minimllriJ. specific energy corresponds to the critical state of fio.v. Thus, at the critical state the two alternate depths apparently become one, which is known as the critical depth 1/e. When the depth of ;fiowis greater than the critical depth, the velocity of flow is less than the critical velocity for the given! discharge, and, hence, th:e fiow js sub criticaL When the depth of flow is less than the critical depth, the flow is supercritical. Hence, YI is the. depth of a supercritical flow, and 1/2 is the depth .of a subcritical flow. ' ' If the discrarge changes, the specific energy will be changed· aucordcurves A' B' and A" (Fig. 3-2) represent positions of ingly. The

1wo

1

B."

The concept of specific energy was first ir~troduced by Bakhrneteff [21

id 1912.

42

BASIC PRINCIPLES

,the specific-energy curve when the discharge is less 'and greater, respectively, than the discharge used for the construction of the curve AB. 3-3. Criterion for a Critical State of Flow. The critical state of flow ' has been defined (Art. 1-3) as the condition for which the Froude number is equal to uni~y. A more common definition is that it is the state of flow at which the specific energy is a minimum for a given discharge. 1 A

ro----

At the critical state of ftowthe specific energy is a dE/dy =0. The above equation, therefore, gives

VI D 2g - 2

min~mum,

N\':;" ,~

4

or

(3-11) j Supercritica I

For a channel of large slope angle 8 and energy coefficient for critical flow can easily be proved to be

)

ct

Vl 2g

=

D cos 2

e

ct,

the criterion

(3-12)

where D is the hydraulic depth of the wateI: area normal to the channel bottom. In this case, the Froude number may be defined as ,Fw. 3-2. Specific-energy curve.

(3-13)

theoretical criterion for critical flow may be developed from this definition as follows: Since V = Q/ A, Eq. (3-8), the equation for specific energy in a channel of small slope with a = 1J may be written Q2 217A -

E=y+--,

It should be noted that the coefficient a of a channel section actually ·varies with depth. In the above derivation, however, the coefficient is assumed to be constant; therefore, the resulting equation is il0t absolutely exact. 3-4. Interpretation of Local Phenomena. Change of the state of flow from subcritical to supercritical vice versa occurs frequently in open channels. Such change is manifested in a corresponding ch&nge in the depth of flow from a high stage to a low stage or vice versa. If the change takes place rapidly o'/er a relati velyshort distance, the flow is rapidly and varied and is known as a local phenomenon. ' The hydraulic drop , . hydraulic jump are. the two types M local phenomena, and, may be . described as follows: Hydraulic Drop. A rapid change in the depth of flow from a high stage to a low stage will result in a steep depression in the water surface. Such a phenomenon is generally caused by an abrupt change .in the channel slope or cross section and is known as a hydraulic drop (fig. 1-2). At the transitory region of the hydraulic drop a reverse curve usually

(3-9)

or

Diffe~entiatiilg with respect to y and noting that Q is a constant,

.

dE = 1 _ ~ dA = I _ VI dA dy gA3 dy gAdy . "

The differential water area dll near the free surface (Fig. 3-2) is equal to T dy. Now dA/dy = T, and the hydraulic depth D:= A/T; so the above equation becomes ' dE V2T 'V2 -=1--=1-dy 'gA , gD I Th~ concept of critica.l depth bas'ed on the theorem of minimurh energy introdl.).ced by BOss [3J. '

Wall

~

first

, ,<

,

(j

(3-10)

This is the criterion for critical flow, which states that at the critical state of flow, the velocity head is equal to half the hydraulic depth. The above eq~ation may also be written V!...;gJ5 == 1,which means F = 1; this is the definition of critical flow given previously (Art. 1-3). If the above criterion is to .beused in any problem, the following conditions must be satisfied: (1) flow parallel or gradually val:ied, (2) channel 01 small stope, alld (3) energy coefficient assumed to be unity, If the energy coefficient is not assumed to be unity, the criti~al-flow criterion is

y -----I

~~1

43·

ENERGY AND MOMENTUM PRINCIPLES

{

() I, 1

"

i.,'

I'

j

,

,I

I

44

\I, \

BASIC PRINCIPLES

ENERGY AND

appears, connecting the water surfaces before and after the 'drop.' The point of inflection on the reverse curve marks the approximate position of the cdtical depth nt which the specific energy is a minimum and the flow passes from a "SUbcritical state to a supercritical state. The free ove'rfall (Fig. 3-3) is a special case of the hydraulic drop. It occurs 'where the bottom of a fiat' channel is discontinued. As the free overfall entel'S the air in the form of a nappo, there will be no reverse curve in the water surface until it strikes some object at a lower eleva.tion. It. is the law of nature that, if no energy were added from the outside, the y

v'{ijl,/::~LJ)

'7

f"'~'''1i

MOME~TUM

PRINCIPLES -

45

.~~h~~ the brin~.

T~actual situation i~ that the brink section is thg true section of minimum energy, but it is not the critical section as comjnite(fEyt'te prineiple based on the parallel-flow assumpti~ Rouse [~l ... found that for small slopes the computed critical depth is about 1.4 times the brink depth, or Yo = 1.4yo, and that it is located about 3y< to 4yc "behind the brink in the channel. The actual water surface of the overfall is shown by the full line (Fig. 3-3). ,..-It should be noted that, if the change in the depth of flow from Do high stage to a low stage is gradual, the flow becomes a gradually varied flow

-,

o

-::!~

_ Ii

""c.
Y

Y

o "

" "" o.tl: /Theore.licOi water surfoce

---"""",-""-,,,,- ==:::t' ~s~~nLP~~e.!...! I~ _ _ _ -'Actual WQter surface - -- - -'\. _ _ _ _ _ _ _ _ _ ~

E

Ir.ilial deplh

Specific-ene'g)'

FIG. 3-4. Hydro.ulic

ju~np

CUrye

Hydraulic jump

Specific-·force

Culye

interpreted by.specific-energy and specific-force curves.

~

FIG. 3-3. Free overfa.ll interpreted by specific-energy c~rve.

\

'

1

water sl1T~!:\cewould seek its lowest possible position corresponding to the least possIble content of energy dissipation. If the specific energy at an upstream section is E, as shown On the specific-energy curve, it will COl1ti~u.e to be dissipated on the way downstream and 'Will finally reach a mllllmum energy content E",,,.. The specific-energy curve shows that the section of minimum energy or the critical section should occur at the brink. r~~~~tie less_tha!Lth.~,J~titical depth _because !Er~he:~ecl'eas.e," In deJ2th woul£.!eguire a.TI, increo.se in sgecific energy, ~.pos~ll>l~o.!ll.p~nsating external en~,ls sup~lie
having a prolonged reversed curve of water surface; t.his.phcnomenoll may ~Clllled ~ grad'unl hud7'a1t~ic drop and is no longer a local phenomenon~ Hydraulic Jump. When the rapid change in the depth of fiow is from a low stage to a. high stage, the result is usually an abl'llPt rise of water ~e. (Fig;. 3-4, in which the vertjc!l,l scale is exaggerated). This local' phenomenon is known a~J.he hydraulic j1tmp. It occurs frequently in a canal below a regulating sluice, at, the foot of a spillway, or at the place where a steep chann!)l slope suddenly turns jlat.' If thejump is low, that is, if the change in depth is small, the water will not rise obviOtisly and abruptly but will pass from the low to the high stage through a series of undulations gradually diminishing in size. Such a low jump is called an uooular jtimp. . When the jump is high, that is, when, the change in depth is great, the jump is called a direct jump. The direct jump involves a relatively large amount of energy loss thro,ugh dissipation in the turbulent body of water in the jump. Consequently, the energy content in the flow after the jump is appreciably less than that before the jump. . It may be noted that the depth before the jump is always less than the

46

BASIC PRINCIPLES

depth after the jump. The depth before the jump is called the initial .depth y 1 and that after the jump is called .the .sequent depOt Y2; The initial and sequent depths VI and Y2 are shown on the specific-energy curve (Fig. 3-4). They· should he distinguished from the altern!J.te. A~p~hs YI and Y2'! whi(J~~e the two nossible depth~ fo!:....the same specific ener~l' _.~~itial and s~lent depths are actual depths before and after a jump in which ~rgy lOss b.E is invoh-ed. In other words, the specific -energy E 1 at the initial depth Vl is greater tha.n the specific energy at the sequent depth .y~ by an amount equal to the energy loss AE. II there were 110 energy losses, the initial and sequent depths would become identical with the alternate depths in a prismatic channel. 3-5. Energy in NQnprismatic Channels. In preceding discussions the channel has been assumed prismatic so t.hat one specific-el~ergy curve could be applied to evil sections of the channel. For non prismatic channels, however, the channel section varies along the length of the channel and. hence, the specific-energy curve differs from section to section. This cbm'plication can be seen in a three-dimensional plot of the energy curves along the given reach of a nonpl'ismatic channel. For demonstrative purposes, a nOllprismatic channel with variable slope is taken as an example, in which agradually varied flow is carried from a, stibcritical state to a supercritical state: (Fig. 3-5) .. The vertical profile of the channel along its center line is plotted on the Hx plane with the x axis chosen as the datum. For a variable-.slope channel, it is more convenient to plot the total energy head H = z + y + V Z/2g, instead of the specific energy, against the depth of flow on the By plane. For simplicity, the pressure correction due to the slope a.ngle and curvature of flow is ignored in this discussion. An energy line is then plotted on the Hx plane below a line parallel to the x axis and passing through the initial total head at the H axis. The exact position of the energy line depends on the energy losses along the channel. Four channel sections are then selected and four energy curves for these sections are plotted in the Hy planes ~ shown. The initial section 0 is an upstream section in the sUbcritical-flow region. The two depths corresponding to a given total energy H 0 can be obtained from the energy curve. Shice this section is in the subCl'itical-flow region, the high stage yo should be the actual depth of flow, whereas the low stage is the alternate depth. Similarly, the alternate depths in other sections can be obta,ined. In the downstream sections rand 2, the low stages Yl and Y2 are the actual depths of flow singe they are in the supercritical-flow region. The critical depth at each se ytion can also be obtained from the energy curve at the point of minimum energy. At &ection C the critical flow occurs, and the depth y, is the critical depth. On the H x plane, varioUs lines can finally be' plotted; showing the channel bottom, water surface, critical-depth line, and

ENERGY AND MOMENTUM PRINCIPLES

47

alternate-depth line. At the critical section, it is noted that the three lines, namely, the .yater surface, the critical-depth line, and the alternate-depth line, intersect at a single point. It is seen' tho.t, in passing through the critical Reotion, t,he water surface entei'S the supercriticalflow region smoothly. The· three-dimensional plot of energy curves is complicated. The description given here is used only for helping the reader to visualize the problem. In actual applications, the energy C'lrves may be constructed

1 \

II

I I i I !

I

I

! II

X

FIG'. 3-5. Energy in a non prismatic channel of variable-slope, carrying gradually va.ried flow from sllbcritical to supen::ritical state.

separately on a number of tWo-dimensional Hy planes for the chosen sections. The data obtained from these curves are then used to plot the water surface, critical-depth line, and alternate-depth line on a twodimensional Ex plane. For simple channels, the energy curves are not necessary because thi;l critical depth and alternate depths C3.n easily be computed directly, Eumple3-L ·A rectangular cha.nnell0 ft wide is narrowed down to 8 ft by a contrMtion 50 ft long, built of straight wa,lls and a horizontal fiooc. If the discharge is . 100 cfB and the depth of Bow is 5 ft On the upstream side of the transition section, determine the flow-surface profile in the contra.ction (0.) allowing no gradual hydraulic drop in the contraction, Ilnd(b) a.llowing a gradual ,hydraulic drop ha.ving its point of inflection a.t the mid-sectioll of the contra.ction. Th!Jrip.tionalloss through the contraction is negligible. .

,

.

{ \

:l

48

BASIC PRINCIPLES

ENERGY AND MOMENTUM PRINCIPLES

. , . Sol1ttiDn. From the given data; the total'energy in the approaching flow meas~red above the channel bottom i~ u: = 5 {100/(5 ?< 1O)1'/2g = 5.062 ft. This energy is kept constant throughout l,lIC contractIon, since energy losses are negligible. A horizontal energy line showing the elevation'of the tobl head is, therefore, drawn on the channel· profile (Fig. 3-6). . "

49

., critical depth at t}Jis section is equal to the total head divided by 1.5 (Prob. 3-3), or 5.062/1.5 = 3.375 ft. By :r::q, (3-10), the critical velocity ts'eqllnl to V, = V3.375g = 10.4.5 fps. Hence, the width of this' critical section should be 100/(10.45 X 3.38) = 2.83 fi. ' With the size of the mid-section determined, the side walls of the contraction can bo drawn in ,with straight lines. The lm~ and high stages at each section are then computed by the equation previously'given, As the flow upstream from the critical section is subcritical, lts water surface should follow the high stage. Downstream from the critical secti6n, the flow is sllpercritical and its Burface profile 'follows the low-stage line. The criti.::al-depth line is shown to sepai'ate the high from the low stage or the subCl'itical from the stipercritical region of flow. On the basis of Eq. (3-10), the critica.l depth can be computed from the equation

+

(lOO/by,),

2g

y,

=

'2

. y = {/lO,OOO , "gb%

or

where' b is the width of the channel, which can be measured from the plan, It should be noted :that the vertical scale of the channel profile is greatly exaggerated. Furthermore, the outline of the gradual hydraulic drop is only theoretical, based on the theory of parallel flow. In reality, the flow near the drop is more or less curvilinear, and the .actual profile would deviate from th/'! theoretical one.. . This example also serves to demonstrate 11 method of designing a channel transition (Arts. ll-5 to 11-7), The designer may fit any type of contraction walls he desires to suit a given flow profile, or vice versa, .

3-6. Momentum in Open-channel Flow. As stated earlier (Art. 2-7), the momentum of. thefiow passing a channel section per 'unit time is expressed by pwQV /y, where p is the momentum coefficient, w is the l.mit and V is the mean weight of water in lb/ft i , Q is the discharge in velocity in fps, ,-----?'?>- According to Newton's second law of motion, the change of momentum per unit of time in the body of water in a flowing channel is equal to the . resultant of all the external forces that are acting on the body,Applying this principletO channel of large slope (Fig. 3-7), the fplfo~~ing expl:ession. for the momentum change per unit tirnein;the body of water enclosed betv.:een sections 1 and 2 may be written:

FIG. 3-6. Energy principle applied to a channel contraction (a) without gr:adual hydraulic drop; (b) with gradual hydraulic .(hop. '

cfs,

The alternate depths for the given tot ..l energy. e..n be computed by Eq. (3-9) as follows: . ' " 100' 5.062 = Y + 2g(by)' or

. , - 5 06'> " y . ~Y

+ 155.25 b'.

=

a

0

This is a cubic equtl.tion in which b is the width of the channel. At the entrance sec-' tion, where b = 10 ft, its s61ution gives two positive roots: a low stage 'YI = 0.589 ft, which is the altemate depth; and ahigh stage Y2 := 5.00 ft, which is the depth of flow. At the exit section ,where 11 = 8 ft, this. equation gives a low ati\ge YI =. 0.750 ftiand a high stage Y. = 4.964 ft. ! When no gr~dual hydraulic drop is allowed in'the contraction (Fig. 3-6a), the1depth of flow at the exit section.should be kept at the high stage, as shown, The high stages for other interm£ldiate sections are then compute~ by the above equation, whicli giveB the flow-surface ptofile. Similarly, the low stages are computed by the aboye procedure and indicated by the alternate-depth line~ ; When a grodua:~ hydrauiic drop is desired in the contraction (Fig. 3-6b), theldepth of flow at the exit ~ection should be at the low stage, Since the point of inAection of the drop or 11. critipal section is maintained at th~ mid-section of the c~ntracti
.

'~l 0(

(3-14)

J(.. 1 •.;.. "'>l·~)(qt

-

cit!

)>1'

_, Yn

I

,I

I !

!,

C(

, . , where Q, w, and 1:::' are :as.. previously defined, with subscl'ipts refe1'l'in'g to ' sectionr-l~nd P! and P 2 are the resultants of pressures acting on the two sections; W is the weight of water enclosed between the sections; and F! is the total external force QL friE,tion and reslstanc~~ing-,ilong the lLUliace of conta,Qt bet\V'een.the water and the cha;nl)el.The above equa-' tion. .is known as the m;omentum equation, l I · ,

2;

1

The application of the inomentum principle was first suggested by Belanger [5J.

,

50

r-

51

ENERGY AND )';IOMENTUJI{ PRINCIPLES

BASIC PRINCIPLES

rectangular channel of small slope and wi~lth b (Fig. 3-7),

For a parallei or gradually varied flow, the values of PI and P 2 in the momentum equation may be computed by assuming a hYdrosta.tic distribution of pressure. For a cur:-vilinea.r or rapidly varied flo"l,' however, the pressure distribution is no longer hydrostatic; hence the ~alues of PI and P z cannot be so computed but must be cOl'rected for the curvature effect of the streamlines of the flow. For simplicity, P 1 and P2 may be replaced, respectively, by {)!'P 1 and f1~'P2' where {:Jt' and {)z' are the correction coefficients at the two sections. The coefficients are referred

PI ;'~Wb1l1:l. P z = 7§.wb yz2 FI = wh/by

. and Assume

L

where h/ is the friction head and ii is the avera.ge depth, or (YI+ Y2)/2. The discharge ~hrcillgh the reach may be taken as the produot of the average velocity and the avera.ge area, or

I

Q

I

~ ~i(Vl

+ V~)bii

Also, it is evident (Fig. 3-7) that eha weight of the body of water is W = wbfjL

and

sin

{j

=

Substituting :111 the abo,'e expressions for the corresponding items in Eq. (3-14) and simplifying, .

.\

(3-16)1

I Co t1r I

1.

Fro. 3-7. Application of the momentum.principle.

to as pressure-dislribution coefficients, Sjnce and P2 are forces, the. coeffi(Jients may be specificall~r (JaIled force coefficients. It can be shown thu,t the force coefficient is expressed by {:J' =

1-:: fA AZ)o

hdA= 1

+~ fA AzJ~

cdA

(3-15)

where z is the depth of the centroid of the wa.ter area A below the free surface, h is the pressure head on the elementary area dA, andc is the pressure-head correction [Eq. (2-9)1, It can easily be seen th'1t pI is !.Treater than 1.0 for ooncave flow, less than 1.0 for and equai =~~ to 1.0 f <2.r...1lllJ:allel flow: It can be shown that the momentum e.quation is similar to the energy equation when ~pplied to certain flow problems. In this case, a gradually varied flow.is considered; accordingly, the pressure distribution in the sections may be assumed hydrostatic, and (1' = 1. Also, the slope of the . channel is ass~ed relatively small. 1 · Thus, in the short reach of a 1 If the slope a.!l~le 8 is large, then PI = }iwdl~ cos 0 and P. = ;fW~1 cos O,where ell and el, are the Beptbs of flow section and cos IJ is a. correction factor (Art. 2-10). , ,a. tI

/::}. A j'(,h .{- c)id JI. = " j.Ad4 . 1 <.l(cd .4 A >(:i:.;. ! c dA

() j:

•

IC!cr-'T(~ 0

'tl.

t'Yltkfr; ~t·-Z,a:k.~ 1.iJ" ~ e"" t Jt-l'!' ~.~~?}

i

i

This equation appes,rs to be practically the same as the energy equation (3-5). The.oretically sp'3aking, however, the two equations not only use different velocity-distribution coefficient,s, although these are nearly equal, but al~o invol;e different meanings of the frictiollltllosses. ,II!. the energy !ill.!ll!-t!.on, the Item hr mea::;yres t.he intl?rna.l energy dissipat.ed in the whole ~.~s of the water iI!tbe reseh, whereas the item hi in the momentum equatirJll measures the losses due to external forces exerted 011 the water by the walls of the channel. Ignoring the small difference between the coefficients a and fJ, it seems that, in gradually varied flow, the internalellergy losses are practically identical with the losses due to external forces. In uniform flow, the rate with which surface forces are doing work is equal to the rate of energy dissipation, In that case, therefore,. a dist,inction between hI and h/ does not exist except in definition. The simila.rit,y between the applications of the energy and momentum principles may be confusing: A clear ullderstandingof the basic differences in their constitution is important, de.'lpite the fact that mmany instances the two principles will produce practically. identical results. The inherent distinction between the two principles lies in the fact that energ'J is a i5calar quantity whereas momentum is a vectC)r quantitT also· the ~nergy equation contains a term for' internal losses, where:s th~ momentum equation contains a term for external resistance. Generally speaking, the energy principle offers a simpler and clea~er I;

r +~ (colA 1/ z

) '"

I

52

~.

BASIC PRINCIPLES

53

ENERGY AND MOMENTUM PRINCIPLES

explanation than does the momentum principle. But the momentum principle has certain advl.I.ntages in application to problems involving high internal-energy changes, such as the problem of the hydraulic jump. If the energy equation is applied to such problems, the unkriown internalenergy loss represented by h, is indeterminate, and· the omission of this term would result .in considerable errors. If instead the momentum equation is applied to these problems, since it deals only with extemal forces, the effects of the internal forces will be entirely out of considerfltion and need not be evaluated. The term for il'ictionallosses due to ex~ernal forces, on the other hand, is unimportant in such problems and can safely be omitted, because the phenomenon takes place in a short reach of the channel mid the efiect due to external forces is negligibl~ compfkreq with the internal losses. . Further discussions on the solution of the hydraulicjump problem by both principles will be given IELter (Example 3-3). An example showing the application ·of the momentum principle to the problem of It broad-crested weir is given below. Example 3-2. Derive the discharge per unlt width ofa broad-crested weir across a rectangular channel.

approach section 1 and the downstream section 2 at themininium depth of the weir, the following equation may be written:

qw g

(!1.Y2 _!J_) YI

= HWYI' -

MWy2 2

-

OIl

the top

Hwh(2y, - h) .

.

where q is the discharge per unit width of th~ weir'. /' Experiments by Doeringsfeld and Barker 181 ·have shown that, on the/average, VI - 11, = 2Y2. III that case the above equation can be simplified ani1 solved for q, q = 0.433 .

V2Y (-4-h ) l~ H%

I

·Y'T

(3-17)

//.

Conslderillg the limit.of h from ,zero to Infinity, this equation variesjfo~'!l'lR.~L tv q = 2.4.6RH. It is interesting to note that the practical range·1lf tfie coefficient to H~" obtained by actnal observations' is from 3.05 to 2.67. In applying the momentum principle tiJ"this problem, .it can be seen that knowledge of the internal-energy losses due to separation of flow at the entrance and to other causes ·is not needed in the ,analysis. .

3-7. Specific Force. In applying the momentum principle to a short horizontal reach of fl. prismatic channel, the extern:al force of friction and the weight effect of water can be ignored. Thus, with (J = 0 and F, = 0 and assuming also {31 = {3z = 1, Eq. (3-1.4) becomes

=

Qw (V2 -- 17 1) g

PI - P2

The hydrostatic forces PI and 'P 2 may be expressed as

PI = wi 1 A. 1

a.nd

P 2 = wi 2 A 2

where i l a~d i2 are the distances of the centroids of the respective wa.ter areas Al and Az below the surface of flow. Also, Y I = Q/ Al a.nd V 2 = Q/ A z• Then, the above momentum equation may be written

ir. + ilAl = il.... + ZZA2 gAl

FIG. 3-8. Momentum principle ~pplied to flow over a broad-cregted weir. Boilltiun. The assumptions to be made in this solution (Fig. 3"8) are (1) the frictional forces Fr' and F," are negligible; (2) the depth y. is the minimum depth on the weir; (3) at the channel sections under consideration there is parallel flow; and (4) ~~~r ,pressure £w'on ~e weir surface is equal to the total hydros~.tic p~.!!.!!re , meD.S\.Ired below the upstreamwatei: surface, or..

p ..

=

Hwh[y,

+- (YI

- 11,)] =

Hwh(2YI - h)

The accuracy (If the last assumption has been checked ~xperimentally [61. If the momentum equation (3-14) is ·applied to the body of water between the upstream J

..

r0 0 1">'1 e'Y°;t~,- [1""< a-"1,..""; 7 (~~4li/M ::

,

. '- 'Y)

c'Af:i

j .. /.::

.r-?

,rgf1lJ,o/,J<.

(){;.:aty/'b .d:./n Ct"7fd-t. 'c.~ t5 If) f3 L

/, .ve-.le, i:-~ ' r · »( , I
g.fi 2

(Qr //.,-c 1-1,) (3":18)

'The v:llue of the coefficient actually depends on many factors: mainly, the rounding of the upstream corner, the length and slope of the weir crest, and the height of the weir. Many experiments on bl'oad~crested weirs haye been performed. From several of the wen-known experiments King [7J has interpolated the data and prepared tables for the coefficient uncle·r various conditions. A comprehensive analysis including more recent data and a presentation of the results for practical applications were made by Tracy [8J. The well"known experiments all broad-crested weirs are (1) Bl].zin tests performed in Dijon, France, in 188.6 [9]; (2) U.B.D.lV.Bi Cornell tests performed at Cornell University in 1899 by the U.S. Deep Waterways Board under the direction of G. W. Rafter, and U.S.G.B. Corncllle.sts performed·by the U.S. Geological Survey under the direction of Robert E. Horton in 1903 [101; (3) Michigan tests performed at the University of Michigan during H)28-1929 [11]; and (4) MinneBola and· Washington tests performed,'respectively, at the university of Minnesota and Washington State University [6]. For some formulas and coefficients of discharge developed in the U.S.S.R., see [12]. For an analytical treatment of the problem, lIee· '[13].

55

BASIC PRINCIPLES

ENERGY AND MOMENTUM PRINCIPLES

The two sides of Eq. (3-18) are analogous and, hence, may be expressed for ahy channel section by a general function " Qz'

For a minimum yalue of the specific force, the first derivative of F with respect to iJ should bl') zero, or, from Eq. (3-19), '

54

.

F

- i t zA

dF

"(3-19)

u4'_

' This function cOMists of two terms. The first term is the momentum of the flow passing through the channel section per unit time per unit weight of water, and the second is !:.l!.e for.J<.ELrillr....unit weigb.:Lm w~t.!lr. Since , both terms al~e essentially force per unit weight of water, their sum may , , be ,c~lled the 8PlJcifjE.l(jr.c~.1 ,Accordingly, (3-18) may be expressfld

Q2 dA ,d(zA) dy T --elY

gP

'

,

For a change dy in the depth, the correspo,nding change 'd(iA) in the static'moment the water area. about the free surface is equal to [A (2 + dy) + T(dyP/2] - zA, (Fig. 3-9). Ignoring the differential of higher degree, that is, assuming (dy) 2 = 0, the change in static moment becomes d(iA) = A dy. Then the preceding equation may be written

of

dF

_ Q! dA

gA2 dy

B

Since dA/dy reduced to

c· o

~=-:-_. 45° for ()

0

= T,

Q/ A

V,and AfT

+A =

i"

!i

Ez '£lEiE:, channel of ,......, zero or small slope (0]

l.bl

3-9. Specific-fofce curve supplemented \~ith specific-energy curve. energy curve; (b) channel section; (e) specific-force curve.

(a) Specific-

'

as F 1 = F 2. . This means that the specific forces of section,s 1 and 2 are equal, p.r:ovided t~t the e~rnal forces and the weight effect of water in the..rea<;h between the two s~ctiQns can be igno];ed. By plottillg the depth against the spec~fic force for a given. channel section and discharge, a ~pecific-jorce CU'fve is obtained (Fig. 3-9). This curve has two limbs AC and BC. The limb AC approaches the horizontal axis asymptotically toward the right, The limb BC rises upward and extends indefinitely to the right. For a given value of the specific force, the curve has two poss(ble depths Yl and Yl' " A13 will be shown later, the two depths constitute the il1it~al and sequent depths of a hyqraulic jump. At point· C on the curve the two depths become one, and the specific force is a minimum. The following argument shows that the depth at the minimum value pf the specific force is eqv,al to the critical depth.~ , 1 This has been variously called the "force plus momentum,'~ the "momentum ftux," the "total force;" or, briefly, the "force" pf a stream (see pp. 81 and 82 of [14J). The {unction represented by Eq. (3:'19) was formula.ted by Bresse [15J for the study of the hYdra.ullc jump to be described in Example 3-3. : i The conoept of critic!!.1 depth based on tbe theorem of momentum :is believed to ha.ve:been developed by Boussinesq [16).

I,

D, the above equation may be (3-10)

i

I I (

,I

= 0

This is the criterion for the criMeal state of How, derived e~rlier (Art. 3-3). Therefore, it is proved that the dep~h at the minimum va-lueaf the specific force is the critical depth.I It may also be stated that at the critical state of flow the specific'jorce is a minim.'um for the given discharge.

NQw, comp£1,re the specific-force curve with the specific-energy curve, (Fig. 3-9). For a gi;iell specific energy Ell the specific-energy curve indicates two possible depths, namely, a low stage YI in the supel'critical flow region and a high stage yz' in the subcritical flow region. 2 , For a given v~lue of F1, the specific-force curve also indicates two poosible depths, namely,.ail initial depth lit in the supercritica.l region nncla sequent depth ljz in the sub critical flow region. It is assumed that the low stage and the initial depth are both equaJ to YI' Thus, the two curves indicate jointly that the sequent depth Y2 is a.lways less than the high stage 1/2'. Furthermore, the specific-energJ' curve shows that the energy content E2 for the depth Vi is less than the energy content El for the depth Y2'. Therefore, in order to maintain a constant value of F1, the depth of flow may be changed from Yl to Y2 at the price 'of losing a certain amouilt of energy, which is equal to El - E, = I:J.E. One example o( this is the 1 It should be noted that the above proof is based on the assumptidrul of para.liel flow and uniform velocity distribution. However, the concept of !lritico.l depth is: a genera.l concept, tl:..a.t is valid for aU flows, whether derived from energy or from momentum considerations. This validity has beeJ;!. proved by Jaeger [14,17,18), and the proof is known llS the J a.eger theorem {I9]. ' ~ In order to make 2. clear distinction between the sequent depth and the high of tile alternate depths, the sequent depth is designated by y. and the ):ligh sta.ge lit. In some other places in thLs book, however, both are designated by 1/1'

'.. ,I "'i

f

56

!'

BASIC PRINCIPLES

hydraulic jump on a horizontal floor, in which the specific forces before and after the jump are equal and the loss of energy is a consequence of the phenomenon. This will be explained further in the following example,' It may be noted at this point, however, that the depths YI and yz' shown by the specific-energy curve are the alternate depths; whereas the ?epths. Yl and Y2 shown by t.he specific-force curve are, respectively, the mitinl depth and the sequent depth of !1 hydraulic jump. ' Example 3-11. Dedve a relationship b~tween the initial depth and the sequent depth of 8: hydraulic jump on a horizontal floor in a rectangular channel. ' , Sob/ion. The el:ternal forces of friction and the wet'lH effect of Wil.Wr in the hydl'~ulic jump o~ a horizontal floor are negligible, because~ the-jump't.;;:kes place in a relatively short distance and the slope angle of the ,horizontal fioor is zero. The' specific forces of sections 1 and 2 (Fig. 3-4), respectively, before and after the jump, can,therefore be considered equal; that is,

i \ I

ENERGY AND MOMENTUM PRINCIPLES

57

section.' In applying the mornentur~l principle to nonprismatic channels, therefore,a three-dimensional plot similar to that shown for the application of the energy principle (Fig. 3-5) can be constructed. For practical purposes, however, this is rarely necessa.ry. Where there is no interyention 'of external forces or where these forces are either negligible or given, the momentum pdriciple can be applied to its best advantage to problems, such as t.he hydr.alllic jump, that deal ;"!I"':';...~;.tJ,,""'!j12!JiJ."'J="_<".bJ'-JlJ""'''''' th.at..!:~l!!!i~~. jUhe energy following example shows how the momentum design of a channel transition in which a hydraulic jump is involve(i. Example 8-4., A rectangular channel Sit wide,.earrying 100 cfs at a depth of 0,5 ft, is connected by a str:oight-wall transition to Il. channel 10 it wide, flowing nt.s. depth.

(3-18)

For ll. recLanguinr channel of width b, Q = V,A, = V.A.., Al = by" A, = bYt, = yi/2, and ii' - y~/2. Substituting these relations IlJld F, = V d v'Uih in the above equation and simplifying,

2,

- (2F,' + 1) (J!J)+ 2F,' 0 (:1!!)' y,. +. ~ - 2F,'] (~ - 1) - 0 [(1Lt)' YI YI YI '

Factoring, Then, let

~V + 1l! _ ( Y.:/ y,

(3-20)

=

~

2F • = 0 I

TIle solutjon of this quadra.tic equation is (3-21) .

For a. J:!.iv:? ,F'roude ?u~ber FI of the apprQaching flow, the xatio of the sequent. depth to the IlUt1l3l depth IS given by the above equa.tion. It should be understood that the momentum principle ill used in this solutiou because the hydraulic jump involves a high amount of internal-energy losses whioh cannot be evaluated in the energy equation., ' .The jOilit. UIlB of the specific-energy curve aud the specific-force curve helps to d~ter­ ~ne gra.plllcally the energy loss involved in the hydraulic jump for II. given appro8.chmgflow. For the given approaching depth 11., points P, ana. P,' are located on the spe~itic-force curve and the spec~c~energy curve, respectively (;Fig, 3-4). The point Pi'!gives the initial energy content E , . Dralv a vertical line, passing through the P?int PI and intercepting the upper limb of the specific-force curve at point P" which gives the sequent depth 1)2. 'Then, draw a horizontal line passing through the point P a and intercepting the specific-energy curve a.t point P.", which: gives the energy content E, after the j';lmp. The energy loss in the jump is then equal to El - E" represented by / l E . : '

a-8.Momentum

priIicipl~

Applied to

Nonprisrnati~

Channels.

The

sp~cifj.c force, like the specific' energy, varies with the snape ofthechannel

FIG. 3-10. (mnergy and'momentum principles applied to II. channel expansion (a.) with hydraulic jump; (b) without hydraulic jump. . .:

o~ 4 n (Fir;. 3~lO). Determine the flow; profile in the transition if th~ frictional loss through tl;\e tra.n&'ition is negligible. If, a hyqraulic jump occurs in the transition, how C!l.n i~ be eliminated? -:' , S"l'Uti"n~ From the given data, the tptal energy with respect to the channel bot~' tom in the approaching flow is E ~ 0.5 + [IOO/(O.S X 8))'/2g = 10.207 It, and in the dowiistrea.i:n, E - 4,0 + (100/(4 X 10)Ji/2y = 4.097 ft. ,It is appnhnt that this , ,

.

58

'BA.SIC PRINCIPLES

ENERGY A.ND MOMENTUM PRINClPLES

IlMrgy difference 01 6.110 It must be dissipated through the transition by ~ome means, since the frictlontll lru;s is negligible. Furthermore, the Fro.ude numbers 6.24 and 0.22 of the 8.pproaching a,nd downstrea.m flo'YS are, respectively, greater and less than unity, indicll.ting IJ. change of the flow from supercritical to Bubcritica.l. Therefore, a hydra.ulic jump can be expected to occur to dissipate the energy difference and to effect a change in the flow st,ate, Whether this jump will occur within the transition or in the upstream or. the downstream channel is, however, to be disclosed by fur~her analysis. .

for instance, by bolting orOM ,!limbers to the bottom. of the transition. It can be !LSSumed ill this a'(ample that the energy diff-crcnce of 6.1 10 ft is dissipated uniformly in tlte transition by artificial roughness. TllUB, the energy line in the tl'ansition is simply a straight line joining the total heads of the two end sections (Fig. 3-lOb). For. deSign p1l1-poses, it is convenient first to assUlne the lIow profile and then to proportion the dimensions of the tranllition so 'that the jump can be eliminated, In proportioning the transition, the jump is eliminated either by varying the wid~h or by raising the bottom of the transition. In this example. it is assumed that the bottom is to be raised, 01' "humped" (Fig.3-10b). The subsequent procedure of the compuGlttion is to (1) a,!lsume the flow profile; (2) compute the velocity head, which is equal to the difference. between the total heo.d B.nd the water-surface eleva.tion, at a llumbe!' of sel",cted sections; (3) compute the velocity and then the water area and depth of flow foreayh section; (4) determine the elevation of the bottom or the transition, which is equal to the devation of the water surfar.:e minus the depth of flow; (5) compute the o.lternate depth, ~ince the bottom of the transition is fixed; and (6) compute F'l and F, lines fo!' the low and high stages, an.d plot them on II convenient Sl'ale. It can be seen that the two F' lines inter3ect and' become tangent to each other a.t a critical section, where the :flow cha.nges from low to high stage, that is, from 8upercritical to subcritical stale. If the critic31-depthlinc is plotted, it will intersect the alternate-depth line and th~ water surface simultaneously at the critical section. Bl\Scd on the critica.l-depth line, a line of minimum speCific energy can also be construc,ted. This line should be tangent to the total-energy line at the critical flection.

TABLE 3-1.

CObfPUTATION FOR A CHAI'1N1:lL EXF.Ai'/SlON

EXAMPLE

Section width b, ft

3-4

Low stage 'II" ft

PI

0.500

78.6 78.7 78.8 78.8 78.8

for E = 10.207

8.00 8,50 9.00 9.50

0.419

10,00

0.398

o 470 0.4'.1:3

DESCRIBED

IN

., High stage 1/2, ft for E = 4.097

3.940 3.960 ,3.979 3.987 4.000

F, 71.9 75.9 79.9 83.6 87.8

Take 'five sections of the transition with their widths shown in Table 3-1. For the total approaching enel'gy of 1O.20i ft,.the low stage YI for each section call be computed by means of Eq. (3-8) or (3-9), or "':"""-1::--=::'-

where b is the width of the section. 4.097 can be computed from

+ VI,:""

59

PRODLEMS

3-1. With reference to a channel of small slope !l.ud a section shown in Fig. 2-2, (a) construct a family of specific-energy cllrves f(or Q = 0, 50, 100, 200, 300, and 400 ds, (b) draw the locml of the critical-depth point on these curves, (c) plot a curve of the critical depth against the'diilcharge, and (d) plot a family of curves of alternate depths, Yl vs. V~, for the given discharges_ . 8-2. Construct the specific-energy curve for a 36-in. pipe carrying 3:n open-channel flow of 20 cis (a) on It. flat slope, and (bj on a 30 slope. S-B. Show that at the critical state of How tha specific-energy head in a rectangular channeli;; equal to 1.5 times the depth of flow, assuming zero slope and a = 1. 3-4. Derive the equations for the lOCI15 of the critiMl-depth point on the specificenergy curve and for the curve of critical depth vs. discharge, as oota.ined in Prob. 3-L 3-6. Prove (3~12). 3-6. PrC/ve Eq. (3-13). 3-7. Prove that at the critica.l state o( flow the discharge is II ma::dmum for a given specific energy. 1 . 3-8. Show that 'the relation between the altc;rnate depths Y1,and 1/. in !1 rectangular channel ca.n be expressed by .

10.207

Similarly, the high st,age 1/2 for a total ene1'gy of

0

+ V2 =

4.097

Thelow- and high-stage linea are then construeted slong with t·lle energy lines (Fig. 3-lOa). After these stage and energy lines are determined, the specific forces P! and F. for low and high stages, respectively, at ea.ch section are computed and plotte~ to any convenient scale and datum. The hydraulic jump must, occur where the specific 'forces for the low and IUgh stages are equal, or at the intersection of the F lines. At. this
as

(3-22) where Y. is the critical depth. Using values of y.!Yc as ordinates !tud of Yz/Yt as abscissas, .construct a dimensiouleSiil graph for the above equation rand study its characteristics. ' The .concepG of critical depth based. on the theorem of maximum diacharge was first introduced by Belanger [20J. 1

r ~ '1

I \'

.1

\

".!

60

!!ABIC PRINCIPLES ,

ENERGY A.ND MOMENTUM PRINCIPLES

3-9. Solve the problem given in Example 3-1 (a) if there is a. total energy 108s of 0.60 ft unifol'Jnly diStributed throughout the lel!gth of the contraction, and (0) if a gradual hydraulic drop is desired with its point of inflection at !l. diStance 20 it upstream from tlie exit section. 3-10. Applying the momentumpl'inciple and the continuity equation to the analysis of a submerged hydrllulic jump which occurs at the sluice outlet in a rectangular channel (Fig. 3-11), prove that

where y. is thesubme!"ged depth; y, is the height of sluice-gate opening; y, is th~ tailwater depth; und F,' g~/gy!3, IJ being the discharge per unit width of the channel. Neglect the channel-bed friction i,.

FIG. 3-11. A submerged hydraulic jump at sluice outlet.

3-11. Prove tlmt the energy loss in a. horizont,.l hydraulic jump is

AE

()J2

-YI)' ./'

4j/lY2

_(3-24)*

3-12, If !I. hydrAulic jump is forllled on the ,horizontal floor at the toe of the spillway descl'ibed in Prob. 2-11, determine the sequent depUl and the energy loss involved in the jllmp. 3-ill. ,With reference to a Ilha.nnel of small slope and Ll. section shown in Fig. 2-2, (a) construct afamily of specific-force cUI'ves for Q = 0, 50, 100, 200, SOO, and 400 cfs, and (b) plot a family of curves of initial depth against sequent depth for the given discharges. 3~14. Construct the specific-force curve for a 36.-in. pipe c,nrying an open-channel flow of 20 cfs on Ii small sIope. 3-lD. Prove Eq. (3-15). 3-16. Using the momentum principle, show th!Lt the Froude number of a parallel or gradually Vllried flow in a cha.nnel of slope !Lngle IJ may be defined by

F

=

V VgD cos 8/13

61

'

(3-25)

* This formula wliU, shown by Brcsse early in 1850 115J. At the slime time Bresse illtrodu,ced the concept of critical depth, as a. dllpth at which the subcritical flow chl>nges to BUpel'criticai, or vice versa.

where V is the mean veloility, D is the hydraulic depth of the section, and t3 is the 'momentum coefficif;nt for nonuniform velocity distribution. ' 3-1'1. For eliminating the hydraulic jump in Example 3-4, the flow profile is assumed to be composed of two Feversed circular curves tangent to el1ch other at the middle aection of the transition and also to the water surfaces "in the connecting channels at the two ends of the transition. Verify the computation (shown in scale on Fig. 3-100). 3-18. A frictional loss of 1.0 ft is p,Ssumed to be uniformly distributed along the, ' length of the tra.nsition in EXMllple 3-4. Determine the flow pro.file in the tmnsition. REFERENCES

1. 'Huntel' Rouse and' Simon Ince: "History of H:rdraulics," Iowa Institute of Hydraulic Research, Iowa City, IOlVa, 1957. 2. Boris A. Bakhmeteff: "0 Neravnomernoln Dvizltenii Zhidkosti v Otkrytom Rusle" (U V9.r!ed, Flow in Open Channel "), St,. Petersburg, Russia, 1912. 3. PItHI Boss: "Berechnungder Wasserspiei\'ellage beim Wechsel des Fliesszustu,udes" ("C'Almputation of Water Surface with Change of the Flow Type"), SpringerVerlag, Berlin, 1919, pp. 20 and 52. 4. Hunter Rouse: Discharge characteristics of the fre.€; overfall, Civil E1I{}ineering, vol. 6, no. 7, pp. 257-250 .. April, 1936. 5. J. B. Belanger: "Resume de lec;.ons" ("Summary of Lectures"), Paris, 1838. 6. H. A, Doeringsfeld and C, L. Barker: Pressure-momentum theory a.pplied to the broad-created weir, Transactions, American .societv of Civil Engineers, vol. 106, pp. 93~-946, 1911.' .r , 7. Horace William King: "Handbook of Hydraulics," 4.th ed., revised by Emest F. Brater. McGraw-Hili Book Company, Inc" New York, 1954, pp, 6-1 to 6-Ht 8. H. J. Tracy: Discbar~e characteristics of broad-crested weirs, U.S. Geologica! Sm'vcy, Cir~ltlar 397, 1957. 9. H. Bazill: Experienceli nouvelles sur l'ecoulement en deversoir {Recent experiinents on tile flow of water over weirs}, 111"tmO'ires et Dor:u:mlmts, Anna/as des pants III cha1IssJ.S, 2e semestre, pp, 393-448, October, 188B. English trllnslatioll by Arthur Ma.richal and John C. Tl'antwine, Jr., Proc~ed:in.gs, Engingers' Club of Philadelphia, voL 7, no. 5) p.p. 259-310, 1890; Vi.)!. 9, no. 3, pp. 231-24~, and no. 4, pp. 287-319, 1892; and vol. 10, no. 2, pp. 121-164, 1893. 10, R. E. HOI·ton: Weir experiments, coefficients, and formul!J.s, U.S; Geologi.cal Survey, Waler S1I'pply and Irrigation Paper 150, 1906; revisedas Paper 200, 1907. 11. James G. Woodburn: Tests of broad-erested weirs, Trarlsuctian.1, American Society of Civil Engi1lee,s, vol. 96, pp. 387-416, 1932. 12. M. A. Mostk~w: "Handbuch der Hydraulik" ("Handbook of Hydraulics"), VEB Verl£1,g Technik, Berlin, 1956, pp. 188-195. 13. L. J. Tison: Le deversoir l\.seuil epais (The broad-creste'd weir), La' Houille blanche, Grenoble, 5th yr., no. 4, pp. 426-439, July-August, 1950. 14. Charles Jaeger: "Engineering l;'luid ~[echanics/' translated from the German by P. O. Wolf, Blackie & Son, Ltd., London and Glasgow,·pp. 98-112, 1956. 15. 'J. A. Ch. Bresse: "Cou~s de mecanique appliquee," 2e partie, Hydraulique (" Course in Applied Meeha:nics," pt. 2, Hydralllics), M"tlet-BacheUer, Paris, 1860. 16. J. V. Boussinesq: Essai sur Ill. theorie des eaux eonrantes (Essay on thli theory of water flow), Memoiresp1'sslmtes par diver~ savants a l'Academie des Sciences, Paris, vol. 23, ser. 2, no. 1, pp. 1-680, 1877.

62

BASIC PRINCIPLES

17. Cll!irles Ja.eger: Contribution a l'etude des courants liquides a 8urf8.c.oe libre (Contributil1nto the St11dy of Iree-surface liquid flo;,vs), RIWWi genlrale .de L'hydrauliq1l.{l PI'tTis, yol. 0, no. 33, pp. 111-120; no. 34, pp. 139-153, 1943. 18. Charles Jae~e.l·: De l'impulsion toto.le at de ses rapports a,yac I'energie totaled'un cou:u,nt liquide a surface libre (The tota.l impulse and its relations with the toto.l energy of a fl"ee-Ilurface liquid flow), RlWue gtntrale de l'hydravliq1l.e, Paris, voL 13, . no. 37, pp ..12-1\); no. 38, pp. 86-87; no. 39, pp. 143-151; no. 40, pp. 191-197; llO •. 41, pp. 257-2iH, 1947. HI. Etienne Crausse; "lIydr3ulique des canaux deco\lverts en regime permanent" ("Hyciro.ulio5 of Open Ch!lJlneL'! with Steady Flow"), Editions Eyrolles, Paris, 1951, pp. 111-112. . , 20. J. B. Bela.nger: Notes sur Ie cours d ' hydraulique (Notes on the course in hydrl:lulies), llHmoire, gmt" Na.tionale de$ Pont8 e~ ChaU$88I3$, Paris, 1849-1850, pp~ 32-33.

CHApTER

I~-

4

CRITICAL FLOW: ITS COMPUTATION AND AP:PLICATIONS

4-1. Critical Flow. As described in the previous chapter, the llritic!1! stf1.te of flow through a channel section is characterized by severa.l important conditions. 1 Recapitulating, they are (1) the specific energy is a minimum for (l. given discharge; (2) the discharge is a maximum for a given specific energy (Frob. 3-7); (3) the specific force is a minimum for a discharge; (4) the velocity head is equal to half the hydraulic depth in a channel of small slope; (5) the Froude number is equal to unity; and (6) the velocity of flow in a channel of small slope with uniform velocity distribution is equal to the celeri~y of small gravity waves in shallow wat,el' caused by local clisturbances. Discussions 011 critical state of fiow have referred mainly to a particular section of a channel, known as the critical sectiQu. If the critical state of flow exists throughout the entire lellgth of the channel or over 11 reach of. the cha.nnel, the flow in the channel is a criticaL flow. Since, as indicated by the critical-flow criterion Eq. (3-10), the depth of critical floW' depends on the geometric elements A' and D of the channel section when the discharge is constant, the critica.l depth in a prismatic channel of unilorm slope will be the same in all sectio1l3, and cl'itteal flow in a prismatic 'channel should, t.herefore, be uniform flow. At this condition, the slope of. the channel that sustains a given discharge at a uniform and critical depth is called the critical slope S~. A slope oithe c;tiannelless than the critical slope will cause a slower flow of subcritica,l state for the given discharge, as will be shown later, and, hence, is called a mild or subcritical . slope. A slope grel1ter th~\n the critical slope will result in a faster fiow of super critical state, and is called a steep or supercritical slope. A flow at or near the critical state is ullstable. This is becau~e a minor in specific ellergy at. or close to Cl'itical state will cause a major chl111ge in depth. This.fact can o,lso be recognb>ed in the specific-energy As the curve is almost verticalliear the critical depth, . curve (Fig. a slight in energy would change the depth to a much smaller or much greater alternate depth corresponding to the specific energy after l

For a historic~l account of the theory of critical £low, see [1J. 63

.,)

I.

(

'.

64

I , \ I

BASIC .PRINCIPLES

the change. It can be observed· also that, when the flow is near the critical state, the water sUl'face appears unstable and wavy. Such phenomena are generally caused by the minor chnnges in energy due to variations in channel roughness, cross section, slope, 01' deposits of sediment or debris. In the design of a channel, if the depth is found at or neart,he critical depth fOl" a great length of the chinnel, the shape or slope of the channel should be altered,if practicable, in order to secure greater stability. The criterion for a critical state of flow (Ar~. 3-:3) is the basis for the computation of critical flow/which will be explained in subsequent articles. Two major applications of critiClll-fiow theory nre flow control and flow measurement, which will also be discussed in this chapter. 4-2. The Section Factor for Critical-flow Computation. Substituting V Q/ A in Eq. (3-10) and Simplifying,

·11 . II

tJ .

(4-1)

...g.

.
~~,..

. Q

z=v'g

"'1 .., c

--:.~

0,-,

'"!"

-0

";l '-'

:3

... 't :5 ..,... . ..c::..," 'w .5 ... ..,..':! .~... Q

When the energy coefficient is nO& assumed tobe unity,

tj)

. Q

z=-v'g!cY.

('1-2)

"'0 ... "OJ

'0

In the above equations, Z A.v' D; which is the t;ection factor Jor critical-flow compu.tation [Eq. (2--3)]. Equation (4-2) states that the section facto!: Z for a channel section at the critical state of flow ~s equal to the discharge divided by the squaJ.~e root of U/Ol. Since the section factor Z is !l. function of the depth, the equation indicates that there is only one possible critical depth for maintaining the given discharge in channel and similarly that, when the depth is fixed, the~'e can be 'only one discharge that maintains a critical flow and makes the depth critical in the g'iven channel section. Equation (4-1) or (4-2) is ll. very useful tool for the computation and analysis of critical 'flow in 1111 open channel. When the discharge is given, the equation gives the critical section factor Zc and, hence, the critical depth Yo_ On the other hand" when the depth and, hepce, the section factw' are given, the critical! disharge can be compu~ed by Eq. (4-1) 'in the following form:

....o

Vg

fIl

0

-<

.j,

ci ~

(5

g

(4-3)

or by Eq. (4-2) in the following form: .

N;'

.s

OJ

I

Q=z:(2

1>

a '" ~... dt5;;:.. ::>

a.

Q =Z

"...

N

'0

(4-4) ci OpI ~ puo' q/,( lO SGnlOI\

65

66

. . CRITICAL :FLOW: ITS C.OMPUTATION AND. APPLICATIONS

BASIC PRINCIPLES

A subscript 0 is sometimes used to specify the oondition of oritical flow. Formulas fOI' t,he section factor Z of seven common channel sections are given in Table 2-1. The Z values for a circular section can be found either from the curve in Fig. 2-1 or from the table in Appendix A. In Ol·aer to simplify the computation of critical flow, dh:nensionless curves sho.wing the· relatiOll between the depth and section factor Z (Fig .. 4-1) have been prepared for rect<:.ngular, trapezoidal, 'and circular channels. .These self-explanatory; curves ·help to determine the depth y for a given section factor Z, and vIce versa.

This equation indicates that the value of M for the trapezoidal section is a function of z and y/b. For values of z = 0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, and 4.0, a family of curves for 111 versus y/b are constructed (Fig. 4~2). These curves in.dicate.that the value of JI.f varies in a range froni 3.0 to 5.0. S.C , . . - - -.......--.,-----,---7""',--,"'7'1rm-"----,

5.0 4.0 I-----.... ~-'-.-.----- --'-----+.........,f'--I'--7'-HI+-+------j

will

1,0

o.a 0

~ ,.. '0

c 0

......a,..

...

(4-6)

dOn Z)

3 l'

--ay- = 2

J.-

= JL .A

.2

~

=

3[1

+ 2z(y/b)P - :2z(y/b)[l + z(y/b)J . [1 + 2z(y/b>:J[1 + z(yjb)]

·r

I

o,oe O.os 0.05 0.04

vi A/T, and

0.03

(4-8) 0.02 . 25

\

3.0

3.5

4.0 Volues of M

4.5

5.0

5.5

,!

FIG. 4-2. Curve:; of Mvalues.

(4-9)

A curve for a circular section with M plotted against 1Ildo, where d u is t,he diameter, is alsoflhown (Fig. 4-2). This curve .w~ developed by a similar procedure but constructed from a much more complicated formula. The curve shows that ~he value of M varies within a rather narrow range for values of yjd D less than 0.7 or so, but increases rapidly as the value of y/do becomes greater than 0.7. The significance of this

This is a general equation for the hydrauUc exponent M, which is a func~ tion: of the channel' section a,nd the depth of flow. For a trapezoidal section, the exptessions for A and T olJtained from Table 2-1 are SUbstituted in Eq. (4-9); the resulting equation [2] is simplified arid Mcomes M

0.2

0.1

(4-7)

1 dT 2T

(31' -'- .:1-TdydT).

0.3

."

Equating the r~ght sides of Eqs. (4-7) an~ (4-8) and solving for M, M

0.6 0.5 0,4

0

.,.

where C is a coefficient and M is a parameter called the hyd"a-ulic ·expo- . nent Jor critical-flow comp-utation. Taking logarithms on both sides of Eq. (4-6) and then differentiating wi th respect to y, . d(ln Z) IvI Now, taking logarithms on both sides of Eq. (2-3), or Z = A then differentiating with respect to 1/, .

1

2.0

4.,.3 .. The Hydraulic Exponentfo.r Critical-flow Computation. Since the section factor Z is a function of the depth of flow y, it may be assumed that

2y

!

5.0

Example 4-1. Derive arl equatioI! showing critic:u dillcharge through a rectangular cha.nnel section in terms of the cha.nnel width and the total head. Sol~tion. For the rectangular section, Tal?le 2-1 gives the sQcti6n fantor Z = by!.J;. At the critica.l state'of flow, the depth y HI1.5 (see Frob. 3-3). Substituting these expressions in Eq. (4-3), using· g = 32.16, and simplifying, we find that the cri~icar discharge is Q. 3.087bHu . (4-5)

dif" "'"

67

. (4-10)*

I z(1/(b) may be constructed. It is obvious that this curve: be identical with the 1 in Fiilj. 4-2. Fo, convenience in application, however, a. family of eurve for z

• This equl).tion was a.lso developed indepe~dently by Chuga8v 13J. In this eq~e,­ tion, M ca.n be regarded as a function of z(ylb):; a.ccordingly, a single curve aiM versus

ourves of M versusy(b an shown, using "

I.

~ I

If.

as

&

pammeter.

·1

58

BASIC,P~INClPLES

CRITICA.L FLOW: ITS COMPUTATION .AND APPLICATIONS

characteristic is that, when the depth of flow in a circular section approaches the top of the circle, the section factor and with it the critical discharge, as shown hy Eq; (4-3) , become indefinitely large. In other words,it is practically impossible to maintain a critical flow in a circular . conduit at a depth approaching the top of the section. In fact, the wavy, surface of the c!'itical flew will touch the top of the conduit before it ' act\lally comes so near as to approach ~he top. A similar characteristic and phenomenon occur also in other types of closed conduit with graduI! ne pial bee om .. curved' "~.n Ihe depth approach.. I ally closing crown, when the water the Q1oduo II}' converging crown of a closed conduit P surface approaches the crown of the i:~g<~~~;o conduit. ot P For channel sections of other than trapezoidal or circular shape, exact 1.1=2. lon9 values of M can be computed di~ rectly by Eq. (4-9), pi'ovided that' N the derivative dT jdy can be evalu'" ~ ated. Approximate values of Mfor any channel section, however, may ,be obtained . hom the following equation

M

69

4 ...4. Computation of Critical mow. Computation of critical flow' involves the determination of critical depth and velocity when the discharge and the channel section are known. Three methods illustl'ated by simple examples will be given below. On the other hand, if the critical depth and channei section are known, the critical discharge can be determined by the method described in Art. 4-2. A. Algebra·ic Method. For It simple geometl'icchannel section, the critical flow can be determined by an algebraic computation using the basic equations. The method has already been used (Example 3-1), but the follOWing e.:1-:.ample is given for further illustration: Example 4-2. Compute the critical depth and velocity of the trapezoid"l channel (Fig. 2-2) carrying ll. disoha.rge of 400 ofs. ' Solution. The hydraulic depth and water area of the trapezoidal section arc eXIueflSed in terms oithe depth lras D

= 1[(10

10

+ 1/)

+ 2y

and

A = V(20

+ 2y)

The velocity is

Substituting the a.bove expressions for D and l'in Eq. (3-10) a,nd simplifying, 2,484(5

+ Y)

[!I(10

+ II)]!

Solving this equation for y by a trial-and-error procedure, Ye '"' 2.15 ft. This is the critical depth. The corresponding area is A. 52.2 iV, !l.lld the critical velocity is V< = 400/52.2 = 7.66 fps.

(4-11)

where Z 1 and are section factors for any two depths Yl and Yz of the given section. This equation can eesily be derived from Eq. (4-6). In applying Eq. (4-11), a graphical m~thod is recommended instead of direct computa.tion. This involves a logarit.hmic plotting of Z as ordinate against the depth as a.bscissa (Fig. 4~3). For most channels, except for closed conduits with depth approaching !J, gradually closing crown and some channels of peculiar shapes, the plot takes a more or less straightline fOl'm. The hydraulic exponent equal to twice the slope of the plot.ted stra.ight line. For a. depth approaching the gradually closing ci'own of a closed conduit,the plot becomes a curve, and the hydraulic exponent of a given depth is equal to twice the slope of the tangent to the curve at that depth, The hydraulic exponent M is described here only as a characteristic v.alue of a channel section under the condition of critical flow. The a:pplication of .this exponent will be further described in the computation of gradually vari~d flow (Art. 10-2), log y'

B. Graphical Method. For!l. complicated or natu~al channel section, a graphical procedure for critical~flow cOll)putation is genernlly employed. By this procedure a curve of yversus Z is constructed. The value of Q/ Vg is then com.puted. . Using Eq. (4-1), the critical depth may be obtained directly from the curve, where Z == Qjv'g.

FIG. 4-3. Graphical determination of the M value.

Example 4-3. A 36-in. concrete circular culvert carries a. dischD,rge of 20 cfs. Determine the critical depth. Solution. Construct a ~tlrve of Y VB. Z (Fi.g. 4-4.). Then compute Z == Q/-vii == 20/-vii 3.53. FrOIXl the curve the critical depth for this value of Z is found to be '!Ie 1.44 ,ft. The dimensionless curve (Fig, 2-1) or the table in Appendix A for the geometric , elements of a circular section might· also be used to solve this problem. Sinee d. 3.0 ft and· do" s == 15.6, Z/d. u = 3.53/15.6 = 0,226. From the dimensionless curve or from the table, Vida = 0.48, ,and so y. = 0.18 X 3 == 1,44. ft.

is

C. Method of Design Chart. The design chart for determinin.g the critical depth (Fig. 4-1) can be used with great expedienCY. In Example 4-2, Z = 400/ v'i "" 70.5 (Eq. (4-1)J. Thtl value of Z IbM is 0.0394this value, the chart gives '!I/b = 0.108 or '!I. ":' 2.Hi ft.

1:<'01' f,

I t

70

BASIC PRINCIPLES

. CIUTICAL FLOW: ITS COMPUTATION AND APPLICATIONS

,

!

71

If the channel has a critical slope (middle sketch in Fig. 4.5), then the flow is initially uniform and critical throughout the channel. In the presence of the dam, however, the flow through the pool Will be subcritical and the pool surface will approach the horizontaL At the dowllstream end a so-called drawdown ettrve will be developed, extending upstream Flo'..,. oon
"I

rr---_ ,

j

BOCkWO!il(

curve

~

Add~d depth due h:I backwater effect

I

1 Drowdawn

Zld~'"

= 0.226.

For this value the clHLrt. gives yld

= 0.48

~

---.

---~

FlO: 4-4. Curve Df y versus Z for a. cir(lular 5C)ction. In Example 4-3, y. = 1.44 ft.

r

Depth of subcriticol flow without dam

or

1

L

l.critIC
4-5. Control of Flow. The control of fim:v in 3,11 open channel is defined loosely in many v'ays. As used here the tertn means the establishment of a definitive flow condit.ion in the channel or, more specifically, a definitive relationship between the stage and the discharge of the flow. When the contror of flow is achieved at a certain section of the channl?l, this section is a control section. It will be shown later that the control section controls tbeflow in such a way that it restricts the transmission of the effec~ of changes in flow qondicion either in an upstream direction or in a downstream direction depending on the state of flow in the Chlll111el. Since the control section holds a definitive st,age-discharge relationship, it is always a suitable site for a gaging station and for developing the dis.charg6 raturtg Gwrve, a curve representing the depth-discharge relationship at the gaging station. At the critical state of flow a definitive stage-discharge relationship can be established and represented by Eq. (4-1). This equation shows that the stage-discharge relationship is theoretically independent of. the channel roughiless and other tlllcontrolled circumstances. Therefore, a critical-flow section is a control section. , The location of the control section in a prismatic channel is generally governed by the state of flow, wmchin turn is determined by the slope of the channeL Take for an example a long straight prismatic channel in which a pool is created bya dam across the chalUlel and the water flows. over the dam through an overflow spillway (Fig. 4-.5). Three flow conditions in the channel are shown, representing the subcriliical, critical, and supercritical flows, respectively. The slopes of the channel in the three cases are, correspondingly, mild or ilubcritical, critical, and steep or 8upercritical.

I

,\

I

.

L

Row condition contrnl1ed Qt lhe upstream enc

Depth of supercrilicol flow wilhoul dam FIG,

4-5. Flow conditions in a long prisma.tic ohaullel.

!rom a section near the spillway crest and becoming asymptotic to the pool level. . If the channel has a sub critical slope (top sketch in Fig. 4-5), the flow is initially subcritical. In the presence of the dam, the pool surface will be fllrther raised for a long distance upstream from the pool in a so-called backwater curve. The additional depth of water is required to build up enough head to give the increased velocity necessary to pass water over the spillway. This effect of backing up the water behind tb.e dam' is

i

t

.~

I

J

·{l

72

i

BASIC PRINCIPLES

known as the backwater effect. At the downstream end the backwater curve is conllected. with a !':imooth drawdown curve which leads the water over the spillway. If the channel has a supercritical slope (bottom sketch in Fig. 4-5) 1 the flow is initially supercriticaL In the presence of the dam, the backwater

r

1J .~

i ~

Ii ,

><:

* ''.N

'

S;:-MMc<:)M:r.I~MP)(o)Mp')

,...r:OOOMp')~¢I'lO~tO
'
r:- . . .

ae

Q)

co

QO Q

000'0

----roI.<;q~:1N~

..

..100C::OOOOQOOOOO

~_-:-l_."C"')C'}(t)~.:";ImMI.'I?

N-N ~ 0

;; " " 0 0

0

t: : : ;) ""'" -

C'l N

~

<:;I 0

C'1

0: 0

~'~

0- <:)

("1 cq

~

~oo(Oooooooooo

SECTION L-L

I

i

a:-t;~~

FIG, 4-6, Plan, elevation, ana dimensions o.f ,he Parsholl flume. (U.s. SlIil Conservation 8811>ilie [26J.} Plan and elevation of a concrete Parshall measuring fIl\me showing lettered dimensions as follows.:· W - size of flume in in, 01' ft; A = length of side wall of converging seetion; ,%A = dista.nce back"from end of orest to gage point: B Ilxiallength of converging section; (} = wid.th ofdownstl'ea.m end of flume; D = width of upstrea.m end of flume; E = depth of flume; F length ofthl'oat; G "" length of diverging section; K = difference in ele.... ation between lower eml oj' flume and crest; ,yI = length of approach /lOOI'; N ~ depth af depression in throat \:lelow crest; P = width between ends of curved wing wa.lls; R radius of curved willg wall; X = hol'izontn.l distanoe to E. gage point from low point in tlu'Oll.t; Y "" verticil.l dilltance to. Eo ga.ge point from low point in throl1t. See the tlible on the next page for actual dimellsions for various sizes of flume."

(I'lC'lVlr')MC<')(?tnC")

. '"

..:; ..... ;:;-CO)oc)OOOQOOO ~Q..-t ..... NC'-lC')"Iq'IV)~,....OOCl! ,

~

q;)

~. ~

Q

S

~~~~~~;::~~ 'otI t--<,

2 '" ::.•~ e ~ S'

C"ol C'.J "'tt "ttl
~ CC .....

1:--

I~""~~~------------------I

~

'"

effect originlj.tillg from ~hepool will not extend far upstream. Instead, the flow in the upstream channel will eontinue in:the downstz-eam direction at a supercritical state until the flow-s]lrface profile is actually below the poolle'Tel;1 the~ .it will rise abruptly to the pool elevation in a hydrau-

.:~i:f

~

.=~M.~O

.• O.

l~ .... ·,......-tto':l(#)MM~"~"'')LI':l

ii~~~

~tO

:~

1 Itshoul:l: be noted that; the pool level in this case is ~ot. horizontal but curved. The curved water surface has an 81 profile, which will be :described later (Art. 9-4).

73

l

:::fOOlQIt)OIOOf;!)Q

..:. Col ,,' 'ttl

~

l.O·tn 0: 10 ..,;. l'-<

co

74

BASIC PRINCIPLES

CRITICAL FLOW: ITS COMPUTATION AND APPLICATIONS

lie jump. The backwater effect will not extend upstream through the hydraulic jump. The fiowupstream from the jump.is governed entirely by t.he Upst.l·cam conditions. The above example explains the important fact that on subcritical slopes the effect of change in water-suI:face elevation downstream is transmitted upstream by a ba:ckwater curve, whereas on supercritical slopes the effect cannot. be transmitted far upstl:eam. The flow condition in a sub critical channel is affected by downstream conditions; but, in a supercritical channel, the flow condition is dependent entirely upon the condition upstream or at the place where water enters the channel. Accordingly, tile control of flow is said to be at th,~ downst.ream end for channels ,,,ith sub critical slope and at the upstream end £01' channels with supercritical slope. When t.he challnel is on a subcritical slope a control section at the downstream end may be a critical5('1ction, sunh as that created on the top of an overflow spillway. On a supercritic~l slope, the r;ontrol section at· the upstream end may also be a critical section, 2.S shown in the figure. A sluice gate or an orifke or other control structure may also be used to create a control section. It should be noted that whether the channel slope is critical, subcritical, or supercritical will depend not only on the measure of the actual slope.but also on the discharge 0; the depth of flow. 4-6. Flow Measurement. It was mentioned in the preceding article that, at a critical control section, the relationship between the depth and the discharge is defillitive, independent of the channel roughness and other '.Incontrollable circumstances. Such a definitive stage-discharge relationshipoffers.r. theoretical basis for the measurement of discharge in open channels. Based on the principle of critical flow, various devices for flow measurement have been developed. In such devices the critical depth is usually created either by the construction of a low hump on the channel bottom, such as a weir, or by a contraction 'in the cross section, such as a criticalflow fiwne. The use of a weir is a simple method,. but it causes relati vely high head 109s. If water contains suspended particles, some will be deposited in the upstream pool formed by the weir, result.ing in a ,gradual change in the discharge coefficient. .These difficulties, however, can be overcome at least partially by the use of a critical-flow flume. ,The critir.al-flow flume, also known as the Vent~ri flume, has been designed in various forms.l : J+, is usually operated with an unsubmerged

or free-flow condition having the critical depth at a contracted section and a hydraulic jump in the exit section. Under certain conditions of flow however, the jump may be submerged. . . ' One of the most extensively used critical-flow flumes is the Parshall flume l (Fig. ,1-6) which was developed in 1920 by R. L. Parshall. The depth-discharge relationships of Parshall flumes of' various sizes as calibrated empirically, are represented by the following equations: '

,. I'

ThrQat width

Equ.a.lion

3"

Q = 0.9921I.u4T

(4-12)

6"

Q= Q= Q~ Q=

{4-13} (4-14)

9" 12" to 8' 10' to 50'

J

75

2.06H.ua 3.07 H.l.63 4WH.uuw'·'" (3.6875W -I- 2.5)H.1.G

(4-15) (4-16)

In the above equations Q is the free discharge in cfs Wis the width of ft, ~nd Ha is the gage reading in ft. Whe:l the ratio of gage readmg Hb (FIg. 4-6) to Ha exceeds the limits of 0.6 for 3-, 6-, and 9-in. flumes, 0.7 for 1- to 8-ft flumes, and 0.8 for 10- to 50-ft flumes, the flow becomes submerged. The effect of submergence is to reduce the discharge. In this case the discharge computed by the' above equatiollS ~ust be correc~~d by a negative quantity. The diagrams in Fig. 4-7 gIVe the correct1Ons for submergence for Pa.rshall flumes of various sizes. The correction for the I-ft flume is made applicable to the larger flume's by multiplying the correction for the 1-ft flume by the fador given . below for the particular size of the flume in use.

.thro~t in

Size of flu.me W, ft

1

1.5 2 3 4 6 8

Correction factor

1.0 1.4 1.8 2.4 3.1

4.3 5.4

Similarly, the correction for the 10-ft flume is made applicable to the I Experin).ents on this type of mea.suring. device, then called the Ventu.ri flu rite, were began by M. Cone at,the hydraulic laboratory of the Colorado Agricultural Experiment Station, Fort Collins, Colo.' The initial studies were reported ill [20[ and {211. ~he name ." Parshall measuring flume" was adopted for the device oy the Execu~Ive. Commlttee of the !rrigation Division; American Society of Civil Eq.gineers, durmg 1~.Dec~mber meetIng of 1929. Further developments on the Parshall flume arc d~scflbed by R. L. Parshall in [22] to [26J. :

y.

The critical-flow flumes mentioned in the text are those developed and studied in 'the United States. Outstanding designs of critical-flow flumes were aIso developed and tested by ,Jamcson [4,5J, Engel [ft,7], and Linford f8] in England; byCrump [9] and Inglis [10] in India; by De Marchi [11,12], Contessini[ll], Nebbia [13-15], and Citrini [16,17J in Italy; by Khafagi [18] in Switzerlandj and by Balloft;et [19J iIi. Argentina. I

.i

..;. ~.

"

·r

1 t

r

~l

]

76

BASIC PRINCIPLES

GRITICAL FLOW: ITS OOMI'iJTATIONAND APPLICA.TIONS

i

U

1.9 ZD

Discharge, cIs

I

77

Correction, cIs (tt)

(OJ

Discharge, cIs Correction. cis

( b)

(e)

Fro. 4-7. Diagrams ior computing Bubme.!"ged flow through ParshaU flumes of various sizes. (Colorado Agricullural Ezperime:n.t Sta.tion [25J a.nd U.S. SlYil Conservation Service [26).) (a) Diagram showing the rate of submerged flow. in cubic feet per ~econd,through a 3-in. Pa.rshall measuring flume. (bl Dia.gram s!lOwing the rate of submerged flow, in cubic feet per second, through a. 6-in. Pars!lall measuring flume. (el Diagram showing the rate of submerged flow, in cubic feet per second, through a 9-in. P!lrshall measuring flume. (d) Dia.gra.m for computing the rate of $ubmerge.d flow, in eubic feet per second, through a loft Parshall measl.lring flume,' (e) Diagram for determining the correction in cubic feet per second Plll': 10 ft of crest for subDlergedflow discharge.

b

'" aa

f

c.

4.0 Oischarge, cIs (~)

FIG. 4~7

4.5

5.0

5.5

6.0

CRITICAL FLOW: ITS COMPUTATION AND APPLICATIONS

BASIC PRINCIPLES 78 larger flumes by multiplying the correctioll for the lO~ft flume by the factor given below for the paJ,ticular flume ill use.

Size rJ/ flume W, f/,

CorrectiOlt factor

10 12

LO 1.2

15

1.5

20

2.0

25 30 40

2.5 3.0 4.0 5.0

50

.

;;

~

:

79

\

1 I

I I

[

j

1

II ,

It is desiiable to set the crest of the Parshall flume so that free flow will occur. If conditions do not permit free-floW operation, the percentage of submergence Hb/H" should be kept, whenever possible, below the practical limit of about 95 %, since the flume will not measure dependably if the submergence is greater. The size and elevatiori of the crest depend upon the discharge to be measured and upon the 5i;(;e of the flume and, consequently, upon the loss of head through the flume. The loss or head can be determined from ~he diagrams in Fig. 4-8. A practical example (Example 4-5) will be given' to show the deterrcinatiol1 of the size and elevat.ion of the flume crest. Because of the contraction at the throat, the velocity of water flowing through the flume is higher than that of the flow ill the channeL For this. reason any sand or silt in suspension or rolled along the bottom call. be carried through, leaving the flume free of deposit. When a heavy burden of erosion debris is pl't;lsent in the stream, ho\yever, the Parshall flume will become invalid like the weir, because deposition of ths debris will produce undependable results. For use under such circumstances, a modified Parsh.aLl flume known as the San DimGSfiume (27,28] has been developed, which has the advantage of a self-cleaning mechanism for heavily debris-laden flows in the stream. For measuring open-channel flow in closed conduits, such as sewers and covered irrigation canals, critical-flow flumes of special designs have been proposed. Palmer and Bowlus [29-31J have developed several of these flumes, including one'which is simplr a flat slab on the bottom and has no. side contractions, one with a rectangular croSs section, and several witp trapezoidal-shaped throats. Stevens [32], recommended a critical-flow flume in which he used a blister-shaped' hump control on the bed of the conduit to produce a critical flow over it. The frictional.1osses in this design are believed to be very small. Like. many measuring devices, the critical~flow flume has certain disadvantages. The flume CD.nnot be used directly with or, combined with a head gate. "It is more expensive to build and requires more accurate

I,

II l

,

I

.:~.

"\'..:,

~~:s' 4-8;cD()lio!l.Tgar;~sAf{)O~ del~ermalirunE' 'g th.s los,s h~!l.d t(hrough Parshall flumes of various Seroi~e ~61.) .ur Slat~{)'11 251 and U.S. Sml Ccm.ssnJl1tion of

lW

• • ell

,:z:pmmen

-\

"

80

BASIC PRINCl;l'LES

CRITICAL FLOW: ITS (TO¥PUTATION AND APPLrcATIONS

workmanship in its construction than other commonly used devices, such as weirs or submerged orifices. Technical information on other kinds of open-channel measuring devices and methods can easily be. found in ml1ny textboQks and handbooks on :hydraulics (such as [33J to [35]). . Many culverts along modern high ways can be used as or converted to critical-flow flumes for measuring runoff from the adjoining agricultural lands. This idea was first suggested by Mavis [361 and others and was later studied at the Oklahoma Agricultural Experiment Station [37,381 by carrying out extensive experimental tests on rectangular highway culverts. The results of this experimental study indicate that the culvert can be used as a flow-rate measuring device if it flows part full and has free outlet fall. A weir sill should be installed, however, to improve the accuracy ofthe measurement in the low-flow ,range. In this investigation, a standard Villemonte-type weir silP was developed and its location on the culvert floor determined. Head-discharge relationships were also determined for various flow ranges.

,

{

=

l

=

V.y, =%H.

V%i'if.

= 3.09H.1.S

,

Similarly, try 2" and 3-ft flumes. It is found that the respective crest elevo.tion~ are L53 and 1.23 ft and that the respective upstream water depths £1.1'(> 2.98 and 3.12 ft. In deciding them,!st practical aize of flume to use, it will btl necessary to examine the freeboard or the ch!LIInel and the effect of rise of the water surface upon the flow through the h~adgate. If these conditions arc satisfactory, the 2-ft flume will be the.

I

I j

,

4:-9.

FIG. 8.9ction of a Parshall Ilume illustrating ,th,e determination of the' proper crest elevatlOn [26]. most eoonomical because of its small dimensions. However when the width of the channel is considered, the final selection may be in bvor of th~ 3- or 4-ft flume because moderat~ or long wing wall.s may be required for a smeoll structure. Usually, the throat wldth of the flume Will be from one-third to one-h!'\lf of the channel width.

(4--17)

This is a theoreticru discharge equation i1) which H. is uncertain since the critical section is usually difficult to locate. For' practical purposes, however, the t\quation is generally written q = CH'Jl, where H is the elevation of the upstream wa~er surface iLbovethe weir c~est. This is the form described earlier (Example 3-2). If an ...!'sated free overfall exists at the downstrea~end of the weir, the a\'ove equation ca.n be expressed in termso! the brink depth Y" which can easily be measured. Since y, :. l.4y~ (Art. 3-4), the equ!l.tion required is

q ,

=

9.38y ol.l

PROBLEMS . 4-1. Prove the following critical-discharge equations for the triangular trapezoidal ' aud circular' sections: I ' , , Channel 8e£!lion Triangular

(4-18) ,

Experiments h9.V~ shown that, when the head on the broad-crested w!:!r is greater than about 1.5 times the length of the crest, the nappe of the free overfall becomes detached and the weir is in effect a sharp-crested weir. Example 4-6. Design ft. Parshall flume for handling 20 cfa of now in a.: chann'cl of moderate slope when the water depth in the channel is 2.5 ft." Solulimt. The discharge given can be m!"llsured by flumes of several si~es, but the best selection' is the flume of most practical' a.nd economical size. i ' Assume W .. 4 it ,and H,/H. 0.7, For Q = 20 cfs, Eq, (4-15) gives' H. ~ L15 ft. Hence, H, = Q,81 f t . ; , ' At 70% submergence, the 1vater surface ih the throat, at Hb gage. is essentially level with the surface of the tailwater. Under :this condition of flow, sllown lin Fig. 4-9, 1 The Vilhimonte weir sill consists or two; tri!LIIgular. tapered cotlvergin~ sills placed on the culvert Hoor with an opening left between them (39}. ' ' • This example is sdopted from [26J.

,

~+~=~~.

Example 44. Using the theory of criticaillow, derive an equation for the discha,ge over a broad·crested weir. Solution. COllsiaer the section on the weir crest where critical flow occurs. At this section, y, = 2(V,'/2g} HJ1.5 or V, "" VUH,/1.5, where H. is the specific-energy head at the section. The discharge per foot width of the weids, therefore, equal to

g

81

the tailwater depth D ~ 2.5 ft, and the elevation of the c~est above the channel b;t;.. torn is X 2.5 - 0.81 = 1.69 ft. . . ' From Fig. 4-8, the head loss corresponding to H~I H u = 0.7. Q = 20 cfs, and W = 4 rt is 0.43 ft. .Therefore, the depth of water upstrcuim from the flume will be

i

,

Equation Q,

=

2.295zH..·•

Trapezoidal

Q _ 5.671{(b + zy)yJu , - -(I) + 2zy)',$ -

Circular

Q.= 0.251(0' - sin OJ''' d

Parabolic

Q.

(sin

=

Yz 9) 0,6

2.005TH.L5

(4-19)

(4-20) U n

(4-21) (4-22)

In the above equation~, IX = 1 and H, is the specifiiJ..energy head; other notation' . follows that of Table 2-1; ; ' ' 4-2. C~mpute the hydr£lulic exponent M of the trapezoidnl cha.nnel section (lfig: 2-2) ~avlDg a flow dept!h of 6 ft, using (a) Eq. (4-10), (0) Fig. 4-2, and (e) the graphlCru method based on Eq. (4-11).. • 4.-3. Compute .the hydraulic. exponeD.t M of a 36"1n. circular conduit having a,; flow depth of 24 In. above the Invert, uSing (a) Fig. 4-2 and (b) the graphical method; based on Eq. (4-11). ' ' 4-4. Prove that the critical depth and velocity for a rectangular channel are'

82

BASIC FRINCIFl;IllS CRITICAL FLOW; ITS COMPUTATION AND ,APPLICATIONS

expressed by

83

(4-23)

and

(4-24)

where Q is the. discharge, b is the channel width, and "dll the energy coefficient. 4-5. A rectangular channel, 20 ft wide, carries g, discharge of 200 cfs. Compute the critical depth and velocity. 4-6. Solve Example 4-2 by various methods if the discharge is 300 cfs. 4-7. Solve Example 4-3 by various methods if tr.e discharge is 15 cis. 4-8. An' a.pproxhnate but practiclJ.I fonnula for the critical depth of a circular section of
is y.

= 0.325

(£t +

[).083d.

I

(4:-25)

I

is

which accurate only when 0.3 < y./d Q < 0.9. ' Solve Example 4-3 and Prob. 4-7 by this formula, 4-9. Referring to the naturalchaonel given in Frob. 2-5, construct a CUrve of critical depth aga.inst discharge, ranging from 0 to 400 crs. 4-10. Prove that the section of a cha.nnel in which the flow is critical at any st'lge takes the form expressed by

I

(

-{ !

J

(4-26)

whhle z is half the top width and II is the di5~ance of the water surface below the energy line. Draw II skeLch of the section and d!lscripe its properties. Is this channel possible? If not, how could it be made pOllSible? Is this channel practicable ind the flow stable? 4-11. Verify the computa.tions for the 2- and :HtPa.rshal! flumes tried in Example 4-5. 4-12. Determine the dischltrge through the 4i-ft Parsha.ll flume described in Example 4-5 if the percentage of submergence is .:l00/". 4-13. Determine the discharge measured by a 10-ft Parshall flume if the gage rea.ding H. is 3Al ft at a. free~flow condition. . 4-14. Design a Parshall f!IJme to mell.sure 10 ers of flow in J1 channel ha.ving a depth of flow equal to 1.5 ft. ' 4-16. A uniform flow of 300 cfs occurs'a.t a depth of 5 ft in a long rectan!!;lliar channel ,10 ft wide, Compute the mini,mum IH~ight of a. fiat-top hUmp that ca.n be built on the floor of the channel in order to produce' a critical depth. What wilLresult if the hump is lower or higher tha.n the computed minimum height? 4-16. If the critical depth in the above problem is produced by a contraction of the channel, wha,twill be the maximum contracted width? 4-17. A low da.m 5 ft, high having a broad horizontal crest is built in a. t"6ct~ng\llar channel 20 ft wide. Assuming that a depth df 2.5 ft measured on the crest is the critical depth, compute the discharge and the depth of flow upstrea.m from the dam. 4-1B. On the basis-of the theory of critical flow, Stevens [32] has derived the rating curves for the blister-lShaped critical-flow flume tlll~t he proposed for use in circular conduits (Fig. 4-10). In the deriva.tion, it is as~umed (1) that there is no energy loss from !ll to !It, (:~) that the !;I.pproachingvelocity in the pipe is eq ual to the discharge I

'

1 FIG. 4-10. Rating curves of ;l. critical-flow flume proposed for a clos"d conduit. J. C. Stevens [32J.) d. ""diameter of the cor.duit. '

I I

I 1 I f

I

I

(After;

dh'ided lJy the water area. corresponding to th'e energy head instead or'the actual area.: corresponding to Y" and (3) that the critical-flaw section i3 at the maximnm ~leight of. t~e cDntrol "hump." The second assu.mption elimina.te,s a. trial procedure In determmmg the velocity hegd of the a.pproa.ching flow !l.nd, furthermore, tends to , co,"?pensatc for the error involved in the first a,'55umption. Verify anyone of the , ra tmg curves.

REFERENCES 1. Charles Jaeger: "~ngineering Fluid Mllchanics,~) trrulsu;,ted from the German by P. O. Wolf, Bla.ckle & Son, Ltd., London and Glasgow, 1955, pp. 93-119. ' 2. Ven Te. Chow: Integrating the eq1!aticns of gra.du!l.lly varied flow, paper 833, Proceedtngs, Am.erican, Society of Civil Engi1l.llers voL 81 pp. 1-32 November 1955. ' , , ' ,

3. R., R. C~uga~v: Nekutorye vop.osy neravonomernogo dviphenna. vody v o tkrytykh pnz~tlchesJ{lk~ ruslakh (About some questions concerning n'Jnuniform flow of wat~r m op~n pnsma:i~channels), I zVcsliia. V sesoi11.Z1logo N a.uChno-l ssledClValel' alcaga Imt';.llila Gidrlliekhntk~' (Tramactians, All-l"lnian Scientific. Research Institu.te of Hydra.ulic Engineering), Leningra.d, vol. 1, pp. 157-289, 1931.

.1

84

CRITICAL FLOW: ITS COMPUTATION AND APPLICA~ONS

BASIC PRINCIPLES

4. A. H. Jameson: The Venturi flume and the effect of contractions in open channels, 1'ransactions, Institul-iolt of Water Engineers, vol. '30, pp. 19-24, June 30, 1925. 5. A. H. Jameson: The (il.rvelopment of the Venturi flume, Wawr and Water Engineering, London, voL 32, no. 375, pp. 105-107, Mar. 20, 1930. 6. F. V. A. E. Engel: Non-uniform flow of water: Problems and phenomena in open ehannels with side contrn.c.tions, The Engineer, vol. 155, pp. 392-394, Apr. 21; pp. 429-430, Apr. 28; pp. 45[':'457, May 5, 1933. 7. F. V. A. E. Eagel: The Venturi flume, The EngincliT, voL 158, PP: 104-107, Aug. 3; pp. 131-133, Aug. 10, 19&4.' 8. A. Linford: Venturi flume flow meter, Civil Engineering and Publ'ic W01'ks Rev-iew, London, vol. 36, no. 424, pp. 532-587, October, 19'11. An abstract is given in Journal, Americar;. Waler Works Association, vol. 34, pp. 1473-1475, September, 1942: 9. E. S. Crump: Moduling of irrigatioIl channels, Pt"dab Irrigation Brar4ch Publications, Paper Nos. 26 and 30A, Lahore, India, 1922 and' 1\)33. . lO. C. C. Inglis, NoteB on standing wave flumes and flume meter baffle falls, Pl:bUc Works Department, Government of Bombay, Techn·ical Papers, No. 15, India, 1928. 11. Giulio De Marchi (anthoT, pis. I and III) and FraMesco Contessi~i (author, pt. II): Dispositivi per la misura della pOl·tatll dei canali con minime perdit,e di quota: Nuove ricerche sperimentali slii misuratori a risalto !draulico (C'l.llnli Venturi); Part,e I, Esame del pro(;esso idmulico; Parte II, Descri,.ione delle esperienze; Pal·te III, Risultati delle esperienze [D.evices for measuring discharge in cannls with minimum loss of level: New expaTimentnl researches on standing wave flumes (Venturi flumes); pt. 1, AnnJysis of the hydralilic process; pt. II, Description of the experiments; and pt. lII, Results of the experiments], L' Enfn'gia . elettrica, Milano, vol. 13, no. 1, pp. 6-15, Jil.nnary, 1936; vol. 13, no. 5, pp. 236-244, May, 1936; vol. 11, no. 3, p'p. 189-214, March, 1937. Reprinted as lstituto di Idraulica e Co,~truz'io7'li JdJ'auliche, .Milano, Memorie e sludi Nos. 17, 25, and 26, 1936-1937. .12. Giulio Dc Marchi: Nouvelles recherches experimentales sur Ie jaugeur !I. ressaut hydra.uliqne, canal Venturi (New expel'imental researches' on standing-wave .flume, Venturi flume), Ministry of Agriculture, Pa.ris, France, 1937. This is an abstract of pt. I of \11]. .' la. Guido Neuhia: Ventul'imetri-per canali a sezioni di forma generica (Venturi meter for canals with cross sections of general forms), Acqua e gas, vol, 25, no. 11, pp. 2'rD-291, November, W36. . 14. Guido Nebbia: VeJ,lturimetri per canali a sezioni di tipo monomio (Venturi 'meter for canals with cross sections of monomial type), Acqua e gas, vol. 25, no. 12, pp. 326-333, December, 1936. 15. Guido Nebbia: Venturimetri per can ali a sezione di forma. generica: Primi rj;,'Ultat~ sperimentali (Venturi meter for cano.ls with cross sections of geneml forms: Preliminary experimental results), Aequo e (las, vol. 27, no. 5, pp. 155-181, May, 103S; vol. 27, no. 6, pp. 199-214, June, 1938. 16. Duilio Citrini: Miauratori a risalto (Standing-wave flumes), L' Energia elettrica, Milano, voL 16, nu. 10, pp. 758-763, October,1939; reprinted as Istitutodi Idra1,&lica e Costruz'ioni Idrauliche, Milano, .Memorie e studi No. 35, 1939. -17. Duilio Citrini: Modellatori a risalto: Guida al progetto (Standing-wave meters: Direction~ for design), Centro studi per Ie. aPl?lica.zioni dell'ingegneria all'agricoltura, Sindaeato ingegn.eri di :A{ilano, Se;oorate Paper No.5, Milan,. 1941 ; reprinted as I stituio Ji I draulica e Cos!ruzione' I drauliche, Milano, JIIlemorie e studi No. 44, 1941. lB. Anwar Khafa.gi: Der Venturikanal: Them'ie und Anwendung (The Venturi flume: theory and application), Eidgenossisehe Itchnische Hochscitu/e Zurich, Mitteilungen der Versuchsallslaltfv.r Wasserbau und Erdbau, No. I, Zurich, 1942.

85

19. Arma~do Ball.offet: C:i~ical fl~w meters (Venturi flumes), paper 743, Proceedings, Amencan Soc.ety of Cwtl Engmeers, vol: 81, pp. 1-31, JlIly, 1955. 20. V. M. Cone: The Ventnri flume, Journal of Agricultm·aIResea·rch, voL 9, no. 4, pp. 115-129, Apr. 23,1917. . . . , . 21. Ralph L. Parshnll and Carl Rohwer: The Venturi flume, Coloraclo Agricultural' Experiment Station, Bulletin No. 265 February 1921 .. 22. R. L. Parsho.ll: The improved Vent~ri flume, Trans~clions, AmeJ";cr!.n SlJciety lJf Ciuil.Enyineers, vol. B9, pp. B41~B51, 1926. 23.R. L. Parshall: The Parshall mell-suring flume, Colorado Agricultural Experimen l Siation Btdlet'in No. 423, March, 1936. ' 24 R. L. Parshall: Measuring water in irrigation channels, U.S. Department lJf Ag"i. ~ulh!rB, Parmer's Bulletin No. 1883, January, 1932; revised, October, 1941. . 25. R. ~. Par.sl;al~: Parshall flumes of large size, Colorado A(Jricul/ural Experiment Statton, BttLettn No. 3R~, May, 1932; revised as Bulletin No.· 426A, March, 1953. 26. H. L. Parshall: Measurmg i.vater in irrigation channels with Parshall flumes and s:nalt weirs, U.S. Soil Consenlation Senlicc, Circula?' 843, May, 1950. This Circular supersedes [24], 27. H. G. Wilm, John S. Cotton, and H. C. Storey: Measurement. of debris-laden stre~m flow with critical-depth flumes, Transactions, Ami-rican Society of Civil Eng-meers, vol. 103, pp. 1237-1253, 1938. . . 2B. K. J. Bermel: Hydraulic influence of modifications to the San Dimas criticaldepth measuring flume, Transactions, American Geophysical Union, vol. 31 no. 5 pp. 763-768, October, 1950. . ' , 2fl. Harold K. Palmer and Fred D. Bowlus: Adaptation of Venturi flumes to flow measurements in conduits, Transactions, American' Society of CilJU E7tginee;s vol. 101, pp. 1195-1216, 1936. . ' 30. John H. Lu~wig and Russel G. Ludwig: Design of Pltlmer-Bowlus flumes, Sewage ~ndltst"al Wastes, vol. 23, no.,g, pp. 1096-1107, September, 1951. 31. Edwl~ A. Wells, Jr., I'.nd Harold B. Gotaas: Design of Venturi flumes in circular condUits, Transactions, American Society of Civil ElIginee1's, vol. 123, pp. 749-771, 1958. 32. C. Ste;ens: Discussion on Adaptation of Venturi flumes to flow measurements III c~ndUlts; by Ha.rold re. Palmer and Pred D. Bowlus, Transactions, American Society of Ciuil Enginee1's, vol. 101, pp. 1229-1231, 193fl. N 33. Herbert Addison: "Hydmulic Measurements" JohnWile v & Sons In . ' J . ' c., ew York, 1941.

a;:d

!.

34. "Wate'l' 'tvIell.sunnnent Manual," U.S. Bureau of Il.eclamo.tioll May W53 43-58. ' , , pp.

, I

l

35. Horace William King: "Handbook of Hydraulics," 4th ed., revised by Ernest F , . Brater, Mct;lraw-Hill. Book Company, Inc., New York, 1954. ' '.. ;;6. F. T. MaVIS: Reducmg unknowns in small culvert' design, Engine~rin(l NewsRecord, vol. 137, no. 2, pp. 51-52, July 11, 1946. 37. W; O. Ree and F. R. C~ow: Measuring runoff rates with rectangular highway culverts, Oklahoma AIlT!CUU'Ural Experiment Stat'ion, Technical Bulletin T-51 November, 1954. . ' 38. W. O. Ree ~nd ~. R. Crow: Culverts as Wo.teT rUIloff measuring' devices, AgriC1!/tural Eng~neenng, \"01. 35, no. I, pp: 28-31 and 39, January, 1954. 39. James R. Villemonte: New type gaging station for small streams; Enginee1'ing News-Record, vol. 131, no. 21, pp. 748-750, Nov. 18, 1953. 40. C. D. C. Braine: Draw-down and other factors relating to the design of stormw2.ter outflows on sewers, Journal, Institution of CiviL Engineers, London vol. 28 no. 6, pp. 136-163, April, 1947. . ' "

.PART II

UNIFORM FLOW

j 1 i.

1 I

'1

;

. \

\

. j,

CHAFTER

5

DEVELOPMENT OF UNIFORM· FLOW AND ITS FORMULAS

.

I

J 1

!I

,I

II

I

i

I

~

89

~,

l

I~~, '

'"

,.: ~..

.)

15-1. Qualifications for Uniform Flow. The uniform flow to be considert:d has the following mn.in features; (1) the depth, water area, velooity, und discharge at every section of the channel reach I.1r,e constant; and (2) the energy line, water surface, and channel bottom are allparallei; that their slopes are all equal, or Sj = S'" = Sn = S. For practica.l pm-poses, the requirement of constant velocity may be liberally interpreted as requiremel1t thtl.t the flow possess a constant mean velocity. Strictly speaking, however, this should mean that· thB flow possesses a constant, velocity at every point on the clHmnel section within the uniform-flow reach. In other words, the velocity distributIon across the channel section is uualtered in the I'each. Such fl. stable pattern of velocity distribution call be attained when the so-called" boundary layer" is fully developed (Art. 8-1). ' Uniform flow is considered to be steady ouly, since unsteady uniform flow is practically nonexistent. In natural streams, even steady uniform flow is rare, for rivers and streams in natural states seanleiy ever experience a strict uniform-flow condition .. Despite this deviation from the truth, the uniform-flow condition is frequently assumed in the computn:tiOD of flo\v in natural streams. The results obtained from this assumption are understood to be apPl·oximo.teallq. general, but they offer a relatively simple and satisfactory solUtion to many prnctico.l problems. As turbulent tmiform flow is most commonly encountered in engineering problems, it will be discussed e,.'(tensively il1 tbe following chapters. Laminar uniform flow has limited engineering applications, and will be described only in Art. 6-10. " It should be noted that unifor:m flow cannot occur at very high velocities, usually described as ttltTarapid. 'This is because, when uniform flow reache~ i certain high velocity, It becomes very unstable. At highel' vetoci~ies' the fl~ "'1ill eventually entrain Edr and becbme unsteady. The criterion fOl: instability of u,niform flow will be discussetl1i:1':A:rt:1f-8: 5-2. Establishment .of Uniform Flow. When flow OCCtll'S in an open channel, resistu.nce is encountered by the water as it flO\vs downstream.

,

'

DEVELOPMENT OF 1}NIFORl'rl FLOW AND ITS FORM,'()'LAS

This resistance is generally cO,unteracted by the compon~llts of gravity forces acting on the body of the water in the direction of motion (Fig. 5-2). A uniform flow will be developed if the resistance· is balanced by the gravity forccs, The magnitude of the resistance, when other physical factors of the channel are kept unchanged, depends on the velocity of flow.

may again be exceeded by gravit:r forces •.apd the flow may become varied again. For purposes of explanation. a long channel is showll with three different slopes: sub critical, critical, and supercritical (Fig. 5-1). At the subcritical slope (top sketch in Fig. 5-1) the water surface in the transitory zone appears tmdulatoi:y. The flow is uniform in the middle reach of the channel but varied at the two ends. 1 At the critical slope (middle sketch in Fig. 5-1) the water surface of the critical Row is unstable. Possible undula.tions may occur in the middle reach, but Ql1 the average the depth is constant and the flow may bf'. considered uniform. At the sl.lpc,rcritical slope (bottom sketch 'in Fig. 5-1) the transitory water surface passes from the subcritical stage to the supercritical stage. through a gradual hydl'aulic drop. Beyoml the transitory zone the flow Is approaching uniformity. The depth of a uniform flow is caned the nonnaJ depth. In all figures the long dashed line represents the llormal-depth line, abbreviated as N.D.L., and the short clashed or dotteclline represents the critioal-depth line, or C.D.L. The length of the transitory zone depends on the discharge and on the phj"Sical conditions of the channel, Buch as entrance' condition, shape, slope, and roughness. From a hydrodynamic standpoint (see Art. 8-1), the length of the transitory zone should not be less than the length required for the full development of the boundary layer under the given conditions. 5-3. Expressing the Velocity of a Uniform Flow. For hydraulic computations the mean velocity of a turbulent uniform flow in open channels . is usually expressed il,pproximately by a so-called unifol·tn-flow formula. Most practical uniform-fiow form1.lia,s ca:n be expressed in the following general form:

90

Ivorled~w

I

Tronsitor

zone

un[f~rm

flow on the ove''::Qe ,J

yt--I I .. ' . ' I

I

r~~;::~1 t;"ifOftl1

fl 0 "'_ _ _-I.1

I

i

I I

I I

I

I

t

I

!

"

1 l

1 )

.\ t

(5-1)

FIG. 5-1. :&!ts.blishment of uniform flow in

;1'

91

UNIFORM FLOW

Il

where V is t,he mean velocity in ips; R is the hydraulic rf;1,dh15 in ft'; S is the energy slope,2 x and yare exponents; and, C is a factor of flow resist~ ance, varying with the mean v:elocity, hydraulic channel roughness, viscosity, and many QtheJ: factors. For practical purpos~s, the fiow in a natural channel may be assumed l,miform under normal conditions, that is, if there are 110 flood Rows or markedly varied flows caused by. channel irregularities. 'In applying

long ohe.nne1.

If the water enters the channel slowly, the velocity and hence th(;l resista~ce are, small, and the resistance is outba,lanced by the gra;~ity forces, resulting:in an accelerating flo~v in the upstr~am reach. The velocity and . the resistance will gl·adually 'increase until a balance between resistance aDd gravity forces is reached. At this moment alld' a.fterward the flow becomes uniform. The upstream reach that is required for the establishment of uniform flow is known as the transilory zone;' In this zone the flow is accelerating and varIed. If the channel is shorter than the transitory length required by the given conditions,. unIform flow cannot be.' . attained. Toward the downstream end of the channel the resil5tance

1 Theoretically speaking, the varied depth a.t each end approaches the uniform dept.h in the middle llSymptotically and gradually. For praotica\ PUfilvS\lS, however, the depth ma.y be considez:ed constant if the variation in depth is within a certain margin, say, 1 % of the lwerage uniform-fiow depth. Z In. uniform flow f S :'" Sf =' S,. = So. When the uniform-flow formula is applied t.o the computation of energy slope in a gradually varied flow, the energy slope will be denoted specifically by Sf instead of S. .

t

. .l·.

I

. i

-\

92

93

UNlF'ORM FLOW

DEVELOPMF.NT OF UNIFORM FLOW AND 1'1'13 FORMULAS

the uniform-flow formula toa natural stream, it is understood that the result is very approximate' since: the flow condition is subject to more imcertnin betors th}l[l would be involved in it regular artificial channeL As pointed out by Schneckenberg [1]; a good uniform-flow formula for an alluvi.al channel with sediment transport and turbulent flow should take equal aecount of aU the following variables;

. 5-4. The ChezyFormula. As early as 1769 the French engineer Antoine Chez)" was developing probably the first uniform-flow formula, the famous CMzy furmula l which is usually expressed as follows:

A .V V ma P R y SUI

n

Q. Qb 'IL

the water area the mean velocity the maximum surface velocity' the wetted perimeter the hydraulic radius . the maximum depth of w-ater area the slope of the water surface a coefficient representing the channel roughness, known as the coefficient of 7'oughness 1 the sllspended sediment charge the bed load the dynamic viscosity of the water' the temperature of the water

T There have been dev,eloped and published a large number of practical uniform-flow formulas, t but none of these formulas meets the qualifications of a good form'lIla as defined above. The best known and most widely used formulas are the CMzy and Manning fOl'mulas, which will be described in the' following articles alid used extensively in this book. Theoretical uniform-flow formulas have also beell derived on the basis of a theOl'etir.AI velocity distribution across the channel section, wl1i~h will he discussed later (Art. 8-5). A different approach to the determina.tion of the velocity in a natural channel has been attempted by Toebes [5]. In this approach amultiplecorrelation analysis is applied to the following significant factors affecting the velocity in a gi.ven alluvial channel: water area, maximum surface velocity, wett~d perimeter, maximum depth,slope of water surface, coefficient of roughness, and temperature of water. By this method it is , possible to e"aluate the independent individual influence of each variable on the magnitude of the velocity. When such an evahULtion is made, the velocity under any givell condition of the variables is simply equal to the algebraic summation of the individual contl'ibut,ions as <\ffected by each variable. However, this method applies only to the streams in the geographical region for which the analysis is made; hence, its applica.tion . cannot be generalized. In British literature the term "l'llgosity coefficient IJ is used. , A number of weH-known uniform-flow formulas are given Md discussed in [2] to [5J. . 1

(5-2)'

where V is the mean velocity in fps, R is the hydraulic radius in ft, S is the slope of the -energy line, and C is a factor of flow resistance, called

ChezY'$ G.

..

,FIG,

. .\,.-,',

, '

,

5-'.l. Derivation of the CI~ezy forml;!a. for uniform flow; in open channel.

The CMzy formula C.:l.n be'derived mathematically from two assumptions. The first assnmption was made by CMzy. It states that the foi'ce resisting the flow per unit, area of the stream bed is proportional to the square. of the velocity; that ls, this force is equal to X"V2 where ]( is a constD.nt of proportionBlity. The surface of contact of the flow with the stream bed is equal to the product of 'the wetted peri~eter and the length of the channel l'efLClh, or PL (Fig. 5-2). The total foroe resisting the flow 2 is then equal to ](VSPL. 1 The source of this famous formula. is.not mentioned in most hydraullcs textbooks. In fact, this knowledge has long been sought for. In 1876, the German engineer Gotthilf Heinrich Ludwig Hagen mentioned in his 'work [7] that Geapard de Prony , had stated tha.t Chezy set up this formula in 1775, on the occa$ion of a report that Chezy made on the Canal de j'Yvette in conjunction with Jean-Rodolphe Perro net, . "But," says Hagen, "I have sought in vain for further information on the' Bubject." Then, in 1897, the American engineer Clemens Herschel through the assistance of a friend in Paris traced the original Canal de I'Yvette report to its hiding place, then translated the portion relating to the formula, IlInd published it in [8]. Chezy's report revealed that the formula' was developed and verified by experiments made on an earthen canal, the Courp!1let Canal, and on the Seine River in late 1769 .. • Thi~ channel resisting iorce may alao be ,explained by the principlea of fluid dynamics. The open channel can be coneeived as a. ftnt plate warped into a oylinder

i

94

i

UNIFORM FLOW

The second asst1l~ption is the basic principle of uniform flow, which is believed to have been claimed first by Brahms [9] in 1754. It states that,' in uniform flow, the effective component of t~e gravity force caus~ng the flow musl', be equal to the total force of reslStance. The effectIve gravity-force component (Fig. 5-2) is parallel ~o th~ chal~nel bott.o~ and equal to tuAL sin e = wAL8, where w is the Ulllt weIght Ol water, A lS the water a;'ea, fJ is the slbpe angle, and 8 is the channel slope. l Thence, wAL8 = KPPL. Let, AlP = R and let vw/K be replaced by a factor C; then the previous equation is reduced to the Chezy formula, Of

V

=

v(wIK){AIP)S

= C vRS.

.

DEVELOPMENT OF UNIFORM FLOW AND ITS FORMULAS

j !

i

i I

I

the formula appears cumbersome, it usua-lly produces satisfactory results. It has been so widely used that many tables and charts are available for its application; so the use of the, formula itself is seldom found necessary in engineering offices. Figure 5-3 gives a popular chart for the solution of the G..K formula. .~.~ B. The Bazin Formula.. In 1897, the French hydraulician H. Ba'lin l proposed a formula according to which Ch~zy's C is considered a function of R but not of 8. Expressed in Engli8h units, this formula is C

I

attempts have been made to de~ermine the value of Chezy's C. Three important fOI'mulas developed for this purpose will be given in the next article , 5-5. Determination of Chezy's Resistance Factor. Three importltnt formulas for the determination of CMzy's C are given as follows: . ~ ~ICk'l11Jj,la~_ In 1859, tw~ Swiss engineers, ?[mguillet and . Kutter [10], published a formula expressmg the value of C m terms of the slope 8, hydraulic radius R, aild the coefficient of roughness n. In English units, the formula is .0.00281 + 1.811

M~my

.41.65

C -

1

+

8

-n°

n + ( 41.65 + 0.00281) S vB,

(5-3)

The co~ffi~ient n in this formula is specifically known as [(utter's n. The value of n will be discussed in.Ads. 5-7 and 5-8. . The G. IL formula was derived elaborately from flow-measurement data in channels of ..various types, including Bazin's gagings and the gagings of many European rivers and of the Mississippi River. 2 Although but unclosed on one si.de which corr'osponds to the free surbce of the. ~~en-channel flow. A fluid flowing ill the uncloaed cylinder will create a drag or resls"mg ~orce on the inside surface.. This force is equal to tha drag created by the flow of .flUld along a flat plate whose tl\,O surfaces offer resistance to th.e flow. The la~ter' IS equal.t,) CdP V'PL/2, wherp. Cd is tbe coefficient of drag and P IS the ma~s de~lty _of the flUid .. Thus;the factor CdP/2 is equivalent to the constant. of proportlonabt;' .K.. . . 1 The' slope under consideration is defined as the sme of the angle 0, inclinatIOn, or S = sin 11. . , The Mississippi River gagings were made by Humphreys and Abbot on the I.ower Mississippi RiYer between 1850 and t860, and the data thus o.btained were published in a report submitted to the U.S. Army Corps of Topographica'J.Engin~ers in 1861 (11]. The term containing S was introduced into the G. K. formula Simply m order to make the formula ~gree with the Humphreys and Abbot d.ata. Thi~ se.ems somewhat ridiculous nolY, .becllu.se these data are known to have been qUite maccurate (see pp. 133-136 of (2]). Some authors have suggested that the slope term 0.00~81/S of

95

= __ 157.t'_·_ 1

+ m/vR

1

(5-4)

>;".here m is 9. coeiIicient of roughness whose values proposed by Bazin are given in Table ·5-1. TABLE

5-1.

PROPOS!lD VALUES OF BAZIN'S 171.

Description of chanael

Bazin's.1n

Very smooth cement of planed wood ............... . Uuplaned wood, concrete, or brick: ............... . Ashla.r, rubble masonry, or poor brickwork .......... . Earth channels in perfect condition ........... : .... . Earth channels in ordinary condition .............. . Earth channels in rough condition ................ .

0.11 0.21 0.83 1.54 2.36

The formula was developed primarily from data collected from small experimenta.l channels; hence, its general application is found to be less satisfactory than the G. K. formula. The Miami ConservancY'District [2] has made a study comparing the v!l.riations ill Chezy's C, Bazin's m, and I{utter's n for Bazin's experimental data and several naturaL streams. The results based on this study are showll in Table 5-2. The values of the average variation indicate that Bazin's formula is not as good as Kutter's even for his own measurements.

.~ C. The. Powell Formula. In 1950; Powell [14) suggested a logarithmic. formula for the roughness of artificial channels. implicit function of C, is C =

-42 (4i + It) log

This formula, an (5-5)

-----------------------------------

the G . .K. formula be omitted in order to simplify the aP?lmranCe of the formula. a.nd even to make -the general results more satisfactory. 1 From 1855 and 1862 an exten.~ive seiie:s of experiments on open-ch,mnel flow were first begun by H. Darcy and then completed by Bazin. The results were published by Bazin in 1865 (12). On the basis of the accumulated date., Baziu finally proposed the formula in 1897 (13).

-\ 1

. I

J j

OJ

.S. tQ

.> ,..

"'

'0

u..

0

'"s<
i'l

c

~

DEVELOPMENT-OF UNIFORM FLOW AND ITS FORMULAS

1l 'tl

where R is the hydraulic radiusin H; R is the Reynolds numbel';and E is a measure of the channel roughness, having the tentative values shown in Table 5-3. For rough channels, the flow is generally so turbulent that R beoomes very large compare~~th C; thus, Eq. (5-5) approximates the. form

.~

"'" .S 0

t1. t:i A

A

Q)

r:I

'"

A, j;:

il:

~-

p"

~

~

..,:il

'"

...'"

til

T.tUILE

'"

Q)

.c: .,

c := ;i!

0

OJ

..,

1:j

,S tQ

:S'"

-

Average vaiue3 C

Bazin's Series 6 .. , .......... " .. "'...

" "' .~

.... Q)

A

o

., c

oil

~g'O

~ l:t:1 ill' ",.",

'"""""

:El~

en

.q

'" '0 .!!l

~

'" :El

S.AZ~H:>

2~!!!~g.~E~i.~ii!~~

m

C

. . . . 0.185 O. 0121 ..

8 ........ " ........ " ...... , ..... 9 .................. " .... ' ..... 10 ............................... 11., ............................ 12 ............ , ................. Ill ........................... 14., ............... , ........ , .. 15 .............................. 16 .............. ·~ ............... 17 ....... :: ............... 1.... 32 .......................... j.... 33 , ., ..••.......... ,. . .... , .....

Q

~~]

varia.tions, %

7 .... , .................. , ... , .. , 0,156'0.0120 .. .

"I

,~

m,

n

t:i

~~

'"

BAZIN'g

Measurements

0

~

C,

AND KTJT'J'ER'S 11

g tJ" is ... "'-~ .,.,.., .gj

.J:!

;0;:

COMPARISON OF VARIATH)NS IN CHi:z.y's

c

III


5-2,

oil

oil .. :a :a .... ... !5 .., .i1III ....'" .~'" ... .!!l 0-

97

.8

g :El'" ...,'"

'" ill

U en "'-

...

'A

'" .,m-

., 0; " 5.,'" ... ..::; '"~

'0

'"

'-'

Q)

J

I lI

.,

!J

0,1420.0116 .... . 0.1990.0130 ... .. 0.1440.0117 .... , 0.129 O.OU3 .... , 0.324,0.0151. 1.6 OJHI0.0148 .... :j2.7 0.3210.0150 .... 4.4 0.'7150.0209 . . . . 4,2 0,7110.0212 ..... ' 5.7 0.7210.0215 16.1 0.4240.0' 1.8 0.444 0.0 3 .1

1 1 1.0 2,5 1.2 1.4 3.8 1.0 1.2

1.8 1.2

1.5 2.2 0.4 L2 8,8 5.7

. 44, ..... .' ..................... 0.8580. , ... ;1118:6 45 .. , .......... :.; .............. 0.7040. . .... 11.1 Mia.mi River at Tadmor, Ohio, 1915-1916". 167.4* L9S O. 4.0810.9 4.9 Bogue Phalia River, Miss .• 1914 .... " ....... 63.3· 4.09 0.070424.20.35.1 22.2 Arkansas Drainll.ge Onnals, Ark., J915 ....... 165.9· 2.12 O. :3 18 4.8 Mississippi River, Carroltoll, La., J912 ..... , ...... 1.33 1.30 5.4 MiSSissippi River, 'Oarrolton, La., 1913 ... , ....... , 1.46 12.8 IrraWllddy River, Burm!l ......................... 1. 35 23.0 Volgu. River at Sa.mara., R.ussia. ... ,.......... .58 0.0311 1.8713.0 Volga River at Zhiguly, RUl~sia., ........ " . ,76 0.0363 18.8036,5 Average va.riation.. .. . " .:, ... , .. , .. ,

gg

fL~.tG,~I~,LIW.!~I~!~I'L1UT~.ll~ttLI~I~.llk'~~~~~~WW~~WU~~~~o S3l'iNlmlO .:Ici SIXV

• Values a..veraged by the author.

I

C = 42 log (R/e). For smooth channels, the iurface roughness may be: . so slight that e becomes negligible compared with R; then the formula approaches the fbrm,C = 42 log (4R/C). Sinoe Chezy's is expressed implioitly in the Powell formula, the solution of 'the formula for C requires: !l. trial-and-error procedure. . . The Powell formula was developed from limited laboratory experiments: on smJoth and rougn channels and from the theoretical velooity distri-;

, (

a

i'

96 .

:1\

."

~

,',

.;

98

UNIFORM, FLOW

DEVELOPMENT OF UNIFORM FLOW AND ITS' FORMULAS

butioll studied by Keulegan (Art. SA). The practical application of this formula is limited, since further investigation is needed for determination of the properva.[ues of E. TAIILll

5-3.

TllNTATlYE VALUES OF POWELL'S.

Powell's. Description of channel

Neat cement surface ............... . Unplaned-plank flmlles .... ; ............ . Concrete-lined channels. . . . . . . .. . ..... . Earth, straight and uniform ............ . Dredged ear~li channels ................. i

New

Old

0.0002 0.0010 0.004 0.04 0.10

0.0004 0.0017 0.006

Example 5-1. Compute the velocity and discharge in the trape~oidal channel described in Example 2-1, hONing a bottom width of. 20 ft, side slopes 2: 1, and a depth of wat,er 6 ft. Given: Kutter's n = 0.015, and S = 0.005. Solution. From Example 2-1, A = 192.0 it' and R = 4.10 It. Using the G. l{.. formula, the value or Ch6zy's C i"

0= Then, by the

Chc~y

41 65 + 0.00281 + 1.811 0.005 0.015 . . '41 6- + 0.00281) 0.015 +( • 0 0.005 V4J.O

=

1242 .

formula,

V

=

124.2 V4.10 X 0.005 = 17.8 fps

Therefore,

Q = 192.0 X 17.8

=

3,420 ds

5-6. The Manning Formula. In 1889 the Irish engineer' Robert Manning I presented a formula, which was later modified to its present 1 Manning first presented the formula in a paper read on December 4, 1889, at a meeting of the Institution of Civil Engine:ers of Ireland. The paper ,...-as Io.ter published in the Transa.ctions of the Institution [15J. The formula \V'as first given in a complicated form and then simplified to V '= CR?~S'\, where V is the mean velocity, C is a factor of flow resistance, R is the hydraulic radius, and S is the slope. This was further modified by others and expressed in me.tric units as V = (1ln)R~\SH. Later, it was converted back again to English uni~,resulting in V = (1.486/n)R;~S'~. In this conversion, as iIi the conversion of the: GanguilLet and Kutter formula, tlie numerical value,of n is kept unaffected. Cons:equently, the same value of n is widely used in both systems of units. In the vielV of modern flllid mechanics, which pays much attention to dimensions, the 'dimensions of n become a matter of consideration. Directly from the Manning formul!l, the din).ensions of·n are seen to be TL-H. Since it is unreasonable to suppose that the roughness coeHicient w01..lld contain the dimension T, some, authors assume that the, numerator contains -../g, thus yielding the dimensions cif i>~ for n. Also, for physical reasons, it will be seen that n = ((Rlk)Jk>~ IEq. (B-26)J, \~here k is

99

well-known form (5-6)

where V is the mean velocity in fps, R is the hydraulic radius in ft, S is the slope of energy line, and n is the coefficient of roughness, specifically known as 111anning' 8 n. This formula was developed from seven different formulas, based 011 Bazin's experimental data, and further verified by 170 observatipns} Owing to its simplicity of form and to the satisa linear meas,ure of roughness and (Rlk) is a iunction of Rlk. If
-1

j l.

\

I

J

,

100

i

I

UNIFQRMFLOW ,

DEVELOPj\HlNT OF UNIFORM FLOW A.ND ITS FORMULA.S

i

factory results it lends to practical applications, the Manning formula has become the most widdy used'of all uniform-flow formulas for 6pen-channel flow computations.; A nomographic solution of the formula is given in Appendix C. Within the normal ranges of slope and hydraulic radius; the values of Manning's n and Kutter's n are generally found to be numerically very close. For practical purpose~ the two values may be considered identical when the slope is equal to or greater than 0.0001 and the hydl'aulic radius is betwe~n 1.0 and 30 ft. Typical values good fer both Kutter's nand Manning's n are shown in Table 5-0 and illustrated in Fig. 5-S. Comparing the Chezy formula with the Manning' formula, it can be seen that

c

)

-I

(5-7)

This' equation provides an important relationship: between Chezy's C and :Manuing's n. The exponent of the hydraulic radius in the Manning formllla is !'.ctnally not a constant but varies ina range depending mainly on the channel shape !l.nd roughness (see 11 previous footnote). For this reason, some' hydraulicians prefer to use the. formula with a variable (lxponent. For example, the unifomi-flow formulil. widely used in the U.S.S.n. is of this type; this is the Pavlovskiiformttla [21], proposed in 192£$.* This formula in mei)'ic linits is (5-8) (5-9)

. where

and where C is'the resistance factor in the Chezy formula expressed in 'metric units. The exponent y depends on the Ioughnesscoefficient and hydraulio radius, Tile formula is valid Jor R between O.i'and 8.0 ill and ,

"

basis of other later studies, some authors Stiggested ll. value oi % [20], and others suggested a variable depending on Rand 11. [21J. . l The Manning formula was suggested for intern~tionaJ use by Lindquist [3j at the Scandinavie:Sectional Meeting oi the World Power Conierenee in 1933 in Stockholm. The final recommendation for such use was mlide by the Executive Committee at the 3d World Power Conference in 1938 in Washington, D.C. . 2 On account of this relationship, the Manning formula is sometimes oonsidered a variation'oi the CItezy fortimla with Chezy's C defined by Eq. (5~7). *' The Pa.vlovsIl.il formula. was published in severa! editions of Pll.v!ovskiI's "Ha.ndbook of Hydraulics" [211_ An article o.bout this formula. entitled Ft)1'l1m!a dlia kt)effitlrien/.ll Chazy (Formula fur a Chtzy coefficient) is given in pp. 140-149 of the 193T edition or the book. A footnote in this article reads: "'1'.he formula. was proposed in 1925."

101

for n between 0.011 and 0.040. For practical purposes, the following approximate forms of Eq. (5-9) are generally suggested for use: Y= 1.5 Y"" 1.3

vrn

vn

for R for R

< >

1.0 m 1.0 m

15-7. Determination of Manning's Roughness Coefficient, In applying Manning formula or the G. 1C. foi'Inula, the greatest difficulty lies iIi" the determination of the roughness coefficient n; for there is no exact of knowledge, to method of f'electing the n value. At the present select !1 value of n actually means to estimate the resistance to flow in a given channel, which is really a matter of intangibles. To veteran engineers', this means the exercise of sound engineering judgment and experience; for beginners, it. can be no mOl'e than agness, and different individuals will obtain different results. . .III Qrder to give guid::mce in the proper determination of the roughness coefficient, four general approaches' will be discussed; namely, (1) to understand thefa,ctors th!l.t affect the value of n and thus to acauire a basi~ knowledge of the problem and l~arrow the wide range of gues;work, (2) to consult a table of typical n values for channels of various types, (3) to examine and become acquainted with t.he appearance of some typical ehannels whose roughness coefficients are known, and (4) to determine' the value of n by an rmalytical procedure based on the theoretical velocity distribution in the channel cross section and on the data of, either velocity or roughness measurement. The first three approaches will be given in the next three articles, and the fourth approach will be· taken up in Art. 8-7, 5-8. Factors Affecting Manning's Roughness Coefficient, It is not uncommon for engineers to think of a channel as having a single value of n for aU occasions. Ii1 reality, the value of n is highly vELdable and depends on a number of factors. In selecting a propel' vaiue of n for various design conditions, fl,basic knowledge of these factors should be found very usefuL 'The factors that exert the greatest influence upon the coefficient of roughness in both artificial and natural channels are therefore described below. It should be noted that these factors are to a Certain extent int/ilrdependent; hence discussion abou,t one factor may be repeated in connection 'with anothel'. ' A. Surface. Rm€ghness, The surface roughness is represented by the size and shape of the grains of the material forming the wetted perimeter and producing a retftrding effect on the flow. This is often considered the only factor in selecting a rOllghness coefficient, but it is actually just one of several major factors. Generally speaking, fine grains result in a relatively low value of n and coarse grains, in a high value of n. In alluvial streams where the material is fine in grain, such as sand, t~e

DEVELOPMENT OF UNIFORM FLOW AND ITS FORMULA.S

UNIFORM FLOW 102 clay, loam, or silt, ,the retardiI).geffect is much less tha;n whe:e t?e material is coarse such as gravel;; or boulders. When the materIal IS fine, the value of ~ is low and relatively un.affected by change i.n flow stage, 'Yhen the material consists of gravels and boulders, the value of 71 is generally high, particularly at low or high stage. Larger boulders usually collect at the bottom of the st.ream, making the channel bottom rougher than the. banks and increasing the value of 71 at low stages; At high stages, a . portion of the energy of flow is used in rolling t.he boulders downstream, thus increasing the value of n. A theoretical discu,ssion of surface roughness will be 'given in Art. 8-2. B. T1 egetation. Veget.ation may be regarded as a kind of surface roughness, but it also markedly reduces the·capacii~ ~f the c~annel. an~ retards the flow. This effect depends mainly on heIght, denmty, dlstnbutiol1, and type of vegetation, I;1nd it is very important in designing . small d[ainl;1ge channels. At the University of Illinois an investigation has been made to determine the effect of vegetation on the coefficient of roughness [22]. On one of the drainage ditches in central Illinois under investigation, an avera~e n :value of 0.033 was measured in March, 1925, when the channel was ill good condition .. In April, 1925, there were bushy willow? a~d dry w~eds on the side slopes, and n was found to be 0.055. ThIS ll1crease m. 71 represwts the result of one year 's growt~ of vegetation. D~ring the summers of U)25 and 1925 there waS a thICk growth of cattaIls on the bottom of the channel. The n value at medium summer stages w~s about 0.115, l,tnd at a nearly bankfull stage it was 0.099. The cattaIls inihe channel were washed out by the high water in September, 1926; the average value of 71 found after this occurrence was 0.072 .. The conclusions. drawn from this investigati'on were, in part, as follows: 1. The minimum value of 11, that should be used for designing drain~ge ditches in· centml Illinois is .0.040. This value is obtainable at high· stages duri!1g the summer months in th~ most carefully n:aintained ch~n­ nels where the bott.om of the channel IS clear of vegetatiOn and the. Side slop~s are covered with grass or low weeds, bU,t no bushes. This low value of n shonld not be used unless the channel IS to be cleared a:nnuaUy of all weeds and b u s h e s . . . 2. A value of n = 0,050 should be used if the channel is to be cleared in alternate years only. Large weeds and bushy willows from 3 to 4ft high on the side slopes will produce this value of n. 3. In channels that are not cleai'ed for a number of .years, the growth may become so abundant that values of n· ~ 0.100 may. be found. ,. . 4. Trees from (3 to 8 in. in diameter growmg on the Side slopes do not impede the flo\v so much as do small bushy growths, provided overhang. ing branches are cutoff,

'.'

j

i ",1\.,'.,..

."'

'"

103

The U.S. Soil Conservation Service has made studies on flow of water in small shallow channels protected by vegetative linings (Chap. 7, Sec ..0), It was found that n values for these channels varied with· the shape and cross section of the channel, the slope of the channel bed, and . the depth of flow. Comparing two channels, all other factors being equal, the lesser average depth gives the higher n value, owing to a larger proportion of affected vegetation. Thus, a triangular channel has a higher n value than a trapezoidal channel, and a wide channel has a lower n value th11D. a narrow channel. A flow of sufficient depth tends to bend over and submerge the vegetation and to produce low n values. A .steep slope causes greater velocity, greater flattening of the vegetation, and low n valu,<::s. . The effect of vegetation on flood plains will be discussed later in item H. . C. Channel Ir:regularity. ChanneUrregularity comprises irregularities in wetted perimet.er and variations in cross section, size, and shs.pe along the channel length. In na.tural channels, such irregularities are uSl,lally introduced by the presence of sand bars, sand waves, ridges and depressions, and holes and humps on the channel bed. These irregul9.rities definitely introduce. roughness in addition to that caused by surface roughness and other factors. Generally speaking, a gradual and uniform change in cross. sectioll, size, and shape will not a.ppreciably affect the value of n, but abrupt changed or alternation of small and large sections necessitates the use of a large value of 71. In this case, the increase in n may be 0.005 or more. Changes that cause sinuous flow from side to side of the channel will produce the same effect. D. Channel Alignment . . Smooth curvature with large radius will give a relatively low value of 71, whereas sharp curvature with severe meandering will increase 71. On the basis of flume tests, Scobey [23] suggested that the value of 71 be increased 0.001 for each 20. degrees of curvature 111 100 ft of channel. . Altholagh. it is doubtful ,..hether curvatllre ever increases 11, mOfe than 0.002 or 0.003, its effect should not be ignored, for curvature may induce the accumulation of drift and thus indirectly increase the value of n. Generally speaking, the increase of roughness in unlined channels carrying water at low velocities is negligible. An increase of 0.002 in n value would constitute an adequate allowance for curve losses in most flumes containilig pronounced curvatures, whether built of concrete or other materials. The meandering of naturai streams, however, may increase the 71 value as' high as 30 %. E. Silting and Scouring. Generally speaking,.8ilting may change a very irregular channel into a comparatively uniform one and decrease n, 'whereas scouring may do the reverse and increase 71 • . However, the dominant effect of silting will depend on the nature of the material deposited. Uneven deposits such as sand. bars and sand waves 'are

\

1

-1

1

I 1

I

.

\

104

i

UNIFORM FLOW

l'

, channel irregularities and will increase the 'roughness. 1 The amount · uniformity of scouring will depend on the material forming the wetted · perimeter. Thus, a sandy or gravelly bed will be eroded more uniformly than a clay bed. . The deposition of silt eroded from the uplands will tend to even out the irregularities in:a channel dredged through clay, The energy used in eroding and carrying the material in suspension or rolling i~ along the bed will also increase the n value, The effect of scouring is not significant as long as .the erosion on channel bed caused by high velocities is.progressing eyenly and uniformly. F. ObstrucHon. ,The presence of log jams, bridge piers, and the like · tends to increase n. The amount cif increase depends on the nature of the obstructions, theil' size, shape, numbel', and distribution. G. Size and Shapl'J of Channel. There is no definite evidence about the size and shape of a channel as an important factor affecting the value of n. An increase in hydraulic radius may either increase or de.creasen, depending on t,he condition of the channel (Fig. 5-4). H. Stage and Di,scha1'{}e. The n value in most strearas decreases ,vith increase in stage and in' diseharge. When the water is shallow, the irregularities of the. channel bottom are exposed and their effects become pronounced. However, the n value may be large at high if the banks are rough and gra.ssy. and When the discharge is too high, the stream may overflow its a portion of· the flow will be along the flood plain. . The n v3,lue of the flood plains is generally larger than that of the yhannel proper, and its magnitude depends on the surface condition or vegetation. !f the bed and banks of a channel are equally smooth and regular and the bottom. slope is uniform, theyalue of n may remain almost the same at all so a constantn is usually assumed hi the flow computation. happens mostly in artificial channels. On flood plains the value of n usually varies with the stage of submergence of the vegetation at low stages. This cnnbe seen, for example, from Table 5-4, which shows the· n values for various flood stages according to the type of cover and depth TAriu:l 5-4.

n

VALUES OF

FOR VARIOUS' STAGES

m

f

!

!I i

I

I

I

;DEVELOPMENT OF UNIFORM FLOW AND ITS FORMULAS

105

of .inundfltion, as obseryed in the Nishnabotna River, low-a,fof the average gfowing season [24]. It should be noted, however, that vegetation has a marked effect only up to certain stage and that the roughness coefficient can be considered to remain constant for pl'actical purposes in determining overbank flood discharges.

a

50 o

Mississippi River oetweenMemphis and Fulton, Tennessee . o U,S, Geologicol Survey doto .. U.S, Corps of Engineers dato

i!:

~ ~O

.'2

g"

'"

20

;:;:

o

I

O.k02;;;5.----:O':::-:03~O"..----;:O,..,.Oi,.3.",5-.---,;,.,!-,=---=~::,! n value

Tennessee River at Cholloooogc, Tenn,

• 0.045·

n value

FIG. 5-4, Va.riations of the n value with tile.me!tn stage or depth.

THE NI5HNJ..BOTNA RIVEI!.,

IOWA, FOR THE AVlllRAca; GROWING SEl.SON.

Flood-pilli~

Depth of water, it

Channel section

Under 1 1 to 2 2 to 3 3 to 4 Over 4

0.03 0.03 0,03 0,03 0,03

Corn

Pasture

Meadow

i

0,10

0,06 0,06

0.07

I

0,07 0,05

O,O~

0.04 0.04 0,04

cover

0.07

0,06 0.05

Small gt'ains

Brush and waste

0,10 0.09

0,12 0.11 0,10 0.09

0,08 0,07 0,06

O,Oi

I I

!

1

Curves of n v!\lue versus stags (Fig. 5-4) in streams have been given by Lane (25\; showing how vahle of n varies with stage three river channels. For the roughness of large canals, it study in connection with the design of the Panama Canal ,yas made by IVley~r& and .Schultz [26).1 . The two most important conclusions reached from this study were (1) that the n value fol' a river !channel is least when the stage is at 01' somev;rhat above normal bankfull stage, and tends to in~rease for both 1 A tJble of ; vJlues for eleven lar~e cbanneJs at tbe most efficie[lt depths and tlle curVel, showing th.e variations 01 11 value with hydra.ulic radius in eight river channels are also given in this reference.

DEVELOPMJilNT OF UNIFORM FLOW AND ITS. FORMULAS

UNlFORM FLOW

:106

In selecting,. the. value of nl, the degree of irregularity is considered smooth for surfaces comparable to the best attainable for the materials involved; minor for good dredged chm1l1els, slightly ei'oded or scoured side slopes of canals or dra.inage cha.nnels; moderate for fair to peor dredged channels, moderately sloughed or eroded side slopes of canals cr drainage .channels; and severe for badly sloughed banks of natural strea.ms, badly eroded or sloughed sides. of canals or drainage channels, and unshaped, . jagged, and irregular surfaces of channels excavated in rock. III selecting the value ofn2' the character of variations in size and shape of cross section is considered gmdlLal when the change in size or shape occurs gradually, alternating occasionally when lal'ge and small se·::tions alterna.te occasionally or when shape changes r,ause occasional shifting of main flow from side to side, and alternating frequently whim iarge arid small sections alternate frequently or when shape changes cause frequent shifting of main flow from side to side, The selectioll of the value of n3 is based 011 the presence and characteristics of obstructions such as debris deposit", .stumps, exposed roots, boulders, and fallen and lodged logs. One should recall that conditions considered in other steps must not be reevaluated or' double-counted in this selection, In ,iudging the relative effect of obstructions; consider the following: t.he. extent to which the obstructions occupy or reduce the average water area, the character of obstructions (sharp-edged or angular objects induce greater turbulence than curved, smooth-surfaced objects), and the position and spacing of obstructions transversely and longitudi. nally in the .reach under consideration. . In selecting the value of n~, the degree of effect of vegetation is considered' . (1) Low for conditions comparable to the following: (a) dense gl'owths of fla"{ible ~urf grasses Qr weeda,of which Bermuda and blue grasses are examples, where the average depth of flow is 2 to 3 times the height of vegetation, and (b) supple seedling tree switches, such as willow, cottonwood, Or salt cedar where the average depth of flow is 3 to 4: times the height of the vegetation. (2) 1l1edi11.?n for conditions comparable to the following: (a) turf grasses where the average depth of flow is 1 to 2 times the height of vegetation, (b) stemmy grasses, weeds, or tree sCl'ldlings with moderate cover where the average depth of flow is 2 te 3 times the height of vegetation and {c) brushy growths, moderately dense, similar to willows 1to2 ~ean; old, dormant season, along side slopes of a channel with no significant vegetation along the channel bottom, where the hydraulic radius is greater than 2 ft. . '(3) High for conditions comparable to the f;llowing: ea) turf grasses where the average depth of flow is about equal to the height of vegetation,

;higher and lower stages; and (2) that the bankfulLn values do Dot vary . greatly for rivers and canals in different kinds of material and in widely. separated locfl.i;ions. For circular conduits, Camp [27,28] was able 'to show that the n value for 11 conduit flowing partially fuBis greater than that for a full conduit. Using measurements on clean sewer pipe and drain tile, both clay and concrete, from 4 to 12 in. in size, he found an increase of about 24% in the n value at the half-depth (Fig. 6-5).l The n value for the pipe flowing full was found to vary from 0.001:15 to 0.011. Taking an average value of 0.0103, the nvalue at half-depth should be about 0,013. This is identical with the usual design value,which is based largely on measured values in sewers Hov\e'ing partially fulL . I. SeasonoJ Change. Owing to the seasonal growtih of aqus,hicplants, grass, weeds, willow, and trees in the channel or on the banks, the value of n may increase in the growing seasOl]. and diminish in the dormant season. This seasonal change may cause changes in other factors. J, Suspended Material and Bed Load. The suspended material and the bed load, whet.her moving or not moving, would consume energy and cause head loss or increase the apparent channel roughness. All the above factors should be studied I1nd evaluated wit,h respect to conditions regarding type of channel, state of flow, degree of mainten~~ce, and other related considerations. They provide a basis for determmmg the proper value of n for a given problem. As a general guide to judgment it may be accepted that conditions tending to hldtlce turbulence and c~use retardance will increase 1. value and that those tending to reduce turbulence and retardance. will decrease n value. Recognizing several primary £actors affecting the roughness coeffic,el1~, . Cowan [32J developed a procedure for estimating the value of n. By thIS procedure, the value of n may be computed by <

n

=

(no

+

",1

+ n2 + nl + n()m5

(5-12)

where no is a basic n value for a straight, uniform, smooth channel in the natural ~aterial:; involved, nl is 8, value added to no to correct for the effect of surface in'egularities, 'lt2 is a value forvf.tl'iations in shape and size of the channel cross sectiol1, Tt3 is a value for obstructions, n4 is a value for vegetation and flow conditions, and mil is a correction factor fllr meanderiIlg of channeL ' Proper values of. no to n, and m, may be . selected from Table 5-5 according to the given conditions. . I The n/n~ curve was based O.D measurements by Wilcox [29] on 8-in, clay and con~ ~rete sewer pipes a.nd by Yarnell a.nd Woodwa.rd 130] on open-butt-joint concrete . ahd cla.y drain tiles 4. to 12 in. in eiZe. For depths less thaD about O.I5d., the eurve was verified by the data of Johnson [311 forlarge sewer!!, .

., ,j

I !

t

t

~""---'"~--'.".-~.-.-,'~

.

r-

1

,-I

J --\ \'

-\ ,1 ~!

:~'

,:i:; 'r.~.'

107

'.

~~.

I

'I

'\

,

!

4;1

I'

UNIFORM: FLOW

I,

(b) dormant season-willow or cottonwood trees 8 to 10 years old, inter-

I

grown with .some v,l:eds and brush, none of thp- vegetation in foliage, where the hydraulic radius is greater than 2 ft, and (c) growing seasonbushy willoW's about 1 year old intergrown with some weeds in full foliage along side slopes, no signifieant vegetation along channel bottom, where' hydraulic radius is greater than 2 ft. . (4) Very high for conditions comparable to the following: (a) 'turf . grasses where the average depth of fio~ is less than one-half the height of vegetation, (b) growing season-bushy willows about 1 yellr old, intergrown with weeds in full foliagealon.g side slopes, or dtmse growth of eat tails along Ghannel bottom, with any value ofhydratilic radius up to 1001' 15 ft, aIld (c) growiIlg season:-trees intergro,vn with weeds and brush, all in full foliage, with any value of hydraulic radius up to 10 or 15 ft. Ir:. selecting the value of ms, the degree of meandering depends all the ratio of th.e meander length to the straight length bf the channel reach. The meandering isconsidel'ed minor for ratios of 1.0 to 1.2, appreciable for mtios of 1.2 to L5, and severe for ratios of 1.5 and greater. In applying the above method for determining the n value, several things should be noted. The method dQes not consider the effect of suspended and bed loads. The values given in Table 5-5 were developed from a study of some 40 to 50 cases of small ftnd moderate channels. Therefore, the method is qU
fi

108

" )

L The minimum value lor Lucite was observed in the Hydra.ulic Engineel'ing La.bol'aLory nt the University or Illinois [33). Such a low n value Inay perhaps be obtained also for smoot\) brass o.nd glass, but no observntions have yet been reported. t A table showing n values and other elements hom 269 observations made on many existinga.rtifiCia.l cha.Ilnels is also given by King [35J.

DEVELOPMENT OF UNIFORM FLOW AND ITS FORMULAS 'fAllLl!l

5-5.

VALUES FOR THE CO]>tl'UTATlON OF THE ROTlGHNESS COE·FFICI.!!lNT

:BY

I,

EQ. (5-12)

Channel conditionB

I

Values'

Earth M.ateriul in vo I.... ed

.,

109

Degree of irregularity

0.020

Rock cut

0.025 nQ

Fine gravel

0.024

Coarse gravel

0.028

Smooth

0.000

Minor

r.,

0.005

Modernte

0.010

Severe

0.020

Gradual Va.ria.tions. of channel cross Alternating oceaSionally section Alternating irequently

0.000

j

n2

O.010~.015

Negligible Relative effect of obstruutions

0.000

Minor 1\...."

. 0.005

O. 01O-{}. 015

..

'11.. ~

0.030-{}.030

.~

0.040-0.080 Low

0.005-0.010

Medium

0.010-0.025

High

0.02S-{}.050

Very high

0.050-0.100

VegeLfLtion

Minor Degree of meandering

Appreciable Severe

1.000 m.

1.150

1.300

I

) 110

UNIFORM FLO>'i

Typ~ of channel and description

I Minimum I Normal

----~~~~~~~~~-I

II

Maximum

. I

CLOSED CONDUITS FLOWING. PaRTLY FULL

A-I. Metal a. Brass, smooth b. Steel 1. Lockbar and welded 2. Riveted and spiral c. Cast iron 1. Coated 2. Uncoated d. Wrought iron 1. Black 2. Galvanized e. Corrugated met.al 1. Sub drain . 2. Storm drain A-2. Nonmp.tal. a. Lucite

b. Glasll c. Cemen't 1. Neat, surface 2. Mortar d. Concrete 1. Culvert, straight and free of debris 2. Culvert with bends, connections, and some debris 3. Finished 4. Sewer wit.h manholes, inlet, etc., strl!-ight 5. Unfinished, steel form 6. Unfinished, smooth wood form 7. Unfinished, rough wood form e. Wood 1. Stave 2. Laminated, treated f. Clay 1. Commondrain8.ge tile 2. Vitrified sewer 3. Vitrified.sewer with manhol,~s, illlet, etc. 4. Vitrified subdrain with open joint g. Brickwork 1. Glazed 2 .. Lined with cement mortar h. S'anitary sewers coated with sewage slimes, with bends and connections i. Paved invert, sewer, smooth bottom j. Rubble masonry, cemented

111

DEVELOPMENT OF UNIFORM FLOW AND ITS FORMULAS

. TABLE 5-6. VALUES OF THE ROUGHNESS COEFFICIENT n (Boldface figures are values generally recommended in design)

A.

-(

I

0.009

0.010

0.013

O.OlD

0.013

0.012 0.016

0.014 . 0.017 .

I

0.010 0.011

0.013 0.014

0.014 0.016

I

0.012 0.013

0.014 0.016

0,015 0.017

0.017 0.021

0.019 0.024

0.021 0.030

0.003 0.009

0.009 0.010

0.Of3 .

0.011

0.011 0.013

0.013 0.015

0.010 0.011

0.011 0.013

0.013 0.014

O.OlD

O.OlD

I ! f

iI

0.011 0.013

0.012 0.015

0.. 014 1).017

0.012 0.012 0.015

0.013 0.014 0.017

0.014 0.016 0.020

O.OlD

0.015

0.012 0.017

0.014 0.020

0.011 0.011 0.013

0.013, 0.014 Q.015

0.017 0.017 0.017

0.014

0.016

0.018

0.011 0.012 0.012

0.013 0.015 0.013

0.015 0.017 0.016

0.016 0.018

0.019 0.025

0.020 0.030

J. f

!

. TABLE 5-6. VALUES OF THE ROUGHNESS COEFFICIENT n

Type of channel and description

-----D.

Minimum

(continued)

Normal Maximum

1

LINED OR .BUILT-UP CHANNE1.S

. B-l. Meta! a. Smooth steel surface 1. Unpainted 2. Painted b. Corrugated B-2. Nonmet.al a. Cement 1. N el1t, surf!lce 2. Morta-r. b. Wood L Planed, untreated 2. Planed, creosoted 3. Unplan(ld 4. Plank with battens 5. Lined with roofing paper c. Concrete 1. Trowel finish 2. Float finish 3. Finished, with gravel on bottom 4. Unfiliished . 5. Guuite, good section 6. Gunit8, wavy section 7. On good excavated rock 8. On irregular excavated rock d. Concr'lte bottom float finillhed with sides of 1. Dressed stone in mortar 2. Random stone in mortar 3. Cp,ment rubhle masonry, plastered 4. Cement rubble maSQnry 5. Dry l'Ubble or riprap e, Gravel bottom with sides of 1. Formed cOllCrete 2. Random stone in mortar 3. Dry rubble or riprap f. Brick 1. Glazed 2. In cement mortar 11. Masonry 1. Cemented I'Ub ble 2. Dry rubble h. Dresse.d ashlar i. Asphalt 1. Smooth 2: Rough . j. Vegetal lining

0.011 0.012 0.021

0.012 0,013 0.025

0.014 0.017 0.030

0.010 0.011

0.011 0.013

0.013 0.015

O.OlD

0.012 0.012 0.013 0.015 0.014

0.014 0.015 0.015 0.018 0.017

0.013 0.015 0.017 0.017 0.019 0.022

0.015 0.016 0.020 0.020 0.023 0.025

0.011 0.011 0 .. 012 0.010 0.011 0.013 0.01.5 0.014 0.016 0.018 0.017 0.022

0.027

0..015 0.017 0.015 0.020 0.020

0.017 0.020 0.020 0.025 0.030

0.020 0.024 0.024 0.030 0.035

0.017 0.020 0.023

0.020 0.023 0.033

0.025 0.026 0.036

0.011 0.012

0.013 0.016

0.015 0.018

0.017 0.023 0.013

0.025 0.032 0.015

0'.030 0.03.5 0.017

0.013 0.015 0.030

0.013 0.016

O.OZI)

J

'-,

0.500

·fI

, .·'h·

I:

I

.

t,

:.::."

,.,,,""::".:

~; I

112

UNIFORM FLOW

DEVELOPMENT OF UNIFORM FLOW' AND ITS FORMULAS

'l"\13LE. 5-6: VALUES OF TliE Ro'l1GllNESS COEFFlCllilNT Type 01 channel and description

i'

I

·Minimum

C. Exe.WATIilD OR DREDGED a. Ea.rth, straight and uniform .1. Clean, recently completed 0.016 2. Clean, e.fter weathering 0.018 3. Gravel, uniform section, clean 0022 4. With short gr!l.8S, f.ew weeds 0.022 b. Earth, winding and sluggish 1. No vegetation 0.023 2. Grass, some wGeds 0.025 3. Dense weeds or aquatic plants in 0.030 deep channels 4. Earth bottom lind rubble sides 0.028 5. Stony bottom and weedy banks 0.025 6. Cobble bottom nnd clea.n sides 0.030 c. Dritgline-eltcavated or dredged 1. No vegetation 0.025 2. Light brush on banks 0.035 d. Rock cuts 1. Smooth and uniform 0.025 2. Jagged and irregular 0.035 e. Channels not maintained, wc.eds and' brush uncut . l.Dense weeds, high as flow dept.~ 0.050 : 2. Clean bottom, brush on sides 0.040 3. Same, highest stage of flow 0.045 4. Dense brush, high stage 0.080 D. NATuML S'IREAMS' D-l. l\llnnr streams (top width at flood stage <100 it) a. Streams on plain Clean, straight, fun stage, no rifts or 0.025 deep pools I 2. Same as ..bove, but more stones and 0'.030 I weeds 3. Clean, winding, some pools and 0.033 shoals . 0.035 . 4. Same ail above, but llome weeds a.nd stones . , 1 5. Same as above, lower stages, more 0.040 ineffective slopes and sections 6. Same as 4, but more stones 0.045 7. Siuggish reaches, weedy, deep pools 0.050 13. Very weedy rea.ches, deep pools, or 0.075 ftoodways with heavy stand of tim- 1 ber a.nd underbrush

.

n (continued) Normal Maximum

I

I:

I I

I II

1·

I J

I

! I

I'

0.018 0.022 0.025 0.027

0.020 0.025 0.030 0.033

0,025 0.030 0.035

0.030 0.033 0.040 .

0.030 0.035 0.040

0.035 0.040

0.028 0.050

0.033 0.060

0.035 0.040

0.040 0.050

0.080 0.050 0.071iJ 0.100

0,120 0.080 0.110 0.1<10

0.030

0.033

0.035

0.040

0.040

0;04:>

0.045

0.050

0.0'48

0.055 .

0.050 0.010 0,100

0.060 0.080 0.150

0~050

TABLE5c6. V,u,UES

OF THE ROUCnl'NESS COE)"FICIENT 7i

ll3

(continued)

description ~inimum Normal Maximum' --~--------------~---I·----~I b. Mountain strea.ms, ·no vegetation in cha.nnel, ba.nks usually steep, trees and brush along banks submerged· at high sta.ges . 1. Bo~tom: gra.vels, cobbles, and fetv 0.030 0.040 0.050 boulders 2. Bottom: cobbles with large boulders 0.040 0.050 0.070 D-2. Flood plains 1.00 it). The 11. vallie is Jess than that for minor streams of similar ,description, becll.use banks offer less effective resistance. a. Regular section with no boulders or /1 0.0215 0.060 brush

I

____b_._I_f_re_g_u_la_r_a_:n_d_ro_U_g""h_s_e_ct_i_on_ _ _ _...i:-_O_,0_3_5_'11___L_0 .100,

114

fl'

UNIFORrk FLOW

from up-to-date information collected from various sources ([34,36,38]' and unpublished data) ; hence it is much broader in scope than the Horton table. 5-10. Illustrations of Channels with Various Roughnesses. Photo-, graphs of a number of typical channels, accompanied by brief descriptions of the channel conditions and the corresponding n values, are shown in Fig. 5-5. These photographs' are collected from different sources and arranged in order of increasing magnitude of the n vslues. They provide a [eneral idea of the appearance of the channels having different n values 8.nd so should facilitate selection of then value for a given channel condition. The n vaJue given for each che-nnel represents approximately the coefficient of roughness when the photograph was taken. The above type of visual aid is also employed by the U.S. Geological Survey. The Survey has made several determinations of channel roughness in streams, mostly in the northwes~ern United S~ates. These include measurements of cross-sectional area.) width, 9.epth, mean velocity, slope, and computation of the roughness coefficient. The reaches were photographed in stereoscopic color, and the photographs have been circulating among the district offices 'of the Survey as !!- guide in evaluating n.

l

I

-l

I

J

I

I

I

I

i

I

1 -
I 1 5-5. Typical channels ~howing different n values. (These photo h ' ft~m [3~J and [38J with the permi,nion of the U.S. Departme~~~f ~ ~~l~!t~:,:­ ~1~1~rtlj.m~1 rc~ure~ used Jo~ reprodu.ctil]ll purposes were supplied thTUugh th~ courtes; ~, •.rr" C' E' ";0 ey tofr ph,otogrhrkphs 1 10 14 rkncl phol()gl'aph 19, rknd lhrou(lh the courtesy FIG.

duced ~

,,,.

•

. ,.am.!er or l!e 01 !Jrs.)

,

1

-'

1

.) 't '. : .~

115

,-1

~l' !I I

I

t

1 i j I

(1)

!

(4)

)

1i I i

I! !

t

\'

(

I ;

,

I

.\

~ii)

!

I

I

I

(6)

FIG. 5.-5 (1-3) 1. n = 0.012. Canal lined with concrete slabs havir;lg smooth neat cement joints and very smooth surface hand-troweled a.nd with cement wash on concrete ba.!Ie. 2. n == 0.014. Concr~te canal poured behind s~reeding and. smoothing platf?rm. 3. n _ 0.016. Small concrete-lined ditch, straight ~nd u;Uf~rm, bottom slightly dished, the sides and bottom covered with 0. rough depoSIt, which Increases the n va.lue.

ll6

.

FtG. 5-5 (4-6) 4. n = O.OlB. Shot-concrete lining without smooth trea.tment. Surface covered with fine a.lgae and bottom with drifting sand dunes. S. n = 0.018. Earth channel excavated in a clay loam, with deposit of clelln sand in the middle and slick silty mud near the sides. 6. n = 0.020. 'Concrete lining made in a. tough lava-rock. cut, clean-scoured very Tougb, and deeply pitted. . '

.

117

\

J

(8)

(9)

.

IL :; ,. '\.

~ ~1

'I

I·

7. 1'1 = 8. 1'1 the earth 9. 1'1 =

.

1"10. 5-5 (1:"'9)

.

.0.020. Irrigatiollcanal, straight, in hard-packed smooth sand. of' .0.022.. ~ent-plaster lining applied directly tp the trimmed suria.ce channel. With weeds in broken places and looStl ssnd on bottom. 0.024. Canal excavated in silty clay loam. Slick and hard bed. US

FiG. 5-5 (10-12) 10. n "" O.024.l:!i~c~ lined on both sides a.nd bottom with dry-laid un~hinked rubble. Bottom qUite lrregular, with sca.ttered loose cobbles. 11. 1'1 = 0.026. Canal excavated on hillside, with upper bank mostly of willow roots and lower bank with well-made concrete wall. Bottom co:"ered with coarse gravel. 12. 1'1 = .o.O~8. Cobbl~bottom channel, where there is. insufficient silt in the wa.ter Or too hIgh a velOCIty, preventing formation of a graded smooth bed. . . lW

(13) (16)

(17) (14)

!

I

I

l i

(18)

I,

t (15)

t.

I,

!

!

13. 11 = ().029. Ea.rth c~nal excavated in alluvial silt 'soil, with deposits of sand on. bottom and growth of gi-ass. 14. 11 = 0.030. Canal with large-cobbleswne bed. 15. 11 = 0.035. Natural channel, somewhat irregular side slopes; fa.irly even, clean a.nd regular bottom; in light gray silty cla.y to light tan silt loam; very little varia.tion . in cross section.

12Q

FIG; 5-5 (16-18) 16. 11 ,= 0.040. Rock channel excav!\ted byexpl~sives. . 17. 7L;-= 0.040. Ditch in clay and sar.dy loam; irregular side slopes bottom !Lna cross section; grass on s l o p e s . : :' , 18. 11"" 0.045. Dredge channel, irregular side slopes .and bottom, in bla.ck, waxy clay at}op to yellow clay at bottom, sides covered with small'saplings and brush slight and gradual variations in cross section. ' 121

(22)

(19)

./

I

(20)

(23)

(21) (24)

I

I

I

I ~

.'{1

i

!

Flo. 5-5 (19-21) '19. 11. = 0.050. Dredge channel ~ith very irregul~r side slopes ap.d. bo~tom, in dark-colored waxy clay, with growth of weeds and gra.ss. Slight vanatlOn In shape of cross section for variation iin size. · 20. 11 = 0.060. Ditch in heavy silty clay; irregular side slopes and bottom; practi· cally entire seotion filled with large-size growth 'of troos, principally willows and cottonwoods. Quite uniform cross seotion. 21. 11. = 0.080. Dredge dianne! in black slippery clay and gray silty clay loam, : irregulrtr wide slopes and bottom, covered with dense gro\vth of bushy ~illOWIl, some · in bottom; remainder of both slopes c017ered with weeds and a soatteflng growth of willows and poplars, no' foliage; some silting on bottom... . . . . . 122

I

I i,

.

FIG.

.22. n = 0.110.

5-5 (22--241

.

8~me as (21), but with much foliage and covered for about 40 It

WIth growthresembllllg smart weed. . . 23. n = 0.1~5. Natural channel floodway hi median line sand toj fine clay none sId~ s!oPe:'; fairly e17ell and regular bottom with occasional Ba~ bpttom sl~ughs' l~ariation Hi depth; practically 17irg;in timber, very little undergrowth except occa~ slUnal dense patches or btlllh~d a.~d small trees, some logs and dead fallen trees. . 24. n ... 0.150. Natur:J.1 river In sandy clay soil. Very crooked course, irregular sl~e slopes and uneven bottom. Many roots, trees .and bushes, large logs and otb r drift on bottom; trees continua.Uy falling into ch",rmel due to bank caving. e .

123

f

.;;." Jt

124

DEVELOP!'.{ENT OF UNIFORM FLOW AND ITS FORMULAS

UNIFORM FLOW

PROBLEMS 5-1. Expiain why a uniform flow cannot occur (a) in Il frictionless channel, and (b) in a. horizontal channel. ' , ' 6-2. When Chezy's C determined by the G. Ie formula becomes independent of the slope S, show that the value of R 3.28. Find the corresponding relation between C all.:! Kutter's n. ., li-S. For tbe conditions given in Example 5-1, compute the values of Ba.zin'i! m and Powell's ~. 6-4. Compute the velocity and discharge of flow in a new etu'th canlll having the same shape, si;;e, slope, and depth of flow as the channel given in Example 5-1. Use (a) tile G. K. formula, ,assuming'Kutter's n "" 0.022; (b) the Bazin formula, selecting !l p'roper value of 11>; and (e) the Powell formula, selecting a proper value of •. 5-6. Taking Manning'S n as the given value of Kutter's n, solve E):ample 5-1 by the Manning formula. , ' 5-6. If the coefficient of l'oughness 11 is unknown for the channe.l ill Example 5-1, but a discharge of 2,000 cfs is observed under the given conditions, compute the v!Liues of l{utter's 1~ and Manning's 11. 5-7. From the MllJlning formula (using a consta.nt of 1.486 instead of 1.49 for theoretical a~curacy) and the Chazy formula, determine the reilltion betweeu CMzy's C and Manning's n for the condition de:;ocribed in: Pl·ob. 5-2. This \\'iii show that the G. K. formula and the Manning formula are theoretically identical at the condition when ,Cl,ezy's 'C is independent of the slope S. , 6'-8. Prov~ that the friction factor f in the Darcy-Weisbach formuir.,Eq. (1-4), is related to .Manning's n by f = 116n~/R~~. 1i-9. Run 12-4 of Bll.zin's teats [12J was made on a rectangular plank flume ,fi.±4 ft wide with wooden strips 1 em thick and 2.7 cm wide nailed crosswise on the bottom and ~ides at a spacinll; of 3.7 em center to center strips. This fiume gave, a mean velocity 0[3.33 fps at a flow depth of 1.02 ft and a slope of 0.0015. The t.emperature rending WIl.S 8.5·C. Determine Manning's n, and compute (a) CM.2y'S C, (b) Kutter's n, (c) ,Bazin's m; and, (d) Powell's •. " 6-10. Run 15-·1 of Bazin's tests was the san1e as run 12-4, described in the preceding problem, ex~ept that the spacing of stripa was increaaed to 7.7 ern. 'Using the same disc,harge as that of run '12-4., the depth of flow wn.s found to be 1.33 :it. Determine Mnnning's IL, and compute (a) Chez)"s C, (b) Kutter's 7'" {~} Bllzin's m, and (d) Powell's ..Oomptu'e the values of • obtained from runs 1274 and 15-4 with the height of the strips, and explain I,he effect of roughness in both cases. '6-11. U!iing the Manning formula., construct 0. discharge-rating curvel for the natural chlll\uel section given in Prob. 2-5. The slope ii! 0.0016, and, 11; ;= 0.035, Extend the sides of the channel by ~traight lines at high stages if necessary. '6-12. Tile actual rating curve of the channel section in Prob. 2-5 is described below. Construct a CUTve showing the' variation in Manning's r. with respect to the stllg'l above the datum. '

Discharge, cis

0.3 0.4 0.5

1.'0 2.3 4,.6 7.8 1l.0 15.0 20.0 25.0

'0.6

0.7 0.8 0.9 1.'0 1.25

Stage,

It

Discharge, cfs

1.50 1. 75 2.00 2.25 2.50 3.00

50.0 62.0 75.0 88.0 10'2.0

132.0 164.0 199.0

3.50 4.00

38.0

6-1S: By the Cowan method, estimate the n value for channel 21 of FIg. 5-5. '

!l.

slightly curved rellch ill

REFERENCES Schnack~nberg: Slope discharge formuiae for alluvial streams and rivers, Proceedings, New Zea.land·lnsWt!lion of Enginesrs, vol. 37, pp. 340-409, Wellington, 1951. Discussions, pp. 4.10-4.4.9. IVllJl E. Houk: Calculation of flow in {)pen channels, Miami Conservancy District, ' Technical Report, ·Pl. IV, Dayton. Ohio, 191B. Erik l,indquist: On velocity formulas for open channels and pipes, Transactions of the World Power C01~feTimce, Sedional Meeting, Scand'inavi(l, Stockholm; vol. I, pp. 177-234, 1933. ' Philipp FOfchheimer, "Hydraulik" (" Hydraulics JI), Teubner Verlagsgesellscha.ft, Leipzig e.nd Berlin, pp. 139-163, 1930. Zivko Vladislavlievitch: Apen;u critique sur le~ formules pour la predetermination de Ie. vitesse moyenne de I'ecoulement uniforme (Critical surveyor the formula() for predetermination of mea.n velocity of uniform flow), Transacli(]11)1 of th~ lst Congress, InternationaL Com'mislrion on Irrigalion and Drainage, New Delhi, voL 2, rept. 12, question 2, pp. 405-4.28, 1951. Oornelis. Toehes: Stres.mfiow: Poly-dimensional treatment of vari!).ble factors affecting the velocity in alluvia.l streams and rivers, Proceedings, Inslittttion of Civil Engineers, Lut,do'/1., vol. 4., no. 3, pt. Ill, pp. 900-93S, December, 1955. , G. H. L. He.gen: "Untersuchungen fiber die gleichformige Bewegung des Wassers" ("Research'!S on. Uniform Flow of Water"), Berlin, l~7f.L , ' Clemena Herschel: On the origin of the Chezy formula, J(l,.rnal, Association of Engineering Sooel:ie3, vol. 18, pp. 363-368. Discussion, pp. 368-369, JalluaryJune, '1897. A. Brahms: "Anfangsgriinde dcr Deioh- und Wasserbaukunst" ("Elements of Dam and Hyd!,aulic Engineering"), Aurich, Germa.ny, 1754 and 1757, vol. I,

L E. C. 2. 3. 4. 5.

or

1 It should be noted that the synthetio rating curve thus obtained is very approxima.te, parti~ularly for a natural channel, beoause the n vDJue is a.ctually not !l. constant but a function of th.e depth (see Art. 5-8). '

Stage, ft

125

6.

7. 8.

9.

p.l05.

,

'

10. E. Ge.nguillet llJld W. R. Kutter: Versuch zur Aufstellung einer neuen allegemeinell. Formel fUr die gleichfllrmige Bewegung des Wassers in CaniLlen und Fliissen (An investigation to establish a new general formula for uniform Bow of water in canals a.nd rivers), Zeitsclirift cIes Oeslerreichischel~ Ingenieu.r- u.nd Architekten Vern1les, vol. 21, no. 1, pp. 6-25; no. 2-3, pp. 46-59, Vienna, 1869. Published as e. book in. Bern, Switzerland, 1877; translated into English by Rudolph Hering a.nd John C.

126

UNIFORM FLOW

Trautwine; Jr., 1.\5 "A general Formula for the Uniform Flow of Water in Rivers and Other Glumnais," John 'Wiley &; Sons, Inc., Nevv York, 1st ed., 1888; 2d ed., 18!H and i\)Ol. 11. C:.\pt:.\in A. A. Humphrey.s and Lieut. H. L. Abbot, U.S. Army Corps of Topographical Engineers: "Report upon thB physics and hydraulics of the Mississippi River; U!Jon the protection of the alluvi!>l region a,gainst overflow; and upon the deepening. of the mOllths; based upon surveys and investigations . . . ," J. E·, Lippincott Company, Philadelphia, 1861; reprinted in Washington, D.C., in 18157, and as U.B. Army Corps (Jf Ellllineel's, PtofC8sionai Pap.,. No. 13, 1875. 12. H. DI.\!'cy a.nd H. Ballin: "Recherches hydra.uliques,'· lre partie, Recherches experimenta.les sur l'ecoulement de Feau dans ies canaux decouverts; 2e partie, Recherches experiment&les rel().tives aux ramous et il. In. propaga.tion des ,andes ("Hydralllic Resea.rches," pt. 1, 'Experimental research on flmv of water in open channels; p~. 2, Experimenta.l research on 'bMkwater and the propagation of 'wave:;), ACMiemie des Sciences, Paris, 1865. 13. H. Bnzin: li:tude d'one nouvelle fonnuls pour ca.lculer Ie debit des canaux deoollverts (A llllW formula.' for the calcula.tion of discharge in open channels), Mtmoire No. 41, Allnal6~ deB ponti I~t cha!,~~ees, vol. 14, seT. 7, 4me trimestrs, pp. 20-70, 1897. . 14. Ralph W. Powell: Resi~tance to flow in l'ough cha.nnels, 'l',u,naadion'5, American Geophysical Union, vol. 31, [lO, 4, pp. 575-582, August, 1950. 1.5. Hobert Manning: On the flow of water in open channels and pipes, Transactions, ·I1/.~titution of Civil Ellgine(Jl'8 'of Ireland, vol. 20, pp. 161-207, Dublin, lSIH; supplement, voL 24, pp. 179-207, 1895, 15. Ven Te Chow: A note on the Manning formula, Transadidns, American Geophysical U'~ion, vol. 36, no. 4, p. 688, August, 1955. 17. Allen J. C. OUlmingham: Recent bydrl'luIic experiments, Proceedings, Institution . of Civil Engineel's, London, voL 7l, pp. 1-36, 1883. . 18. Ph. Gauckler: Du 1l10UVeme[lt d~ l'eau da.::ls les ()onduites (The flow of water conduits), Annates de~ pOJ~ts .1 chriu.ssees, vol. 15, ser. 4, pro 229-281, 1868. 10. A. St.ridder: J3eitriige zur Frage der Geschwindigkeitsformr.i und der Rauhi,gkeitsia.hlen fur Strome, !{a,nii.le und ge.schbssene Leitungen (Some contributions to the problem of velocity fonnul" and roughness c(lI;lfficient for rivers, canals, and dosed conduits). Jfillcil!l.nge'l des eidgen6:isischen Amteofiir }Vasserwirtachajt, Bern, Swit~erland, no. 16,1923. 20. Thomas Blench: A new theory of turbulent fto,v in liquids of small viscosity, Journal, b'stittlt'ion of einil Engineer., Lolkion, vol. 11, no. 6, pp. 611-612, April, • I~~ . 21. N. N. Pavlov'skiI: "GidravlkheskiI Spravochnik" ("Handbook of Hydraulics"). This book ha.~ many editions: (1) "Giclrl\vlicheskiI Spravochnilc," Put, Letlingrad, 1924, 192 pp.; (2) "Uchebnyl GidravlicheskiI Spravoclwik'" (for schools),.!~ubuch, 'Leningrad, lng, 100 pp.; 2d ed, W3l, 168 pp.; (3) "Gidrnvlicheskii Spravochnik," Onti, Leningrad and Moacow, H}37, 890 pp; !lnd (4) "Kratki! GidravlicheskiI . Sprflvochnik," (concise version), Gosstrolizd'at, Leningrad and Moscow, 1940, . 314 pp. , 22. George W; Pickets: Ru·n-off investigat.ions in central IllinQis, University of Illinois, Engineering Experim.ent Sta.iion, Bulletin 232, ,vol. 29, no. 3, September, 1931. 23. Frederick C. Scobey: The flow of water in flumes, U.S. Departm.ent of ;Agriculture, Technical Bulletin No. 393, December, 1933. 24. Methodology for cmp and pasture inundation damage appraisal: "Training manual for hydrologists on watershed protection and flood preventiQn work pIau

DEVELOPMENT OF UNIFOIUI{ FLOW A~m ITS FORM (JLAS

127

parties," prelimi~ary ~raft, U.S. Soil Conservation Sel'vice, Milwaukee, Wis., 1954. 25. E. W. Lane: DISCUSSIon on Slope discharge formulae for alluvial streams a.nd by E. C. Schnackenberg, Pr(JcllJldin(J~, New Zealand 1~lItitu.l.ion of Enqillee!'s vol. 37, pp. 4.35-438, Wellington, 1951. '. ) 25. J•. S. ~ey~!,~ :'I,nd E. A. Schultz: P~na;Ha Canal: The sea-level pI'oject, in A symposIUm. Tlda.l currents, Tran,~act!OnB: Ametican Society of Civil Engineers voL 114, pp. 668-571, ~g49. ' 27. Thomas R. Camp: Design of sewers to facilitate flow, Sewage 'Work.:; Joumal, voL 18, pp. I-HI, .January-December, 194.6. 28. 1i"lomas It
1

..~

. r

COMP!UTATION OF UNIFORM FLOW

(6-6), this factof may be expressed as

CHAPTER·

6

6-1. The Conveyance of a Channel Section. The discharge of uniform flow in a channel may be expressed ns the product of the veloc:ity, represented by EC[. (5-1), and t.he water area, or Q = VA = CAR-S· = KS"

(6-1) (6-2)

where The term J{ is known as the comeY(1,nce of the channel section; it is a measure of the carrying capacity of the channel seDtion, since it is directiy proportional,to Q . . . . . I· When either the Ch~zy formula or the Mmmmg formult\ IS used D.s t"le. uniform-flolv formula, i.e., when y = 72, the discharge by Eq. (6-1) becomes

Q= K',IS

(6-8)

and, from Eq. (6-4),

COMPUTATION PF.UNIFORM FLOW

(6-3)

Primarily, Eq. (1)-8) applies to a channel s~ction when the flow is uniform. The right side of the equation contains the values of n, Q, and S; but the left side depends only on the geometry of the water area. Therefor'}, it shows that, fora giv~n condition of n,Q, and S, there is only one possible depth for maintaining a uniform flow, provided that the value of AR~i always increases with increase in depth, which is true in most. cases. .This Q~h is the normal depth. Wyen nand S are known at . a .()1~~!l!!!)1 section, it can 'be seeii1l1ilXl.f1Ji}]6-8) that there can beomyr!ne discharge for maintainin a unjform flow through th~ sectlOD.~provlaect ..-:Ui8:tARW'a ways increases withincrease of depth.l ThTs discharge 18 . 'tfi'e-normal d~8~'i.a.r.gr----·· .,._ ....,-.--.. :-_._-.... ,._-,-.' ...------.--:.-

''Equation'T6-8) is avery useful tool for the computation and analysis of uniform flow. When the discharge, slope, and roughness are known, this equation gives the section factor A"R"H and hence the norml\.l depth Yn' . On the other hand, when n, S, and the· depth, hence the section factor, are given, the normal discharge Q" can be computed from this equation in the following form: .

and the conveyance is l(

=

Q.yS

129

~

Q = 1.49 AR%

(6-4)

. n

v'S

(6-9)

.

'

where C' is Chezy's resistance facto::. Similarly, when the Manning formula is used, (6-6) K = 1.49 AR~ n

Thisis essentially thc product of the water area and the velocity defined by the Manning formula. The).'UP-.?cript ~_g,--.§ometill!!lL1L~.d t
The above two equations are used to compute the Conveyance when the geometry of the water area and t1')e resistanc~ factor or rou~hlless coefficient are :given. Since the Manning formula IS used extensively, mos~ of the folldwing discussions and computations will be based on:Eq. (6-6). 6-2. Th~ Section Factor for Uniform-flow Computati<:n. '11h: ex~re:­ sion AR~i is called the section factor for uniform-flow. eomputatwn; It IS an important element in the computation of uniform flow. From Eq.

This is true for channels in which the v'alue of AR3, al'ways increasllII. with increase of depth. since Eq. (6-8) will give one value of AR~,. which in turn gives only one depth. In the CMe of a closed conduit having a gradua.ily closing top, the va.lue of Ami will first increase wit~ depth and then decrease with depth when the full depth is approached, becaUse a n;taximum value of AR3i usually occurs in such a conduit at a depth slightly less than :the full depth. Consequentlj, it is possible to have two depths for the 'lame value pf ARH, one greater and the either le5J3 than the depth for the maximum value of Ami. For further discussion onthie subject see Art. 6-4.

This equation can Qe used to compute the convey!mce when the discharge and slope of the channel are given. . When the Chezy formula is used, Eq. (6-2) becomes K

=

CAR~~

(5-5)

I

.

1~8

1

.

COMPUTATION OF UNIFORM FLOW

131

6-3. The Hydraulic Exponent for UniformAiow Computation. Since the conveyance !{ is function of the depth of flow y, it may be assumed that

a

(6-10J

where C is a coefficient and lV is a pn.rameter called the hydra.ulic l3:cponent for uniform~jlowcompulalton. Ta.king logari~hlTIs on both side.$ of Eq. (6-10) and then differentiating with respect to V, d(ln l()

N

dy

2y

(6-11)

Now, taking logarithms oil both sides of Eq. (6-6), K lA9AR~~/n, alld then differentiating this equation ,... ith respect toy under the assumption that n is independent of y, .

..; ...

..s... . Cl,

~

d(1nI{) dy

= .!. dA: A dy

+~!

dR 3 R dy

(6-12)

0

d '0 c; Q

~

co

.0

Since dAjd'b' = T and R

....'"

~

~(111 K) =

hO

~=

dy

.S

?- s

-

N .:~ (5T - 2R ~:)

...

,..; I

<0

ci

I!i;

8

This equation indicntes that the value of N for the.tmpezoidal section is a function of z and y/b. For values of z = 0, 0.5, 1.0, 1.5, 2.0,.2.5, 3.0, and 4.0, a family of curves for N versus y/b can be constructed (Fig. 0,,2). 2 These curves indic;lte that the value of N varies within a range of 2.0 to 5.3. The curve for a circula.r section with N plotted against y/d Q, where do is the diameter, is also shown in Fig. 6-2. This curve shows that the

o

j

lI

I

I This equation (IJ was "Iso developed independently by Chuga.ev [2) through the use of the Chez.y lormula. . S Similar curves to tho'le in Fig. 6"2 for trapezoidal channels were constructed by .Kirpich [3J and al~o prepared iudependently by Pa.vlovskil [4] alJd R~khml\noff (5).

.

N.

d

130

1 l>.

'A

- - -~-·--:---~~~----~-·-'------~~"--., ...-..-,-"..---.c~---.~=...~-=-,_....-: ....=_ .. _... '-" .-.~~-----. ~

-,.~-.,-.~.'

':"

(6-14)

This is the general equntion for the hydraulic exponent N. For a . trapezoidal channel section having a bottom width b (l.lld side slopes 1 on z, the expressions for A., T, P,. and R may be obf,aind frorn Table 2-1. Substituting them in Eq. (6-14) and simplifying, the resulting equatioll1 is 10 1 + 2z(y/b) 8 (y/b) N (6-15) '3 1 + z(y/b) --3' 1 + 2 ~1:=::+=z=~-(y-/-b)

!:l

<3

0p/" puo q/" }OsanlOll

(6-13)

Equating the right sides of Eqs. (6-11) and (0-13) and solving for N,

:>

0

2R

'0

l
0

0

(5T _ dyd!.)

.,.

"CJ

.,'"

d

J.-

3A

'"tt:
I

. A/ P, the above equation becomes

.""-~-.-"--"

132

UNIFORM FLOW

value of N. decreases rapidly as the depth of flow approaches the top' of the channel. Further mathematical analysis has reveo.led that the value of N will be equal t~ zem at Vida = 0.938 and will then become negative at greater depths. The significance of this fact will be discussed \ later. in this article and the next. 6.0

5.0 >.4--1-_-1---1_-'- -.'---'-4.0 I-+--/----J 3,0

and abscissa are interchanged. If a constant n value is assumed, Eq. (6-6) indicat~ that K IX he~ce, the~e curves for AR~~ should show the same characteristics as if the curves were plotted for Ie From (6-10), it can. be seen that the hydraulic exponent for the straightline range of the plot is equal to twice the slope of the plotted straight line. Thus, if any' two points with coordinates (K1,Yl) and (K 2;Y2) are 150r,--,---",-,-,. , - - . , - - . , - . , - " taken' from the straight line, the ap100 proximata value N may be computed by the following equation: 50

or

I~+-'-----J

2.0

N' -1

§

0.3

::: "

0.2

~

'J

~

log (!(dK~) log (Yl/V2)

i

III

0.<1

<;

.-

(6-16)

+ I

1,0 0.8

.0.6 0.5

133

COMPUTATION OF UNIFORM FLOW

I

{)'•• Kll

I

./ The cutved' plot when 'h. depth opprooches

1 Ille

I I

~I

/

l

crOllln

K

ot 0

closed conduit

0,;

0.06 0.06

0,q5 0.04

log Y

0.0 3

I .1,

0,02 2.0

Fw. 8-3. Graphica! determination of N by

logarithmic plotting. 2.5

4,0

4.5

5.0

5.5

VOlues of N

FIG. 6-2. Curves of N v.alues.

Fot channel sections other than the rectangular, tnpezoidal, nnd 'circular shapes, exact values of N filly be computed directly by Eq. (6-14), provided that the derivative dP/dy can be eVllluated. For most chan-. 'nels, except for channels with abrupt changes in cross-sectional form and for closed conduits with gradually closing top, a logarithmic plot of K as ordinate o,gainst tIle depth as abscissaJFig, 6-3) willappear approximately as a straight line. This can also be seen from tll!') dimensionless curves for ARH in Fig. 6-1, which are plotted similarly except that the ordinate

':I

Fro. 6-4. Typicltl channel sections having appreciable variation ill N value with respect ro depth. (After R. R.

C}~ugaev

[2].)

When the cross section of a channel changes abruptly with respect to depth, the hydraulic exponent will ohange accordingly .. Several typical sections are shown in 6-4. In such cases the logarithmic plot of N against y rna.y appear as a broken line or an evident curve, For the nearly straight portions of the broken line or curve) the hydraulic exponents may be D.Ssumed constant. When the depth of flow approaches the gradually closing crOWl1 of a closed conduit, the logarithmic plot will appear as a curve. The hydraulic. exponent in the range of the curved plot is equa,l to twice the slope of the tangen't; to the curve at the given depth (Fig. 6-3). For practiaalpurposes,

i34

I

! i

i,

J.

i

I I· (

UNIFORM FLOW

the curve may' be divided into 11 number of short segments', arid eachsegment may be considered a straight line having,a constant slope orhydraulie exponent. N ow take the circular section as an example, The dimensionl€&s log:l.rithniie plot of ARH against depth is shown in Fig. 6-1. Assuming a const,ant '('Llue of n, this curve will show the same characteristics as if the depth were plotted against K. As the depth' increases, the curve deviates gradually from a straight line and finally reaches a prononnced curvature at Yldo = 0.938, where the value of A RH/dij~'J is a maximum. Since the n value is assumed constant, t,his ratio yl d Q = 0.938' also corresponds to the maximum value of the conveyance K. The slope of the tangent to the curve at this depth, according to the gl'aph in which the ordinate and abscissa a,re interchanged, is hQrizontal, and thus the hydmulic exponent N is equal to zero. For depths with ratio greater than 'lild o = 0.938, the curve shows a decrease in t.he value 'Of A RH!do'~ and, hence, a decrease in the conveyance K if r:. is ai$Bumed constant. The slope of the tangent to the. curve and with it the hydraulic exponent will thus beconie negative. 6-4. Flow Characteristics in a Closed Conduit with Ope-.l~channel Flow. Taking the circular section as an example, the dhnenslol1less curves for ARHI AoRoH and R 3i l Roy" are shown by the full lines in Fig. 6-5. The subscript zero indicates the full-flow condition. 'If the n value is assumed constant or independent of the .depth variation, these" two curves will represent the variation of the ratios of the discharge and velocity to their corresponding full-flow values (Le., QI Qoand YITT 0)' Both the discharge and velocity curves show maximum values, which occur at about Q,938d o and 0.81d o respectively. Mathematically, the depth for the maximum discharge, or O.93ado) can be obtained oimply by equat.ing to zero the first derivative' of ARH with respect to y, since the discharge computed by the Manning formula is proportional to ARH for consta,nt nand S. Similarly, as the velocity by t,he Manning formula is proportional to R*, the depth for the maximum velocity, or 0.81d o, can be obtained by equating the first oerivative of R"" to zero. Flu;thermore, the dimensionless curve of QIQo shows t.hat, when the depth is greater than about 0.82do, it is possible to have t'l'IO different depths for the same discharge, one !J.bove and one below the value of O.938d o. Similarly, the cut've of V /1'0 shmvs that, when the depth is gn:&ter than the half-depth, it is possible to have two different depths for the same velocity, one above . and one below the value of O.81do. The above discussion is based on the assumption that the roughness . coefficient remains constant as the depth che.nges. Actually, the value of n for average clean sewer· pipes and drain tiles, both. clay and concrete,

135

COMPUTA.:!'ION 'OF UNIFORM FLOW

for example, hilS been shown to increase by as much as 28% from 1.00d n to O.25dc, where it appears to be a maximum (see Fig. 6-5 and the disas a factor affectiIlg n value). cussion in Art, 5-8 regarding t.he This effect callses the actual maximum discharge and velocity to occur . at depths of about O.97d o and O.94do, respectively. The corresponding. curves of Q/Qo and ViTr Q are shown by the dashed lines in 6-5. According to the assumption of constant n value, the velocity would be the same for a half-full pipe ru; for a full pipe; whereas, if the n value is . Subscri:>t "d' indicolas the lull tlow condition

r----:r:....-.-,---,--.-----r-- . . r--r~-.-__.~._-r--~ I

II 0.6

0.7

0,8

0.9

Votues of 0/0 0 , vivo • AR
FIG. 6-5. P'low characteristics of a circula.r section.

(Alter T. R. Camp, [27J 0/ Chap. 5.)

taken to vary with the depth, ru; shown, the veloc.ity at the half-depth is only 0.8 the full velocity. The discussion for the circular conduit applies also to any l:losed C011duit with a gradually closing top.. The exact depths for maximum discharge and velocity, however, will depend 011 the shape and roughness variation of the specific conduit section. Since the ma.x;imurn discharge and velocity of a closed conduit of gradually closing top do not occur at the full depth, this means that the conduit WillllOt. flow full at the maximum capacity as long as it maintaimr an open-challlHil flow 9n a uniform grade free from obstructions. For praetical purposes, however, it may sometimes be assumed that the maximum discharge of a cir('.ular cOllduit or similar ,closed conduit with gradually closing top does occur at the full I

I

136

UNIF,ORM FLOW

137

COMPU'l'ATION OF UNIFORM FLOW

depth, because the depth for maximum discharge is so close to the top . that there is always.a possibility of slight backwater to increase this depth closer to and eventtu:IUy equal to the full depth. ' . 6-0. Flow in a Channel Section with Composite Roughness. In simple chann81s, the roughness along the wetted perimeter may be distinctly different fl'OJU part to PE1l't of the perimeter,but the mean velocity can stiil be computed by a uniform-fl,uw formula without actually subdividing the section.. For example, a rectangular channel built with a wooden bottom and glass walls must have different n values for the bottom and the walls. In applying the Manning formula to ~uch channels, it is sometimes' necessary to compute an equivalent n value for the entire perimeter and use this equivalent value for the computation of the flow in the whole section. . . For the determination of the equivalent roughness, the water area is divided imnginatively into N parts of which the wetted pel'imeters P 2, ... , PN and the coefficients of roughness nl, n2, • . . ) nN are known. Horton [6]
where ~l' Hi, ... ,R/'f are hydraulic radii of the subdivided areas. simple c~!\.nnel sections, it may be assumed that

For

RI = R2 = . . . = RN = H . Raugnne88 of Ice-covered Channels. When a. channel is covered with the wetted perimeter of the flcrw is greatly increased. . The bottom surface of the ice cover may be either as smooth as a finished concrete sUl1fa:ce or as rough as the natur&'l channel bed v,rhenclrifting ice blooks .exist. Tabb 6-1 gives the 11, values for drcdg;ed channels covered with ice, as proposed by Lotter [13]. .

.lee condition

Velooity of flaw, fps

n Value

1.3-2.0 >2.0

0.01D-0.012 0.014-0.017

Smooth iee: Withol\t drifting iee blookB

With drifting ice blocks

1.3-2.0

0.OI6-0.01E

>2.0

.0.017-0.020 0.023-0.025

Rough ice with drifting ice blocks

Let nand n1 be the roughness coefficients for channds with and without ice cover, respectively. By means of Eqs. (6-17) to (6-19)' it is possible to compute the roughness coefficient 7l-1 of the ke cover. However, the coefficient thus computed may sometimes be a negative ,falue . , which is, of course, unrealistic, . In order t.o develop a realistic approach to the problem, Pavlovskil [14) assumed that the total force resisting the noW' is equal to the sum of the resisting forces due.to the channel bed and' the ice cover. Thus, from Art. 5-4, (5-20)

(6-17)

There are many other assumptions for the determination of all equivalent l'Oughness.Pavlovski! (9] and also Muhlhofer [10] and Einstein and Banks [111 assumed that the total force resisting the flow (that is JCV'"PL; see Art. 5-4) :is equal to the sum of the forces the flo~ developed in the subdivided 3reas.By this assumption, the equivalent roughness coefficient L'3

where the subscript 1 refers to the channel bed and 2 to the ice cover. Since Chezy's C = -vu;jR OJ' J( = W/C2, the above equation b~comes

(Plnl 2

n

,I J

+ P2n2~ + ... + PN'1i~!)H P}i

(6-18)

Lotter [12] assumed that the total discharge 'of the flow is equal to thei sHm of the discharges of the subdivided areas. Thus, the equivalent roughness coefficient .is

PRH'

n = ...-----'-.-

(lH9~

P

C1

i

II

! i

=

P1' P! C1 2 Czl

+

Let: the wetted perimeter P z = aPi or P = PI then 1 a 1 a = C12 C2 2

+

Sin(:e,: by Eq. (5-7), C

+ P2

(1

+ a)P1i (6-22)

t'/n,

1. 49R , (1 a),n2

+

(6-21)

RH!

(6-23)

138

UNIFORM FLOW

COMPUTATION OF UNIFORM FLOW

It is further assumed that the total hydraulic radius R is made up of . two parts: t.he hydraulic radius Rl due to the channel bed and the hydraulic l':1,dius R~ due to the ice cove'r; that is, that R '= Rl + R 2 • Now, let El = Rl/R2 a,nd E2 = nt/nz. Then Eq. (6-23) is reduced to

(1 + a)~2

=

11,2

(1 + ~)H

2

(E22

+ aEif,)

Tt2

VT+li

(a'' (T'

0Z''S)2~ 7

For wide channels, it may be a,ssumed that PI Thus, E22 = ~l~', and

:i

channel section is equal to the total discbirge divided by the total water "1 area. Owing to the differenr.es that exist among the velocities of the subsections, the velocity-distribution coefficients of the whole section are different from those of the subsections. The values of these coefficients may be computed as follows: Let Vt, V2, : • • , vI! be the mean velocities in the subsect,ions; let al .. a2, . . . 1 aN and fJI I fJ~, ... ,fJN be the velocity-distribution coefficients for the corresponding subsections; let .6.A 1 , .6. A 2, • • " .6.AN be

(6-24)

For the condition of maximum discharge, Pavlovskil. postulated that the relation !.LlUOJ;lg R I , R 2, and n is such that dnjd~i = O. Thus, from Eq. (6-24), Ez z =aEI)', a.nd n = -

139

(6-25)* =

P z, that is, that

a = 1.

Tt =

..72 (1 + ~2~l)%

Fw. 6-6. A channel con8isting of one main section and two' side sedions.

(6-27)

the water areas of the corresponding subsections; let K l , [(2, . • '. , KN be the con veyances of the corresponding subsections; let V be the mean velocity of the to~al section; and let A be the total water area. From the continuity equation Emd Eq. (6-3), the following can be written:

The roughness coeffiCient for the ice cover is, therefore, n~ =

(1.68n n -

nl~I)~l

(6-28)

Now let the discharges wit-hand without ice covel' be Q and Ql1 respectively. Then, using the Manning formula and assuming R = RI/2, where Rand R 1' are the hydraulic radii with and without ice cover, respectively, the discharge of an ice-covered channel is Q

nl

= 0.63Ql 11,

VI =

Q = VA

=

(6-29)

=

Eq. (5-7).

Sf~

_ 1(/'1 SH VN -

.6.AN

+ VN dAN

+ ](z + .. + ](N)S~~

0:: IC.r) N ,

,

Sfi

and Incorporating the above expressions with Eqs. (2-4) and (2-5) and simplifying, the velocity-distribution coefficients of the entire section are N

I a

R~/n instead of Eq. (5-7), obtaining

,~ I

f ~

i !

X.I,

. .t .

=

(aNKNJjdA N')

(J

(6-30)

0; [(NY/ A'

l and

1.

1

~-N~--~--~-

1

N

i

"

us~d

1(2

Ll.,A z

= VI .6..4. 1 + Vz dA 2 +

;

(6-26)

It was Belokon [lSj whQ

{Kl

V2 -

I

Channels of Compound Section. The cross section of a channel may be composed of several distinct subsect.ions with each subsection different in roughness from th~ others. 'For example, an alluvial channel subject to seasonal floods generally consists of a main cha.nnel and two side channels (Fig. 6-6). The side channels a.re. usually found to be rougher than the main channel i so the mean velocity in the main channel ia greater than the mean velocities in the side channels. In such a case, the Manriing formula may be applied separately toearih subsection in determining the mean velocity of the subsection. Theni the discharges in the subsections can ,be computed. The totaL discharge is, therefore, equal to the sum of th~se discharges. ' The mean velocity for the whole .. PavlovskiI [11] used the relation. C

[(~ Sf.

.6.A. I

(f3N[(N

2

jdA N )

= -J-'N'-~--~--

(Il(NY/ A I

"

(6-31)

:' ' I

140

UNI~ORM

I

FLOW

Example 6-1. ' Compute the velocity-distribution coefficients at a peak flow in a natural stream channel consisting of a main sectioll and an overflow side section. The data obtained at the peak flow stage are:

A, ft'

Subsection Main section.

. - . - - I · --,3-6-0-1\.

S 5,710

I

{J I ~5 11~0-.-O-3-5~JII-1-.-1-0-'--1-'0-4-

P, ft

1

Side section. .. .. .. .. Sol'u/'i01l:.

J

405

0.040

1. 11

1. 04

The computations are given below.

S"b~"tio"

AA

I-~

R

J. "

I /3K2/I!A

K

..

-I---!

I

",K'/I!.tl'

10' 16-::;4 10'1".93

10"

Main section. 5,360\22523.88.290.035 I. 892 X X X 1.244 X 10' 12.82 X 10' B,56 X 1O" I 3.136 X 10' i g. 76 X 10 8 I 32.49 X 10"

Si~::~~i~~.,:: ~:~~~I~~y~·15:~~ ~:~'~~

I

32.49 X 10'0, (3.136 X 10')'/11,070' = 1.29 9.76 X 10' {J = (3.136 X 10°)'/11,070 =1.10

and

6-6. Determination of the Normal Depth and Velocity. The normal depth and velocity may be computed by a uniform-flow formula. In the following COlnplltations, the M.anning formula is used with threedi.fferent methods of solution.· ./ A. Algebraic Method. For geometl'ically simple channel sections, the un'Hol'm-ftow conditIon may be determined by an algebraic solution, as ' i!lustrated by the following example: Example 6-2. A trapezoidal channel (Fig, 2-2), with b = 20 ft, Z = 2,S. = 0.0016, Ik"ld n = 0.025, carries a discharge of 400 cfs. ComIJ.!!te t~rmal 'depth and velocity. ... Sol,lt/ion 1: The Analytical Approach. The hydraulic raclius and water area of the given section are expressed in terms of the depth y as

, !

R

=

+ y)_ + y vS

y(lO

10

and

A = y(20

+ 2y)

The velocity is

I

, II

'100 ./ y(20 + 2y) ./

200 y(10 y)

+

."

Substituting the given quantities and the ILbove expressions in the Manning formula • Besides the methods described here, there are other methods for the computation of uniform flo~, suc.h as the use of hydraulic tables., Popular tables for this purpose can be found in [16] to [20].

=

1.49 0.025

[Y(lO 10

+ Y)-J~~ O.0016H

+ y V5

7,680 +:1,720y = [y(lO

+ !:::)J2.5

Solving this equation for Y by trial and errol', V. = 3.36 ft. This is the normal depth. The corresponding arelL is An = 89.S ft' alld the normal velocity is V. = 400/89.8 = 4.46 fps. From EXlLmple 4-2, it is known that the critical depth for the same discharge in the channel is 2.15' ft. ,Since the normal depth is greater than the critica! depth, , the flow is subcritical. Soh;/ion 2: The Trial~and-e,.~or Approach. Some engineel's prefer to solve this type of problem by trial and error. Using the given data, the right side of Eq. (6~S) is 1IQ/1.49 ...;s = 167.7. Then, assume a value of y and compute the section fnctor ''AR%: Make several such trials until. the computed value of AR3; is very' closely equal ......t.u-liiJ....7.; Lhen the a.ssumed ii-i~;~ th~-~l~sest trial :e. the.llormaCdepth:-·This tl'ial-and= error computation is shown as follows:

y

0'=

.

'.rT"

,',

[md simplifying,

I

By Eqs. (6-30) a:nd (6-31), the.coefficients are

/71

COMPUTA.TION OF UNIFORM FLOW V.:=: LJ"'L ~ J../l

or

n value

~

of

/r

J

-----1 3.00 3.50 3.30 I 3.35 , 3.36

I

I

A

78.0 94.5 87.7 89.5 89.8

-~II II

2.34 2.65 2.53 2.56 2.56

Ii I

I

RH

Am~

Remarks

1.762 1.915 1.852 J .870 1.870

137.4 181.0 162.6 167.2 158.0

Y too sll~all Y too large

i

I

I

The closest

The normal depth is,therefore" Yn = 3.36 ft.

B. Graphical 111ethod. For channels of complicated cross sel}tion and variable flow conditions, a graphical solution of the problem is found to be convenient.. By this procedure, a curve of y against the section factor A RYJ is first constructed and the value of nQ/IA9 VS is computed, According to Eq. (6-8), it is evident that the normal depth may be found from the V-A R~1 curve where the coordinate of A RH equals the computed value ~f nQ/l:49 VS. When the discharge changes, new values of nQ/1.49 VS are then computed and the eorresponding new normal depths can be found from the same curve. ' Example 6-3. Determine the normal depth of flow in a 36-in. culvert (Example 4-3) laid 011 a slope of 0.0016, having n = '0.,015, and carrying a discharge of 20 cfs. Sollllion. Construct a curve of y vs, Am, for the given culvert (Fig. 6-7). Compute nQ/1.49 V8= O.DlS X 20/1.49 YO.0016 '" 5.04. From the y-Am; curve, find the depth corresponding to the value of 5.04 for AR3!. This depth i~ the required norma.l depth, or y" = 2.16 ft. Since this depth is greater than tlie critical depth determined in Example 4-3 under the same condition, the flow is subcritical. The table in Appendix A for the geometric elements of a circular sect,ion may also be used for the solution of this problem. Since do = 3.0 ft and do" = 18.75, AR,;/ d.}; = 5.04/1S.75 = 0.269. From the table, lI/do = 0.72, or V = 0.72 X 3 = 2.Hl ft.

-_.,

142

UNIFORM FLOW

COMPUTATiON

The design. chart for determining the normal depth (Fig. 6-1) ~all be used with great expediency ... C. Method of Design Chal·t.

In Exalllple 6-2, ARn =< '167.7. The v",[ue of AR~fi/M~ is 0.0569. the uhnrt gives yllJ = 0.168, or 11" = 3.36 ft. In Example 6-3, AR15ldo~1 - 0.269. For this va.lue, the chart or 11 = 0.72 X 3 = 2.16 ft.

I

UNIFORl\I FLOW

143

b. Determine the critic.!tl slope a.nd the corresponding norma.l depth when the dis" charge is 400 cis. ' '

c.D~term!ne the critir.;a.1 slope at the n01'l:11al depth of 3.35 ft, a.nd compute the ~or­ respondmg dlscharge. So:1ttion. ' . (v.) h From the given data it is found that R = 2.56 it and V '= 446 ' f ps.. Su bstltutmg t ese va.lues; in the Manning formula and .~ol-dng for Sn,

For this value,

yla., =

OF

il.72,

A,

Ad -

1.49 256"'8 I" 0.025 . ~.,,'

s.

0.0016

•••• C'

6-7. Determination of the Normal and Critical When the discharge and roughness are given, the Manning formula can be used to determine the slope of Ll. 'prismatic channel in which the flow is uniform at a given normal depth Yn; The slope thus determined is sometimes culled specifically the normal slope Sn.

or

-

This is the slo~e that will maintain a unifO~'m flow i~ the gh'en channel at a depth oI 3.36 ft and a discharge of 400 cfa (see Example 5.2), b. From the given data t.he critl!:al depth is found to be 2.15 ft (see Example 4-2) The corresp:ndin g. v:J.lues of R ~nd V are R 1.97 f~ and V 7.60 Ips. Substitutin~ the known ~ A.lues m the MnnnlIlg formula and solvmg for Sa, '

1. 97HB.l."

7.66 or

S.

J I ·'l.

!

,I I

0.(1067

t

Th~s is the slope that will maintain a. unifornl and critical flow in the givell cha.nnel for

a dis~harge of 400 cfs. The depth of flow is 2.1& ft. c. For the given normal depth of 3 .•36 it, it is found that R = 2.68 ft, Ii = 89.8 fV D :" 2.~8 ft, and, by Eq. ~I-ll), the critical velocity V. 9.3'fps. Substi~ tutmg tne known values m the Manning formula and solving for 13 m , 1.19' 9.3 = 0.025 2,56~~S",,~ or

o

This is ~he slope that will maintain a uniform and critical flow in the given channel at the glvp,n normal clepth of 3.36 ft. The disoharge is equal to 9.3 X 89.8 = 835 ds.

FlO. 6-7. A curve of !I vs. ARJ~ for n. circula.r section.

Example 6-6. petermine the limit slope of !I. rectangular channel (Fig. 6-8) with 10 ft and n = Q.0l5. .S~ll'tion. .Since the limit slope is the :;mall~.st critical slope, its value' maybe determmed gra.phlc.\J.lly.rro~ a Cll::ve ~f.the critical slope plotted aga.inst discharge. ~or the determmatJOn of a cntlcal slope, the following two cOlldition~ should be satisfied: L The first condition, from Eq. (6-3), is

By varyillg the slope of the c:hannel to a certail1 value, it is to change the normal depth and make the uniform flow occur in a cl'it,ical state for the ghren dischatge and roughness. The slope thus obtained is the critical slope 8., and the corresponding normal depth is equI11 to the critical depth. The smallest critical slope for a channel of shape and roughness is called the l~:1nit slope SL. Furthermore, by adjusting the slope and the discharge, a critical uniform flow may be obtained at the given llorinal depth. ·slope thus obtained is known as the critical slope at the given normal depth So... ,The folloH.'ing examples will illustraie the above discussion. Example 6-4. A trapezoidal channel has a bO,ttoro w'idthof 20 H, side slopes of 2:1, and u = 0.025. a. Determine the normal slope at a normal depth of 3.S6 it whf:)n the discharge is 400 cfs.

S.n = 0.0070

b

=

.Q = K

vB.

(6-32)

or, 'when the Manning formula is used,

(6-33)

or, for the rectangular channel,

I

1

i I.

,t .

(f}-34)

i

'I

:2. The second condition, from Eq. (4-3), is (6-35)

Q =Z,...;g or, for the .ectangulul' chmnel,

Q=

1O....;g y'"

(6-36)

By using Eqs. (il-34) and (6-36) .iLnd eliminating 11, the relation between Q and S, can be established. This relation is expr;lssed, however, as an implicit function, and

a direct solution is mathematically complic!l.ted, A practical solution of the problem i~ to assume different values of ii, substitute 11 in Eq. (6-36) and solve for Q, and then substitute lIll.!ld Q in Eq. (6-34). anti solve for So. Following this procedure, the rel!l.tion between Q and S< was compute.d ..nd plotted as shown ilt Fig, 6-8. The plotted.

Supercrltlcal

flow

2.

Slope in 10-3 (al F!G~

145

COMPUTATION OF UNJl<'ORM FLOW

UN)FORM FLOW

144

:3 4 5 Slape In 10- 3

6

7

. (b)

t>-8. Curves of critica.! slope vs. discharge.

curve MLN indicates a minimum value·of S, O.004Il.t L, which IS the required Iimii slope. ' , . : ' . Assumil1g that the maximum expected depth of flow in the ch$.IlneLis 5 ft, a dlScha.rge cUl:veOM (Fig. can be cOIl1itrueted according to Eq. (6-9). It becomes evident tha.t within the sha.ded area. betweEn the curves OM and MLN, a.ll expected flows wHi b~ subcritical. On the right side of the Cllrves, the flows will be, supercritical. Since t1:,e point L is below the cUr'le 'OM, the limit slope is possible in the exj:lected flLnge of flow. . . . Similll.dy, the maximum expected depth of flow is !\.Ssumed to be 1.5 ft, and the curves ar~shown in Fig. 6-8b. In this case, the point L is above the Cl.lCVe·01l1; there-. fore, the limit slope cannot be expected to occur in the realm under cQnsideratioll.

6-8. Problems of Uniform-flolw COII}putation. The computation of uniform flow may be performed: by the use of two equations; the· continuity i equation and a unifodn-flow. formula. When :the Manning formulli. is used as the uniform-fl6w formula, the computati/;m will involve ." the following six variables:'

1. The normal discharge Q 2. The mean velocity of flow V 3. The normal depth 'Ii 4. The coefficient of roughness n , 5. The channel slope S " B. The geometric elements that depend on the shape of the channel section,such as A, N, etc.

When any four of the above six variab'lesare given, the remaining two unknowns can be determined by the two equations. The following are some types of problems of uniform-ftow computa~ioll: A. To compute the normal discharge. Inpractical applieD.tlons, this comput.ation is required.for the determination of the capacity of a given channel or for the construction of a synthetic rating curve of the channel. B. To: determine the velocity of. flow. This computation has many appli.cations. For example, .itis often required for the study of scouring and silting effects in a given channel. C. To compute the normal depth. This computation is required for the determination of the of flow in a given channel. D. To determine t.he channel roughness. This computation is used to ascerta.in the roughness coefficient in a given channel; the .coefficient thus determined may be used in other similar channels. E. Ta compute the channel slope. This computation is required for adjusting the slope 0(0, given channel. F. To determine the dimensions of the channel section. This computation is required mainly for design purposes . Table 6-2 lists the known and unknown variables involved in each of the six types of problems mentioned above. The kllown variables are indicated by a check mark (v') and the unknowns required in the problem by a question mark (?). The unknmm.val'iables that can be determined The last from .the known variables are indicated by a dash TABLE

Type of problem

t>-2.

Discha.rge Q

A

?

B

v v v' v

c

D E F

SOME TYPES OF PROBLEMS OF UNn'ORM-FLOW COMPUTATiON

VeIoeity V

,

?

-

.-

.-

I De th p 11

v v 1

v v v

Roughness

n

v v v ?

v v

Geometric elements

Slope S

v v v v ? ,

v;

I

v v v 'I. v ?

. Exa.mple

: Prob. 5-5, (Ex, 5-0 Prob. 5-5, (Ex. 5-1-) Example 6-2 Prob.5-6 Example 6-4a Example 7-2

145

UNIFORM FLOW

column of the tabie shows the example given in this book for each type of problem. The examples shown. in parentheses are solved by the. use of the Chl:z;y formula. It should be noted, however, that Table 0-2 does not include all types of problems. By varying combinations of various known and unknown variables, more types of problems can be formed. In design problems, the use of the best hydraulic· section and of empirica.l rules is generally introduced (Art. 7-7) and thUl'J new types of problems are created . . 6l.9. Computation of Flood Discharge. In uniform-flow computation it is understood, theoretically, that the energy slope Sf in the uniform-flow formula is equal to the slope of the longitudinal water-surface profile and also to the slope of the channel bottom (Art. 5-1). In natural streams, however, these three slopes are only approximateiy equal. Owing to inegular channel conditions, the energy line, wB,tel' surfa.ee, and channel bottom cannot be strictly parallel to one another. If ·1;he change in velocity within the channel reach is not appredable, the energy· slope may be taken roughly equal to the bottom 01' t.he surfs.ce slope. On the other hand, if the velocity varies appreciably from one end of the reach. to the other, the ·energy slope should be taken as the difference between the total heads nt the ends of the reach divided by the length of the reach. Since the total head includes the velocity head, which is unknown, a solution by successive approximation is necessary in the discharge computa~ion.· . During flood stages, the velocity var~eS greatly,. and the velocity head should be included in the total head for defining the energy slope. Furthermore, flood flow is in fact varied and unstean'y, and use of a uniformflow formula for discharge computation is acceptable only when the changes in flood stage and dischurge are relatively gradual. The direct use of a uniform-fl.ow formula for the determination of flood discharges is known as the slope-aJ'ea method. The flood discharge may also be determined by another well-kno".,n method called the contractedopening method, in which the principle of energy is applied directly to a contracted opening in the stream. Both methods l require information about the high water marks that are detectable in the flomded reach. Good locations for coll~cting such information may be found not only on main streams but also on smaller tributaries; but they ml.\st be either comparDtjvely regular valley channels free from bends and thus well suited to the slope-area method or else contrll-cted openings wiyh sufficient constriction to produce definite indrease in head and velocity and thus suited t.o the contracted-opening method. The foilowingis a description of the slope-area method. 2 The conI

~

For a comprehensive description of the metholis, see [21]. : It shoul.d be noted tha.t the slope-area method actually deals with gradually var.ied

COMPUTATION OF UNIFORM

:now

147

tracted-opening method is related to rapidly varied flow and, therefore, will be descri I:>ed later, in Art. 17-6. . The Slope·-area Method. The following inform:ttion is necessary for the slope-area methqd: the determination of the energy slope in the channel reach; the measurement of the a vel'age cross-sectional area, and the le.ngth of the reach; and the est.imation of the roughness coefficient applicable to the channel reach, so that frictional losses can be calcu!il.ted. When this information is obtained, the discharge can be computed by a uniformflow formula, such as l'vIa.nlling's. The procedure of computation is as follows: 1. From the known values of A, R, and n, compute the conveyances Ie, andKd,respeetively, of the Epstl'eam and dO\\'llstrealll sec~ion8 of the reach. 2. Compute the average conveyance K of the reach as the geometric mean of 1(u and F:.o,' or (6-37)

3. Assuming zero velocity head, the energy slope is equal to the fall F of water surface in the reach divided by the length L of the reach, or S

F'

=£

(6-38)

The corresponding 'discharge may, therefore, be computed by Eq. (6-3), or Q= K

vB.

(6-3)

which gives the first approximation of the discluirge. 4. Assuming ~he disclul.rge p.qltal to thefil'st approximation, compute the velocity heads at the upstream [1,nd downstream sections, or au V,,2j2g and ad V d Z j2g. The energy slope is, therefore, equal to

s=!!:!. L where

i

j I

I !

I i

J1,

(6-39) (6-40)

and kis a factor.' When the reach is (;ontracting (Vu < V d ), k = 1.0. • When the reach is expanding (V" > V.), k = 0.5. The 50% decreru;e in the value of Ie for an expanding reach is customarily assumed for the recovery of the velocity head due to the expansion of the flow. The corresponding discharge is then computed byEq, (6-3) u.'ling the revised flow, but it is believed that at this stage of reading the rcad~r should be able to follow the procedure.described here. This method shows how the uniform-flow formula can be applied to gradually vllded flow and thus paves the way for a more cOinprehensive treatment on the subject of gradually varied flow in Part Ill.

·l

i

,

,

.~

148

149

UNIFORM FLOW

COMPU'l'ATlON OF UNIFORlI'I FLOW

slope obtained by Eq. (6-39). This gives the secolid approximation of the discharge. 5. Repeat step 4 for ~he third and fourth approximatiOJls, and so on until the assumed and computed discharges agree. 6. Average t.he discharges computed for several reaches, weighting them equally or as circumstances indicaie.

surface flo~ occurs mostly as a result of natural runoff, and is called oo'erland flow. Uniform flow may be turbulent or laminar, depending upon such factors 11..'> discharge, slope, viscosity, and degree of surface roughness. If velocities and depths of flow are relatively small, the viscosity becomes a domina.tiQg factor and the flow is laminar. In this case the Newton's lD.w of viscosity applies. This law expresses the ~~at~.9l1_p~tween the

Example 6-6. Compute the flood discharge through a riv.er reich of 500 it ha.ving· known values of the ws.t"r areas, conveyances, and energy e,oelficient8 of the upstream and downstream end sections. The fall of water surface in the reach was found to be 0.50 ft. Solution. ';rI.e. wl).ter areas, conveyances, and energy coefficients for the two elld sections of the' reach are: ," Au Ad

11,070 10,990

= =

K, = 3.034 X 10 6 K. = 3.103 X 10 6

.J,~,~Ofti~T!~ ~V~~~,~" V?",,-~ ~ ..... 1,;:\.~j··

~"

'r-

.~ f' "'"(i"', r

w(Y"l~'Y)S ( _ • . t~F"'~~101l

=

~

.

au = 1:13" O'd

1

" Ii!

.(

....

.~~ ..,

\

;.:.;

) ' ..,~

,/

··'·1:':··~7

,., AifC

~~1:;;t·

The a.verage K = Y3.034 X 10' X 3.103 X 10' = 3.070 X 10'. FOI' the first approximl,tion, assume h, = 0.50 ft. Then S = 0.50/E,00 = 0.0010, v'S = 0.0310, and Q = K VB = 3.070 X 10' X 0,0316 = 97,000 ds. For the second approximation, assume Q = 97,000 cfs. Then the velocity heads at the t\~O end sections are: .

V,,·._ 1'13 A (97,000/11,070)" =

«. 2(/ ad

.""

2(/

V d ' _ 1177 (97,000/10,990): 2g - . 29'

=

1.354 FHl. fr9. Uniform laminar open-channel flow.

1,42,1

-0.070 .

dynamic viscm:ity i" and the shear stress bOlmdary surface (Fig. 6-9)" as follows:

Since l'u is less than V d, the flo," is contracting, and k = 1.0. Hence, h, = 0.500 0.070 = 0.430, S = 0.430/500 = 0.00086, "IS = 0.0~93, and Q = 3.070 X 10' X 0.0293 = 90,000 ds. Similarly, other approximations are made, as shown in Table 6-3. The estimated discharge is found to bl! 91,000 cfs. TABLE

6-3.

COMPUTATION OF F1.OOD DISCHARGE BY THE SLOPE-AREA METHOD FOP. EXAMPLE

Appr.oximatlon

IAs~umed

1st 2d 3d

4th 5th

Flo. ~

Q

97,000 90,000 91,200 91,000

I

I

2g

a~ ~l 2U

6-6

h,

I

S

v'S 1Computed Q

-~----I-!-----

0.500 . . . . . . , .. \°.50° °.0010000.0316 97,000 0.500 1.354 1.424 0.430 1 0.000860 0.0293 90,000 10.500 1.165 1.~25I0.44010.0d0880 0.0297 .91,200 10.500 1.H.l5 1.258 0,437i 0.000874 0.0296\ 91,000 1 0 . 500 1.190 11.253 1°.437\ 0.000874) 0.0296 91,000

6-10. Uniform Surface Flow.

When water flows across a broad sur-

face, so-called S1J.1"!ace flow is produced. The depth of the flow may be so thin in comparison with the width of flow that the flow becomes a wide-' open-channel flow, known specifically as sheet flow. In a drainage basin

=

r

,.,j

I>

r

at a dist.ance y from the

du dy

(6-41)

For uniform laminar flow, the component of the gravitational force parallel to the flow in any laminar layer is balanced by the frictional force. In other words, thc shear stress T per unit arel;\ of the flow along 'the laminar layer PP (Fig. 6-9) is equal to the effective component of the gravitational force, that is, r = tV(Ym - y),c;. Since the unit weight tV = pg and 1>/ P = v (Art. 1-3), T = gl>(Ym - y)S/v. Thus, from Eq. (6-41), .

dv

=

g8 (Ym - y) dy J'

Integrating and noting that v = 0 w;hen Y = 0, v

=

g8 (YYm _ )I

•

.

t) 2

fr.

J,;.iii:.

i n7 .... , ~, .

'\\

' . . •• . ,

_,(.

.'

1.177

~,'

.

.' 'flop'li' 't~o

(8-42)

This is a quadratic equation indicating that' the velocity of uniform, ·laminar flow in a wide open channel has a parabolic distribution. Inte-

COMPUTATION OF UNIFORM FLOW

UNIFORM FLOW

150

grate Eq. (6-42) from Y average velocity is

=

O. to Y = Ym and divide the result by Yrn; the

V = -1 Y••

10 °' vdy 11

PROBLEMS

(6-43)

'y

a. A rectangular section 20 ft wide b. A triangular section with a. bottom angle equal to GO" c. A trapezoidal sectiC'u with a bottom width of 20 It and side slopes of 1 on 2 d. A cil'cular section 15 ft in diameter . e. A parabolic section having a width of 16 ft at the depth of 4 ft .

(6-44)

where CL = gSj3v, a coefficient involving slope and viscosity. Uniform surface flow becomes turbulent if the surface is rough and if the depth of ftmv is sufficiently large to produce persisting' eddies. In this case the sui'face roughness is a dominating factor, and the velocity can readily be expressed by the Manning formula. Thus, the discharge per unit width is (6-45) where y", is the average depth of flow and where CT = 1.49So·~/n, a coefficient involving slope and roughness, The change of state of sheet flow from laminar to turbulent hn.s been studied by ffi8.ny hydraulicians. The transitional region was found variously at R = 310 by JeffreY
'where C is a coefficient a~d where the exponent highly turbulent. flow and 3.0 for mixed flow.

,,-'~

6-1. Detern:ine the normal discharges in channels ha.ving the following sections for .

= 6 ft, n = 0.015, and S = 0.0020:

0

and the di:;;charge per unit width is

:r ." ..... -,

151

:j:

varies between 1.0 for .

6-2. Prove the following equ1Ltion for the dischuge ina. tl'iangular highway gutter (Fig. 6-10) having one side vertical, one 3ide sloped at 1 pn z'. MallTlilig's n, depth of

I

I I \

FIG. 6-10. A highway gut:ter section. flow y, and longitudinal slope S: (G-48)

where . 6~3. Compute the discharge in the triangular highway gutte!' described in the precedmg problem when z = 24, n = 0.017, Y = 0.22.ft, and S = 0.03. 6-~. U~ing the Manning formula, determine the hydmulic exponent N for the [01lowing channel sections: Ca) a very narl'OW rectan(!;ls, Cb) a very wide rectangle, (c) a Y~ry wide panbola for which the wetted perimeter i~ practically equal to the top Width, and (d) an equilateral triangle with a vertex at the bottom. . 6-5. UBing the Ch6zy formula,' ~how that the general equation for the hydmulic exponent N is .

N

=

1:..

A

(3 T _ U dP) dy

(/H9) .

6-6. Solve Prob. G-4 if the determination of the hydl'!l.)llic exponent is based on the Chezy formula. Compare the results with those obtained in Frob. 614. I The G. K. formula shows that Chezy's C is a f~nction of the hydrauiic radius and hence of the depth ii. Thus, the Chezy formula has hot been found very convenient for determination of the N value. Forcanal~ in earth and gravelly soil the N value is generally found to havBan increase of 0.30 to 0.50 due to the v!l.ri!l.ti~n in Chezy's C with respect to the depth. Thi~ i~crease, however, brings the N valuc closlJr to that bMcd on the Manl1ing formula.

1, (

-\

;;1

'152 .

C01-IPUTATION OF UNIFORM FLOW

UNIFORM FLOW

r

6-7. Compute the l)ydraulic exponent N of the trapezoidal channel section (Fig. 2-2) having a normaldep.h of 6 ft, using (al Eq.(6-15), (b) Fig. 6-2, and (el the graphical method based on Eq. (6-16). 6-11. Compute the hydraulic exponent N of a 36-in. circular conduit having a n~r­ mal depth of 24 in. above the·invert, usi,ng (al Fig. B-2 and (b) the graphical method based on. Eq. (6-16). ' 6-9. Using the Manning formula, show that the depths for a maXimum discharge and velocity in a aircular conduit are, respectively, 0.938d. and O.Sld..· . 6-10_ On the basis of the Chezy formula, determine the respective depths for maximum discharge and maximum velocity in a circular conduit. . 6-1~_ A~ what depths will the maximl.lin discharge and velocity occur in a square cOlldUlt laId flat on one side? . ' . 6-12. Prepare the curves of dis~harge: andveloeity variations with respect to the depth in ll. square conduit laid on one side. 6-1S. A channel is assumed to have a constant hydraulic radius R for any dep'th of flow. .Prove that the CrOss section of this channel can be defined by y = R[ln (x

+

vx' - R')

II

i

Q.. The cross sect·ion of the stream at 11 typical upstrel1m station in the uniform Tea<:h is given hy the following cQordinates:

I

StrLUr:f(1.

l

Elev. m.s.!.

Sta/·ian

Elev. m.s.l.

590.0 580.7 578.2 582.0 581.0 580.0

6. + 00 8 + 00 10 + 00 11 + 00 12 + 00 14 + 00

543.7 540.0 572.2 573.2 568.5 590.0

Left b8lnk: 0 1 1 3

+ 00 + 00 + 50 +00 4 +00 5 + 00

b. The value of n for the main channel is estimated as 0.035, for the side channels 0.050.. e. The nl1~ural slope of the stream is about 1 it/mile. Construct a synthetilJ re..ting curve. It is suggested that th", wl1ter areas of the '. main channel and the side channels be seplJ.l·ated by the extended sides of the main 1l.S

ch~.nnBl.

- In R]

(6-50)

= R when y = o. Draw the sketch of this section and discuss its properties. (~IN;~: Frrm~ the giv~n condition, R = A(P =dA/dP "'" "dY/Vd:!"' + dy'. Solve

where x

thIS mfferentlill equatlOn, and evaluate the mtegration constant by the condition that x = R when 11 = O. MathematicallY, the section is formed by two catenaries as sides. For practical purposes, an. artificial bottom should he provided ~iilce the .theo~eti.cal section is bottomless. A.uniform-flow fOl'mulfl, such as the Manning formula, lIlfiIcates that the hydrfLlrlic radius is the sole shape parameter for the velocity. The adequacy of this indication can bo verified experimentally hy testin'l' a channel built of tha section oi constant hydraulic radius. If the indication is tru:, then, once this cl~annel is designed for a safe velocity; it should I;>e nonscouring and nonsiit,ing bver a WIde mnge of stages. In .earthen canals, however, the large variation in water surface during the change of stage wouldetode the sides very easily.) . . 6-14. Verify Eqs. (6-17)to (8-19). 6~16. A rectangular testing channel is 2 ft, wid'e and laid on a slope of 0.1035%. When the channel bed and walls were made smooth· by neat cement, the measured normal depth of flow was 1.36 ft fOT a discharge of 8.9 cfs. The same' channel was then roughened by cemented sand grains, and thus the measured norrne,! depth became 1.31 ft for a discharge of 5.2 cfs. a. Determine tile discharge for a normal d~pth of 1.31 it if the bed were roughened and the walls were kept smooth. , . ' . b. Detel'mine the discharge for a normal depth of 1.31 it if the walIs were rOl.]ghened and the bed were smooth. _ .' c. The discll!ir~es for the conditions described-in a and b were actually measured and found to be' G.50 'and 6.20 cis, respectively'. Determine the corresponding n; values, and compare these values with those computed by Eqs. (6-17) to (6-19). ' . 6-16. A chann~l consists of a main section and two side section.s (Fig. 6-6) .. Compute the total disbhfLrge, assuming that the main section and the two side sections are separated Ca) by vertical division lines and (bl by extended sides of the main channel Given: n· = 0.025 for the main channel. n = O.OBOfor the side channels, andS i= 0.001: 6-17. The hydrographic survey of a stream jndicates tha.t the hydraulic p¥operties of the stream llre relatively unifOl'm for a length of over 2' miles. The data 0btained by the survey" are: '

6-18. Compute the discharge in I1n overfliJ~ved highway gutter (Fig. 5-10) having Il. depth of flow of 3 in. and a longitudinRl slope of 0.03. The gutter is made of conprete \vith n = 0.017 and has a t~iangular section with a vertical curb side, a sloped side of z = 12, and a top width of T = 2 ft. The overflowed soil-aggregate pD.vement has a crosg slope of z. = 24 and n, = 0.020. 6-19~ For all equal amount of discharge, an ice-covered .channel should have greater depth of fl(HV than an uncovered channel, lor two reasons: (1) thc wetted perimeter is greater in an ice-covered channel and thus results in greater resistance or less velocity. and (2) the thickness of the ice cover is greater than a depth of wa~er oi equal weight, since the spGCific gra\'ity Qf i~e is about 0.917. Sbow that the increase . in depth due to resistanee in' an iue-covered wide open channel may be expressed by . I:>y

=

[1.31! (nn.)% - 1 ] Y

(6-51 )

where n, is the roughness coefficient of the channel with' ice cover. n is the roughness coeflicient of the channel without ice cover, and y is the depth of flow in the channel carrying the same discharge but without ice cover. . 6-20: Compute the conveyauce and velocity-distribution coefficients of a channel ser.tion 500 ft downstream f!"om the section described in Example !l-l. The survey data at the secti.on 'fol' the same flood are:

Subsection

A., ft'

P, ft

n

'"

(3

5,320

205 408,

0.035 0.040

1.12 1.10

1.05 1.0-1

----------Main section ........ Side section ....

"':1

5,670

6-21. Solve Example 6-2 by the G. K Form.ula, 6-22. A rectangular channel with 20 ft width, S = 0.006, and n = 0.015, carries a discharge of 200 cfs. Compute the normal depth and velocity. 6-23. Using the l\'Ianning formula, determine the norm!,1 depths in channels having the following sections when Q = 100 cfs, n = 0.015, and ,8 =' 0.0020:

UNIFOll.iVI FL01-V

COMPUTA'l'lON OF UNIFOR1I1 FLOW

155

a. A rectangular section 20 H wide D, A tril1ngulltr section with the bottom angle equal to 60~ c, A ~rapezoidal section "'ith a bottom width of 20 ft and side slopes of 1 on Z d, A circuliI' ~cction 15 H in diameter e. A parabolic section having a wicl'th of 16 ft at the depth of 4 ft

REFERENCES

6-24. Solve Example 6-2hy the gr'aphkal method. 6-26. A re.ct'o.ngular channel 20 it wide has a. l'oughnells coefficient n = 0.015. a. Determine the normal slope at a normal "depth of 1.23 i~ when the discharge is 200 cfs. . D. Determine the critical slope o.lIa the corresponding nonna.l depth ",hen the discharge is 200 cis. c. Determine' the critic!,l slope at tbe normal depth of 1.23 ft, and 'lr>mpute the corresponding disch'al'ge. 6-26. Show that the critic!!.l slope at n given norma.l depth !i4 may be expressed by (6-52)

a.nd that

thi~

slope for a wide cluumel is S

= 14.5n> crt.

(6-53)

lIH

·6-27. Deterrnhie the limit slope of the channel described in Exa.mple 6-4, 6-28. Construct the critical-slope curves of the cha.nnel described in Exampie 6-5 for bottom widths b == 1 ft, 4 It, 2() ft, and "'. 6-29. Determine the criti·cal-slope. curvl',s oi the channel desi::ibed in Example 6-4 for side slopes z €I, 0.2, 0.5, 1, 2, 5, and "'. 6-30. A cha.nnel reac)11,OOO ft loug h
Subsection Upstream: Main channel. , . ....... Side channel .... .. , ....... . Downstream: channel ... , . ...... Side channeL. ....... ... ~

~

..

4,250 25,620

2,050

5,760 25,610

l,gOo

320

11.

Of

~

0.038 0.038

1.10

1.0'1

1.20

L08

0.042 0.038

1.10 1.18

1 ..04 1.06

i

I,.

6-:11. Prove Eq. (6-46), 6-32. Using Eqs. (1-5) and (6-43), determine the value of ]( in Eq. (1-8}. 6-33. Compu~e the discharges ·per unit width of a ghect flow on a. surflLc€ with 11 0.01 and AS = 0.035 when the depth of flow is (a) 0.01 ft and (b) 0.004 ft. The temperature of water is 6soF. . .' 6-34. Compare Horton's criteria. for sheet flow in Prob. 6-33 with those shOW\l by . ~he cha'tt of Fig. 1-3. " .... . 6-35. Show that the velocity-distribution coeffici(mts for la.minar uniform flow in wide open channels are a "" 1,.54 and ~ 1.20. . 6-36. Using the Blasius equation (1-6) for turbulent flow in open cha.nnels, ;ahow that the cOI'nsponding exponent, in Eq. (6-47) is:c = 1 77.

i

I !I

I

I"

.

.'!

"> ,,/.-:'

1. Yen Te Chow: Integrating the eqltation of gradually varied flol\', paper 838, ProceediniJs, American ,"!ocietl! of Civii Engineers, voL 81, pp. 1-32, November, . 1955. 2. R. R. Chugaev: NekotorY(l voprosy neravnomemogo dvizheniia vody v otkrytykh prizmaticheskikh ruslakh (Abollt some questions concerning nonuniform flow of w!).ter in open channels), /zv6stiia. l'Besoiuznogo Nauchno-Issierlov(Ltel'skoIlO [nstitllta. aidrotekhniki (Tri.lnsa:clioM, AU-Union Scientific Re6eard~ institl,/e of Hydraulic Enyincerinp), L~ningl'!ld,·vol. 1, pp. 157-227, 1931. 3. Phillip Z. Kirpich: Dimeru;ionless cons~ants for hydr>l.ulic element·s of opellcha.nnel cro~s-sections, Civil Engineering, vol. 18, no, 10, p. 47, October, 1948. . 4. N. N. PavlovskiI: "Gidra.vlicheskii Spravochnik" ("Handbook of Hydmulics"), Onti, Leningra.d and Moscow, 1937, p. 515. 5. A. N. Rakhmanoff: 0 post.roenii krivykh svobodnolpoverlthnosti V. prir-maticheskikh i tsilindriche.~kikh ruslakh pri usta.novivshemsia dyil!henii (On the construction of curves 01 free surfaces in prismatic nnd cylindrical channels with established flow), hvestiia V sesoiuznoyo Naudmo-Is:sledouaf.eI'.kO'lo bl.lititula Gidrotekhllim (1'ranS(lctions, All-Union Scienli.fi.c ResC(Lrcfl InstituteDf Hydraulic Engineering), Leningrad, vol. 3, pp. 75-114, 1ll31. 6. Robert E. Horton; Separate roughness coefficients for channel bottom and sides, Jj7'dl""'.""di News-ltccord, vol. 111, no. 22, pp. 052-653, Nov. 30, 1933. 7, H. A. Einstein; Del' hydraulische odsr ProJil-Radius (The hydraulic (11' cross section radius), Sch",ei$~rische Ba.uzeilv,1!g, ZUrich, vo!' 103, no. 8, pp. 89-91, Feb. 24, 1934. 8. Ahmed M, Yassin: Mean roughness coefficient in open cho.nnels with different, roughness of bed and side walls, Eid(jelll!,~.rische technische H oCMchllle Ziirich, Mitieilungen ruts der Vcnu.chsa.nst(lltflIT Tfr(lsse!'oau una Erd:bau, No. 27, Verlag Leemann J ZUrich, 1954. 9. N.N . .Pavlovskii: K voprOStl 0 raschetnoI formule dlia ravnomernogo dvizheniia y yociotoka.hk s llcodnorodllymi stenkami (On a design formula. for uniform move· men~ in channels with nonhomogeneous walls), Izul1s€iia VsesoiUZllogo Nau.cir.lIoI 8sledovatel' skolJo Instiluta. Gidratekhnikt (T"(LMrtClions, ,111- Unio;~ Scienlifu; Resear~h Inslill1le of Hyd1"altiic Erl9ineeril~g), Leningrad, vol. 3, pp. 157-164, 1931. 10. L. MUhlhofer: Rauhigiteitsuntersuclmngen in einem Stollen mit betonierter Soble und unveddeidete~ Wand en (R(Jughne5S investigations in a shaft with concrete bOUom and unlined wo.lis), Wass8rlcra!t u.na Wasserwir/,sdtafl, Munich, vol. 28, no. 8, pp. 85-83, 1933. . ll. H. A. Einstein and R. B. Banks: Fluid resistance of composite roughness, Trans- . actions, Am.erican Geophysical Union, vol. 31, no. 4, pp. 603-6jO, August, 1950, 12. G. l{. Lott.er: SOQbrazheniia k gidravlicheokomu raschetu cusel s l'!>tlichnoI sherokhovatosliiu stenoI!: (Collsidera.tionson hydraulic design of channels wi~h different roughness of \ir8.1I8), /zu.estiia. V~e~oiu2nogo Nauchno-!ssledvvateJ,'skof/Q Instiluta. ·Gid1'otekhniki (Transa.ctiqna, .till-Union Sde1\tijic Reseprch In.stilutc of Hyd~at;lic En(Jineering), Leningrad, voL 9, pp. 238-241, 1933. . 13. G. It Lotter: Vliianie uslovii ledoobrazovaniia. i tolshchinY l'da naraschct derivatsionnykh ka.nruov (Influence of condi~ions of ice formation and thickness on the design of derivation ca-nlLls), IZTleiltiia. Vsesoiu.znogo Nav.chno-I:ssled()va.t~l'slco(jo. Ins/itu/a Gidrolekhltiki (Tra.nsactions, All-Union Scimtiji.c ReseO;rch 17Ultitute of H1!d~G7j.lic Engineering), Leningrad, vol. 7, pp. 5&-80, 1932. 14. G. Ie LoLter; Metod akademilca N. N. Pavlovskogo dUo. ojJcedeleniia koeflitsienta

I·

i

I

i

156

. UNIFORM FLOW

sherokhovl1tostl rusel·, pokrytyl,h I'dam (Method by Academy Member N. N. Pavlovskil for der.crmination Df roughness coefficients of ice-{lovered channels), I TllVest{ia V seso;wm
CHAPTER

7

DESIGN OF CHANNELS FOR UNIFORM FLOW

Channels to be discussed in this chapter include nonerodible channels erodible chann.els, and grassed channels. For erodible channels, the discussion will be limited mostly to those· which scour but do not silt (see Preface). A. NONERODIBLE CHANNELS

7-1. The Nonerodible Channel.' Most Hned channels a!ldbuilt-up channels can withstand eroslonsatisfactorily and are therefore· considered nonerodibZe. Unlined channels are generally erodible, except those excavated in firm foundations, such as rock .bed. . In designing nonerodihle channels, such' factors f:t:;, the maximum permissible velocity (Art. 7-9) and the permissible tractive force (Art. 7-13) are not the criteria to be considered, The designer simply computes t.hedimensions of the channel by a uniform-flow formula and then decides the final dimensions on the basis of hydraulic efficiency, or empirical rule of best pi'acticability, and economy [1,21. The factors .to be considered in the design are: the: kind of mat.erial forming the channel body,which determines the roughness coefficient; the minimum permissible velocity, to avoid deposition if the water carries silt or debris; the channel bottom slope and side slopes; the freeboard i and the most efficient section, either . hydraulically or empil'ically determined. 7-2. Nonerodible Material and Lining. l The nonel'odible materials used to form the lining.of a channel and the body of a built-up channel include concrete, stone masonry, steel, cast iron, timber, glass, plastic, etc. The selection of the material depends mainly on the availability and cost of thf; material, the method of construction, and the purpose for which the channel is to be used. The purpose of lining a channel is in most cases to prevent erosion, but oacasionally it may be' to check seepage losses. In lined channels, the ma:r;imum permissible velocity, i.e" the maximum that will not cause erosion, can be ignored; provided that the water does not catty sand, gravel, or stones. If there are to be very high velocities over a lining, however, it should be remembered that there is a tendency for the rapidly 1

For detailed information on channel lining, aee [3). .

157

./

'~.

UNIFOR\I1 FLOW"

158

DESIGN OF CHANNELS FOR UNIFORM FLOW

159 a.now the pradicable Use of just about any type of lining or lining tl'eatment now or in tho future ant.icipatcd by the Blireau. 7-6. Freeboard. The freeboard of a channel is the vertical distance fi'Oill the top of the channel to the water surface nt the design condition. This distance shouid be sufficient to prevent wnves or fluctuations in water surface from overflowing the sides. This ftwtor becomes imp or. tant particularly in~he design of elevated flumes, for the Rume substructure may be endangered by any overflow. There is no universally accepted rule for the'determination of freeboard, since wave action or wat~t-8urface fluctuation in a channel ma.y bi;: created by many uncontrollable causes. Pronounced Waves and fiu.ctuatioll of water surface are generally expected in channels where the velocity is so 11igh and the slope so steep th~t the flow becomes very unstable, or on curves where high velocity and large deflection angle may cause appreciable superelevuted water surface on the convex side of a curve, or in chnrmels where the velocity of flow appt'oaches the critical state at which the water may flow at alternate depths and thus jump from the low stage to I;hc high stage at the least obstruction. Other natural causes such as wind movement und tidal adion may also induce high waves and Tequire special considemtion in design. Free boards va.rying from less than 5 % to grea,ter tha.n 30 % of the depth of flow are commonly used in design. For smooth, il1terior, semicircular metal flumes on tangents, carrying water at velocities not, greater than . 80 % of the criticnl velocity with a maximum of 8 fp.'l, experience has indicated that a freeboard of 6 %. of the flume diameter shouid be used, For flumes on curves with high velocity or deflections, wave Ilction will be produced; so freeboard must he increll.sed to prevent water from slopping over,

. water to pick up lining blocks and push them out .~~ ??SlulOn. movmg .. h Id be designed against such POSSibilItIes. Accordingly, the, Immg SOli, 'bi e VelO~l'ty h M' urn PerrnlSSI ' - . The mim:mum pel'missible . 7 -3, T e mlm "t'n is the lowest velocity that Will not" velocity, ?L' I'.he ~onS'O" ~d ~ duc~ the growth of aquatic plant and mo~s. start sedlmentatlOn an IJ1 t ' . d its e"act value cannot be er.sIly 1'1 . I ' ty is very uncer aJl1 :Ln l\. h' liS ve. OCI ' ' no ISlt load d For water carrymg , or for desiltecl flow,, t IS . , t for .its effert fdetermme t . h s little si"'nI'fi canee excep - onplant growth.h Gen'h . ac IIOJ a kino'' .a mean t> l ' t ' f 2 t 'Hps may be used safely w en" ve OCI y 0 0 • . 't ef era y spea "". , t' the channel is small, and a mean veloci y 0 percentage of sIlt preseDillln. t a growth of vegetation that would not less than 2.5 fps W pi even,. ' th ,. i'lg capaCity of the channel. , sedously decrease longitudinal bottom Glope of cha.nnel is 7,.4. Channel Slop , ra hv and the energy head required for generally governed by the topog , P', h slope may ,depend also on the h f! f ater In many cases, t e . , t e . ow 0f the w channe , 1 F'" 'or exanlple , channels' used for W:Lter-dlstrlh d r ' pur~ose o. " ' b se used in irrigation, water supply, y :au I~ butlOn plll poses, such as t o , . h' , level at the pomto! . . , d h droDower proJects reqUIre a Ign . mll1Jl1g, y . a smaL1 b-I' '~d eSI'rable in order to keep the los3 III delivery;~J.l1 therefore, ope I::>

v~locit'lJ

::allihe

a

elevation to a minimfum' h 1 deIJend mainly on the kind of materiaL, The side slopes 0 a e anne . h . " 7-1 give:, a genera1 I.d ea o f the slopes Table . suitable for use Wit 'l'fLrJOUS ' TABLE

7-1.

' sLOPES

SUl'rABLE SID;;;

FO R CHANNELS BUILT IN V,UllOU.S I{INDS

OF MATElRIALS

. 111 aterial Rock .... ,.,.. . .. , ........ ', ... , * • • • • • • • • • Muck and .peaL soils . .. ", .. . .... : .'........ , ... ~ ........ . Stiff clay or earth with, concrete Immg ... , , . ' , .......... . " Etlft.h ,Vlth Btone I"Wing, or earth . for large eila,nnels ....... . Firm clay or earth for small ditches ........... , , .......• Loose sa.ndy earth. , ....... , ... , .... , .... , , .. , ... , : . , .• Sandy I06.m 'or porous clay .. , . : .. : ... , ...... , . , .. , .... , ~

w

•••••

Side slope Nearly vertic al ~:i: 1

H':1 to 1:1

'an

1:1

1%:1 2:1 3:1

. . erodible material, however, a more accurate kinds of h'd b e checked [wainst the criterion of . m~LterJal. , f th For lopes SOUl " "0 determll1atwn 0. . e S l i t (Art 7-10) or by the principle of tractive maximum permli5s1bb~~e. °f:c~ors to' be considered in determining slopes ,~1. ~dition of seepage loss, climatic change, force (Art. 7-14). are method of constru<..; lon, COl . . 'd 1 hould be made as . t Generally speakmgj SI e s opes s channel Size, .e c. d Id·b designed for high hydraulic efficiency steep as practICable a~ s ou 1 e th US Bureau of Reclamation [4] and st:Lbility. For 11l1ed can~~, e'~'5'1 slope for the usual sizes has been considering standarfdlzhl~lg ~n a .. that it is sufficiently flat to. of canals. One advantage a t IS S ope IS ,

~.

h

I I

1;,

Freeboll.rd in unlined cann.'l OJ' lateral will nornu;.lly be governed by considerations of canal size and locatiOJ1, storm-water inflow, and watertable fluctuations eaWled by checks, wiud action, soH characteristics, percolation gradients, operating road requirements, and availability of excavated material. According to the U,S. Bureau of Reelamation [4J, the approximate range of freeboard frequently used extends from 1 ft for smalliateral"/ with shallow depl;hs to 4 it in canals of 3,000 cfs or more capacity with reheively large \vater depths. The Bureau recommends that preliminary estimates of the freeboard required under ordinary conditions be made according to the following formula: (7-1)

where F is the freeboard in ft, y is the depth of water in the canal in ft, and C is ll. coefficient varying from 1.5 for a canal capacity of 20 cfs to 2.5 for a canal capacity of 3,000 cfs or more~ This approximation is

.\

II

160

I

I,

I

UNIFORM FLOW

be noted that a best hydraulic section is the section that gives the minimum area for a given discharge but not necessarily the minimum excav!1tion. The section of minimum excavation 'occurs only if 'the water surface is at. the level of the bank tops. Where the water surface is below the bank tops, as frequently occurs, channels narrower than those of the best hydraulic section will give minimum excavation. If the' water surface overtops the banks and these il.l'e even,with the ground level, wider channels will provide minimum excavation, TABI,E 7-2. AArea

Cross se<;tion

II_~__·. I'

Tmpezoid, half I of a hexagon I Rectangle, half of a square I Triangle, half of a :;quare Semicircle

T=2V2y Hydro~tatic

catenary Capacity. cIs

T_I_D~ _ _Z_ _

il, _ _

.:

.

4y

,II

y'

I 2 V2 11

,

Section . factor

R-I_ _

P

,13 yl: 2 .../3 11 I 2y l

Hydraulie depth

%'

%,y

!

~YI

Y

1. 39.58Sy'

2, 9836y

'%y

2YI

Y

%y'" 2:\1',6

"I ~' ~y"

I

Try

y'

v3" 11

2111

20YI

4-y

-4 y'"

%YI % y'3y u

0.46784y 1. 917532yIo, 72795y 1.19093y ' ·&

.' The pri~ciple of the best hydraulic section applies only to the desig.n of nonerodlble channels. For erodible channels, the principle of tractive force must be used to determine an efficient section (Art, 7-15). . •

(U.S: Bureau

of Reclamation.)

7-6. The Best Hydraulic Section. It is knO\vn that the conveyance of ar chn.nnel section increases with increase in the hydraulic radius or wit.h .decreD,se in the wetted perimeter, ]'rom a hydr~ulic viewpoint, therefore, the channel section having the least wetted peri!lleter for a given area has the maximum conveyance; such a section is known as the best hydrauNc section, The s~micircle hrLs the least perimeter among all sec,tions with the same area; hence it is the most hydraulically efficient. of . all sections, • The geometric elements of six .best hydraulic sections are listed in iTable 7-2, but these seytions may not always qe practical owing to 'difficulties in constructioJ;! and in use of material. : In general, a channel :section should be designed for the best hydraulic. efficiency but should be : modifi~d for practicability. From a practical point of view, it should

Wetted /' Hydraulic' Top perimeter radius width

~Y' i I "~YI 72YI " I 1% V'2 )%'V2 HYt ! r

.)

P,arabola,

BEST HYDRAULIC SECTIONS.

I

I

FIG. 7-1. Retommended freeboard and lleight of bank of lined channels.

161

DESIGN OF CH .... NNELS FOR UNIFORM FLOW

based upon average Bureau practice; it will not, however, serve for all conditions. . For lined canals or h1.teralS, the height of lining abo"';'e the water surface will depend upon a number of factors: size of canal, velocity of water, curvature of alignment, condition of st0rm- and drain-water inflow, fluctuations in water level due to operation of flow-regulating structures, . and wind action. In a somewhat similar mannel', the height of bank. above the water surfitce will vary with size and location of canal, type. of soil, amount of intercepted storm or drai.n water, etc. As a guide for. lined-canal design, the U.S. Bureau of Replaination [3] has prepared curves (Fi~. 7-1) for average freeboard and bank heights in .relation to capacit.ies.

Example 7-1. Show that the best hydraulic trapezoidal 8ecti~n is one-half of a hexagon, Solutio ll . Table 2-1 gives the water area. and wetted perimeter of a tl'apezoid as A = (b

+ zy)y

and

P = b

+ 2 VI + .' y

whe~e y is th~ depth, b is the bottom width, and .: 1 is the side slope,'

.. Flrst, conslder A and z, to b,e constant. : Differentiating the above two equations with respect to y and solvmg Simultaneously for dP Idy, . . dP dy = 2( v'f+Zi

-

b y

z)

I

f t I

I

1

For a minim~m wetted perimeter, dP Idy ~ 0, or b = 2Y(VI

,+ ZZ

-

2)

Substitl!ting this equation for b/n the previ~us two equations for A ELnd Pand solving

11

simu1t,a.neDusiy for: P, p

=z

Now, find tile va.lue z tha.t makes P th~ least. z, equcting dP Ida to zero, and sol viug for z,

of

z =

163

DESIGN OF CHANNELS FOR UNIFORM FLOW

UNl.F'ORMFLD'W

1,62 .

Diifer:entiating P with respect to

where x. is the width-depth .ratio b/y and z is the horizontal projectkm of t~e SIde slope corresponding to 1 ft vertical. Similarly, engineers in IndIa [61 uS,ed an enlpirical formula: V = VA I 13 '=' 0577 V: r7A h'ch . . have I .ti, 'IV 1 IS equlva ent to x = 3 - z for trapezoidal sections' alld Ph'II'lP' , . . . f7] El' . . , pme el1gJ.-: ne61S t use q. (,7-3) WIth z 1.5, or z ~ 2.5, far earth canals,

~3 ~ ta.n30·

35

This means that the section is a half he"agon,

7 .7. Determination of Section Dimensions. The determin8,tion of section dimensions for nonerodible ch:u1l1eis includes the followin.g steps: 1. Collect all necessary information, estimate n, and select S. 2. Comput~ the sectioll factor ARH by Eq. (6-8), 01'

AR'i=~ 1.49 vIS,

practicability. 5 .. For the design of irrigation channels; the channel section is some~ times proportioned by empirical rules such as the simple rule given by·the early U.S. Reolamatiau Service [51 for the full supply depth of water in

-vA

25

'"<>.

..

'0 <:

20

"

of Art. 6-6. . 4. If the best hydn.ulic section is required directly, substitute in Eq. (6-8) the expressions for A and R obtained from Table 7-2 and sol verOr the depth. This best hydraulic section may be modified for

y= 0.5

'-

.= ~

(6-8)

3. Substitute in Eq. (6-8) the expressions for A B,ud R obtained from Table 2-1, and solve for the depth. If there are oth('J' unknowns, such as band z of a trapezoidal section, then assume the values for these unknowns and solve Eq. (6-8) for the depth. By assuming several values of the unknowns, a number of coltlbinations of section dimensions can be obtained. The finaldimellsions are'decided on the basis of hydraulic efficiency and practicability. For lined canals, the trapezoidal section is 'Commonly a.dop~ed, and the p.S. Bureau of Reclamation [3J has developed experience curves (Fig. 7-2) showing the ayerage relation of bottom. widths and water depths to canal ca.pacities. These curves can be used as a guide in selecting proper section dimensions. • The determination of the depth for the computed value of AR7~ can be simplified by use of the design chart (Fig. 6-1). Some engineers prefer a. solution by trial and erro!", similar to Solution 2 for Example 0-2

feet ..

30

== 15 ,,,!

"E 2 0

OJ

:>

Q! 10

100

IPOQ

'1,000

'!,pOD

FUIG ' 7B-2. Experience cur:re.sshowing bottom, width and depth of lined· channels ( .S . .ureau of RG~aJltatum,,) . '

6. Check the minimum permissible velocity if the watel' carries silt. 7. Add a proper freeboard to the depth of the channel .section. b :~ample 7-Jl. A trf,pez~idal channel carrying 400 ds is ,built with nonerodible e", Zlavm g aBslope of 0.0016 and n =- 0.025. Proportion the section dimensions ,,0 !!t,on. Y' Eq. (6-8), . 025 AR~1 = _0: _ ~ '" 167.7 1.19 VO,OOIS

ex~~!~I~t~ting

A, = (b

+ zy)y

a,nd R

=

(t,

+ zV)y!CIJ + 2 Vi + z' y)

-::::---'-":......:..~~;.--'

y)%

(7-2)

where A is the water area in ft2. For a trapezoidal section it can be shown that this rule may alsabe expressed by a simple formula x= 4- 2 (7-3)

ZOO 300400 600

capodry ill cis

Assuming b "'" 20 it and z

== 167.7

in the

~ove

[

.

= 2 and simplifying, 7,680

+ 1,720y = (y(1O + y)Ju Y,= 3.36 It

\

I

164

i'

UNIFORM FLOW

It should be noted that thia solution is exactly the same as the computation of the normal depth given in Solution 1 of Example 6-2. Accordingly, the solutions by trial il.nd error and by the graphical method described in Exo.mple 6-2 can also be applied to the present prublem. Similarly, assume other suitable values of band z, and compute the corresponding depths. The final' decision on dimensions will depend on practical considerations. If the 'values of band z are decided o.t the 'beginning of the compuio.tion, the depth will be c,)mputed only once. Suppose that b =20 ft, 2 = 2, and y = 3.36 ft are the final vo.lues. Assign a freeboard of 2 ft; the total depth of the channel is, therefore, 5.36 ft and the top width of the channel (not the width of the ,vater smface) is 41.4 ft: The water area is 89.S ft', and the velocity is 4.46 fps, which is greater than the minimum permissible velocity ' for inducing silt, if any. When the best hydraulic section is required, substitute A, = vl3y' and R = 0.5y, obtained' frem Table 7-2, in AR~~ = Hi7.7 !!p..d simplify; the depth is found to be y = lUI it. Add 3 ft freeboard; the total depth is 9.1) ft. The corresponding: bottom width is 7.6ft, the top width of the dUlllnel is 18.7 ft, the water area is 75.2 ft', afld the velocity is 5.32 fps. Since the, best hydraulic trapezoidal section is the half hexagon, t,he side slopes are 1 on

r

L

of Reclamation and is tentatively recommended for design of erodible channels. It should be noted that either method at the present stage will serve only as a guide and -\vill not supplant experience and sound engineering judgment. 7-9. The Maxim'urn Permissible Velocity. The max-im1tm permissible velocity, or the nonerodible velocity, is the greatest mean velocity that. will not cause- erosion of the channel body. This velocity is very uncertain and variable, and ,can be estimated only with experience and judgment. In gencml, aId and well-seasoned channels will stand much higher velociT.~llLE·

7-3.

MAXIMUM PERMISSIBLE VELOCITIES RECOMMENDED' BY FOR'rIEll

,AND SCOBEY ANi. THE CORRESPONDlNG UNIT-TIIAC"l'IVE-FORCE VALUES

U.S. BUREAU OF RECLAMATION" (For straight channels of small slope, after aging)

CONVERTED BY THE

,

va/a.

Material

I

)

I It has been noticed that certain channels are erodible whereas others v:ary similar in cha,nnel geometry, hydraulics, and soil physical properties are not. As a further step in investig~tioIl, the chemical properties of the material forming the channel body should be explored, It may be that an ion exchange between water and soil or hydration of the, material is providing a binder in some places and thus affecting the erosion. For a general discussion of the compiexity of this problem, see [81 and [91.

!

Clear water

_I

'7\

B. ERODIBLE CHANNELS WHICII SCOUR :BUT DO NOT SILT

7-8. Methods of Approach. The behavior of flow in an erodibl~ channel is influenced by so many physical factors and by field conditions so complex and uncertain that precise design of such channels at the present· stage of Imowledge is beyond the realm of theory.! The uniform-flow formula, which is suitable fOr the design of stable nonerodible channels; ,provides an insufficient condition for the design of erodi.ble channels. This is because the stability of erodible channels, which governs the design, is dependent mainly on the properties of the material forming the channel body, rat.her than only on the hydral..lllcs of the flow in the channel. Only after a stable section of the erodible channel is obt,ained can the uniform-flow formula be llsed for ,computing the velocity of flow and discharge. Two methods of approach to the proper design of erodible channels are described here: the method oj perrn1;sSible velocity and the method of tractive f01·ce. The method of permissible velocity has been used exteu- . sively for the design of. earth canals in the United States to ensure freedom from scour. The method of tractive force hOes sometimes been used in Europe; it is now under comprehensive investigation by the U.S. Bureau

165

DESIGN OJ!' CHANNELS FOR UNIFORM FLOW

v,

I

fps

ilb/ft2

Water transporting colloidal silts

V, fps

rD,

TO,

Ib/ft'

--'1---------

'1

I

iI i

J

Fine sand, colloidal ... , .. , . , .. ,. " ""'1 Sandy loam, non colloidal. . .,. . . , , . . .. Silt loalll, nOllColloidal .. " ", ... .", ''', Alluvial silts, noncolloidal. ......... Ordinary firm loim,., ..... , ....... , .. Voleanicash" ...... " . . . . . . . . " .... Stiff clay, very colloidal. , ...... " " , .... " Alluvial silts, colloidal.. " " " ' ' ' ' " ' ' - ' 1 ShalesandhardPl1ns .... : ......... ".: ... j

i

0.020 1. 50 O. (127 0,020 1. 75 . 0,037 0.020 2.00 0,048 0.020 /2.00 0,048 0.020 2.50 0.075 0.02012.50 0.075 0.025 I 3.75 0.26 0,025 3.75 0.26 0.025 6.00 0.67

I

I

1

2,50 0.075 2.50 0,075 3.00 0, n 3.50 0.15 3.50 0.15 3.50 0,15 5.00 0.413 5.00 0.46 6.00 10.137

~~:~:tl:~m' ~~. ~~b'b'l~~ ";h~~ :~~~~~li~i~l~l: :~:~~~

i: ~~

~:~~5 ~:~~ I g:~~

Graded siits to cobbles when colloida.l. , , . 0.0;10 Coa-rse gmvel, ~0IlC0I10idlll .......... , .. ; ... / 0.025 CObblesandshmglE'3, ... , ................ 0.035

4.00 4.00 5,00

0.43 5.50 0.30 .16,00 0,91 5.50

,.1

,

•

i

0.80

I 0.67 1.10 I

I!

• The Fortier and Scobey values were recommended for use in 1926 by the Special Committee on Irrigation Research of the American Society of Civil Engineers.

f f

ties tha.n new ones, becau$e the old cha.nnel bed is usually better stabilized, particularly with the deposition of colloidal matter. When other conditions are the same, a deeper channel will convey water at a higher mean velocity without erosion than a shallower one. This is probably because the scouring is caused primarily by the bottom velocities and for the same mean velocity, the bottom velocities are greater in .the shal~ lower channel.

i' i

f J

t

IL !~

, I

UNIFORM FLOW

LG6

DESIGN OF CHA.NNELS FOR UNIFORM FLOW

Attemp~1 were made early -to define a mean velocity that would cause neither silting nor Scoill:ing. From the present-day viewpoint] however, it is dOHbtful whether such a velocity Mtually exists. In 1915,

[

167

2

J _Ir j

! ,g 0.7 :

~, ~

'"

.~

.2

'0

g 0.5f----j--·,--+-..·---/---';-',,\},c-:-+---f-1--i

.)

0.4

,." :is '" "iii

I

I

j o~~,-=l==I=t:I=l=

·t

0,.8

()'5~H-++--t

-{

O.4!-H-++-fflt-f--+

:3

o.:;f--f-+-++-+H-H--t Q.~

Per"r.issible va loci!i es. fps

!-,-,-.---l...--'---r",-,--,--W.-..c Cloy

!

FIG. 7-4, Curves showing' U.S.S,u, da.ta on permissible velocities for cohesive soils.

FIG. 7-3. U.S. and U.S.S.R. data on ]lermissible veloCities for no,nco,hesive soila,

Etcheverry [2Gl published probably the first table of maximum meIl.n velocities that are safe il:gaillllt e1'Osion. . In 1925, Fortier and Scobey [27J pli bUshed the well-known to ble of "Permissible Canal Velocities 11 shown ill Table 7-3. The values in thiE table are for lVell-seasoned channels of small sfopes and for depths of flow less .than 3 ft. The table also shows 1 The first famous fo,rmula for this nonsiiting and no,neroding vciocit,y fo,r silt-laden water was published in 1895 by Kennedy [lOj. From a stl\dy o,f tne dischargo and depth of 22 canals of the Upper Bari Doab irrig.!l.tion system in Punjab, India, the Kennedy fOrImll", WD-S developed as .

(7-4)

where V, is the nonsilting and noneroding mean velocity in fps; y is the depth of flow in ft; C - 0.84, depending primarily or. the firmness of t,he material forming theqhannel body; and z "" 0.64, an exponent which varies only slightly. Ba.sed on later studies by other engineers, the valu~s of C generally recommended are 0.,55 for extremely fille soils such a.s tho,se found.in Egypt; 0.84 for fine light sand soils such as those found in the Punjab, India; 0.92 for coarse light sandy soils; 1.01 for sandY loamy silts; and 1.09 for coarse silt or hard-soil debris. For clear water, a "alue of z = 0.,5 has been suggested. For the design of canals carrying sediinent-laden water, the Kennedy formula is now practically obsolete and is being replaced by methods based on Lacey's regime theory [11-16], Einstein's bed-load function [17), a.nd Maddock-Leopold's principle of channel geometry [18J. There are voluminous writings on these methods. Comprehensive bibliographies can be found in [19] to [25].

,, l I~ l~

. I

.suitable n values for various materials and the converted values for the corresponding permissible tractive force, which will be discussed later (Art. 7-13). In 1036, a .u.~l'''''.<1iH magazine [28] published values of m[l.ximum permissible velocities (Figs. 7-3 and 7-4) ahove which scour would be produced in noncohesive materiai of a wide range of pa.rticle sizes and various kinds of cohesiVe soil It also gave the variation of these velocities with channel dep!;h ~ (Fig. 7-5). The maximum permissible velocities mentioned abo Ire are with reference to straight channels. For sinuous channels, the velocities should be lowered in order to 'reduce scour. Percentages of reduction suggested by Lane [29] are 5% for slightly sinuous canals, 13 % for moderately sinuous canals, and 22 % for very sinuom; canals. percentage val tieS, however, are veJ;'Y approximate, since no accurate data are available Carreclion foctor at the. present time. FIG. 7-5. Curves showillg 7-10. Method of Permissible Velocity. Using U.S.S.R. corrections of the maximum permissible velocity as a criterion] permissible velocity fo,r the design procedure for a channel section, as- depth for both collesive silmed to be trapezoidal, consists of the following and noncohesive materials. steps:

I

I

r i

-j

168 I, l

It should be noted that the unit .tractiv~ force in channels, except for wide open channels, is not uniformly distributed along the wetted perimet.er .. Many attempts have been made to determine the distribution of the tractive force in a chann~l. Leighly [31] attempted to determine this ,distribution in many trapezoidal and several rectangular and tl'iangular . channels from: the published data on the velocity distribution in the

1. Fot' the given kind of material forming the channel body, estimate the roughness coefficir.nt n (Art. 5-7), side slope z (Table 1-1), and the maximum permissible velocity V (Table 7-3 and Figs. 7-3 to 7-5). ' 2. Compute the hydraulic radius R by the Manning formula. 3. Compute the water area required by the given discharge and permissible velocity, or A = Q/V. ' 4. Compute the wetted perimeter, or P = A/R. 5. Using the expressions for A and P from Table 2-1, solve simultaneously for band y. The solution may be expedi~ed by the charts given in Appendix B. ' 6. Add a pl'0per freeboard, and modify the section for practicability.

.

i 1.5

'

Example 7-3. ,Compute the bottom width and the 'depth of fiow ,of !l. trapezoidal cha.nnel laid on a slope of 0.0016 and carrying 8. design dischfl.rge of 4{)O cfs. 'rhe channel is to be excavated in earth containing noncolioidal coarse i1;rllvels and pebbles. Solution. For the given conditions, the following rm!estimated: n 0.025, z = 2, , and nlll:llili1um pe;l'missible velocity = 4.5 fps. Using the Manning formula, !wlve for R. 4.5

=

O:97QwyS

:FIG. 7·6. Distrihution of tra.ctive force in

!1

trapezoidai channel section.

1.49 -0.025 R~ ";0.0016

R ." 2.60 ft

or ThenA = 400/4.5

=

88.8 ft', a.nd P

A = (b

and

169

DESIGN o.F CHANNELS FOR UNIFORM l"L~W

UNIFORM FLOW

P

=

b

+ zy)y =

=

(b

AIR

+ l!y)y

+ 2 v'l+ ii Y =

(b

88.8/2.60 = 34.2 ft. ~

Now

88.8 ft'

+ 2 y'S y)

Solving the above two equations simultaneously, b

)\4.2 it

18.7 ft and y

3.46,ft.

.

'I -11. The Tractive Force. When water 'flows in' a channel, a. force is developed that acts in the direction of flow on the channal oea-:-"This fm'cb, which is simply the pull of wilter on the wetted area, is known as the tract~'ve force. l In It uniform flow the tra()tive forceis apparently equal to the effective component of the gravity force acting on the body afwater, parallel to the channel bottom and equal to wALS, where w is the unit weight of wa.ter, A is the wetted area,' L is the length of the channel reach, and S is the slope (Art; 5-4). Thus, the a.verage value of the tractive fOl'ce pel' unit wetted area, or the so-called tmit tractive f~rce TO, is equal to wALS/PL = wRS, where P is the wetted perimeter and R is the hydraulic radius; that is .

.

. [TV

=

w~·

V

Ii,

l- fl--+--!--+---l- -

123'156789

-I

I

rmm]

7

B

9 10

!4-+-1-+--+=1t++i~ :-.-JI:~

o o

I

2

3

4

5

6

bh

bh

On sid s S 01 channels

On bcttom of cl1annels

FIG. 7-7, Maximum unit tractive forces in terms of wyS. '

channels. U nfoi·tunately,. to deficiency of data, the results of his study were not very conclusive. In the CS. Bureau of Reclamation, . Olsen and Flol'ey[32] and other engineers ha,ve used the membrane analogy and analytical and finite-difference methods for determining the distribution of tractive fOl:ce in trapezoidal, rectangular, and tria!1g ular a trapezoidal chanchannels. A typical distribution .of tractive force Itel reSUlting from the m~mbrane-analogy study is shown in Fig. 7-6. The pattern of distribution varies with the shape pf the section but is

q-5)

In a wide open channel, the hydraulic radius is equal to the depth of flow y; hence TO '" wyS. 1 'l"his is also known as the shear force or the drab fOl·ce. The idea of tractive force is gellorally believed 1.0 h!we been first :introdue~d intohydraulia litera.ture by du Boys in 1879 [po 149; of 30]. Howelrer, the principle of balancing this foroe wit~ the

in

channel resistance in 11 uniform flow was sta.ted 1;Iy Brahms earIy in 1754 (see Art. 5-4).

I .Ii".

170

.

UNIFORM FLdw

DESIGN OF QHANNELS FOR UNIFORM ;FLOW

.

practically unaffected by the size of the section. Based on such studies, curves (Fig. 7-7) showing the maximum unit tractive forces on the sides and bottom of various channel sections have been prepared for use in cu,ilal design. Generally speaking, for trapezoidal c.hannels of .the ilhapell ordinarily used ill canals, the maximum tractive force on the bottom is close to the va.lue wyS, .and on the sides close to 0.76 wyS. 7 -12. Tractive-force Ratio. On a soil particle resting sm the sloping side of a channel sect.ion (Fig. 7-8) in which water is flowing, two forces are acting: the tractive force aT, and the gnlvity cfol'ce component 'YV; sin ¢,

171

is equal to the force tending to cause the motioil. The resistance to motion of the particle is equal.to the normal force W, cos ¢ mUltiplied by the coefficient of frictioil, or tan (J, where e is the angle of repose. Hence, (7 -(3)

Solving for theullit tractive force T, tlmt causes impending motion on a sloping surface, . .. W . / tan" tP (7-7) T, = cos rb tan II '\;1- tan' (1

,c!

,.

J

Similo,rly, when motion of :1 pa.rticie on the level surface is impending owing to t.he iractive force an, the following is obtained fromEq. (7-6) with
-

a

tan

.

e

(7-9)

The ratio of T, to TL is called the tractive-force ratio; this is an important ratio for design purposes. From Eqs. (7-7) and (7:'9), the ratio is

Simplifying, ) PIG. 7-8 .. Analysis of forces ac.ting on a pa.rticle resting on the surface of a chn.nnel bed.

which tends to cause the particle to roll down the side slope. l The symbols used are a = effective area of the particle, T, = u\lit tractive force on the side of the channel, W, = submerged weight of the particle, and", = angle of the side: slope. The resultant of these two fO;'ces, which are at right angles to each other, is

VlV,2 Si112

'"

+ oh,2

When this force is large enough, the particle ~vill move. By the principle of frictional motion in mechanics, it may be assumed that, when mot.ion is impending, the resistance to mqtion of t.he particle 1 The concept of the three-dimensional analysis of the 'gravity and tractive forces acti]lg on a particle resting on a slope at the Btate of impending motion was first given by F'orchheimcr [33J. A complete analysis of a chann"l section using this .concept Wa.'l first developed by Chia.-Hwa Fan [34J. The a.nalysis ,vas also developed independently by' the U.S. Bur~a.u of Reclamation under the direction of E, W. Lane [29,35].

=

~

tan 2 ¢

f

T

K

= COli '"

J( =

~ 1 - tan 2 0

/1 _

'\;

~

Si.ll '" Sill" ~

(7-10)

1

(7-11)

It can be seen that this ratio is a fUllctioll only ~f the inclination of the sloping side", and of the angle of repose of the material O. Fot cohesive and fine noncohesive materials, the cohesi vc fOI'ces,even with comparatively clear .vater, become so great in propprtion to the gravityforce component causing t.he particle to roll down that the gravity force wn safely be neglected. Therefore, the angle of repose need be considered only for coarse noncohesive materials. According to the U.S. Bureau of Reclamation's investigation, it was found in general t.hat the' angle of repose increases with both size and anguiarity df the material. For use in design, curves (Fig. 7-9) were prepnred by the Bureau, showing values of the angle of repose for nOli cohesive material above 0.2 in. in . diameter for various degrees of roughness. The diameter referred to is the diameter of a particle than which 25% (by weight) of the material is larger. 1 Equation (7-10) was presented by the U.S. Bureau of Reclamation [35,361 and Eq. (7-11) by Fan [341. The two equations are mathematically identical.

.

1

172

UNIFORM FLOW

0.1

. 0.2

L-----.rL--;

42

~-

I

I

, I

I

3/16 1/4 lIB 112 ~j

• .....

-T--[I

l

I

I I I I

I

3/4

I

8

""

,6

-1--

:3 4

2

..,'"

I !,

.

'"c:

~<~I ;;"'/ /

.If': ~/ ~

b'"

Il

I.;;:,~

'0 .!?

t1

$O~

'2 6

f--- ... 24

2. 2

'ti

V if

J~Ci

V."O- V ~ ~0° .~~ L ,'") r

I

---V~.' // /// V / / V/ V//V L/

V

~ .'"...."

3.0 4.0

173

DESIGN OF CHANNELS FOR UNIFORM FLOW

particle size for noncohesive material and upon compactness or v:o~ds ratio for cohesive materiaL Other' soil properties such as the plastIcIty index 1 or the chemical action may probably also be taken as indexes for defining permissible tractive force more precisely. However, sufficient data and information on these indexes are lacking_ The U.S. Bureau of Reclsmation has made 'a comprehensive study of the problem, using data fol' coarse noneohesive material obtained from the Sall Luis Valley

VI/';I/I

/

Ii t7-

~o

0

.;

'"c.e ''"

~~"'/ ~'"

."

//

~~i/I

N

o ..,

V , V

I

2.0 I

H/2

I...

r

r

40

-;:;

,i .

Panic 1e size in in clles O.l 004 0.6 o.e 1.0

;VL

.---;----

·Z

1 "

!

\

I

I

.I

/ .<1r-+~I+H Recommeoded value for

"-:;

i---

'b

canals in coarse l

I

noncohe~ive material~

si.e 25% lorg er

;

j

,

V:o."':;" 20

I)

I

.FIG. 7-9. An .. Jes of repose of nonconesive material.

(U.S. Bureau of Rec/a.1l1alion.)

7-13. Permissible Tractive Force. The pe?'m-issible tract1've'01"ce is the • • J maXImum Ulllt trnctive force that will not cauge seriOlls erosion of the material forming the channel bed 011 a level surface: This unit tractive fol'c~ c"n. be determined by laboratory experiments, and the value thus obtamed IS known as the critical t1'uctiz,c JOT'ce. However, experience has shown that actual canals L.'1 coarse nUl1Cohasive material C[l.n st!].nd substantially higher va.lues than the critical tractive forces measured in the laboratory. This is prohab!y because the water {md soil in £l.ctUtll canals (J?nt~in slight amounts of conoidl1.1 and organic matter whiGh provide a bmdlllg power and also because slight movement of soil particle" can be t.?lerated in P:a.~tical designs. without endangering channel stability. B.mce the permISSIble tractive force is the design criterion fOl' field. condi- ' tlOl1S, the permissible value may be taken less than the critical value. Tb,e determination of permissible. tractive force is now based upon

50

100

FIG. 7-10. Recommended permissible unit tr.active forces for canals in noncohesive (U.S. Burea:!' of Rec!amalit.m.)

Ino.terial.

canals [37), values converted from permissible velocities, given by Etche-, verry and by Fortier and Scobey,' the U.S,S.R. values, etc. (Alt.7c9). As a result, values of permissible tractive force recommended for canal design were developed as follows:

I

-\

I

I

A

1 The pla~ticity index is 'the difference in .percent of moisture between plas~ic limit and liq:uid llmit in Atterberg soil tests. This index has been investigo,ted by the U.8. Bureau of Reclamaiioll as a soil characteristic that can be lL~ed to indics.te resistance to SCour for cohesive materials. For cs.nal deSign. a plasticity illdex of 7 may be. taken tentatively as the criticnl vll-Iue, with SCOill' OCCUrring for moder:.te 'tractive forces below this value. However, acours a.re still observed in many cases where the index is above 7. Research'shows that determinat.ion of the plasticity index in (:00" junction with consolidated-shear testa may possibly be necessary.

174

UNIFORM FLOW

For coarse noncohesive matel'ial, with sufficient factor of safety, the Bureau recoIilluends tenta.tively a value of permissibletracti Ire force in pounds per square foot equal to 0.4 times the diameter in iilches of a particle than which 25 % (by weight) of the material is larger: This .recommendation is shown by the straight line in the design c'hart (Fig. 7-10). For .fine nOll cohesive material, the size specified is t.he median size, or size sma.ller than 30% of the weight. Three design curves (Fig. 7-10)

\'0

., E==E=E~j=R+q:B=t==+=a r\

,.....

Sondy cloys (sand < 50 %) _~

Llilil! I I

DESIGN OF CHANNELS FOR UNIFORM FLOW

175

7 -14. Method of Tractive Force. The first step in the design of erodible channels by the method of tractive force consists in selecting a11 approximate channel section by experiene.e or from design tables,l collecting samples of the material fo;:ming the channel bed, and determining the'required properties of the samples .. With these data, the designer investigates the section by applying tractive-force analysis to ascertain probable stability by reaches and to determine the minimum section that appears stable. For channels in noncohesive materials the rolling-down effect shouid be considered in addition to the effect. of the distribution of tracth-e'forces; for channels in cohesive material the rolling-down effect is negligible, and the effect of t.he distribution of tractive force alon:e is a criterion sufficient for design. The final proportioning of the channel section, ho\-vever, will depend on other nonhydl;aulic practical considera!;iops.· The analysis for tractive force is best described by the following example: Design a trapezoidal channel laid on a slope of 0.0016 a.nd carrying The cha.nnel is to be excavated in earth containing noncollnidal coarse gravels and pebb['~s, 25 % of which is 1.25 in. or over in diameter, Manning's n. = 0.025. . SoLution. For trapezoidal channels, .the maximum unit tracth'e force OIl the slopillg sides is usually less th!>ll that Oil the (,ottom (Fig. 7-7); hence, the side force i~ the controUing value .in the analysis. The design of the cha.nnel should therefore include (It) the proportioning of the section dirnensioIlll for the maximum unit tractive force on the sides. and (b) c.heckiilg the proportioned dimensions for the maximum unit tJ:active force on the bottom, . a. PTopol'lioning the. Section D!:men,~ions. Assuming side slopes of 2: I, or z = 2, 3,nd a base-depth ratio bly ",. 5, the maximum unit tractive force on the sloping sides (Fig. 7-7) is 0.775wi;s = 0.775 X 62.4 X 0.0016Y = 0.078y lb/ft'. Considering a very rounded ma.terial 1.25 in. in diameter, the angle of repose (Fig. 7-9) is 8 = 33.5". 'Nith fJ = 33,5' and z ~ 2, or cJ> = 26.5", t,lle tractive-fol'ce ratio by Eq. (7-11) is K = 0.587. For a size of 1.25 in., the permissible tractive force on a level bottom is TL = 0.·1 X 1.25 = 0.5 lb/ft' (sam.e from Fig. 7-10), and the permissible tractive force on the sides iST. ~- 0.587 X 0.5 = 0.294 lb 1ft'. For a state of impending motion of the particles. on side slopes, 0.078y = 0.294, or y = 3.77 ft. Accordingly; the bottom width is b = 3.77 X 5 = 18.85 ft. For this trapezoidal section, A = 99.5 ft' and R = 2.79 ft. With n = 0.025 aud S = 0.00113, the discharge by the Manning formula is 470 crs. Further computation will show that, for z = 2 and bIy = 4.1, th" ~ection dimensions are y = 3.82 ft and b "" 15.66 ft and that the discharge is 41"4 cfs, which is close to thO} design discharge .. Alternative section dimensions may be obtained by as~uming other '(alues oI z or side slopes. . b. Checking the Proportjalled Dimensions. With z = 2 and bly = 4.1, the maximum unit tra,ctive force on the channel bottom (Fig. 7-7) is 0.97wyS = 0.97 X 62.4 X 3.82 X 0.0016 = 0.370 [b/ft', less than 0.5 Ib/ft', which is the pel'mu;sible tractive . force on the level boLtom. Example 7-4.

a discharge of 400 cfs.

Voids ratio

FIG. 7-11. Permissible unit tl'l1ctive forces for canals in cohesive material as converted from the U.S.S.It. data on permissible velocities. .

are tentatively recommended (1) for canals with high content of fine sediment in the water, (2) for canals with 10',11 content affine sediment ill the water,
I Typica.l average earth sections of irrigation ca.nals and laterals, constructed or proPQsed by the U.S. Bureau of Reclamatiotl. and selected for the flows required on the basis of econom.v and stability, are given in Fig. 5, paragraph 1.120, of [4].

!

176

DESIGN OF CHANNELS FOR:UNIFORM FLOW

UNIFORM: FLOW

transfer (if tractive force has little effect on the results and can safely be

7-15. The Stable Hydraulic Section. The section of an erodible channel in which no erosion will occur, at a minimum water area for a given dischilrge is called the stable hydraulic 8ection. Empirical pi'oJiles, such as the ellipse and the parabola, have been suggested as stable hydrauUc sections by many hydraulicians.. The U.S. Bureau of Reclama.tion [38] has employed the principle of tractive force t o · a theoretically stable section for erodible channels carrying clear water in noncohGS'ive .materials.· . In designing trapezoidal sections, as described in.the preceding article, the tractive force is made eqnal to the permissible value over only a part pf the perimeter of the section, where the forces are close to maximum; on most of the perimeter forces are less than the permissible value. rn other words, the impending instability occurs only over a small pr.rt of the perimeter. In developing a stable hydraulic section· for maximum efficiency, it is necessary to satisfy the condition that impending motion shall prevail everywhere on the channel bed. For material with a given angle of repOSE> and for r~ given discharge, this optimal section will provide noi only the chl1Ilnel of minimum water area, but also the channel of minimum top width, maximum mean velocity, and minimum excavation. In the maihematical·derivation of this section by the Bureau, the follow'ing assumptions are made: .. . L The soil particle is held against thechanllel bed by the component of r,he submerged weight of the particle acting normal to the bed. 2. At ,and above the water, surface the side slope is. at the of repose of the material under the action of gravity. 3. At the center of the channel the side slope is zero and the, force alone is sufficient to hold the .particles at the point of incipient instability. 4. ·At points between the center and edge of the channel the particles are kept in a state ,of incipient motion by the resultant of the gravity component of the particle's submerged weight acting on the side slope and the tractive force of the flowing water. 5. Thc tractive force act.ing on an area of the channel bed is equal to the weight component of the water d,irectly above the area acting in the direction of fiow. This weight component is equal to the weight times the longitudinal slope of the channel. If assumption 15 is to hold there. can be no lateral transfer of tra.ctive force between adjacent currents moving at different velocities in the section-a situation, however, that never actually occurs. Fortunately, the mathematical analysis made by the Bureau I has shoWn that the actual I

ignored. . . '. It· y . According to assumption 5, the tractlve force actmg on any e em en ar. n.rea AB OIl the sloping side (Fig. 7-12a) per unit length o~ the channell~ equal to wyS dx, where w is the unit weight of water, Y is the depth 0 'water· above AB, and S is. the longitudinal slope. , . Since the area AB is VCdX)2 + (dy)2, the unit tmctive force is equal to

t

V

tal

where rP is the slope angle of the \ tangent at AB. . • 29.5'--·-----1 The other assumpti(;ms stated above have been used previously to develop the equation for the trac. tive-forc~.ratio K (Art. 7-12). The unit tractive force all the level bottom at the channel center is 1'L wYuS, where yo is the depth of flow at the center. The corresponding unit tractive force on the sloping area ~B is,therefore, equal to wyoSIL , . III o~der to achieve impending 7-12. Andysis and desigll ?f sto.ble mo~ion ·over the entire periphery of. FIG. hydraulic secti0!1' (a) ~heoretLcl'l.l secthe channel bed, the two forces tion {or given soil properties. and eh!l.l1ll:l mentioned in the above paragraphs . slope, providing Q - 220 cis; (b) mod~fred section for Q" 400 cfs; (c) modIshould be equal; that is,. fied section for Q' 100 efs. wyS cos

+ (lL)£ tan (dy)t dx Yo •

2

. At the center of the channel, y

Takiflg the effect of lateral tnl.litive force into accoUnt, an a.!ternative assumptioll

was ina.de by the' BUreau, which states that the tractive force acting on a particle is prop~rtiona.1 to the square of the ll1ean velocity in the channel at 'the point where the

177

·1

I i

(I

=

ya' and

:t

tan~

fJ

= O.

With this condition

pa.rticle is loea~d. This assumption gives ~ solution .that agrees very clqselY w~th the solution bailed on assumption 5. Therefo.re, neglect of the trB?"fer 0,1 .tractwEl force in the a.~!l.lysis will give equally sn.tisfactory results, Bad wll:.h cOJ;lslderably less ,,'ork.· .

.. ~ .178

1

the solution of the above differential equation is tan (7-12) Y = Yo cos ( - -8 x) yo , This equation shows that the shape of a stable hydra,ulic section under the specified assumptions is a simple cosine curve. From the results of the Bureau's mathematical anaiysis, the following properties of this stable section can be stated ~ TO

Yo

o.97wS

l' A

=

1.35 - 1.19 tan 8 yo%Sl! n 2.04Yo 2 tan e

I

(7-14)

·11

(7-15)

a

= 0.99

(1 - ~ T

Example 7-6. Determine the profile of the stable hydro.ulic section to replo.ce the tra.pezoidal section of the channel described in Example 7-1. SolUlian. For the given conditions, TO = 0.5 lb/ft l , S = 0.0016,. e.= 33.5°, a'J.d n = 0.025. By Eq. (7-13), the center depth is Yo = 0.5/(0.97 X 62.'1 X 0.0016) 5.16 ft. From Eq. (7-12), the shape of the theoretical section is

which

is

y = 5.16 cos 0.128z

plotted as shown in Fig. 7-12a..

II l I

I I

\

(7-16)

On the other hand, if the channel is to can'y more than the theoretical section will carry, it is necessary to add a rectangular section at the center. Suppose the. discharge to be carried is Q", which is greater than Q, and the top width of the added rectangular area is Til. The value of T)) may be computed by /I _ n(QI! - Q) (7-17) T - 1.49Yo~~SI~

.

I

(7 -13)

where 70 is the permissible t.ractive force in IbjfV, V is the mean velocity in. the section in fps, A is the water area in ft~, 8 is the angle of repose for the material or the slope angle of the section at the water edge of the channel, T is the top width, and the rest of the symbols are hS previously defined. The discharge of the theoret,ical section is equal to Q '= FA. If the channel lS to carry a discharge less than Q, it is necefisary to remove vertical portion of the section at the channel center. S'upposc the dis- . charge to be carried is Q', which is less than Q, and the top widths of the designed section and the removed area are T and T f , respectively. The value oLT' may be computed by

T'

179

. DESIGN OF 'CHANNELSFOR UNIFORM: FLOW

UNIFORM .FLOW

It should be noted that the angle. of ~he

~o~inc function is expressed inradiansj it may be converted to degrees: by multLplymg

I

I

I

l [ I!

.

I

.

it by 180/11" or 57.3. The top width mllY be computed by Eq. (7-12) with if = 0, or cos 0.128::; = O. Thus, 0.128::; = 11"/2 and x = 12.3 Tli.e top width is T = 2::; = 24.6 ft. By Eq. (7-14), the mean velocity is V= (1.35 - 1.19 tau 33.5') 5.151' X 0.0016}'/ 0.025 = 2.69 fps. By Eq. (7-15), the water Mea is A = 2.04 X 5.16'/tan 33.5° = 82.2 fV. Hence the discharge is 220 cfs. Since the design discharge is 400 cfs, it is necessary to add a rectangular area at the middle (Fig. 7-12b). The width of the rectangle may be computed by Eq. (7-17) as

n.

. ,,_ ·0.025{400 - 220) _ T - 1.49 X5.l6~~ X 0.00W1 - 4.9 ft TherefCJre the top width is 24.6 + 4.9 = 29.5 ft. If the channel is 'designed to carry 100 cis, it isnecesso.ry to remove !l. vertical area ir.:im the middl~ (Fig. 7-12c). The top width of the removed a.rea may be computed by Eq. (7-1il). T' = 0.96(1 - V 1 0,%20)X 21.6 = 7.7 ft Therefore the top width is 24.6 - 7,7

=

lB.\) ft.

C. GRASSED CHANNELS

7-16. The' Grassed Channel. Presence of grass or vegetation in channels will result in considerable turbulence, which means loss of energy and retardance of flow, For earLh channels used for carryin~ water on farm lands, however, a lining of grass is often found to ndvantageous and desirable. The grass will stabilize the body of the channel, consolidate the soil mass of the bed} and check .the erosion on the channel su1'face and the movement of soil particles along the channel bottom. The U.S. Soil Conservation Service (39-411 has conducted a series of experiments on channels lined with various kinds of grass (Fig. 7-13) .. The . results thus obtained under different testing conditions and the procedure suggested for the design of grassed channels will be described in. the following articles. 7-17. The Retardance Coefficient. The Manning coefficient of roughness for grassed channels is specifically known as the relardance coefficunt. According to the investigation by the Soil Conservation Service, it was found that Manning's n for just one kind of grass varied over a wide range depending on the depth of flow and the shape and slope of the channel. Thus, the selection of a desJgn value for n would be nearly· impossible. Fortunately, it was discovered that the retardance coefficient n holds a.certain relationship with the product of the mean velocity or flow V and the hydraulic radius R. This relationship is charact~ristic of the vegetation and practically independent of channel slope .and shape. As a result, therefore, a number of experimental curves for n vers4S' VR (Fig. 7-14) were developed for five different degrees of retardance: very high, high, moderate} low, and very low. For very low retardance

be

\

-1

1

. . ---'i

180

UNIFORM FLOW DESIGN OF CHANNELS FOR UNIFORM FLOW

lSI

TAllLE 7-4. CLASSIFICATION OF DEGRE1!l OF RE'l'ARDANClll FOR VAIUOUS KINDS OF GRAS/;*

. Cover

Condit.ion

A Very high

Weeprng love grl\SS ...•.•....• Yellow bluestero ischa.emuln ...

Excellent stand, tllll (av 30 in.) Excellent sta.nd, trtll (av 36 in.)

B. High

Very den'sE. growth, unout :Kudzu ............. , ..... . Bermuda: gra.ss ......... , .... . Good sta.nd, taU (av 12 in.) Native grass mixtur~ (little bluestem, blue gramat a.nd othr:r lon.g .l!.nd short Midwest grtUlSes) . . . . . . . • .. . ., ... . Good stand, unmowed Weeping love gra.sS ........... . Good stand, ta.ll (av 24 in.) Lespedeza ~ericea., ........... . Good sta.nd, not woody, tall (av. 1 \J in.) Good st!tnd, uncut (av 11 in.) AHal£a ........ . Weeping l.)ve grass ...... ' ... . Good stll.nd. mowed (av 13 in.) Dense growth, unout Kudzu...... . .......•.. Illue granla ............•...... Good stand, uncut (av 13 in.)

Reta.rdo.nce

Moderr.te

C

\

.l

Good stand, uncut (6 to 8 in.) dense cover (av 6 in.) Good stand. headed (6 to 12 in.)

D

I.ow

E

Very low

Bermuda gras.s....... . ....... Good stand,. cut to 1.5in.heighj. Bermuda grass. . . . . . . . . . . . . .. Burned stubble

;.

!

Fair stand, uncut (10 to 48 in.) Good stand, mowed (~W 6 in.) Good sta.nd, uncut (!loV 11 in.)

Bermuda grass .........•...... Good stand, cut ~o 2.5 in. height Comm·:lo let;pedeza ......... . Excellent stand, uncut (av 4.5 in.) Buff.Jo grass.... . ........... . Goodstand, uncut (3 to 6 in.) Grass-legume wixture-fall, spring . (orchard grass, redtop, ltaliar, rye grass, and common le5pe·iez.ll.).. ..• . . . . . .. . . .... . Good stand, uncut (4.to .5 in:) Lespedeza. sericea, .... After cutting to 2 in. height., very good 'stand before clltling

Fm. 7-1~. CenLipede grassed channel. (Courtesy of W. O. Ree, U.S . .'igricultura! Research.Smn:ce.) .t A ) Before experiment; (B) after test a.t t1 flow equal to 15 cfs for 4~ mm; (C) dunng test at a flow equal to 30 cf.5; (D) at completion of the whole experiment.

only the average ctll've is shown, together with the curves for low retardance. The classification of degree' of l'etardance is based on the kind of vegetation and the condition of growth, as described in Table 7-4. The term" stand" used'in the tabie refers to the density of grass, or the count of vegetation, which is sometimes expressed as the 'number of stems per square foot. The n- VR curves thus developed may also be applied to other kinds of gmss, provided that their characteristics and degree of I'etardance can be identlfied. For this purpose, Table 7-5 is provided as a guide in the selection of the vegetal retu.rdance for diffetent conditions of stand and average length of the grass. .

Crab grass. . .. . .. .. .Bermuda grass ............... . Common Jespedezo. ... , ....... . Grthss-leg'.1me mixture_ummer (orah1lrd grass, redtop, Italian rye grass, and common le8pede:z.a) ..•....... , ....... . Centipede gr3.5s . .' ........... . Kentucky bluegrass .......... .

U.~.

Soil Conservation

[41J.

182

UNIFORill FLOW·

DESIGN OF CHANNELS FOR UNIFORM FLOW

183

.sl---l--t--........f.--+-++-+-!-l-f---+ ~1---4--4--.--+---+--~-~+++----r--+--~~~-+-+~~+---+--+---~

H

r:: .1 : .09 .OS .O~

.00

.O~

!

...

-.I

VR

VR

(0) Curves for A or very high vegetal retardance.

(e) CurYe.s for C or modera~e vegetal retardancB.

i

.\,

VR . (b) Curves for B or high vegetal reta.rdllnce. FIG.

7-14. Experimental n-VR curves.

(d) Curves for· D or low vegetal reta.rda.nce, a.nd a.n .e.vernge curve for E or very low vegetal reta.rdll.nce.

(U,S. Soil Conservation Service.)

l

'f I

i .1

I

~I

184

UNIFORM FLOW

TABLE

7-5.

GUIDE I,:! SELECTION OF VEGE'L'AL RE'l'ARDANCE*

Stand Avcrage length of grass, in.

Good

Foil' .

Degree of retarciance

>30 11-;-24 8-10 2-6 <2

A . Very high B High C Moderete D Low E Very low

>30 11-24 6-10 2-8 <2

B C

D D E

High Moderate Low Low Very low

* U.S. Soil Conserv!!.tion Service [411. 7-18. The Permissible.Velocity. The permissible velocity of Row in a gl'assed channel is the velocity that will prevent severe erosion in the channel for a reasonable length of time.' Permissible velocities for different veg.etal ?overs! ch~nnel slopes, and soil conditions, recommended on the baSIS of mvestlgatlOn by the Soil Conservation Service, are shown in Table 7-6.

, I

!

DESIGN OF

1

7-19. Sel~ction of Gra~s. The selection of grass for the channel lining depe.nds mawly on the clImate and soil in which the plant will grow and surv.I~~ under ,the given conditions. From the hydraulic viewpoint, stablhu y . and other f~ctors should also. be considered. In general, a higher dlsch:.l.l'ge requu'cs a stronger or better lining. On steep slopes ?unch grasses; such as alfalfa, lespedeza,and kudzu, will develop channel~ mg of the flow and" hence, are unsatisfactory for lining. For slopes greater than 5 %, only fine and uniformly distributed sod-forming grasses sueh as Bermuda grass,. Kentucky bluegrass, and smooth brome ar~ re~om~nended fOl' lining· where the main flow occurs. Because' of the obJec.tlOnable spreading nature of sod-forming grasses, the top portion of the. Sides and the berm may be planted with gras.:;es that do Ii.ot.spread easliy, such as weeping love gl'B,SS. 11'01' fust establishment of the lining, Bermudagl'assand weeping love, grass fLl'e recommended. Sometirne~ an~uals are used as temporary protectibn until permanent coveb by native gras~e~ are ~stablished. Silt deposition in channels may be con~rolled by Iml?g With bunch grasses, which will develop channeled flow lnCl'ease velOCity, and thus reduce silting.: , ' . 7-20. Proced?re of ~esign., After the !kind of grass for channel ]ining IS selected, the qegree of retardrmce can be determined from the condition of the stem length and the density of growth. During the period of

I !

CHANN~LS

~85

FOR UNIFORM FLOW

establishment, tht; grass will grow and the channel will be stabilized under a condition of low degree of retardance. The channel will not reach its maximilm capacity until' the. grass 'cover is fully q,eveloped and well established. Therefore, it is suggested that the hydrll:ulic design of a gras:;ed channel ,consist of two stages. The first stage (A) is to design the channel for stability, that is, to determine the channel dimensions under the condition, of a lower degree of retardance. The second stage , TABLE

7-0.

PERMISSIBLE VELOCITIES FOI! CHANNEI.S LINED WITH GRASS*

,

Slope IMlge,

Cover

%

----

Bermuda grass

0-5 5-10 >10

Buffalo g'rass, Kentucky bluegrass, smooth brome, blue grama

GrllSs mixture

,

I. I I

Permissible velocity, fps Erosion-resistant soils

Easily eroded' soils

8

6 5 4

7 8

0-5 5-10 >10

7 5

5 4 3

(}-5

5 4

3

5-10 II Do not use

6

all

4

slopes steeper ~han _10~%_ _ __

I

I

0-5 ,3.5 2.5. Lesp,edeza sericea, -w-e-ep-i-~~g-lo-v-e grass, ischaemuffi (yello.v blue- I Do not use on slopes steeper than 5 %, except lOr stem), kudzu, aHalf!\., crabgrass side slopes in a combination channel

I

------

Annuals---used on mild slopes or as~ --~-=5---1 3 ,5 2.5 temporary pr"tection until per- Use on slopes steeper than 5 % is not reCOffimanent CQyel'S al'e established, mended common lespedeza, Sudan grass

I

REMARKS. The values apply to average, uniform stands of each type of covel'. lIse velocities exceeding 5 fps onlr where good covers and proper maintenll.nce cll,n be obtairied.' , . * If.s. Soil Conse.rv!l.tion Service 1411.

(B) is to review the design for maximum capacity, that is', to determine the increase in depth of flow necessary to maintain a maximum capacity under the conditiun of' a higher degree of retardance. For instance, if. common lespedeza is selected as the grass for lining, the comn-lOn lespedeza of low vegetal ret!l.rdar\.ce (greell~ average length 4.5 in.) is used for the first stage in design. Then, in the second stag~, the cummon lespedeza of moderate vegetal retal'dance' (green, uncut,: average length 11 in.) shou1.d be used. Finally, a proper freeboard is added to the computed'

186

I

UNIFORM

depth of the channel, which will further increase the maximum design capacity. The design procedure I is described as follows: A. Design Jor Stabilt·ty. Given the discharge, channel slope, and kind of grass, the Ilrst stage of design may proceed in the following stiOlPs: 1. Assume a value. of n, and determine the corresponding values of VR from the n- VR curve (Fig. 7-14). 2. Seler:t the permissible velocity from Table .7-6, and c()mpute the value of R. 3. Using the Manning formula, compute the value of L49R%SH

n

and check this value against the value of V R obtained in step 1. 4. Make other trials until the computed value of V R is equal to the IT R value obtained from the n- V R curve. 5. Compute the water area, or A Q/Y. 6. Since the correct values of.A and R have been obtained, the section dimensions may be determined the procedure described in Art. 7-7, The sections generally used for grassed channels are the trapezoid,' po,rabola, and triangle, named in order of increasing depth required in excavation. Owing to the norl11al action of channel deposition and erosion, trapezoidal and tLiallgular sections, if selected but not well maintained, will generally become para.bolic after a long period of service. Example 7-6. Determine the section of a coannellined with grltSS miX~ure, laid on erosion-resistant soil at a slope of 0,04, and carrying a discharge 0; 50 ds. Solution. In designing for stability, the grass mixture that offers a low vegetal retardance; j,e., that of the dormaut ~eaaon, is considered. Therefo~e, the corre~pond­ ·jng n- 'V.n curve should be used in the c.omputation. . From TablCl 7-5, the permissible velocity for design is taken as 5 fps. Using the n~l'R curve (Fig 7-14) for grn.s8 mixtute for fall, winter, and spring, the trial computations involved in tho design procedure !lore ¥ follows:

Trial no.

R

1.49R~5SH

Side slope z:1

Bottom width b, ft

Depth

17.0 12.5 0

0.53 0.62 1.00

Top width T,ft

!I, it

....

3;1 6:1 Triangle ........... Para.bola ..........

10: 1.

1

187

DESIGN OF CHANNELS FOR UNIFORl!vl FLOW

1

---~-

20.18 19 94 20.00 2().00

o.n

B. Design j,'}r lI1a:rim~tm Capacity. The second sta,ge in design. is to determine the additional dept.h necessary to sustam the maXImum capacity of (1, fully developed lining. The procedure is as follows:. . 1. Assume Lne depth y, and compute the water area A and the hydraulIc radius R. . 2. Comput.(J the velocity Y by Y = Q/ A and the value of V R. 3. From the n-V R curve of a higher degree of retardance for the selected lining, detcrmiac the value of n. 4. Compute the velocity by the Manning formula, and check this value of V against the value obtai!led in step 2. 5. Make trial computations unt.il the computed V in step 4 is equal to the computed IT in step 2. ·It should be note~ that this velocity is a.lw.''\ys les5 than the permissible velocH,y assumed 111 the first stage of deSign, that is, in design for stability, since the cross section has heen enlarged in the second of the design. 6. Add proper freeboard to the computed depth. Example 7-7. Modify. the selected sections in Example. 7~6 for maxi,mum ca,pacity. Soh,tion. For determining the proper depth for maXlnlum. capacIty of t.he proposed sections, the grass mixture of the growing. season, whIch offers a X?oderate vegetalreta.rdallce, is considered in the computatlOn. TnI' n-VR curve (FIg. 7-14) for grass mixhure in summer is therefore used. Other known da.ta are Q = 50 cfs and S 0.04.. . For the tl'apezoidal section with 3:1 side slope and b = 17.0 ft, the trial computations are given below:

11.

'Trial 1

2

' L35 0.34

3

4.72

4

2,50

The correct va-lues for the determination of sections o.re.R = 0.50 It and A = 3% = 10 ft'. Several cha.nnel sections meeting these requirements are proposed as follows (using charts in App~ridb:: D ) : ' , I

For an eXl).mple .of the pra.ctical

of a. grassed channel, see £42J.

noQ.

!I'

A

R

V

l'R

n

I

V

1.491mSH 11.

-

1

2 3

0.70 0.60 0.65

13,4 11.3 l2.3.

0.G3 0.54 0.58

3.73 4,42 4.07

2.35 2.39 2'.36

0.051 0.050 0.051

1

4.2.2

3,96 4.07

1 1

The correct depth is 0.55 ft. Adding a freeboa.rd of 0.2 tt,.the total depth is 0.85.ft. . For the trapezoidal ::;ection with a side slope of 6: 1 and b 12.5 ft, t~12 computat~on results in a total depth of 0.94 ft. Simila.rly, the tot!).l depth of the trlangula.r sectIOn with a side slope of 10: 1 is found to be 1.33 ft,.

.1 ..;....

-!~

" •

188 For the parn.bolic :3ection of y as follows: Trial no. ---

y

0.80 0.90 0.85

I

2 3 ·1

DESIGN OF CHANNELS FOR UNIFORM FLOW

UNIFORM FLOW

I

0.85

I I

=- 0.73 ft and T = 20 ft, the trial computn.tions are

T

A

21.0 22.2 21.6 21. 7

11.2 13.3 12.2 12.4

I

I

R

V

0.52 0.60 0.57 0.58

4.406 3.76 4.10 4.03

I- - -I- - VR

n

2.32 2.25 2.34 2.34

0.051 0.051 0.051 0.051

I

I

1.49m1S~~

n

3.78 4.15 4,03 4.07

It should be noted that the computation for the parabolic section is simplifi
, i

7 -1. Show t~at the most efficient rectangular or triangular section is one-half of a square. . 7-2. Explain (a) that any section formed by a p'olygon which can be inscribed by !\ semicircle with the ccnter in the water surface will have its hydro,ulic radius equal to one-half the radins of th~ inscribed circle, and (b) that such section will have the best hydl'll.ulic effiuiency. . 7 -3: Detenuine the best hydraulic section of the channel in Example 7-2 if the section is (n) rectangular, (b) triangular, (e) cirl!ular, (d) parabolic, and (e) in the form of It hydraulic catenary. . . 7-4. Solve Example 7-2 by the empirical rule of Eqs. (7-2) and (7-3). 7 -6. Design a nonerodible channel carrying 200 cfs with n = 0.020 and S = 0,0020. Use your own judgment and assumptions. 7-6. Based on the p.ractice of t1u~ U.S. Bureau or'Reclamation, det.ermine (0) the freeboard of the channel designed in Example 'r-2 when the channel is unlined and (b) the heights of the lining and benk if the channel is lined. ' 7-7. Solve Example 7-3 if the material forming the channel body is fine silt having an average particle size of 0.006 mm. Estimate the permissible velo,~ity with the aid of (a) Fortier and Scobey's table, (b) U.S.S.R. data, and (e) the Kennedy formul~. modified for clear ,vater. . . 7-8. SOlVE Example 7-3 if the mn.terial forming the channel body is fairly compact heavy clayey soil with a voids ratio of l.0. 7 -9. Solve Example 7-3 if the channel has !l. pa.rabolic section. 7 -10. Design the section of a canal to carry a discharge of 200 cfs throllgh a land of erodible soils with n = 0.020 and S = 0.0020. Assume other necessary data and use your o~vn jlldgment. . . 7-lt. 111e All-American Canal is desigr:ed to divert 15,155 cfs of desilted water from the Colorado River to irrigate the Imperial Valley in southern Califpmia. This canal is 80 millls long. . The typical maximum section has a bottom width of 160 ft, width at wo.t¢r surface of 232 it, water depth of 20.6 ft, minimum freeboard of 6 ft and bank width of 27 to 30 ft. The terminal capacity is 2,600 cfs. The canal was 'excavated

189

mostly in n.Uuvi!ll soil, ra~ging from light sand~ or silty [oams to adobe and having an average particle size of 0.002b in. Review the hydraulic design of the channel section. 7-12. Review the stability of the section dimensions obtaincd in Example 7-3 by the method of trn.ctive force, !l.Ssuming that 25 % of the material forming the channel bed is L2b in. or over in diameter. . 7-13. Solve Example 7-4 for the following conditions, respectively: (a) If the side slopes are. assumed as 1 on 1.5. . (b) If the mateJin.1 forming the channel bed contains fine noncohesive particles, 50% of which are larger than 1 mmin diameter. The water is clear. (e) If the m~.terial forming the channel bed is cohesive compact clay, having !L voids ratio eqllal to 0.5 .. (d) Ii the channel is moderately sinuous. 7-14. The conversion from the maximum permissible velocity to permissible tractive force (Table 7-3) is based on 0. flow depth 0'£ 3 ft _and an avernge channe.l section ha.ving a bottom width of 10 it and side slopes of L5: 1. For a.llllvial noncolloidal silts and. clear-water floW, the m!lximum permissible velocity recommended by Fortier and Scobey is 2.00 fps r.nd the n value is taken as 0.020. Compute the corresponding permissible tractive foree. 7-15. Compute the maximum tractive force per unit area on the section of the AllAmerican Canal described in Prob. 7-11. . 7-16. Determine the cross ser.I,ion and discharge of the st~ble hydraulic section of 0 .channel excavated in a' noncohesive IDr.terial ho.ving To = 0.1 lblft', S = 0.00040, o "" .31", and n = 0.020. . 7-17. Determine the modffied profile for the channel section obtained in the preceding problem if the channel is tOCllUY (a) 75 cfs ILnd (b) 300 efg. 7-18. Design a waterway lined with Bermudn. grass on- erosion-resistant soil and cn.rryillg a discharge of 200 cfs .. The n.vernge slope of the channel is 3 %. Use the curve for modemte vegetn.l rel,ardance. 7 -19. Determine the total depth for maximnm capacity of the channel section propo,,,ed for the preceding problem. Allow 11 fre,eboard equal to 20% of the computed depth. .

REFERENCES 1. Victor L. Streetel': Economical conal cross section~, Transactions, American Society of Civil Engineers, vol.. 110, pp. 421-430, 1045. 2. I van E. Houle" Irrigation Engineering," vol. 2, "Projects, Condllits, und Structures," John Wiley & Sons, Ill~., New York, 1956. 3. ",Linings for lrrigrttion Canals," U.S. Bureau of Reclamn.tion, Jl1ly, Hl52. 4. Cll-nals !lnd related structures, U.S. Bureau of Reclamation, Design and Constnlclion Manila!, Design Supplement No.3, Apr. 17, 1952; vol. X, 'pt, 2,. ghop. I, paragraphs 1.15, 1.8, and 1.18. . . 5. B. A. Etcheverry: "Irrigation Practice and Engineering." vol. II, "Conveyance of Water," iVIcGmw-HiIl Book Company,Jnc., New York, 1st ed., 1915, p. 122. 6. G\lilford L. Molesworth: "Pocket-book of Engineering Formulae (Useful Formulae and Memoranda) for Civil·and Mechanical Engineers," E. & F. N.Spon, London, 7th ed., 1871, p. 175. ' 7. Isidro D. Carino: A graphical solution for flow in earth channels, paper 1360, Proceedings, American Society of Civil Engineers, J oumal, Irrigation and Dra.ina.ge Division, voL 83, no. IR2, pp. 1-9, September, 1957. .

190

UNIFORM FLOW

DESIGN OF CHANNELS FOR UNIFORM FLOW

8. 'Pete W. Terrell [LIld Whitney M. Borland: Design of stable canals and channels in erodible ma.terial, Transactions, AnLerican Society of Civa Engineers, vo!' 123, pp. lOl-U5, 1D£i8. . 9. E. W. Lane: Stable channals in erodible material, Tmnsact·irms, American Society of Cilia Engineel's, vol. lO2, pp. 123-142, 1937; and discussions. 10. Robert G. Kennedy: The prevention oI silting in irrigation canals, Proceedings,. Inslit:l.tiolL c,f Civil Engineers, London, vol. 119, pp. 281-290, 1895. 11. Edward S. Lindley: Regime channels, Minutes of Proce,ed£ngs, Pun}ao Engtneering Congress, Lah01'e, India, vol. 7, pp. 63-74, 1919. 12. Gerald Lacey: Stable channels in alluvium, Proceedings, Inst·itution of Civil Engineers, London, vol. 229; pp. 2.'i9-384, HJ30.' 13. Gerald L"c"y: Regime flow in incoherent alluvium, Cenl,.al Board of h1'1:galion, . PlLblication No. 20, Simla, Iudia, 1940. 14. Gerald Lacey: A general theory of flow in alluvium, Journal, Instit'ution of Civil Engineers, Londan, vol. 27, pp. 16-47, 1946. '. 15. Thoinas Blench: "Bydrs-ulies of 'Sediment-bearing Cano.ls and Eivers," Evan~ 'Industrie$, Ltd., Vancl)uver, B.C., Canada, 1951. 16. Thomas Blench: Regime theory for oclf-formeci sediment bearing'channels, Tmnsactions, American Bociety oj Ciliil En.gineers, vol. 117, pp. 38.1-400, Hi52. 17. H. A. Einstein: The bed-load fUllction for sediment tran~Jlorta~ion in open channet flows, U.S. Depa"bILent of Agriculture, Technical BuUelin No. 1025, 1950. 18. L. B. Leopold and Thomas Maddock, Jr.: The hy.draulic g~ometry of stream cha.nnels and some physiographic implictttions, U.B:G'eological BUTVP,Y, Professional Paper 252, 1953. 19. ,K W. Lane: Sta.ble channels in erodilJle material, Tra1tSdctio,is, American Society of Civil Engineers, vol. 102,pp. 123~142, 1937. .' 20. Serge Leliavsky: "An Introductiqn to F1uvial Hydraulics," Constable & Co., Ltd., London, 1955.' 21. Instit\ltion Research Committee: Recent developments hydraulics, Proceedings, In.stilution of Civil Enginp,cr.s, London, pt. III, vol. 4, pp. 990-1049, December, 1955. 22. Ning Chien :'The present status of research on sediment transport., Tl'anoactions, A lILerican Society of Civil Engineers, va!. 121, pp. 833-868, 1956. 23. Ning Chien: Graphic design of olluvial channels, Transactions, Amer':can Society of Cilia Enuineers, vol. 121, pp. 1267-1280, 1956. ' 24~ Ning Chien: A concept of the regime theory, Transactions, American Society of Civil Engineers, vol. 122, pp. 785-793, 1957. 25. Eno~ J. Carlson and Carl R. Miller; Research needs'in sediment hydraulics, paper 953, Procced£ngs, American Society of Civil Engineers, J oumal, Hydraulics D£visio1];, vo!' 82, no. 11 Y2, pp. 1-33, April, 1956. 26. D. A. Etcheverry: "Irrigation Practice and Engineering," va!. II, McGraw-Hill Book Company, Inc., New York, 1915. 27. S. Fortier and Jf. C. Scobey: Permissible cana.l ;relocities, Tran8~tions, AlILerican Society of Cillil Engineers, vol. 89, pp. 940-956, 1926. , 28. The maximum' permissible ,mean 'velocity in open channels, Gidrotekhnichesko·ie Stroitel'stvo, (Hydrotechnical Constmction),I\{oscow, no. 5, pp.5-f, May, 1936. 29. Emory W. Lane: Design of stable channels, Transactions, Am.erican Society of Civil' Engineers, vol. '120, pp. 1234-1260, 1955. . 30. P. du Boys: Etudes du regime du B,h6ne et l'action exercee par les eaux su~ un lit ~ fond de graviers indetiniment affouillable '(The Rhone and streams with movable •bedsj, Annales rles ponts et Cha1L8S~eS, ser. 5, vo!. 18, pp. 141-195, 1879.

31. J. B. Leighly; Toward a thaory of the morphologic significance of turbulence in the flow of water in stream, Uni~ersily of Ca.17/orlliCL, Publications in GelJgraphy, vol. 6, no. 1, pp. 1-22, Berkeley, 1 9 3 2 . , . 32. O. J. Olsen and Q. L. Florey (compilers): Sedimentation studies in open channels: Bo.und~y shear and velocity distribution by membrane 'ano,logy, ana.lytical and fil11te-dlfference methods, reviewed by D. McHenry and It. E. Glovel' U.S. Bu,rea.u of Reci.rultation, LabOl·atory. RepOl·t, No. Sp-34, Aug. 5, 1952. ' 33. Ph.ili~p Forchhei~eT: "Hydraulik" ("Hydraulics "), Teubner Verlag:sgesellschaft, Lelp,ag and Berlin, 1st ed., 1924, p. 495; 3d ed., 1930, p. 551. 34. Chi~-Hw~ Fan:.A study.oI stD,blc channel cross section (in Chinese), Hydraulic En(]~neer.~1!!l, Chtne.se SoCtely of Hydraulic Enginse'rs vol. 15 no 1 pp 71-79 Nanking, 1947. ' ",. " 35. E. W. Lane': Progrl:'ss report on results of studies on design of stable channels U.S. Bureau of [leclamation, Hyd'·(L!J.lic Laboraiot']J Repol·t, No. lIyrl-352 , J\1l1e' 1952. ' ,

in

191

36. A. C. ~~rter: Critica~ tractive fones on channel side ~lo]Jes, U.S. Bgrea.l! of Rcclmrt(lMTc, Hydrav.hc LaboratoJ'Y Report No. Hyd-3G6 (supersedes Hyd-295) Feb. 18, 1953. '. . . , 37. E. W. Lane ~tl1d E. J. CarIllOn: Some factors affecting the stn.bility of canals coustructed in coarse granular materials, Pr'oceedings of the Minnesota [nlernalional IIydmulics Convention, Sept. 1-4, 1953, Joint Meeling of Inte1'1wtional As.sociation for H ydt·l}.ulic Research and HydmldiC5 Divi.sion, American Bociety of Civil Eng,;rwers, pp. 37-48, August, 1953. . 38. R. E. Gloyer and Q. L. Florey: Sta.ble channel profiles, U.S. But'eau of Re,clama,ti01l, HydTflHl'ic Laboratory Report No. Hyd-325, Sept. 27, 1951. The work was started by R. G. Conard and revie.wed by E. \Y, Lane. 39. W ..0. Ree: Hy~raul.ic characteristics of veget.ation for vegetated waterways, Agnculttlral .Enqmeenng, vol. 30, 110. 4, pp. 184-187 and 189, April, 1949. ' 40. ~'. O. Ree and V. J. Palmer: Flow of water ill channels protected by vegetat.ive lmmg, U.S. Boil Co,~seTVation. Service, Technical Bulletin No. 9lH February 1949. 41. Stillwater Outaool' Hydraulic Laboratory: Handbook of chandel desig;ri' for soil an~ water conservation, U.S. Soil Conservation Service, SCS-TP-Gl, Marcll., 1947; reVised, ,June, 1954. . . 42. V. D. Fredcnhagen and E. H. Doll: Gmssed waterways, Agricullural Enuineming, vo!. 35, no. 6, pp. 417-419, June, 1954. .

.,\

.j

I ·1

-\ 1, (

·1

I

'l'HEORET1CAL CONCEPTS RELATED' TO 'UNIFORM FLOW

. CHAPTER I

lI

8

THEORETICAL CONCEPTS OF BOUNDARY LAYER, SURFACE ROUGHNESS, VELOCITY DISTRIBUTION, AND INSTABILITY OF UNIFORM FLOW

This chapter presents assorted theoretical concepts which have been developed in the mechanics of open-channel flow. These concepts, though not to be thoroughly discussed, may shed some light upon the solution of many practical problems. 8-1. The Boundary Layer. When watel' enters a channel, the velocity' distributionncross the channel section, owing to the presence of boundary

193

bution; (3) the depth of Row is indefmitely 1111'ge,so the depth of ftow cun be considered constant as the water enters the channeL 1n th!'i,~ha!!neJl the effect on the velocity di!J.trib!Jti®~t9. boun.1.::I£Y-XQ.hlghnes~ is indicatedbythe- ii~~" IlHf,- Outside the surface represented by ABC, --7'- th~ ;~I~city7 d'i;tribution is practically uniform. Near the channel surface and within the region ABC, v~locjty"'y£l.ries l\.ccording_tQ_ 4isww.e iromthe channel surface. . The region inside ABC, though not distinc'ti;;;,i;k~~~~ boundary layer 1 and its thickness is deSignated by o. Since the bouhdary layer is not distinctive, jiB. thi A common definition is that the lrei&-C,;11 _ ' ._~ thickne;:;s 6 is the magnitude of the ':((5 w/,bu l( ~n normal distance from the boundary ;ll~ft1,ceat ,~hi()h th;Vclocity ;; is ' equitnQ99%Of theli:rrd~~: -n:y Vo, ''I'hTcli.the velocity:'distribu1ion-:-b~ur:ve1n the boundary Tayer

1

approaches

asy!llptoticall~(Fig.

. g:<~---~-~ ....--_~

The effect of the bounda.ry layer on the flow is equivale!lt to a fictitious upward displacement of the channel bottom toa virtual position by an amOl1!l,t equal to the so-called displllcementthiclmess 0* (Fig. 8-2), which is defined as '

The e,1lranCE condition is hypolhelicol

\

r-

~

FlO. ~L Development of the boundary layer in an open chl1nnel with an idea.l entrance condltlOn. . .

i

i

roughness, will vary with the distance over wllich the water t!'avels in the;. channel. (Fig. 8-1).1 If· the flow is uniform and stable and if' the channel is prismatic and of constant roughness, the veiocity rustdbution will eventually reach a definite pnttern. For simplicity of discussion the following m'e assumed; (1) th~ fiow enteX'ing the channel is lamil1al' and of uniform velocity distribution; (2) no restriction exists at the entrance that will cause abrupt disturbanc~ of the water surface and the velocity distri. ,1 For· the sake of simplicity, the two-dimensiona.l profile of:a. wide open channel With exagger~ted verticll.l sc.ale isahown.

192

r",-

. '

(8-1)

"-

FlO. 8-2 .. Distribution of velocitv over a smooth channel Sur!l1c9{not in sC!lle). / '

where l' ill the velocity at any elis" / £-,.. . .. _--"-" tance'li from the channel surfnce ,!!-nAvl-is the 'yJ;!lGtci~t~Jit~!toftE.E!. _~9.1!.~l.g3lry.l~r.. The value of t.he displacement tl1ickness generally varies . from one-eighth to. one-tenth of the thickness of the boundary layer, depending on the magnitude of the Reynolds number. At the beginning of the now in the channel (Fig. 8-1) the ftow iii entirely laminar and a laminar boundary layer is developed along the channel surface, .as shown by the curve AB.· The velocity distribution in th~ layer is. approximately .parabolic. As. the water travels farther along in the channel; the ftow in the boundary byer will eventtmlly change to turbulent. (rhe point where the challge tn,kes place is indicated by B. Downstre~m from B a tl1rb1dent boundary layer is developed, 'as shown I

For a. compl'ehensive treatment of this shbject see !ll to [4].

; . I-I/:. .,. .If {J. ,(.r C> -=- vI(.')j"~oJ.fCL 1'r"G<:.-' .tId .;)' l.tfol V{~U_1. .d~ ~/..

_'

~-"

.t:d _ . i

/'/-t . . . ~7;1'.,A;..

<¥ /i

194

THEORETICAL CONCEPTS RELATJi!D TO UNIFORM FLOW·

by BC. The velQcity distribution in this layer can be shown analytically to be approximately log:1l'ithmic (Art. 8-4). If the channel suds.ce is relatively smooth, the velocity close to the channel surface is low; thus, a very thin stable film offiow known as the la:minar sublayer will be develope-d on the surface. Within the laminar sublayer the flow is kept laminar.l The top surface of th~ laminar sub~ ___ '-_~ layer corresponds to the transitional zone of flow from laminar to t.urbu-' lent (Art. 1-3) and, hence, cannot be precisely defined.

195

irregularities forming the roughness elements is called the 1'oilgl!1i68.s height! k. The ratio kl R of the roughlless height to the hydraulic radius known as tha relative roughne.ss. --_~""''''' If the roughness height is less t,han a certain fraction of the thickness j( -< So of the laminar sublayer, the s~rfa.ce irregularities will be so small that all ronghness elements will be entIrely submerged in the laminar sublayer (Fig. 8-3a).' Under this condition the roughness has no effect upon the fio'w ou.tside the lam.inar sublayer, and the surface is said to be hydrau_._.;;;> liGally sn;ooth. A hydraulically smooth surface is said to be wuu!LiUh.e aVer~iW.8Jl.....l1UrY:e (Fig. 8-3b). ---. . In connection with flow in pipes or on flat plates at zero incidence, Schlichting (see p. 454 of [ID . the following condition for a surface to be hydraulically smooth:

is

V,k

<5

or

k

<

51'

1

(8-2)

)I

where V f a term known as the friction lJe/ocitu (Art. 8-4). Using the Gh6zy formula. it can be shQ\rn~ above condition that, for a surface to be hydraulically smooth, the roughness height must be less than a critical rotlghne.ss expressed by

k.

'~-----Y-

5C ;

(8-3)

where C is CheZ~I's C, I' is th.e kinem(\tic viscosity, and l' is the mean velocity. The above conditionis·giveri for r0l1ghi1ess obtained with sand . having \'alue;s of C greater than 100, prob"bi,r. For the avcri1ge COlldi~ion, Schlichtillggives kc = 100"/1', which corresponds to C '" 113.5. ail approximD.tion, (8-3) U1:;ty be applied to channels. If the roughness height is gl"eater tha~l th'l critical va111e..d~y EJi: . (8-3) (Fig. 8-3c), the roughness elemeuts will have sufficient magnitude ~ angu!a.ritY..to extend...tht'lis~ffects beJ:::9l!.c!. the !l).mil~[1,r subl~r..i1ncl. thus to disturb the flow in the chi1nne1. The surface is therefore said to bC"ro1Lg/i:--in rough ch~i;,nJS, ~ the-~eI;;~ity distribution will depend 011 the form and ~ize of the roughness projections, .!1l1d a stable laminar subli1yer can no longer be formed.

If the conditions for ul1iformflow exist throughout the channel, the turbulen t boundary layer will be fully developed at section CD; thereafte! the velocity distribution w~e patt.er.!!,:. In a laporutory ·cha.nnel, the laminar boundary layer AB _gan be elh?inated easily by placing a..l."ou~hness el_e!l~e!l~ el1t.!~lce: Thus, the turbulent bounC!=ary layer will be developed very beginning of the channel, an~ the totallen&h of the :/lone for the full development of boundary la~l' c!tllbe slloi'·tel1~~ SinGe the flow in ordinary chanl1els is usually turbui~nt:the f9UO;:;;g articles will deal only with the turbulent boundary layer. . 8-2. Concept of Surface Rough~ The concept of the existence ofa laminar sublayer in the turbulent· boundary layer offers a picturesque explanation of the behavior of surface roughness. When the surface profile of a channel is enlarged (Fig. 8-3), it can be seen that the surface is composed of irregular peaks ~l1d valleys. The ejJectiue height of the i A refined concept of the Ia.minar subla.yer will consider that there exi~ts in the s\lblayer a small a.mount of eddy which dec~eases very ra.pidly to zero· at the boundary surfll.ce. .

=

,

, i

lI

I It should be noted that the roughness height is mel'eLy a measure of the linear dimension of the rQughness elcmeuts but is not necessarily eqoaL to t.he ac~ual, or ev~n an average, height. For example. two roughness elements ll13.Y have diffctcllt linear dirnr>,nsions, but, owing to the difference in shape and orientation, they may produce identical roughness effect and, thlls, their roughncsscs will be clesignated by the same roughness height. 2 The position from which the rOllghness hcight should he measured is IL disputable mli\tter. It is a.'
I

~I 196

UNIFORM FLOW

r->'I'h~

average roughness height for a given surface can be determined by expenment. Table g··1 gives averaged from manv The con~ept, of l"~ughness in conduits was further advanced by' Morris [5]. Morns assumed that the flow over. a rough surface is due largely tl) roughness element. The o.f

I,

I ;

I I

.i

THEORETICAL CONCEPTS RELATED TO ·UNIFORM. FLOW

i97

of abnormal turbulence. 1 °In 'such a flow,therefore, the ratio yl).will be an importn.nt correlating parameter.. . ~si-~m.ooth ~o~..2.ccurs when_ the roughness elements. are.~.Q.~... together tb.!--'l!i..~' ski.ms the _cre5t..§.2l..the_.e!e~nt£, . The grooves between the elements will be filled with dead water contall1mg

!i

!.~etermiM,

and TA1l!.E

8-1.

ApPROXIMATE VALUES

OF

ROUG.lNESS HE1(lHT

. Mc.lerial

B:ra.ss, copper, lead, glass ... ". , , , ,. Wrought iron, steel. ...... ' ....... Asphalted east iron, . . . . . . .. . Galvanized iron ... ,., ........ '. Cast iron ............ , .... " ...•.. Wood stave ....... , ............. Cement. . . . . . . . . . . . . . . . . . . . . . . . .. Concrete., ........... , ...... , .... Dra.in tile .......... ,.............. Riveted steel. ..... , .............. Natural river bed. . . . . . . . . .. . . . . ..

k,

It

it

0.0001:"'0,0030 0.0002-0.0080 0.0004-0.0070 0.0005-0.0150 0.0008-0.0180 0.0006-0.0030 0.0013-0.0040 0.OU15-Q.0l00 0.0020-0.0100 0.0030-0.0300 0.1000-3.0000

tudinal ~f the r~~ss el~.ments·is the roughness dimension of paramount Importance in rough-conduit flow. Under this concept, flow ~~vel' IOtlgh surfa.ces can be classified into three basic types (Fig. : I 'L~olC!!:e.?::..~:01J.2hJ1.es.s flow, wake-interference flow, and quasi-smooth (01'

(0)

~

I

.-'--;:;,

. , 1" .

slcimmt'ng) flow.

---------

'

~ solated-roughne~~Jow.

prevails when the roughness elements are so far apart that the wake and VOl'tex a.t each element are completely developed and dissipated before the next element is reached. The apparent roughness would thereforere8ult from the form on the roughness elements, represented primarily by the height of projection k of the element, in addition to the friction drag 011 the waH surface between elements, ,which depends on the spacing of the elements. In such a flow, the. ratIO kl}.. may be taken as a significant correlating parameter influencmg the apparent friction factor in the flow. . Walee-interference flow :esults when the roughness elements are placed ,s~ clo~together tha.t the ~nd .YQrJj)x at each element wiUjnterfere WIth those developed !l:L~tJ resulting in intense and c.!?~pl~x."yor~i~ and tu~·bulen~g. hi such ~ flow, the height of the e~em~nt is. rehlthr ely unimportant, but the spacing is obviously of major ImpOl·tance. Also, the average depth y of flow above the crests of the elements will control in part. the vertical ex tent of the surface region

( c1

FiG. 8-4. Sket.ches showing concept of three basic typ;;s of How: (a) iso~ lll.ted-roughness. !low; (b) walJ:e-interf~rence flow; (c) quasi-slnooth tlcw.

_~ stable eddies, creating a£:?~1}. Large projections Illoe ~tisent from this pseudo wall, and the surfuce £!'cts hydrauli~l\.ll-y smooth. In such a flow, the ratio k/A (or .ill., where j is the groove width) will again be a significant parameter. Quasi-smo.Q.thJiy.:.:Y.....ba~ factor than flow ovel' a true smootll surfa£ft because the eddies in the groove~me a certain. am~f~gy. The nbove concept ap'pears to be substantiated adequately by experi1 Morris llsed the pipe radius instea.d of the depth in defining the parameter because he was ',;oncerned primarily· with pipes instead of channels;

198

UNIFORM FLOW

THEORETICAL CONCEPTS RELATED TO UNIFORM

mental 'data from many· different sources. The· concept can also be extended to surfac€s of variable roughness by u5ing average values of the roughness dimensions that vary or by combining the friction factors for· each flow type to give all over-all apparent friction factor of the flow. 8··3. Computation of Boundary Layer. For the development of a turbulent bound~ry layer in wide channels, ·an-;:pprQximate but practical meth~2!L h~_~.E!:Q12Q~_Qy_Jl~lJ&t.f.§1, This method waS' developed primarily for flo\v i.,1l; ~.han~els.oI large ~l'?.P.e, but it has been found applicable also to channels of small slope; provided the flow is . -.::- -.... ,-...... -- .--:'

. computing the varied-flow surface is used. In the case of tIle boundarylayer development of a uniform flow, the application of Bau61:'s method is just the same except that the water surface requires 110 computation, for it is simply parallel to the ehanneLbottom. Example 8-1. Aooncret.e overflow spillway of indefinite length has a. surface slope angle 11 = 53°8' (Fig. 8-5) and a roughness snch that k 0.005 ft. When the discharge is 3130 cfs per foot of spillway width, compute the length for boundary-layer dev!:')opment, the profile of the boundary layer, and the water surface, Solution.. The computation is shown in Table 8-2; the headings are explained as follows:

~

"

.j

now

\

,j

Col. 1. Arbitrarily assigned length of x in ft, !DeMured from 0 Col. 2.. Values of xlk, where k 0.005 ft Col. 3. V",lues of lJ/:z, computed by Eq. (8-4) Col. 4. Values of ~ in HI obtained by multiplying x by 8/x CoL 5. Velocity head vo'/2g in ft, equal to x sin·e = 0.80x Col. G. Velocity in fps corresponding to the velocity head in the preceding column CoL 7. Potentia.l thickness of Row in it, equal to the dischal'f!;e 1160 cis/it divided by the yelocity fl. . CoL 8. Actual thickness of Lhe flow in ft, equal to the potent.ia.l thickness plus the displacem<:lntthickness, which is assumed w be 10 % of the thickness 0 of the boundary la.ye!'

Nomin.o\ limit of boundory'loye.r

TA.BLE

k

Dam

8-2.

= 0.005 it

x/k

BOtiNDARY-LAYER COMPUTATION (J

53°8'

q

=

u. i /2g

li/x

360

cfs/n v.

glv,

y

I

( 4)

F" •.

50

Fm. 8-5. Growth of bounda.ry layer over the surfac<:l of an overflow Rpillway.

either accelerating or. uniform, and, if accelerating,

~~M t;-~l1!use boulldary~layer S~~~llS. -

100 200 400

th.~l£~~~idlr

600

.

The st,udy of boundary-layer de.velopment by Bauer was made on concrete overflow spillways (Ii'ig. 8-5). In this case the transition from the la~illar to the turbulent boundary layer usually occurs far upstream from the zone under consideration; hence the laminar boundary layer can be ignored, fo'r it is all insignificant part of the .problem. From the results of Bauer's investigation, the following equation may be written:' l) _

1 X 10; 2 X 10·

4 X 104 8 X 10' L2 X (05

0.0073 0.001313 0.0060 0.0055 0.0052

0.36

40

50.S

0.136 1.20 2.20 3.12

SO

71.8

7.09 5.00

7.13 5.07

160

101.5

3.~

3.65

320,

113.5 175.8

250

2.72 2.35

480

2.04

The computed water surfa.ce a.nd t,he bounda.ry layer are plotted on logarithmic paper (Fig. 8-6). The point of intersection of the two profiles indica.tes the location of the section where the bOlmdary layel: reache.s its maximum, or where it is fully developed. The lengtb of the development is shown to be approximately 460 ft. It shOUld be noted that the simplified method of 'computing the water-surface profile of a varied flow is justified in this problem because the flow is relatively thin and the slope is very steep.. As the frictional loss in this case is :negligibly small, the potential energy of water is practics.lly entirely converted into the lunetic energy expressed by the velocity head. For the computation of surfl:\Ce proj\Ies of varied flow in general, various methods will be described in Part Ill. To illustrate the effect of change in roughness upon the growth of a boundary layer, the computation has been repea.ted using ~he8amc da.ta but with a rOl\gbness twice as large, The line marked with It 11 = 8, k = 0.01 itO (Fig. 8-6) is the result of this computation. The change in water surface ispractic.!Llly negligible. It can be seen tha.t the difference between the values of the development lengtll for the two cases is

0.024.

x - (x/k)o.13 where 8 is the thickness of the turbulent boundary layer at distance x from 0 in the diiectionof flow (Fig, 8-5) and where k is the roughness height, Bauer has shown the application of this method by the 'following ·example. It should be noted that the flow on the spillway surface to be described in this example is varied and that a sirtiplified method of

1 I

! I

I I

.. I


i

J

200

~I',

UNIFORM FLOW

10

:FEfIi ! I t

9

,~ ! ~u{j ,

s

i'.r-.,

5

I,

t-

~>'er.....

~.,

~

I

"

"/.bf:

I

JiV

....

'+-

i

,

OJ U

,,'

~

i5

"..'"

't>'

"\<-

1

~I

O.

,

0.4

v: /

I r

[/1/

/" V ,

i-

{j

I

I

200 DistallCe x, f j

300

I

400

600

I

BOO

P,y./('·UViJ.f-

vgRS

(8-8)

Vr

. 8-~., Veloc!ty Distribution in Turbulent Flow. The velooity distributiOn In a ulllform channel flow will become stable when the turbulent ",-_Q.9un~ar~ la!el' 1l3i'!!!.v ,develoE~d. In the turbulent boundary layer, the, clistnbu:lOn can be shown to be approximately logarithmic. "," The shearing stress at any point in a turbulent flow moving over a solid surface has been given by Prandt.l [7] as T

= pl2

(::)2':

L~~:::) .~

0:.

tVl ~

(8-9)

Thili equation indicates ,that t~l:9,ci.~.J~he turbulel:t re~~~ logarithmic funotion of the distance .y. It is commonly known as the -. - . -~~-~ .. ~--~~ Prandtl-von Il..(irmdn· univeJ'sal-velocity-dist1-ibution Zaw. I This la.w has been VCl,jti.ed by several experiments [10J.' The results indica.te a striking i'imilarity betvveen observed and computed distributions and, therefore, offer reasonable justification for use of this log;arith' mic law in practical problems., When the sul'face is smooth, the constant yo in Eq. (8-9) has been found to depend solely ,on the friction velocity and the kinematic viscosity; that is, ' my (8-10) Yo = Vr

8-6. Solution of Example 8-1 for the growth of a boundary layer.

about 30 ft, or a variation of less than 10%. This shows that an increase in rough,ness has a tendency to speed up the boundary-layer growth or to',educe the developmenL l e n g t h . ' ,

(8-5)

where m is a constant equ~l to about

mass ~el1sity = wig, wh~re w i~ the unit weight of the fi.tiid and g IS the gravitational acceleration ': l ':'" a characteristic length known as; ~he mixing length ,........ , dvldy velocity gradient at a normal distance y from the solid surfp,ce

where

=

The quantity represented by V r has the dimensions of a velocity. Since wi~h the boundary friction To, it is known as the ~n.i]i;;jii1i or shea;' velocity. Thus, Eq. (8-7) may be written . vr'

'1-

.

-L-J

,

(8~7)

yo

-----iY' it varies

0.3 c,. 40 50 60 70 80 100

FIG.

I

I~ In JL

Vp

·where Yn is a constant of integration. From Eq. (7-5) and w = pg; it. CD,n be shown that

I .~

V,

v = 2.5

O·

'/

.,~

I

//"'~

I

o.61--:- f-. 0.5 f--.

'++

Lbo..:\) . '

t

.. I

oP~c:7 epV:l

J.QLth~n near the acEd surfac\l, ,Prandtl introduced two assumptions:' (1) that themixh}g 'length is proportional. to 'Ii, and (2) that the shearing stress is constant; Since the shearing stress at the, channel surface is equal to the unit tractive force (Art. 7-11), the second assumption 'gives T = TO.:, From these two assumptions,Rq. (8-5) may be . . wrItten (8-6)

where K is a constant for the proportionality between L~nd }I:, The value of K has been ·determined py many experiments [8] to be about 0,40. 11ltegr'ating li I '" ~ "3'

I

'b'

o. 9 0,7

r

/'

,\

201.

THEORETICAL" COl-iCEPTS RELATBD TO 'UNIFORM FLOW

r

p

% for

smooth surfaces. 2

,

For wavy

1 Von Karman .[9J also proved this law by II similarity hypothesis which assumes a linear shearing-stress distribution, the mixing length being proportiona.l to (dv /dy) /

(d~/dy·).

!

• Thill value is derived from Nikuradse's experimental data.

I

011

smooth pipes [111.

20~

! i

UNIFORM FLOW

surfaces, the' value of m wHI depend on the shape of the wa.ve profiles. Substituting Eq. (8-10) for Yo in Eq. (8-9). and simplifying, v

=

9yVI

575V . r log - v -

for smooth surfaces .

if:

= 5.75YI

log

30 -!Co

~Q'

for rough surfaces

=

P - 'YY

(8-15)

where P is the wetted perimeter and l' is a function depending on the shape of the section. Thus, the water area is equal to

(8-11)

'k

11

=

(

}o

.

B d.y = Ph - 2: 11,2 2

(8-16)

Substituting in Eq. (8-14) the value of v from Eq. (8-9), that of B from Eq. (8-15), n.nd that fO!' A from .Eq. (8-16) and then integra ling and

(8-12)

where the const.ant l m is equal to approximately Eq. (8-12) for YQ in Eq. (8-9) and simplifying,

203

proportional to its vertica.l distarice Y from the boundary i that is,

B

This equation gives the velocity distribution in turbulent flOlv OVer smooth surfaces. When the surface is rough, the constant Yo is found to depend on the roughness height ; that is,

v

THEORETICAL CONCEPTS RELATED TO UNIFORM FLOW

.<

Substituting

(8-13)

lli

equation' gives th~ velocit.y difitribution in tru:l:mlellt flow ml£,r surfaces. It should be noted. that thexoughness height k in this equation is the mean dilolmeter of the sand grains used by.NikIJm.d.®; ---~ . it is Imow!LSpecifically as the Nikurad136 8and rough!.!§M. It is to be noted that the constant.s nsed in the above velocity-distribution e@ations for smooth ®d rm~'faces were derived from data on pipes. It is assumed, however, tha.t it is permissible to apply the llniversal-Velocity-distribution h~w to other cases of turbulent flow, using the constants determined from tests on circular pigcs. • -. 8-0. Theoretical UnIfOrm-flow Equations. Using' the Prandtl-voll Karman universal-velocity-distribution law, KeulElgan (13)2 has derived equations for mean velocity of turbulent flow in open channels. . :I;<"o.r the sake of clarity l;1.nd simplicity, Keulegan's derivation will be modified below. By the cQntiuuit.y equation, the. total discharge through an ordinary channel section (Fig. 8-7) may be written

~h

Q

=

FA

t V=k v dA

=.

1.= O .

=

lch vB dy , 0

.

i'LG. 8·7. Channel

simplifying, the following equat.ion is

v=

.

or

-V = VI {5.75 log

1

[m~ exp ( -1 -~~) + 5.75 lug n~~}

=

(8-17)

0

+ 5.75 log mR) Yo

r

(8-18)

This is the general theoretical equation for the mean velocity of uniform . flow in open channels. For. smooth channels, Yo mn:y be represented by Eq. (8-10). Also,

1

[n

i

I

lf ...... .

obtl~ined:

[li!!:. exp (-1 _1'h~)l Yo R . 4A J

V Yr (A

where h is the depth of water, A is the vlater area, B is the length of the curve of equal velocity, and y is the vertical depth measured from the boundary to the curve of equal velocity. Since the laminar sublayer is relatively very thin, 00 can be assumed to he zero. It is further assumed that the maximum velocity is at the free surface and that the length B is

",,,',

2.5Vr In

to illustrate notation.

. In thp, above equation t.he quantity represented by the first term on the right-hand side is a function of· the shape of t.he chn.nnel section. However, the variation of this quantity with different shapes of the section is relatively small. F~r the sake of simlJlicity, the quantity may be represflnted by an ov~r-all constant" 11 0• This constn.nt will include not only the shupe function but also other uncertain factors, such as the effect of free ~urfn.ce and the effe.ct of nonuniform distribution. of the tract,ive force at the boundary. Accordingly, Eq. (8-17) may be wribten

(8-14)

This' value is derived from Nikuradse's experimental data on Tough pipes [121. Nikuradse's experjments, sand grains were cemented to the inner walls of pipes to . simulate roughness. Such roughness is known as artificial T"Ougimess. 2 Other references on this subject fLre [141 to r21J~

~ec~iol)

I t

* It is entirely feasible to simplify the derivation by assuming this constant at the very beginning. However, the procedure given here is aimed to give a logical demon·stration of the basic principles involved.

f

.J

-I

204

THEORETICAL CONCEPTS RELATED TO UNIFORM FLOW

UNIFORM FLOW

from ICeulegan's study "of Nikuradse's data [111,' Ao is found to be 3.25. The theol'eticaluniform-fiow equation for smooth channels is, therefore,

Y VJ (3.25 + 5.75 log R:/) " =

for smooth channels

"d' ber functions 0 f t h e F. rou e num . as follows: IT = VI

(8-19)

l'

Umng Bazin's data [22] for wavy surfaces, Keulegan obtained a value of Ao = 1.3 for smallwood channels and -3.0 for large wood channe1s. For rough channels, yo may be expressed by Eq. (8-12). From Keulegan's study of Bazin's data [22], the valu~ of Ao was found to have a wide range, varying hom 3.23 to 18.92. Thus; a menu value of 6.25 for Ao may be used. The theoretical uniform-fiow equation for Tough channels is, therefore,

v=

V, (6.25

+ 5.75JOg~)

for rough channels

=

205

Let" E<,qs. (8-1 g) and (8-20) be expressed

I _ _ RY /) "A, + ~.7u log--;-

for smooth channels

"I (AT + 5.7510g ¥ ) r o r rough channels

(8-2·1) (8-25)

·t'"

.. , .' f' ns of the }'roude number .. In the above eouatlOns, A. and £l~ ale .mc 10 . ft. The 1· d t' f" ct~ of other mlllOI' ac OIS. These constaill.s may al,;o ll,/; u e ne e Ie·" .' L

•

(8-20) 7

From the CMzy formula F = C ,IRS and from the definition' of friction velocity 1', = ',fiRS, it cltn be shown that

v VI

C

=

Yo

6

5

(8-21)

.;

4

"c:c

Substituting this equation ahd the Reynolds Il'lmber R = RV / ~ in Eqs. (8-19) and (8-20), the following express:olls fo\' Chez}"s C are obtained:

20.8R C -- 32"61 " ogc;-

for smooth channels

(8-22)

C

for rough (!hannels

(8-2:3)

3

4:

2

f

0

= 32.6 log 12}!!:

-I

After Keulegan's analytical study on laws of turbulent flow in channels, Powell [16-18J experimented with small rect~tnglllar sills used as artificial roughness and arrived at Eq. (5-5) for Chezy's C. Similarly, Robinson and Albertson [19] used baffle plates as artificial roughness and developed another empirical formula of C for rough chaunels. :Moore, Rand, and Hama [3] £\lso conducted similar studie:3, using transverse-bar roughness of various sizes. A further" study was made by Iwagaki [21] on experimental data obtained from many sources. The results of the study have disclosed that resistance to turbulent flow i;l open channels becomes obviously ·f--.»arger than that in pipes with increase in the Froude number. Iwagaki ~easonedthat this is due to the increa:led instability of the hee surface at high Froude numbers. Using the data presented by Iwagaki, therefore, it is possible to introduce. the effect of free-surface instability into Keulegan's equations by assuming that the constants in the equations are

j

-2 -3 0.2

0.3

0.4

F FIG.

8-8. Rel~tions amoog A"" rL, and F.

. . . F' 88 by the lotted data are found to be much scnttered(as shown III 'Ig" I - I "1 . . t " l ' ffects "When t 1e c lu,nne shaded boundaries) probably OWlIlgo t le:;e e . I h slope becomes too large, the flow will be unstab~e (Al:t. 8-8).. l~ suc cases, the \u,ws of turbulent flow described in tIllS o.rtlCle are no onger

P

valid." .. , R hess Coefficient. S-S. Theoretical Interpretation of M.an?m g s oug. n .. he theoIt should be interesting to study Mannmg s n by relo.tl~g l~ tfn\he preretical channel roughness for rough channels, as ~e(v8e ~~e Ma ning's n ceding article. Eliminating C from Eqs. (5-7) [1.n -~, n

-="

206

THEOBETICAL CONCEPTS RELATED TO UNIFORM FLOW

UNIFORM FLOW

may be expressed as follows:

n where

~ (~)k;6

(8-26)

(R/k)'A 21.9 log (12.2R/k)

(8-27)

The plot. of t.his equation, (Fig. 8-9) indicates that, for a wide range of R/ k, the variation in
A. M etlwd of Velocity M easuremellt. With reference to the logarithmic law of velocity distribution expressed ih Eq. (8-13), it can be seen that

the velocity distribution depends on the roughness height,- which may be related to Manning's n by Eq. (8-26). : In other words, the roughne8s in terms of Manning's n can be taken as a dominating fa,etor that affects the velocity distribution. If the distribution is known, the value of Manning's nean, therefore, be determined, This concept has been used by Boyer (24} and others [25-28] in estimating the value of 11 from the vertical velocit,y distribution ill a stream, Let VQ.2 be the velocity at two-tenths the depth, that at a distance O.8y from the bottom of a wide rough channel, where y is the depth of flmv, By Eq. (8-13), the velocity may be expl'essed a.s .

,

...---o~':-

S J feems in No,lhweslern U.S. (I

~ ..;,n.. _ _ _ _ II

"

'\.

Sfricklerts __ _ ~

~

=

,'"

Similarly, let

UO.s

be the veloeity at eight-tenths the depth; t.hen Yo.s =

R/l<.

5.75Yr log

Eliminating VI from the above two equations,

FIG. 8-9. Functiqn ",(Rlk)

I

median sieve size of the material. Data [24J· collected from sever",] streams in the l1orth\ycstcl'll United States and from the 'Mississippi River are also shown in. Fig. 8-9 for purposes of comparison, If
----

1 This constant has been converted to ft-Ib-sec units. in ft.

The value of k is expressed

Y

og'k =

O._.~77~8~x~ __.1....,. 3~8_1

x

(8-28)

!la.dvo.s.· Substi.tuting Eq. (8-28) ill Eq. (8-20), with R ~ y, "there x and simplifying, V 1.7&(:1: 0.95)

+

f='-~x-l

(8-29)

From Eqs. (5-7) and (8-21), with R = .1(,

V Equating the right-hand siues of

J

)

24y

5.75V r logT

Uo.t .

;;.

207

(8-30) (8-29) and (8-30) and solving for 1'1., (8-31)

This equation gives the va.lne for 1'1. for a wide rough channel with logarithmIc velocity distribution. When this equation is applied to a.ctual streams, the value of it may be taken as the mean depth. A pli1t of this equ:1tion (n/y;f, vs. x) has been made to compare with a similar curve developed by Boyer [24Jand also with actual observations taken . from several streams in the northwestern United States and the Mississippi Rivet;. The. comparison leads to the belief that a general relation-

208

, UNIFORM FLOW RELATED TO UNIFORM FLOW THEORETICAL .CONC""FTS '"

ship exists between Manningls n .and the velocity distribution. However, more data, are neceSsary to! verify this theory more convincingly and to d~linea.te the rela.tiol1!!hip that it can be used extensiv e1y for practical purposes.

2U\:l

'.1,000 eooR~~~~~1!~flE~~~!~!~~f~~~~~~t~~

so

600

j

The simple measurement. of stream flow is llsually made by taking velocity mf)asl,lrements in several vertical!! at 0.2 and 0.8 depths (Al-t. 2-6). The.')e measurements, which are averaged to give the meatl velocity in. the vel'tical, can be used to estimate Manning's 1"1, by means of Eq. (8-31). If this methqd can be shown to be satisfactory for practical applications, it will Pl'ovidean easy way of determining roughness in streams wherEJ velocity observations have been made. . . r--~B. ~lethod of Rfl.u(Jhness lI;[eal1'!m'ment. In this method it is assumed I that Eq. (8-27) for the function
400

\.

. . rl'1

This method can be' extended to strea.ms containing moving sediment beds, A{lcording to Eiu..,tell1 and BrLrbal'ossa [29], the hydraulic radius R for such stt'eams may be I1Ssumed to C011Sist of two parts: the hydraulic radius RI due to surface roughness and the hydniulic radius R - R' due to roughness ce.l.lsed by moving sediment beds. For surface roughness, the roughness height is represented by k'5 in feet, which is the grain size just coarser than G5 % of the materifl.1, as obtained from an average curve of mechanical anatysi.~. This curve is to be prepal'ed from the material sampled from the wetted perimeter within the chosen reach of the stream. T1~e rongnness height for ~oving sediments is reprepented by the size /,;s; in feet, which is the size just coar,sel' than 35 % of the sedim(:nt grain. This size is taken from a mechanical-analysis curve obtained from sedi. ment samples collected within the reach of the stream. Following this concept, Doland and Chow [30] hlJ;v:e shown that the function Ib(Rlk) for the combined effect of the surfice roughne3B D.nd moving sediments is 0.0342 (WIR)3~

20

I'"

~ '0

'"

'" E 0

10 B

6

:> 4

0.8 L -_ _ .;.._+0.6 O. 4

-l---l++-i4+t·~-+·-·".ci--f';

L_·_-.l--~4--+-\-+-H+-~~:---'l--"r'": ,-l~4~'Il-~n:-1

0.2.

(8-32)

~here the value of R'I R depends on the hydraulic radius R, the slope S, and the grain sizes k65 and ka•• By computing values of (R/k G.)'/.! and k361 RS, the value of R' / R Can be obtained from the semiempirical curves of Fig. 8-10. These Curves are based on data taken from seven typic",l rivers in the United Sto.tes. 1 When (Rlk) is computed by Eq, (8-32), the va1ue of n can be determined by Eq. (8-26), with k = k65. When

0.1 L~__-l._-L-_L") .. L l.1..L.._ _..L..--.L~_'-.'. 0.1 0.2. 0.4 0.6 0.81 2. .

Values of

x35

'liS

FIG. 8-10. Relationship het,+een (R/l.:iS)H anel 1.:3&/~B •.

. there is little moving sed'lment 'In th ~ .St' 1'eam , or R - R' = 0, it can . . be seen that Eq. (8-32) is reduced t.o Strickler's co~stant .(Art. 8-~), The method described above doeS not comudel' kmds of !IOu~hness other than :surface roughness an d t h:e rong hness d ue "to moving-sedIment 1 beds. Therefore, it is applicable on1y to problems 111 whlch the bUS1IY-

':Missouri River at Pierre and Ft. Randa.U, S.D., and Omaha, Nebl'.i Elkhorn Rhier at Waterloo, Nebr.; Big Sioux River at Akron, Iowa; Nioprara Ri~er at Butte. Nepl'.; Platte River at Ashl!l.nd, Nebr.; SBlina.~ River at San Lll,c"s and Paso Robles, Calif.; and Nacimiento River.at Junction,' CElIif. . j I

J

,

I\

UNIFORM FLOW

,,

bank friction is negligible and in \vhich the active channel, even though split by sand ba'rs and possibly by islands) is free of debris and vegetation. 8-8. Instability of Uniform Flow. Uniform floW' will become unstable 'when the velocity of How is very high or the channel ,slope is very steep. When this happens, the instability of the free surface is characterized by 'the formation of a series of roll waves. This phenomenon was first reported by Cornish [31J in 1910 when he observed it in open :channels in the Alps. The roll Wave is a phenomenon of unsteady flow and its nature will be described ill Art. 19-9, Many attempts [32-45J ha.ve been made to develop a criterion for instability of uniform flow. In 1945, Vedernikov [35], employing certain approximations of Saint-Venaht, developed a criterion which may be called the Vedemikov number V. Later, Craya (41} and Iwasa [43} studied the initiation of continuous tilliegrowth or decay of an infinitesi1110.1 di5tUl'bo.nce wave 011 fluid surface, and developed similar criteria. The Vede1'llikov number may be expressed as

II

210

V

=

I

1

I

I

xyV V",- V

where x = exponent of the hydraulic radius in the general uniform-flow formula Eq. (5-1). Thus, x = 2 for laminn.r flow (Art. 0-10), x OJ) 'for turbulent flow if the Chezy formuln. is and x = H for turbulent flow if the Manning formula is used. V = mell.n velocity , V", = absolute velocity of disturbance waves in channel ~ = a shape factor of channel section, defined by dP 'Y=l-RdA.

= xyF

providing the value of X,. For lack of such a formula, the Manning or Chezy formula is used as an approximation only. Since the channel slope is us,tally large when instability of [Jaw develops, the Froude number ill (8-35) should be computed by Eq. (3-13), which considers the slope effect. PROBLEMS 8-1. Show that Eq, (8-0 is deriv~d from the condition that the presence oi'the boundary Lg,yer raises the bottom of the cha,nnel to a virtual distance equal to the dispLa.cement I;hiclmetls. 8-2. A laboratory [·ecta.ngular channel 1.5 ftwide carries a unirorm flow of 0.34 cis at p, depth of 0.15 ft and ll. slope of 0.0009. The roughness height of the channel surf!>ce is 0,00215 ft. Plot the profiie of the turbulent boundary layer, 1).0(\ est.im!l.te the length -of channeL required for a fun development of the boundary layer \vhich begins tobe turbulent u.t the entrance. 8-S. Determine whether the c.hannel'described in the preceding problem is hydrauli' cally smooth or rough. 8-4, A trapezoidal channel (Fig. 2-2) with b ,.. 20 H, z 2, and S = 0.001 ca.rries a uniform Row at a depth of 6 ft. Compute the unit tractive force :Lnd the fric~icm velocit,y developed in the channel, Wha.t are maximum friction velocities on the sides and bottom of the dUlIInel? 8-li. Snow t.1U1.t

where f is Darcy's friction factor and 11 is Mll.rming's roughness coefficient. 8-6: Using Eq, (8-13) for expressing the theoretical velocity distributi.on in wide rough cha.nnels, (a) show that the I:iverag8 of the velocity of 0.2 depth alid the velocity at 0.8 depth gives the velocity at 0.6 depth, and (b) compute the position of the mean velocity below the free surface) and compare the result with that determined by the rule of the U,S. Geological Survey (Art.. 2-6)., 8-1. A wide channel carries a uniform .flow a~ a depth of 5 in. on a slope of O.OOOL The channel surface is rough, having a value ork = 0.25 in. Compute and construct a curve showing ~he theoretical velocity distribution in the channel section. What are the mean velodty and its position? . 8-8. Determine the values of k in Probs. 5-9 and 5-10. 8-9. Eq, (8-13), show that the theoretico.l velocitJ:"-distribution coefficients in wide channels can be expressed as

(8-34)

<:t.

~

1

+ 3.'

- 2.'

(2-6) (2-7)

(1=1+.-

(8-35)

When the Vedernikov nuniber V is less than unity,any wave in the chalinel will be depl'essed!ll1d the flow can be stable.' But when V is equal to or greater than unity 1 waNes will amplify so that stableflow will become impossible, unsteady flow will preva.il, and roll waves will form. In, computing V for a turbulent ft.ow, it should be noted that a suitable uniferm-ftow formula for flow in chul1nels of large slope shoul(i be used in

211

i,Il

(8-36)

where R is the hydraulic radius) P is the wetted perimeter, and A is the water area: Thus,"( -= 1 for very wide channels, and 'Y 0 for very narrow· channels. It will be shown that V", - V is equal to the celerity c of the waves (Art. 18-6») or to the critical velocity V.. Since ,the Froude number F =- VIV., Eq. (8-33) may be reduced to V

THEORETICAL CONCEPTS RF.LATED TO UNIFORM FLOW

Vi

I r 1>;

r

where. 2.5V';V. Also show that ( = H.2/C ~ 0.883 = 9.5n/RH .. (v~r/V) - 1 where C is Ch6zy's resistance factor,:j is Darcy's friction factor,:n is Manning's roughness coefficient, R is the hydraulic radius, VA( is the maximum velocity, and V is the mea.n velocity. 8-10. Plot a curve showing the relationship between a a.nd {J defined (2-6) and (2-7), respectively. 8-11. Comput,e the velocity-distribution coefficients. of the flow in the channel described in Prob. B-7.

1\

1:

212

UNIFORM FLOW THEORETICAL CONCEPTS RELATED TO 'UNIFORM j,'LOW

18-~~:

A wide st.ream carries approximately uniform flow- at a depth of 12 ft Th ve OCI les at 0.2 and 0.8 depths are found to be 1.85 o'nd 1 32 f ...e mate (a) the roughness coefficient (b) th I . . ps, respectIvely. Estl' ' n, e mean ve oClty (e) the slope f tJ h I an d ~d) tn8 dis,:harge PCI' unit width of the channel. ' , o l e C anne . 8-LS In a wIde' t ' . moving . sediment bed the f I' " . s ream IlaVing 1 • able: S = 0,003 k, = 050' d 1. , 0 . 0 " mg (ata are aVO,IJJ,"' • mm, an ,,,~ = 0040 mm TI e I d· r d' al'ea al'e found to be It = 0.85! _ 1 8, d A _ _. I ,Iy lau lC ra IllS and li'ater , • J ' . an - 2/3y - 797 where R d ' ano, A IS in [('.' Construct tl. e disc lIar t" ' an yare m ft " . , ge-ra mg curve of the strean' , b-14. Show that the Vedernilcov' number in wide channels is V ~ 2F '. Row, V = O,67F fo), turbulent flol'v: 'f th "T " f ' for lammal' 1 • e ,',-annmg orm'll- IS us d dV 0 turbulent flolV if the Che~y formula is used. ,~e ,an = ,SF for ' 8-11i. Using t.he Vedernikov' criterion expla'n tl ' . I"" • . . fIIl.W- I 00 ompany I K Y Ltd:, Lor~don, and Verlag G. Braun, ICarlsrulle,n c . ew ork, Pergamon Press l 955 2. J. W. ~elieur: The boundary layer development in open channels a ' ,

Pro~e~dm!!s, .f1merican SOCiety of Civil Engineers Journal Engi" .'

P

/;e\ ll~:-;,

D~!Jts!?n, voL 83, no. EM1, pp. 1-24, Janu!l.ry 1957. ' n.enng. ec antcs 3. Franms n Hama' Bound I' h ' '. , ,. : ary- ayer c aractenstll:s for smooth unu rOll h surf 'Z1anSaclwlIs, 80c'lety of Naval Archiiects and 11[ an'ne I ; L • I ~~ aces, """,gmeers, VO , u"" pp. 333351, 1954. 4. Yoshiaki Iwasa: Boundary layer grow'h of oPen ch"nnel fl ", ct '[ 'b' .•. ~ OWS on a sn'iooth bed ;,n I s contl'l utlOn to practical application to chp.nnel ciesiO'n JI'{.n'~·r. 0 'I r J' aClilty a En' . }' U' . ", "'v. u 1, ,e ~ gtneenng" yoto ntvers~tll Japan vol XIX no III' 0')9 JUly, 1057. ' ,. " , . ,pp. J~ -254, 5. Henry M. Morris, Jr.: A new'conce t of fl " _' . A. ,.' _, S · " . . P O I , m .ough condUIts, Tran,;rJ,cl'io1Ls , ~le"ca,. aClety of C~j,,1 En{}tneeTs, vol. 120 pp.373-398 19-- D; . " VIctor L, Streeter Walt R d H ,,1)0. .SCUSSlOns by Jr., pp. 399-'410. '.' er an, arry " Ambrose, and Henry M. Morris,

H

6. 7 .

" ' , William J, Bauer: -Turbulent boundarv layer on 3teep slopes T A me"iC(nI S C'·, ( C' '1 E , , ransaC,1.OnS L . . 0 ~ .. y 0, ngmeeu, vol. 119, pp. 1212-1233, 1954. ' P udwIJ,:P:tandtl: Uher dIe ausgebildeie Turbulenz(On fully developed turbulen ) TOCte mgs of the 2d Inte?natianal Congress Of Applied Mechan'''cs Z" 'h ~e , ~ " Hone, pp. 62-74, 1926. '

tv:

8. pBoris Ap' Balchmeteff: "The Meahanies of Turbulent Flow" Princeton Un' 't ress, rmceton, N.J., 1936, p.'06, ' Ivers! y 9.

;h~~d~r

von

I{~rm.!in:

Mechanische Aehnlichkeit und Turbulenz (MeChanical

~ml:rlt,Y anSu t~h:bulence), Proceedings 01lhe 3d International Congress 01 AppHed

,ec ames, toc,. aim, vol. 1, pp, 85-92,1930. : ' 10. VItO A.Vanoni:, Velocity distribution in open channels Civil En' " 'I no, 6, pp, 356-357 June HJ41 ' gtneertng, VO . 11, 11. J Nikur'adse' Ge 't .. ' . I: d f' " se zm~sslg.(elten el' tUl'bulenten Str5mung in glattC!l Rohren ' (LaIVs 0 turbulent flow In -'Smooth p' ) V ., I h It N ""6' B ' ~ Ipes, eretn ( e!ttscher I"geniel"'e, Porsclnm(JSe o. "", erhn, 19,,2. . ' 12; J. ~i,kurlldse:'Str5mu~gsgesetze in rauhen R(\~ren (Laws of flow in rough i es Vele'n deutscher IngemeuTe, Forschungsheft No. 361, Berll'n, 1933. p p ), J

,

213

'

13. Garbis H. Keulegan: Laws of turbulent flow in open channels, Research Paper RP 1151, Journal of Research, U.S. National Bureau of Standards, vol. 21, pp. 707-741, Decemher, 1938. 14. Erik'Lindquist: 00 velocity formula.s for open channels and pipes, T"ansacl-ions of the "World Power Conferer.ce, Sect'ional M feting, SfX!ndinavia; Stockholm, vol. 1, pp. 177-234, 1933. 15. A. E. Dretting: A set of practical hydraulic formulae based on recent experimental research; comparison with older formulae, Appendi:r; 24, 2d !Jleeling, International Association of Hydraulic Str'!J.ctuTes Research, Stockholm, Apr. 7-9, 1948, 20 pp. 16. Ralph W.,Powell: Flow in a channel of definite roughness, T,ansaclions, American Society of Ci'j'il Engineers, vol. 111, pp. 531-566, 1946. ' 17. Ralph "'. Powell: Resistance to !low in smooth channels, Transactions, American Geophysical Union, voL 3D, no. 6, pp. 875-878, December, 1949. 18. Ralph L, Powell: 'Resistance to flow in rough chs.nnels, Transact~ons, American Geophysical Union, vol. 31, no. 4, pp. ,575-58:2, August, 1950. 19. A . .fi.. Robinson and M. L. Albertson: Artificial roughness standard fOr OP€ll ~hannels, Transa,ctions, A.merica" Geophysical Union, vol. 33, no. 6, pp. 881-888, December, 1952. " 20. Yuichi 11vagaki: OIl the laws of resistance' to turbuleut !'low in open smooth channels, lI1emoil's oj the Faculty of Engineering, Kyoto University, Japan, vol. 15, no. 1, pp. 27-40, January, 1953. 21. Yuichi Iwagaki: On the law of resistance to turbulent flow in open rough channeia, Proceedings of the,4lh Japan National Congress for Applied M echanies, pp. 229-233, 1(l54., 22. H. Darcy and H. Bazin: Recherches hydrauliques:I, Recherches cxperimeotales sur l"ecoulement de I'eau dans les canaux decouverts (Hydraulic research: I,. Experimentftl research on',the flow of water in open channels;, ],f emoires present~s par divers savants a l'Acud~mie d~s Sciences, vol. 19, no. 1, Dunod, Paris, 1865, 23. A. Si,rickler: Beitriige zurFrn.ge der Geschwindigk'eitsformel und der Rallhigkeits~!l.hlen fiir Strome, Kanale und geschlossene Leitungen (Some contributions to the problem of the velocity formula a,nd roughness factors for rivers, canals, !l.nd closed conduits), Mitteilungen des ddgenossisehen Amtes 1iir Wasserwirtiieha1t, :Bern, Switzel'land, no. 16, 1923. 24, M,. C. Doyer: Estima.ting the Manning coefficient from ~_n average bed roughness in o[3en channels, Transactions, American Geophysical Union, voL 35, 00. 13, pp. 957-9&1, December, 1954 . 25. Boris A. Bakhmeteff and Nicholas V. Feodoroff: Discussion on opeu cha.nnel flow, Transactio1'1s, Americcm Society of Civil Engineers, vol. 1'08, pp. 492-502, 1943. 213. M. P. OiBrien: The vertical distribution of velocity in wide rivers, Tra.nsactions, American G~ophysical Union, voL 18, pt. 2, pp. 4137-470, 1937. 27. F, C. Scobey: The flow of water in il'rigation channels, U.S. Department of Ag,'iculture, Professional Paper, BI"Uetin No. 194, 1915. 28, W. R Langbein: Determination qf Manning'S n from vertical-velocity curve, Transactions, American Geophysical Union, pt. II, pp. 618-620, July, 1940. 29. Hans A. Einstein !lnd H. L. BarbuTossa: River channel roughness, Transactions, Amen:can SocieLy of Civil Engineel's, vol. 117, pp. 1121-1132,1952'30, James J. Doland and Ven Te Chow: Discussion of River channel roughness, by Hans A. Einstein and H. L. Ba.rbaross3., Trr;msaclions, American Sociely 01 Oivil Engineers, yol. 117, I'p. 1134-1139, 1952. 31. Vaughan Cornish: "iVaves 'of the Sea and Other Water Waves," The Open Court Publishing Comp!l.ny, La Salle, Ill., and T. Fisher Unwin, Loudon, 1910.

214

l,

UNIFORM FLOW

I

32: Harold Jeffreys: The flow of water in an inclined channel of rectangular section, London, Edinburgh and Dublin Philosophical i1!agazine and Journal oj Science, ser. G, YfJL 49, pp. 79:3-807, May, 1925. . 33. G. H. Keulegan and G. W. Patterson: A oriterion for instability of flow in steep channels! T,'o,nsactions, American Geophysical .llnion, vol. 21, pt. II, pp. 594-&96, July, 1940. ·34_ Harold A. Thoms,s: The propagation 'of waves in steep prismatic conduits, Proceedings oj Hydraulics ConJere1J.ce, Universit·y oj Iowa St'~dies in Engineering, Bulletin 20, pp. 21!l--229, 1940. 35. V. V. Vedernikov: Conditions at the front of a translation wave distributing a steady motioll of a real fluid, Camptes renclHS (Doklady) de I' Acad~mie des Sciences de t' U.l?.S.S., vol. 48, no. 4, pp. 239~242, 19i5. 35. V. V. Vedernikoy: Characteristic featur!)~ of a liquid flow in an Open channel, Comptes rendus (Dok.lady) de l'Accidemie desScience~ de l'U.R.S.S., vol. 52, pp. 207-210, 1946. 37. RalphW.Powell: Vedernikov's criterion for ultnHapicl flow, Transactions, Ame,icanGeopilysical Union, vol. 29, no. 6, pp. 882-886, December,1948. Discussions by V. V. Vedernilwv a.nd Ralph W. Powell, voL 32, no. 4, pp. 503-507, August, 195138. Robert F. Dressler: Ma.thematical solutions of the problems of roll waves in inclined open
\

I

PART III

GRADUALLYVARffin FLOW.

f Ii

~k'~

l

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I.

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2.

.Y!r~~ IJ
/'''' e-c..

~·J:7 !'Wut.(.f.'q'

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CHAPTER

!n~ for '/, Jir""tJy nt~;tv' ai"r£- c;(,,"" nel .

.

1

~l'il1·i;;~·~ &~"Rtff

THEORY AND ANALYSIS

•

, l/}~ctr~ a:lt/ -~~ ~~"7~7~~~1/2-prn#.J 11-.I~ [af,;d~..td on -Ilu hMCd'9- tJ7Ur/&-'n1 t!c# t.".-.ru;dfL-!

lV1301HOd qn'gt.:;~

9

.

9-1. Basic Assumptions. The gradually varied flow to be discussed in Part III of this book is the steady flow whose depth varies gradually dong the length of the channel (Art. 1-2), This definition signifies two conditions: (1) that the flow is stea,dy; t.hat is, that the hydraulic characteristics of flow remain constant fqr the time interval under consideration; and (2) that the streamlines are practicaily parallel; that is, that hydrostatic distribution of pressLlre prevails ovel' the channel section, The development of the theory of gradually varied flow dates back to the eighteenth .century. Mn.ny early hydraulicians L have contributed to this development. The theories thus developedprac.tically all hinge on the following basic assumption: -----p A. The head loss at a section-is the same as for a Hnijo'(m flow having the velocity and hydra111'icraciius of the section. . According to this assumption, the u:r.iform-flow fonmila may be used to evaZ1/ate the enngy slope of a gracl1wlly va-riecl flow at a giveTI. channel sect'ion, and the corresponding coefficient of roughness developed primarily f01' wd.form flow is applicable to the va'ned flow. This as~umption has never been precisely confirmed by either experiment 2 or theory, bnt errors arising . from it are believed to be small compared with those ordinarily incurred in the use of a u;1iform-flow formula and in the selection of the Toughness coefficient. Over years of use this assumption has proved to be a reliable basis for design. The assumption is undoubtedly more correct for varied flow where the velocity increases than where _the velocity decreases, . because in a flow of increasing velocity the head loss is caused almost. entirely by frictional eff~g~, whereas in a flo,"v of decreasmg -velocity-~here may.E(;l!~t.];~:s.c::~l~_e~_dy_.lo..~~I Bela.nger ll] is believed to be the outstanding contriblltor. Also El.mong early contributors are Bernoulli, Bresse, Poncelet, Saint-Yenant, BOllssiuesq, El.lld others [2] to [5]. _ . 2 Using the experimentlll data from the .Sunderland Technical College lind ICing's College in England and from the University of lllinois, Bettes [G] has dClL'ived an j-R relationship (Art. 1-3) for gradually variedftow in smooth open cblHlels, which WIIS found to agree very closely with the relationship for uniform ftow obtained by Allen [7]. Also, the computlltion of backwater curves based on this assumption ho.s been verified satisfactorily by many experiments. These experimental verifications, though not VEry rigorous, indicate the validity of the assumption for practical purposes. 217 .

218

'l'HEORY AND ANALYSlS '

GRADUALLY VARIED FLOW

In addition to the above basiQ assumption, the following aSEum:ptions will also be. used where further simplification is necessary in subsequent discussions: . B. The slope of the channel is small; so that: 1. The depth of flow is the same whether the vertical or normal (to . the channel bottom) direction is used. 2. The pressure-correction fact,or cos IJ [applied to ·the depth of. the flow section" Eq. (2-12)] is equal to unity . . 3. No air entrainment occurs. 11'1 case of notable air entrainment, the computation may be carried out assuming no entrainment and then corrected approximately, at the end, using Eq. (2-15). C. The channel is prismatic; that is, the channel has constant alignment and shape. D. The velocity distribution in the channel section is fixed. Thus, the velocity.-distribution coofficients are constant,. E. The conveyance K (Art. 6-3) and section factor Z (Art. 4-3) are expOl18ntial fUllctions of the depth of flow. . F. The roughness coefficient is independent of the depth of flow and constant throughout the channel reach under consideration. 9-2. Dynamic Equation of Gradually Varied Flow. Consider the profile of gradually varied flow in the elementary length dx of an open channel (fig. 9-1). The total head above the datum at the upstream section 1 is

. H

I Ii

!

T,

V2

= z + d cos e.+ a 2g

(3-2)

,4

where Ii is the total hea,d in ft i z is the vertical distance of the channel bottom above the datum in ft; d is the depth of flow section in ft; (J is the bottom-slope angle; ex is the energy coefficient; and V is the mean .. velocity of flow through the section in fps, It is aSllUmed that 8 and ex are\ constant throughout the channel reach under consideration. Taking the bottom of the channel as th!3 x axis and differentiating Eq. (3-2) with respect to the length :1.: of the w[.;ter-sUl'face profile, which is measured along the x axis, the following equation is obtained: . dH dz + cos e-dd d -(9-1) -dx = -cl.t Ct -dx _ ..:.::.Jb:. 2g

+

It shouldJ?e n(jte
and the slope of th~ ch~,l1nel bottom So = sin 0 = -dz/dx. t,hese slopes in Eq. (9-1) and solving for ddjdx, dd

dx = cos ()

.=

r

I . 'I~ ,

i

".,

.

So - Sf

(9-2)

+ Ctd(V~j2g)jdd

'(

This is the general differential equation for gradually varied flow, referred to hereafter as the'dY1iamicequa:t~Q'n;of gradually variedflow,or simply as , the gradual/y-va7'ied:'Jtow eqttation. It represents'the slope of the water) surface with "espect /0 the bottom of the channel. The depth d is measured from the bottom of the channel, lJ.rld the channel bottom is taken as the x axis. Thus, the slope of the water surfcLCe is equftl to the bottom slope r So if ddjdx = 0, less than .so if dd/d.'C is positive, and greater than So 1 if ddjdx is negative. In other words, the water surface is paranel to

l It should be notEld that the frictional loss dE is always a negative quantity itl the 'direction of flow (unless outside energy is added to the course of the flow) and that the change ill the bottom elevation dz is a negative quantity when the slope descends.

. :";

Substituting

(

sl~le

-'!""'--~ .. -....;;:.;...=~~---....:---'~

,

. " t

FIG. 9-1. Derha.tion of the gradually-varied-fiaw equation.

(V2)

a1~~ssuEled po§.itive_if i~e8cena~n the direction of flow and negative if it aSC§Jl..ds. . Hence, 1 in Fig. 9-1, the energy slope Sf = -dH7dx,

.

:)

.f ""

:.5 <1>

54J<.~~

Y'~>3'"

i

i,

~ .

.:; i .

I

220

.

GRADUALLY VARIED. FLOW

the cbannel bottom when. ddjdx. = 0, rising when dd/ dx is positive, and . loweling when del/dx is negative. .. In the abovH equation, the slope angle 0 has been assumed constant or independent of x. Otherwise, a term -d sin (J(d8/dd), where (J is a function of X, wonh:l have been added to the denominator. For small .OJ cos 8"" 1, d "" y, and cld/dx ~ dy/dx. Thus Eq: (9-2) becomes

\

,

THEORY AND ANALYSIS

. velocity and hydra1,llic radius of the section. is used, the energy slope is

So - Sf

.;:rJ,x - 1 + ad(Tn/2Q)/dy

(9-3)

In most problems, the channel slope is smaH; accordingly, Eq. (9-3) will be used in subsequent discussions. The term ex d(P/2g}/dy ill the varied-fiow equation represents the change in velocity head. The coefficient a has been [;,ssumed to be constant from section to section of the channel reach under consiciel·p.tioll. Ot,hel'wise, the change in velopity head would have been expressed as (I(Cl V2/2g)/dy, where IX is a function of x. Since V =Q/ il, Q is constant, and dA/dy '1', the velo~ity-hf.)ad term may be developed as follows: 1

d (V~) a dy

\ Since

21}

Z = VJP/T,

=

aQ2 dfi- 2 2g--dy

aQ2 dA =

-

gA3 dy'= -

aQ2T gfP

(9-4)

the above may be written

d (V2) a

ely \ 2y

aQl = -

gZ2

(9-5)

Suppose that a critical flow of discharge equal to Q occurs at the section; Eq. (4.4:) gives (9-6)

'\

When the Mann~ng formula (9-8)

When the Chezy formula is tiseo, Sf

dy _

221

In: a

Vi =

GZR

(9-9)

form, expressed in terms of the conveyance [(, the energy (6-4), may be written (9-l0)

Suppose that a uniform flow of a discharge equal to Q occurs in the section. ._'!h.~.}~!!~~~~()jJe ,vo.u1d ~~t)9.ual to the bot_tom slope, an~q .. (9-10) may be written (9-11) where K" is the conveYf-llCe for uniform flO'w at a depth y". This Kn should be distinguishedfro}n K in Eq. (9-10). The notation K represents simply the numerical v!),lue of the conveyance at a depth yof the gradually varied flow. The value [(" is t~e conveyance computed for Q at the depth yAas if the flow were r.onsideted uniform. Dividing (0-10) by Eq. (9~1l), Sf

Kn~

So =

](2

(9-12)

Substituting Eqs. (9-7) and (9-12) in Eq. (9-3), ., ~Y = So 1 - (Kn/IC)2 dx . 1 - (Z,/Z)2

where Z, is the section factor for cl',i~ical-fiow computation for discharge Q at depth Ye. The symbol Z, used herein should be carefully distinguished from the Z in Eq. (9-5). The symbol Z simply represents the numerical value of VA3/T, which is computed for the discharge Q at a depth equ~l to y of the gradually vUl'iedfiow. The value of Z. is the section factor.; which is computed for Q at the depth v, as if the flow were considered critical. Now, substituting Eq. (9-6)' for Q in Eq. (9-5) and simplifying,

This is anotber form of the gradually-varied-flow equation. There are other popular forms of the gradually-varied-flow equation that can readily be derivea, such as

(9-7)

where r So/ S,n, or the ratio of the channel slope to the critical slope at the normal depth of discharge Q (Art. 6-7) ;

The term Sf inEq. (9-3) represents the energy slope. According to..the :first assumption in Art. 9-1; this slope at a channel section of thegr!ldually ~al'ied flow is equal to the' energy slope of the uniform fiow that has t),e

(9-15)

dy dx

=

. 1 - (Kn/KF So 1 - r(K,,/J()~

(9-13)

(9-14)

222

Since the values of K and Z increase 01' decrease continuously with the depth y, the first case indicates y > y" and Y > Y.. As y > y" the flo'N must he subcritical. If Y > Yn > y" the sUDcritlcal flow must occur in a mild channl!l (i.e., a channel of subcritical slope). On the other hand, if y > y, > Yn, the subcritical flow must oceur in a steep channel (i.e., a channel of supercritical slope). Similarly, the second case indicates Y < Yn and y < yo. The corresponding flow must be supercritical; and it occurs i:n a mild channel if Yn > y, > !I and in a steep channel if y, > Yn > y. For a drawdowll curve, dy/d:!: is negative aild Eq. (9-13) gives two possible cases:

where Q is the given discharge of the gradually varied flow at the actual depth y; Qn is I;he nQrma.1·discharge at ~. depth equal to y; and Q. is the cntic!1r discharg;;:t a depth equal t·o y; [\nd ' dy (IX

;'

So - Q~/C2A2R - aQ2/ gA2D

= ,1

(9-16)

where D i~ the hydraulic depth, C is Chezy's resist~nce factor, and the rest of the notation is as defined in this article. For wide rectangular channels, 1. When the Ma11n.ing formula is used,

1. 1 - (K n /K)2 2. 1 .- (Kn/K)"

(9-17)

=

<:: 1 '-0

(Yn/V) 3 1 _ (yjy)3

(9-18)

9-3. Characteristics of Flow Profiles. The dynamic equation of gradually varied flow developed in Art. 9-2 expresses the longitudinal surface slope of the flow with respect, to the channel bottom. It can therefore be used to describe the characteristics of various flow profile.s or profiles of the water surface of the flow. F'or simplicity, the channel is considered prismat'ic, and Eq. (9-13) is used for discussion. , The values . of I( and Z in this eauatiori 1),re assumed to increase or decrease continuously with the depth' y. This is true for all opeil-chanmil sections except for conduits with a gradually closing top. In such conduits, the value of K, after reaching its maximum value, will decrease as t.he depth of flow approaches the top of the conduit (Art. 6~3). , ' The flow profile represents the surface cUI'Ve of the flow. It 'will represent a backwater cUl've l (Art. 4-5) if the depth of flow increases in the dil;ection of flow and a dmwdown curve (Art.4-5) if the depth decreases in the dil'ecti'on of flow. :Following the description in the preceding article, it can be seen that the flow profile is a backwater curve if dy/dx is positive and a dra-wdown curve if dy/d~: is negative. For a backwater curve, dy/dx is positive; thus, Eq. (9-13) gives two possible cases: 1. 1 - (K n/K)2

2.1- (Kn/KP

1-

(Z./Z)~

(Zc/Z)2

<0

<0

(Z./Z)"

>

and 1

Considering Un 1. Yn

>

y

>

=

Yo

00,

1 The term" backwater curve 11 i$ used primarily to indicate the longitudinal surface curve of the water backed up above .0. dam or into a tributary by flood in the mll-in stream. Ma.ny authors ha.ve extended its meaning to include all types of flow profiles.

dy/ilx is s

(9-19)

~,draWs1oWl~\::::e~ :~~:e~o_yac._."

flow with ve. e second case represents a supercritical ~.- ... a ackwater curve, ~s shown as positive. 0\

0:-----.

1

~I

l·

,

this equation indic,1tes two possible conditions: ~7'd' -<'I "ac-<'O' J,'OY-",", sQor:lo?

T:~ ::s:c:;e:e:rese~L~b~~t~cal

<0

.!

0

dy _ -(Q/K)~ dx - 1 - (Zo/Z)2

> 0 and 1 - (Z,/Z)2 > 0

< 0 and

> 0 and 1

The first case indicates that Yo > Y > Yn. and, thus, that the flow is superCi:itical in a steep channeL 'Slffiiiarly, the second case indici,tes that ~ V > Yo, 9E.. that the flow is subcritical in a rild channel. vVhen the water surface is parallel to the bottom of the channel, dy/dx = 0, and Eq. (9-13) gives 1 - (1(',/ K)2 = 0, or Y == yn, which indicates a uniform now. Tbe flow is a unjform critical flow if y = Yn = Yo, 8, Ulliform subcritjcal flow if y = y ... > Yet and a uniform sllpercritieaIflow if Yo > Yn = y. 70-r purposes of discussion, cha.nnel slope may be classified as sv.stair.ing and nonsusta.ining. A sustaining slope is a chai1l1el slope that falls in the direction of flow. ,Hence, a sustaining slope is always positive and' may also be ,called a positive slope.' A sustaining or positive slope may be critical, mild (sllbcritical), or steep (supel'critical). A nonsustaining slope may be either horizontal or adverse. A horizontal slope is a zero slope. . An adverse slope is a negative slope that rises in the direction of flow. . In a channel of horizontal slope, Qt" So = ~ (9-11) gives Kn = 00 . or ~ince K ... ~ = Q, Eq. (9-13) fo~ hOrizo\ltal channels may be written

2. When the Chezy formula is used, dy dx

223

THEORY AND ANALYSIS

GRADUA.I,LY VARIED FLOW

r

224

Gfu\DUALLY VARIED FLOW

I

j

I

I

Ij

1. A 8ubcl'itical flow in which y > y~ 2. A supercriticaJ flow in' which Y
In the first case, dy/dx is negative, and .the flow profile is 3. drawdown curve. In the second case, dy/dx is positiYH, ami the flow profile is a backwater curve. Since Ie. is imaginary, the value of Yn cannot be easily expressed. 1 In fact, a finite positive .value of Yn is physically . impossible because a uniform flow call never occur in an adverse channel.: (Why'?) The above discussions are summal'ized ill Table 9-1 and in Figs. 9-2 and 0-4. Some special features of the theoretical flow profiles are' as' follo",'s: . A. Discontinuit1{ in Flo'/j) ProjjJ.e. When Y. y., Eq, (9-13) indicates tMt~-= co, that that the flow pl'(jfile will be vertical in crossing the critical-depth line. If the depth of flow is ch!mged suddenly from a low stage to ll. high stage in crossing the critical-depth line, a hydraulic jump will occur, l'epresentinga discontinuity iIi the flow Profile. If the depth changes from a high to low stage, then a hydraulic drop wi.ll occur. It should be noted that, at or nep.l' the critical-depth line, the flow profile is bent to produce such great curvature that the parallel-flow assumption' for ~he definition of gradually varied flow will. introduce large errors. . In fact, the flow may become so curvilinear or rapidly varied that the theory and equations deve~oped in the preceding artiCle become inapplicable. Therefore, Eq. (\)-13) cannot be used to describe or compute accu!'ai-ely the flow profile near the critical depth. . B. Behavior of Flat/) P1'Ofile at Specific Depths. For the subsequent discussion, it is important to recognize the theoretical behavior of the fiow profile at several specific depths. When y = <0; Eq. (9-13) shows that dy/dx SD, that is, that .the flow surface is horizontal. When y = Yn; Eq. (9-13) shows that dy/dx 0, that is, that the flow surface is paranel to the bottom of the channel; th,is signifies unifonn flow, When Y YG, ~ hydraulic jump or drop in flow profile may occur, as noted. When Y = Yn = Y" the flow is uniform and criticaL . C. Points of Inflection on Flow Profile. When y 0, Eq. (9-13) seems to produce. an indeterminate form co /00 for dyjdx. However, 1 Some auth.ors l'egard Un as ~ega.tive [9]. Then the Ch.ellY formula. may be used for explallation. By the Ch.':zy iormlila, K = CAR~' and K' ,= C'A.'R. ThUll, for So' ilegative va.lue of K', Rand llJnce 1/n must be negative. The value Un hw; also been \lssumed to be positive [10J. In this case, the Chezy C is tak~n 1M! an imaginary Y!l.lue in order t.o keep values of A aud.R inY!l.riably positiye. I '

225

THEORY AND ANALYSIS

In a channel of adverse ~lope, or SD < 0, Eq: (9-11) indicates that, for negative values of SD, K" :must be imaginary or [(Hi must be negative [8]. Consequently, Eq. (9-13) gives two possible cases:

TAllLE 9-1. TYPES QF FLOW

i

I Rela.t.io~ I

Deaign:,tioil

Oha.nnel.

:

P ..QFILES. IN

~.I
Zone 1 Zone 2 Zon.a Zone 1

PRISMATIC CHANNlliLS

of,1) to VA and 'JI4

Zone 2

I

,

G.n~raJ typO I of c,u{"ve

Zone 3

TYP8 of fiow

'i :

Non~

None

Horizontal S, 0

H2

fill

'1> Mild

l)

-"12

<1.. < B.

NJ 1.'1 i

Critical

11>

j

C2

8,-8,>0

8.

I

81 182

> 8, > 0

'11>

]

133 .

J.!

None Adver••

A2

!In.

>

y~

yn

>

y.

lin

>u>

y"

">

y,

1i>

Dra.wdown

>y

'I Subcritieal i SupercrhJcal

1:1

I B ...kwnt;,r

! SubcrlU",,1

Drnwdo\itn

Suberitir.t'l

);. >!I

Bllckwn.ter

Supererltjoal

-

I/n

Ba.ckwa~er

Sub.ducal

Parallel to

Uniform~

11-

'lJ~

II,

-

y. > 11

Bll.ckwJlter

Super-critical

1I.

>

Yn

Backwater

Subcrit.icul

y,

>!I

>y.

11•

>

'lin> !I

>

II'

1),:;

C3

Steep

> Y>,'J«

1'.

i None

:=

(lIA)*

channel bQttom

~

I"

.-

en tlcnl

Sup.relitlcal 8 -lpexcri,ti.en.l 1

None 8ubcriticnl

(lin)' >!I > Y,

S, <0 A3

!

11,,~)

'* >

lie

>11

'D,

Supercritical

*11", in parenthe3es is uaumed a. positlve value.

it can be shown that the theoretical behayior of the flow profile at or near y = 0 depends on the tYIJe of uniform-flow formula used in the computa,tion. For a wide rectangular channel, Eq. (9-17) represents the slope of flow profile ii the Manning formula is used. By this equation, it can be shown that dy/dx becomes infinite. when y = O. This means that the curve is vertical at the channel bottom. If the Chezy formula is used, it can be sh~wn that dy/dx == So(y.,,/y.)3 for y = O. This means that the curve \vill make a certain angle with the bottom. It is apparent that there shou;ld be a point of inflection pn the flow profile when y: < y. < Vii' Mathematical investigations by Gunder [11] and Mouret. [12] have ;revealed that this point of inflection is at [l, depth very close to the channel

226

227

THEORY AND ANALYSIS

GRADUA.LLY VARIED FLOW

Downstream pooll&>e i lD

I

None

Yn

,

! r---------------, ,'I,

.l 1

-"

---

,

'

~'0;;W7_';w;wj&a0

Ml

--- ------- --.

_ 2--- -----

FIG.

-----

,r.. ~.:....-·-----_._.

<

II i

'~

!

"0.o

E 1c ~

-----_ .. --

Y

~

r-------

\ _______ "-1..--

Yo

;!1·1~

l.:-~ . . 0-W//////////''''''#~

f-----~-------i~--

I~ 7/%! L~T/lP/7/.i/~

Yo

FIG; 9-2. Classification of' flow profiles. of gradually vari!)d flow.

Zone 1. The space ,,;bove the upper line Zone 2. The space between the two lines Zone 3. Th;:; space below the lowel' line Thus, the flow profiles may be classified into thirteen different; types according to t,he nature of the channel slope and the zone in which the flow surface lies. These types a.re designated as H2, H3; MI, M2, 1113; CI, 02, C3; 81,S2, S3; and A2, A3;rvhere the letter is descriptive oithe slope: H for h/)l'izontal, 11-/ for mild (sub-critical), 0 for critical, S for steep (supercritiCal), and A for adverse slope; and where the numeral represents the zone number. Of the thirteen flow profiles, twelve are for graduJ1lly v~d Bow and one, 02, is for lltl.i£effll: fia w It should be noted that a continuous flow profile usually occurs only in one Llone. The general characteristics of th~e profiles are given in Table 9-1, and the shapes al~e shown in 9-2 and 9-4. Since the profiles near the critical depth and the channel bottom co.nnot be accurat~ly defined by the theory of gradu-

_ , _ _ _ ..sL-

Vi

bottom. 9-a).1 Similarly, Moui-et found that another point of inflection exjsts on the flow profile when V"> y" > y, (Fig. 9-3).~ 9-4. Classification' of Flow Profiles. 3 For the given discharge and channel conditions t.he normal-depth and criticahiepth lines divide the space in aehannel into three Llones: .

I

Yh='ic~---

None

\1-3 ..Theoretical points of inflection on now profiles.

"

I

of

1 It is believed that the question points of inflection was first discussed by Merten (13). • This point of inflection occurs because the profile must have !J. horizont.al slope in crossing the tra.nsitional profile and then bend forward tangent to the downstream . pool level (Art. 9-6). , It is beliP.ived that a comprehensive description a.nd c!llSsification of diff~re.n t flow profiles were first given by Boudin [14].

! 1

r

228

GRADUALLY VARIED FLOW

ally varied flow, they are shown with short dashed or dotted lines, Various flow profiles are discussed below. A, M P1'ofiles. SQ < Se and Yll >Y•. The -"11 pt'ofile the most well.. known backwater curve; it is the most important of all .How profiles from the practical point of view. ThiS profile oeem's when the downstream end of a long mild channel is submerged in a l'eseryoir to a; greater depth than the normal depth of the flow in the channeL This flow profile lies ill zone L The upstream end of the curve is tangent to the normal-depth line, since dyldx = 0 as y = y,,; and the downstream end is tangent to the horizontal pool surface, Rince dy/dx = Sn as y c:Q. Typieal examplea of the 1111 profile are the profile behind a dam in a natural river (Fig, 9-4a) and the profile ill a canal joining two reservoirs (Fig, 9-4b). . An MZ profile occms when the bottom of the channel at the downstream und is submerged in a reservoir toa dept.h less than the normal depth. The upstream end of the flow profile is tangent to the normaldepth line, sinae dyld:r: = 0 as Y Yn. If the amount of submergence at the downstream end is less than the critical depth, the flow profile will terminate abruptly, with its end tangent to a: vertical line at a depth equaJ to the criticald~pth, .since dyJdx = <0 for y = y.. This means the. creation of a hydraulic drop. If the depth of submergence at the down.stream end is than the critical depth, then as much of the profile will form as Jies above the water 'surface in the reservoir. Examples are the profile at the upstream side of a sudden enlargement of a callul cross Stctioll (Fig. 9-4c) and the profile in a cana'! leading to a reservoir, where the pool level is shown both above and below the critical-depth line (F'ig. 9-4d).

The 11£3 profile starts theoretically from the upstream channel bottom, at either a vertical-angle slope or all acute angle, depending on the t.ype of uniform-flow formula used (Art. 9-3), fl,lld terminates with a hydraulic jump ut the downstream end. This type of profile usually occurs when a supercritical flow enters a mild channel. The beginning of the profile, although it cannot be defined precisely by the theory, depends on the initial velocity of the water. The higher the velocity, the farther downstream the profile will begin. 'The theoretical upstream end of the profile will intersect the channel bottom. At this end y = 0, thus the velocity would become infinite. Therefore, the theoretical upstream end of an M3 profile can never exist physically. Examples of the M3 profil.e are the profile in a stream below a sluice (Fig. 9-4(3) and the profile after the change in bottom slope from steep to mild (Fig. 9-4f) . .B. S Profiles. So > S. and y" < Yo_ The Sl profile·begins with a jump' at the upstream and becomes tangent to the horizontal pool level at the downstream end. Examples are the

~l

229

THEORY AND ANALYSIS' (0)

J

/

/"

dy!d ••

--...._~~

+

C'_

---

--:~

/

230

,;1.

d

THEORY

GRADUALLY VARIED FLOW

profiles of flow behind a dam·in a steep channel (Fig. 9:-4g) and in a steep canal emptying into a pool of high elevation (Fig. 9-4h). The 82 profile is a drawdown curve. It· is usually very short and rather like a transition between a hydraulic drop and uniform flow, since it starts upsti·eo.m with a vertical slope at the critical depth and is tangent to the normal-depth line at the downstream end. Examples are the profiles formed on the downstream side of an enlargement of channel section. (Fig. 9-41.) and on the steep-slope side as the channel slope changes from steep to (Fig. The S3profile is also of the tra!Jsitional type, formed between an issuing supercriticoJ flow .and the normal-depth line to which the profile is tangent. Examples are the profife on the steep-slope side as the channel slope changes from steep to milder steep (Fig. 9-4k), and that below a sluic~ with the depth of the entering flow less than the normal depth on a· steep slope 9-4l). C. C P1·ofilcs. So and y ... = y•. These profiles the transition conditioi15 between Ai and S profiles. Assuming a wide rectangular channel,Eq. (9-17) shows that C1 and C3 pl'ofiles Iwe curved and that the Clprofile is asymptotic to a horizon tal line (Fig. 9-4m and n). When hha Ghezy fot'mula i~ used J Eq. (9-18) will show that the two profiles are horizontal lines. The C2 p7'ofile the case of uniform critical flow. D. H P1·ofili3s. 8 0 0 andy" = 00 • . These are the limit.ing cases of ]..f profiles when the channel bottom becomes horizontal. The H2 o,nd H3 profile3 correspond to the 1lf2.and 11{3 profiles, but no H1 profile can actually be est,ablished, since y ... is infinite. Examples of H profiles are shown i!l Fig. 9-40' and p. E. A Profiles. S~ < o. .' The A 1 profile is impossible, since the value of y. is not reaL The /12 and A3 profiles are similar to the H2 and H3 profiles, respectively. In general, A profiles occur infrequently. Examples are shown in Fig. 9-4q and r. F. Profiles in Conduits with a Gradually Closing Tflp. .For nny conduit with a gradually closing top, the normal discharge will increase as the depth of flow increases. It will increase first to the value of full discharge at a depth yo' less' than the full depth yo. Thereafter, the discharO'e will reach a m!)'ximum value Q,m at a depth Yn *. Further increase depth of flow will decrease the discharge eventually to the full discharge at the moment when the flow surface touches the top of the conduit. 9-5a .shows the variation of normal discharge in such a conduit. fll the case of a circular conduit, the depth Yo'·"" O.82yo and y" * = O.938yo where yo is the diumeter of the conduit (Art. 6-4). Within the region of IJ ya' and y = Yo, there a.re two possible normal depths for a given dis- . charge, namely, the lower normal depth y" and the upper or conjugate

A~D

ANALYSm

231

Following the principle used in the preceding paragraphs, it can be demonstrated that four types of How profiles are possible for a given slope [15-19J. Figure 9-5b, c, and d shows these fol' mild and st.eep 8lopes. The positions of the depths y" and y,,.' are assumed constant in

\

I

in

itorrnal depth y ,.'.

.

FIG. 9-5. Flow profiles in a closed condui~.

these figures .. It shuuld be noted that the critical depth in Fig. 9-5d is . greater than .the normal. depths y,.' and Y.., but that its corresponding lower normal depth is less than Yfo' and Yn. Consequently, the corre, sponding critical slope should be less th!l.l1 the normal slope, and the channel slope is considered mild.

(

[

232

~l

GltADUALLY VARIED FLOW

I (a)

~l ~--_.'_.::::=:-

____ _

M;ld slOn . ~e

/

I

~----~-'--'-'--" ,

233

THEORY AND A.NALYSIS·

, em)

--~ ///~~~~-~~~-:. -"....

CI ; ... ~------....

Cr' JlicG/

Milder slope

~ (q} ...~

-"-.

Slo,oe

'!

..

- - -. -~.--~- .:':'.:::.:..-:: .... -- -~ ..--~"-.----~

~'

'::""~~\\':~ Mild. slope

I

I

I

(n)

(s)

-.:.... (P)

~. ___ ."_-/:.~_.~~~~~~i.~~. Critic

Depending

MI

,._:~mp

Dependinc;; on downstream

~ (f) ---'---:::::.-_ _----:.----

~

(tJ

t2.

'lIl

downstrecm

con~?:

~ • P ./oPe

.

FIG. 9-6. Profiles of gradually varied flow in a long prismatic channel with bottom slope.

.

I

C"\

'\:\ .. Adverse. slope

"'-" ~---:);- --1~~'--

LEGEND

'0

1171777Tmn,.,!._

I " 1/

,

I

FIG. 9·6 (Continued).

:3 ,~" ,se s\O?e

(>.a~e

01 s~ /. OPe

Thlck lines indicate water surJoce

",,~y ~".--"';'1:.~.:;:- .. ---- Jump

Sfs e '

Depending on aownst ream

Cl'

11

hreak in'

9":5. Analysis of Flow Profile. Flow'"profile analysis is a proQedure used to predict the general shape of the flow profile. It enables the engineer to leatn beforehand the possible flow profiles that may occur in a given chann~l la.yout. This procedur~ constitutes a very significant part of all problems inchannel design for gradually varied flow. A. Pi'ismatia Channel with Constant Slope. The flow profile in ~ long prismatic channel with a constant slope:has been described in Ar~. 9·4.

Typical examples shown in Fig. 9-4 shduld be helpful in determining the type of flow profile in a given problem. B. Prismatic Cha.nnel with Ii Change in Slope. This channel is equivalent to a pair of connected prismatic channels of the same cross section but with differen~ slopes. Twenty typica.l flow profiles in a long prismatic channel with a break in'slope are shown in Fig. 9-(3, These profiles are· ' self-explanatory. Howevel;, some special features should be mentioned; 1.' The profile near or at the critical depth caml0t be predicted precisely i:ly the theory of gradu.~lly varied flow, since·theflow is generally rapidly varied. . , 2. In passing a critical line, the flow profile should, theoretically, have 0.

234

GRADUALLY VARIED FLOW

vertical slope. Since the flow is usually rapidly varied when the critical linej the actual slope of the profile cannot be predicted precisely by the theory. For the same reason, the critical depth may not occur exactly a.bove '~he break of the ehannel bottom and may be different from the depth shown in the figure. 3. In some cases (Fig. 9-6g and l), the hydraulic jump may occur either in the upatl'eam channel or in the downstream channel, depel1~il1g upon the relative steepness of the two slopes. 1 In case {I, for instance, the jump will occur in tHe dowllstream channel if the normal depth i~ this channel is comparatively smalL When the slope of the downstream channel decreases and,. accordingly, the normal depth increases, the jump will move upstream, eventually into the upstream channel. The exact location of the jump will be disclissed in Art. 15-7. 4. If the upstream channel has an adverse slope (Fig. 9-Bq to t), the discharge is fixed not by upst.ream channel conditions but by the elevation of the upstream pool level, which is the horizontal asymptote of the )12 profile. The procedure of analysis is .to assume a discharge and to determine to which case q to t the profile should belong. Then, compute the flow profile in the upstream direction and determine the pool level. H the computed level does not agree with the given pool level, then repeat the computation with another assumed discharge until the computed level agrees with the given leveL 5. Typical profiles (Fig. !H3) are illustrated for long channels in ,vhich a uniform flow can be established far upstream and downstream. C. Prismatic Chann,el with Seueral Changes in Slope. ;For such channels the general procedure of analysis is as follows: 1. Plot the channel profile with all exaggerated vertical scale. 2. Compute y.,. for each reach, and plot the normal-depth line, shown by dashed lines, throughout the entire channeL ' 3. Compute y. for each reach, and plot the critical-depth line, shown by dotted lines, throughout tile channeL . 4. Locate all possible control sections. At the control section,2 :flow must pass through a control depth which may be the critica,l depth; the normal depth, or any other known depth. There are three types of cont.rol section: a. UPSTREAM CONTROL SIWTION. This occurs in any steep reach at the upstl'eam end, since the flow in a steep channel has to pass through th$ critical section at the upstream end and then follow .either the Sl or S2 profile. Thecritical depth therefore, the control depth (see also Art.

235

THEORY AND ANALYSIS

4-5). If the downstream water surface is very high, it may raise the flow surface at the upstream control. When several st.eep reaches occur in succession, the control section is at the upstream end of the uppermost reach. Upstream control also occurs in long mild reaches, because the lVIl or lIlZ curves will approach the normal depth at the upstream end. b. DOWNSTREAM CONTROL SECTION. This occurs at the downstream end in any long steep reach, because the flow will approach th.e normal dep~h at the downstream end. If the downstream end of a mild channel terminates at a free overfall, the control Section 'may be assumed at the brink where the depth is crit.icaL r C. ARTIFICIAL CONTROL SECTION. This occms at a control structure, sur.h as a v;eil', dam, or sluice gate, at which the control depth either is known or can be determined. 5. Starting D,t the control depth at each control section, trace in each reach a continuous profile. The position of the profile in each reach can be corre~tly located with respect to the normal~ and critical-depth lines. For this purpose, typical profiles described previously (Art. 94) should be found useful. B. When flo'''' is super critical in the upsi:.ream portion of a reach but subcriticaI in the c1ownst,ream portion, the flow profile has to pass the critical depth soniewhere in the reach. In crossing the critical-depth line, a hydraulic jump is usually crsl:'.ted in raising the wat,et surfa<;e from a low depth to its depth. The exact location of the jump will be discussed later (Art. 15-7).

f

,I

"""""ntu,,, 9-1. Th":l normal and critical depths of the flow in a channel have been W'"'!,,'""_'" !Lnd ~,re shown in Fig. 9-7,

Sketch ~he possible flow profiles. First loeste the possible control sections, such n.s tho~e for upstream control U.C., downstream control D.C .. and artificial con~rol A.C. The artlficil.t.l con" trol in this e:
D, Nonprismatic Channels and Chan'n,els wilh Spat£ally Varied Flow. In nonprismatic channels and channels with spatially varied flow, the analJfsis of flow becomes complicated by the fact that the control section . may occur at any section in the channel and its position cannot be determined easily. Consider three different channel slopes for three types of :low: (1) continuous flow in aprisrnatic channel (Fig. g...ga), (2) spatially

1 Also depending on relative roughness and shape of the two connecting chan-' nels. In this discussion factors arc assumed constant. • The term "control section" used here has a. broad meaning. It refers to any' section at which the depth of flow is known or can be controlled to !l. required sta.ge.

1 Actua.lIy

-s.:~

the computed critical depth is somewhat behind the brink

(Ar~.

3-4).

\

236

a.

E

.a

M2 --- ...

a. E

_-

.:!.

Mild slope

Z

FIG. 9-7, Analysis of flow profile for .Exa.mple 9-1. D.C. = downstream control; A.C. = artificia,1. control.

Critical control sedion

U.C.

=

upstream control;

Critlcol control section I [;ubCritical flow

I

I'.

~ ~t Critical'

!S b control section

[s Upercrj~. flow 1------- ------l ,

U

critlca/

flow

f',O, •

%

CrJ t i co I 'C;:~~~".-L conirol section Critlcol

I

1ICoi

.~~ Critical control section

I

Steep

~

varied flow of increasing discharge in a prismatic channel (Fig. 9-8b), and (3) continuous flow in a nonprismatic channel having a constant'slope but varying cross section (Fig. 9-8b). All channels have a free overfall. In the first .type of flow, the flow changes from sub critical to supercritical as the channel slope varies from mild to steep. Accordingly, the critical control section will be transferred from the downstream end to the upstream end. It should be· noted tht1t the t,ransfer of the critical control section occurs almost instant.aneously .. At the instant of transi": tion/ the crit.ical flow takes place throughout the entire length of the chan.nel (middle view of Fig: 9-8a), and any section in the reach is, therefora, a critical sflction .. In the see-ond and third types of flow, the transition of the state of flow will take 'place gradually from section to section as the critical control section moves upstream, At any moment. during the' time of transition' (middle view of Fig. 9-8b), the flow downstream from the control section is supercritical, and the flow upstream is subcritical. The determination of the control section will be described in the next article. 9-6. Method of Singular Point. . The preQeding analysis of flow pl'ofiles was given primarily for prismatic cha,nuels. For a comprehensive treatment of flow profiles, in either prisrnatic or nonprismatic channels with constant or variable slope, use has been made of advanced mathematical approaches. 2 One a,pproach is the use of the theory of the singular point .. This theory was developed by Poincare [22] but ,vas first applied to flow studies in cha.nnels of variable slope by Masse [23]. The method based on this theory has been further discussed. and extended by Jaeger [241, De Marchi [25], Hom-rna [26,27J,Escoffier [28], Iwasa [29], and others. . For simplicity of discussion, channels with small slopes will be considered. Let the numerator and denominator of Eq. (9-15) be represented by two functions, or .

"

dy dx

leD'.~_:~_ l Superr:rir' flo", cal I

237

THEORY AND ANALYSIS'

GRADUALLY VARIED FLOW

---- .......

=

Sol.- (Q/Q,,)2 = F 1(3:,y) 1 - (Q/Q.)2 F2(X,y)

or

(9~20)

I

~,

Then, set each of these functions equal to zero, or ~~

(9-21) (a)

(b)

FI~. 9-8. Location of critica.l c'ontrot section. (a) Flow in a prismatic channel; (b1 flow ill a nOllprismatic channel or spatially varied flow.

and The solution of Eq, (9-21) Fi = 0 will giveQ = Q" or y = y".

(9-22) Hence,

1 At this ~oment, the normal depth coincides with the critical depth, bec.oming a sO-C!tlled transitional depth (Art. 9-6). 2 See another method of anillysis by Merten [20,211.

THEORY aND ANALYSIS

GRA.DUALLY VARIED FLOW 238 Fl = 0 represents t.he normal-depth' line in a prismatic channel. In

Eq. (9-20) gives dy/dx 3 D; that is, a flow profile must be hori:z.ontal'in crossing the transitional profile. Since the transitional profile is defined by the condition that Qn Qr, its position is fixed by the channel characteristics but, unlike the normal and critical profiles, it is independent of change in discharge. At ~his stage of the discussion, the reader must try to see clearly the differences between the transitional, normal, and critical profiles_ At the singular point, y Y.j thl;l,t is, the fiow profile passes through the· critical depth. Hence, there exists a critit.:al section. The slope of the water surface at the singular point is equal to the limi~ing value of the indeterminate formd.y/d';l; = % if this form is COllv"ergent. By the differential calc·J.lus, this limiting va.lue can be cvnJuated by

l

nonprif:lmatic channels, Fl 0 I'esults in a fictitious normal-flow profile. It is fictitious because U1~iform flow in nonprismatic channels is unrealistic. Similarly, Eq. (9-22), P 2 0, represents the critical-flow profile. Since, Q< is independent of the channel slope, the concept of the critical-flow profile is valid in channels, of variable slope.

Yl

A

,C

B

I

'j

I:

I

l

---~--l-... _______ '"

"

. ~I . ~~.-1- -~--;:::;:.::;:::;_.~ •.:__:,.' ..

~~... ---... - . . --:-.. ~ ........ . :. . .~-- ...!........

@.,-'-- -----"-'1'-"" '/ j /

;'I.L ,

tLLZ

'

'

\

\'\

,*""".". ' /

~;,'

/

If'

If"

' "

"

1/,'

/,

l ,

:

/

"

' I 't I

,

,. __ Reoillo w profile passing

,throJgh Pond osymptotic 10

Y

F,·O

) (d dx ,

--Flow p(o/ile pcssinq th(ouoh P . b\;t nc~ asymptotic fa F. =11' . - - - -F ,:Oor . quos,-normal flow ' ' <

profile

---Transitional profile ------otherf!ow profiles

3:,

r II

FIG. 9-9. Study of flaw profile ill

8.

OsHed by Masse [231

(9-24)

ohannel of variable slope.

In prismatic channels"F l = 0 &'ud F2 0 represent two pa:rallel lines. Ii1 nonpJjsmatic channels, however, the two profiles may intersect, say at point P (Fig. 9-9). At point P, Eq. (9~20) dy/dx %, an indeterminate form. Such a point is known in matheraal;ics as a sinfjllla1' . point. When Qn ;; Q., Yn= Y<, which is a depth known as the tra.nsitional depth. 2 The curve representing this depth is called the lransitt'onal profile. A flow passing through this profile at the singular point will change its state from sub critical to supercritical or vice verJ'la. In fact, the' transitional profile must pass through the singular point, since at this point Qn = Q. = Q or y,. = y. = y. At other places where Q~ = Qo = Q, I

(9-23)

The above discussion can be illustrated by a simple example. Consider a wide rectangular channel 9-9) in which the flow changes from sub. critical to supercriticaI according to the change in slope from a subcritical SI in reach I to a supercritical 3 3 in reach III. . The slope in the intermediate reach II is variable and may be expressed by SD = 3 1 + U';I; in which U is a coefficient.. Somewllere in reach II, at which So = a critical section exists. This section can be determined by the simultaneous solution of FI 0 and F~= O. Graphically, this section lies at the intersection of the curves FlO and F 2 = O. The cl'itical depth can be shown to be

----F2:O Of critical-flow profile

I

239

qtw,si-norma! flow.

, Also known as the chq.r.acterntic dept)" whleh was first applied by Mouret [12J und later was discussed' by Lazard [301.

to uniform ch!l.nnel~,

Considering a unit width of the channel, the critical depth is constant t~roughout the length of the channel; hence, the profile F2 = () is equi-

dIStant from the channel bottom. In reach I the flow is subcritical' hence, the profile PI ashould be above F~ O. In reach III the flo\; is supercritical, and the profile FI 0 should be below F\ ~ O. The flow profile under consideration is shown by the heavy full curve in the shaded 0 in Fig. 9-9, area between PI = 0 and F2 Since t,he transitional profile is defined by Qn = Q" its equation can be shown to be

I I

[

y

(9-25)

At the critical section, where 3 1 + ax = t~e above equation bec'omes Eq. (9-24). To determine the slop~ of .the water surface at the singular point in the

,1 \

I

~I 240

GRADUALLY VARIED FLOW

prese~t e~ample by Eq, (9-23), the followingequntion can be obtained (using Manning's formula): 2 dY) ( dx •

108 0 (d Y)

dx ,

1J: ·(~§o) 3

dx

= 0

I I

(9-26)

c

This equation has two roots opposite in sign. The negative root will define a flow profile through the singular point P, wi.th depth decreasing in the direction of flow. Since the r_ow in the present probJeCljl changes frorn. 8ubcl'itical tosupercl'itical; this flow profile is evidently the profile of the flow under consideration. At x = ± this profile is ttsymptotic to the profile of the quasi-normal flow, or FI = O. The other root of Eq, (9-26) "till define a. flow profile also passing through P but not asymptoti'c to £1\ = O. Other profiles, which do not pass through will intersect the transitlona.! profile with a horizontal slope, pass through Pi = 0 with a hydraulic jump, and become asymptotic to the two profiles that pass through P. The flow profile through P but not asymptotic to PI =0 is not the profile of the flow under consideration. However, such a profile may become real under some other circumstances. In 9-10, C() , •

241 Iu3 these operations are carried out, the jump GH is forced upst.ream until THEORY AND ANALYSIS

its height is finally reduced to zero, In this way the profile CP D is realized. SUCh a pI'olile, hm'v'ever, is unstable, because a. very slight increase in level in the profile P D will entail a change'from the profile CP to the proille AP and the space between the two will be filled with water. A general solution of the condition dy/dx FdF2= % for any kind of channel can be aohieved mathematically by the method of singular point [31]. The result will produce four types of flow profile that can be .developed theoret,ically around the singular point.l The four types are known as the saddle) noda.l, spiral, and vorlex. types, as shown in Fig. 9-11, y

n

. Saddle

Spiroi

FIG. 9-10. ·Illustration of flow pro.5.les defined by a. singular point.

for example, I P represents a singular point; APB is'the profile through P and asymptotic to 0; OP D is the profile passing through P but not. asymptotic to F 1 = 0; llnd EFG and HI al'e other profiles not passing through p, The upstream gate has been adjusted to the sti.percri.tical profile EFG and the downstream gate toi give the subcritical profile H I. The tmnsitionfrom one to the other takes place by means of the hydraulic jump GH. upstream gate is now raised slowly to cl).usethe profile EF to app~oach OP as limit. As tpis is done, the pool . is adjusted to keep the discharge constant, Simultaineously, the downstream gate is !ow61'ed to cause the profile HI to apPlioach PD as a limit.

a

: I This exa.mple wiw:! suggested .by Mr, F. F. Escoffier of the U.S. Army Corps of Engineers. For simplicity, the.curv6 for Fl = 0 is not shown)n Fig. 9-10.

Nodal

Vort ••

FIG .. 9-11. Flow profiles around El. .~ingula.r point, P =0 singular point; F flow profile passing through P and asymptotic, to F, == 0; F' = flow profile passing throl.\gh P but not asymptotic to F • ... 0; f ... o~her flow profiles; F, = quasi-norma.l flow profile; F.= oritical-flow profile; and T ... llrallsitiomu profile.· ,

I I ~

I

where the profiles are plotted with the ordinate Y representing the distance above a horizontal datum and the abscissa X representing the distance along the "datum. . . The p0s4ble flow profiles pre$ent~d in the example of Fig, 9-9 are of the saddle type. In this case, the flow profile that passe~ through P and is a~ymptotic to FI 0 indicat 9ll eo continuous How: changing I

For further discussion, see [23]ll.nd [28].

242

, l Ii

GRADUALLY, VARIED FLOW

243

THEORY AND ANALYSIS

from sub critical to supercritical in a channel with slightly convex bed (8 1 < Se < Sa). In the nodal type, the,flow profile that passes through P a!ld is asymptotic to F1 '= 0 indicates a continuous flow changing from super critical to subcritical in f1 channel with slightly concave bed (8 1 > Se > S3). In the spiral type, the flow profile that pa.sseg through P and is asymptotic to Fl ;"" 0 indicates a discontinuous flow 1 changing from supercritical to subcritical in a channel with sligMly concave bed (8 1 > Se > 8 a). In the vortex type, the, flow profile that passes through the singular poiut is the point itself and has no hydmulic significance. A general solution for the transitional profile in all four typ~ h
1 , J

.J

For purposes of dis,cussion, take the ,case of the backwater behind a dam as an example. When So < BL (Fig. 9-12a), it is appal~ent that the flow is always subcritical and that the corresponding b'ackwater profile is of the' M1 type. When So > SL, the condition depends as follows on the relation of Bo to lh.

II I

I -' (

I

(oj

_---"1 -=-..--T,'- __ _

.. "--

1 -.. - __ a

(9-27) ~

Let K" = Q/V8o,K = L49AR%/n, Z, Then, the above equation becomes 2.22RY.iSo

=

= n~gD

Q/VU, and "

Z

=

A·,fIT. '

s

S

//,

UncriticO/.S/o pe

(9-28)

This is a theoretical condition for the establishment of the transitional depth.: It indicates that the transitional depth depends only 011 the channel geometry, roughness, and slope. This equation contains no discharge; therefore, t.he tranllitiOlial depth is independent of the actual discharge. It is logical to say that there is a certain discharge Qt that occurs at the transitional depth Yl./· This discharge may be called the transitional discharge. According to the definition of the tranl?itional depth, the transitional discharge shoilld be a normal discharge and also a critical discharge. Referring to the critical-slope curve discussed in Example 5-5 (Figs. 5-8 and 9-12), the transitional discharge can be represented by a point on the curve. It is evideilt from the curve that, for a given slope B~, whit::h is greater than the limit slope BLI there are two possible critical discharges, say, Q., and Qb, both of which are transitional discharges. The actUal discharge is designated by Q and the corresponding critical slope by Be. '

, (d)

fi---'

PAl

FIG. 9-12. Flow pro.files expla.ined by a critical-slope curve.

A. When the slope is close to the limit slope.

I

Here .the 'upstream normal flow changes to tbe downstream normal flow at an abrupt tr'ansition formed by a hydraul~{l jump. , • See [f2], [II)], [28] to [30]. and [32] .•

In this case, the condition

will depend further on the magnitude of the actual discharge Q with

, I

1:

I

.1·,

respect to the smaller and larger transitional discharges Qo and Qb, respectively. '

\ )

0,,)

244

\"j

I"

i

t' i.

I

J

THEORY AND ANALYSIS

(Fig. 9-12b), tIlen So < Se and Ye < y" < Ya < Vb. Since So < Sc, the flow is sub critical, and the profile should be of the 1VI1 typ~ . However, the profile will contain two points Ta. and Tb at which the slope is horizont.aL Bet-';veen these two points a p(')int of inflection apparently exists. The depths at the two poihts are transitional depths Ya. and y~. If Qa < Q < Qb (Fig. 9-12c), then So > S,and y. < Yn -< y. < Yb. Since So > Sc, the flow profiles are of the S type. However, there will be a point Tb where the slope is horizontal on the Sl profile and a point T a where the slope is horizontal on the 83 profile. If Q. < Qb '< Q (Fig. 9-12d), then So < S, and Y. Qb, the flow will be sub critical and the profile will be of the M 1 type. If Q < Qb, the flow will be supercritical and the profile .will be of the S1 type. C. When Ihe. slope is very large. In this case, the large transitional discharge Qb is considered to exceed the maximum expected discharge (see Fig. 6-8). Thus, the flow is supercritical and the profile is of the S1 type. The highest point of theS1 profile is very close to the downstream end. The above discussion was developed for the case in which the point L of the limit slope is below the curve of maximum expected discharge (Fig. 6-8a) and in which the channel sections !1re rectangulal' or trapezoidal or similar to such forms. If the point L is above tbe curve of maximum expected discharge (Fig. 6-8b), the larger l;ransitional depth of the flow will be greater than the maximum expected depth, or Yb > Ym, and the larger transi~ional discharge will be greater than the maximum expeeted discharge, or Qb > Qm. The foregoing discussion, however, remains valid as long as the actual discharge Q does not exceed Qm. If Q exceeds Qm, the discussion has no practical meaning. Similariy, the flow profiles remain t,he same, but the useful part of the profiles will be where the depths al'e less than y",. If Q

."

GRADUALLY VARIED FLOW

.1 '",,

< Q.

LEGEND:

-------~-

FIG. 9-13. Channels for Prob. 9-7.

EIJ272 :s2 EI.f270

--x-jEU266

..

(""I'/;":~:;::(

.-+----j.500'----~.t
v

[= I

''''11'//''7777,,---,1

FIG. 9-14. A channel profile for Prob. 9-8.

9-8. A rectangular channel (Fig. 9-14), 20 It wide, consists of three reaches of different slopes. The channel has a. roughness coefficient n = 0.015 a.nd carries a discharge of 500 cfs. Determine: a. tbe norma.l flnd critical depths in ea.cb reacb

I )

The vertical scale is exaggerated.

EI.1274

9-1. Show that the wnter-surface slope S" of a gradually varied flow is equaJ to the slim of the energy slope S and the slope d~e to velocity change d(", V'/2g)/dx. 9 c 2. Show that the gradually-varied-fiow equation is'reduced to a uniform-flow formula if du/dx ... 0, 9-3. Prove Eq.' (9·14). 9-4. Prove Eq.: (9-15), 9-5. Prove Eq. (9-16). 9-6. Prove Eqs. (9-17) and (9-18).

I

Criticol-c\eplh line

- - - -Noqnal-deplh line

PROBLEMS

I

245

9-7. f?ketch the possible flow profiles in the channels sbown in Fig. 9-13.

,:1 "

246

GRADUALLY VARIED

THEORY AND ANALYSIS

~'LO"'Y

b. the possible (low profiles c. .the distance ;~ troln the outlet of the channel to the poiht where the backwater curvs terminate~. The .backwat,er curve is assumed. to be a horizontal line.

9-9. Prove Eqs. (9-24) to (9-26). 9-10. A change in slope from 0.0016 to 0.0064 occurs in a wide rectangula.r channel. (Fig. IHI). The length of the transition is 10 ft, and the slope in .the transition rea.ch is So = 0.0016 + 0.00048::, where x is the distance measured from the beginning of the change. 'The channel carries a discharge of 100 crs per unit width. Assume that a = 1 a.nd n ~ 0.02. a. Determine the control section. b. Computethe slope of tho flow profile nt the contrqi section. c. Con~truct the tro.nsftional, normal, and critic!>l proliles .. d. Construct the renl and some other possible flow pI·oliles. 9-11. Show' that the gradually-varied_flow cq\!ation for flow in a rechar.gular channel of va.riabli!. width b may be (lli:pressed as =

S. - Sf

+ (",Q'ylgA~) (dbldz)

1 - ",Q'b/gAJ

...._........

(9-29)

All notation has been previously defined.

REFERENCES 1. J. B. Belanger: sur Ill. solutionnumiirique de quelques problemes relatifs au mouvement. permanent des eallli: courantes" ("Essay on the Numerical Solution of Some Problems Reletiv~ to the Steady Flow of Water"), Cn.rilian-Goe\lry, Pa.ris, 1828. . 2. J. A. Ch. Bresse: "'Cours de mecJl.nique appliquee," 2e po.rtie, Hydraulique ("Course in Applied l\'Iechanics," pt. 2, Hydraulics), Ma.1let-Bachelier, Paris, 1860. 3. Boris A. Bikhmeteff: "Hydraulics 'of Open Channels," appendix I, Historicnl and bibliographical notes, McGraw-Hill Book Cr)mpany, Inc., 'New York, 1932, pp.299-301. 4~ Charles Jaeger: Steady flow ill open channels: The problem of Boussinesq, J Ot,rnal, lnetitution of Civil Engineers,. London, vol. 29-S0,.pp. 338-348, November, . 1947-0ctober, 1948. 5. Charles Jaeger: "Engineering Fluid M.echanics," translated from ~he Germa.n lly P. O. Wolf, Blackie & Son, Ltd., Glasgow, H,56, pp. 93-97. 6. F. Bettes: Non-uniform flow in channels, Civil Engineering and Public Works Review, London, vol. 52, no. 609, pp. 323-324, March, 1957; no. 610, pp. 434-436, April, 1957. . 7.•(-Allen: Streamline and turbulent flow in open channels, The.Lond(}tl, Edinburgh and Dublin Philosophical Magazine and Journal Science, ser. 7, vol. 17, pp. 1081-1112, June, 1934. S. Hunter Rouse and Merit P. White: Discussion on Varied flow in open channels of adverse slope, by Arthur E. Ma.tzke,. Trallsadion.s, American Society of Civil Engineers, voL 102, pp. 671-676, 1937. 9. ShermanM. Woodwa.rd and Chesley J. Posey: "Hydraulics of Steady Flow in Open Channels," John Wiley & Sons, Inc., New York, 1941, p. 70. 10. Ivan M. Nelidov; Discussion on S~face curves for steady nonuniform flow, by Robert B. Jansen, Tra.nsa.ctions, American Society of Civil Engineers, vol. 117, pp. 1098~1l02, 1952. .

0;

:

',~-

247

11. Dwight F. Gunder: Profile curves for open-chruloel flow, Transadions, American Socieey IJj Ci.-il Engineer", vol: lOB, pp. 481-488, 1943. 12. G. Mouret: "Hydraulique: Cours de rneca.nique appliquee" (It HydJ:aulies: Course tn Applied Mechanics "), L'Ecole Nationalc des Ponts et Chaussees, Paris, 1922-1923, pp. 447-458; revised lIB "Hydraulique giinerale" ("General Hydraulics"), cours de 1'Ecole NatiQnii.le des Ponts et CtwLusseesi Paris, 1927-1928. 13. A. Merten: Recherches 'sur Ill. forme des axes hydra.uHques dans un !it prisma.tique (Studies on the form of fiow profiles ill. a. prismatic channel), Anna!e.~ de jl.:l18socic:Ucm de. In(16nieuTS Bortis des ~cole8 8pec.iales de Gand, Ghent, Belgium, '101.5, ser. 3, 1906. 14. M. 'Boudin: De I'axe hydraulique des coms d'eau contenus dans un lit prisma.tique et des dispositifs reaiisll.nt, en p~!l.tique, Bes formes diverses (The flow profiles of w.ater ill a prisma.tic ch!1nnel' and actuI'1.1 "dispositions ill various forms), A nnales des trcwaux pu.bliqucs de Belgique, Bru~s!'
......

'

(

.~

I

248

\

I

! I

r \

GRADUAl;.LY VARIED FLOW

26. l4asashi Horn-fill.: "General Hydraulics" (in Ja.panese), vol.'l of "Applied Hydraulics," cdiLcd by Masashi Hom-mil. and Tojiro Ishihara, Mal'uzen, Tokyo, Bi58, pp. '108-11 I. 27. Masa.hi Hom-ma and Sukeyuki Shima: On the flow'in a. g~adually' diverged open channel" The Japan Science RevifIW, Series I, Tokyo, vol. 2, no. 3, pp. 253-260, 1952. 28. Francis F. ,Escoffier: Transition profiles in nonuniform cha.nnels, Transactions, American Sociqty of Civil Engineers, vol. 123, pp. 43-56, 1958. 29. Yoshiaki lwasa, Hydraulic signifi~ance of tl'B,nsitional be,haviours' of flows in channel transi~imls and controls, M8mdirs ,of Ihe Faculty oj Enftineering, Kyolo Uni~e1'$ity, Japan, vol. XX, no. 4, pp. 237-276, October, 1958. 30. Achille La~ard: Contribution. A I'etude theorique du mouvement graduellement vo.rit! en hydraulique (Contribution to the theoretical study of gradually varied fiow in hydraulics), Annllies des ponls el c.hausseeil, vol. 117, 110. 2, pp. 185-219, March-April, 1947. 31. Theodore von Karman and Maurice A. Biot: "MatltelU!l.tical Methods in Engineering," McGraw-Hill Book Company, New York, 1940, pp. 150-158. en regime permanenV' 32. :E. Crausse: "Hydraulique de.~ (lllnallX ("Hydraulics of Open Channels with Stea.dy Flow Regime"), Editions Enolles, l"aris, 1951. .

CHAPTER

METHODS OF COMPUTATION

i

!

, 1

i

f·

10

The computation of gradually-varjed-ftow pl'ofile7 involves ba.sical~y the oolution of the dynamic equation of gradually vaned flow. The mam obTec Ive o. e com u..,itl.lOn lSO - 8 "ermme the shape of the flow profile. Br~a y elassif1ed; tl1ere are three methods of comput!ttioIli namel I;h. graphical-integration method, the direGt-integra~ion met cd, and the step. method. The development and :proce({ure of several typical meth~ ods will be described iri this chapter. 10-1. The Graphical-integration Method. Tpjs method is to integrate the dynamic equation of gradually varied fiow bya graphical procedure. Consider two · channel sections (Fig. lO-la) at (a) · distanpcs Xl and X2, respectively, from· a chosen origin and with corresponding depths nf flow Vl and Yz. The distance along the channel floor is Xl .

=

['". dx =.lY'dX-dy dy

-,'"

y.

(10-1) o~~-~~~~~~~~~----~Y

Assume several values of y, and (b) compute the corresponding V!11ues FIG. 10-1. Principle of the gl'.!Iphical-inte· .bf dx/dy, which is the teciprocal gration method. . the right-side member of a grad. . ually-varied-flow equation, say Eq. (9-13). A ,Curve of y against dx/dy is , then constructsd (Fig. lO~lb). Accbrding to Eq. (10-1), it is apparent that the value a!. x is equal to the shaded area formed by the curve, the y axis, and the ordinates of dx/ dy corresponding to Yl and' Y2. This area call be measured and the value of x determined.

of

I

249

I

.'-'"

250

1 in

This method has broad applicatIon. It applies to flow prismatic as . well as nonprismatic channels of any a.nd slope. The procedure is straightforward and easy to follow. It may, however, become very laborious when applied to actual problems. A relatively simple example will be given as an illustra.tion. Example 10-1... A trapezoidal channel hnving b "" 20 ft,'z = 2, So = 0.0016, and' !l. discha.rge of 400 cfs. Compute. the backwater profile created by a dam which backs up the water to a depth Ilf 5 n imm'3diately behind the cam. The upstream end of the profile is assumed at a depth equal to 1% greater than the normal depth. The energy coefficient 0/ "" 1.10. Solution. Following the solutions of Examples 4-2 and 6-2, the critical and nonnal depths are found to be·y, ",; 2.22 ft and y.. 3.35 it, respeotively. Since y. is.$reater than y. and the flow startswitb. a depth greater than Yn, the flow profile is of the 21(1 type. The section factor Z, = 400/VIJ/" 74.0 an:! the conveyance 1{n = 400/ ,/0.0016 = 10,000. . For simplicity; the channel bottom at t.he sitE' of ~ho dam is chosen :l.S the origin and the ~c va.lue in the upstreltm direction is t!l.ken as positive. The C'lmp11tation of dx/dy by means of Eq. (1l-13) is given in. Ta.ble 10-1 fOf various values of 11 varying from 5 it

n "" 0.025 carries

T
10-1.

COMPUTATION OF THE FLOW PE!OFlLEl FOR EXAMPLE

10-1

.

251

MlllTaODS OF COMPUTATION

I

GRADUALLY VARIED FLOW'

!

,I

(

K = L49AR% = .::.:..c.:....;;..,~~~__2~.~32_3 = 20,800 . n

!

290.2. 1 - (74.0/290.2)1 . = 760 1 - (10,000/20,800)'

\,

Yalues of yare thon agr:.inst the cor~tlspondi~g values ofax/dy (Fig. 10-2). The inore~nents in area AA are planimetered and listed in the table. According to Eq. (10-1), the cumulative v-a.luea of !lA should give the len6th ;r; of the flow profile. Finally, the backwater profile is obtained by plottingy against x (Fig. 10-3),

"OOT

I

I

\ .

"ot

-

\

j

6,000r

Br-

GJiAPRICAL INTEGRATION

n

400 cfs

Q

v

T

5.00

40.00 ag.20 38.40 37.60 \ 36.80' 36.00 i 35.20 i 34.801 34.40 34.20 ! 34.00 1 33 ,88 33.76

4.80 4.60 4.40 4.20 4.00 3.80 3.70

3.60 3.55 3.50 3.47 3.4\1, 3.42 3.40 3,36

So = 0.0016

= 0.025

R

A

!/n = 3.36 ft. .

2.22 ft

l(

Z

AA

d:t:/dy

0/

= 1.10

I

x

150.00 142.08 134.32 125.72 119.28 112.00 104.88 101.38 S7.92 96..21 \14.50 93.48 92.45 33.68. gl.aO 91.12 33.60 33.44 89.78

I

i

3.54 3.43 3.31 3.19 3.08 2.96 2.84 2.77 2.71

2.68 2.65

2.63 2.61 2.60 2.59 2.56

T 40.00 ft A "" 150.00 fP R '" 3.54'ft 2.323

2.323 i 20,800 290.2 2.274 11t,230 270.4 2.221 17,770 251.5 2.167 16,360 . 232.3 2.117 15,050 214.5 2.052 13,750,197.5 2.006 12,550 181. 0 1.972 11,910 . 173.0 1.944 11,350 1155.0 1.929 11,060! 161 1 1. 915 10,800 157.3 1.904 10,600 155.2 1.894 10,440 153.0 1.890 10,340 151,7 1.886 10,230 150.0 1.872 10,000 147,0

For insta.nce; when !I

760 79:! 155 155 836 1(;3 318 913 ' 175 493 1,000 191 684 1,140 214 8913 1,430 257 1,155 1,750 l!,\} 1,314 2,260 201 1,515 2,770 126 1,64.1 3.480 /156 1,797 4,520 120. 1,917 5,990 i 158 2,075 7,930 139 2,214 10,760 187 2,401

I

I

"

5.000[-

-

to 1 % greater than Ii. or 3.40 ft. columns of the table are

rm

R~;,

y,

I I

d, d)'

I

4.000~ i

'MO~ I

200i

I

I 1 I I II

',OOor

. 151011

IINI::!I 11 II I I I

5.00 ft, the values in other

o

!

0

(\J

t
<.0

r'l

",I t:::1 ~I ~! 4

I

I I

-

..

5 Y

r
. FIG. 10~2. A curve of y vs. dx/dy.

\ 1

;:;., 252

GRADUALLY VARIED FLOW

10-2.

EXISTING METHODS OF IN1.'EGRATING THE G&A.DUALLY-VARIED-l'LOW EQUATION

T,uiLE

Ye6.r of puhll-

Type 01 cba"nel

InvOBtigo.tor

catioo

1848

Dupui.

1850

Bresse

1875

Gr... hof

l!;il!(!

R1iblm~nn

1896 1900 1912 (1932) 1914

FIG, 10-3, An .llf1 flow profile computed by the graphical-integration method,

10-2. Method of Direct Integration.

The differential equation of gradually varied flow cannot be expressed explicitly in terms or y for all types of chn.nne! cross section; hence, a direct nnd exact integration of th., e~llation is practically impossible. M..fl.PY. ~~~.!;l,!IlpJ.s. Jt~YJL_hrum_ma.dlL _i;l.~r to solY~t..~.~~ig.g,!Cl.r a few special cases or to introduce as~­

_.Ji2ns~.:"!l.l:l~~..~.!!~~tioll ame~a bl~jQJnath~ma.ticaLintegrati.o.n.

Table_~~~2.~iv:~s_::\...~~.~~~X ..~xi~Hrlg methods of dii'eJ:;i.integ.I:aJ:.i.c>n, .. /

arranged chronologically.'" Although the list is incomplete, it provides !l generarJile}iofUie deveTopment of the dil'ect-in tegration method. Note that most of the early methods were developed fOl~ channels of a specific cross section but that later solutions, since Bakhmeteff, were designed for channels of all shapes, Most early methods use Chazy's formula, whereas later .methods use Manning's formula. . In .the Bakhmeteff me~hod [8] the channel length under consideration is divided into short reaches, The change in the critical slope within the small; l'o.nge of the va.rying depth in each reach is assumed constant,1 and ,

1S

I In the Bakhme(eff method, Eq. (9-14.) is used. The coefficient r in this equa.tion a.ssumed cOlUltllnt in the reach. Thus, it ca.n be shown tha.t the ra.tio of the change

Broad'tee. tangle Broad rso· . tangle Br?ad ree-

I

Effect or change iu kinetic energy

1928

1930 1938 1947 1~51}

1954

Veloeity formula

I

N - 3, M = 3 JV"", 3. fyf = 3

i2J

Con.id,..."d

ebhy·

N = 3, ftf - 3

[31

N-3,M=3

l~]

22'.1~'1I:·uSil>lJ

Ignored

CMzy

Broad reoConaidered C'1lQ ,?3"lj tangle Seboldiaeh IIras.d r~e· Ignored CB m 8' . tangle Mononobe AU.hape. 1 COlloli'idered OIl~S'" Le. Ail Conoidered Manning VQn 6egll'Olrn All Considered Manning Keifur-Clm Ci.rc~lnr. but Con.idered M .. nning .he method mAY be ex· tanded t.o other 3hB.pea

I

tiJ

Ohby eM.y

'"ngl.

Approximate trApezoid

KotallY

Refer-'cnce

Ignored

Cb~.y Ignored Brond ,... •• tan;:le Tolkrult~ Considered Cilhy I Broad para· bola Masoni Common Tee"' C"".idered CM.y '"ng)e Bakhmetefl All eh"pea Con.idero.d CMz:!, by.teps C1l{)·nBo.~ Senafl'nnak. Broad reo19:nored 23.7~'1IFJ.TlIS~~HI!J Ehrenberger ~angle

BIltic16

Mnu mptions rOr· . hydra.ulio expon.ents.

Oon.idered

I

1921

It should be noted tllll.t, when the depth approaches the no~mal depth, the incremental area varies so greatly with the change in y value that it becomes difficult to pirmimeter. In such Il. c.tl.Se, the area. may be computed by assuming it is a trapezoid. For instnnce, the incremental area between y '" 3.42 a.nd 3.40 is t,,1 - (7,930 + 10,7(0)(3.'12 3.40)/2 187. .

253 ..

METHODS OF COMPUTATION

N = 4, M - 4

lSI

N=a.1I1=3

(5]

.It:; c: UH

Ii,S)

N ~ 3.5. M = II N = 3.552. M n= 3 N = 3. HI, M: = 3 [{:t .- ol" A,.1R. where - is a varilll}le and dll/dz = eonet N - 3.;1. 1>f -.8

I~J

1111

N=2+2m.~1-3

(12]

0:::

•

P

1£

UI;OD"\ A.!c :rv~

K:

tt

K.:

ct lI'~1

None

tt

tf°M:t

AI

cc lI CO I.l,at

Z2

1%

1JJl

[IOJ

lIS1 [14.15] [16] (17)

the integration is carried out by short-range steps and with the aid of a varied-flow function. In an attempt to improve Bakhmeteff's method, Mononobe [13] introduced two asllumptions for hydraulic exponents. By these assumptions the effects 9f velocity change and friction head are taken into account' integrally without the necessity of dividing the channel length 'into short reaches. Thus, the Mononobe method affords a more direct and accurate computation procedure wherebyresuH.s, can be obtained without recourse.

to

in kinetic energy the friction slope, or r in Eq. (9-14), i3 a.ssUI;ned cOMbnt in each rea.ch. Since an increa.se or decrease in depth will cha.nge both these fa.ctors in the sa.me direction, their ratio is relatively stable a.nd can-be IloBsumed constant for pralltical purpose!!.

254

to successive st.eps.· In applying this method to practical problems, it has been found !.ha,t the first assumption (see Table 10-2) is not very satisfactory in many cases. Another drawback of this method perhaps lies in the difficulty of using the accompanying. charts, ."hich are not sufficiently accurate for pl'actical purposes. Later, Lee Tl4] and Von Seggern [16] suggested new assumptions which result in more satisfactory solutions. Von Seggern introduced a new varied-flow hmction in [t,dclition to the fUllction used by Bakhmeteff j hence, aD additional table for the new functicn is necessary in his method. In Lee's method, however, no new. function is required. The method [18] described here IS the outcome of a study of many existing methods.' By this method, the hydraulic exponents arE; expressed in terms of the depth of flow. From Eqs. (6-10) and (4-6), /{,,2 = Gly"N, /(2 = G,yN, Z.2= C2y,M, and Z2 = C2 y M, where G, and C2 are coefficient,s. If the;3e expressions are substituted in Eq. (9-13), thegradually-v.aried- .' flow. equation becomes (10-2) =

~ ~: [ 1 -

1

~

ltN

+ (~:)'{ 1U~-:;N ] dtt

(10-3)

This equation can be integrated for the length ,; of the fio,v profile. Since the change in depth of a. gradually varied flow is generaIJy small, the hydraulic exponents may be assumed constant within the range of the limits of integration. II:_. ?-~I1~e ..U!.e _hY'dra~lic eXQonentL1l
x= sy"o[u - )0f" 1d~t N+ (1I,)'1f ("1·UN- UN duJ + canst - 1~. . Yn }o - u

(10-4)

The first integral on the right side of the abo,re equation is designated by F(u,N), or ' F(u,N) =

,

fu ~ 1 -:- UN

(10-5)

)0

which is known as the varied-flow- function. The second integl'al in Eq. (10-4) may also be expressed in the form of . the varied-flow function. Let v = U N1J and J = N I(N - Ai 1); this integral can be transformed into

+

fv. )0

U N- M

1 -

.'

UN du

=

J f' N )0

dv 1 - vJ =

.J

N F,v,J)

( 10-6)

)0

-

(10-7)

J

V

This is a varied-flow function like F(u,N), except that the variables u and N are ;'eplaced by v and J, respectively.l . Using the notation for va,ried-flow functions, Eq. (10-'1) may be writt.en

or where

A

x

=

x

=

=~:,

t

[ 11 -

F(u,N)

A[u - F(u,N)

_ (y,)M J -,

B -

y"N

, 'M J 1 + G:) N Fev,J). + canst

+ BF(u,J)] + 1t =

't,

L

= x~

-

\

I.

(l0-8) ( 10-9)

const J

Yn.

=

N

. .-

N - M+ 1

and where F(u,N) and F(v,J) are varied-flow functions. By Eq. (10-9), the length of flow profile between tVTO consecutive sections 1 and 2 is equai to

yly,,; the above'equation may be expressed fOl' dx as dx

fCv,J) ';", . f" 1 dv

where

=

Let u

255

METHODS OF COMPUTATION

GRADUALLY VARIED FLO","

. 'j

Xl

A! (u, - u,) - [F(u2,N) - F(1I:"N)]

+ B[F(U2,J)

- FCli"J)] I (10-10)

\

-I

where the subscripts 1 and 2 refer to sections 1 and 2, respectively. Equation (10-10) contf~illS variedLflow functions, and its solution can be simpiified'by the use of the varied-flaw-function table, which is given in Appendix D.2 This table gives values. of F(u,N) for N ranging from 2.. 2 to 9.8. Replacing values of u and N by cOlTesponc!lng values of v and J, this table also gives values of F(v,J). This transformation W:l.S also performed independently by Levdlg]. 1,. " , .The prep:l.ration of ~uch a table wr,s undertaken !LId performed for the first ;time dUrIng 1914-1915 by Lhe Hasearch Board of the then Russian Reclamation Service under the direction of Boris A .. Bakhmeteff, then Profes:;;')r of General and Advancc'd . Hydraulics at Polytechnic Institute Emperor Pet.er the Great, St. Petersburg, Russi:l.. It was s8.id that the work had involved a long and tedious procedure [8]. In the turmoil of the Russian Revolution in 1917, the table so computed becaIne umwailabl e; so the task·of computing was done over :l.ga.in by Professor Kholodovsky and partly by Dr. Pestrecov. The reeomputed table was more precise and complete, covering a range of N from 2.8 to 5.4. This table was published in 1932 [8l When Bakhmeteff becELme Professor of Civil Engineering at Columbia University. In the meantime, in the U.S.S.R. iIi 1928, the unavailable table,oopied in hand-written form, had been published in a second edition of [71 with values of N equal to 2.0, 2.5, 3.0, 3.25,3.5,3.75, 4.0, 4.5, 5.0, and 5.5. Methods of computing the varied-flow-function tahle a.re explained in pp. 303-305 of Bakhmeteff's book [8J. An error in printillg should be noted in that. book; that is, a nega.tive sign shoulci be placed in front of the ent.ire right-side member of the equation printed neat the bottom of p. 305. Table D-1 given in Ap'pendix D is an extension of Bakhmeteff's table to almost triple its original 5iz!). It was prepared by the author in 195~1954 for teaching purposes at the University of Illinois, and was published in 1955 [18]. 1

._.

(

1

\

./

.. ''1'.--

I

I

L ,

~.,,-..:~ In computing, a flow profile, f,!rst the flow in the channel is analyzed J

(

I ,

l \

Example 10-3.

b

(Art. 9-5), and tho channel ,is divided into a numherofreaches. Then, the length of each reach if! computed, by Eq. (10-10) from knmV'll or assumed \lepths 8.tthe ends of the reach. Tile procedure of computation is as follows: '1. Compute the normal depth y" and critical ,depth Yo from the given data Q and So (see Arts. 6-6 and 4-4). ' ' 2. Determine the hydraulic exponents Nand M for an estimated average depth of flow in' th~ reach uuder considerl1tion (see Arts. ~3 and 4-3). It is assumed that the channel sectivn under consideration has approximately constant hydraulic exponents. 3. Compute J by J = N/(N - M + 1). 4. Computc values of U = Y/Vn and ,1> = uN/J at two end sections of the reach. 5. From the varied-flow-function table in Appendix D, find values of F(u,N) a.nd F(1),J), ,6. Compute the length of the reach by Eq. (10-10); The above procedure is illustrated by the following examples:

. 'U

l.488

5.00

3.40 DiL ...

},012 : 0.476 •

I

I I

I

v :

I

1 . 0,1.5

}i'(''',N)

,

1 025 -0.B77

I

1'-

Q =,400 cis n

=

FI..OW

O.O~- 0.0036

1_

I

y -'-~! :

PROFILE FOR EXAMPLE

10-3 BY TIiE

'" - 1.1Q

y;

2.22 fty. = 2.57 ft

2.22 2.14 1.B7 1.60 1.33 1.07 0.80 0.53 0,27 0.00

11

F(1_,N)

, F(v,J)

x

L ~--

0.831 0.800 0,700 0.6')0 0,500 0.400 0.300 0.200 0.100 0.000

I

0.792 0.755 0.638 0.525 0.420 0.;315 0,219 0.132 0.055 0.000

0.979 0.917 ' 0.756 0.627 0.511 U.404 O.3D1 0.200

0.100 0.000

0.962 0.888 0.699 0.552 0.431 0.319 0.219 0.132 0.055 0.000

20:6

0

20'40

2

188 161 134 102 71

IB 45 72 10i 135 i63 1BB 200

43 18 0

to dischllrge 400 cfs with a depth equal to 0.55 It at the vena contr~ta: C~mpute the flow profile. If a hydraulic i ump occurs at the downstream end, suartmg Wlt~ a depth of 1.6 ft, determine the di,stancl; 'from the vena contro.cta to the foot, of the Jump, SQlulion. From the given data, li. = 2.67 ft and y. = 2.22 ft .. Smce y.. > y" the

channelsiope is mild. As the depth of flow issUing from the slUIce !!;ate ls.1ess than the critical depth, the flow profile is of the M3 type. . T ' Considering aflilverage depth of 1.61 Ct, the hydrauhc exponents are I\ 0= 3.43 and M = 3.17. Thus, J .. 2.72, N IJ = 1.26, lind (Ycly~)MJ IN = 0,442,. ' : . Table 10-3 'shows t.he computation of the flow profile. For convenUl[lce m mter-

-

-..:..-. - - ---..: ___ !!.D~L. ~-­

- ... - .... -

F(v,J)

[.293 -1,080

,

+ 0.Hl7

TliE

----

5, The .varied-flow funct.ions k(lI,N) und F(v,J) are obtainbd from the table in Ap~(lndix D ancl given in the fourth and fifth colmnns of the o.~ove table. 6, In Eq. (l0-9), A = y,./So =: 2,100 and B = (voly,,)MJ IN F 0.197. The length of the backwater profile is" thereCbre,' i . L = 2,100[0.476 - (-+0.877)

Water flows from under a sluice into a trapezoidal channel having '" = 1.10, andn ,= 0.025. The sluice gate is regulated

DIRECT-INTEGRATION MET,BOD

-1-.6-'2-5- ---0~ -1-~0-'-2-13-

I

,

= 20 ft, z =2, So = 0.0036,

TABLE 10-3. COMPU'i·.uroN o~

Example'1O-2. ,i,rith reference to the chs,1l11EJI described ill Example 10-1, compute the length oC the backwater profile extending frorn the dam site to an upstream .'3BCtion where the depth of flow is 1 % grea.ter than the normal depth. Sohaion. The given data are Q = 400 cis, b = 20 ft, Z = 2, So = 0.0016, a = LID, and. n = 0.025. , 1. FOlIo'wing Example 6-2, y,.= 3.36 £to Following Exa.mple 4-2 with a = 1.10, y, = 2.22 ft. i '2. The depth at the downstream end of the backwater profile is Y2 =,5 ft: At the upstream end, the depth is Yl = 1.01 X 3.36 = 3.40 ft. The average depth may be to.ken as 4.20 ft'3.nd ylb = 0.21. From Figs. 6-2 and 4-2, the corresponding hydraulic exponents are fonnd to be N = 3.65 and 'llf = 3.43, 3. The vnJue of J = 3,65/(3~65 - 3.43' + 1) = 2.90. 4. For each section, values of H and v are computed, as given in the second and third columns of the following table:

'Y

257

METHODS OF COMPUTATiON

GRADUALLY, VAllIED FLOW

256

X (-LOBO)) ;" 2,395 ft l

I

1

\

--- --- ---

258

GRADUALLY VARIED FLOW

x =

~

[

(1 "-:,::)

tI -

+ const

1"(u,3) ]

F(u,3) = )0

1

_ 1-:1'

du

-113 -

n

7,

11'

+ g +1

(u - 1)'

_ _1_

.H,2l~

V'§ cOu

As the depth of flow varies from 5.00 to 3,4,0 ft, an average value of'y = 4.20 ft may be assumed for the evaluation of Chezy's C. For a wide' rectnngular channel, Eq. (5-7) gives C = 75.13. Since y. = 3.36 ft a.nd So = 0.00113, A = 2,100 and B = 0.715, Eq. (10-17) gives . x == 2,100[u - 0.7151"(n,3)] + const (10-19) Assuming a constant of zero in Eq. (10-19), the computation of x is as follows:

+1

-Y3

I F(t':3~1

y

X

.

-------I---~---I

5.00 1.488 ) 0.260 \2,720 3.40' 1.012 t 1. 360 . 80 ---'----

The length of thB backwater curve is, therefore, equal to 2,720 - 80 = 2,640 ft, about 10 % larger than the values determined by the previoLls methods. .

Channels of Nonsustaining Slopes. When the n,bove procedure is applied to cImnneJ:; of adverse slopes, the slope of the channel bottom may be taken as negative. Thus, Eq. (9-3) becomes

. (10-11)

where 1"(1<,3) can be found i'rom the varied-llolI·-function t.able in Appendix D. matically, 1"(u,3) is integrable, or .

(u

259

METHODS OF COMPUTATION

polating values af F(u,N) from the varieti-flOlv,function.tablervalues of u are assigned at regular inter',:[LI:<. Values of x are thereby computed by Eq • .00-8), in which the const-ant is assun1cd equal to zero. The last column gives the length of the profile measured from t.he section untler considera.tion to the downstream enu, where the profile terminates theoreticlLily at the critical ciepth. . The computed flow profile is 'plotted as shown in Fig. 10-4. The a~tual profile betwee.n the bluic(!-gate opening and the computed profile is uncertain and,hence, is . fitted in by eyc. The distance from the gate opening to thllsection oi the vena contmcta is known as thecontraciion distance. For a sharp-edged sluice gate, this distance has been assumeu equnl to approximately the hl:igl:t of the gate opening (Art. 15-7). If a hydraulic jump starts off at It depth of 1.6 ft, the distance from the jump to the vena contracts. would be about 130 ft, as shown. Example 10-4, Determine the profile of flow in a wide rectangular channel, using the Chezy formula. Solution. Fol' a wide rectangular channel, Fig. 4:-2 gives M = 3 and Eq. (6-49) gives N = 3. Thus, J = 3, v, = tI, and Eq. (l0-8) becomes

Mathe-

dy

dx = 1 +

(10-12)

-S~ 0'.

- s

(10-20)

d(V2j2g)/dy

The corresponding equation for the flow profile can be shown to be This integra.tion Wi,S lirst performed by Dresse [2]. A determination of the flow profile by this solution is,therefore, \videly known as the Ere.sse method. The critical and no~mal depths in· a wide re~tangular ohannel m ...y be expressed, respect·ively, by .

.

'\J-g

F(u,N)_s,

and

Thus,

G'So

.

or

~0 T u L x

=

-

(1 - C'So) 1"(",3) -I +const g

,.f[u - B1"(tt,3)]

UN

_

+ const

(10-16) (1U-17)

Example 1.0-6.

where 11 =y,,/S, and B = 1 - C·So/g. The lengdh of flow profile between twoconsecutivc sections of dcp'ths?l' and y, is L = A.{(u. - UI) - B[F(u.,3) - 1"(uI,3)J)

(10-21)

.

F(v,J)_s,

=

=

r

j.~

a u lf - N }o 1 + uN du

.

1

d1t

+ UN J

=

(10-22)

r v

d!,'

N }o 1 -+ vJ

(10-23)

where v u,Y!f and J = N/ (N'- 111 + 1). --For evaluating these functions a t.able [18jl has. been prepared which is Table D-2 in Appendix D. Accordingly, the length of flow profile betwe~n two sectiqns 1 and 2 may be·expressed by Eq, (10-10), where A = -'y,,/So, B = -(y,jYn)MJjN, and the varied-flow functions are replaced by those for' adverseslo pes.

(10-15)

= -g-

Substituting this·expression for Y;/Yn', Eq. (10"11) may also be written x =

+ 1tN

ra

where q is the dischargl'\ per unit width of the channel. YR3

I

(10-14)

Yn = "C'So

Ye

(l

N-MduJ + const (y,)M r"u Yr.,}o 1 +

(10-1:3)

.317

3

[~ - j(U ~ _

where the varied~fiow functions for adverse slope~ are

31qi

Ye =

and

x = - y" S~

Derive an expression for the flo;w pl'ofile ina horizontal channel.

z.... Solution. For horizontal channels, So .= 0, and'the differential equation ~,., ..... '1<

(10-18)

c7t/.

Example 10-0. Solve El(ample 10-2 by the Bresse method. Solillion. , The Bresse method is derived primarily for an infinitely \Vide rectangular channel. . When this metho9. is applied to channels of other cross-sectiona.l shapes, the solution is therefore very approximate. .

\

'- 1

dy

d2

-(Q/K)' (Z,/Z)'

= 1 -

is (9-19)

Since the critical slope S,.is defined as the 'slope that will produce a discharge Q :at

(

Il.

.!

I Tables of varieq.-flow functions for adverse slopes. with limited range af the hydraulic exponent N hiwe also been prepai'ed by Matzke [20J and others [2IJ.

j

I

I

!.,

-

\ \

260

for the range of Yl to Y2; let Me be the M vaille at the critical depth Ye; and let M be the average M value for tpe range Yl to Y7. Thus, Eq.

normal depth equal ' to the critical. depth !I' (Art• 6-7) , the d'ISC1~arge may be ' expressed , !IS

Q

1

=

VB,

K,

' (10-24)

Substituting Eq. (10-24) i(l Eq. (9-19) and letting (K.jK)' = ( /)1'1 (Z,/Z)' (yJy)Dl, and p = y/y" . " . y, y ,

r

\

~:!! '" S dx

pM-N ,

(9-13) may be written

=

(10-25)

'1 - pM

Let 1t = y/y"Nn/N and, hence, ,dy

Integratinll a.lld solving for z, _ y, ( X -

R

berBduced to pN-.lf+1

N _ M

'

+1 -

pH+-\)

N

+ 1 + ~onst

(10-26)

dx

This equation can be llsed for the computation of the length of fJ/)·v fil' h' zontal cha.nneL ' . ~ , pro e In a Orl-

It sh ou~ld, b e llotec1 t h at , the assumptIOn of COllstnnt hvdraulic exponent<: £0' • d'ISCUS. J . ... ~ l'n I-he v • legomg N

y N"IN [ 1. -

x = ~

'"

o

log K

l

N"2tonli

18 ,.,/

------~:-~

I I I I

I I

! I

I

I I

,.," ". 2

I

!I

I

I

I I I

I I

_

I

I

N

"''''''' 0>". 0>

!2E£

.

log Y

r

l

1 1t~r-,I! 1 ___ _, -;- (y/1'IM)M -- - du ' 1 - UN y"N .. !N 1 - 'UN j

(10-28)

F(u,N)

J '] , + (y,M'IM)H ~>.fN N F(v,J) + c.onst

(10-29)

(10·8). The length of the flow profile between two consecutive sections of depths Yl and Yz can be computed by Eq. (10-10), except where A YnN.fN ISo and B = (YeM,IM IYnN.IN)!':J IN. ChanrLeis with Gradually Closing Crown. For channels with gradually closing ero\vn, the hydraulic exponents are variable near the crown, and the m'ethod proposed r.bove may be used. For more accurat_e results, however, the integration of the dynamic equation maybe performed by a procedure of numericp.l integration. Such a procedure has been applied to circular c,onduits hy Keifer and Chu [17]. Let Qo be the discharge of a circular conduit, flowing full at a depth equal to the diameter do of the conduit and having the energy gradient equal to the bottom slope So, and let Ko be the corresponding conveyance.

I I I log Kn

I

Then, Eq. (10-27) may'

+

I I I

,8 el

y,/J,IN duo

where u = Vly"N,,{N, V = u NIJ , J = N /(N - M 1), and F(u,N) and F(;),J) are varied-flow functions. If the hydraulic exponents are consbant, or N" = N and Me = 111, Eq. (10-29) obviously becomes Eq.

r I I

log Ze

__ y"N"IN __ So

=

Following a procedure of integration and transformation simibr to that applied to the solution of Eq. (10-3),

Channels with l'a'I'iablc Hydrauli-vETponen'" • . ".

,

261

METHODS OF COMPUTATION

GRADUALLY VARIED FLOW

I

I

'-----.l.:-c--J_--LL"'>.

.>..>.'»

01

CJIo OJ. OJ.

. ..2

log y

2. .2 52

~~~'r;u~i~'e~;~~::~~~~iC plots of depth against Z and lIf, respectively, for vari[,ble

Thus,

)

sion ~s sa~isfactory in most rectangular and trapezoidal channels. -As ~escnbe~ mArts. 4-3. and 5-3, the hydraulic exponents may vary appreciably with res~ect to the depth of flow ~vhen the channel section has abr~pt changes m cross-sectional geometry or is topped with a gradually closmg crown. In s~ch cases, the channelhmgth should be divided "into a number of reaches III each of which the hydraulic exponents appenr to be constant. ' ' Referring to Fig. 10-5, it is assu~ed th:;Lt the hydro.ulic exponents in the range o~ depth from YI to Y2 of a reach ~re practicaJlyconstant. Let N n be the N value at the normal depth y,,;'let N be the average N value

(10-30)

For a uuiform flow in the circular conduit with a discharge equal to Q of the actual flow, Eq.(9-11) gives (10-3'1) Q = Knv'So From the above two equations the following may be developed:

i 1

\

263

GRADUA"LI,Y VARIED FLOW

METHODS OF COMPUTATION

where (ICo/KF is evidently a function of y/d o and, hence, can be represented by fl(u/do). Ji'r"omEqs. (\)-4) and (9-7), the following may be written:

applications. The direct step method I is a simple step method applicable to prio:matic channels. Figure 1O-6illustrates a short" channel reach of length I1x. Equating the total heads at the two end sections 1 and 2, the following may be written:

262

0:(.221' ZC)2 = 'qA3 (Z

T /d o "

aQ2

=""Ci7 q(A/d o2)l

=

aQ! (Y) -([;Sf. C4

(10-3.'3)

where (1'/ d o)/ g(A/d 0 2 ) Sis appa,rently afunction of y/d o and, hence, can be represented by hey/do). Substituting Eqs. (10-32) and (10-33) in Eq. (9-13) and simplifying,

,

(10-40)

(10-34) (10-41)

dey/do) 1 - (Q/QoFh(y/d o)

x _ do [ (v/do

- So}o

aQ2 (v/e.,

-~)o or

Ia(y/do) d(u/d o) ] 1 - (Q/QoFfl(y/d o)

.(x _aQ:~ y.) +"const do' (y Q) .(Ii/d. -dey/do) X FI do' Qo =}o 1 - (Q/QoFh(y/d o) Y = F? (X, R) = -f.(u/do) dey/do} - do Qo )0 1 - (Q/Qo)2!t(y/uo) x = - do So

where and

+ const

T

."

=

(v/d

where E is the specific energy or, ass1_unillg cr.l = a2 = a,

(10-35)

V2 E=y+cr.2q

(10-36)

(10-42)

FIG. 10-6. A channel reach vation of step methods.

fOl"

the deri-

In the above equations, y is the depth " of flow, 'V is the mean velocity, cr. is the energy coefficient, So is the bottom slope, and S,is the friction slope. The average value ofB! is denoted oy Sr. When the Manning formula is used, the friction slope is expressed by

. (10-37)

o

(10-38)

(9-8)

These are the varied-flow functions for circular conduits, depending all y/d o and Q/Qo. They can be evaluated by a procedure of numerical integration, say Simpson's rule. A table of these functions for positive slopes, I prepared by Keifer and Chu, is given in Appelldix E. The length of flow profile between two consecutive section3. of depth "Vl and Ya, respectively, in a circular conduit may be expressed as

The direct step method is based on Eq. (10-41), as may be illustrated by the IDliowingexo..mple. .

(10-39) where A = -do/So and B = cr.Q2/d o• 10-3. The Direct Step Method. In general, a step method is chal"acterized by dividing the channel into short reaches:and carrying the computation step by step from one end of the reach to the other. There is a great variety of step methods. Som.e methods o..ppear superior to others in certain n~spects, but no one method has been found to be the best in all 5

, t ;

If So = 0, then Q, = 0, Q/Q, = "", and the varied-flow fund-ions hecome meaningless. If S, is negative, Eq. (10-30) shows .that Qo> is negat,ive. Since the actual discharge Q must be positive, (Q/Q.)2 bee"omes" negative. Thus, the integration procedure must be done for negative values of (Q/Qo)l in the two varied-flow functions.

"

1

i

!

Example 10-7. Compute the flow profile required in Example 10-1 by the direct step method. . Solution. With the data given in Example 10-1, the step computations are carried out as shown in Table 10-4. The ,,&lues in each column of the table are expln.ined as foll
£.

,

f", >, ,

,~-

.: -,

\

.. --/

METHODS OF COMPUTATION

1 (

f

1

21

~

~

~

~

;g ~ ;g

I.(')

-..:tt 0

~

e,p 00·

r-...

...-I

C"I .D C'l

NMl!"Jt-...~;:S~~~~8~g~c::. ~ r--: u:cr: ~ c ~ 00 00 00 J:-..l"-..;! ~

'""""!

~~~~~~~~~~~~~~~J

I

<01 !:::;:E~~~~g;$~::g~...-IN(QC'-) t

~

~!:::, ~•~ ~ ~ MO'~ ~~ ~ • ~ ~, . C'I') M::'t;l '~r ~•~• • I 000000000000000 _

~

€I ~ ~ ~ ~ ~ ~ ~ ~ ~.~ ~ ~ g;; ~

Jl:'""'i

3/

rt;

;;01 ~I

! ./1 _

~

•

•

•

•

•

• ".

~

•

'- ... C"I;I

M

Note that, in either tpe direct step method or thE'l standard step method which w:ill be described in the next article, the step computation should be carried upstream if the flow is suhcritical and downstrea.m if the flow is supercritical. Step computations carried in the .wrong direction 'tend inevitably to make the result diverge from the correct flow profile . 10-4. The Standard S!~Jl_~et~9d. Thill method is applicable also to nonprismatic channels. In nonpril'lmatic channels, the hydraulic elements are no longer independent of the distance along the channel. In natural channels, it is generally necessary to conduet a field survey to conect the data required at all sections considered in the computation. The computation is carried on by steps from station to station where the hydraulic charactel'i3t~cs have berm determined.. In such c'ases the distance between stations is given, and the procedure is to determine the depth of flow at the stations. Such a procedure is usually carried out by trifl.l and error. In explaining t.his niethod it is convenient to' refer the position of the water s\:rface to a hQrizontal datum. In Fig. 10-6, the water-surface , elevations above the, datum at the two end sections are

gj "/

'"

C'\tNC'lMC"t')C"t')MM~~~''';~j;'';~

I

!

I J

'
J )

{,

--.1 ~'

~I

I ."

--.1I

.~

~I

I

265

Col. 8. Change of sp.ecifi~ energy in ft, equal to the difference between the E value in col. 7 and that of the previous step CoL 9. Friction slope computed by Eq. (9-8) with n "" 0.025 and with V as given in c'o1. 5 and R~i.in col. 4 Cbl. 10. Aver~ge friction slope between the steps, equal to t.he arithmetic mean of the friction slope just computed ill col. 9 and tha.t of the previous step Col. 11. Differe,nce between the b0ttom slope 0.0016 and the average friction slope CoL 12, Length of the reach in ft between the consecutive steps, computed by Eq. (10-41) or by dividing the value of b.E ic col. 8 by the vahle in col. 11 . Col. 13. Distance from the section under consideration to the dam site. This is equal to the cUlmllativ~ sum of th~ values in coL 12 computed for pr\lvious steps. Th~ fiow profile thus computed is practically identical with that obtained by graphi~l integration (Fig. 10-3). . .,.,... 'Example 10-8. A 72'-in. reinforced-concrete pipe culvert, 250 ft long, is raid. on a slope of 0.02 with a free outlet. Comput,e the flow profile if. the. culv;;l't dischs.rges 232 ds, n = 0.012, and" = LO. Sol'utian . . ,Fronl tl:e data, V' = 4.35 it and Yn = 2.60 ft .. Since y, > y", the chr.nnel slope i~ steep. As shown in Fig. 10-7, the control section is at the e.ntrance; water will enter the culvert at the critical depth !lnd .thereafter flow: at a depth less than 'Y, but grer.ter than y •• . The flow profile is of the 82 type. Table 10-5 shows the computat,ion of. the fio\v profile, which is self-explanatory. The computed profile is plotted Il.S shown in Fig. 10-7. Plotted also in the figure is the energy line indicating the variation of energ:' along the culvert. The comp\,tll.tion has be~.n car~ied to exceed the length of the eulvert, so. that the depth of flow at the outlet can be interpolated. This depth is fOHnd to be 2.81 ft, and the corresponding outlet velocity is 19,4 fp3. It should be noted that, if the. pipe were fiowing full at the outl'et, the outlet veloCity would be only 10 fps.

264

Zl

..

=

So t.x

+ Yl + ~2

(10-4:3)

METHODS OF COMPUTATION IJ) C'J 0

o

0

~

1""""1

1""""1

1""""1

(10.,.44)

and

cno:.

l .... ""dI 1""""10':1 LO 0 0 cqr..oCtlO)"'IfIO

267

The friction loss is

NM

(10-45) CO"'lflCON~I""""ICOO

H

<1

where the friction siope S/ is taken as the average of the slopes at the two end sections, or as Sf. Substituting the above expressions in Eq. (10-40), the following may be writt.en: . V 12 )12 2 (10-46) Zl al -2-- = Z2 + 0:2 -2 + 11,1 + h. g g .

O(O
O':IM"'lflt-O"lCCflr-

o:JcqO~LO-tK:.Otn Lf') "'IfI C"I C"J to ~ ~""""'I""""II""""IOOOO

r-

tn

00000000

00000000

+

v

where h. is added for the eddy loss. which ma,Lbe appreciable III nOI1..NiBma~ic~. No rational met,hod of evaluatin edd' loss is available. r~~epends mainly on the velocit,y head change a.n . lllay be expressed as 11 part Of It, or k(~aV2!2!l) where k is a coefficient. For -graduzHy COl1vel'gmg and-dive-rgll1g reaches,!, = 0 to 0.1 andO.2, respectively. For abrupt expansions and con}.r.actions, k is about 0.5. For prismatic ~uTai'Chailiiels) the eddYloss is practically ~'Ie = O. For convenience of computation, h. may sometimes be cOll03idered part of the friction loss and Manning's 11. may be properly increased.in computing _!}l> I~.~PI h~.i~£9 in the cor,nputatiol1. The total heads at two eUd sections are

000000000

0')00<1:><0<0000'"

1""""I0l~U"J"'IfIC'-I~t­ OON~t-M"l~ •• * •• ••

00000000 OO,.......(cnl.O-.;.t-tOtO~

~ ~ ~ ~,~ ~ ~ ~ ~ ~ <0 t..OcO

jC'~ 00

;"'..... "'1'1

j

(0

1'""'1 0':1 11"; 1""""1

0

.qt

t-t"-o 0000 00

1""""1

0

t-' to to ~

f.O

CO

Q ...-I

(10-47)

c.o 1""""1 C"J O. '<"' t- 1""""1 ~ ",,,lN~~.q;.;;t.OcO 0

('>

~

..

·and 1100 <'> <0 "" ..., \I') <0 0 ...., .... ~O)OC"lC:OCf.lt-C*:loC() ,r- ,....;....; N ~ ~ 00 ~ c;)

to

I

1""""11""""1.1""""1'_. 1""""1

I I""~,

A:

.... ....

""

t-COCOO

.....

.....

0'>

t-

~I

t:'-

t;-

C'~

t-

00

.... .... ....

.

HI = H 2

....

°° '" ..... .... .... .....

::g~g~~

I

<'l

I

\OMtnHMC'O~t:OO OlI""""l~t-O':I....-l~Ot-

~ ~

. . .;

jN N

~

r--:

~ ~ M~

C-l

1'""'1 1""""1 ....-l'.I""""1....-l .-t'....-l

I

'1"""0000000<:'1'"' M cq, en co CI'J 0 00 c.c l'-o ,"'IfI "dt r:v'l cr:,

I'"

C1:I.~ C\l N

C'l

.

QC'lOmo\l')oaor-
........

"r-r-<1:>,","'\t)~"""
~

(

"

;

\

J

Therefore, Eq. (10-46) becomes

-,1""""11""""1

io,-,tI"",",,,,r-oor-,.:>
A:

~

-

(10-48)

I

000000000

Z56

I "

..f

,

+ hI + h.

(10-49)

This is the basic equation that defines the procedure of the standard step method. The standard step method is best suited to computations for natural ...£l1annels. 4Jl.£is!!latic channel, howevelJ.~ill be used in trw following 'example in order to ~ the iI1ustration and to' aUows_oml?J!.risoll with reSiiItSobtained by the other methods that have b.een: described. For apphcatiOilto natural channels, .an example will be shown later (Art. lO-G}. Example 10-9. Compute the flow profile required in Example 10-1 by the standard str.p method. As~ume that statiollS along the channel are j1x~d at the distances determined in the solution of Example 10-7. The elevation at, the dmnsite is 600 m.B.!.

(

--I

268

GRADUALLY VARIED FLOW

Solution. The step computations are arrauged in tabular form, as shown in Table, 10-6. Values in each column of the table are explained as follows: CoL 1. Sectionidcnbiiied by stiJ,tion n'lmber such as "station 1 55:" The location of the stations is fixed at the disto:nces det~rmi!led in Exampie 10-7 in order to compa.re t.he procedure with that of th~, direct step method. Col. 2. ,Vfl.ter-surface elevation at the station. A trial value is first entered in this column; this will be verified or rejected on the basis of the computations made 'in the remnining columns of the table. For the first step, this elevation .must be given 'or assunled. Since the elevation of the darn site is 600 m.s.1. nnd the height of the dam is 5 ft, the first entry is 605.00 m.sJ. When the trial value in the second step has been verified, it becomes the basis for the verifieation of the trial valu~ in the 1I1lxt step,and so on. CoL 3, Depth of flow in ft, conesponding to the water-s'll'fll.ce elevation in col. 2. For instunce, the depth of flow at station 1 55 is equal to \Vater-surface elevatioh minus ehvation at the dam site'minus (distance.from the clam site times bed slope), or 605.048 - GOO.OOO - 155 X 0.0016 = 4.80 ft. ' CoL 4, Water area corresponding to y in coL 3 CoL 5. Mean velocity equ!l.l to the given discharge 400 cf" divided by the wa.ttoi' area'in col. 4 CoL Ii. Velocity head in ft, corresponding to the velocity in col. 5 Cnl. 7, Total head computed by Eq. (10-47), equal to the sum of Z in col. 2 and the velocity head in col. 6 . Col. 8. Hydraulic mdius in ft, corresponding to 1J in coL 3 Col. 9. Four-thirds power of the hydraulic radius Col. 10. FYiction slop~ computed by Eq. (9-8), with n = O.O~5, l' from col. 5, and R~~ ]rom coL 9 CoL 11. Average friction slope through the re~.ch between the sections in each' step, approximately equal to the arithmetic mean of the fdctloll slope just comp';lted i.n col. 10 and that of the'previous.step . Col. 12. Length of the reach between'the sectioll.S, equal to the difference ill station numbers between the st-ations CoL 13. Friction loss in the reach, equal to t,he product of the values in cols, ] j and 12, Col. l~. Eddy loss in the reach, equal to zero Col. 15. Ele\·ation of the total head in ft, This is computed by Eq. (10-49), that is, by adding the values of hI and h, in cols. 13 and 14 to the eleva~ion at the lower end of the reach, which is found in col, 15 of tli __ previous reach. If the vallie so ohtained does not ng'ree closely with that entered in col. 7, Il. new trial value of the water-surbee elevation is assumed, Ilnd so on, until agreement is obtained. The value that le.1ds ' to agreement is the c!!rrect 'water-surfuce elevation, The computatiDn may then proceed to the next step. The computed now profiie is practically identical with that obtained by the gl'aphical-integmtion method shown in Fig. 10-3.

+

+

i )

I

,I

10-5. Computation of a Family of Flow Profiles. In previous articles methods were described for, determining a single fio,;' profile. Frequentl:)', several flow profiles,or a family offiO\v P1'0 files, are desired fOl' various cOnditiolls of stage and discharge. An example of this type of pro~lem is the determination of the economical height.of a dam, where the initial elevation is indeterminate and, hence, a number of flow profiles may have to be computed for the same discharge with different assumed

. 0

0

0

09°.0

a

0

0

0

000

:. f6 ~ '[3;.~'~ ~ ~ ~ ~ ~ ~ g ~ ~ 0 0- 0 N C'l C'1.1""""1 C'1 N N I""'"'f

-I

_

1""""1

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i

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~!::;~~~~g~f2~~~~t;::;~ u-; L~ ~ ~ ~ ~ ~ M M ~ M ~ M M M C'f':l 1""""1' m co u:; ~ t- ..... IX) Lr.J C'o':: ..... 0 0') It)~M'''''OOl;:Qt-t-~t.D~tO~U'::I

-.::tt

M~MMe..)~~NNC'1~C'i"l~~

. . ;tt

~

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C"l ~ ..... 1- a:: ~ C"l ~ 10 C"t c::: a a a C'l 00 co tn r- ~ co tf) 00 ~ I..Q ~ ",,:1'1 -*. M ........ ......-I C-l M ...q-. (!:I co o-C"t ~ to co 0 C'\ lO It:I u-.) lfJ tn L.":J. IQ lfJ <0 c.o to lO lO r- 1- r0000000 00 '=100000 (O(()tD~ot.O(QtOtO~tOtOt.D~tO

---~----------------------t-cot-c.oV":l~O-e::tl~CO[-......-I~~C"'-\

."

:;;"" i:>-

is

II

----

~~;;;~~~~~t2~~;3~~~

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~

(.'-1 C'l cr"

C"";)

M

C"";)

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000000000000000 C") a.: ~ ....,.. C"t ~ co Lt':l O"J M 00 ~ r- co ~ ~ t- U') L"':) t- !"""'4 ~ co lfJ M t- C"l U") co ~ co O"l ....... , eo"J: lfJ co OJ a ......-I ,,1 ~ ~ CC (r"

l-

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t....,~.;1JC'I.....-I....,oIoocn~O":IC":IO":ImCJ) ~......-I......-I_"""-f~......-I

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£!

a

J

'';;

,,$

!

.[

I

ell

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g :g~' c; ~ 8'~ ~ g ~ f::: ~ ~ ~ ~ + + + + + + ,+ + + + + + + + + o .....

M

~ to 00'::: ~ ~ ~ ~ ~ ~

269

;i

~

271

. GRADUALLY VARiED FLOW

METHODS OF COMPUTATION

initial elevations. AnothEll' example is the tracing of flow profiJes in a tributary :;;trm1m for different stagp-s and discharges in the main river, or in a river with a tidal estuary for different tidal elevations, or in ll, cana'! t,wo reservoirs for changing reservoir elevations and 1lariable The following are some time-saving methods which may help in the computation of a family of flow profiles. A. C'um'Js of Geo'l11£tric Elements. When 11 number of flow profiles are desired fOI' different stage and discharge conditions, it is best to construct curves showing the geometric and hydraulic elements (Figs. 10-8. and

and the friction slope vary with the square of the discharge. Therefore, after the curves for one value of Q, say 400 cIs, in Figs. 10-8 and 10-9 have been computed, CUI'ves for othol' discharges, say 200 and 800 cfs, may be obtained t'ithel' by shifti!lg the curves horizontally (Fig. 10-8) or by shiftthe abscissas (Fig. 10-g). The distance by which either the curve or the abscissa should be shifted is determined by the square of the ratio of the !lew to the original discharge: .

270

-l -j

l

0.05

0.1 Veloci~y

FIG.

0.5 need, ft

1.0

5

1O~8. Plot 'of velocity head aga.inst water-surface eleva.tion.

10-9) required in the colnputatiol1. Considerable time can be saved itl. the computation by interpolating values fnim these cur'Tes, provided the curves are not extended needlessly beyond the expected range of the water-surface elevation. When the discharge of flow in a channel varies, the velocity hea.d and friction slope will change, and the water-surface elevation will be affected. Thus, for general use in the computation, the velocity head in logarithmic scale can be plotted against the water-surface elevation (Fig. 10-8), and the friction slope in logarithmic scale can be plotted against the wateL"surface. elevation (Fig. 10-9). From cOllsideration of the Manning formula and the continuity of flow, it call be shown that the velocity head

Friction slope for Q" 400 cis FIG.

1O-1l. Plot of friction slope IJ.gainst water-surffICe elevation.

B. The Leach Diagram. When a hrge number cif flow profiles is required for the same discharge but diifel'ent stages, a diagram develop~d by Leach [24J may be used to advantage. Such a diagram (Fig. 10-10) may be prepared after a few flow profiles within an expected range have been computed by any method. In the diagram, each curve represents the rehttionship between the water-surface elevations at thebeginnillg and end sections of a reach. For instance, when the elevation at section 1 is plotted against the elevation at section 2 for the first reach 1-2, then

,/"

l<

272

GRADUALLY VARIED FLO'lT

the cur~e fOl'. renc? 1...2 as shown in the figure is obtained. For simplifythe IdentdieatlOu of the curves, elevations at sections of oddnurnber are . represente~ ,by abs~i8sas and those at sections of even number by ordinates: . ':Vltn the ,dlagram thus prepo.red, the flow profile for any a,ssumed 1mbal elevatlOll can be determined, Taking the initial elevation as 605.00 and fonowing the dotted line in the direction of the 'arrows the corresponding water-surfaue elevations Il(; subsequent sections can b~ obtained easily, The diagram is const,ructed for a discharge of 400 cfs. For any other discharge under investigation a diffe;'ent diagmffi must be prepared. GQar

~ .i 607~ J

,

u

:; C

,2

606 ,-

.'" Q

"

" ~ ,"

...

505

I

'" '0 if:

606404~----c*~----~!~____~I~____~ 606

607

608

W'ater-3urfoce ele".crion ot !iclioo$ 1~~. 5, ere. "

FIG. 10-10. The Le!Lc~ diagram for flOW-pl'ofile computatlon when a large numbel' of flow profiles are reqUIred for the same discharge.

C. The E'l:m lIfet!.wd. The Ler;.ch diagm,m is recommended if flow profiles Iorthe same dlScharge are required fOl' val'ious initial st!Lgcs. When flow profiles for various discharges are desired for different initial stages, a method developed by Ezra; (25) may be used. Similar method':! have also ' been developed by others (26-281. • ~ul:lstituting Eq. (10-45) in Eq. (10-46), the following equation may be wntten: ','

I.

(10-50)

'o/here !l.nd

(10-51) (10-52)

MlllTHODS OF COl\:fPUTATION

!r

!

I

n V1 '8 1 = 2.22Rl~~ 2 V 22 ~ ~2 = 2.22R2~i 2

and

81

I

,82 =

a.ud

"

273

2

n ( Q)2 1(1

(10-53)

(10-54) (1O~55)

(!l)2. 1(2

It can be seen that functions F(Zl) and F(Z2) are proportional directly to the squares of.the veIQc1ti~s or of the discharge Q, For any other disQs, the corresponding values of these functions can be obtained by, muit.iplying the functions by a factor (Q./ QF. The Ezra method gives a graphical solution of Eq. (10-50). There are two major parts of this solution: L COMPUTATION ,o!..ND CONST1H)'C'l'ION OF Z + F(Z) CURVES. For each chosen section in a charulel under consideration, several values of \Vi:J.tersurface elevation are selected, and necessary geometric elements A and R for each of these water-surface elevatiofL'> are determined ,and tabula~ed, 'This procedure is the same as in the standard step method. For given values of Q and n, values of C/ P /2g and SI are determined at each section for each of the selected elevations. In artificial prismatir. channels, the friCtion slope Sf 1l10,y be computed by Eqs, (10-53) and (10-54). In irregular r..hannels, the v(,lull of 1C is determined first and the value of Sf is then computed by Eqs. (1()"'55) and (10-56). This will be shown in Example 10-12. ' For each section, t,he corresponding vl1llleS of F(Z) are then computBd by (la-51) and (10-52). It should b€! noted that two values, F(Zl) , and F(Z~), should be computed for each section. In computing F(Zl) of the section, the value of ..6.x is the value ..6.Xd measured downstream from that section, In computing F(Z2), the value of..6.x is the value ..6.x,. measured upstream from that section. In the computation, this rule ensures that values, of ax will common to sections at both ends of any reach, since it will be shown that the same value of ..6.x is used on both sides of (10-50), In other words, at each section Zl = Z7 = Z., Now, fol." each section' with Z = ZI = Z21 values of ZI + F(Zl) and Z2 + FCZ2) are computed, and curves of Z1 + F(Z'l) and Z2 + FCZ2) l1re plotted against Z for each section (Fig. 10-14). , 2. DE'rEP.MINATlON OF WATER-SVRllACE, ELEVATIONS. The effect of eddy losses may be included in the value of the roughness coelficient n, as described in Art. 10-4; thus, the term h. in Eq. (10-50) isi zero. The resulting ~alue of Z2 + F(Z2) for a \:l.ownstream section is the:refore equal to Z 1 + F(Zl) for the next upstream section of the same reach, and vice versa..

be

- 1 "

274'

GRADUALLY VARIED FLOW

METHODS

, For subcritical flow, the flow profile is determined in an upstream direction. Star!.hlg with a given initial water-surface elevation at a down:;;tream sectioll, the value of Zz F(Z2) ClLn be obtained from the appropriate Z2 + F(Z3) curve. Entering the curve of ZI F(Zl) for the next upstream section with this value, the corresponding, water-surface elevation is determined directly, This procedure is J'Elpeated from section to section, tracing the desired flow profile. For supercritical flow, the flow profile is traced in a downstream direction. Starting with the value of .?I Ji'(Zl) at the initial section and taking this value to the Zz F:(Zlz) cw've for the next downstream section, the corresponding water-surface elevation is determined. If eddy losses are not included in the roughhess, the term h. may be expressed as k{O'V2/2g), where k is a coe.fficient described in Art. 10-4. Accordingly, the procedure of computation should be modified. First of all, it necessary to plot curves of k(a V1j2g) the elevation Z for eil,ch section. . For subcritical .flow, water-surface elevations for two consecutive sections are determined in an upstream direction by the procedure described O. Values of k(aV2/2g) may, therefore, be obtained previously for h.. from the plotted k(a V2/2!7) curves for these elevations .. The difference ka:{VN2g. T'~2/2!l),or he'in the reach, is then added to the value of Z2 for the lower one of the two sections. The resulting value is taken to the + F (Z 1) cun'e for the hight'll' of the t.wo sections, and the corresponding corrected water-surface elevation is determined. For supercritical .flow, the procedure is similar. However, the flow profile should be traced in a dowllstreant direction, and the correction for eddy losses should be deducted from the value of ZI F(ZI) before this value is taken to the Z2 + F(Zz) curve. The application of the Ezra method will be illustrated by examples ill Art. 10-8. 10-6. The Standard Step Method for Natural Chll1lnels. For flow in most natural channels at a normal stage, the flow profile in short reaches is very close to that of a uniform flow, but slightly modified by local channel irregularities .. For apparently gradually varied flow, an approximate solution may be obtained by either the di!'ect-integration method or the direct step method, assuming a prismatic chp.,nnel having the average geometric and hydraulic characteristics of the natural channel. For a practical and precise Bolu tiol1, however, the standard 'step method is recommended. . The standard step method has many obvious advantages in application to natural cfU'.llllels. When the velocity head is small, ~he step method can be carried even in t e wrong direction wH~ resulting in serio~ ~rr~rs, al~ is always advisa Ie 0 ca;ry the computation upstream

+

is

+

I

+

~~---.-:--

COMPUTATION

275

. jf. the flow is subcritical and downstream if it is supercrit~ The wa,ter-

+

+

OF

, !

i

I

surface elevation of the iiiltJ.al section, where a flow-profile computation should start, may not be knovm ina natural stream; use of the step computation in this connection offers a special advant:1ge. If the step computation is started at an assumed elevation that is incorrect for the given discharge, the resulting flow profile will become mOl'e nearly correct after every step, provided the computation is carried in the right direct,ion, Thel'efore, if no elevatiol1 is known within or near the reach under consider~tion, an arbitrary eleyation may be assumed at a distant section far enNlgh away, above or below as the ca'se may be, from the initial sect.ion. By t,he time the step computation has been carded to the inith,lsection, the elevations will be correct. A check may be made by performing the same computation with another as,'lumed elevation at the dista,ntsection. The comput.ed elevation at elle initial section is the correct elevation if the second computed value agrees with the first value. The two values usuall;v ag~ee if the distance from the distant section to the initial section is sufficient. In computing a flow profile, th~ following information is generally required: 1: The discharge for ",ihich the flow profile is desired, 2. The water-surface elevation at tho control section. If this is not available, the qomplltt,tion may sta.rt from an assumed elevation at a sectiOl) far enougli a~.vay from the initial se(ltion through whieh the profile is decircd. . 3. The geometric elements at various channel sections along the reach for all depths or flow within the range expected. These dat.a ma.y be obtained by a hydrographic surveyor from a contour map of the channel bottom, A convenient method of recording these data is shown in Table 10-7.' In the table, the cross sections are identified by number and river mileage in conformity with the map in Fig. 10-11. Other dat.a include the reach length between sections, channel width, wetted perimeter, and water area. The column for remarks is provided to indicate the limiting features of the cross section, such as the side slopes, making it pos~ible to estimate the water area and wetted perimeter within a moderate range of wat.er-sUl~face 'elevation. . If flow profiles for different discharges are contemplated, it will be convenient to construct curves of the geometric elements (Fig. 10-12) and to interpolate their values at different elevations. 4. The channel roughness and eddy losses at various sections, In flow-profile computation, it has been fOlll1d that the smaller the :value of Manning's n, the longer will be the profile, and vice versa, Hence, a smallest possible n value should be selected in the computation if knowledge of the longest possible flow profile is required. This knowledge is

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tj

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277

METHODS OF COMPUTATION TABLE

10-7.

TAIJULJ'.1'ION OF DATA FOR.

MlSsounr Mo.-"

AND KANSAS RrvEns AT

KANSAS CITY,

<0 '-

'Q

'Q

.:;""

:: "0

"

I'il

.E

E

'o"

0

""t::

"

'0.,>

">

:>

-"

.0 0

\:.)

~

"

~

~ ~c:

..

"
cor

r:Q

~,~

",,,

o·~

,,0 '""'~~ :0; ::: ~ c: - " ]~ .E:>:

l-

I-

I

I

I

I

~t

...'"""

e ...

"E

.I J

o~

Cl

92

Ch.a.nnel

1 + SO Hi t ' 2 1,520

1.550

38,aOO DaL
H:'L9 Channel

2 +IH 23 + 72 2.0S0

~'.

ltO

48,50(;: Lev •• 1 on;l i BUllk 1 on "~f

720

7l!.D

3,020 Levee 1 on 4

2,040

2.080

47,SOO Bank 1 on 1

720

720

0+·61

DVc.r ..

1 +W

bnr.. k

~

t:

4

a ~

l,il00


>::

il

q

:< ...,

7.16.2

37S.U5"

Lef~

over-

Channel I, S8il

6

>-

I

746.2 Loft. eyer· br.nk

378,95

~

Ch3une~

880 1.320

01.

.

'-

~

u

:;:

"

OJ

'"

...J

1

t

i.,

'"

0

U

'"

...

ovet-

lD

0-

ol

b.nk Channel Dike

E X

OJ

Lovee IonS bo~h sides

:12 + 80 4,4 +

10n3

44 + 50 5.7'+40 1.29()f 460, 57 + 40 62 -I- 29

1,310 490

34,600 5,700 Levce 1 on-4

640

5,010 1 levee ,ur· {iLea. 3 pier slIrtt\eas

2,430

<=:

>-<

·Fllirfax 384.00 Bridge

,...; ,....

0 ,....

'750.3 Left over-

5GO

b~nk'

Chllnn~1

" r£;

RigMoverbank 11

3SS.57

i

216

840 It,·!iS50utL RivQr

0 '-

"t:l

t

Le\·c. 1 0:1 3 but.h slue3

;:;

"

8.27G Levee 1 .00·4 700 Levee 1 on 3

3~.

54()

";:: 0

<1;

Riv~r

ol

,~

,~

I{ansfls

"0 >::

0

c:

;.

~

'C

3,020 Le"'eg-l on

2,080' 47,300 Bauk 1 on 1

~ ~

""E c: "

"'

bank

..,

<:>

B9

745.5 Left

:3

f;

W ..J

1.200

Chfll\nel 84.5

.:I

Z

w

178

b~nk

<5

I

I

Channel 1,060 .". 1 745 . 3 Left ovor·

2

b '-'

I

I f

Ri\~3r

0

'"

.!!~

Missnuri

E-

.1 1

.. U.S. Army Corps of Engin.er~ (291. t See Fig. 10.. 11 for location or iltu~1':l. seot.ions. 1: Distance tlP,ttcam from l1i!etion O.

1,400 · ...... 1

125

•

278

I

j

GRADUALLY VARIED. FLOW 0

:!:

important in cert.ain engineering problems, such as the determination of the bu,ekwatel' effect due to a dam. . On the other hand, ,,,hen knowledge of a'shortest possible flow profile is required, a largest possible n value should be used, for example, in the problem of improving a ehannef for navigation. Since the navigable depth should be greater. than a oertain minimum value, the shortest possible profile will indicate the lowest depth of Howat a given channel section. The eddy losses may be estimated separately and included in the computation:· Some~imes, however, it is convenient to raise the value of n and thus cover the effeet of the losses.

~ ., .,

.,;

."

....

- - . --

.,

..-!::-

c:0

0

;e

;; ,,;

,,;

c:;M "":.

o

co

~

....

--~-

'"

;:;

!;

'"0

0 0

c:0

'"

0

--Oil

---;;;-

..... 0

'"8'"

Example 10-10. Compute the flow profil;; in the Missouri River near Kansas City, Mo., for a. design discharge of 431,000 ds. The plan indicating the locations of the cross sections is shown in Fig. 10-11. The geometric elements of these cross sections a.re shown in Table 10-7 for given water-surf!t.Ce elevations .. For other elevaLions, the geometric elements are to be estimated from such data. l . Solution. The computat.ions are tabulated as sho',vn in Table lO~p. The column headings in the table are explained below: Col. 1. . Section number in conformity with the plan of Fig. 1O~1l Col. 2. Subsections, for which M.C. designates the main channel section and L.O. designates the left overbank section Col. 3. River mileage above the mouth of Missoud River Col. 4. Water-surface elevation. The initial elevatiol1.of 752.25 at. seotion 1 was estimated from the rating curve of the Kansas City gaging station on the Hannibal Bridge at Missouri River mile 3.77.58. . Col. 5. Water area. For instance, section 1 is subdivide'd into le;:t-overbank and main-channel areas. At elevation 752.25, the area is determined for each subdivided area from Table 10-7 or from a curve prepared as shown in Fig. 10-12.

r

0 0

'" '" -_....'"'"-

0

,,;

0 --0-

~

-'

... 0

'"t";... ..'" ~

'"o.

.!!

00

8

0

"

0

'"M

Ii X

...'" ~

~

~

h'

§§§ xx X

">

<1l0"O:;

0>

p;j

.;::

'" ~~

" 0

gj .~

:;;"

.."'""


~

'"

iii

--",-

0

II 0>

~

j

0

~

;:!

:6 :l .£ " p..

~"'I~

-0-

I

'" '":

000

""'" '" .,;

:::: I <0 ""

...... ~I

--;0-

(

",

J

",.

'"

...!::-

",.

A in 1,000 fi2

",.

FIG. 10-12. A and R vs. \'liater-surface elevation for channel section No.1 of Missouri Riyel" at Kansas Cit:!;, ¥o. (U.S. Army Corps oj Engineer•. ) 1 The

plan and data used in this example were obtaine.d from [291. values, however, have been modified to suit the present purpose.

Some numerical·

o· ~l

d ~

;:

.0 ~..:l

I'"

I""

1.5. ·0 ~,j

t

-I

280

GRADUALLY VAlli,ED FLOW

Col. '6. Wetted pcdmeter. :For elevation 752~2p a.t section 1, the wetted perimeter is determined from' Table 10-7 or from Fig. 10-12. Col. 7,' Hydmulic radius, obtained by dividing the area in col. 5 by the wetted perimeter in col. 6. , ' Col. 8. Two-thirds power. of the hydraulic radius in col. 7. Col. 9. The value of Manning's n. It is assumed ~hat the general rosses due to contraction, expansion, find bend are included in the friction losses coniputed from the selected n values. ' Col. 10. The conveynnce K = 1.49 A R* /n Col. lL The value of I(} / A' Col. 12. Energy coefficient for nonuniform velo~ity distribution. From col. 5 to col. 12, the procedure of computation is the same as that described in Art. 6-5. Tne coefficients for the .subdiviLh,d channel sections are assumed equal to unity. CoL 13. Mean velocity, which is e~luar to the section discharge of 431,000 ds divided by the water area in col. 5. Above section 5, the dischaJ;'ge is divided between' the Missouri River (350,000 cfs) and the 1{ansas River (81,000 ds). The velocities shoold be computed for the divided discharge.s accordingly. The division of discharg;e is balled on a hydl'ologic study of the drainage basins of the two rivers. Col. 14. Velocity head ' Col. 15. Total head, whi~h is equal t.o the sum of thc elevation in col. 4 and the velocity head in col. 14 Col. lB. Friction slope, which is equal to (Q/ K)'. The 1C v!Llue is the total value for the section under consideration. Thus, for section 1, S, = [431,000/ (290.1 X 100)]' = 0.000220. Col. i7. Average friction slope through the reach between the two sections, that' is, the arithmetic mf!fLn of the friction slope just computed in coL to and that for the previous step Col. IS. Length of the reach between the sections, that is, the difference in river miles between the sections as converted into ft Col. 19. Friction loss in the reach, th!lt is, the product of the slope in col. 17 and the reach !ength in col. 18 ' Col. 20. Eddy. loss in the reach. The general losses are included in the friction loss cOlIlputed in ,col. 19. However, at the entrance of the KansaB River to the Mi,~. souri River, an additional eddy loss at the C1mfillence is expected. This is estimated at, 10% of the increase in velocity head,or 0.10 X (1).63 - 0.13) = 0.05 ft. CoL 21. Total head, which is obtained by t1.dding the SUIll of the losses hl in col. 19 e.nd h, in col. 20 to the total head in the same column for the previous section. If the value so obtained does not agree closely with that entered in col. 15, the watersurf!lce elevation in col. 4 should be assumed again until a fair agreement is reached. For the computation ot' water-surface elevations at sccHons1K, 21(, 6, and 7, see Art. 11-10.

METHODS OF COMPUTATION

281

, The friction slope Sf in a short reach of length Linay be expressed as - F S, - ,

+ h~2L -

hvI

(10-57)

'where F is the fall in water surface and h'J2 -' h~1 is the change in velocity. head. If hvi - hVI is zero~r negligible" then S, = F /L, and the normal disch[J,rge of a uniform flow by the Manning formi.tla is

Q

= l.~~

_4RH

(J)'Il

(10-58)

For a gradually vlLl:ied :flow with backwater effect having a' discharge Qz and a corresponding F r in the same re!wh, it may be assumed ,that ft. form similar to Bq. (10-58) may be written, i.e., that

.Q~

=

1;9

AR>1

(~ys

(10-59)

effect are also n where the velocity- h ea d ch n.nges' d ue t 0 back"'ater neglected. From Eqs. (10-58) u.nd (10-59) I '1." L'r

(~)Z

= Q/v'F

(10-60)

10-7. The Stage-fall-discharge Method for Natural Channels. When flow profiles of a stream in its natural state, without backwater effect, are available for a number of discharges, thestage-fall-discharge method ~ay be used; this method has the advantages of simplicity and economy [29.]. Similar rnethods'have been 'n.lso developed' by others [30-34].1

, ~vhere Q/ y'F is called the discharge for 1-ft fal!.1 !his equationC8,l: be used ill the flow-profile eomputntion if the stage-fall-discharge relatlOnshtp for uniform flow in the reach is known. The stLlge-fall-discharge relationship f.or 11 selected reach may be determined froni. records of observed stages and discharges Cfable 10-9). The stages or water-surface elevations at the beginning section _of the reach are plotted as ol'dinates, and corresponding values of Q/y'F are plotted as ~bscissas, resulting in a sta.(je-ve1"!:;us~Q/....IF C'urve (Fig. 10-13). When any water-surface elevation ~\t the_beginning section of the reach fS given, the corresPQnding value of Q/ v'F can be read from the curve, and the fall for a dischrLrge Q. be computed by Eq. (10-60). The computed fall, when added to the water-surface elevation at the beginning section of the -reach gives the water-surface elevation at the end section of the reac.h, whicl~ is also the water-surfaee elevation at the beginning section of the next reach. The procedure is repeated for each reach until the complete ,required flow profile is ob,tained. : The stage-versus-Q/v'F curve is generally cOl~structed as an average 'curve for varying river conditions; such as rising and falling of stage,

1 Reference (30) describes the so-called Grimm Tllethod. It requires a trial computation which, however, can be avoided by using nomQgraphs, !lIl suggested by Stron. be.rg [311. '

,lIn a similar method devel~ped by Rakhmanoff [341 a term F /Q' is. us 7d in lieu of Q/ ~F. This term has the nature of /J. resistance factor and therdore IS given !I IIame of resistance'modulus by PavlovskiI [21, p. 1151.

282

fluctuating stream bed, and effects of wind, aquatic growth, ice, .and overbank flow .. Owing to these varying conditions, the plotted points 20re often scattered; and a smooth line, giving consideration to the varying conditions, should be drawn through the points, representing the average condition of the-channel. Where sufficient measurements are available, data of doubtful accuracy should be rejected. In general, the more recent meaSurements should be given greate: weight, as reflecting recent channel changes. Other factors that should be considered in constructing the curve are the relative accuracy of individual discharge measurements; the ,flow condition during the measurements, whether rising, falling, or stationary; conditions affecting the stage-fall-discharge relationship, such as the changes in channel roughness, levee bteaks, and shifts of cha,nnel controls; and the existence of substantial local inflow between the stations. The stage-yersus-Q/VF curve may qe extrapolated !].bove or below t.he range of the observed data by extending the curve at its ends in accordance with the general trend of the curvature. Howe-.:er, any abrupt change in hydraulic elements of the channel section will produce a, eorresponding change in the curvature of the curve. In this case, a correc.tion for the change, if known, should be made in extrapolating the curve. This method is used most advantageously when a number of discharges corresponding to known stages, or vice versa, a.re desired in a stream. By making proper allowance for variable conditions, satisfactory results ca.n be obtained for reaches of large rivers 50 to 100 miles from the measuring station. The data required by the methOd are often .les;:! expensive the,n those required by the standard step method. However, this advantage.is usually.offset by the inaccuracy of the-results, because the effecl of the change in velocity head 1s ignored in the present method. For this' reason, the stage-fall discharge method is more· satisfactory for problems . iriwhich the velocity is well below critical end decreases in the downstream direction. E:x:amplelO-ll. Compute the water-sudace elevation at section 1 of the Missouri River problem in Example 10-10 by the sto.ge-fall-discharga method. The reach from section 1 to section 5 istakea as the first reach. The water-surface elevations are available from stage records for gages located at sections 1 and 5. The discharges have been observed at the A.S.B. Bridge located about 3,000 ft downstream from section 1. These data are tabulated in Table 10-9.' . Solution. The data arid cornputa tions for the sta.ge versus discharge for a I-it-fall! curve are given in Table 10-9, which contains the following headings: Col. 1. Recorded water-surface elevations at section 1 Col. 2. Recorded wo.ter-surface elevations at section 5 Col. 3-. Fa.ll in ft, which is equal to the differ~nce between elevations entered in cols. 2 and 1 1

e.;.-,···

This example is taken from [291 with modifications.

283

METHODS OF COMPUTATION

GRADUALLY VARIED FLOW

·Col. 4.0bserveddischarges,at.the, A ..8J3. Btidgej i!lds Col. 5. Discharge per I-ft faIl, or Q/ y'F, where Q:is,th'e discha.rge in col. 4 and F is tilE; fa.ll in col. 3 . Using water-surfa.ce cle\'ations at section 1 as listed in col. 1 of the ta.ble and the CQl:responding values of Q/ ..\/ifi col. 5, construct 0. sta.ge-versus-Q; vF curve (~'ig. 10-13).

in

755 752,25

~ 750 crt

:

---I-~-lf--,-~-

745

,

! ~;

··1- '

\

I 740r--r-

..,

I~

II

i

~i----~

I

'0

'0 735

L~

c

.12 0;

>

-[- w-

~

'" 730

."'g" . 725

"": ~ 0

o

3;

720

1

715 0

I

-r~! 50

I

100

I

ISO

I

I

I

I

I

200

I

250

I 1

h _ _I

300

,I I

1

.,,350 ".,

".,

Values OfQ/IFinl.o00

lJ~its

FIG. 10-13. The stage-vs.-Q(V'F curve for Example 10-11.

For a we.ter-surfaceelevation of 752.25, a value of Q/ -../F =33~,OOO i5 obtained by extrapolation. By (10-60), the fall between sections 1 ancH is equal to (431,000/ 355,000)' = 1.65 ft. Ad'ding- this va.lue. to the elevation at section 1, th~ required water-sufface elevation at section 5 is 753.90. This is about half a foot lower than the elevation. compu~ed by the standa.rd step method; the difference results primarily from the neglect of velocity-head changes in the present-method. The computation may be continued for subsequent· reaches. A tabulation,. as 5hown in Table 10-10, is suggested for the computation if a. complete flow profile is requil·ed. .

Eq.

"

\

284

GRADUALLY VARIED FLOW

285

METHODS OF COMPUTATION

If .d~ired, the ,,:,ater-surface elevati~m; at the intermediate sections 2 to 4 rna be obtaln~d by breakmg up the reach 1-5 Into four short reaches The profile 1 .~ at th . t d' . • e eva,.lons e III, erme la/,,", sectlOns may be obtained by in'erpel-t' " Th.e stage-versus, 1, } Ion. Q/ . v F curves can be drawn for each section and the cbmputat'o b . d fol' the subdivided reaches. ' . I n can e carne out

effects of velocity-head changes and eddy losses, is required, the Ezra method described in Art. 10-5 should provide more satisfactory results.

B

TAllLE 10-9., DATA, AND COMPUTATIONS FOR S·rAGE-vs.-Q/YF CURVE USED IN EXA~IPLE 10-11 (Missouri River at E:ansas City, Mo., sections 1 to 5)

Water-surface elevation, m,s.l.11

,

TA.BLE

Section 1

Section 5

(1)

(2)

724.8 725',3 729.13 727.4 727.8 730.2 730.8 731.3 734.6 735.8 736,6 745,0 722.2 724.6 725.0 725.3

725,7 726.2 730.2 728.3 728.8 731.2 731.7 732.3 735.6 736.7 737.7 746.6 723.1 725 6 726.0 726.4

.

"

I

10-10. ConiPUTA'I'I~N

(Missouri River Sea. no.

I

River mile

D'ISCh al'ge, cfs

ft

I

(3)

0.9 0.0 O.G 0.9 1.0 1.0 0.9 1.0 1 0 0,9 1.1 1.6 0.8 1.0 1.0 1.1

THE

(4)

I

.'

I

Length of reach

377.58

.....

5

378.65

5,655

. ....

Q

VF !~5)

33,600 36,100 66,100 69,500 76,000 97,200 105,000 113,000 141,000 157,000 104,000 326,000 22,900 45,400 49,900 52,300

35,40',) 38,000 85,30U 7'6,200 76,UOO 97,200 111,000 113,000 141,000 165,000 156,000 ' 258,000 25,600 45,·100 49,9UO '19,800

FLOW P'ROFILE FOil. EXAMPLE

THE ST,lGE-FA.LL-nrSCHARGE II-fETHOD at Kansas City Mo' ~ections 1 to 5 f)

1

......

OF

FII,11 ,

".'

~rater-silrface elevation

, '., =

I

10_11

Example 10·12. Determine the w!),ter-surface elevations a.t sections 1 to 5 of the Missouri River at lCa-nsas City, M.a., as described in Example 10-10. The da.ta required for the computation by the Ezra method are given in Table 10-7. The discharge is 431,000 cis. The initia.l water-surface elevation at section 1 is 752,25. It is assumed that eddy losses are included in the friction losses. S:)/u.t-ior.. The first step is to compute value of Z + F(Z) from the given data. The computation'is tabulated in Table 10-11 with the following column headings: Col. 1.' Channel-section number Col. 2. River mileage CoL 3. Length of reach in ft. The upper value f>:Cd is the length of the downstream reach from the selected section, and the lower va.lue Llx. is the length of the upstream reach. ,' CoL 4. '\Vater-surfa.ce elevations. Three elevations are given for each section. Generally, at least, three elevations are seiected for eaah section to provide at least three points for plotting each Z F(Z) curve. Cols.5 to 14. These columns correspond exactly to those in Table 10-8 for the standard step method. The values in the top row for each elevlltion al'e for the mllio cha!lnel, and those in the bottom row arc for the left-overbank section. CoL,15. }'rictioii slope, which is equal to (Q/[()~, where Q = 431,000 cis and lC is from col. 10 Col. 16. Value of .-8 , ClXd/2, where S. is the 'value from col. 15 and ClXd is the upper value in col. 3 Col. 17. Value of S, t..x./2, where S. is the value from col. 15 and C.x" is the, lower value in col. 3 ' Col. 18. Value of F(Z,), which is equal to the sum of the value in coL 14. and the value in col. 16 _ ('..01. 19. Valu~ of F(Z2), which is equal to the sum of the va.hl"- in col. 14 and the value in col. 17 Col. 20. Sum of the values of Z in col. ·1 and F(Z,) in coL 18 Col. 21. Sum of t.he values of Z in col. 4 and F(Z,) in col. Ii) The second stP.p is to plot curves of Z F (Z) against Z for each oross section, using , values from cols. 4, 20, and 21 of Tl1ble lCl-11. The resultiug curves are shown in Fig. 10-14. . The third step is to determine the water-surface elevations from the Z F(Z) curves. At section I, for a.n initial water-surfaae elevation of 752:25, the value of ~. F(Z~) is found from t.he appropriate curve· (Fig. 10-14) to be 754.14. Taking this value to the Z, + F(Z ,) curve for the next upstream section 2, the corresponding water-slUiace elevation is found to be 752.72. Contiuuing the procedure for other sections, the values are trD.Ced in the direction shown by thc dashed line in Fig, 10·14. The rcsults of the IV ater-surfac,e-elevation determinat.ion are tabulated in Table 10-12. They are in very close agreement with those obtained by the standard step method, Example 10-13.. Solve the problent in Example 10-12 for a discharge of 500,000 ds, The corresponding initial wuter-surface elevatior. &t section 1 was estimated from the rating CUrve to be 752.30, Solution. The values of F(Z ,) and F(Z2) for Q = 500,000 cfs may be obtailled by mUltip1ying the corresponding values in Table 10-11 by (500,000/431,000)' = 1.34.' T~,1e values thus obt~jned are tabulated in.cols. 3 and 4 of Table 10-13, respectivel}', and the values of ZI + F(Z.) and Z, + F(Z.) in cols. 5 and 6, t:,espectively.

+

BY

431,000 ds)

+

+

Q/-./F

F. =~-2 (Q/ -./F)

i

752.25 1.65 r 335,000 1.65 753,90 .... ' .. To be continued if desired' ...

I

10.,8. The ~zra Method for Natural Channels.' If flow profiles for a numbel' of discharges or stages are desired, the stage-fall.discharge metho~ can be use.d most advantageously for a simple and economical but approXlrnate solutlOn. However) if a precise computation, including the

+

10-11.

TA.BLE

R I R%

p

A

COMPllT.AT10N OF Zl + F(Z!) ANn Z2 +F(Z,) FOR E:u.MPLE 10-12 (Missouri River at Kl\llsas City. Mo.; Q - 431,000 cfs)

".

K

Ie'

A'

+F(Z.)

0.12

.... 11,&3

753.93

0.11

.... 11.80

754.80

I'

~

co

0>

!;;;:~• . 3:~c-.-_-.t

I~i

I.

f

0.10

...

0.13

1.46 1.70

753.46

753.7il

(l.l~

1.39 1.65

154..39

754.66

1.3111.55

755.31

755.M

~

1.71

7.55.71

J -0.151

-~'.~,--

;;<

TAllLE

10-11.

COMPUTATION OF

Zl

+ F(Z,)

AND

z. + jt'(Z.)

FOR l:i:::CHtI'LR

10-12 (conli'n"lJ.ed)

8ee.

+ F(Z.l

no,

0.24

0.21

.t

.

"" co

1.111 1,45\

753.11

753."'5

l.oJ 1.341

754.04

754.34

-0.08

0.. ~g

1.00 1.27

75S.00

755: 27

-O.Hi

O.1.~

0.5J

0;90

M2.61

752 ..90'

-0.15

0.12

0.5·, 0.83

753.5u

753.g3

-·0.13

0.11

0,~4

751.54

754.78

-l

i

0.78

752,62

0.62

H'OUI,,·u;onr'" x

In~.LJII)<.orovl:t'U!;~I""·.1 7,900 7411,10.6 4.B3,O.Q50 11

I

0.58

0.54

,.....---.

... 1

753.58

288

GRADUALLY VARIED FLOW

TABLE

10-12.

COMPUTATIOl\ OF THE FLOW PROFILE

FO~

EXAMPLE

10-12

TABLE 10-13. COMPUTATION OF Z + F(Z) ~'OR EXAMli'LE 10-13 . (Missouri River at.Kansas City, Mo., Q = 500,000 cfs)

DY

THE EZRA METHOD

(Missouri River at ICan~as City, Mo., Q

I

I

Sec. no.

2

3 4

5

River mile

Z,

• 377 58 377.78 377.9,1 378.33 378.65

+ f(Z.) z, + F(Z,)

. . . .. . 754.14 754.41 754.G8 754.93

754.. [4 754.41 754.68 754.93 .

= 431,000

cfs)

-~

'ffater-surface elevation Z, It

no.

752.25 . 752 72 753.38 7154.15 764.43

Z

F(Z,)

(2)

(3)

2

752

;::

3

.:o

:;:: .~

0;

754.43

'-'

75~5~=====

.,

~ 754

4

"'",

F(Z.)

z. + F(Z,) z. + F(Z.)

. (4)

752 753 754 75:1 754

OJ

289

METHODS OF COMPUTATION

(5)

754.59 755.41 75G.20

2.59 2.41 2.29 1. 96 1. 86

1. 78

(6)

2.36 2.22 2.08

753.{16 754.8G 755.78

75 1L36 755.22 756.08

1.94

753.49 754.40 755.31,

753: 94 754.80 755.70

752.82 753.75 754.73

753 21 754.11 755.05

752 753 754.

1,49 lAO

1.34

1.80 1. 70

752 753 754

0.82 0.75 0.73

1.11 1.05

752 753 754

0.83 0.78 0.73

1. 21

~

OJ

~

5

753

75ZL-____~_____L_ _~~L-~ 756

755

FIG.

754 Zl+F(Zll,ft

10-14. 'Curves of Z

753

+ F(Z)

.754 755 Z2+F{Z21.ft

756

for Example 10-12.

--,-----~----~--756,,----_.----_..------._--~

752.83 7fi3.78 75<1.73

The Z + F(Z) curves a.re then plotted (Fig. 10-15). These can o,lso be'obto.ined by a simple graphical metbod. In Fig. 10-14, straight lines Z = Z may be drawn along with the Z + b'(Z) c;urves. For example, referring to the curve for section 2, the intercept AB between the line Z "" Z and the curve for Z+ F(Z) is "qual t{) F[Z) for Q = 431,000 cfs at the water-surface elevation of 754.00. This intercept A. B multiplied by 1.34 gives the wrresponding intercept A'B' in Fig. 10-15 for the same elevation but different discharge. In this manner, the point Bf Bud, hence, the Z F(Z) curves for Q = 560,000 cfs can be plotted. From these curves, the required watersurface' elevations are determined, as shown in Table 10-11.

+

TADLE

10-14.

10-13 BY

COI.'iPUTATION OF THE FLOW PROFILl:l FOR EXAMPLE

THE EZRA

METHOD,

EDrn LOSSES BEING INCLUDED IN 'I'HE FRICTION LOSSES

(Missouri. River at I(an5fis City, Mo., Q Sec. no.

River mileage

1

li77 .58. 377.78 377.94 378.33 378.65

.

755

FIG. 10-15. Curves of Z

+ F(Z) for Example 10-13;

756.

'2 3 4 5

Z.

+ F(Zd z. + F(Z,)

= 500,000 ds) I Water-surface elevation Z; ft ~

I

754.83 755.20 755.56 755.83

1

754.83 755.20 755.56 755.83

752.30 752.97 753.84 754.80 755.18

,290

GRADUALLY VARIED FLOW ",,-,jI"'N-,jI

Example 10-14. Solve th(, problem in Example 10-12 by treating the eddy losses separately. It is assumed that the values of the roughness coeffident n do not include the effect of eddy losses and that the latter is to be computed ILS 50 % of the change in velocity head when the velocity head is decreasing downstream or as 10 % of the increase in velocity head when it is increlLsing downstream. Solution.. Curves of Z + F(Z) constructed for Example 10-12 (Fig. 10-H) are also applicable to this solution. The computation for the determination of t.he wD,tersurface elevation is tabulated in Table 10-15, in which the numbers in parentheses 755

Nt-~~~

~ C"l Cr') -.;fI ...q.I llJU")I.{)UJtI"J

t-t---t.... t-t-

\fJ~OOOM

C"Ir-~..-I-::t'

NNM-,jI ....

to l..() L.r:! lO l- t- t- t- t4"J

*;::;r::-~G~

-.- ........ -C'lM

I

i-f;)r O.f.,Vz/2g r~

'-""-"~

For O.5aV z/2g

__

,

l-

I

·11"

c

u

u

t

(/)

(/)

'"

VJ

~ ;8

c

.8

a.,,.

,,<

c

c

+ N'

-"-------.,

I

754 I

-

.;; c

0

.9-

u.,

"

rJl

~

~

tOt-lQ"ItI ..-I'd'tt-G -,j<",-,j<-,jI It;!1.()1J)'U')

t- t- t-.t-

+

N'

- - ; - - .-;:N~.n.--::-,;<..--:...... ::;-I -00'00 · 0000

~\,I

0;

:g'" ~

.'., '"

Il'l '0

C

C1

'0

753

vl~ NI~ ~ c 5 c0

:r; :fi~~ '" ",,, (J)(/)Ul

"

U)

I-

752

o

"

"

0.2

0.4

~ 0.6

\

• <,0"

]~

o.a

ka VZ /2 9

FIG. 10-16. Eddy-loss correction to be applied to Z,

...... • tL.,l

o~

• . C"')

.. ..

~

~t-O>

""..,. .... -.1'

~c.:.

IJ)

If)

It)

t-t-t---r--

g+

+ F(Z.).

t'- LC

.-----,........... C'l OJ ......-t

:8~£3~ --


indicate the order of the computation steps. The curves ~f 0.5,. V'/2g and 0.1,. Y'/2g are then plotted (Fig. 10-16). The:;e 'curves are used to correct for eddy losses, as ' , explained previously (Art. 10-5). Starting at section 1 with a water-surface elevation of 752.25 (see step "1 in Table 10-15), the ,value of Z. + F(Z,) is found from the Z, + F(Z,) curve for section 1 as 754.14 (see step'2). The corresponding values of 0.5,. Y'/2g and 0.10: Y'/2g are found from Fig. 10-18 to be 0.89 (see step 3) and 0.18 (see step 4), respectively. Taking the value of 754.14 to the Z, + F(ZI) curVEl for the next upstream section 2 (see step 6), the corresp6nding water-surface elevation is 752.72 (sEle' step 7), and: the value of " 0.5",Y'/217 is 0.78 (see step 9) and that of 0.laY2/217 is 0.16 (see step 10). Since the

.,.;

......,

S

~

-.!'..,;t-,jl-,jl

r..

~:I,CJlQtClJ)

+

..-~ ~~

t- t-.. t- t-

": ~;;a ~ I----~~--=-=-=--I ~ ~

e:5~~~:g

~

t-~l-or-cooo

a

~~~~f.;

JS

I----~-----------l~ .,,; 0

lJ.la

291 ~.

METHODS OF COMPUTATION

292 )

GR.4.DUALLY VARIED FLOW

c. The dire~t step method . d. The standard step method " . he section properties described in Example 10-9. If the sl~pe of a challn~l ha~I~~; 001G on the upstrean1 sidl', to 0.01G9 011 the 10-1 has a break 10 grade changlllg hOI • th downstream side from the l;rcak to tt BI downstrenmside, compll:e the. flo~ ~ro ~he ontheenorInal depth. Use: section where the depth IS 1 % glea er an .

velocity is in~ren.sing downstream (or decreasing upstrenm), the eddy- los8 is 0.18 0.16 = 0.02 (see sLep 11). In order to include the effect of this eddy los5, add 0.02 to 754.14 (see step 2) which was previously found to be the value of .2. + F(Z.) a.t section 1. The resultIng v!l.lue is 754.16 (see step 13). Referring to the Z, + F(Z,) Curve for section 2 with this value, the "'ater-surface elevation is found to be 752.74 (see step 14). . Repeating thll !l.bove procedure a.nd correcting for the eddy losses accordingly, the ,vater-surfllce elevations at all sections over the reach nre obtained.

\

The gr[1.phical-integra t iol1 Ineti10d . d b. The direct-integr!l.tion metno. c. The direct step Inethod d. The standard step method . . h' ectioiJ roperties described in Exa.mple· 10-tO. If the slope of p- ch~nnel..h~vm~ to;6~ on th~ upstream side, to 0.00 tG on t,\le 10-l hll.S " bre~.k ln grade c!U\,lglll., hOffi · the downstream side frOIll the b:'eak to" fii downstream Sl. d e, com pute the flow pro f; on h el is !UlSLlllled uniform. U ~
a.

PROBLEMS 10-1. Show that the flow 'pro file in a wide horizon tal channel. may be axpressed ns

\.

:c =

I

"'uC' (v - '!y" ~) + const

(10-61)

"there C is Chezy's resistance fa'ctor. 10-2. Show that the condition for the existence of a point of inflllction on the flow profile may be expressed as

\

I

(l0-G2) 10-3. Complete the computation of the flow profile in Example 10-2. , Complete the computation of the flow profile in Example 10-5, nnd compare the result with that of Prob. 10-3. 10-a. Show Lhat the .flow profile in a frictionless rectangular channel ma.y be expressed' as 10-~.

:c =

_V, 2,1).

r_ 2 -

3M: + Y

10-6. Solve Example 10-1 by assuming a of c< on the shape or, the flow profile. Use:

=;

(it:)3] v

graphical-integration method direc t-in tegration method direct step method standard step method

10-f)' A free fail instead of the dam controls the depth at the downstream end of the channel described in Example 10-1. Compute the flow profile from the control to an upstream section where the depth of flow is 1 % less. than the normal depth, using:

1

\

\

.10-7. Compute the flQw profile in the chl\unel described in Example 10-1 if S. = 0.0159 .. Assume that the profile starts Ilt the dam site at a depth of 5 ft and terminates upstre~m at a .critical depth. Use: '

.a. The graphical-integration method' .0.' The direct-integration method

I

1~O, and exnmine the effect of the vahle

b. 'Tlte direct-iutegration method c. The direct step nwthod d. 'The standard step method

a. The b. ,The c. The d. The

\

i

a. The gro.phical-integratioll method·

a. The,graphical-integration method b. The direct-integration method c. The direct step method. d. The standard step method 10-11. Solve Example 10-3 if S, = O.OOlo,using:

,(10-63)

+ const

293

!

\ i,

The The c. The d. The

(t.

o.

grE1phical-integration method direct-integration method direct step method standard Ilt,~.p method

10-12. Solve Examp:"- 10-3 if S.

=

0.OlG9,using:

a. The gra.phical-integration method b. The direct-integra.tion method c. The Llirect step method . d. The stancl.ard step met,hod

10-13. Solve Example 10-3 if So = a. b. c. d.

a,

ustng:

The The The The

gmphical-i·ntegr!l.tion method direct-integratioll method direct step method stand~rd step method " , , . ' P b 10-8 is horizontal, compute the flow profil" 'h'~ the depth of ~0\V is 10 n. U~c: 10-14. If the cha.nnel cescnbed III rO. t' from the control section to an upstreu.m sec lon'~ e o l , , •

a. The gmphicai-integr.ation method b. The direct-integration method c. The direct step method d. The sta.ndard step method

10.16. Solve Prob. 10-14 if So = -0.0016: 10-16. Solve Example 10-8 if a = 1.10. of numerical integration, using the 10-17. Solve Example. 10-8 by .the method varied-flow-function table (AppendiX D) .

294

METHODS OF COMPUTATION

10-18. A prisma.tic trapezc;ida.1 earth spillway with 3: 1 side discharging. 1,500 cfs. The bottom of tlie spillway is horizontal and is 200 ft long a.nd 75 it \i'ide in the reach between a downstrea.m critical control section and the upstream reser. voir. Manning's n is estimated to be 0.035. Using any method given in this chapter, determine:

ll. Josef Kazeny: Berechnung der Senkungskurve in regclmassigen braiten Gerinnen (Computation of surface curve in uniform broad channels), Wasserkrafltmd IVasseT'lUirtschafl, Munich, vol. 23, no. 16, pp. 232-234, Aug. 15, 1928.

12. Armin Schoklits!:h: "Wasserbau" ("Hydra.ulic Structures"l, tmnslated from the Germ2.n by Samuel Shulits, American Society of Mechanical Engineers, New York, 1.934, vol. 1, p. 105. 13. NagahQ MonoDobe: Baclt-water and dnJp-doWD curves for uniform channels, Tra11.s(!clioM, American Society of Cilri! E71gineer., vol. 103, pp. 950-989, 1938. 14. Ming Lee: Steady gradually va.ried flow in. uniform channels on mild stopes, Ph.D. thesis, University of Illinois, Urbana, 1947. 15. Ming Lee, Harold E. Babbitt, and E. Robert Baumann: Gradually varied flow ill uniform channels on mild slopes, University of IUinoUl, Engineering Ex-peri'lnenl Statio'!), Bulletin Se7ifUl No. 404, vol. 50, no. 28, November, 1952. 16. M. E. Ven Seggern: Integra.ting the equation of nonuniform flow, Transactions, American Society of Civil Engineer~, yol. 115, pp. 71-88, 1950. 17. Clint J. Keifer and Henry IIsien Chu; Backwater functioIls by numerical integrll-tion, Transac&·'or.a, American Sodelv of Civil Engineers, vol. 120, pp. 429-442, 1955.· 18. Ven Te Cho~': Integrating the equn.Liol1 of gradually varied flow, paper 838, ProceedillflS, American Society of Civil Engineers, voL 81, pp. 1-32, November, 1955. Discussions by Clint J. Keifer a.nd Henry Hsien Clm, Robert Y. D. Chuu, iV[3.Sashi Hom-rolL, Allan N,,::wman, anel Kolupo.ila, pa.per 955, Journal, Hydmulics Division, vol. 82, no. HY2, April, 1956; corrections on p. GO. Discu'isions by R. Silvester and Alfred pa.per lOlD, vo\. 82, no. HY3, pp. 13-21, June, H)56. Closing disoussioll by Buthor, paper 1177, vol. 83, no. HYl, pp. 9-22, February, 1957. 19. I. I. Levi: GidraV'licheskie pokaz!1teli rusla i i1th prilozhenie k teorH nerl).vw.merno· go dvizheniia zhidkosti v otl,rytykh kanalakh i r\lslllkh (The hydraulic exponents of channel;:; and their application to the theory of nonuniform flow in open channell, 'Vestnik lrrigat.sii (Herald of Irrigation), Tashkent, U.S.S,R., no'. 2, pp. 35-49, 1925. 20. Ar~hur E. MatzkfJ: Varied flow in open cha.nnelsof adverse slope, TnL7l,.saclioM, Ame"ica'il Society.of Civii Ell!Jineers, vol. 102, pp. 651-660, 1937. 21. M. D. Chertousov; "Gidravlikll.: SpetsiahlyI Kurs" ("Hydraulics: Special Co Ill'se i,), Gosenergoizd3.t, Moscow, 1957. 22. V, J. CharnomskiI: Zadachi na ustallovivsheiesi" neravnomerlloie tecbenie vody v otkrytykh ruslakh s priamolineinym i trapetsoidalnym pOJlerechnym secheniem (Problems on stea.dy nonuniform flow of water in open channel> with straight,lined trapezoidal cross dec~ion), Trudy Va.J·shauskago Polittkhni.ch~kago lnslitu.l.a, Warsaw, 1914. . . 23. Alva G. Husted: New method of computi\lg bo.ckwater and drop-down curves, En(rineering News-Record, vol. 92, no. 17, pp. 719-722, Apr. 24, 1924. 24. H.R. L"o.ch: New metl10ds for the ~olution of backwa.ter problenL~, Engineering News-Recul'd, voL 82, no. 16, pp. 768-770, April, 1919. 25. Arthur A. Ezra: A direct step method for computing. water-surface profiles, Transactions, American Society of Civil vol. 119, pp. 453-462, 1954. 26. Fra.ncis F. :&Scoffier: Graphic ·calcula.tion of ba.ckwater eliminates solution by trial, Engineering News-Record, vol. 136, no. 26, p. 71, June 27, H)46. 27. N. Raytchine a.nd P. Chatelain: Determina.tion graphique de 11\. ligne d'eau et calcul des remous (Graphical determination of backwater curves), La Houil.le blo.ncke, Grenoble, 5th yr., no. 3, pp. 373-379, May-June, 11l50.. 28. Josef Frank: Graphische Berechn'lng von Wasserspiegel-Linien (Graphicll.l calcu-

a. The flow profile between the reservoir a.nd the control section b. The friction loss through this spillway measured in it c. The elevatio11 of the pool level in the reservoir 10-19. A prismatic tntpe20idal earth spillway with '3: 1 side slopes and a bottom width of 75 Jt is discharging 1,500 cfs. The spillway h!ls Il horizontal bottom upstream from a. critical control section for a distance of 90 ft Itnd an advene slope of 10: 1 foro.n upstream distance of 100 n. from the reservoir. Manning's 'II. is ,0.035. Using any . method given in this chapter, determine:

a. The flow profile in t.he spillway ,b. The energy line in the spillway

,\\

., ,

295

GRl..DUALLY VARIED FLOW

10-20. Solve the problem in Ex::o.mple 10-10 by the standard step method, io~ a design discharge of 500,000 ds. 10-21. Carry out 'the computation in Example 10-11 by brep.king up reach 1-5 into four short reaches at tho intermediate sections. 10-22. SOlve the problem in Example 10-11 by the stage-fall~discharge method for a design discharge of SOO,OOD cis. 10-23. Solve Example 10-1 by the stage-fall-discharge method. 10-24. Solve Example 10-1 by the Ezra method. 10-25. Solve Example lO-i'by the Ezra method if Q = 500 cIs.

REFERENCES 1. A. J. E. J. Dupuit: "Etudes tht\oriques at pratiques sur Ie mouvement des eaux" ("Theoretical and Pmctical Studies on Flow of Wo.ter)J), Paris, 2d ed., 1863'. 2. J. A. Ch. Dresse: "Cours de mecat:.ique appliquee," 2e j;:artie, Hydraulique, (tr Course in Applied Mechanics," pt. 2, Hydraulics}, Mallet-Bachelier, Paris, 1860. 3. F. Grashof, "Theoretische Maschinenlehre" ("TheoreGic~,l Course on MD.·' chines"), Leipzig, 1875, '(01. 1. 4. M. Ruhlmann: "Hydromechanik" (" Hydromechanics "), LeipZig, 1st ed., 1857 ; Hahnsche Buchlumdlung, Hanover, 2<1 ed., 1880. 5. G, Tolkmitt: "Grundlagen der Waaserbaukunst" ("F'llndamentals of HydrauHc Engineering"), Ernst & Sohn, Berlin, 189S. 6. U. Masoni; "Corso .di Idraulica Teoriea e Pratica" (" Course of Theoretical o.nd pro.ctica.\ Hydraulics"), 2d .;d., Naples, 1900. 7. Boris A. Bakhmeteff: "0 Nern.vnomernom Dvizhenii Zhidkosti v' Otkrytom Rusle" ("Vo.ried F10w in Open Channels "l, St. Petersburg, Russia, 11112. 8. Boris A. Bakhmeteff: "Hydraulics of. Open Channels," McGraw-Hill Book Company, Inc., New York, 1932. 9. Philipp Forchheimer: "Hydmulik" (" Hydraulics"), 1st ed.} Teubner Vcrlags. gesellschaft., Leiptig and Berlin, 1914. 10. E. Baticle: Nouvelle methode pour la d6termination des combes deremous (New method for the determination of. backwater curves), Le Genie civil, vol,71l; no. 23, Dec. 3, pp. 4.88-492, 1921; no 24, Dec. 10, pp. 515-516,1921.

1, ~.

.1

-j

.j

i I

I·

296

GRADUAT,LY VARIED FLOW

.

]ation of water flow profile ) S h ' ,; , : 65-71, Aug. 5, 1!J33. s, c we1Zen/iche Bauzeitung, Zi\\-jch, vol. 102, no. 6, pp .. 29. Hydrologic and hydraulic analysis' Com t · r channels, U:S, A rmy Corp~ of En'gine/?7'S p~atlOn 0: backwater curves in river CDnsinlction,.pt. CXIV ehap. 9 M 19~2 nqmeenng klanual for Civil lYorks 30 C I -" " 1Iy, D. , . . . urJmm;.Baclcwll~er slopes above dams E ' . . no. 23, p. 902, June 7, 1928 . ' nqmeermg News-Record, vol. 100, 31 . I . H. Stemberg: ' . Thc nomograph as a aJ'd . . . . E7!.{Jineeririn vol 9 110 6' pp 365· 36 (ln J In. computing backwa~l' curves· Civil .""'" -;ulle 1939 ' . 32. Sherman M. Woodward and Che~ter 'J Po' '. ,: .... Open Channela," .John Wiley & Sons I~c y Hydraulics of Steady Flow in .' ., ,e\~ ork,1941;'PP.103_107. 33. L. E. Jones: The QIy'Fteoh' . - mque in open c.nnnnel hydrau'·· P d' 2d M 1'd western Conference of Fluid.M h " T· . lIes, rocee 111fT;; ·of the ,. E . . de a7IJ~, he 01lto Stale. U' . E . ., lny xper'ment Stal'io."-, B'ullet;n 119· 8. , ',nwers1ty, ngmeer· 34; A. N. Rakhmanoff: 0 po
;;Y:

0; R

I

~t

I

Ii

i

I'

CHAPTER

rI I

11

PRACTICAL PROBLEMS

! t

11-1. Delivery of a Canal for Subcritical Flow. When a cana.l connects two reservoirs .. having varyill;g levels, the discharge of the can'al under variou~ conditions of reservoir level is called the del1'uery of the canal; this problem was discussed by Bakhmeteff [lJ. Bakhmeteff has treated this subject for prism:3,tic canals with subcritical flow under thr,ee general cases. The Cl1SI3S are clMsified according to the condition of three v2.l'iables, namely,the depth of flow Yl at the upst~eam end of the canal, the depth of flow Y2 at the dovvnstream end, and the discharge Q of the canal. A. Case of Constant Yl. This is the caSE in which th~ water level at the upstream end oUhe canal does not change (li'ig. 1l~1). The depth Yl is

FIG. 11-1. Delivery of a canal with subcriticl!.l flow IJ.nd C0l15tIJ.Ilt YI, . .

,

assumed to remaincon~tant, owing to a constant pool level A i while Y2, determLned by level B,fiuctuates. Also shown in Fig. 11-1 are the flow profiles for various conditions of Y2 and the corresponding dischat'ges Q. The relationship betwe~n Yz and Q is shown by a eo-called del1'very cW've Q = f(Y2). Several flow conditions may be desci'ibed as follows: 1. UNIFORM FLOW. "Yhen Yz = YI = ij", the ~ow is uniform, with Its smface represented bya straight line an parallel to the channel bottom. The corresponding normal discharge Q" is indicated on the ddivery curve. 297

I I

298

299

PRACTICAL PROBLEMS

l \

GRADUALLY VARIED FLOW

The value of thi~ discharge is Qn = /(" v' So, where I{,. is the conveyance of the cross section with the depth Yz equal to YI and where So is the bottom slope. 2. FLOW OF MAXIMUM DISCHARGE. When Y2 is equal, to the critical depth Yo of.section 2, the discharge will reach its ma.ximum possible value, since Y2 cannot be le~s than Yo and tlw head is at its mfLximum. If t.he downstream pool level B falls below the depth Ye, a free overfall at that depth will occur. This discharge, as indica.ted on the delivery curve, is equal to the critical discharge at section 2, or Q, = z .. vg, where Zo is the section factor of cross section 2. For the determination of the maximum discharge, a trial computation is required. The procedure is first to take a series of discharges, beginning from Q" and going upward. Then, making Yz = Ya in each case, determine the corresponding YI. The discharge that makes YI equal to the given d'epth at the upstream end is the Qm ... required .. 3. FLOW OF .l1.fl PROFILE. When Y2 > 11", the flow profile is of the 1];I1 type. The upper limit of this curve is a horizontal level, indicated by az; at this condition the discharge is zero, since the head, or difference between the pool levels, is zero. For Y2 > Y., the fiow will l:everse its direction .. The lower limit of the 1111 profile is, apparently, the uniformflow surface an. For any intermediate flow between these two limits, the depth Yz and thp. corresponding discharge can be determined by a trial flow-profile computation. The procedure is first to assume a. discharge less than Q,. and then to compnte the corresponding depth Yf. Consequently, the delivery clirve can be plotted. : 4. FLOW OF 1112 PROFILE. When Y2 < y", the flow profile belongs tQ the .1112 type. The lower limit is, evidently, the critical flo\y surface ac. The Y2-Q relationship can he determined by the procedui'e described above for the lIfl profile. From the delivery curve, it becomes evident that the portiori NC of the curve' is very steep, so that Qm.., exceeds Qn by only a very small amount. It will be seen that this is true in most practical cases except for very Ehort or very fiat canals. Referring to Fig. 11-2, it is assumed that the end point d of the limiting 11{2 curve is loeated at a depth O.99Yn. If the length L of the canal is greater than the length L' of the limiting J1'2 profile,' then no change in Yz between Yeand Vn will affect the condition upstream from the point dj that is, the discharge will rema'in the same. As long as L > L', the maximum discharge QJD.x will be pra.cticaUy equal to Qn, that. is, the portion NCoi the corresponding delivery cllrvewil! be practically vertical. The flow-profile equation indicates that the length of the flow profile is :inversely proportiOlial to the bottom slope; the smaller the slope,the longer the profile, and vice versa. For this reason, reducing the slope will have an effect similar to that of making the canal

~ , r I

-1

I

,.

1,

z

Deli,ery

! ---'-------:--------,---1----- .--~-

cu"~:

~,=

I.

I

!

. conslonll

d

0""':'"

MZ"p'orile \

J

-. -----..---

~ ~--T-i--

~,

----------I c

I ~%~ -----.:.... -.__ l ,/_ _ ,_UJ__01 -.-- l ---__ ' - , r-- 0

---..,.

[1,

N

.

! :

k'

11-2. Delivery of

0

I

0

'0"'°< -~

cane.l, the lellgth of which exceeds the length

df

tbe falling

.

ID~~

. shorter. Therefore, for practical purposes, it may be assumed tha\ the maximum. possible discharge in a long canal or 0, canal of not-tao-small SoOpl! . is eqtwl 10 the normal discharge. .' . B Case of Constant yz. This is the case In whICh t.he watel le~el at 'the 'downstr'eam end of the canal, or the depth Y2, is const~nt, whlle ¥l fluctuates. The corresponding delivery curve Q = fCYI) IS shown III Fig. 11-3.

Fro. 11-3. Delivery of

3.

canal with subcritical flow and constll-nt y,.

1. UNiFORM FLOW. When :111 = 'yz = Yr., the flow is u.niform, the flow profile. nb is parallel to the channel bottom, and the dlschar~e Qn corresponds to point N on the delivery curve. The value of Qn IS equal to

) '!'

r, I

~I'

"1

'I ~-

J _ -t

FIG.

o;/(yzl

I

.1

... 300

,

'-

I

GIL'l.D1:ALLY VARIED FLOW

K" .where Ie, is the conveyance of cross section 1 with dept.h Y1 = Y2 and wher~ Su is the bottom slope. 2. }I'.fow of Maximum DischaTge. When YI reaches a depth Yn< that corresponds to a clitic111. discharge at section 2, the discharge becomes maximaL Any depth Yi > Yrn is out of consideration, because it would sill1ply raise the flow profile mb to the position shown by the dashed line m'b' and, consequently, require an increase in the downstr~am depth ·Y2. The~o.lue of (J •.• ~ is equal to the critical discharge at section 2, or Zc -iii, where LIe is the section factor at section 2 for a depth to Y2. 3. FLOW OF M"l PROFILE. For Rny depth 1:/1 < Y", the flow profile belongs to the Ml type, and the is less than Qn. The lmve"t possible limit for YI is y.; at this condition the flow profile is horizontal and the dL':!charge is zero. 4. FLOW OF 1\-12 PROFILE. For any depth Yl varying between Ymand Yn, the flow profile. belongs to the 1112 type, and the discharge is less than Q,,,"x b:llt greater than Q~. C. Calle of Constant Q. This is the case in which the delivery of the canal is constant,while the pool levels at the two extremities of the canal fluctuate. 1. UNIFORM :now.. Referring to Fig. 11-4, various possible flow pro- . files are sketched.. "Vhen YI = y., = y", the flow is uniform, and the surface is a straight line ab pal-aIlel to the char~nel bottom. This normal depth 11", cu.n be determined by the Manning formula for a con~tallt discharge Q. 2: .FLOW OF 1111 PROF'ILE. For positions above ab, the flow profile belon'gs to the 1\{1 type. The upper limit of the 1111 profile is a horizontal line; at this. position 112 approaches Yl + SoL as a l~mit. As this c0nditiol1 ilS approached, the difference between pool levels and hence the head, or velocity of flow; decrease, However, the water area increases as the depths increase; whereas the discharge, as a product of them'e[j, and the velocity, cnn still be kept consts.nt and equal toa given value. . 3. FLOW .OF .312 PROFILE. For positions below ab,the flow profile is of the JlII2 type. The 10we.'3t possible position. of the JJ12 profile is a'b'; at this position y~ is equal to the critical depth corresponding to the given discharge Q. . The Q-constant Curve. The relationship behr~en the depths Y1 and 1'2 for const~nt Q can be plotted (J:"ig. 11-4). The resulting curve CN P is known as the Q-constant curve. Several auxiliary e:urves have also been constructed to make clearer cert,ain chamcteristic features of the Q-constant curve. . The N line is a straight line drawn from the Qrigin of the coordinates and inclh1ed at an angle of 45 with the coordinate axes. This line is the locus of the normal depth for all discharges. For any point on this line, Q

301

PRACTICAL .PROBL'EMS

_ ,_

. The Q-constant curve intersects this line at t?e point N

1/1 - 'Y2 .- V,,· ~ which is the norml1,l depth for the given discharge Q.. where Y1 V2 :- Yhn, wh'lch y is equal to the critical depth y. of The curve 15 t e curve on 1 · r g

at-

2 for

cdrosshse~ IOn L tion ept . a se,c ..

agiven discharge and on which Y1 is the corresponc :n

It is apparent that 112 cannot be less than Yo 0,£ sectIOll . Q H"'l1ce the Q-constant curve termmates at 2 for the given dIscharge, ~, ' t ' ,2 fthe discharge the point C 011 this curve that makes V2 = y. at sec 101. or

Q.

. FlO, ll-? Delivery of l\. canal withsubcritical flow and constant Q.

rr.

The Z line is a straight line drawn parallel to the lh~e fl;om a point on . . d' t S T from the orilrin O. ThIS hne represents the the Y~ aXIs at a is ance I)J.J. c . fil H + SJ." or the upper limit of the lJtil pro e. enee, ' . d con ItlOn Y2 - Y l , t f 11 f m the left , the Qcconstant curve apPl'oaches this line asymp 0 1(\a y 1'0 when both YI arid Y2 become vel'Y large. . , The coordinates Vl and Y2 of any point P on the Q-collstant curve 101 [l.

-. I 302

\

303

GRA.DUALLY VARIED FLOW

PRACTICAL PROBLEMs

given discharge Q ·call. be .determined by a .flow-profile computation. Geilerally, when points C and N and one or two other points are locatcd, the Q-constant curve can be drawn in smoothly. By plotting a series of Qcconstant curves for various discharges, sueb as the claShed lines, a general chart can be obtained, representing. all possible flow conditions in the given canal. 11-2. Delivery of a Canal for Supercritica\ Flow. When the slope of the canal is steep, that i:o, greater than the critical slope, the flow in the channel becomes supercritical (Fig. 11-5). In practical applicf.tions, 8teep canals are usually short, such as the raft and log chutes th~t are used ~LS spillways. If t.he canal is too steep, so that an ultrarapid flo\\" develops, then the flow is no longer steady. A study of unsteady How in canals is beyond the scope of this chapter.

3. As the tailwatel' level rises further, the jump will move upstream maintailling its height a~~d form in the uniform-flow zone nb, until' reaches point n. From there on, the jump will move upstrecun on the en CUl've, gradually ciiminishing in height. The height, of the jump becomes zero when it reaches the critical depth at e. In the meantime the flow profile reaches its theoretical limit, cb l l of the SI profile.. Be~ond this limit the incOiriing flow will be directly affected by t.he tailwater, and the entrance acts as a submerged weir. III practical calculations, the'hol'izontallil1e Cb'" may be +,aken as the practical limit of the tail water stage. This wi\] avoid the compntation of bUlb" and will also provide a margin of safety; 11-3. Problems Re.1ated to Canal Design. Knowledge abollt the delivery vf a c
-I

it

t ______! __ ~----'~z

o -

i=.:::::.:::::::~ ~

, /'

I

t:.y

,

. -=.--=

_I

J

-·-,.ily'

c~~-·=tt- /-=~~~f-~:-=~'F r~\~: -,_,,// / I~_ ----,.-.-- _'

~

Conal bof!om of jncrec!;ed slope

FIG. 11-5. Delivery of a cana.l with supercritieal flow.

_

.[

~

U

!

!

Omin Ome.·

11>-

0

Service discharge = Q

.. [

FIG. J 1-6. Canal design for variable sel'vice discha.l'ge.

/1. Dischm·ge. Asthe control section in a channel of supercritical flow is at the upstream end, the delivery of the canal is fully govel'ned by the critical discha.rge at section 1, which is simply equal to the discharge through a weir: B. Flow. PTr).file. The type of flow profile developed in a steep canal depends on the tailwater situation. 1. When the taihmter level B IS less than the outlet depth at section 2, the flow in the canal is unaffected by the tailwater. The flow profile passes .through the critical depth near point e from a convex to a· concave shape and approaches the normal depth by mea,ns of a smooth drawdown curve of the S2 type. As a yule, the drawdown curve en is comparatively short, andYn is the lowest possible stage in the canal. In designing such canals, the normal depth made equal to the depth required for passing floating craft 01' for allowablescouririg effect. 2. When the tailwater level B is greater than the outlet depth, the tail-water willl'aise the water level in the downstream portion of the canal to form an SI profile between j and b' , producing a hydraulic jump at the end j of the profile. However, the flow upstream from the jump will not be affected by the tailwater.

is

,i I

,t I

II

II I

I

::.pplications in the hydraulic design of cana.ls. Several important problems related to such applications are described below. A. Change £n Depth Due to Changing Delivery. In designing 11. canal, it is often necessary for the engineer to anticipate t.he fluctuation ill depth of flow due to any possible change in delivery. This fluc(,uation in depth can be estimated easily from the delivery curve of the canal for the given range of fluctuation in discharge. In most cases, except where the canal is very short or where the bo~tom of the canal is unusually fiat, the canal can be designed for a uniform-flow condition, because 'the maximum discharge will be practically equal to the normal discharge. The procedure of canal design for uniform flow h::),s been discussed in Chap. 7. The relationship between the depth and discharge can be obtained easily on the basis of any uniform-flow formula. such as the Manning formula. When a canal is designed to deliver water from a reservoir of constant pool level to a service chanilel at the downstream end, the discharge in th·e canal should meet the variable demand, when a,nd as required by the service channel (Fig. 11-6), This; wi~h a subcritical fiow, falls into the

.[

I .)

1

-\ .•o:.~'

.[ I

'304

P~ACTICAL

GRADUALLY VARIET,> Io'LOW

• case of constiUlt Yl, A. in' Art. 11-1, in which the l~pstream depth is kept constant while the dOWllstream depth fluctuates. The delivery curves in Fig. 11-6 will explain how the fiuetuation in the downstream end depth, caused by the variation in the service discharge demand, can be redlacec1. The method is 'simply to increase the bottom slope of the canal. ,This , increases th~ normal diseharge, and the delivery curve will change in position from ZNC to ZN'C'. It is apparent that, for the same range of Q",.~ to Qmi., the fiuclu:1tion in depth fJ.y' is reduced and becomes lestJ tha~ fJ.y.

PROBLEMS

305

upstream-pool-level position A. The ditTerE)llCe fJ.y between YA and YI mny beset by judgment or may depend on 'the design of the entrance. , 3. FREE ENTRANCE; When water enters a mild-slope canal freely (Fig. 11-7d), the depth YI is related to the static pool level A by the law of energy. The relation between the depths Yl and YA can be expressed by ,

,VAl

YA '= Yt

,

+ h. + ayg

(11-1)

where aY A 2/2g is the velocity head of the npproaching fiGw, which is uSltaliya small quantity arid can be ignored. Hence,

=

Y...

Yl

+ h.

(11-2)

For subcritical flow, the term h, is the head loss due to friction and may be expressed in terms of thE) velocity head at section 1, that is, TI i ~

(al

(11-3)

= C'-2 g

h,

where. C. is a coefficient which has an average value of 1.25 for a wellrounded entrance. l Solvili.g for TIl from the above equation, VI

1

, -vc: v'2Yh. =, ~-

(11-4)

The delivery of the cana] is equal to

Q = V 1 .4 1 = .

(b]

FIG, 11-7. Outlet and entrance conditions.

B. Outlet and Entrance Conditions. In the pl;eceding artides, the delivery of the canal was related to the depths '!it and Y2 at the ends of the canal; but the conditions that accompany the'inflow or outRow of the water were not c.onsidered. 1.. OUTLET. "Yhen the canal empties into a reservoir, an amount of kinetic energy equal to a V z2j2y, carried with the flowing water, is expected tO'be restored as a potential energy. TIlUS, the pool level should be higher by this amount than the depth at the outlet of the canal (Pig. 11-7a). This energy, however, is usually dissipated entirely in eddies and whirls. In pl'fl.ctical computations it may be ignored, and Yz may be taken equal to YB, If the o,utfiow is accompaiied by a hydraulic drop (Fig. 11-7b) and if YB
Jc:

,~

Al ;;2(Jh.

=

.

~ Al 'V2g(-V.{ -'~ Yt)

v~

(11-5)

In most pract.ical problems, the depth YA, instead of YI, is given. For example, in the C[l.se of the constant upstream depth, A iJ:l Art. 11-1, the depth YA would he given as constant. For any given condition of YA, the relation between Q and Yl can be established by Eq. lI1-5). A curve representing this relRtion can therefore be constructed. By means of this so-called inflow-discharge-rating c'Urt1e, the reiations.among YA,YI, o.nd Q can pe detei-mined. For supercriticalflow, the flow, at section 1 is criticaL, The problem is simplified by the fact that the rf',lation between Yl and y.{ is practically fixed, irrespective of the entrance friction loss. C. Elimination of Hydraulic Jump in a Steep Canal . .. As pointed out in Art. 11-2, when the tail water-pool level is higher .tho.n th~ critical depth in ~ st~ep canal, a hydraulic jump will develop in the callal (Fig. 11-5). Such a;hydraulic jump is objectionable and dangerous, par,t.icularly when , ,

Dat~

the losses in entrance structures may be found in some hydraulics textbooks, handbooks, or other literature [21. 1

OIl

l

306

I

GRADUALLY VARIED' FLOW ,

the canal is a raft chute or some other structure intended to transport a floating raft from I,he upstreail1 reservoir to a downstr~am pobl. Bakhmeteff [1] suggested that the design of a neutTalizinO' Teach (Fig. 11-8) might be a solution. In this reach the bottom slope of the channel is made equal to the critical slope. According toa corresponding case of Cl profile in Fjg. 9-2, the tailwatcr lcvels will be approximately horizontal lines l \-vhich intersect the surface of flow in the canal without causing any disturbance. At the point of intersection, theoretically, there is a jump o'f zero height.

s'

B

Tailwaler

Fro. 11-3. Elimination of hydraulic jump by Bakhmeteff's neutralizing reach.

11-4. Computation of Flow Profile in Nonprismatic Channels. The theory and analysis of gradually varied flow ii1 nonprismatic channels has been discussed previously (Art. 9-5). The integration of the differential equation for the flow profile of such a :fl.ow is mathematically complicated. If the control section is uncertain in a given problem, its position may be determined by the method of singular point. For the computition of the flow profile, the step method is recommended. The computation should proceed upstream from the control section if the flow is sub critical and. dOWllstremn from the control section if it is supercritical. The procedure of computation is practically the sa,me as that applied to a prismatic channel, as will be seen in the following example': Example 11-1. A spillway channel, as shown in Fig. 11-9;has been designed tentatively for La Tuna Canyon Debris Basin at Los Angeles County, Calif.' ' According 'When the Chez), formula is applied, the lines are theoretically horizontal. The data for this example are taken from [3]. For additional information, the following is extracted from this reference: The spillway was designed to pass the maxi-, mum probable flood H),OOO cfs with the maximum water surface 5 ft below the top of the dam. 1;'he wide inlet sill would provide maximum sill elevation for retention of debris with ininimum height of dam. The invert slope was designed such that control, or critical depth for 19,000 cfs, would occur at the relatively narrow base width of.55 ft at stati?n II + 22. Establishin control at t~is narrow w.idthwould mini9 mize the formatlOn of large waves produced by the rapidly convergmg walls. For a design flood of 6,200 ds, waves would have:no adverse effect, because a relatively large freeboard WQuid be availo.ble. The spillw~y transition was developed to' carry 19,000 cfs with 2.5 'ft of freeboard to st.ation 13 -+- 95. This station was considered to be a 1

2

I

i

!

i

II

PRACTICAL PROBLEMS

307

to this design, the channel starts with a 140-ft-wide rectangular concrete section. The 140-ft ,base width extends from statio'n 10 -/- 00 to the sill at station 10 + 45. From station 10 + 45 to station II + 45 the walls converge on circular curves to a base width of 60 ft. From station 11 + 45 to st.ation 15 + 82.50 the walls convei'ge in a straigh t-line transition to a. base wid th of 25 ft. The channel below the transition consists of a,'rect.angula,r concrete section with a constant width of 25 ft. Compute the liow profile in, the spillway channel for' a design discharge of 6,200 ds. The control for this discharge was designed to occur at theinlet sill station 10 + 45. Use ex ,= 1 and n = 0'.014 in the computation. Solution. In t.his probiem the flow is supercritical, for the COIl trol sectib,n is placed at the upstream stat.ion 10 + '15., The critical depth RI, this section is equal to Yo ;", ,'l' (6,2007140)'/32.2 = 3.93 ft. The critical depths at other stations may also be computed. If the' computed depth happcns to be greater than the corresponding critical depth, the flow will be subcritical. The coinputation of the flow profile is given in Tnble 11-1, with the fo!lcwing ' headings: Col. 1. Station number Col. 2. Length of reach in ft, w11ich is equal to the difference between the two station llumbel's at the two ends of the reach CoL 3. 'Width of the channel inft Cols. 4 to 16. Same ns the st.eps from cols. 1 to 12 in Table 10-4, except t.hat an extra column for So (Col. 14) is provided since So is not constant throughout the whole channel length under consideration. The computation is arranged in a forin similar to that used for' the direct step method, but it is performed by trial and error. This procedure is introduced because an additional variable for the channel width is involved. In this computation,the depth of flow y is assumed and entered in col. 4 at each step. The assumed depth is considered correct when -,he resulting value of AX entm:ed in col. 16 agrees with t·be length of reach in coL 2. It. shouid be noted that the depth of flow computed in this eXD:mple has been car.ded'to more decimal places than would be necessary for practical purposes. The flow profile thus computed should be corrected for air entrainment for highvelocity flow occurring at the downstream end of the channel.! 'When a high-velocity supercritical flow occurs in a noriprismatic channel, it is likely that standing waves would appear as a result of the lateral-boundary changes (Arts: 17-3 and 17-4).

11-5. Design of Transitions. The transition in a channel is a struct.ure designed to change the shape 01' cross-sectional H.rea of the flow. Under normal desiglt and .installation conditions, practically all canals and flumes require some type of tramition structure to and from the waterways. The function of such a structure is to avoid excessive energy losses, to safe distance downstream from the dam for the release of spillway flo"/s. Do\vnstream of station 13 -/-' 95, the tr:l.!lsition and the channel were designed to carry the design flood of 6,200 cis with a minimum freeboard of 1.5 ft. The channel alignment below the transition would provide for transition spirals, and the channel invert would be banked through all curved reaches to maintain uniform depth of flow. 1 A procedure for computing the flow profile in steep chutes with a correction: for air entrainment has been proposed by L. S. Hall [4]. For a simple correction, :the Douma formula, Eq. (2-15), may be usad.

j

-I -I

-I

-I

g.j?;~~= ~'-

0,1>'"

'

SPILLWAY

.~

~

C1>

'~"

>

~I; ~~

~I;' .~.'" ~I~ ~

.

..ci PROFl~£

FIG.

\. SPILLWAY

11-9. Tentative design of the spillwa.y for a, debris basin.

,-.

TA,llLE

II-I.

COMPUTA,'l'ION OF THlil FLOW PROFILE IN A NONl'RlSMATtC CHANNta, FOR i!:XA.MPLE

(La Tuna. Canyon spillway, Lo~ Angeles County} CaliL, Q Station (1) ,

~~

I

b

)j

A

R

l/}i

V

a:V'/2g

3.72 3.36 3.29 3.36 3.45 3 . .')0 3.70 3.87

5.78 5.04

3.88

(UO

11.27 13.48 114.85 15.75 10.71 17.80 19.00 20.35 21.62 201.14

1.97 2.82 3.42 3.S5 4.33 4.92 5.60 6:44 7.26

So ., Sf i AX

,(2!_~L

10 + 45 550.0 14U.0 :l.9~ 10 + 51 {50.0 6.0 I I:~O.O 3.54 10 + 5S 417.5 7.0 120.0 3.48 10 -I- 65 7.0 1l0,0 :1.58 393.9 10+73 8.0 100.0 3.71 371.0 10.0 CJO 10 + 83 1)0.0 3.87 348.2 ~ 10 + 95 12.0 320.2 80.0 4.0S 11+10 70.0 4.35 304.5 15.0 11 + 22 12.0 . M.O 4.41 I 2~6.7 11 + 45 2:J.0 60.0 4.28 1200.8 12 + 01.5 62:5 55.0 .3 123 J71.1) 12 +70 62.5 50.0 2.855 142.8 40.0 13 10 46.8 2.817 1:1l.8 13 +'32.5 22 5 45.0 2.()58 133,1 13 + 95 62.S 40.0 3.399 135.!l 14 + 57.5 62.5 35.0 3.948 138.2 15 + 20 62.5 [ 30.0 4.050 lag 5 15 + 82.5 1 62.5 25.0 .5 05Z 141.3 1

I

I

+

I

11-1

6,200 cfs)

3.75 2.81 2.57 2.51 2.61

4.90 5.04 ,).22 5.44 5.72 6.08

5.83 3.97 3.52 3AI 3.60 2.!)1 . 4. 1G 3.22[4.75 3.55 5.42 3.89 I 6.11

6.. 36 0.90 7.43 8.04 8.n)

9.68 10.79 11.67 9.05 13.33 3(l.09 20.21 23.33 43.41 29.27 32.13 47.04 :34.38 37.20 46.61 33 .74 . 36.70 45.62 32.32 35.72 44..89 31.28 35.23 44.45 30 09 3&.34 43.87 I 2~.8B 35.53

I

0.4610.0032 0.54 0.0042 0.5:1 0.0043 .0.61 0.004.6 0.75 v.D052 0.89 0.0056 1.11 0.0060 0.88 0.00G8 1. GO 0.0088 10.00 0.0290 8.80 0.0473 5.07 0.0572 -O.GO 0.0532 -0.98 0.0442 -0.'19 0,0374 0.11 0.0322 019 0.0278

O. 0026 I O. 0800 I 0.0774 0.0764 0.003°10.0800 00758 0.0042 0.0800 o .0045 [ 0.0800 0.0755 0.0751 0.0049 0.0800 0:0054 0.0800 . 0.0.746 0.0742 0.0058 0.0800 0.0064 0.0800 0.0736 0.0722 0.0078 0.0800 0.1599 O.OUl9 0.1788 0.1407 0.0381 O. li88 0.1788 0.1266 0.0330 -0.0222 0.0487 0.O~30 -0.0157 0.0408 0.0330 -0.0078 0.0018 0.0348 0.0330 0.0300 0.03301 0.0030

5.9 7:1 7.0 8.1 10.0 11.9

15.0 12.0 23.0 62.5 62.5 40.0 22.5

62.4 62.8 61 2

G3.3

- :1 :no

GRADUALLY VARIED :FLOW

PRACTICAL PROBLEMS

eliminate cross wn.ves and other turbulence, and to provide safety for the structure and waterway. "Yhen the transition is designed to keep streamlines smooth lind nearly 'parallel and to minimize standing waves, the theory of gradually varied fiow may be used in the design. The essence of such a design has been di~cu8sed earlier (Arts. 3-5 and 3-8) in connec-

in terms of the change in velocity head between the entrance and exit seetions of the structure. For inle t; structures, the entrance velocity is less than the exit velocity; hence, the water surface must always drop at least a full difference

tion with the applicf.ttioll of the ellergy and momentum principles.

inlet

311

betvveen the velocity heads, plus a small oonversion loss known as the

In

'~his

mticle, emphasis is on design practice. The form o{ transition may vary from straight-line headwalls normal to the flow of water to very elaborate s!:,reamlin\3d warped structures. Straight-line headwalls are usually found Ratisfactory for small structures or where head is not valuable. For. the sake of economy, the U.S. Department of Agriculture [5] has tested a cylinder-quadrant trnllsition as a substitute for the expensive warped structure. The cylinderquadrant transition is essellti~Jly a pair of circular wings or vertical walls, tailgent to the flume sides n.nd curving through 'a qual'ter~urn to mClet the . sidei'! of the canal. For the same 1'e<1.30n, the trend. of practice in the U.S. Bureau of Reclamation [5J has been toward simplification, with the top edges of walls and the intersections between wa.rped walls and floors , designed in straight lines. Further simplification of an ela.borate form is permitted, however, only when a close control of water areas and velocities is unnecessary and when excessh:e wa. ve action or turbulence ~ould not be developed. The comnion types of transition are inlet and outlet transitions between canal and flume, inlet and outlet transitions between ca.nal and tunnel, and inlet und outlet transitions between canal and inverted siphon. . It should be noted tho,t appreciable change in depth of flow generally occnrs in all t,ypes al tr'<)'llsition. If the change in depth or width is very rapid the flow may become rapidly varied, and standing waves may occur. 11-6. Transitions betwee.n Canal and Flume or Tunnel. On the basis of the performance of existing structures, the 'following features have beim found important in design. For more complete information, the reader should refer to [2], [6], a.nd [7J. A. Proportioning. ·For a. well-designed transition the following rules for proportioning should be considered: . 1. The optimum maximum angle between the channel axis and 11 line connecting the channel sides between entrance and exit sections is 12.5". 2. Sharp· angles either ill the \vater surface or in the structure that will induce extreme standing Waves and turbulence Should be avoidcd. B. LO/Mes. The energy loss in a transition consists of the friction loss and the conversion loss. The friction loss may be estimated by means of any uniform-flow formula, such as the Manning formula. This loss usually has very little effect on the transition flow profile and may be ignored in preliminary design. The conversion loss is genemlly expressed

lOllS.

The drop Lly' in water surface for inlet structures may there-

.

fore be expl'essed as (11-6) wher~ £'h.

is t.he difference in velocity head and c, is a coefI1cietlt of inlet

loss. For outlet struGtures, the velocity is reduced, at lea.st in part,in order to lift the water surface. This rise in water surface, known as the recovery of velocity head, is usnaily s.ccompanied by a conversion loss known as the ouae/loss. The rise t:.y' in·water surfnce for outlet structures may. be I?xpressed as (1l-7)

'oNhere co is a coefficient of outlet loss. The average safe design values of Ci and follows:

Co

that are recommended are as

Type of Transition Wal'ped typ'e, ......... . Cylinder-quadrant type ........ _.. . Simplified straight-liM type ....... . Straight. line type .. _.............. . Square-ended type. . .... _....... .

, i

J

'"

0.10 0.15 0.20 0,30 0.30+

. 0.20 0.25 0.30 0.50 0.75

C. FI·€f;boQ.'l·d. ApproXImate rules for freeboard estimation for lined arId unlined canaLs (Art. 7-5) may be used. For depth of How over 12 ft, the freeboard in tl':J.l1sitiol1 should be given special consideration. Figures 11-10 and 11-11 show, re.'5pectively, typical designs for an inlet transition from canal to flume and for an outlet transition from flume to canaL The design of the inlet transition \viJ( be described in EX2,mple 11-2. The design of the outlet transition may follow the same general steps. However, the expanding flow in the outlet transition often presents special hydra.uIio behavior that should not be overlooked. In all expanding flow, the distribution of Yt;locity in the cross section can be extremely uneven; so the velocity-distribution coefficients ma.y become appreciably gren.ter than 1.0, and their values should be calculnt.ed or properly assumed in the design. Furthermore, uneven distribution of . ,'elocity may cause asymmetry of flow and thus develop SCOllr at places of highly concentrated velocities. .Caution in this respect, therefore, should be taken when the transition is designed in erodible channels.

--1 ....... !

-I I

(

,r,

312

313

PRACTICAL l'ROBL:fj)M!;

f

GRADUALJ;.Y VARUJD FLOW

!

c

'~ 51.40~~~~-~--~~~-.~-.,~~~~--~f~~~~~-,~_Je

"

'" 51.30 '"

Q

:;; 57.2 0 ~-f.---+-7"'-I~'

'"

."

I

o

I

o

:- -,

:+o o

+

'"

+

+

+

on

~

+

" +

r'1

WATER SURFACE PROFILE

0

+

ll'l

"

.n

+,

T

+

0oJ

'" oJ

+

+

o

cO

o

'"

1'1

'"+

+

+

WATER SUR.FACE PROFILE

HYDRAULIC PROPERTIES Hume Canol

-L-J" '"

U"l

C

5,0.0009 ,,'°,01 4 $'0,00039<'- "'0.018

+

"'6.0

0.315

V'3,24

Ci'315,

HYDRAULIC PROPE RTIES FJuma

5'0.0009 ,,-0.014 V'5.98 Q'314.5

PLAN

PLAN

I-

\ "

CD

~~~~~~~~~~~~ iii< N '"

SECllON ON CENTER

W

line

Top 01 bonk

SECTION ON CENTER LINE

SECTION D-O

FlO. ll-lL'Typica! design of au outlet transition.

,Rip-fop if r~~ui,ed

(Alter J. Hinds [2].)

)

(After J. Hil14s [2].)

315

GRADUALLY VARIED FL01V

PRACTICAL PROBLEiHS

D. Elimination of H ya7'auUc Jump. Existence of hydraulic jump in a transition may beeomeobjectionable if it hindel's the flow !\.nd consumes useful energy. When the transition leads [rom 'a supercritical flow to a sub critical flow, the, hydraulic jump may be avoided by carefully propor-, tioning the transition dimensions (Example 3-4). In an example illust.n~ted by Hinds [7], a transition- from a segmental canal to a circular tunnel wus desigrled for flow from one subcritical flow stage to anothej· (Fig. 11-12). In the libel'!11 altowance \v'as made

. head from l' - 2.715 fpl! to 5.98 fps is equal to tlk. = 0.553 - 0,117 = 0.436 ft. Neglecting the chnpllel frietjoll fol' the time being, the total drop in water surface is, thet'Afol'e, LlO X 0,436 0 ..480 ft. . For a. smooth Itnd continuous .flpw, the theoretical flow profile'may be u.sswlled 11.[ , two equal pll.rn.bolas, !.IJ.ngent to ea.ch other o.t point B ,\nd hori~ollt,al, respectively. a,!; A and C, Strictly spellking, the parabolas should be tangent to the water surfaces' in the canal and the flume, but !l. small deviation of these water surfaces from the horiwnt.at is not important in ~he present example. A number of sections are then aele(;t~d a!.:mg the transition, where the /lOll' prolile will be computed and the stnlctund dimensions determined. 3~ Cc.mplilatio'll of the Flow Pmfilc Inch,diug Friction. The computation ]" shown in Table J 1-2 with the following heo.dings: Cd. 1. Number for stD-tioll1; eqtHI.l!y spaced every 5 ft aud lneiLsured in the direct.ion of flow, ' Col. 2. Drop in water surface. The tolal drop in \I'D-tel' surface from A to C, neglecting friction, is OA80 ft. The drop from it to t.hemid-point B of the antisymmckicill reversed p~\,l'abola is made equal to half th-e drop, or 0.240 ft. CoL 3. Change in velocity heo.d. Assuming thnt the conversion loss i~ distributed over th" entire l~ngth of the transitiQl). in proportion'{'o the change in velocit,y head, vo.l:Jes of tll~, are obtaine.cr by dividing value.<; of t!.lf' in col. 1 by 1.10. Col. 4. Tot.[11 velocJ;,y heD-el, equal to the cumulative v?Jue of t!.h, ent~ring the preceding coiumn . , , ' Col. 5. Veloci(" in ips corresponding to .the velocity head in the preceding column Col. 6. Water area in ft~, equa.l to th~ d~"charg~ 314.5 cfs divided by tbe velocity in the column CoL 7. Half the' top width in ft, obtained from the cross sections or the sketched plan 11-10). The piau m~}' be chos@ either ~rbitmrily or by trial untilsatisfactory reslll!;s !l.re obtained., The choice of a proper shape for the plan is a matttlr of judgment .. Col. 8. Half the bottom width in It, obtained f"om the, ~ketched plan tV.)!. 9. The dt'pth of flow ill it, equal to A/CO.aT + 0.5/;) Col. 10. Hydraulic nlrliu& in ft CoL 11. Til'.! friction slope, computed by Eq. (9-8) with n = 0.014 for all sections the tmnsition If Col. 12. The friction head, equal to 'the dist.a.nce between stationli, or 5 ft, multipliedby the avernge of the friction slope of the section and that of the preceding section Col. 13. Cumulative frir:t.ion head in ft Col. J 4. The water-surface elevation, including the Ilffect of the ch~,l1ncl friction, - 'Zhf. The flow profile thus obtained should be free from equal to 57.41 objectiona.ble irrllgularitics;' if not, the plan ma.y ')B t\:tered. It shOuld bG lloted, hQwever, that, a slight change in th
314

~Qer9Y

energy

line

rne

energy

line

d water .st.,foce

Tunnel

FIG. 11-12. A faulty transition design.

(Af!,r J. Hind¥ [7].)

for tramitions and friction losees, and a safe coefficient of roughness was used to determine the depth of flow in the tunnel. After construction, however, the transition losses were found to be practically negligible; so the actual normal depth of the floY! the tunnel entrance was cOl1sirler::.bly less than the assumed vahle. As a result, an objectionable hydraulic jump was observed inside the tunnel. Actually, the transition' should have been proportioned to avoid the jump on the basis of negligible t.ransition losses. Since the structure had already been constructed, the hydraulic jump was iinally eliminated by bolting cross timbers to the channel bottom, thus increasing the friction and bringing the normal depth up to above the critical depth. Example 11-2.' It is required to design an inle:t st,ructure connecting an earth ca.na.l huving 8, bottom width of 18 it and side slopes of 2: 1, to It rectangular concret'l flume 12 it {} in. wide, The hydrauli9 propert.ies of the cana.l and the flume arc given (Fig. 11-10). The design discharge is 314.5 cis. Solulion. The design procedure~ involves thll following steps: ' L Delermimr!ian of the LmgU; of the Tra:ll..'l1:tion. The length of the transition is detllrmined so that B, straight line joining the. flow line !It the two ends of the transition will ma.ke an angle of a.bout 12.5" with the 'axis of the strnetlll·e. This length in the design is found to be 50 1 . 2. DeterminatiDn. of fhe Flow Profile Neglecting Frict.icn. For the type of stl'uctul'e under contemplation, the inlet loss be safely assumed to be 10% of the change in velocity head, or 0.1 ail-". ,The total in water surface is, thel'efOl'e, equul to L1 ah. pillS the drop necessa~y to overcome channel fridiOn. The change in velocity

ft:

* Thi.~ example is taken :from [2J.

in

-I I I

I

PRAC'l.'ICAL PROBLEMS

317

CoL 18. Height of lining; in It, equa.l to ZL -Zo Col. 19. Computed value of half the width atthc top of the lining, equal to O.5W ZHL

+ 0.50

Col. 20.

OI.l?CIt:lO\,/')OI.t)0U10 OO __ (\l<:,-~c-;tf;l"';l'l""1t:I

+++++++++++ 00000000'000

316

.

O.5W to nearest 0.5 in.

11.:.7. Transitions between Canal and Inverted Siphon. Figure 11-13 shows a typical design of siphon inlet and outlet tra)1sitions between canals !1Ild v.n inverted siphon. The method of deaign is similar to that for the transitions between clmal and flume. Howe vel', the following special features of design are recommended by the U.S. Bureall of Reclamation [6]: 1. In the design of an inlet transition, it is generally desirable to hll.ve the top of the 8iphon opening set slight.ly below the ,cpprQacliing normal water surface. This practice will minimize possible reduction in siphon capacit,y caused llyl;he introduction of ail' into the siphon. The depth of stlbmerge~ce of the top of the siphon opening is ImolVn 11.8 the walel' seal. The recommended value ~f the water seal is between a minimum of L111hv and a maximum of 18 in. or 1.5 6h., whichever is greater. It should be noted that use of the minimum value in a well-designed transition theoret.ically allows the flow barely to touch the top of the siphon opening; whereas use of lal'ger values up to the maximum provides :\ senl of water above the top of the opening. An u.deqllate nmount of seal depends upon the slope ilnd size of the siphon barr~l. Generally, a large and steep barrel requires l1 large seal. In the design illustrMed (Fig, 11-13) It seal of 18 in. is u3ed. It shotlld also lie noted that the seal IT\ay make it impract.icable to construct. the lower end of the tmnsition strictly in aCGo!'chnce wi th the hydraulic computation. ,"Yhen this is the case, the computed bottom elevation a short distance upstream from the inLet headwall nuty be altered arbitrarily to meet the practical requirement. In the design illustr"ted, computations ha'.:e beeI~ carried to the end of the transition; beyond tl;lis the conduit flom is simply extended smoothly to connect with the floor of the siphon ba,rreL For long siphOlls, ul).der certahl conditions, the inlet may not necessarily be sealed. Consequently, 11 hydraulic jump may occur in the siphon barrel, and the resulting operating condition will be un'favorable. 2. After the seal is determined for the inlet structure, the velocity at the headwall is computed, and the totai drop in water surfo;ce, neglecting friction losses, is taken as 1.1 t.h.: A smooth flow profile is thEm assumed, . tangent to the water sUl'fac~ in the canal at the beginning of the transit.ion and passing through the point at the headwall set by the above computation. There no data ll.vailable for determining the best form of the flow profile. In the illustrated design, a slmpJ:e parabola is assumed.

are

PRACTICAL PROBLEMS

0.'

~

3. In the design of the crutlet structure, the theOl
t

... h~

c,

'"(; J

t;:;

;:' ::> 0

a.

0

,-,

Z 0

:.:: 9!. c..

U; W

I-

UI

tr

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"

:§

r.I 0Z 'I:

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65 i=5: +0

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"0

c

..,m

,2;)

e

I

318

319

1

Sec. ISland [9) for further discussion.

- I

{

--I

-\

"\

320

GRADUALLY VARIED FLOW

11-9. Flow Passing Islands. When flow in u stream is divided by a long island (Fig. 11-14a), the division of flow between the two channels may be determined \Viti). the aid of f1o-w~profi)ecomputations. In the case illustrated it is considered that the flmv throughout ·all channels is subcritic!Ll. The pl'Ocedure is to assume nrsta set of discharges Ql and Q2 for the divided flows such that the Sum ~f the discharges is equal to total discharge Q.. Then, compute flow profiles in the two ch:mnels each side of the i.;;land to u point A where the flow is divided. Since the flow is subcritical, the computation should

(a)

, ,,

,,

,

I·

i

Wo!er-surface elevation at

poinl A from compuletior. tor channel Na.2

FIG. ll-l·t Solution

Discharge in chonnel No.1 , Q

i

id

(bj fOI'

321

PROELE:&!S

flow passing an island.

proceed upstream from the d~wnst,eam point B Where the divided flows unite again. The initinl water-surface elevation at puint B may be determined from the curve at this statiori for a total discharge Q. The computed water-snrh1ce elevation at point A tor channel 1 is then plotted against the sa,me for channel 2. A ct,U've may, therefore, be dl'a\vn (Fig. 11-14b) fo1' several assumed sets of :dischal'ges Ql and Q2 of different proportions. :Since the flow is divided ~t point A, the two computed water-surface elevations at this point fOl~ the channels should be \;!qual if the assumed qivisiol1 of flow is correct.: Thus, the elevation at point A corresponding to this correct division of flow may be obtained

from the plotted c.urve at which the curve is Intersected by a.dashed line bisecting the coordinate axes. The dashed line represents the condition that the two computed elevations are equaL In the meantime) the computed elevation at point Afar chfml1el 1 is plotted against the discharge Q1 (Fig. l1-l4e). From this curve the correct discharge Ql may be obtained for the correct elevation. The corresponding discharge in channel 2 is, therefore, equal to Q2 Q Q1. If the dividfld flows are supel'critie~\l, the control point will be at A; hence, division of flow will depend on the entranee condition of the divided channels. During the normllJ-flow eonditioll, it may be assumed that nil flows ~\re uniform, and division of flow may be roughly determined from the following I'elations: Q1 1(1 vs;., Q2 = K2 -VS;, and Q = Q1 + Q2. 11-10. River Confluence. SVhen the flow-profile computation is carried upstream through the confluence of r.· river and its bributary, it is neeessary to determiQ.e the water-surface elevations immedIately uostream hom the confluence. The procedure for solving this problem is' illustrated in 10-10 fOl'the confluence of the Missouri and Kans&s Rivers. In this example, a dischi.rgeof 81,000 cfs from the Kansas River is combined with 350,000 cfs from the Missouri River to give'a total discharge of 431,000 cfs immediately below the confluence. Cross sections lK and {) are located immediatelyupstl'eam fromthe confluence of the two streams (Fig. 1O-1l). In Tr.ble 10-8, the hydraulic elements, ;'elocity heads, ancl total heads are computed separately at the two sections. The velocit)~ head at cross section 5, is, compnted for 11 total discharge of 431,000 cfs. This value is divided between the cross sections 6 and lK corresponding to th9 dischal'ges of 350,000 cfs f.l1d 81,000 cis, respectively. The friction slope at each cross section is computed fol' the discharge of 81,000 cfs at cross section IK and fOl' 350,000 ers at cross section ().The friction head loss hJ is then computed, using thA average of the friction from cross sections 5 to lK in the Kansas River and from cross sections 5 to 6 in the Missouri River. At the confluence, eddy loss is usually high, In Example 10-10, this is estimated as 10% of the increase in velocity head from cros~ section II{ to 5, or 0.10 ({l.G3 0.13) 0.05 ft. The total energy in cross section .5 is, therefore, equal tohJ h~ + H = 0.10 0.05 755.0,1: = 755.HJ. Subtracting the velocity head' from this value gives 755.19 - 0.13 = 755:06, which should be equal to the assumed watel'-sul'faceelevation ab cross section lIe. , Th~ above method should bli:flpplied to sub critical flow of relatively low velocities l1Qt exceeding about 10 fps. At high velo'cities, the eddy 10sses'al'e high, and the error il1~olved in the estimation may become quite appreciable. The problem of riYer confluence may be further illustrated by an exam~

+

+

+

322

,323

GRADUALLY VARlED FLOW

PRACTICAL PRDBLE.IS

pIe given by Sto~;:m' [10], using the Junction of the Ohio and Mississippi Rivel's (Fig. 11-15). The initial conditions of the uniform flows in the Upper l\::Ii.~ssippi, the 'Ohio, and the Lower Mississippi Rivers are, respectively, as follows: the normal depths, Ynr, Yn2, and tina; the channel slo'pes, SOl, S02, and Soa; and the roughness coefficients, nI, n2, and n" Fwm these da,ta the normal Qnl, Qn2. and Q,,3 call be computed. Now, ,a flood wave is assumed to initiate in the Ohio River at a place L

flow condition is expected to develop; at thifi time the depth at the pla.ce of flood initiation becomes Y2, but ~he depth far upstream in the MissisRiver remains Ynl. As a general rule, the backwater effect in a long sr,ream resulting from even fairly large discharges of its tribntaries aoes vel"y far ,upstream. Therefore, the backwater curve in the not Upper Mississippi Riyer should be l'elatively short. At the final steadyflow condition, steady b:l(',kwater curves will be formed in the Ohio and Upper Mississippi Rivers. Downstream in the Mississippi River, the depth will change from Y1l3. to Ya, but the flow may beasslimed to remain uniform. ThE:; flow involved ill the [I.bove-mentioned problem is considered to be subcritical, which is the usual COl.'le and can be verified easily by a camput,~tiol1 of the Froude numbers. At the final steady-flow condition, the following conditions are evident: (1) the discha.rge in the Upper Mississippi River remains the same, or Ql = Qnl; (2) the depths of the three channels at the junction n.re all equal to Yi at x = 0; and (3) the Slim of the from the Ohio and Upper Mississippi Rivers is equal to the discharge in the Lower Mississippi River, or QI + Q2 Q3. By assuming a value of Vi> Q3 may be comp:lted un1t.hell

initiation of flood

Upper Mississippi River

Lower Mississippi River

I .. I

Point of

floDd

Since YiJ Q~, and Y2 ~are ilow known,. the leng!;h L 01 the backwater curve' can be computed. If the computed L agrees with the given L, the assumed Yi is the correct value. Otherwise, new vulues of, Yi should, be assumed unt.il a correct valLle is obtained. This is a trial-and-error solution:. Ii solution may 0,180 be obtnin<3d:byassurning Q2 at the beginning l1l1d ~heclr.ing finally for either Vi or L. PROBLEMS

FIG. 11-15. A junctiDn problem of the Ohio a.nd Mississippi Rivers,

miles upstl;eam from the junction and to such that' the Ohio River rises rapidly at that point from the initialnormal depth Y,,'J. to a final maximum depth Y2. The point of flood initiation is shown at:l; -L on the coordinate axes, where:r; is the dista~1ce .measured from the junction along the channel. The wave of the flood thus originated will move down the Ohio River to the junction and CI:eate new waves, which travel both upstream and downstream in the JVlississippi River, and also a reflected wave which travels back up the Ohio. After n certain time, a eteady-

11-1. A rectangular channelS ft w'ide and 500 'ft long connects two reservoirs of va,rylng surface levels (Fig. 11-16). Assuming 3. frictionless channel, construct the Q-constant curves of Yl J(y~) for Q = 10, 50, 100, 150, a.nd 200 cfs, The loss at t,he channel ent.rance is negligible,

~T y.

1 FIG. 11-16. Profile of

~T Ya

11 It CMlld

for Prof,.. 11-1.

')\

324

II,

GRADUALLY VARIED FLO,"

.yzg

d. Slope of the neutl':J.lizing reach e. Length of the neutralizing reach f. Length of the !'iteel' reach 11-14. COl'rect the flow profile computed in Example 11-1 for air entrainment, Eq. (2-15). 11-15. Solve EX:ll.mple 11-1 for Q = 19,000 cfs. 11-16. A trapezoidal earth spillway with 3: 1 side slopes is 190 ft long from the reservoir at Sta.. 0 + 00 to the downatream critical control ·seetiol'!. at Sta. 1 + 90. Th~ bottom of the'spillway at Sta. 0 + 00 is 95 It wide and conV'erges unifol'luly t.~ a width of 75 ft c.t Sta. 1 + 00. The bottom width of 75 ft ill constant from Sta. 1 + GO to Sta. 1 + 90. The bottom slope of the spillw!l.Y between Sta. 0 00 and Sta. 1 + 00 is ad verse, and is - 3 % between Stu. 0 + 00 and Sta. 0 + 8!J ane! - 20 % between Sta. 0 + 80 and St.... 1 + 00. The bottom of the spillway is horillontal . between Sta. 1 + 00 and Sta. 1 + 90. Manning's n = 0.03.5. Determine:

+

n. The flow p::-ofiles for discharges of 600, 800, 1,000, 1,200, 1,500, and 1,800 crs in the spillway , /), Curve showing dis(.\harge \'5. elevation of pool level in the res,ervoir c. Friction loss in spillway between the reservoir and the control sectioll ror I.l. discharge of 1,500 cfs . 11-1"{. Revle\v the design of nil O\lt.let tl'll.l)sition from Ilume to canal (Fig. ll-ll). The flume r..nd canal are the same as those described in Exnmple 11-2. 11-18. Review the design of ihe siphon inlet and outlet transitions (Fig. 11-13). GiYen th.e hydraulic properties: A = 01.52 it', R = 2.89 ft., SQ = O.OOOS, nnd n = 0.0225 fa! the canals; and A = 38.48 ft2, R = 1.75 ft, So = 0.0012, and 11 "" 0.014 for the circular siphon ba:reL 11-19. Compute and construct li bnckwater envelope curve for the backwater' caused by a 5-ft dam in the channel described in Example 10-1,

n. Assuming the end point at a depth equal to 1.01Yn ~.t a depth equal to Yn plus a.

b. Assuming the end poiilt

...,.,,---l(.-8'

? ~----'-:-~ FIG. ll-17. A raft chute for PI·C'b. 11.13. Determine the following items by the Manning formula and by any method of flowprolile'computation as described in this book: a, Slope of the steep reach

b. The entrance depth y. c, Length of the flow profile in the trnnsition teach

325

PRACTICAL PROBLEMS

11-2. Construct the Q-constuut curves of YA = f(vB) for Prob. 11-1, The entrance discharge ,nt\y he eomputed by Q = 3bf]1.l where b is the channel width a.nd H is the tot.a.l hettd. 11-3. A tmpe;;oidal channel connecting two reservoirs 2 miles Ilpart has b = 50 ft, z = 2, fr = 1~ n = 0.025, and So =; 0.004. The upsireamdepth V, is mninto-ined consLant !).;ld equal to 6 ft (Fig. 11-1). Construct the d'olivery curve Q = f(y.), 11-4. COllstruct the delivery curve Q = f(ytl for Prob. 11-3 if the downstream depth:v~ i.; lTIaint(Lined constant and equal to 6 ft (Fig. 11-3). 11-0. Assuming a variable y, in Prob: 11-3, construct !l. curve of Y, against Q", .. for 0. range of diEcharge varyiag from 0 to 2,000 cfs. il-6. Assuming v1l.riable depths !/t and y, in Prob. 11-3, construct the Q-constant curves for·dischal·ges having tile normal depth equp.l to 2, 4, G, and 8 ft, respectively. 11-7. Sulve Prob. 11-3 if tht: reservoirs l\re 2,500 it ap:.i,rt. 11-8. Solve Prob. lJ-4 if the reservoirs Me 2,500 ft c.part. 11-9. SolvePI·ob. 11-5 if the reservoirs !lre 2,500 ftapart. 11-10. Solve .Frob. 11-6 if the reservoirs are 2,500 it llpRI't, 11-11. Construct the delivery curn Q = f(YB) for Prob. 11-3 if the upsLream reservoil' depth Y II ia Dlaintained con~tant and equal tc B ft. The fzee ·entranoe is wtlllrounded, or C, - 1.25. 11-12. Construct tbe delivery curve Q = f(Y,J) for ·Prob. 11-4 if the downstream resp.rvc-ir depth YB is maintained constant and equal to 6 ft. C, =1.25. 11-13. A rectangu~ar ra.ft. chute 20 ft widc, as descril::ed by Bllkhmeteff [lj, is to be built between levels A. and B(Fig. 11-17). The design conditions ,,,'e: L The'ta.ilwatm· fluctuates by 8 ft. 2. The depth ill Lhe steep reach is kept to a minimum nav'igl1ble depth ~f 2.5 ft. :l. The (lvCl'nge velocity should not exceed 14.7 £p8. 4. The dischal'ge should be kept below 750 ers, 5. The roughness coefficient n = 0,03. 6. The entrance discho.rge is computed· by L\ weir formula Q _ 0.4 b·y", l.~.

frcl~board

of 5 in.

11-20. A dil!charge of 1,000 cfs is divided between two rectangular channels exoavated in rock. The chnnneis are later joined !J,gain (Fig. 11-14). Channel 1 is 10 ft wide and 200 ft long. Channel 2 is 15 ft wide lend 150 ft long. The·bottom of channel2 is on the llverage 'about ::l ft lower than that of cho-llnel 1. Assuming n = 0.035 aml a tot'll drop of water surfa.ce between the dividing !lnd joinip.g poillts of the chan7 nels equal to 5 in., compute the divided uniform flows .. 11-21. Wi~h reference to the problem shown in Fig. 11-15, the following data are assumed: Yr.' = Yn~ = 1/ .. 3 = 20 ft, SOL = So. = 0.5 ft/mile, S03 = 0.49 ft/mite, n, = n •. = no o.oa, the widths of the Ohio and Upper Mississippi Rivers = 1,000 ft, the width of the I,ower l\{ississippi Riv~r = 2,000 fti II> = 40 ft, and L = 50 miles. All rivers are assumed to have rectallgular channels. Determine the junction deptl\ and the flow profiles ill the rivers after the flood flow approaches a .stead); condition.

REFERENCES 1, Boris A. Bakhliletefi: "Hydraulics of Open Channels," McGraw-Hill Book

Compo.ny, Inc., New York, 1932, pp. 143-215.

326

GHADUALLY VARIED FLOW

2. Julhtrl Hinds: The hydraulic design of flume and siphon ~ransitions, TrcT.nsadions, America" Society of 'Civil Engineers, vol. 92, pp. 1423-1459, .1928. ' '3. "C;vil Works: Flood Con~rol in the Los Angeles Area," TIl[) Engineer School, For~ Beh'oir, Virginia, 1950, E206.00 (4-50) ML, pp. 22-28 and plate 10. 4. L. S~alldi6h Hall: Open channel flow at high velocities, in Ent.rainment of air ill flowing \\'flter: a symposium, Transactions, Ameriwn Sodetv oj C;~il Engillec",3, vol. lOS, pp. 1394-1447, 1943. 5. Fred C. Scobey: The flow of water in flumes, U.S. DepC!1'lmenl oj Ag"ic7l!t1l7'e, Techniml B1t/lelin No. 303, December, 1933. G. Hrdraulic design d!\1;a, appendix I of CanalS and related structures, U.S. Bure(1.(L oj Reclamalion, De~ign and Cons/rllction .liJ(l.n1Lal, Design Su.pplwtent No.3, 1952, vol. X, pt,. 2, paragro.ph I-13. ' 7. Julian Hinds: The hydraulic jump and erit,ical depth iu the design of hydraulic S~r\)ctllres, Engineering Ncws-RsClJrd, vol. 85, no. 22, pp. 1034-1040, Nov. 25, 1920. 8. WallD.ce M. La.nsford and WilliD.1n D. Mitchell: An investigD.tion of t.he backwater pro(ile for steady flow in pi'isnl:t~ic channels, University oj Illinois, En!1ineering EXpcTl:lI1cnl Stalion, Bulletin Series No. 381, vol. 46, no. 51, lViD.1'ch, 1949. 1.1. Willi"'ffi D. Mitchell: Sf,age-fall-clisoharge reiatioils for steady flow in prisma~i~ ehfl.Tln<:.ls, U.S. Geolo!!ica.18urvey, Wa.ler Su.pply Paper 1164,1954. 10. J. J. St.oker: "Witter Waves," vol. IV of "Pure and Applied Mathemat.ics," Interseience Publishers, Inc., New York, 1957, pp. 45G-461.

CHAPTEIt

12

SPATIALLY VARIED FLOW

i

i

12-1. Basic Principles and Assumptions. Spatially (gradually) varied flow, ~'s previously defined (Art. 1-2), has a nonuniform discharge resulting from the addition or diminution of water along tbe eourse of flow, The added or diminished water will cause disturbance in the energy or moment.um content of the flow., As ftres'ult, the hydraulic behavior of a spatially va,rlcd flow is more complicated than that of a flow of constant discharge. Furthermore, the hydraulic behavior of spatially varied flow, with increasing dischai'ge is different in certain respects from that of s'imilar flow with decreasing discharge, Therefore, the two types of spatially varied flow will be discassed separately. A. Flow with Increas~'ng Discharge. In this type of .spatially varied flow, an appreciable portion of the energy loss is due to the turbulent mixing of the added water and the water flowing in the channel. In most caEes, this mixing is of relatively.high magnitude and uncertainty. Because of the resulting high and uncertain losses, the momentum equation will be found more convenient than the energy equation in solving this problem. From a practical vie~vpoint, the high energy loss seems to mz.ke channels designed for such Bpn,til1lly variedflbw hydraulically inefficient, but physicl1l circumstances sometimes make the lise of lltlch structures desirable. A substantially correct form of the fundamental differential equation for spatially varied fiowwith increasing discha.rge was probably first established by Hinds [1] for the design of lateral spillway channels. A more complete equ:Ltlon, ho,vever, was developed by ,Favre [2,3], including a friction term and a componentof.lriflow velocity in the direction of the I1xis of the channel. The methodG developed by Hinds and Favre are applicable to any channel, prismatic and nonprismatic, but the procedure requires 8, step computation with successive approximations. For prismatic rectangular channels with ulliform 'illflow throughout the channel length, the differential equation of the flow has been \ntegm,ted by Camp [4] and Li [5J. Li also treated prismat.ic channels of sloping walls. Theoretical and experimental studies of the ,flow were also made by De Marchi [6], Citril~i [7], Forchheimcir [8], Schoklit.'3ch [9], and others. In practical applications, the theory has cOi'ered a variety of problems, 327

.j

,

-I

I

!

,

'. 328

329

GRaDUALLY .vAniED FLOW

SPATIALLY VARIED FLOW

from the study of flow in roof gutters [lOJ to the design of wash-water troughs in water-treatment plf1nts [11,12] and of side-channe1 spillways on dams (Fig . .12-1). B. Flow with Decreasing Discharge.. Fundamentally, type of spatially varied flow may be treated as a flow diversion where the diverted. watel; does not Ilffect the energy head. This concept has been

problem' anll-Iytically by assuming the energy line to be parallel to the spillway crest and to the channel bottom and also by assuming the flow profile illong thl'l spillway crest to be linear. Theoretica.lly, De l\farchi [16,17-J proved that the energy head alt.ng the spillway crest is essentially constant and that the flow profile is curved, rising in sllbcl:itical flow and dropping in supercritical flow. This theoretical investigation was fur~her vedfied experimentally by Gentilini [18]. Theoretical and practical studies of the floYI were also performed and advanced by Favre [2,19], Nimmo [20], Noseda [21-23], Schmidt [24-26], Mostkow [27,281, Ackers [29], Allen [30J, Collinge [31], Frazer [321, and many others, In the derivation of the spa,tially-varit;d-flow equation in the next ::uticle, the followin,g assumptions will be made: 1. The flow is unidirectiona1.. Actually, there are cross currents present in the form of spiral flow, particularly in l::tteral spill way channels, The effects of these cm'rents and of the accompanying turhulence cannot be easily evaluated, but will be included in computations if the momentum 'pl'illCiple is used. The lateral unevenness of the water surface, as a result of cross currerlts, can be ignored. 2. The velocity distribution across -the channel section is constl:U1t and uniform; that is, the velocity distribution coefficients are taken fiB unity. Ho\\'ever, proper values of the coefficient£! may be introduced, if necessary. 3. The pressure in the flow is hydrostfl.tic; that is, the flow is parnlleL The flO\y at the o,utlet, however, may be curvilinear and deviate greatly from the parallel~fio"T assumption if a hydraul;c drop occurs. In such. cases, proper values of the pressure-distribution coefficients may be introduced, if necp,.<;sary .. 4. The slope of the channel is relatively small; flO its effects on the pressure head and on the force on channel sections are negligible. . If the slope is appreciable, corrections for these e~ects may be applied. 5. The Manning formula is used to evaluate the friction loss due to the shear developed along the channel wall. 6. The effect of air entrainment is neglected. A correctioll, however, may be applied to the computed result when necessary. 12-2. Dynamic Equation for Spatially Varied Flow. The discussion is given sepo.rately for flow with increasing discharge and flow with decreasing discharge. .. A. Flow with Increasing Discharge. Refening to the lateral spillway channel in i2-2, the momentum passing section 1 p~r unit time is

)

\

1

i

FIG. 12-1. The side-channel spiilway (·n the Arizona side of Hoover upstream. (U.S. BUTeau of Reclamation.)

verified by both theory and experiment,. Therefore, the use of the energy equation will be found more convenient in solving this problem. The theory of spacially varied flow with decreasing was probably employed first in the design of lateral spillways or side-spillway wehs. This type of structure is usually a long notch installed along the side of a cha.nnel for the purpose of diverting or spilling excess flow. Laboratory tests on such structures w~re first made by Engels tI3] and by Coleman :and Smith [14J.I Forcl~heimer [15J has approac.\1ed th~ 1 Engels's ex~eriments indi~f.lted a rising and those by Coiemnn and Smith, on the contra.ry, a dropping flow profile along the spUlwa.y crest. This confusion \vas lEIter expla.ined by De :!vI.arahi [16] as resulting from the f8.Ct thu.t flow was sllbcritical iil Engels's but supercritical in Coleman and Smith's experiments.

i

1»

g

QV

where w is the unit weight of water, Qis the discha.rge, and V is the veloc- ,

330 ity.

Similarly, the momentum "' ..,""',' ... section 2 per unit time is

~{] (Q

+ dQ)(V + dV)

where dQ is the added discharge bet\veen sections 1 and 2. The momentum change of the body of water between sect.iolls 1 and 2 is, therefore, equal to '

~ (Q (J

+ dQ)(V + dV)

-

:!£ QV f!

~o [Q elV g

+ (V + dV) dQJ ,

Let W be th€ weight of the body of water between the sections. component of W in the direction of flow is

W sin

u;So(A

f) = ::

+ J1: dA) dx

=

331

SPATIALLY VARIED FLOW

GRADU.'\LLY VARIED FLOW

, The total pressure on section 1 in the direction of flow is ,equal to the ullit hydrostatic pressure at the centroid of the water area A multiplied by the area, which is equivalent to the moment of A about the free surface multiplied by W, or = wiA where z is the depth of the centroid of A below the surface of Row., Similarly, the total pressure on section 2 is

P2

w(z

+ dy).t1 + ~ dA dy

The

w8 QA d,o;

where dy is the difference between the depths of the two sections 1 and 2. Neglecting the term containing diffel',mtials of higher order,

w(z +dy)A

-~.

where slope So is equal to sin IJ and th.., term containing the product of is dropped.

The resultant hydrostatic pressure acting on the body of water between sectiolls 1 and 2" is

p. - P z

=

-

w A dy

Equating the momentllln change of the water body to' all t;he external forces acting on the body, w

' (V g '

- [Q dV

+

+ dY) dQ]

" =

P1

-

P2 .

+W

sin

f)

FJ

(12-1)

Neglecting dV dQ and substituting in thc above equation all expressions for external forces expressed previously, dy =

Fro. 12-2. Analysis of spatially varied flow.

The friction head between the two l3ect.ions is equal to the friction slope SI mUltiplied by the length dx, or . hf

Sf

ax

where the friction slope may be represented by the Manning formula as

The frictional force along the channel wall is equivalent to the pressure due to friction head multiplied by the average q,rea (see Art. 5-4) J or

Ff

=

w(A

+ Y2 dA,)Si dx = wASf dx

where the product of the differentials is dropped,

.Jo

If.-

S/ne'

1(

"f! . V dV

Since V = Q/A and V tion becomes

+ dV

(12-2) +. AV dQ ) + (Se - S,)dx (Q + dQ)/(A + dA), the above flqua-

, _. V(2A dQ-A +Q dAAdA+ dA d'"-Q) +

ely -

-

{J

2

(So - Sf) dx

( 12-3)

Neglecting riA in the denominator and dA. dQ in the numerator; and sjmplifying, . dy Se Sf - 2QqjgA 2 (12-4) dx 1 - Q~/gA~D where q. = dQ/dx, or the discharge per unit length of the channel, !Lnd D is the hydraulic depth, Ifnonunlform velocity distribution iu the channel section is considered, an energy coefficient can. be introduced in the equation, or . dy So - Sf - 2aQq.lgA 2 (12-5) dx, --1= aQ2/ qA2D

( I

.1

:

i

'

I

333

GRADUALLY VARIED FLOW

SPATIALLY. VARIED FLOW

This is the, dynamic equation for spatially varied flow with increasing discharge. Theoretically spealdng, a momentum coefficient should be used in the equation. However, the energy coefficient is' used because the friction slope Sf is evaluated by a formula for energy loss, such as the MaIming formula. l When q. = 0, this equation becomes the dYnlimic equation for' gradually varied flow of constant discharge. B. Flow with Decreasing Discharge. ,For t.hc analysis of this type of spatially varied flow, the energy principle is directly applicable. Let z .be the distance of toe bottom of the channel section above a horizontal datum (not shown ill I~i.g. 12-2); tIH') total energy at a channel sectiou is

Adding this term to the right side of Eq. (12-6) and differentiating) the resulting equation will be identical wit.h Eq. (12-5). 12-3. Analysis of Flow Profile.. In discussing the analysis of the flow profile, a simple example will be given first.

/

I

H = z

CiQ2

+ y + 2g112

(12-6)

.Example 12~1. A horizontal rectangula.r lateral-splIlway chan,nel hus a free-overfall outlet. The inflow is ulliformly distributed along the channel with a rate of fl. per unit length of the channel. Derive the equation. of the . flow profile, ignoring the friction [ass. Soltttion. In this problem the ru.te of inflow is '1. Q./L, where Q, is the dilscharge at the outlet an.d L is the channel length. ThUlS, dQ =
Differentiating this equation with respect to' x, dlI _ dz dx - dx

Noting that dJI/dx

=

+ dy + dx

a (2Q dQ 2Q2 dA) 2g A2 dx - AJ dx

-Sf, dz/dx

=

-So) dQ/dx

(dA) (ddxY) -_ dx,dy

~,! =

=

(12-7)

q., and

This ill a linear differential equation of the first order. equation is gb'll' x' - - - +1:11 2q.'

dy _ So - Sf - aQq.luA2 dx- - 1 - CtQ2/gA~

!

1

?, '

(w dQ dt)(a V2) IJw(A d:c)

This is merely a practical interpretation, which lul.s no theoretical basis.

b'y.3) 1(LZ + lJ2'1.

c

(12-8)

which is' the dynamic equation jorspat'ially varied flow with deer'easinu _ _-;,.. discharge. It should be noted that this equation differs from (12-5) only in the coefficient of the third term of the numerator. Now it is interesting to know [33] that the momentum principle can also be used for the derivation 'of Eq. (12~8). In 11. spatially varied flow, with decreasing discharge, no momentum is added to the water. Followa ,procedure similar to the derivation of. Eq. (12-5), the term containing dQ may be dropped from Eq. (12-1); the resulting equation will be identical with Eq. (12-8). _~ Likewise, the energy principle can also be used for the derivation of Eq, (12-5). In applying this principle to spatially varied flow with incre~sing discharge, the energy due to the added discharge dQ per elementmy lellgth d:r: should be added to the total energy the course of the flow during the time interval dt. This kinetic energy per pound of water is equal 'to mass X velocity2 g X unit weight of water X volume

(12-10)

where I: is an integration con.taIlt which may be determined by the bOUl,da,y conditions of the flow profile, At the outlet, x = L lIud 11 Yo, Thus, Eq. (12-10) gives

TdiY dx

the above equation may be reduced to

r

The genera! solution of this

(12-11)

~

11.

and Eq. (12-10) becomes (

where

1 +_1 2F.1

).JL_ y.

('J!)3 y.

(12-12)

(12-13) '.

F,' =

It is apparelit that F, is in the form of n Frqude number oi the flow at the outlet. When free overfali occurs at the outlet, the :!low is critic",l; therefore, F, = L The equation of the flow pre file becomes 11 (11)~ %v. -H -y.

(12-14)

where depth y. is the critical depth at the outlet for Q,. When the outlet is submerged, ~he depth Yo ill determined by the downstream aurfa.ce elevation, and Fa can be computed from tbe knowledge of y, and Q,. For each value of ~/L, Eq. (12-14) wilIgive two real positive solutions. However, only values of y/y. ;?; 1 are true solutions, because the energy in flow rous~ deal-ease a.s the. flow' proceeds dov.;uatream andy aallllot be less than'll •.

The above example illustrates spatially varied flow in horizontal channels of rectangular -cross section. Similar analyses can be made for channels with parallel side walls having irregular bottoms and for channels with sloping side walls. For channels with sloping bed, hbwevel', a

SPATIALLY VARIED FLOW

334

. GRADUALLY VARIED FLOW

general explicit '"rtuatio:1of thefhw profile such as Eq. (12-12) cannot be obtained. If no hydraulic j\lmp occurs in the channel, the flow profile l cn.n be computed directly by the method of numerical integration (Art. 12-4)) starting from a known control depth of flow. Li [5] hns performed an analysis of. spatially yaried flow in channels of sloping bed ,by means of numerical integration. The results of, this n.nalysis can be, summarized in a general diagram (Fig. 12··3). For channels wIth parallel side walls, the clbgra,rn i.~ shown in Fig. '12-4. 1

I

335

by the dashed line. , In region A, F. 1, and the flow condition is governed by the upstream depth YU' In ot,her words, the water 8urfac~ recedes in the downstream direction, and only the value of y" is of interest in determining the channel dimensions, Values of y,./Yo for region A have been computed by numerical integration and have been plotted as solid lines (Fig. 12-4). The computation was made on the condition that y"jy, ~ 1 at x/L = 1It cnn be proved that, when dF/t1:x ;:> 0, G < %(1 + 2F.'). This is represented by the line dividing, regions A and B in the diagram (Fig. 12-3),

( I

i

Value of G

G

FIG. lZ·4:-SO"luiioDs for channels with sloping bed and par:>ilel side walliJ. W. H, Li [5].)' , .

FIG. 12-3. Types of spatia.lly varied flow as determined by F, nnd G.' The diagram is shown for [\, channel of an arbitntry sec~ion. Fvr rectangular channels, the line dividing regions Band Cis G "" 1 + F.; for trio.ngular channels, it is G '" 2. The dashed line in each channel is the critical-depch line.

(After

Region B. This region: represents the condition' where the flow is subcritical throughout the channel but where the value of F will first· increase as the flow proceeds downstream, reaching a maximum 'value . less than unity, an,d then decrease. It been found that the line di:vidregions Band C can be represented approximately by G = 1 + F •. This line indicates all the cases in which the maximum value of F reaches unity. ' Regicrn C. This region represents the condition at which there is supercritical flow in the downstream portion of the channel and a hydraulic i:ump in the channel. It h(l.s been found that super critical flow occurs when G is greater than approximately 1 + F.. The hydraulic jump will fo:-m only if the outlet is sufficiently submerged. As the jump occurs, the control section will be shifted into, the channel, and the elevation

This diagr8;m indicates the relo,tionship between the Fl'oude number F, and the value of G SQL/y.. It consi.sts of fOUl" regions, representing four conditions 01 flow. Region A.This region l'spresent.s the condition where flow is sub7 critical throughout the channel and where the value of F increa.se!; as the flow downstream; that is, dF / dx > O. The symbol F l'epresents the Froude number of flow at a. section a distance x from the upstream end of the channeL The value of F can be computed from Eq. (12-13) by replacing F 0 with F and L with x. Since the flow is subcritical; the depth of flow at any section is greater than the critical depth, as shown , L The profile can a.lso be computed bra gra.phica.l method developed by Ca.mp ;4], which requires trial adjustments~

I~

336

,( \

GRADUAIJLY VARIED FLOW

SPA.TIALLY VA.RIED FLOW

of the \,rater surface at the outlet will not affect the entire flow profile. 'The flow profile upstream from the . jump cannot be determin!ld froni the : value of YO) but it can be determined from the critical depth yaand the position of the critical section Xc. A dimensionless flQ~" in the sllpercritical reach ('[;~ig. 12-5, ill which A" and A .!we water areas, respectively, of the critical section and the section a distance x from the upstream end of the ch~nnel) has been computed by numei'ica[ inte~ gratioll. This curve can be used to FlO. 1,?-5. Dimensionless flow profile in compute the flow profile below the Btipercriti!\a~ reaches in a sp .. til1!lyvaried-liow cht,nnel of sloping bed and critical section and. above the hyparallel walls, (After W. H. Li [5),) , draulic jump. . The positi'on of a critical section in the lateral-spillway channel can be determined by the method of singular point (Art. 9-6), Regio-n D. This region represents the flow,condition Fo.t which there is supercritical flow throughout the downstream portion of the channel but where the depth of submergence at the ouMet is not enough to create a hydraulic jump in the channeL Thus, the value of Fa is not determined by the depth of su'bmergence. The dividing line between regions C and D (Figs. 12-3. and 12-4) was obtained by numerical integration on the condit,ion of a millimum depth of required to prOdtlCe a hydraulic jump at the outlet. This Jnllllmum depth 'is . equruto the depth neeeSS1J.ryto set the downstream pool level at a seq).lent depth. A depth of submerge'nee greater than this. minimutn depth wlll force the jump to move upstream into the channel, and the oondition of flow Ifill be represented by region C. When the slope of the channel is extremely steep or when the value of G is very large, the flow ,vill become unsteady. The limiting value of G that will keep the flow in a steady condition has not determined. In the above a.nalysis, the' effect of friction has This has been verified as justifiable for the design of wash-)vater troughs and side-channel spillways. For effluent channels around 6ewnge-treatment tanks. however, the effect of friction may increase t~e upskeam depth Yu ~s :much as 10%.* '

'For an advanced theoretical analysis of spatially varied flow, the method of singular point (Art. 9-6) may be

$LJ 15] has computed curves repre:;enting the increaSe of !/Y horizontal channels. .

),

I

relmitof friction in

337

Example 12-2. Analyze the flow in iii rectangular ch:wuel .of .small slope with a bottom mck (Fig. 12-6) and derive the equation of the ,flc\v profile.' SoluiiOn. The flow in a channel with bottom rack is a case of spatially vaded flow .with decreasing discharge. The ra.ck is usually made of parll.lJel bars 01' perforated screen. There arC va.rious application;s of such 0. device. For exo.mple, the channel may be' an "intake" to withdraw'w{l;ter from, say, a mountain torrent, or a "skim, mer." to reduce the ,volume of wa~er required to transport, say, fish. A~s;lming '" = 1 !l.nd 8 "" 0, the speoific Bn"rg'y at any section of the channel (Fig. 12-6) is E

V'

=. '11 +" ~g

=

11

Q'

+ 2gb'11

2

(12-15)

For a spatially vaxicd flow with decreasing discharge, the specific energy can be considflrcd constant aiong the channeL' Thua, dE/dx = OJ or, from Eq. (12-15),

dll, Qy( -dQ/dx) dx = !lb'y' Q1

(0'

(12-16)

where -'--dQ/d:!; is obvio'Jsly the discharge withdrawn through a length dx of the rack. Equatil:m (12-16) is the general dynamic equation for the flow under consideration. i~l The disc.harge through'the rack depends upon the ei'fective head on the rack. FIG. 12-6. Channel with a bottom rack . When the direction of liow through the (a.) Partin.! withdrawal; (b) complete withdrawaL ra.ck openings i« nearly vertical (A, below), the energy loss in the process is negligible' and, thus, the effEctive head on ro.ck is practically equal to the specific ellergy E. Mostkow 128J found that this is true of racks such as those composed of parnllel bars. On the other hand, when the direction of flow through the ra.ck openings makes ..n appreciable with the verticai (E), the flow will impinge all thfl sides of the openings, in a loss of energy e.nd a change in direction of the flow from inclined eyentually to vertical. From experiments, :MostkQW found tha.t this is true of as those composed of a perforated screen and that tile corresponding energy is approximately equal to thE velocity 1 There hav~ been many investigations o~ this problem. Th.e actual iflaw phenomenon is lather complicated, particularly *'hen the slope of the rack is tll.ken into consideratiqn. : For details, see [21J to [23J, [26j to, [28), and [34J to [381. ' • This assumption was found to agree with the experill}ents [21J.

338

SPATIALLY VARIED FLOW

GRADUALLY VARIED FLOW

hea.d of the flow over the rack. It may, therefore, be Msumed that the effective head on the racle is [lqual to the !!btic hee.d, or the depth of flow. over the rack. A. }I'or VeNieal ,"'low lhrough Ih~ Rack. In this case the discharge through a length ax of the rack may be expressed by (12.17)

where ~ is the ratio of the opening area to the total area of the rack surface and c is the coefficient of discharge through tb:e . From Eq. (12-15), the (Iischarge is

Q

(12-18)

by

Substituting Eq. (12-17) for -dQldx and Eq. (12-18) for Q ill Eq. (13-16)~nd simplifying,

'1

l

SUbstituting'Eq, (12-24) for ~dQldxand Eq, (12-18) for Q in Eq. (12-16) and simpliiying, dy 2ee ',/iJ[1?=-Tj) (12-25) ;h "'" 3y - 2E Integra.l.ion of this aqUlltion gives the eq1!ation of the flow profile as l

I

x

!

=

~ [~i sin-I EO

~ }~

1

(1 - 2lt) - %~ "VE -/!J. (1 1!_)] + 01 E E

-J~ - % ~~ ( 1 .. ~) ] + ~2

(12-2B) (12-27)

I

or

1

The integration eon.stants in the above equ:1.tion may be evaluated by the condition that 11 - !II and x = O. Then, when 11 = 0, Eq. (12-26) wi!! give the length of the rack required for!!. complete withdraw!l.1 of the main flow througll the I'l'l.Ck, or

I

x =

(12-19)

[

-E

.c

tntegratioll of this equation gives tlle equation of the flow profile o.s

C05-

[v ~1I1E /1!

-

'Ei!l) - ..,',.sm- (. 1

( 1 - -

~

'~

1

2Y ,)

E

+

(12-28)

A and B described above, the entraDce to t11";' reach of the rack as (I. broad-crested weir. Thus, Q, = c'bE"', where " may h!l.ve value of 2.80. Also, Eq. (12-13) gives Q, = bYt V2g(E 1h) and Thus, the discharge of. a partial withdrawal from the :nain Q.. = Q, Q" or .

TM'J>Trjm"

J1

x= For !I

C

=

=

y, and x

(12-20)


= 0, the integra.tion

a.ll

Q.

cOI~stant is determine<;i from Eq. (12-20) as

flow

(E/ec)(ydB) \./1 - 1/dE. Thus,' (12-29) x =

~ (!L! .,; .E

y

E

(12-21)

Wilen y = 0, Eq. (12-21) gives the length of the rack I'equire,d for e. complete with~ (\iawal of the main flow through the rack. or . E EC

(!L!E

(12-22)

By Eq. '(12-18), the above equa.tion may be reduced to

L,

(12-23)

. where QI is the discharge through the ent.ranceto the reach of the rack and is also equal to the wit.hdrawal discharge Q" through the rack. B. For Incl:ined Flaw IArough Ill.e Rack. In this case the discharge through a. length dx of the mck may be expressed by

_dQ_.eb

ax

1 This equS:tion may also be derived by:means of Eqs. Eq. (12-17), ~he discharge through the ratck of Eq. (12-18), Qw = Q,,.- Q = by, V2g(E -7 !II) - by two expressions for Q.., Eq. (12-21) is obtained.

(12-24) .

and (i2-18). By .cbz V'iiE, By Eq'tating these

For a complete withdrawal of the main flow through the rRck, it is evident that Q, = c'bE,·6, from which .E = (Q,/e'b),', Thu8, E may be computed if the incoming discharge Q" b. and c' are given. , 'The v~lue c of the coefficient of dischll.rge throug;h tho mcl. cpenillgs actually varies considerably along the rack. For fJx"mple, typical values dcf,ennined experimentally were found Lo vary from 0.435, for a grade of 1 on 5, to 0.497, for a horizonkl.l slope of the racks of parallel bars; and from 0.750, for Il. grade of 1 on 5, to 0.800. for a horizontal slope of the racks of perforated scrsnns [271. III general, the value is higher for racks of screens than 'for racks of parallel bars. The value is higher for horizontal racks than for inclined racks. The local value increases as the; ftow depth on the rack increases if the bars are parallel to the direction of the main flow bilt decreases with the depth if the bars are in transverse direction, ' The n.nalysis of this problem may be further extended by considering t,he effects of the streamline curvature, the nonuniform velocity distribution, and the bottom slope, and by classifying various types of the flow protil€s thus created. In general, there are five types of flow profiles, which are similar to those to be described in the neXt exampie, It may to note that, when a critical state of Row exists OIl th·" upstream side of the rack, the critical depth will occur u.t a. section somewhere upstrea11l frOl:n Q",

~ Equat~on (12--26) is given by Mostkow [27J and Eq, (J 2-27) by Noseda [2Il. THe tw.,' equatIOns are·mc.tbematically idellticll.l. The relatitlnship between the integra- . tiol! constants ill 0 1 = Ot ,..EIS.c,

+

'. 340

GRADUALLY VARIED FLOW

the entrance section. This phenomenon is, therefore, similar to that of a free overfall (Art. 3-4). The ratioYliV" of the entrance depth to the critical depth -decreases \I'itll increase ill the value of • and of the rack slope. Typical average :atios vary from 0.70 to 0.90, which correspond approximately to the valuGs of VilE frGm 0.47 to 0,60. ' Example 12-3. Analyze the ftow through a side weir in a prismatic horizontal rectangular channel.l Solution. The tlow th;:cugh a side weir is a case of spatially varied flow wit.ll decr~asing dischD.rg~ (Fig. 12-7). According to Frazer [32j, the foHowing five typm; of flow profile can be produced:

_--'-'T~

-

{OJ

consta.nt, or S, = So. Since the channel is horizontal, So = Q. a = 1, Eq, (12-8) gives an equatiou identical with Eq. (12"16), or

Type c. Type a flow at the beginning of the weir selJtion \vith a hydraulic jump occurring in the w'Jir sect,ion, and type b flow after the jump at a lower- specificenergy level owing to jilin p losses (Fig.

===tl ~----:'. YIJYJ=:=::::J~J2 .

Qy( -dQldx) dx = .Ilb'y' ~ Q'

--,-'--

12-7e)

'l'ype e. Type d flow at the entrance section with a hydraUlic jump occurring in the weir section and type b flow aiter the jilmp a.t a lower specific-energy level owing to jump losses (Fig. 12-7f)

The last two types of flow are possible if the approach fiow is 5upercriticaL Iu a conventionai analysis, it is assumed that tt,e velocity thro-ugh the side w~ir is in general at right !Ingles to the weir. This aSaumption is more satisfactory fo'r FIG. 12-7, Variolls flow pl'ofilesalongaside subcritical flow than for supercritical weir. flow. In supercritical flow the velocity will be high and the angle that the overflow' makes with the weir will be smidL Consequently, types d and e canno-t easily be analyzed succe~sfully. Furthermore, the discharge in types d ~nd. e flow is controlled upstream, where additional consideration in the analysis is required. In the present problem the specific energy along the side weir may be assumed 1 For detailed studies of the side weir, see [241 to [271 and [2D I to [32].

(12-16)

The discharge over ailY given length of tho weir can be computed by a weir formula,

~..!!!. dx

=

_

dQ "" cV2a (V - s)1.' dx

(12-30)

where c is the discharge coefficient and s is the height. of the weir sillll.bove the bottom of-th~ channel. The sill of the weir is pa,'allel to the bottom -of the channel. EquatiQn (12-18) also is applicable to the present problem. Tl1us, the discharge at -any section is (12-18) Q = by -/2U(B - 0 where b is (·he width of _the channel and E b the specific energy. With Eqs. (12-30) and (12-18), Eq. {12-16} becomes 2c-V(E - y)(y - s)3

ay

ax

=

II

(12-31)

311 - 2E

Integrating Eq. (12-31) nnd solving for ." x=

12"7d)

7'ype d. Depth of flow less than critical at the entmnce with supercritical flow in the \veir section, the depth of flow decrell.Sing along t-he weir section (Fig.

Thus, assuming

dy

12-7c)

(e)

j

'J'ype a. CritilJ~j conditions a.t or near the entrailce with supercritical flow in the weir section, the depth of fiow decreasing along the weir (Fig. 12~7b) Type b. Depth of fiow greater than critical at. the entrance with subcl'itic,ll flow in ttl''! weir section, the de-nth of flow increasing along the weir se~tion (Fig.

341

SPATIALT,Y VARIED FLOW

where

P

(1) \E

=

~c F (1) + canst E

(12-32)

2E - 3~ ..JE - y _ 3 sin-1 - IE - Ji E-s V-s'V y - s

(l2-33)

F(y/E)'is a varied-flow function which

WiLS

first solved by De Marchi [16J.

12-4. Method of NumerIcal Integration. This method wj1J be applied fir.st to fl, flow with increasing discharge and then to a flow with decreasing . discharge. A. Flow with Increasing Discharge. Considering the differentials as finite increments, Eq. (12-1) may be written

~ [Q 6.V + (V + t..Y) 6.Q]

-w

lo-"Y

Ady

+ wSo !o~%

A d:1;

- wS, (Il.'l: A d:c

.

-wA 6.y

+ wSoA 6.x

}o

- wB,A 6.x

(12-34)

,,,here A is the average area. Since the discharge varies. with the finite increment of the channel length, the average area may be taken as A = (Ql + Q%)/(V 1 + V 2). Also taking Q = QI and V + 6.V = Vz and simplifying,

342

The dl'OP in water-surface elevation between sections 1 and 2 (Fig .. 12-2) may be expressed by

dy'

=

-dy

+ Sodx

TABLE

+ So c.x

COMPl.rTA'l'ION OF CRITICAL DEPTHS FOR EXAMPLE

II

:;

(~;

I

4 48 6 I 78 8 112 10 150 12 192 14 238 16 288 18 "342 20 400 22 462 24 528 26 598 28 572

(12-37)

Example 12-4. * A trn.pe~oidlLl lateral spillway channel 0100 fl long is designed to carry a varying discharge of 40 ·ds/ft. The cross section has a bottom width of 10 fl and sid~ slopes of H: L The longitudina.l slope of Lhe channel is 0.1505, starting at an upstream bottom elevation of 73.70. Assuming n = 0.015 and a = 1, compute the flow profile for the design discharge. Soltdion. The Erst step is to determine the control section frQf\l which the Row profile c'omputation can start. The control section !nay be determined by the method of singular point. (Art. 9-6). In this example,. however, a method developed by Hinds [11 is·.employed. The corr.putation is shown in Tables 12-1 and 12-2. Table 12-1 shows the computation of critical velocities and discharges correspond~ ing to a number of arbitrarily ASsigned depths, shown in col. 1. The critical velocities in col. 5 correspond to the. critical velocity heads in col. 4. The hydraulic mdii are also computed and reco.rded in this t~ble for use in computing friction lo;;~e.s. Table 12-2 shows the computation of the drop in'water surface necessary to ma.intain a flow at the critical depth throughout the full length of the channel. The column headings are explained as follows: . Col. 1. Distance of the station alO1'lg the channel, in ft Col. 2. Increment of the distance . Col. 3. The inflow discharge, equal to x times 40 cfs/ft . Col. 4. Sum of the discharges Ql of the previous station and Q, of the station under consideration 1 The use of an energy coefficient instead of a momentum coefficient has the same reason given in Art, 12-;J. ... This example is taken from [11.

I

I

(12-38)

This equation can be used to compute the flow profile of a spat,iully varied flow with increasing discharge. On the right-hand side' of the equaUon, the first term represents the effect of impact loss and the second term represents the ·effect of friction. It is interesting to note that, if c.Q and Sf are zero, or QI = Q2, then this equation ,vill be reduced to l1y' = a(V22 - V I 2)/2g, which is the energy equation for flow of constant discharge, neglecting friction. The procedure of numerical integration is iaustn,.ted by the following example. .

I.'

I

1. 7 t 2 44 18 '3.11 20 3.75. 224.36 24 4.95 25 5.541 28 5.11 II 30 6 67 32 7.22 34 7.76 36 8.31 38 8.84 1-l.. 16

I

II

10.49 12.52 14.15 1.5.53 16.75 17.86 18'.88 19.82 20 .71 21. 55 22.34 23.12 23.84

504 978 1,585 2,330 3,216 4,252 5,440 5,780 8,284 9, 960 11,800 13,820 16,020

----~------~--~~----~------~-----

TADLE

12-2.

Q

I'Q·+Q,ly, I

v,'!V,+v,

(4)

400 l,pOO 2,000 4,000 8,000 8,000 10,000 12.0001 14.000 10.000 1

I I

--I

8.77 9.25

--~-------

1

(5)

(8)

400 3.4 '10.01 1,4GO 5.2 12.5 3,000 9.2 ' 14.91 5,000 13.5 1 17.6 10,000 16.9119.31 14,000 1 19.7 20.61 18,000 '122 I 21.B: 22 ,000 242/22.4'[ 26.000 25.2 23.2 30,000 28.0 23.8

(7)

10.'0 22.5 27.4 32.5 36.9 39,\1 42.2 44.0 45.6 47.0

12-4

AQ

(3) I ----.--------j-.----

,0 (2) -:-:~~I-·

0+ 10' 10 0+25 1.5 o T liO 20 1 + 00 50 1 + 50 50 2 + 00 50 2+50 50 3+00150 3' + 50 50 4+00 50

I

1.52 2.52 3.33 4.01 4.63 5.22 5.76 6.29 6.82 7.31 7.81 8.29

COMPUTATION FOR THE DETERMrNAT!9N OF THE CONTROL S::;:CTION FOR EXAMPLE

" ""I

12-4

(7)

- - 2 - ---;;-)1--'-1-'-2·-jl--0'-.,..9-2- ---'-7-'-6-8- 1 - - -- - 1 69

Substituting Eq, (12-35) for l1y in Eq. (12-37) and introducing !l,n energy coefficient a for nonuniform velocity distributbn,' the drop in water surface is

&~'=O:~(I~~r~+Q~2)(6V+ ~:6Q)+Sf~.1;

12-1.

(~) (~) I (~) A:~T

(12-36)

COllverting the differentials to finite increments,

t;.y' ,." -l1y

343

SPATlALLY VARIED FLOW

GRADUALLY VARIED FLOW

(8)

11·:1 R, hi --.-------oV (9)

A'l'l:lly'

(10) 1 (11)

400110.0 .... 2.25 RuO 2.5 4.2.5 3.41 1,000 2.4 4.9l 4..i0 2,000 2.7 ·B.77. 5.63 2,000 1.7 5.21 5.53 2000 1.3 4.33 7.23 2'00011.0 8.74 7.S3 2:000 0.813.30 8.H 12,000 0.8 3.04 882 ' 12,000'10.612.7319.25

(12)

(13)

Q.03 0.05 0.08 0.16 0:16 0.15 0.15 0.15 0.15 0.15

4.30 4.30 4.99 9.29 6.93 16.22 5.37 21.59 4.48 26.07 3.90 29.97 3.45 33.43 3.1.9 3fi.52 2.88]39.50

1

(14)

I

-

)

I

1

1

\

Co\. 5. Critical depth in it, interpolated from Table 12-1 corresponding to the discharge in col. 3 Col. 6. Critical velocity in fps, interpolated from Table 12-1 corresponding to the discharge in col. 3 Col. 7, Sum of the velocities the previous sta.tion and in the station under con·· sidera.tion Col. 8. Increment. of discharge tlQ = Q, - Ql Col. 9. Increment of velocity t. V = V, - V, Col. 10. Drop in water suriace due to impact loss, or

.·1

t

in

(12-39)

1

,i

,

J'

344

GRADUALLY VARIED FLOW'

Col. 1 L Criticl\\ hydraulic radius in ft, interpolated from Table 12-1 corrc;:;pond'ing to the di.schmge in col. 3 Col. 12. Friction loss, be.sed on :&t. (9-8) with n. 0.015, V from col. 6, and R from coL (ll), Since this is ll. minor item. compared with tlie impact loss, it may be ignored if desired. Col. 13.. 'The I,ctal drup in water surface 6y' Ay,: + h, CoL 14. Cumulative drop in water llurface

0""U'.l"''''' ....

000::>""0> ....

t-r-t-l.Ce'to

~<6u:ice>~

~C'lC'40000

C"l'~

~

CJ)

....... M (,,', \0 ~?"'"' M 1""""! ........~ 0 0

:::g:s~~g

000000

oeoco

..

.

..

COOMC~OO

1..... ~'l..... C'\I N

~<.OtOtOu?-.i

c,ot-t-t-C()

C'3 .......

Thecumulati:"e drop in water surfaee is plot~ed as the heavy dashod line in Fig. 12-8, starting from !l.U a.rbitrary eleva.tion 120' ft from some station at x = 10 ft. The critical depths from col. 5 of Table 12-2 are then ptotted from the dashed line?s ShO'Uil by the dotted line.. It is a.pparent that this dotted line represents the bcttom of II

~

rl~O{,O

~~~~c:

Ci)C:~"""'~(D

~~t"CeoOCJ

NQ ....... OOCO 1_ t""'

r-

~~ ....... .-;~

\1:1 C'l 0.

"'t'tlCO
....... 1.C"OO\Qot'-

a

~~~L.)~c..i H

N¢'l~N-C-'1

co r-

~' ct)

r..:::;

en

r:Q

(0

O'J ':"')

....

lOC";It.OC'fllO

.

Oischor~e In cfs.

'

00000

~_8~~~g_ ....... N

100

,

C"1 'C'1

c,~,

OM¢'tt-(H

90

en

~'0l'Q)

~

l"':'c-.i

00

~ C')

.... "'lOU'.lCO

I

L

t!l

I

00000 00000 L..":)... 0,.. 0_ 0_ "'00,""

~

c:.

,$

...... _N

m

jt\-~--~------~ O~LOC"lt'-OO...-i

. . . .

· .

l",",OOCi)t"--('I':>~r-

:;i'>o\;l'j"'I':TI ....... - .......

M

~ '1""""1

~

lJ")

~

N

000

-0

a

. N

.......

-0

a

0

~

I-l""""'.MtOr¢I~ «: 'ZO t'- ..-! t- '-di

DislonCi '.1:0ng

,

c~onnel

F~G. 12-8. Computation of tlow .

~C)C'1'C'-,C'll"""'i,....c

length In It

t:;:;

e

1--'--1

:3

'§

....c

J3

~

'.;s

t.Ol"--o:c:r;C'!~~

W.

CO-..:ttM""t'O'l,....tC'1~

~

- ' t-

OiIct~r-­ ,....~NN

I--
~-

g8g~~~~~ r...: ~ L..j ~. c-i ~ ~ cO

,..... ............................ .,. . . 1-0--'-0-0-""-"'-,.....-00.--,.....--1

oo(>?.;f:1

l'_,

e'8"

] 1----·----------------, ~

;rome for EXllmple 12.4.

. fictitiolls channel in which the flow at the givl'..rt dis~ht\rge condition is critical at e~ery section throughout the full length of the channe!. The c!l.I.Shed·line is. the corresponding water surface. A tangent parallel to the bottom of the actual channel can be drawn to the dotte~ bottom line of the fictitious' critical-ftow channel. The point of tangency, at Iyhich the two bottoms have the same slope, gh'es the location of,the critical'section, which is found at Ste.. 1 + 64. It is evident that the slope required to ma.inta.in critica.l flow to the left of this section i~ greater than the actulll slope ~and that to 'the right it is)esa, which is the con<;lition ne'cessary for the formlLtion of.!l. control. If more than qne point of tangency is possible, the one giving th'e lowest position of the ta.ngent ,viii be likely to control., It is:also p08~ible to ha.ve two or more control sections with'hydraulic jumps 'between. : Having located th~ control section, the tlow-profife comput.ntion can be carried: ou,t as shown in Table 12-3. The computation proceeds upstream from the control section for the subcritical flow in the upper part Qf the channel and -downstrenm for, the

'" g ',B .,.,'" >.

,.0

.....-

;....

~

~8:;;;;; M~M~

1--------------: Ol.OCf')"I:t' 1..... (,0 CD

a

1;'-

CO ~

....-i

.,.... ~ ....... C'1

::l.

'"

<:or-r-.~t-C).jooco

·OOt-OOI'..Ot-~

~~~~c:~~~

"'t-t-r-LOC'
000000 ....... 1;'-¢'.1r-c-,1LC";l

O~~OO~M

~,""cQ¢'lN1"""f

· '" • to

'<"0 '" 0

++

OlQCO tOC'l"""'O'

++++ 0000 345


0000

0

L..":l 0

(Q

tQ

+

++++ C'lMM~

346

SPATIALLY VARIED FLOW

GRADUALLY VARIED FLOW

347

ally, this method can be applied equally well to any type of varied-flow equation for flow in prismatic as well asnonprismatic channels. 12":6. Spatially Varied Surface Flow. An important type of surface flow encountered frequently in engineering, problems dealsvrith runoff from a plane surface as the result of rainhdL Apparently" this is a problem in spatially varied flow with illcreasing discharge and can be treated as such; it may, however, be very complicated, becoming a threedimensional problem if the surface is CUl'vedin space, as in the case of a road pavement that has a cambered transverse profile and a longitudinal slope. The theory of SlHl,tially varied flow was first used in sudace flow by Keulegan[40], and the equation thus derived wa.'3 applied

supel'l~ritical flow in the lower part of the channel. ' The procedure of computation is the same.as thnt explained for Table 12-2, except that the wllter-ourface drop /J.y' in col. 4 is finally (Obtained when it agrees with the computed Ay' in col. 17. This is shown for the computation at SLa. 1 + 00. In cols. 3 and 5 are elevations of the channel bottom and the water surface, respectivelJ. 'The value of Ay~ between :z: = 1{l and x = 0 cannot be computed, but it is arbitrarily fLssumed to be twice the velocity head at X = 10 ft. The final flow profile is constructed as shown in Fig. 12-8. The accuracy of the computatior.. will depend on the length and' number o(subdivisions assum
B. Flow 'with Decreasing Discharge. For spa,tially varied flow with decreasing discharge, an equation for numerical integration similar to Eq. (12-38) can be obtained. Referring to Fig. 12-2, the velocity and disr:harge at section 1 are as.sumed as V and Q and at section 2 as 11 - t. V and Q - t-.Q. The momentum lost because of diminished discharge may be taken as w t-.Q (V - Q V /2)/1}. Adding this lost momentum to the momentum at section 2, and following the procedure described for the flow with increasing discharge, the equation for numerical integration can be shown to be

-1

1 II

(12-40) Owing to the variable velocity distribution in the channel cross section, the value of the energy coefficient may be very high. According to Schmidt [251, values up t.o 1.30 have been observed at the beginning of the spillway, and even higher values were found at the end of the spillway crest. By experimental study, Schmidt Was able to develop an adjustment procedure to correct fOI' the effect of the nonuniform velocity distribution. The value,of 6.Q in Eq. (12-4:0) is the discharge over the spilhvayper 6.xof the crest length. :Nfany formula.'3 have been proposed for its determination. For practical. purposes, the formuh\ for. the regular weir of similar crest shape may be used if the corresponding discharge coefficient is reduced by 5 %. 12-5,' The Isoclinal Method, For a simple but approximate computation of a flow profile, a graphical method suggested by Werner [39J may be used. By this method the spatiaIly-varied-fiowequation iIi any form is plotted with y against z for different values of dy/dx aspararneters, i'esulting in a numb'el' of isoclinal curves (Fig. 12-9). Starting from the depth at the control sectioll C, aline is drawn with a slope (= 0.03) equal to the average value of dy/d.'1; (= 0.05) indicated by the isoclinal curve passing through the control depth and dy/dx (= 0.01) of the next isoclinal curve, which the line interdects at P. Starting at P,' repeat the procedure to determine P', and determine similarly other P9ints of intersection. ,The flow profile is the curve joining all the points of; intersection. Actu-

.'.

.~

..

') i x, II

FIG. 12-9, Comput,2.tion of flow pro51e by the isoclinal method.

to experimental data by Izzard [41]. For flow on a Toad surface, a comprehensive analysis was performed by Iwagaki [42]. For practical purposes, all approximate equation for discharge of surface flow is generally assumed, s11ch as

q

=

leY'"

(12-41)

whereq is the discharge per unit width of the flow, y is the depth of flow at the point of outflow, aild k and, m are constants. At equilibrium conditioll, the discharge q at a point x distance below the drainage divide is q = xq*

(12-42)

where q* is the constant inflow dne to rainfall excess, or s,upply rate, per unit area. The rainfall excess is equal to rainfall minus infiltration and other losses that will not become surface runoff. Combining the above

\

i Iy. 348

SPATIALLY VARIED FLOW

GRADUALLY VARIED FLOW

3. The depth is theoreticnJly constant when the transverse profile of the road is a parabola with its vertex at the crown, that is, when n = 2. When the road surface is formed by straight lines connecting the crown and the' that is, when n 1, the depth becomes smaller near the crown and greater toward the sides. 4 .. The effect of the longitudinal slope on mean velocity and friction velocity is to increase the l11.ean velocity and the friction velocity of the fl0W. This effect is greater near the crown of the road for larger nand greater neal' the sides for smaller n.

two equations and simplifying, y

(12-43)

This is the eguation for the flow profile, which is generally applicable when x is not too iarge. The value of k has to be determined experimentally since it depends 011: the surfaoe charaoteristics, slope, type of flow, and viscosity (in the.case of laminar flow). The value m depends on the type of flow; it is approximately % fqr turbulent and 3 for laminar. For tmbulent flow, the differential equation for the surface flow, from Eq. (12-4), may be written . dy dx

(12-44)

'I"here F2 V 2 /gY. If the raindrop momentum is ignored it can be shown that the coefficient 2 in the numeraotor will becom~ 1 [of. Eq. (12-8)]. For analytical studies, the profile of the surface flow can be computed by the method of numerical integration. The control section of the flow profile can be uetermined by the method of singular point or by a criterion developed by Keulegan [43]. For laminar flow on a road surface, I wagaki [<12] has performed a.n elaborate matherna~ical analysis, in which the continuity and momentum equations are applied to n, three-dimensional element of the flow .. By consielering a, general case in w.Uich the velocity D,nd depth of the flow do not change in the longituuinal direction of the road surface. he was able to derive a differential equation as follows: '

ely where and

1 I· I j

~.~.-------

L -----

;Fm. 12-10. Cross section of road for surface-flow analysis.

5. The effect of the longitudinal slope is practieally negligible \vhcn this say, less than 0.002 under the normal condition of slope is very H/L 0,02. 6. In order to minimize erosion due to raindrops on unpaved road surf!:we, the -iongitl2dinal slope should be kept as small as possible. To maintain a unifol'm gra.de of erosion, a cross section witl1 n = 1 is ·IJl'eferable; PROBLEMS

Fl

ilx - F~

349

(12-45)

(r)"-l

p

6q.2X

- --g;;2

The notation is given in 12-10. The transverse profile of the road -H{x'/L)1t. Equation (~2-45) was then surface is represented by y applied to a numerical example, and fio,,, profiles were computed by the isoclinal met.hod. From this investigation, the following conclusions were obtained: 1. 'The flow profile is independent of the longitudinal slope of the road surface. 2. In case of natural runoff, the flow profile is approximately representeel by the .curve dy/dx· 0 except for the part neal' the crown of the road.

!.

12-1. A rectangular wash-water trough 20 it long a.nd 1.32 ft wid~ l)<1rries a discharge at· a slope of 0.065 to a. free-fall outlet. If the measured upstream depth is 0.34 ft, I)ompute the discharge by means of the chart in Fig. 12-4. 12-2. An formula for calculating the discharge capacity of rapid sandfilter wash-water tronghs,has been developed by Miller [Ill by asSuming a parabolic flow profile at a maximum discharge. The formula is

Q "'"' 1.91b(y..

-+ L tan 8)

(12-46)

where b is lhe width of the rectangular channel in ft, y .. is the upstream depth in H, L is the channel length in ft, and IJ is the angle that the channel bottom makes with the horizontal. Using this formula, compute the discharge required in Prob. 12-1. 12-3. A rectangular wash-water trough 30 ft long is required to ·cal'l'ya discharge of 8 cfs, having a free fall at the outlet. Design the trough for the least o.mount of material required fOT theconstl'uction·(neglecting the end wall and making the total wall and bottom wall of the channel a minimum). Assume:

a. A horizontal channel a. That the channel has a slope

-I GRA.DUALLY VARIED FLOW

350

12-4. Compute el;e flol\' profile in Prob. 12~1.

SP_~TrALLY

Assume

a. A horizontal channel b. A channel with slope equal to 0.0(35 12-5. Derive the equation of the flow profile in Example 12-2 if a is not equal to unity. 12-6. A horizontal bottf)In rack made of perfor:ttcd screr.r, is. designed to divert water from a channel . . Determine the length of the rack req1:lired to withdraw the total main .fioll' of 26 ds from the channel. Given:. = 0.5, e = 0.8, c' = 2.80, b = 3 ft, and y,/E = 0.60. 12-7. Solve tb.e preceding problem if the rack i~ Illade of pam!lel bars. 12-8. A side weir is ased to ciivert the excess of a storm flow of 75 cfs from >\ 'IS-in.diameter sewer. The sel'!er has a grade of 1 in 400,a iull·flow eapacity of is cis, and an unrestricted outlet. The dry-weather flow is 5 cis. Detimnine (a) the height of the weir sill, and (b) the lengt.h of the weir, assuming a = 1. It is ,.Iso assumed that tile top width of the water s.re:t is const2.nt and equal to the diameter of the sewer; so the equations derived for recta.ngular channals can be applied. 12-9. Solve the preceding problem if '" = 1.20. 12-10. By converting increments to differentials, show that Eq. (12-35) is ident.ical with Eq. (12-4). . 12-11. Determine the control section in EX!l.mple 12-4 by the method of .~ingular point. . . 12-12. Demonstrate analytically tha.t Rinds's method for the determmatlOn of control section is identical with the method of singular point. 12-13. C~mpllee the flow profile in the channel described in Example 12-4 carrying a varying discharge of 50 cfs per foot of ch!l.nnellength. . 12-14. Compute the flow profile in Example 12-4 by the isoc!inalmethod. 12-15. Verify Eq. (12-40). 12-16. Shol~ that the flow in a rectangular prismatic channel with a lateral spillway may be expressed by Q

,----zy

-= b \) (Hy' --

1/3) ~

(12-47)

where b is the channel width, y is the depth, H is the constant energy head in th.e spillway section measured above tile channel bcittom, and a is the energy coefficient: 12-17. Artificial rainfall of a constant intensity "qual to 3.6 in./hr is applied on a concrete pavement ha.ving a slope of 0.0'.1 and a roughness coeffkient n = 0.025. Compute the flow profile, ignoring the raindrop momentum. Assume that:

a. The lower end of the pavement is a free-fall outlet. b. Ther~ is a dam % in. nigh at a dista.nce of 5 ft from the lower end of the pavement. REFERENCES

.

1. Julia.n Hinds: Side channel spillways: Hydraulic theory, economic factors, and e)cperiment.al determination' of losses, TTansactions, American Society of Civil Enr;ineers, vol. 89, pp. 881-927, 1926. 2. H. Favre: "Contribution o.l'etude des courants liquides" ("Contribution to the Study of Flow of Liquid "), Dunod, Paris, 1933. . . 3. E. Meyer-Peter and Henry Favre: Analysis of Boulder Dam spillways made by Swiss laboratory, Engineering Ne1J)s7Record, vol. 113, no. 17, pp. 520-522, Oct.

25, 1934.

VARIED FLOW

351

4. Thomas R. Camp: Lateral spillway channels, Tmnsactions, Am.ericJenieuTwe~ens, Berlin, nos. 200 and 201, 55 pp., 1917. 14. G. S. Coleman and D(~lilpstcr 'Smith: The discharging c~"pacity of side weirs, lnstil.ution of Civil Engineers, LondO'll, Seleeied Engin~,;ring Papar~, No. G, 1923. 1.5. Philipp Forc·hheim.~r, "Hydraulik" ("Hydraulics"), Teubner Verla.gsgesdlschJ.ft, Leipzig and -Bm'lin, 3d cd., 1930, pp. 406-409. . 16. G. De Marchi: Saggio di teona del funzionamento degli stramazzi lateraJi (Essay ·of the performance of lateral lVelrs), L' Enc,.gia electrica, 11:[ ilano, vol. 11, no. 11, pp. 849-860; November, 1934; reprinted as I stitllto di ldraulica e Coslruzioni ldrauliche, Milano, Mamorie e studi No. 11, 1934.. 17. G. De. Marchi: Profili longitudinuli della superficie lib era delle correnti permanenti lineari con port,a.ta progressivamente crescente 0 progressivumente decrescentf! entro can ali dt sezione costante (Longitudinal flow profiles of linea.r steady flow with increasing discharges or d"creasillg disclIarges in prisma.tic cha.nnels), Ri~erca scieniifica. e ricostruzion.e, Rome, nos. 2 and 3, pp. 202-216, February-March, 1947. Also published as Des formes de la surface libre de courants permanents avec debit progressivement croissant (ItI progressivement decroissr,nt dans Ull canal de section constante, Revue g~ntrale de I' hydraulique, Paris, vol. 13, no. 38, pp. 81-85, 1947. 18. B. Gentilini: Ricerche sperimentali sugli sfioratori longitudinali (Experimental researches on side weirs), L' Energia elctt1:ic~, .Milano, vol. 15, no. 9, pp. 583-595, September, 1938; reprinted as lsti/'4/0 di I draulica t; Costruzioni ldrauliche," :Milano, Memorie e studi No. 65, 1938. .

1

!

1I.

·1

1 I

I!

~-

3;)2

SPATIALLY VARIED FLOW

GRADUALI,Y v ARum FLOW

19. H. Favre: Sur ip.s lois regissant Ie nlouvement des flu ides dans les conduites en , charge avec }l.
J. Orth, E. Chardonnet, and G. 1'I~ynareli: Etude des gri1~es paul' pris~s d'eau cl.u 37. type "en dessous" (Study of bottonl'type W[I,ter-intake gmls), La Houdle blanche, G'renoble 9th Yf .. 110. 3, pp. 343-·351, June, 1954. , . 38. Josef Fr~nk: Hydraulisch~ Untersucbungen. fUr das TiTOler Wehr (Hyc\J'aulic B.nalysis for the Tirol weir), DeT Bm!~llgenieuT, BerliIl, vol. 31, no. 3,pp. gO-lOl] 1956.' i b' I' I ' . 39. P. Wilh. Werner: Wasserspiegelberechullg von Kana~en el g elC lmagSlg?l' Bewegung lind veranderlichcl' Wassennenge (ComputatIOn of .water surface Il1 channels wi.th steady flow aad variable discharge), Dte Bautechmk, Berilll, vol. 1[), 110.23, pp. 251-252, May 30, 1941. ' , . " . 40. G. H. l{euleg[l,u: Spatially variiLble discharge over a slopmg plane, 'lransacltons, Arnorican Geophysical Union, pt. VI, pp. 955-959, 1944. . '. 41. C. }'. Izzard: The surface-pn)file of overllJ.nd flow, Transaclwns, Amenca.n Geo. ' .,. physical Union, pt. VI, pp. (J59-968, 1944. 42. Yuichi Iwagaki; Theory of flow on road Eurface, Jl;ie7ll0lrS of the Faculty OJ EngmcBring, Kyoto Ur..il,ersity, Japan, vol. 1~, no. 3, pp. 139-14:, JulY~ 1951. . , ~,3. Garbis H. '1Celliegan; Determination of critical depth III sp[l,tl.all y varw,b~e flow, Proceedings of the !lid Midwestern. Conference of Fluid Mechamcs, The Ohto State University, ETl.gineeTing Experiment Station, Bulletin 149, Septem.her, 1952, pp. 57-80.

~

>

"d

S l:"'C ~

~ ~

>-I

~

t1

."

)~

>-:l

~

~

l.""l

0

~

._ _ L

. , ----

). ',.

CHAPTER

13

INTRODUCTION

13-1. Characteristics of the Flow. Rapidly varied flow has yery pronouncedcurvaturB of the streamlines. The change in curvature mn.y become so abrupt that the flow profiie isvh'tua.lly broken, resulting ill a state of high turbulence i this is ro.pidly varied flow of discontinuous profile, of which the hydn\ulic jump is an example. rvi.ew of the contrast 'vith-g~aried flow, the following characteristic features of rapidly varied flow should be noted: 1. The CUl'vatUl'e of the flow is so pronounced that the pressure distribution cannot be assumed to be hydrostatic, 2. The rapid varir.tion in flow regime often takes place in' a reln.t.ively short Accordingly, the boundary fric:tion, which would play r. primary role in a gradually varied flow, is comparat.ively small Lmd in most. cases lU~"iS1UU''''''1"" 3, \Vhen rapidly vllJ'ied. fiow occurs in a sudden~transition structure, characteristics of the flow are basict\lly fixed by the boundary of the st.ruct.ure as well as by the state of the flow, When changes in wat.er ~uea occur in l'fI,pidly vD.ried flow I t.he far ""'""Lr.,,·

. _. ___..::;;-

I'

I

O. 'Eieparation zones, eddies, a.nd rollers that may occur in rapidly varied flow tend to complicate the flow pattern and to distort. the actual velocity distribution in the stream. In such cases, the flow is actually confined by one or more se.e..aration zones I:ather than by solid boundari!""" 13-2. Approach--tontheProbiern. Th~th~ that assumes a parallel flow wit.h hydrostatic preSSUl'e distribution is known in classic hydraulics as the Breslie theory;1 such .a theory is used for uniform flow and gradun.lly varied flow. This theory, of course, does not n.pply to rapidly varied floW', even wit.h continuous flow profile, For rapidly varied flow of continuous fiow' profile, cln.ssic hydraulics has shown that a mathematica.l equation of the flow can be established on the hasis of an inviscid- (i.e., frictionless or nODviscous) n.nd potential-flow condition [1-4]. A direct 1 Thill is 50 called because of. the early contribution by Bresse to the solution of the d,rnamic equutioll for gradually varied flow (see Exa.mple lO-4). 357

I 358

INTRODUCTION

RAPIDLY VARIED FLOW

solution of the mathematical equation will require further knowledge of· the CUl'vatUI'C of the flow. In·the classic theorll of Boussinesq [5], the curvature is assumed to illcr'ease ~arly from the channel bed to-.th.e . cllt'ved flow 8Urrace~ and the problem IS sOlved by the moment.11ill prir_ ~-Elt-he theory of Fawer [6], the curvature is assumed to vllry exponentially with the distance from the channel bed to the free surface, and the problem is solved by the energy principle. Modern approaches to the solution of an inviscid potential flow often resort to a graphical method ~r to a ilUmericnl method of approximation. A popular graphical method is Row-!let £malysis [7,8}, which was first suggested by Prasi! [9] and later genel'alizerl BarilIon [10]_ There are many numeTical methods; of these the method of relaxat.ion [11) is hi~quently used. The above-mentioned theories and methods of am,lysis for ra.pidly varied flow with continuolls profile can be found in many hydmulics textbooks an::! in the references listed here. Despite such developments, a satisfact.ory general solution of this type of problem has not yet been obta.ined. Practical hydraulicians, ther.efore, have long ago come to l'ego.rd the various phenomena of mpidly varied flow as a number of isolated cases, each requiring its own specific empirica;l treatment .. In the following chapters, trpicai problems of rapidly varied flow are treated mOl'e or less in this. way. In most cases, the experimental results are to be used empirically. The physical aspects of the flow will be int.el'preted fJ.ualit~f;ivelYJ whenever possible, according to thl') principles of energy, momentum, and geometry, and sometimes by dimensional. analysis.

7. 8. 9.

10.

11.

359

some permanent flows with curV'sd filaments), thesis, Universite de Lausanne, Lausanne, Switzerland, 1937. . . H. Alden Foster; Construction of the fiow net for hydraulic design, Transa.ctions, America,n Soci-.ty of Civit Engineers, vol. 110, pp. 1237-1252, '1945 . E. W. Lane, F. B. Campbell, and W. H.· Price: The flow' net and the electric analogy, Civil Enqinael'in{/, pp. 510-514, October, 1934. Fnmz Prasil: Uber FHlssigkeitshe\\'egungen in RotationshQhll'i!.umen (On fluid motion in rotational vacuums), Schweuei-isc/le Buuzeitung, ZUrich, vol. 41, no. 18, pp. 207,201); no. 21, 233-237; no., 22, 249-251; ·no. 25, 282-283; and flO. 213, 293295, May-June, 1903. ·Also, '''l'echo.ische Hydrodynarnik" ("Technical Hydrodynamics "), Springer-Verlag, Berlin, 1913, p. 61; 2d ed., 1926, pp_ 2!il-:236. E.G. Barillon: Note sur les ravons de courbure intervenant d&ns la construction des reseaux hydrodynan~iqlles (Note on the radii of CUl'V'ature invol~ed in the COIlstruction of hydrodynamic networks), R"uue ge.nerdZe de l'hydrauliqu.e, Paris, vol. 2; no. 8, pp ••111-415, 1936. John S. r.-IcNown, En-Yun Hsu, and Chio.-Shun Yih: Appli~tions of the relaXll.tion .technique in fluid mechanics, TTil1tsQ.c!i01tS, .'imerican Society of Ciliil Enqinee"s, voL 120, pg. 650-669, 1955_

I

.i

,i

J,

REI<'ERENCES

1. Charles Jaeger: "Engineering Fluid Mechanics," translated from the German by P. O. \:\rolf, Bln,ckie & Son, Ltd., London and Glasgow, 1956, pp. 120-130. 2. Philipp Forchheimer:" Hydraulik" (" HydraUlics "), Teubner Yerlagsgesellschaft, Leipzig and Berlin, 3d ed., H)30. Sectioll B2, pp, 230-237, is on Boussinesq's backwater curve with uniform bo·ttom slope,. taking the curva.ture of the flow filaments into consideration; sec. 83, pp. 237-242, is on Boussinesq's backwater curve with varying bot,tom slope, surface with undulating bot.tom. 3. Prangoi:; Serre: Contribution a l'etude des ecoulelllents permanents et variables dans les canaux (Contribution to the study of permanent and nonpermanent flows in channels), La Honine ~:anche, Grenoble, 8th yr.; 110. 3, pp. 374-388,. June-July, 1953; no. 6, pp. 830-872, pecember, 1953. Differential equations for steady flow, taking into account the effects of stream curvature} are developed a.nd a.pplied to the determination of /low profiles. 4. Josef lCozeny: "Hydraulik" ("Hydraulics"), Springer-Verlag, VienllR,. 1953, pp. 46-47 e.nd 229, ,5: J. Boussinesq: sur Ill. thi!orie des. saux courn.ntes (Essay on .the theory of water flow), Mtmoi"es presenMs par di~'er8 savants d l' Academia dss Sciences, Paris,· vol, 23, pp. 1-680, 1877; voL 24, no_ 2, 1878. :6. C. Fawer: :Etude de quelques ecoulements pernHLnents a filets courbea (Study of

-j

~

j.

f

t

1

I CHAPTER

14 (

,i

FLOW OVER SPILLWAYS

whers 0' is the value of y at x = 0; apparently, C' is equal to the vertical distance bet'l-y'een the highest point of the nappe and the elevation of the crest. Eliminating t from the above two equations, dividing each term by the total head H above the cl'est,and expressing the resulting gellentl .equation for the lowel' surJace of the nappe in dimensionless .terms, .

k = A (liY+ B ;

FLOW. OVER SPILLWAYS

14-1. The Sharp-crested Weir. The sharp-crested 'weir is not only a measuring device for open-channel flow but a.lso the simplest form of overHow spillway. The characteristics of flow over a weir were recognized early in hydra.ulics fl.S the basis of design for the round-crested overflow spillway; that is, the profile of the spillway was determined in conformity wi~h the shape of the. lower surface of the flmv nappe over a sharp-crested weu·. The shape of the flow nappe over ashal'p-Cl'ested weir can be interpreted by the principle of the pl'ojectile(Fig. 14-1). According to this Energy liM

!

)

of

principlo, it is assumed that the horizontal velocity co~por,ent of the flO\.y is constant anu that the only fOl'ce acting on tlie nappe is gravity. In time t, a particle of water in the lower surfl1c~ of the nappe will travel a horizontH.l distance ;1;' from the face of the weir, equal to • x

voi cos ( J . .

(14-1)

~"h~reu~ is the velo~ity at the point where Xi = 0, and IJ is the angle of mChno.tlon of the velocity Vo with the horizontnl. In the same time t the particle will t.ravel ~ vertical distance y equal'to ) / .

I

Y = -voi sin !J

+ Yzgt + C'

360

2

(14-2)

+C

(14-3)

where it = gH /21)0 2 C05 2 8, B = - tan 8, and C = c' / H. 8illce the horizontal velocity component is constant, the vertical of the nappe l' m~.y be assumed constant. Adding a term D T / H to the,above equation, the general equation for the upper surface of the nappe is (14-4)

The above nappe equations are quadratic; hence, the nappe surfaces are theoretically parabQlic. . , Numerous tests on nappe over a vertical sharp-crested weir have been made, On the b'asis of data of the U.S. Bureau of Reclamation [1], of Hinds, Creager, and Justin [2,3], and of Ippen [4], Blaisdell [5] has developed the following equations for the constants ill the general nappe equations:

A.

FIG. 14-1. Derivation Of nappp. profiles over S!larp-,jr~stcd weir by the' principle projectile.

351

=

-0425 + (l25

t

B

0.411

1.603

C

0.150

0.45

D

0.57 - O.02(10mp exp (lOrn)

(14-[»)

~ (14-8)

where m h;,/H - 0.208, and h. is the velocity head of the apPl'o,\ch Bow, For high w.;;irs, the velocity of appro[\,ch is relatively small and call be ignored. Thus, the constants become .4. -0.425, B 0.055 , C 0.150, and D = 0.559. Experimental data hays, indicated that th~se equations ~re not valid when x/H is leS8 than ab011t 0,5 and that, for h.IH > 0,2, additional data for verification are needed, For xl}! < 0.5, the preSS1ll'e within the nappe in the vicinity of the weir crest is actually ab9ve atmospheric because qf the convergence of the streamlines. Consequently, forces other tha~ gravity a.t'e acting on ,the nappe, which makes the principle of the Pfojectile invalid. . It should be noted that tJie above theory anq. equations apply ollly if the approach flow is sub critical. For supercritical flow, or F > I, t.he·

1 363

RAPIDLY V AllIED FLOW

FLOW OYER ·SpiLLWAYS

nappe profile becomes essentially a function of the Fl'O(lde number rather than a functiol\ of the boundary geometry as described above [4, p. 633) .. Many experimental formulas for the discharge over sharp-crested weir have been developed.! Most such formulas can be expressed in the general form.2 Q =. CLHl.i (14-9)

nappe, however, usually occurs in overflow spillways and measul'ingweirs. This means a. reduction of pressure beneath the nappe due to the removal of air by the overfalling jet. This reduction of pressure will calise undesirable effects, such as (1) increase in pressure difference on the spillway or weir itself, (2) change in the shape of the nappe for which the spillway· crest is generally rle"igned, (::\) increase in discharge, sometimes ac('.ompanied by fluctuation or pulsation of the nappe, which may be very obi ectionable if the weir or spillway is used fol' measuring purposes, and (4) unstable perfoi'rnance of the hydraulic model. . 01'. the basis of experimental studies on spillways with gates (Fig. 14;-2), Hickox [l1J developed the following equation giving the quantity of air required for aeration in cubic feet per second per foot FIG. 14-2. EXlJerimental of length of weir; setup for studying fi.era.-

where C is the discharge coefficient, L is the effective length of the weir crest, and H is the measured head above the crest, e.tcl1Uling the velocity head. The effective length of the weir may be computed by O.~NH

L = L' -

(14-10) .

where L' is the mea~ured length of the crest and N is the numbet· of contractions. For two end contractions, N = 2. For one end contraction, N == 1. When 110 contractions are present nt the two enus, N = O. According ~o a well-known weir formuia of Rehbock [10], the coefficient C in Eq. (14-9) is approximatety . H

C

=

3.27

+

0.4011:

(14-13)

(14-11)

C

=

(

5.68 1

+ hH)!.6

tion below the n2.ppe. Dashed lines indica.te the condition of fun aeration. (A/tel' G. H. Hickox [Ill.)

where H is the mea.~ured head in ftover the top of the gate ; p is the reduction of peessure in feet of water to be maintained beneaththe nappe; and C is a coefficient depending on the ratio of the discharge beneath the gate to the discharge over . the top of the gate. The ratio is represented by a dimensionless value

",here h is the height of weir. IVIeasurements by Rouse [4, p: 532] indicate that this equation holds up to Hlh = 5 but can be extended to H Ih = 10 with f~Lil' approximation. For If Ih greater than about 15, the weir becomes a sill, and the discharge is controlled by a critical section immediately upstream from the sill. The critical depth of the section is approxim.ately equal to H + h. By· the critical depth-discharge r,~la­ tiol1snip, it oan be shown that the coeffir.ient C is

(14-14) where y is the height of the opening below the gate in it and H" is the head on t.he center of t.he gate in ft. for 1.1ngated weir or spillway, (f = O. The va.lues of C are as follows: .

(14-12)

The transition between weir and sill (betw~en Hlh = 10 and 15), however, has not yet been clearly defined. . Experiments have shown that the coefficient C' in Bq. (14-9) remains approximately constant for sharp-Grested well'S under varying heads if the nappe is' aerated. 14-2. Aeration of the Nappe. In the preceding article the ovedalling nappe is considered aerated; that, is, the tIpper and lower napp'e surfaces are subject to full atmospheric pressure. Insufficient aeliationbelow the

"

I

.-.-c-/

0

0.077

I

0.5 0.135

\

1 0

'-1-.5-1 ..~1 2.5+

0.175.[0.202

I 0.220

0.225

Intermediate values can be interpola.ted from a cUl've constructed with the above va.lues. . 14-3. Crest Shape of Overflow Spillways. Early crest shapes were usually based On :i simple parabola designed to fi~ thE; trajectory or Hie falling nappe, as already described in Art. 14-1. From 1886 .to 1888 Bazin [12] made the first comprehensive lahoni.tory investigation of nappe shapes. The,use of Bazin's data in design will produce a crest shape that coincides with the lower surface of an aerated nappe over a shmp:-crested weir. Theoretically, the adoption of such a profile, generally kilown as the Bazin profile, should cause no negative pressures on the crest. Under

IFor a genera) description of shnrp-crested-weir experiments and formulas, see [6J. For further studies. of the discha.rge characteristics of sha.rp-crested weirs, see [71 a.nd [81. 2 Th.e derivation of Il theoretical weir-discha.rge formula. ca.n tie found in ma.ny hydra.~lics textbooks. The first mathematica.! analysis on discharge of weirs was performed by Boussinesq [9].

..

I

!

\

.I

)

.\

364 (

\ 1

I

I

FLOW OVER SPILLWAYS

RAPIDLY VARIED FLOW

actual conditions, however, there exists friction due to roughnes.'!! on the. surfi1ce of the I;;pillw2,Y. Hence, negative on such a profile cannot be avoided. The presence of negative pressures will lead to danger of cavit<1tion dama,ge. In selecting a suitable profile, avoidance of negativ6 pressureS should be considered an object-ive, along with such other fadors as maximum hydl'aulic efficiency, pr~.cticability, stability, n.nd economy. ConsequeJ1'~ly, the Bazin profile hus been variously modified, and many o~her profiles for design purposes have been proposed. l From 1932 to 1948, ext.ensive experiments on the shape of the nappe over a sharp-crested weir wete conducted by the U.S. Bureau of Recl:1mation. 2 On the basis of experimental data including B".zill'S, the Bureau hn.s developed coordinates of nappe surfaces for vertical and various slopefaced weirs. This information is indeed invaluable for accurate an[\,lys(3 and precise of SpillWity overflow sections .. For practical purposes, how~ver, t,his infol'llll.lGion can bo usedrnore simply without essential loss of ace\U'acy, thrQugh modifications that wiil be described below: On the basis of the Bureau of Reclama.tion data, the U.S. Army Corps of Engineers hn.s rleveloped sever<11 sta.ndal'd shapes at its Watel'ways Experiment Station. Such . 14-3), designated as the WES lltandal'd spillway shapes,3 can be by the following eq llntion: y (14-15)

365

For intermediate slopes, approximate va-lues of I( and n may be obt~ined by plotting the ,11bove values the correspqn~ing slopes a~ld ,mterpolating from the plot the required values for any given slope wlthm the plotted range. . . . The upstream face of the spillway crest may sometllnes be deslgned to Eet back, as shown by the dnshed lines in The shape of the crest

where X and Yare coordinatei.l of the crest profile with the origin ::It t,he highest point of the crest, He is the design head e:cch~din(J the velocity head of the [\,pproach flow, and IC and n nre parmneters depending 011 the slope of the upstream face. The values of J( [,nd n fl.re given as follows: Slope of ups/rem" fave

K

VerticaL ............ 2.000 3· on 1.. . . . . . . . . . . .. 1.936 3011:2 ...........•.. 1.939 3 on :1. . . . . . . . . . . . .. 1. 873

n 1.850 1. 836 1.810 1.776

~ EKamples of othel' well-known pmfile;; a;:8 (1) the CreG.ge,· profile [13) developed from a mathematica.l extension of Ba.iH's data, (2) the modified CnilJ.gcr profile [21 based on t"e U.S. Bmeau of Reclaniatioll dfJ,tn from Di!m-er tests [II, (3) the LClnBDavill projile [l4J Q!lsed on the U.S. Bure!L\\ of Reciall1aLicn data from Fort Collins tesLs'[l] and the datn of Bazin [12] ,l,nd Scimel11i [15], t4) theSci·,.i;emi profile l151, (5) the SmetmtCl Ill)], (!l) the J)¢ Marchi profile [17J, and (7) ~he Escande profile. [18] .. For" good of various well-known profiles, see \19]. • See. [11. The main project WIlS sta.rted in 1936, nnd the test.s were conducted in the Bureau's Hydl·ll.ulic Laboratory in Denver, Colo. Earliee experiments conducted by the Burea.u were performed in H)32 at the Colorado Agricultural Experbllent Station,' Fort Collins, Colo. The former are usually referred to as the Den.ver tests, the btter as the Port Collins tests. 'From [20],· Hydraulic Design Charts: Ill-I, WES 4-1-52; and· 111-7 to 111-9, WES 2-54. The development of the WEB standard shapes is described. in [21].

FIG. 14-3 .. The WES-standard spillway shapes.

(U.S. A"my Engineel's Wa;~e1"lDa.ys

Experiment Slatioll.)

will not be affected materially by such details, provided the modification with at least one-ha.lf the total head H. vertically below the origin of coordinates. This is because the vertical velocities are srrmll below depth and the con:esponding effect on nfl.ppe profile. is negligible. Discharge of the Overflow Spillway. The dl~charge over a spilllVay can be computed by an e~ttation in the fo:'m ~f Eq. (14-9), Ilor spillways designed for the WES shapes, the equatlOn IS . (14-16)

366

RAPIDLY VA.RIED FLOW

FLOW OVER SPILLWAYS

whel:e H. is the total energy head on the crest in ft, 'inclUding the velCl()ity head in the "pproach chani1el. Model tests of the spillways have shown that the effect of the approach velocity is negligible whim the height II of tho spillway is greater than 1.33H,!, where Hd is the design head excluding the approach velocity head. Undel' this condition and with the design

....... 0

1.04

1.3

1.03

1.2

1.02

lot

367

to show the effect of the approach velocity on· the relationship between

H.!~ld and C/Cd for spi.lIways d€;si~l1ed for WES shapes having vertical upstrea,m face. For spIllways havmg sloping upstream face, the value of a can be c.arre.cted approximately for the effect of the upstream-face slope ~y m?-itlplymg C ~y a correction factor obtained from the attached chart 111 !lg. 14~4. ThIS correction was developed from the Bureau of ReclamatIOn da.ta flJ.

I

Hd " 17.7'

u

EJ.

.E

983.:.3i~~,.;..;....."._

r

r

c: 1.01 .o:! of-

3.54 8.95'

v

::'

'.6S

15 1.00

23

u

r.... Axis of crest

0.99

/ 0.98 0.2

0.4

O.S

1.0

....

Point of to;;;e.1CY

1.2

X=35.9' Y=:3-2.7'

0.6 Ha~veloclty

head

Hdd design head e~clucinil Ha H.: total head in~ll.dlng Ha

0.5 Ha

1 --

HdI_

0.4

.

E1. I

0.\ 1---1--:7""'1

a

0.70

FIG. 14-5. Design of an overflow-sp.illway section: 0 •.80

0,85

0.90

0.95

1.00

C/Cd in which Cd 4,03

FIQ. 14-4: Hllad-discharge relation for WES-etandard spillway shapes.

a~x:am~le 14-1: Detcr~ine the crest elevat.ion and the shape of an overJlow-spill. l~ .sect!O? havmg a. vertloal upstren.m face and a cres~ length of 250 ft The d ; discharge 15 75 000 cfs Th t . , . as.gn 1 000 0 d h' . e ups ream water surface at design discharge is at El . . .an t 0 avera~e channel floor is at EI. 880.0 (Fig. 14-~) . , Sol,l!MIl. h' I · fl'l1 0 . r 1'bl8 •;\sSltffiing d a 19l Over ow BpI way, the effect of approach vclocity is n(:goalgx 2"Oa)n ~ d = 4,03. By the discharge equation, H.u = QICL = 75 0001 . 0 = /4,1 and H, = 17.8 f t . ' ) The approach velocity is V = 750001(250 ' 20) . i I ' t h d' H' a. I .,' X ~ = 2.5 fps, and the COrrespondng :e OOlJ as. IS a ':" 2.5'/217 = 0.1 ft. Thus, .the design head is Hd _ 17.8 0.1 - 17. ( it] and the height of the dam is h "" 120 17.7 = 1023 ft Th' h' ht' IS greater thai 1 33H d' . . IS eIg , ' ' I I . . d! an , ll~nce, tIle. effect of approach velocity is negligible. ' Th e ares!; e_evatlO n ls.at 1000.0 .~ 17.7 "= 982.3. . B . . y Eq. (14 15), tbe C,Est shape IS expressed by}' X 1•• s/23 C d' .~ f tl shupe t d b hi . . oor Ina""s 0 1e compu e y T, S equatIon tHe plotted as shown in Fig. 14-5.. The crest shupe

a

. head (that is, h/Hd greater thanL33 and H, Hd, for the approach .velocity head is negligible) the coeffidentof discharge C has been found to be Cd. = 4.03. In low spillways with hiEd < 1.33, the approach velocity will have appreciable effect upon the discharge or the discharge coefficient and, consequently, upon the 1U1PP';) profile.' A dimensionless plot (Fig. 14-4) based on the data of the Waterways Experiment Stl.).tion [20] cn,n be used

I

!

~

I

.

, .

FLOW OVER SPILLWAYS

upstrCftm from the orir~in of ~!le coordinates is constructed accordin~ to the d.imensiorus recommended in the chart of Fig. 14-3. The design of the straight portion ,0£ th~ spillway sudacebdow the crest section depends ~n the stability requireme;ut and 011 the features of the sti;Iing basin at the toe of the spl11w!l.Y. Aslope of 0.0: lis assumed for the straight pOl·tioll of the downstream spillway surface.

head is greater thp.ll about twice the height oLthe spillway above the upstream bed.. Such spillways are often found in earth dams. In using the Br:dley curve, it is necesilary to' know the coofficient of discharge for the design head. If this coefficient is unknowl1, but the spillway shape 18 given, 11 method 1 suggested by Buehlel' (23] may be used. B'T this method,' based on an equation derived by BrudeneU the c~e:ffici6nt of discharge is computed by the ~q'.lation

14-0. Rating of Overflow Spillways. ThE:; profile of an overflow way can be designed fol' one head only. This head is the' design head, which genel'ally procbuces s. lower nappe of flow that 8,grees closely with . the spillway profile. The spillway, however, must a,lso operate under other heads, either lower or higher than the design head. For lower heads the pressure on the crest will be. above atmosph~ric but still les:i! th.anl~ydl'ostat.ic. For higher heads, on the other hand, the pl'essu~e w~n be lower than atmospheric, and it may become so low that separatlOn ~lr flow will occur. Model experiments indicate, however, that the design head may be sufely exceeded by at least 50%; b~yond this, harmful cavi1.B I tation may develop [4, p. 535J . ,6 For spillways designed for WES shapes, the curves given in Fig. 14-4 I can be used to deten:nine the coeffiI I cient, of dischai'ge for heads otherthah , the design head. ThllS, a rating , !. curve of the spillway can be 0.8 I, .--1----17:~-~l-- cbll1puted. 1"0;: spillwo.ys designed for other '" shapes, Bradley [22] has developed ~, ''"" universal curve (Fig. 14-6) showing the relationship between FI ,/H D p.nd 0.2~~~1 C/C D • The term HD is the design I I 015 0.80 O_BS 0.90 Q.95 1.·)0 !.O~ 1.1'; head including the approach velocity. Volue of Ihe 'Olio clc o head, and CD is the corresponding FlG. 14-6, Coeffident of discharge of . coefficient of dil:charge. The term overflow spillways for other than the He is the total head other thaD. the design head: (t1fl~r J. N. Br,~dley . d (22].) , design head, and C i~ ·the corre~pon . ing discharge coefficient this curve i~ well supported by tests of some 50 overflow spillway crests .of variot~s shapes and operating conditions. It can be used to coltlputeapproxlthe'mting Clll'ves of most overflow spillways. The dashe.d curve'in Fig .. 14-6, supported by date, from 29 exIsting spillways, applies to spillwttys with overfal! suppressed. This is rile type ~f spillway the fiow over which is affected by downstream cha~mel conditions' it odcurs when the'downst:ream slope i~ flat and the approach depth is shdllow. The spillway in such a hase is usually so low that the design ::I

-t O,5,~

~/ OAI--~:':-:I~I 73' •

I\ f

I

'

369

RAPIDLY VARIED FLOW

368

.

. (He)V'12

(H-17)

C = 3.97 JJ;,;

where H, is all operating head and HDis the theoretical design head, including tho approach velocity hea-d, for a standard profile havh1g a

.S

Vi

:;: u

a;

-!'. 3<

o

ru

.0

.,

'"co

-;;

'6

40~---4-~-4-----i-~4-~-4----1

HodlOplOi distance from the

cr~st

in U

FiG. 14-7. Stand~rd spillwa.y :3hapes for different valuesof H D •

(After B. Buehler [23J.)

vertical upstl'eam face. it should be noted that II D is the theoretical design head of the standard profile for which the Brudenell eqtmtion was developed; therefore, it may not be equal to the actual design head used by Bradley or defined for other profiles. The valtte of HD may be obtaiu.ed from a chart (Fig. 14-7) showing the standard profiles. The profile of the given spillway as.desiglled or built is firfit drawn on a piece of transparent paper on the same sc:ale as the sto.nd7 ard profiles. This paper lS laid over the chart; then the value of Ii D associated with the standard profile that gives the best fit is selected. F01' a. given profile, the H D values that give the best fit may be different on the upstream and downstream sides. The higher of the two indicated v.alues of lIo $ould be used. FQrexample, the profile of a spillway is shown by the dashed line on the chart.' The higher value of HD is approximately, 45 ft, 0!l the upstream side, of the crest. , 1 A method for the sawe purpose is also proposed by Bra.dley [22]; this Inechod DJs , the giveri profile to existi'ng profiles of known dischll.Cge coefficient.

1

370

RAPIDLY VARIED FLOW

Using the chart and Eq. (14-17), the rating curve of. a spillway of given profile can be cumputed; , According to comparisons with actu,al model tests, both the Bradley and Buehler methods have been shown ~,o give highly'accurate at all but very low heads. At low heads of 5 ft or so, the computed discharge may be as low as 8% below the actual value. For pra.ctical purpOSGS, however, both methods fire sufficiently accurate. 14-13. Upper Nappe Profile of Flow over Spillways. The shape of upper nappe profile of flow over 8 spillway crest is significant in the desigu of spillway abutment Ivalls and for the selection of pivot elevation ot, tainter The WES shapes for high overflow spilhlrays with vertical by the U.S. upstream face have been investigated, using model Army Engineers Waterways Experimant Sta.tiOll. Figure 14-8 shows the shapes and coordinates X r;,nd Y of the upper nappe profile obtained from such tests for negligible approach velocity, fo], conditions with and without piers, and for three different head ratios. The tenn Hd is the design he!'.d excluding t.he velooity head, and H is the operating head other tha:n the design head, also excluding the velocity head. Profiles for intermediate head mtios may be interpolated .. Owing to thecontrac~ion effect of piers, !J. pronounced hump between X/Hu. -0.5 and 0 occurs all the upper.nappe profile along piers when the discharge is high .. Upper nappe profiles fill' three gate bays adjacent to abutments are in Fig. , 14-9, showing the abutment effects on the nappe profiles. It should be noted that the upper nappe surface is exposed to the atmosphere and,hence, subject to alteration due to wind and air CUlTents and the absorption of surroupding air. As s. result, the flow is aerated and the surface becomes wavy and unstable. The upper-nappe-shape only the ideal cases, coordinates given in Figs. 14-8 and 14-9 where air plays little or no part. The upper nappe surface for slt'ping upstream face should have a lower elevation than that for vertic.'1.1 .upstream face. Hence, the given coordinates may also be 1.lsed safely for spillways with sloping upstream face for which the actual data are not yet available. 14-7. Effect of Piers in Gated Spillways. Piers are needed to form the sides of the gates in gated spillways. The effect of the piers is to contract the flow and, hence, to alter the effective crest length of the lSpillways. The effective length of OM bay ,of a ga.ted spillway may be expressed as ' L = Lo - J(NH. (14-18)

1 COO.ft'O!NATE'~ Fan. U1>?ER

H IH;

,

X Ill.

--=!.O

1

N A.PPln

W'J,:Tn

No Pume •.

'0.50 11:1 IRa - 1.00 : HIll. c 1.33 }'iil, ,X/H.', Y/ X/lid Y/H.

1- .4!lol-=t:or::. o.

'

1:210

0 -1. 0 -0.484:,1-0.81-0.9115 -0.8 -1.185 i-OA75 -0.6 -Q.893 -O.G -1.151 '1-0.460 -0.4 -0.865 -0.4 -1.UO :-0.425: -O.21-0.S21i -0.2, ,1,,-'1.01;0 :-0. 3n 0.°,-0.7551 0.0:.,1-1.000 '-0.300, 0.2, -0.681. 0.2 '-0.019 -0.2001 0.'1,-0,580, 0.41-0.821 1-0.075 0,61-0,465; 0.6 -0.71)5 0.015, 0.81-0.320[ 0.81",-0.569 1.0 0.258, LO -0.145 l.O'-OAlI 1.2 0.470 1.2 0.055, 1. 2 0 >220 1.4" 0.7051 0.29{j 1.4 :"'0,002 1.6, 0.972. 1.6 0 ..;!l3, 1.0 0.240 1.8 i 1. 269[ L~~!~l:~ 0.5$1

-O.ll .-0,0 -0.4 -0.2 0.0 0.2 0.4 0.6

l

o,s,'

l

'I

1.-

1"1

• Baaed on CW 801 t •• 1rs for negligible v.looioy .

of a-pprnne h.

COO1lDINATEB FOR UpP£n NAPPE A~ CENTER Ll,i'iE OF BAY W1Tii 'tYPE 2 I'lER5~

-1.0 -1.230 ,-O,g -.. 1.215 ' -0,6 -1.194 -0.4 -1.15'<; -0.2 i-I. 122 0.0. -1.071 0.2 i-1.015 0.4 ;-0.944 0,6[-0.847

-1

I

~:~ I,

3Sfj o 1.4 -0.102 1.2'r. 1.6 0.172 _ _.......:.._ _...L..._---'-_ _-L_I~.8

'* BUild on C\Y 801

0.465

test::. f?r negligible velod~y

of a.PJ;n"'ou.ch.

COORDlNATl!::S FOR Ul'PISE NAPPE Al.QNO PlERS.

-1

-0.6 -I 209 -0.4 1.218 -0.2 -1.244 0.0 -1.103 0g5O 0.4 -0.821 0.6 -0.689 0.8 -0549 LO -0.SS9 1.2 -0.215 1.4 0.011 0.208

0.21-

>

where Lo is the clear span of the gate bay between piers; K is the pier contraction coefficient; N is the number of side contractions l equal to .2 for 1 These tests are d~signa.ted as II General Spillway Tests-OW 801." The information given here is from [201, Hydraulic Design Charts 111-11 to 111-15 WES 9-54.

371

FLOW OVER SPILL'\.I'AYS

FIG. 14..8. Upper. n!\ppe profiles of flow over WEB spillways with a.nd without piers. (U.S. Army En[f
,

.J

-i.5

-10

.., -0.5

"x

)::-

a 0.51------+--++..

1.0

I Left abutment

'-11---I Bay No" 1

LEGEND

I

~===:::J~0.205 Hd Boy No.2 l.078 rid

.

,"

J.

- - - Left side of bonk center line of b\"y Right side of boy Profiles ore bosed on pine flot model tests -"'«""'-

i Soy No.5

-i2=~3--t---- axis OVERFLOW "SPILLWAY CREST

UPPER NAPPE PROFILES 1.0

1.5

2.0

APPROACH CHANNEL AND ABUTMENT EFFECTS

FIG. ]4-9. Upper nappe profiles of flow over WES spillways showing appro!l.ch channel and abutment effects.

(U.S. Army

Engineers WaterWIlYs Experiment 8to.lio:l1..)

x'C ..

o.5

'-

1 . "

r

0.5 BAY" NO. 2

1.0 L _ - L_ _L.....L_·L._ _" _ _-'-_---'J._---'" 1.5

-1.0

-0.5

0 " 0.5 X/lid

LO

1.5

2.0

LEGEND" Left side of bonk

- - Center line atbay Right side (>f boy on pinE>

0.5

f---+---~f-

oxis OVERFLOWSPI"LLWAY CREST

i.a L..-:..-L_ _L...l._..-:.._..-:..L.;..._....l...._->.J_----' -i.5

-LO

-0.5

a

0.5 X/Hd

i.0

1.5

2.0

Fie. 14-9 (continued).

UPPER NAPPE PROFILES APPROACH CHANNEL AND ABUTMENT EFFECTS

374

RAPIDLY VARIED FLOW

each gate bay; and H. is the total head on the crest including the velocity head. In comrl1lting the discharge through gated spillways,the effective length determined by the above equation should be l.lsed. The discharge cpefficient, however, is assumed the same in both gated and ungated spillways. The pier contraction coefficient varies mainly with the shape and position of the pier nose, the head condition, the approa.ch depth of flow, and the operation of the adja.cent gates. The approxim~,te J( value given by Creager and Justin [3, p. 120] r1lnges from 0.1 for thick, blunt noses to 0.04 for thin or pointed noses and is 0.035 for round noses. These values apply to piers having a thickness equal to about one-third th8 head on, the crest when all gates n,re open, When one gate is open n.nd the adjacent gates are closed, these values become roughly 2.5 times larger. The U.S. ,Army Engineers Wat.erways Experiment Station has conducted tests (General Spillway Test cW 801 [20]) on different forms of, pier nose for spill ways of the liVES sha.pe. On the basis of these tests, a round-nose pier is recommer.ded for general use with high heads. The [(value for the round-nose pier plotted against the ratio of He/ Hd with variable distances upstream from the crest is shown in Fig. 14-10. The effect of other nose shapes 011 the contraction coefficient is shown in Fig. 14-11. The he.ight of the test spillways was 6.67Hd, which had negligible velocity of !l.pproach. Under the testing conditions, these da.ta are applicable to high spillways and to the condition that ,the adjacent gates are open. For low spillways with appreciable approach velocity, the pier'contr[Lction coefficient fOl; the round-nose pier with various approach depths is shown in Fig. 14-12. III .. he abs~nce of adequate data, pier contraction coefficients for other nose shapes for low spillways may be obtained by proportioning from the data for high SpillW8,yS (i.e., from Fig. 14-11). 14-8. Pressure on .Overflow Spillways. If the spillway profile is designed exactly in the shape of the lower nappe of a free overflow, the pressure on the spillway crest under the design head should be theoretically nil. For practical reasons, however, such an ideal profile is generally modified so that low pressures will develop under the design head. As the spillway must be operated under heads other than the design head, the pressure will increase under the lower heads and decrease under the higher heads. Assuming a two-dimensional irrotational flow, the pressure on the spillway crest may be accurately determinedanalyticnny by a numerical method, 'graphically by flow-net analysis, or instrumentally by an electronic analogy.l More exact determination of the pressure, however, will depend upon model tests. See references given in Art. 13c2, A pra.ctical procedure for the r!l\axation method appFed to the problem under cDnsideration is given in [25]. ' I

(/)tJ>

f-f-

(/)z

wl!! a:u

(.J

I~

u:: z

3"-

g~

,

~

:r:

W

--1

"-u a:: a:z W wo Q. >ofQ

u

LI..

« ,0

we::

f-f-

I-

«z u e>O U

W

LI.. LI..

<\l

'E.,.. <\l

H

Il:l

'" ... "
:;"

tl ~

t

...."'"

§E5 w i!'.

;r-

n.

~

...~ '

~

0

:::; ~

I,

t I ,i

,

I

I

,I

J

i

I

I

t

~~.~~~~J-J-J-~~~~~~.-L-L-L~O,,-L-L~ ~o~~-L~-L-L-L,~o3 i

!I ,\ .

PH/'H 'poal{ uo!sap o. ~saJJ uo poa~ JO O!'O!,l

I

375

"0

Z :r:.~

-0

.,

Type I

0

Type"

.c: C

.!?

.,'"

1:1

.

J

E ~

~ <>

~

~I

c

""

0

1

"0

''""

.J:.

'0 0

"15 tr

'PIER NOSE SHAPES

Note: Pier nos~ loi:ct~d in somo plone as upSt"eOr.1 face of spillway

HIGH GATED OVERFLOW C"lESTS PIF.R CONTRACTION COEFFICIENTS

Coefficienl of pIer contraction, K

EFFECT OF NOSE SHAPE FIG. 14-11. Coefficient of contrMtion for piers of various nose shapes in high d:l.ms with the 'nose located in the same vertical plane as the WES 4-1-53,) face of the \VES spillway. (U,S. tinny Engineers, Wn.(erwa.yli ExpeMlrtCnl8talian ['2(lj, Design Chart ) 11-:;, ,

ups~ream

Hydt(~u1ic

----.---..~

1,4

1.2-

'5
Hd ~ design head

.J:.

c

.ffi' ~

"

:L

~ 1.0

i

:L

.t:f 0

"'c:

.J:.

'"

"iii

'"

O.S

.2 1ii <.<>

-1

~

u

c:

0,6:

-1.5 r.'.

Il:.

-2,0,

'"

.J:.

0.4

~

r",-" ,

-a,5L-LL

0

0

.0,5 Hd rod. 0,2 Hd rod

~ ,g

0

'0

'0

l' "~''"T'~'

\1\

0

i5 0.5 u '~ -1.0

-0

-:j

0.0

\-1

Axis of

'reO

Ij~

qt=:r Type 2 pier

I

I I ! \ I I I ! O. 0,5 1.0 \.5 2..0 Ratio of horizontal distonce to design head

-'

-1.5 -\.O -O,el

a::

0.2 LOW GATED QGEE CRESTS PIER CONTRACTION COE.FFICIENTS

EFfECT OF APPROACH DEPTH -0.05 ,

0 0.05. 0.10 Coefficient of pier contractIon, K

, • FIG., 14-12. Coefficient of contractIOn for t'] Ie TOUD d - Dose piel' in lbw dams.

[201,

H!ldra1~lic

Design Chart 122-2, WES 4-1-53.)

(u.s. A,rm!l EngineeT8 Waterways Experiment S!atiot~

.. "-.-,-.-

j 378

0.4

0.3

1\

\

r\V

.",1r 0.2 t '" 0.1 ~ 1~

j!i -o.~

"- ~

r<-

-0.3

,.0-..

r V

-0.4 I

!.I I

');.

hJ

-0.2

-",

~

-K

V

i

~:~

\

--+I

f

I

--

I ,."..

i". ,./

r-.- ;-- r--- -

l

c-:- -I~

·t-

-\

(

~----------1~T-YP-e-2--Pi-er-{-F-i9-.1-4-_-ln-----1-·-------J+.-

.\

-, I

-0.2

a

0.2

0.4

0.6

o.s (-1S....)

1.0

1.2

Horlzonto! distonce Dasign head \Hd

i

'-1-- 1---

-

~ »-.

J

i'.,

\

/

1.4

....;-'"

H!Hd"1.33

KIKa" 0.50

H/Hd· I .33 r---."

"./

-0.2

l--

r----., >-...<.;

H/Hd ;'.00

\

0

iO

/v

""'"

1'8

'" ~Ii~ w .", 01

-b~50

!

.... r-'<>H!Hd·i.QO

-0.2

-0.5

-t·

,

l

\ \'''' '\ 0.2

M

~

0.4

--

I

,

o

0.2

_-t--__

0.6 o.S Horizontal dislance (X) Design head Hd 0.4

.\

0.6

The pressure distributions on a spillway crest with and \vithout piel's under three different head ratios, based on OW 801 tests uf WES shapes [20], are shown in dimensionless plots (Fig. 14-13). Pressures for· intermediate head ratios can be obtained by interpolation .. 0.5

379

FLOW OVER i:lPILLWAYS

RAPIDLY VARIED FLOW

1.0

12

I

l.4

.

R~0.2 Hd

~

I

R· 0.5 Hd

., . ,

i 1'"\ 1

-\

I."

l' ~ ~

I~

-\

IV

I Net.: Dota tJosed on CW 60! tests

i'~

FIG. 1'1-13. Cres~ pressur9l! on WES high overflow spillways (continu.ed). (b) Along center line of pier bay. (U.S. Army 1!l11.glneers Waterways Rxperiment Station [20], HydrClulic Desiqn Chart 111-16/1; WES 3-55.) . .

)5

!

wiD result in ovel'design of the spilhvay, particularly for high dams ..' Suc.h overdesign is an acceptcd procedure, providing an additional factor of safety.! . . The pr'essme teduction on the upstream face of a vertical weir has been determined both theoretically and experimentally by Harris [26). On the basis of OW 801. tests for ungatad WES-shape crests of vertical upstream face and of U.S. Bureau of Reclamation tests of pressure on a

Nole: Dala based on CW 801 !esls

14-13. Crest pressures on WES high 'Jverflow spillways. Ca) No piers. (U.S. Army Engineers H'ale)'u'ays Expe"imenl Slation [20], Hyd1'rJ.ulic Design Chart 111-16, WES 9-54.)

FIG.

Because of the conversion of static to kinetic. energy as the water flows. over the spilhvay, the hydrostatic pressure on the upstream face of the spillway crest is actually red\lced. This reduction in pressure is not appreciabl'e, but, where the moment arm is lon·g; as in high: dams, the effect on stability· may be worth considering. The usual method of analysis by assuming straight-linepressllre distribution nea~ the crest.

is

1 Actually, the effect on structural stability due to this pressure reduction COIDpensated to a large extent by the moment of the horizonto.l component of the nappe pressure on the crest surfaqe.

I

.. " '. - ..-,;

.-.~.'

.,'

~,' ," .','

.',

..

-.I I

[

380

FLOW OVER SPILLWAYS

RAPIDLY VARIED FLOW

381

shul'p-crestecl wdI' under the design head, the resultant of the reduced pr~~sure is f?~lnd t~ be approximately 12,9Hd 2 lb per unit length of the spIlLway, actmg hOrIzontally at a distance of O.161Hd it below the top of the crest. . . ,

I

Uj

1:)

'" Ol

'" 0

~12

.0.

'"a:i'"

" c

li:./JJ '" v;

~

-0.'2

104

I ,f

Rotlo, Hr"

!

3.2

,L~

3.4

3.5

Dischorge coefficient, C

FIe,;. 14-14.Cufl'es for determination of tIle discharge coefficient of drum gates. (After J. N. Bradley [27].) Note: dota based on CW . 801

F.!G: 14-13. Crest pressur.es on WES high ovel'fi~\\' spiHwo.ys (co/!t"'~lled). (c) Along pletS. (U.S. Al'Il'Y Engmeers lVate1'7l1aYs Expel'tmellt Sin/ion {20] HI'dl'ClL:iic Dcsif!1l Chm"t 111-16/2, WES 3-55.) . ' U

I

.14-9. Drum Gates. The drum gate is a hinged g!1.te which floats in a chamber and is buoyed into position by regulating the water level in the chamber. Primarily, it is llsed to control the surface eiev!ltibn of the water upstream. As a measllring device, the drum resembles a weir

with a curved upstream fa(;e over the major portion of its triLvel. The rmgle e U;'ig. 14-14) is formed between. the horizontal and a .line drawn tangent t.o the downstream lip of the. g!\te. This angle is considered posiwhen meastive when it is measured above the horizontnl ilnd ured below the horiz.ontaL For positive values of 8, the gate acts as a sharp-crested weir, the control poil'l.t i.~ the downstrea.m lip of the gate, and the head is measured o.bove this point.· For negative values of 9, the gate !lcts as ~ curved-crested weir, the control point is the highest· point of the gate surface, and the head is measured above this po}nt.

382

RAPIDLY VARIED FLOW FLOW OVER SPILLWA·YS

Since the d!'um gate acts as a weir, the discharge through the gate may be expressed as Q = CLH.!.6 (14-16) where C is the coefficient of discharge, L is the length of the gate, and H. is the total head. Laboratory investigations have shown that the flow' ever this type of gate call be completely defined by H., &, C, the radius r of the gate, and the depth of approach. The dept,h of approach. however, has very little influence on the flow behavior when the appro!',ch depth, measured below the highest point of the gate, is equal to or greater than twice the head on the gate.' This condition is well satisfied by most drum.:gate installations, especially when the gate is in a raised llositiou, Therefore, the coefficient C may be considered to be a function of He, fJ, and 1'. . Bradley [27J has made a comprehensive stu.dy of the drum gate, using data obtained from 40 hydraulic models of existing drum-gate structures , of various sizes and scales. . The results of this study are showl1 by a family of curves (Fig; 14-14) where C is plotted against fJ with the r,atio H.lr as a parameter. When H.lr .0, t~e ~ate becon:es a straIg~t inclined weil~, and the cOI'l'espollding dashed Ime III the family of curves IS based on Bazin's data [12J. The curves extend dowmvard to f) =;0 -15°. The discharge coefficients in the regi011 between.6 = -150 and the gate cOl1lpl~tely down clln be obtained by graphical i.ntetpolation of theratiI~g curves of the gate, The comput.ation of the ratmg curve when the gate IS completely down is the snme as that for a spillway with an ungated crest (Art: 14-5). . 14-10. Flow at the Toe of Overflow Spillways. The theoretIcal velocity of flow at the toe of an overflow spillway (Fig. 14-15) may be compllted by .

=--=:---:---=--

V2g(Z,

+ H" -

Yl)

I

600~ 560

I

II

!

L

I

1

440

,II I

.----:r-

I

I

I'

,I

II

360

i'

-",,,l

'l

240

1

--I--

f

II

!

/1

I

tnlr--4.-.-17 I ! if

"

11 -01

:I:

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I

I If I/, !I / Y//

!

XI

12C :---

II

f

80 40

.. I-10

20

30

~ j' I 40 50

,

I---

-~.-

-

--

T

~

z

!

.

I ~

1 i

---1

60

-

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I.

-

II I I 17 / ; W'

I

II J V IV I

N!...:......c.tn!+ ~'--.>L~.

~r-

8f

I 1/ I /1 I , / '/ / IiI/ I /

'''ll~_ ... ' ~1: :r: I

lit 1."1, :.;,,-,:1: 1

.~,

1 '/

NL1/-;

.;::

II

~I:q

..: J.---.

:>n()

0 0

I I

# :'7.1/!i

-

280'

Ii

i I

!

!

I

:

. o,r d- if

JI

i ,I

il

i!

ow..fJ

I

II

II I

I I

. I! ~ l-+-.~ r--i Tr I ;tL ~R,r I--I ~I x .:cr:- i -

T;----it-

i!

.a()n

I

!

J I

-""

f

I

'i

520'

(14-19)

where Z is the fall, or vertical distance in ft from the upstnam reservoir level to the floor at the toe; H .. is the upstream approach velocity head; and Yl is the depth of flow at the toe. Owif).g to the energy 10s8 involved in the flow over the spillway, the actual velocity is always less than the theoretical value, The magnitUde of the actual velocity depends mainly on the head on the spillway crest, the fall, the slope of the spillway surface, and the spillway-surface rouglll1css. 1 By reasoning ~d experi~ellts it is sho\"n that the deviation of the actual velocity from ItS theoretical vahle beconles larger when the head is smaller and the fall is greater. On the basis of experience, theoretical analysis, and a limited amount 'of experimental information obtained from prototype tests oil Shasta and Grand Coulee dams, the U.s. Bureau of Reclamation [29) has studied the

383

relatiOilship between the actual velocity and a theoretical value,l From the result.s of this study, a chart (l~ig. 14-15) was prepared to show the a,ctual velocity at the toe of spillw3,Ys under various heads, fal1<" slopes from 1 on 0,6 to 1 on 0,8, and the condition of avera,ge surface roughness, It is felt that this.. chart is SUfficiently accurate for preliminary-design

70

80

90

100

1/0

I

I

!

r

120 130 140

I I

I

f 150 160 170

Valocity (V), fps

FIG. 14-15. Curves for detennination l)f velocity. at the toe of spillways with slope5 1 on 0.6 to 0.8, .

pUrpi)SeS, although it can be refined by addition~l experimental information which may become available in t.he future. . Experiments by Bauer [30J indicate that friction losses in accelerating the fiow down .the face of a spillway may be considerably less than the . normal friction loss in fl0W with well-developed turbulence. Therefore, the friction loss is not significant on steep slopes; but it woulcJ. become impcrta.nt if the slope were small, For this reason, the chart in Fig.

Bile [28J lor fUrther 'information, I

The theoretical velocity defined by the Bureau is V, =. V2(!(Z ...:. O.SH) .

I - \

384

14-15 i,~ not !1pplicable to slopes finttcl'than 1 on 0.6. For fb,tter slopes, the problem nmy be analyzed by the method described in Art. 11-4. At the end of the sloping spill W[\y surf&ce, the flow cho.nges its direction . rather abruptly and therebjr produces appreciable centrifugal pressures. In order to create a smooth transition of the flow [Lnd to prevent the impact of the falling watar from scouring the foundation, the surface at the spillway toe is usm'l.lly designed as a cUl'ved bucket [31]. To be thoroughly effective the bucket should be t[~ngel1t to the foundation or neady so. The radius R of the bucket, measured in feet, may be estimated approximately by the following empirical formula:

R

385

FI,QW OVER SPILLWa.YS

RAPIDLY VARIED FLOW

10(VH.4H+!&,;{Z.6H+EiJ

. 14~12. Submerged Overflow Spillways. Spillways and weirs are sllid to b~ slJbmerged ,vhen the tail water is higher than the crest.. As subflow. is usually unstable , having considerable disturb::mce immediately do\vnst;eam, such a spillway or weir is unsatlsf3.Cltory for

(14-2u)

where V is the velocity in fps of the flow :1t the toe fl.l1d 11 is the head in ft, exclucHng approach velocity head, on the spillway crest. The design .bucket pressure and the maximum side-wall pressure sh,mld be equal to the centrifugal pressure plus the hydrostatic pressure corresponding to the tai!water depth. The centrifugal pressure may be compnted by Eq. (2-8) for known radius of the bucket and velocity of the flow .at the toe. Determination of the bucket pressure, however, can be made more ~.ccurately by flow-net analysis or by the relaxation method, or, still more precisely, by rnodel tests. The results of such determination ve indicated that theclIect of the bucket ·curve on pressure actually extends even beyond the ends of the curve. . The flow leaving the toe of a high overflow spillway is a high-velocity jet, containing a Hnge amount of energy capable of causillg heavy scouring. Unless the· downstream approach is resistant to such scouring, measures to D.void the danger of scouring should be takeli. in the design . . Popular methods :11'C design of a ski-jump spillway, utilization of submerged taiIwater as a bl'ake, and use of a hydraulic jurr.:p as an energy .. dissipator. 14-11. The Ski-jump Spillway.. In this type of spillway the toe is "'''';;1.",.'''''' in the form of a large specially shaped lip or bucket 'which throws the whole jet flow into the air. P,nt of the energy in the jet is dissipated in the air, but in any case the in.lIs back into the river channel a.t a. safe distance from the dam. This was fin,t proposed by Coyne [32,33]. Figure 14-15 shows a typical ski-jump spillway. : In this design, the spillway-bottom slab is also the roof of the power house which is built against. the downstream side of the dam. The slab is heavily reinforced in order to ta\l:e the lon.ds due largely to the centrifugal pI'essure of the jet acting ori the spillway toe. l

un

of

1 See [34] to [36] for fmther information. \Vell.known ski.jump spillways are ill5tailed in Ca:;telo do Bode Dam in Portugal; !lond the Mar~ges, Bart, l' Ag1e, Saint.fj;tiel'lne-Cantales, ll.\ld Cltastang Dams in France. Each of the last three dami has two ski.jump spillways. 'The jets from the two spillways mllY meet and collide in the o.ir, thus dissipating a large amount of energy.

Fl~. 14-16. The ski-jump spillways of l' Aigle Dam.

-

(CDuT/esy 0/ P. Danel, Els.

NE YRPI C.)

accurato flow meMurements. Studies on submerged Iound-~rested w~irs are useful however, for they will furnish iuformationneeded m the de~.llgn of low ov~rfiow dams which may be occasionally subject to submergence. ;Extensi ve studies on au bmel'ged round-crested weirs have be~n performed by the U.S. Bureau 9f Reclamation [l,37J.I In these studies: ~he tlow is classified into four distinct types according to the fiow condItIOn 1

Other well-known studies are reported in [38] to [401.

'"

b

.,,

\

\

\

\

\

VI

("

...

*"b

\I

\~

II _ I

no

:::00

\Jll

rrln1


'"i"::: ~ ~

Ul"l1 -I:;) Ul~

n1 2. -1

\

I

. @-o,

I

---..

b

--

I

-~

I \

I

I

I, I'

I

..

--~

(J)

-T. ,1

~ •

--

~.-~~*--""~"<"l'<"""']~""",,,='"

- ,

-",~----"""-

I---,'+<~-~"""-"'''i''--~'"""",.....

,

I

~~:d

..

::-;:.~

+d Y-'>-i---I~-+--·-+·-.."..l1.5~

I---+-.:'"~~~~;;j:;;:;=""'--¥~-t---r-c::;~ 2.14 0.513

1.1;> 0.562

l.:i2.

hd

.He 0.201

0.796

1.53

.

2.14 1.75

o

2

:3

4

5

4

6

5

6

X/He (0)

0.049 0.OS3 0.142

2.14

0.$55

-1.0

3

4

.tv)

FIG.

1~-18.

5

4

6

5

6

(d)

Typical pressure (dashed lines) and surface (solid lines) profiles for flow over submerged overflow dams. (Selected (a) Supercriticn.[ flow; (b) flow involving hydraulic jump; (e) ftow with!l. drowned hydraulic jump; (d) flow a.pproaching complete submergence.

from U.S. Bureau of Reclamation data [1J.)

388

389

FLOW OYER SPILLWAYS

UAPlD!wY VAlUED FLOW

secoD~, will be required fo,' aeration beneath the nappe to

prevalent on the dowl1strel1m apron; (1) snpel'critical flow, (2) su bcritical flow involving hydraulic jump, (3) flow accompanied by a drowned jump with diving jet, and (4) flow approaching complete submergence. Submergence of spillway or weir will reduce the coeflicient of discharge of the corresponding ullsubmerged flow. The BUl'eA.u of Reclamation's test results on this reduction, expressed in percenta.ge of the discharge coefficient for unsubmerged flow (Fig. 14-4), have been presented in a chart for the four types of flow mentioned abov€;. This chart in a slightly modified form (Fig. 14-17) was further checked against other datal by the u Army Engineers Waterways Experiment StatiOll. 2 It was found thltt the, chart is also applicable to t·he determination of coefficients for WES ,shapes under submerged conditions. In the chart (Fig. 14-17) h,J is the drop from the upper pool to the tail·· water elevation, H, is the total hend above the crest, and d is the tailwil.ter depth. The genera.! pattern of the curves shows that, for low 'ratios (h.J + d)/ H" the flow is of type 1, or supercritical, and that the r(~duction in coefficient is affected essentially by this ratio o.ndis practically independent of hd/l-I.. The cr-oss section BE in the upper right-hand corner of the chart shows the variation of (hd + d)/H, at hdill, = 0.78. For large 'm.lues oi (h,l + d)/H., on the other hand, the reduction in coefficient is [1ffected esseu~ia.lly by the ratio ad/H,. Under this condition. for values of hatH, less than 0.10, the flow is of type 4, the jet is on the surface, and no Jump occurs. For values of hd/H. greater than 0.10, tho flow of type 3, or accompanied by a drowned jump with diving jet. The cross section AA shows the variations of ha/H. at (h. + d)/H, near 5.0. Subcriticr.l liow, or flow of type 2, occurs in, the indicated on the cimrt. Other regions for transitior1D,l flow conditiollS are also shown. The typical pressure and surfa.ce proiiles for, submerged spillway flow are shown for different values of (hrl + d)/li. and A4/H. for folU' types of flow (Fig. 1<1-18). These were selected from the data of the Bureau of Reclanul.tion. They aTe useful in the design of spiJlwaysfor stability.

(1

pressure reduction of

2 it? Compute the load 0'.1 the weir after this aeratiol1. 14-3. If the channel floor is at El. 975.0 instead of E1. 880.0, and other data remain the s3-me, determine the spillway section required ill Example 14-1. A trial-anderro, procedure is required for the determination of the value of C. 14-4. If the upstream fa.ce has a slope of 3 on 2 instead of veriic!J.I and other data remain the s",ne, determine the spillway section required in, Example 14-l. 14-5. Determine the rating curve of tlHil spillwl!.y developed ill Exarnple 14-1. 14-6, AD o,-erflow spilhvay of unknown profile designed br a total hC!l.d of 14.5 ft has a crest length of 64 ft and a coefficient of discharge equal to 3,48. Determine the rl>ting curve' by the Beadle.)', curVe. ' 14-7. If t,he spillway in the preceding problem has a profile shown by the dashed line in Fig. 1,*-7, detennJne the rating curve. 14-B. CompuU! ~he upperm.ppe profile of the fiow over the spillway designed ill EXl1mple 11-1, assuming"'!):!:! piers. ' 14-9. Determine'the wall lIcigh',s for the overflow-dam seetion designed in Example 14-1, assuming six b!l.Ys formed by WES round-nose piers and a maximum operating head 35% higher than the design head. ' , 14-10. Determine the discharge over the spillway section designed in Example 1"-1 if the spillway has six bays formed by WES round-nose piers. , 14-11. Detecmine the pleasure on the crest of the spillwn.y designed in Example 14-1, fOJ: operating head equal to 0.5u, 1.00, and 1.33 times the design head; assume (a) no piers and (b) six bays fOl'med by WES round-nose piers. H~12., A drum gate (F:ig, 14-11)) 50 ftlong alld 20ft in.radius is installed on top of!l. high overflow spillway., Determine the rating curve of the gate: i.e., discharge VB.

is

FIG: 14-10. The drum gate for P~ob. 14-12, elevation of the hlgh~st point of the gate suna.oe. The position of t.he highest point 'of the gate surface and the value of (J may be determined graphically. 14-13. Estima.te the depth a.nd velocity of flow and also the radius of the bucket a.t the toe of the spillway dp,signec.l in Example 14-1. 14-14. Determine th., discharge of the spiihvay designed ill, Prob. 14-3 if the tailwater elevation is at El. 990. Would a hydraulic jump be possible at the toe of the spillway?

PROBLEMS H-1. A vertic[L1 sharp-crested weir, 20 ft high and 60 ft long, is built as a.n overflow spillway without eud contractions. When the head is 20 ft above the crest of the weir and the nappe i$ completely aerated, compute (a) the nappe profiles, and (/» the hydrostatic load acting on the weir. l!!-2. During te~ts on the o.er!ltioJl of the spillway described in the preceding problem, negative presSure~ under tlie nappe of 9.7 it of wCltel' were observed. Compute the increase ill the t,otalload on the weir. How much air, measured in cubic feet per 1 2

Dat,;). given iI, [41] to [4.-1J. From (::JOJ, Hydraulic Design Chart 111-4, WEB 4-1-53.

RE;FERENCES

t

I

I

1. Studies of crests for overfall da!llll, BO'l,lder Canyon Praject Final Reparl$, pt: VI, U.S. B~Lreat' of ReclamatiDn, Hydraulic Inve.tigalions, Bulletin 3, 1948.

2. Juliall Rin.ds, William P. Creager, alld Joel D. Justin: "Engineedng for Dams," John Wiley & Sons, Il1c., New 'York, 1945, voL 2, pp. 358-361.

Ji i

390

RAPIDLY YARTED FI,-OW

.3. William P. Crea.ger and.Jor.l D. Justin: "Hydroelectric Handbook," 2d·ed., John Wiley & Sons, Inc., New York, 1\)50, pp. 362-363. 4. A. T. IppeJJ: Channel transitions n.nd controls, chap. VIII in Hunter ROllse (editor): "Engineering Hydraulics:" Joh!'. vViley. & Sons, Inc., New York, 1950, pp. 49G-588. ' . . . . . 5. Fred W. Blaisdell: Equ!\tion of the free-fallIng nappe, Proceedtngs, American Society of Civil Enginee1'S, vol. 80, separate no. 482, 16 pp., August, 1954. 6. Horace William I\:ing, "Handbook uf HydrE:.ulics," revised by Ernest F. Brater, 'Hh ed., McGraw-Hill.Book Company, Inc., Hlf,'!, sec. 4, pp. 1:-67; 7. Carl E. lCindsvaterancl Rolla.nd W. Carter: Discharge characteristics of rectangular thin-plate "'eirs, .paper 1453, PI'ocecdings, A. merican Society of C1:vil Enoineers, Journal, Hyd"aulic~ D·iviGion, vol. 8a, no. Hye, pt. 1, pp. 1-'36, December, 1957. 8. P. K. E:andaswamy and Hunter Rouse: Characteristics of fl.)w over terminal weirs n.nd sills, paper 1345, Proceedings, American Society of Civil Engineers, J ollrnal, Hudl'aulics Division, vol. 83, nQ. HY4, pp. 1-13, August, 19.57. 9. J. -Y. BQussinesq: "Theorie approchee de l'ecoulemellt de l'eau sur un d8\'el'.~oi;: en mince paroi at sans contraction latentle" ("Theoretical Approach to the Flow over a Knife-edge Weir wit.hout Side Contraction n), Paris, 1907. 10. Th. Rehbock: Discussion on Precise weir measurements, by Emest W. Schoder and Kenneth B. TlJrner, Transactions, American Society of Civil Engineers, vol. 93, pp. 1143-1162, 1929. . . 11. G. H. Hickox: Aeration of spillways, Transa.dions, Am"'ican Society of Civil Eng·ineers, vol. 109, pp. 537-556, 1944. 12. H. E. Bazin: Experiences nonveiles sur l'ecoulement en deversoir (Recent expcrim·ents on the !low of water over weirs), lIIemo·ires et documents, Annc!le, des pants at chaussees, seL 6, vol. 16, 2d half-yr., pp. 393-148, 1888; .~er. 6, vol. 19, 1st half-yr., pD. 9~82, 1890; ser. 7, 1'01. 2, 2d haH-yr., pp. 445-520, 1891; ser; 7, vol. 7, 1st h~lf-Yl'" pp. 249-357, 1894; ser. 7, ,rol. 12, 2c1 half-yr., ?p. 645-731, 1896; and ser. 7 vol. 15, 2d quarter-yr., pp. l51-264, 1898. English translation of the first part h~ Arthur Marichal and John C. Traut'~ine, Jr., Proceedingl, EngiMel'8' Club of Philadelph.ia, vol. 7, no. 5, pp. 259-310, 1890; VDJ. 9, no. 3, pp. 231-244, 1892j and voL 10, no. 2, pp. 121-164, 1893. Ba.zin's data were reprinted almost eutirely by G. W. Rafter in Report on special water'-Supply investigation, CongresS1;onaZ Documents 41*6 and 4147, WlI.Shington, D.C., pp. 571-950, lQOOj o.nd Hydro[ogy of the State of New York, 'l'few York Sta-/e 1l1~se141n, B1(.lletin 85, Albany, N.Y., 1905. . 13. William P: Cre·ager: "Engineering for Masonry Dams," John Wiley & SOllS, Inc., New York, 1929, p, 106.. • 14. Calvin Victor Davis (editor-in-chief): "Handbook oi Applied Hydraulics," 2d ed., McGra"'-Hill Book Company, Inc., New York, 1952, pp. 259-263; Sec. 7, Spillways and Streambed Protect jon Work~! by Emory W. Lane and Calviu V. Davis, pp. 253-289. . . . 15. Ettore S.cimemi: Sullo. forma delle vene tracimanti (The form of flow over :weirs), U Ene?'gia eleUrica, Milano, vol. 7, no~ 4, pp. 293-305, April, 1930. 16. J. Smetana: Etude de la surface d'ecoulement des grands barrages (Study of flow profile of hrge dams), Revue generaLe ·del'hydrauliqlle, Paris, vol. 14, no. 46, pp. 185-194, July, 1948; vol. 15, no. 49, pp. 19-32, January, 1949. 17. Giu'lio De Mn.rchi: Riccrche sperimentali .sulle dighe tracimanti (Experimental study on overflow dams), Annalide?: lavori pubblid, Rome, vol. 7, 1928. lB. L. Escande: "Barrages" ("Dams"), Hermann & Cie., Paris, 1937 .. 19. Anton Grzywienski: Anti-~vacuum profiles for spillways of large dams, Transa~-

,f.

FLOW OVER SPILLWAYS

20.

21.

22. 23.

25.

26. 27.

28.

29.

30. 31.

32.

33.

34.

35.

3S.

391

tions of the 4th Congress on Lal·gs Dams, vol. 2, Pl'. 105-124, International Commission on Large Darns of the World Power Cbnfcrence, New Delhi, India., January, 1951. "Corps of Engineers Hydraulic D'"sign Criteria," prepared for OlIiGe of the. Chief of Engineers, U.S. Army Corps or Engineers, Wat.erways Experiment Station, Yicksburg, Miss., 1952; revised in subsequellt years. John C. Harrold: Discuss-ion on Equn.&i.on of the free-falling nappe, by Freel W. Blaisdell, P:-oceeding~, American Socie(~ of CivaEngineB1's, vol. 80, sp.para&e no. 624, pp. 16-19, August, 1955. J. N. BradlEY: Discharge cQefficients for irregular overfall spillwaYB, U.S. BIL1'CaU oj Recla.mation, Engincering Monograph No.9, March, 1952. Boo Buehler: Discussio'l on Rating curvp.3 for flow over drum ga&es, by Joseph N. Bl'adley, Tran.~ac1.ions, American Society oj Civil Engineers, vol. 119, pp. 421-428, 1954. Ross N. Brudenell: Flow o.er rouurled crests, Engincering Ne1ll8-Rec01'd, "oJ. 115, no. 3, p .. 95, July 18, 1935. M. K, GanguE and S. K. Roy: On the st.1.ndardization of the relaxation tl'eatment of systematic pressure comput[l.tions for overflow spillway discharges, liTigation and Power, '1'h~ Journal. of ihe Central Board'of Irrigation and Power, vol. 9, 110. 2, pp. 187'--209, New Delhi,India, April; 1952. . Charies 'V. Harris; An analysis of the weiI' coefficient for suppressed weirs, Univen":t!! of W(Jahington, Engineering Expm:"!enl Station, Bulletin 22, ·]:.923. Joseph: N. Bradley; Rating curves for 110w over drum gates, Tmnsa;;Cilms, American SoCiety of Ciua Engineel's, vol. 119, pp. 403-420, 1954. H.obert B. JansGn: Flow characteriRtics on the ogee spillway, paper 1452, P"oceed- . ings, American Society of Civil EnOl:neers, JOIL1'1lal, Hydmulics Diviiion, vol. 83, no. HYS, pt. 1, pp. 1-11, December, 19.57. Research study Qn soilling basins, energy dissipators, and associated appurtp.no.nce~, U.S. Bu.refJ.u of Reclama.tion, Hydraldic Report, No. Hyd-3fl9, June 1, 1955, pp.41-43. William J. Bauer.: Turbulent boundery la.ye\' on steep slopes, Trcnsaction.l, American Society of. Civil Engineel·s, vol. 1U), pp, 1212-1233, 1954.· J. H. Douma: Discussion on De5ign of side walls ia chutes and spillways, by D. n. Gumensky, Tranlluclio1lS, American Society of Civil Engineil1·s. vol. 119, pp. 364368, 1954.· A. Coyne: Latest development of do.ms and hydro-electric power stations in France, paper read before a joint meeting of the Insti&ution of Civil Engineers and the British Section of the Societe des Ingeuieurs Civils de France, Loudon, Nov. 25,194i. Editorial review entitled Development of dams in France, Engineering, vol. 164, no. 4274, pp. 613-614, Dec. 26, 1947. . A. COYM; Barrages-usines de I' Aigle at de Ss.in t-:f;tienne-Can tales (Dams and hydr.oplants of Aigle and Saint-Etienne-Cantales), Trava1~x, Paris, vol. 34, no. 185, pp. 194-215, March, 1950. . R..Mo.itre and S. Obolensky: Etude de quelque~ caracteri,;tiques de l'ecoulement dans Ill. partie aval des evacu!l.teurs de surface (Study of some flow oharacteristics in the downstream part of spillways), La Houille blanche, Grenoble, 9th year, no. 4, pp. 481-511, July-August, 1954. F. Auroy: Les eV2.cuateurs de crue.s Uti barrage de Chastang ,(The spillways. of Chastang Dam), Transactions of the 4th Conqresson Large Dams, vol. II, pp. 661f·8G, International Commission on Large Dams of the World Power Conference, New Delhi, January, 1951. . E. A. Elevatorski: Trajectory bucke~-type energy dissipators, paper. 1553, Pro-

l

I

.-i

I \

I

J,

392

RAPIDLY VARIED FLOW

ceedings, 'Arnerica.n Society of Civil Enrr.ineers, J oul'nal, POlljf!;'l' DiVi8jon. voL 84" no. POl, pp. 1,,17, Febru:::ry, 1\)5B. 37. J. N. Bradley] Studies of flo'\v characteristics, discharge a.nd pressure relo.tive to submerged dams, U.S, Bm'ec.u of Redamalion, Hydm,tiic LabonJ./.lyry Repor! 1-82

HH5.

"

'

38. Report of the Board' of Engineers on Deep Waterways, U.S. Board of Engineers, pt. 1, p. 291, 1900, " ~~\). R. E. Horton: Weir experiments, coefficients and formnl!\S, U.S, Geologica.! S1:11!8y. W(ltel"S'llP1I1V CLnd 111''i~'Q.lion Paper 200, 1907. ' ' 40. Glen Nelson COltl The ~ubmerged weir as a measuring device, Univers'ity of WisEn!/ineeri"g Blation, Bulietin 67, pp. 48-75, 1928, 41. II. J. characteristics of I'lllbn:terged spillways, Master's thesis, Colorado Agl'ic.tll~ul'a1 a.nd Mechanical College, Fort. OolliIlS, December, 1951. , 42. i\I. Bar Shan;)'] Pressure distribution on downstream face of a. submerged weir, Master's thesis, Sta.te University of Iowa, Iowa City, June, 1950. 43. Spillwa.y and lock a.pproach, Jim Woodruff Apalachicola River, Flol'idp": Model investigation, U.S, Army Engineers lVaierway;s Experiment S'(.!lio~ :i'cch, nual Memera1Ull/lI. 2-340, Mal', 1952. , . I 44, Morgantown spillWay, Speciaf tests; lIl).published report, U.S. Army Engineers Wliterwaye Experiment Station, Vicksburg, lVIis"., 19:'*9.

)

1

I

CHAPTER

15

JUMP AND ITS USE AS ENERGY IHSSIP ATOR

15-1. The Hydra1!1ic Jump. The hydraulic jump was first ;,o"D"f'i",~ experimentaU}' by Bidone [1,2], an Italian, in 1818. 1 This led [3] (1828) to distinguish between mild (subcritical) ::md (supercl'itical) . slopes, since he had observed that in steep channels hydraulic jump is fre~ quently produced by a. b!l.lTier in. originally uniform flow. abundant studies 'were made and the. results were quoted by many writers. Outstanding contributors to OUI' present knowledge about the hydraulic jump a,reBresse (1860) [4], Darcy.a,nd Bazin (18fi5) [5], Ferri':' day and Merriman (1894) [5], Gibson (1913) [7], Kennison (1916) [8]. Woodward and Riegel-Beebe (191'7) [9], 1(och and Carstanjen (1926) (lOJ, Llndquist(1927) [I1J, Safranez (1927) [12], Einwachter '(1933) [13,14], Smetana (Hl34) [15,16J, Bakhmeteff and Mat;1ke (1936) [17), Escande (H)38) [18], Citrini (1939) [19], Nebbia (1940) [20], lCindsvatel: (1944) (21], Blaisdell (1948) (22J, Forster and Skrinde (1950) [23], Rotlse, 8iM, and Nagaratl'l.am (1958) [2'1J, and mallY others. 2 The theory of jump developed in early days is for hOl'izontalor slightly inclined channels in which the weight of water 1."1 the jump has little effect upon the jump beha:/ior and hence is ignored in ~he analysis. The results thus obtained, however, applied to most cha.l1uels encountered in engineering problems. For channels large slope, the weight effect of water il). the jump may become so pronounced that. it must be i.nc,luclecl in the analysis. . Practical applications of the' hydraulic jump are many; it is used (1) to dissipate enel'gy in water flowing over: dams, and o,ther hydraulic structures mid thus prevent sCQuriug: downstream from the struotures (Art. 15~8); ,(2) to recover head Dr raisethe water level all the dovvnstream side of a measuring flume El.nd thus maintain high wate,l' level in the ~ The experit'nellt was made ill Paris in 181;8 and re'pul'ted the following year in [11, The hydraulic jump is also known as a, sta:n.di1lg wave. In French, it is called Ie ressm,1 hlldra.tdique . . In German, ,it is der Wass6rsprung. III honor of Bidone, the; hycil'Alllic .;jump in Italian is named il sail a di Bidone (the jtimp of Bidone). =For a, compreltsnsive review of the studies 011 hydraulic jump, see [,25], For a mathematical trea.tment of the sllbject, see (251.' ,

M3

394

RAPID!,Y V AEIED FLOW

HYDRAUUC JUMP AND ITS USE AS ENERGY DISSIPATOR

channel fo).· irrigation or other water-distribution purposes; (3) to increase weight on an a.pron and thus reduce uplift pressure under a masonrystructul'e by raising ~he watel' depth on the apron; (4) to increase the· discharge' of a sluice by holding back tail water, since the effective head will be reduced if the tailwater is allowed to drown the jump;l (5) to indicate

395

'15;..2. Jump in Horizontal-Rectangular Channels,l For supel:critical flow in a hodzo'utal rectangular ch~nnel, the energy of flow is dissipated through frictional resistance [l,long the channel, resulting in a decrease in velocity and an increase in depth i n · ' the direction of flow, A hydraulic ~"'" jump will form in the chan:nel if the ~ .,.-... ..........--..~-.....-___ ..",..--Froude number Fl of the flow, the flow depth. YI; and ado,vnstream :'i7~/'/I/m/ff&///;W/'///'//'/I7$)'/7#/ . , 1"1' 1-1.7 Undular jump 'depth 1/2 satisfy the equation

1

(3-21)

il

. ~ 18 r---t-~--r-

~ 16~-4--+-~--+--~74']

~

14

I/

t / I

1--1----I'----+---+---1I<--t--r-t--r-

r---

, -I I-

r---t---t---"-T--l-V+-+I---I-+-I---1---1. I I 0 r-~I---r-" t-I/--fj+----+-+-t-~I_--I---f--l 12

8

I--+--+-j-Y---+--r

6 f--+-,--!-t'--,f--+----l

.2= ~ (.j1+8F'-1\ Y, z ,

\

F{ V/-/9Y,

4~~-+7L~--l--+--r-'--I---'-''--+--1 2 17't- +----I--I---+--+--+---l---r---t-~"i

V

f

o

L-.--'-_-'--_-'i' _-,-,"_"

o

2

4

5

B

'---"--_~~"_'_~_L __~ 10

12

14

16

18

20

22

15-1. Relation between FJ and y,jy,·for a hydraulic jump in a hQrizontal ~ectangular channer. . .

FIG.

special flow conditions, such as the eJiiistence of supercritical flow or the presence of a control section ::;0 that a gaging station may be located; (6) to m\x chemicals lIsed for water purification~ and so forth [28J; (7)" to aerate water for city water supplies; and (8) to remove air po.ckets from water-supply lines and thus prevent air locking [29J. 1 This principle hRS been applied by Saugey [27] to an interesting device known as fall increaser. The device is intended to increase the effective head, in a 'yater-power plant during periods of flood by holding b!\ck tail water from the outlet of the draft tube by a hydraulic jump.

,J

This equation may be represented by 0'/7/I/,Y//'/II//;/I./)?'/m/////,I,7/#"W~.a'7;'; the curve in Fig. 15-1. This c',lrve' F,'L7-2.5 Weak jump has been ,-erified satisfactorily with Oscilialiog je 1 many experimental data and will be Railer ?---.. found very useful in the analysis and ';J-?~ ,,~&. design for hydraulic jumps. ~ ~: ;:~"";y ~ ,--" -15 7 3. Types of Jump. Hydraulic W~%'?///ff),/m7/./,1?/,'l"///,I//I////&/Ih' F,' 2.5, 4.5 Os.cillali~g jump jumps on horizontal floor are of severai distinct t,ypes. According to t.he studies of the U.S. Bureau of Reclamation (34,35], these types can be conveniently classified according to the Froude nnillbel' F I of the inF,' 4.5-9.0 Steady jump coming flow (Fig. 15-2), as follow's: For FJ = 1, the flow is critical, and hence no jump can form. For Fl = 1 to 1.7,· the water surface showa undulations, and the jump is called an undulm' J·Hmp •. For Fl = 1.7 to 2.5, a series of small rollers develop on the surface of the .F IG. 15• -~. Q V . anons types of hydraulic jump, but the downstream wa.ter sur- jump. face remains smooth. The velocity throughout is fairly uniform, and the energy loss is low. This jUIl'Ip may be called a weak Jump. For FI = 2.5 to 4.5, there is:J.ll oscillating jet enteririg the jump bottom to surface and back again with no periodicity. Each oscillation produces

Il I

I

(

1

I

1 FOT hydraulic jumps in trape1:oidal channels, see [30J and [31J. For jumps :in closed conduits, see [29]- and [.32). For a general treatment of nonrectangular channels, . see [33J.

"'·"--1-

396

I

RAPIDLY VARIED FLOW

HYDRAULIC JUMP AND ITS USE AS ,ENERGY DISSIPATOR

a~large wave ofirregul/1,r~period which, very commonly in canals, can tra,:e! for mi!Gs doing unlimit.ed damage to earth banks and ripraps. TIlls Jump may be called Em. oscillating jump. For Fl = 4.5 to 9.0, the downstream extremity of the surface roller and the pl)int at'which the high-veloci.ty jet tends to leave the flow occur at practically the same vertical section. The action and position of this jump are lea..o:;t sensitive to variation in tai1wate1' depth. The jump is ' well-balanced and the performance iS3.t its best. The energy dissipation' ranges from 45 to 70 This i llmp may be called a steady jmnp. For F 1 = .9.0 and the high-velocity jet grabs intermittent slUgs of \Vater rollmg down the front hce of the jump, generating Wlwes downstrenm, and a rough surface can prevail. The jump action j,~ rougb but effective since the energy dissipation may reach 85 %. This jump may be called a .slronq j11mp. !t should be noted that the ranges of the F'roude number given .above for the various typea of junip are not clear-cut but overlap to a certain extent depending on local conditions. 15~~1:. Basic Characteristics of :the Jump. Several bf1sic characteristics of the hydraulic jump in horizontal rectangular channels are to be dis~ cussed below: E'nergy Loss, . The loss of ellergy in the jump is equal to the LHJ.!"'! " " in specific energies before and after the jump. It can be shown that the loss is (3-24) The ratio 6.E/E 1 is kno,"-u as the relative loss. Effic'ienc'!J·. The ratio of the specific energy after the jump .to tha.t before t.he jump is defined as the efficiency of the iump. It can be shown that the efficiency is Ez (SF l 2 + 1)'1> - 4F 1z + 1 El = BF;2(2 + Fl2) This equation indicates that the efficiency of a jump is a dimensionless function, depending only on the Froude mnnber of the approaching flow. The relative loss is equal to 1 Ed E 1 ; this also is a dimensionleas function F 1. . . He1:ght of J1~mp.The difference between the depths after and before the jump is the he£(!ht of the j11mp, or hi = Ya - Vl. Expressing each term as a ratio with respect to. the initial specific energy,

of

Yl El

where hi/El is the relativ8'he1'ghl, VdE 1 is the 1'elativeinitial depth, and

397

ydEJ, is .the relative seqtwnt depth. All these ratios can be ,shcnvn to be dimensionless functions of F 1. For example, hi El =

,II

+ 8F 12 +2

F12

3

(15-2)

Since the relative loss, efficiency, rel2_tive height, and relative initial. and sequent depths of a hydraulic jump in a ·horizontal rectangular F 1, resulting in a channel are functions of F 1, thcy c[',n be plotted,

Veiues of ,Ft

Fw. 15-3. Cha!:a.cteristic curves of hydnwlic jumps in horizontal rectangular channels.

set of characteristic curves (Fig. 15-3), 'With reference to these curves, the following interesting features ma.y be noted: 1. The maXinmltl relative height J'4/E1 is 0.507, which OCClll'S at Fl 27~ . 2. The m!:.ximum relaLive depth yz/E1 is 0.8, which occurs at vrf El = 0.4 and F1 = 1.73. Experiments have shown that the tral1sition from annndular jump to a direct jump takes place appro:dmateJy D.t'this point Fl = 1.73. 3. When Fl 1, the flow is critical and YI Yz %E1 • 4. When inol'eMes, the changes in all characteristic ratios become gradual. . The characteristic curves will provide the dcslgnel' with a general idea about th~ range of conditions under which the structure is to be operated. involving a jump below the FOl'instan'ce, in the design ofu sluice gate, such curves will show clearly the formation.of the. jump f~r diifel-el:t gate openings under a given head. The above dlscusslOn applIes to hOrI-

398

l

RAPIDLY VARIED FLOW

zontal rectangular channels. For horizontal nomectangular channels, similar curve,~ may also be pl'epa,red. The theol'e~ical curves for y2/ Eland hi/ E 1 have been verified experimentally by Bakhmeteff and Matzke [17], who found that these curves, give values of yl/ Eland hi / EI about 3 to 4 % greater than the experimental values. I The characteristic curves were also checked with U.S. 7

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Y;

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1

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I

li-'eolOS(;illtlliJiCj

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f.'

s,,'rlacp.·7

jlimp

Sleady lump

f..-Vlo.,,)' ----- f.. --- ---- -- --- --8esl perto~rnOl1ce

lurbulenr.e only

3

o

/.

~

/.

3

6

, b--

,

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r-

=i -r

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-~~

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j

Sirang- jump

--A~''-P'Ob''--T -- £;-p-.n;';';~~i~19b~i;.-'-;;J per lotm:Jnt;;e

T 2

/~/.'I'I//

,

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rough SudClce

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11. IS

IS

20

F"V.!m, FIG. 15-4. Length in terms of sequent depth y, of jumps in l,lorizontal channels. (Bt;!,sed all dala and rer.ommendations of u.s. Bl!rerwof BeclallwUon [34].)

Bureau of Reclamation data [34,35] obtained from six test flumes. Perfect agreement was found between the yJ/E 1 curve and the data. The agreement between the E2/El or t.E/E l curve and the data was f!lirly good except for PI < 2. The experimental curve for t.E/E l recommended by the Bureau is shown by the dashed line (Fig. 15-3). The agreement between the yziE l and h;/E1 curves and the data was good for high Fl values, but the scattered data failed to define the curve8 B.CCUmtely forF l <:; :3. 15-5. Length of Jump. The length of a j-ump may be defined as the distance measured from the front face of the jump to a point on the surface immediately dowIlstream from the roller. This length cannot be deter': mined easily by theory, but it has been investigated experimentally by, many hydrauliciaus. 2 ' I 'It is possible that at least part of this discrepancy is due to, the scale effect of the testing model (see next article). , Among them Safranez at the Technical University of Berlin [36,37J, \Y6ycicki at the Federal Institute of Technology in Ziirich, Swit~erland [381, Araviri in Russia [391, Bakhmeteff and Matzke at Columbia University [17), Moore at the California Institute of Technology 140}, a~d the engineers of the U.S. Bureau of Reclama.tion [34,35J.

r

HYDRAULIC JUMP AND ITS USE AS ENERGY DlSSIPATOR

3[}9

The experimental data Oil length of jump can be plotted conveniently with the }<'roude number PI against a dimensionless ratio, L/(Y2 - yD, L/Yl, or L/yz. The plot of F 1 vs. L/y I is probn.bly the best, for the resulting curve can be best defined by the data. ' For practical purposes, however, the plot of Fl YS. L/Y2 is desirable, because the resulting Qurve.shows , regulariflY or a fairly :flat portion for the range of well-established jumps. A curve of FI vs. L/Y2 (.Fig. 15-4) based on the experimellt11] data of six test flurries has been prepared by the Bureau of Reclamation. In comparing this curve with the well-known Bakhmeteff-Matzke curve [17], pronounced disagreement was found. Investigation of 1;he matter has Jed to the belief that this disagreement is due to the scale effect involved in B8,khmeteff and Matzke's experimental data. This scale effect means that the prototype ::I,etion wn.s nOG faithfully reproduced in the model. The curve shown in Fig. 15-4 was cleveloped primarily for jumps occurring in rectangular cha.nnels, In the absence of' adequate data,' this curve ma.y also be applied approximEl.tely to jumps formed in kapezoidal channels. 15-6. The Surface Profile. Knowledge of the surface profile of a jump is desirable in designing the freeboard' for the retaining walls cif the stilling basin where the jump takes place. It is important also for determining: the pressure for use in structural design, because experiments have shown that the vertical pressure on the horizontal Hoor under a hydraulic' jump is practically the same as would be illdicatedby the water-surface ~fi~, '

,(

I ·1 !

I '.

\

I

On the basis of their e,xperimelltal data, Bakhmeteff and Matzke [17J have found that, the EUrface' profile of a hydTaulic Jump can be rcpresented by dimensi6nle:3s curves for various F 1 vaJues, as shown in Fig. 1,5-5. Moore [40] has developed simil8,r curves for jumps helow a fr'ee overfalL The profiles shown by lVIoore rise more rapidly at the beginning than do Bakhmeteff and lVIatzke's proiile". It is believed that this is because t.he nonhy?rostatic-pressllt'e distribut;ion in the jump wasIlot registered properly by the piezometri"c measurements for Bakhmeteff and Matzke's data. Furthermore, Moore's length of jump was about 20% longer than that shown by the Bakhmeteff-Matzke curves. Since the jump in the latter case ...vas formed downstream from a regUlating sluice, lack of agreement may be caused by a difference in the velocity profile of the shooting flow entering the jump. ' 15:"7, Location of Jump. Hydraulic, jump occurs in a supercritical flow when the depth changes abruptly to its sequent depth. Theoretically speaking, the jump will occur in a horizontal rectangular channel if the initial and sequent depths and the approaching Fl'oude number 8atisfy Eq, (3-21). This theoretical condition is generally used to locate the position of a jump. For a closer estimate of the jump positi()ll, how-

'-"l

i

400

HYDRAULIC JUM? AND. ITS USE AS. ENERGy 'DISSIPATOR

RAPIDLY VARIED FLOW·.

--c

ever, the length of the jump should be considered. The following will illustrate the locaGion of a hydraulic jump in three typical cases (Fig. 15-6) : Case A shows the jump below a regulating sluice in a mild channel. The profiles A Band CD c.an easily be identified as of J\lI3 and M2 type, respeet.ively. The methods of computing these profiles are discussed in Chap. 10. The curve A'B is a plot of the depth sequent to AB. By

"~

1'Ol~·~-n I -J 'i

.............

Fl.

CIJse A

F,o 1.98~

0.9

0.8· .., 0.,

401

A' •

F,=2.92" I F,=4.09, -,~:-...."-+nl)Y--1r---j F , ' 5 .53 ~-A,"--:;,¥,-A"l'---+--+ F l ' 8.63

0.6

~

~------.

0.5 i--j----l'l<---:>f7"-7''b''-0.4 i--j---,/C-I:t'7"-+''<--+---+

0.3 l------tl'--f7'-tr:;r-j---I----; --+--tCase B

.. 0.1

f.-...I"IA--+-~t-

1

3

4 x

5

7

8

/h j

FIG. 15-,5. DimewSlouless snrface profiles of hydraulic jumps in hOl'izontnl channels .. (Based on Bakhmeic.tf-Mal .• ke (lata [17J.)

{ J.,

1

)

the position of pI, the length of the jump can be estimated. By trial and error, a horizontal intercept bet,ween the curves A'B and CD can be found equal to the length of jump. For instance, the horizontal distance EF is equal to the length of the j'Jmp corresponding to the dept,hyz at F. It becomes apparent thHt the jump will fOl'm between G r.nd F, Slllce .the ; depth at F is sequent to the depth at G und the distll,nce EF measures the length of the jump. It may be noted th~t, if the length of the jump were not taken into account in the analysis,the jump would have been considered to form at the upstream ]Joint F', resulting in an error represented by F'F. In case A, it can be s£en that, by. increasing the downstream water depth or raising the curve CD, t~e jump can be moved up~tream. The downstream depth may be raised to such a height that the ju~p will eventually be ;drowned out.in front of t~e sluice. Decreasing the' downstream depth PI' lowering CD will move the jump downstream.' . The above discussion applies also to the lo!!ation of a jump fOI'med at the foot of a weir 01' overflow spillway, .

Case C F1G.

15-6. Location of a hydraulic jump.

When there is a hydraulic jump below a sluice, the issuing flow from the sluice will form 0. jet that possesses a vena. conti"acta, '1'he distance L, from the vena contract,!L to the sluice opening is usually short. Hegitrd. iug this distance, there is.a generally adopted rule which stc'l.tes that the vena contracta is located approximately at a distance h from the sluice , opening. l 1

This rule was first used by Agroskin [41J.

It is baseq. on the assumptions that

~ Row from a sluice correspond,S to one-half the flow from a. drclliar orifice and that the 'lena contracta in the flow fro'm lJ. cil'cular orifice is located approximately at a, distance I)f half an orifice diameter from the orifice, The rule of half orifice diameter was originated by Weisbach [421.

.,

1 402

RA.PIDLY VARIED FLOW ,

B shows the jump in a channel having a break in the bottom slope tha.t changes from steep to mild. For simplicity, it. is assumed that the flow is uniform in the channel except in the reach between the jump and the break. The jump may occur in either the steep channel or the mild channel, depending on whether the downstream depth Y2 is greater or less than the depth y/ sequent to the upstream depth Yl. If the depth Y2 is greater than'YI', the jump ,will Ocettr in the steep region, Then the surface curve OC is of 81 type. The line A I P indicates the depth sequent to the line AR. Now, determine a horizontal intercept IJ between A'P and CO that is eq,ittl to the length of a jump.· n is apparent that a jump HJ will the section containing J. If the depth Yz is lowered. approximately to less than y/, the jump wili start to move into the mild channel: In this case, the jump can be located a.s described in case A, Case C shows the jump behind an ove"fiowban'ier.' Theoretically, a jump will form if the depth at the barrier is greater than .the depth Yl' sequent to the approaching sllpercritical depth Ill. The loca.tion of the jump is the same as that for case B if the jump oecur" in the sLeep region. Increasing; the height of the barrier will' mOYe the jump upstream. eCf'el:\,Sln.g the heighb wiIi move the jump downstream. When the depth at the barrier is less than the sequent depth y/, the barrier will be crossed by a standing swell in the form of a single undular surface rise which will not be followed by further undulations. CIUl6

Example 1.6-1. Lo;)ate the hydraulic jump in Examp.le 10-3 if the flow downstream . ~ , from the jlllnp is uniform. Solution. From the given data, the ,pecific-energy curve E = lJ + C!I. V~/2g and the curve F = 13Q;1gA + fA. of the channel may be constructed as sbown in '15-7, InComputing the spenific-force curve, the va.tue of fJ may be ~tima.ted as 1.04 tor IX = l.10 [using Eqs. (2-6) and (2-7)). Below the sluice, the M3 profile has been computed in Example 10-3, as ShOWll by AGB in Fig. 15-8. Using the curves in Fig. 15-7 and following the method described in Art. 3-7, the curve of sequent' depth A'F'B co~respondirig to tile curve of initial depth AGE ClJ,n be determined,l The curve A'F'B a.nd the downstream flow p::-ofile CF D (equa.! to the nOl'ma.l-depth line in this example) interSect at P'. The initial depth of flow at F' is then found from the M3 profile to be 1.70 ft. The (l/ll'responding F = 1.52 and, from Fig. 15-4, Lly. = 3.6, The length of the jump is therefore equlJ,l to L = 3.6 X 2,67 = 9.0 ft. At this point there is an approximation involved, becs.use the length of jump should be based on F a.t E (instead of F/l, which, however, is as yet unknown .. In this example ,I> is equal to. the OOrmal depth of flow in ,the channel since the flow downstream.is uniform: If the How downstream is not unif,Qrm but gra.dually varied, then the depth a.t the intersection F' of the downstream profile with the curve A'F'B should be taken as 'iI" This i.;;. also an approxima.tion, because the aqtua.l depth lJ1 should be at F, the position of which is as yet unknown. .

.\ JUMP AND ITS 'USE AS ENERGY DISSIPATOR.

II

is

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bR~ ~~Cc»'

.

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<-

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y

2~

if

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10 Scale (Dr E 600 Scole (," F

15·7, The E curve and F curve for the determination of jump location.

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M3-proI1l.e

. I

. lOwing to the difference between," and 13, the computed critica.l depths indicated ;by the specific-energy and specific-force CU.J:VBs are not precisely identica.l. However, the discrepancy is so small that it cnn be ignored. . FIG. 15-8. Location of a hydraulic jump.

404

, "

405

RAPIDLY VARIED ']i'LOW

''''P AND ITS USE AS ENERGY DISSIPATOR HYDRAULIC JU "',

.After the lrmgth of jump is determined, a hOI'izonta,i intercept EF equal to 9.6 ft iq found uetw",m the ourve A'fi"B and ·eF'D. The hydraulic jump will, therefore, occltr between G and fi', A.~ shown in Fig. 15-8, the jump appears to start at a dis. tance of about 140 it fronl the. vena oontracta. Since the location of the jump is determined, the n.pPI:oximations menf;ioned above can be checked, and more exact determination can be made by repeating the procedure if desired. Such verification seems unnecessary, \iowel'er, Oil account of the approxima.tions 'involved in the theory aud other aspects of the prohlem, .

t.han Y2. This means that the tailwateI' depth in' ?l1se '1 is decreased . As a result, the jump will recede downstream to a pomt ",-here ~q, (3~21) is again satisfied. This case must, if possible, be aVOlde~ lD deSign, because the jump, repelled from the . apr~n, will take place either on the ;oose rubble bed or, still m entIrely unprotected

15-8. Jump as Dissipator. From iL practical viewpoint, hydl'iLulic jump is a useful means of dissipating excess energy in l3upercritical flow. Its merit is in preventing possible erosion below overflow spillway:?, chutes, and for it quickly reduces the velocity of the flow on a· paved apl'on to a· point. where the flow becomes incfl.pable of scouring the downstream channel bed, The hydraulic ju~p used for energy dissipation is usually confined pa.rtly or entirely to a chunuel reach that is known as the stilling basin. The bottom of the basin is p~wed to resist scouring, In practice, the stilling basin is seldom designed to confine the entire length ofa free hydraulic jump on the paved apron, because such a basin would be too expensive. Consequently, necessoria!: to control the jump u·re usually installed in the basin. The purpose of surh control. is to shorten the range .within which the jump will take place and thus to reduce the size and cost of the basin. The control has additional advantages, for it improves the dissipation fLU1ction of the basin, stabilizes the jump action, and in some C!lSeS increases the factor of safety. In designing a stil:ing basin using hydraulio jump as energy dissipator, the following practical features should COllflidered. 1 A. Jump Position. There are three alternative patterns (Fig. 15-9) that allow a hydrnulic jump to form dr}wnstream from the source (such source as an overflow ilpillway, a chute, or a sluice): Case 1 represents the pattern iil which the tailwatel' depth Y2' is equal . to the depth Y2 sequent to YI. In this case, the values of F\, y), a.nd Y2' (=: Yl) will satisfy (3-21), and the jump will occur on a solid a.pron immediately a.he~Ld of the depth y 1. For ;;eour-protection PlU'POS~s, this is an ideal case, One big objection to this pattern, however, is tha.t a little difference between the·actual and assumed values of the relevanthydrauliD coefficients may cause the jump to move downstreani from its esti:r;llltted position. Consequently, some device to control the position of the jump is alw!1Ys necessary. Case 2 represents the in which the tailwater depth y{ is less l For sinlplicity, the lellgth of the' hydraulic jump will not be considered in the present discussion. See [251, [341, (35), [4S), and" Hydraulic. Energy Dissipators," by E, A. Elevatorski, Mc'Graw-Hill Book Company, Inc., New York, 1959, for further iniormatioll 011 the design of stilling basins.

CjJSi! I:

Yz

ccse 3: yi ,. Yz FIG. 15-9. Effect of tauwllter depth on the formntion of a hydraulic j lInlp below a w"ir . or sluice.

chm1llel, resulting in severe erosion. .The remedy for the design is to use a certain co~trol in the channel bottom, which will increase the tailwater depth and thus ensure a jump within t.he protected apron. Case 3 represents the pattern in which the tail \V£\.ter depth Y2' is greater than Ya. This means that the tailwater depth in case 1 is increased. As a result, the jump will be forced upstream and may finally be drowned out at the source, becoming a submerged jump. This is possibly the safest case in desi~w, because the position of the jump can be most readily fixed .. Unfortunately, the design is no~ efficient, for little energy will be dissIpated.'

., .

.•r

406

RAPIDLY VARIED FLOW

HYDRAULIC JUMP AND ITS T!SE AS ENERGY DISSIPATOR

B. Tail'W((,ter Conditions. In the above discussion it is assumed that, the tail water has a celtain fixed position, whether its depth YI' is equal to, less than, or greater than the sequent depth Yi. In most practical problems, however, the tai~water fil.!ctuates, owing to changes in discharge of flow in the channel. In such cases, a tailwater rating curve is U!lUaHy available to show the relation between the tailwatcl' stage Yz' and the discharge Q. In a similar way, a jump rating curve may ~e constructed y~;;equent de'plh,

yi'toilwa:er depth

.~<>ilL Coso. I

~

'"m, '" "",'

toilwcler rating

-'~rJU~"p 1

~

Clfin g

: , /; ~ I

~ ~

Discharge Q

Cose :3

jump will occur on the protected apron is to use sills to create a stilling basin (Art. 15-9). Class 3 represents the cot'ditions in which the jump rating curve is always at a lower stage than the tail water rating curve. This means that case 3 in Fig. 1.5-9 occurs at all times (i.e., the tail water is higher than the sequent depth) and that the jump will move upstream and will probably be drowned out at the Source. Consequently, little energy wilt be dissipatr:!d. An effective method to ensure a jump is to build a sloping apron above the channer·bed level (Art. 15-16). The slope of the apron can be such that proper conditions for a jump will be developed on the apron at aJl discharges. Another method is to provide a drop ill the channel floor. alJd thus to lower the tailwater depth (Art. 15-10). Class 4 represents the conditions in 'which the jump rating curve is at a higher stage than the tailwD,ter rating curve at low discharges but nt a lower stage at high discharges, An effective method to ensure n jump is to provide a stilling basin for forming a jump at low discharges and to combine with the basin a sloping apron for developing a satisfactory jump at high discharges. . Class 5 represents the conditiona in which the jump rating curve is at a lower stage than the tailwater rating curve fit low discharges but at a higher stage at high discharges. An effective method to ensure a' jump is to increase the tail water depth sufficiently high by providing a stilling pool, thUd forming a jump at high dischatges. C. J'urnp Types. In~, view of the various types of hydraulic jump df;lf!cribed in Art. 15-3, the U.S. Bureau of Reclamation [34,35J gives the following practical considerations: 1. All types of jump are encountered in the design of stilling ba5ins. . 2. The weak jump requires no baffles or special consideration. The Ollly requirement necessal'Y is to provide the propel' length of a pool, which is relatively short.. This can be determined from Fig. 15-4. 3. The oscillating jump,.frequently encountered in the design of canal structnres, diversion dams, ::mcl eVen outlet 'Yorlcs, i.s difficult to handle. If possible, jumps in the 2.5-to-4.5 Froude-number range shollid be avoided. In many cases use of this jump cannot be avoided but, in other cases, aitering of dimensiot's ma'y bring the jump into the desirable range. Baffle blocks 01' appurtenances are of little va:lue. Waves are the main source of difficulty; hence specially designed wave suppressors may be used to cope with them. 4. No particular difficulty is encountered in steady jumps. Arrange" ments of baffles and sills to be discussed later will be found valuable as a means of shortening the length of the stilling basin. 5. As the Fronde number inereases, the jump becomes more sensitive to taihvater depth, For Froude numbel;s as low as 8, a tailwater depth

Cose 2

"'0

\'

Tailwoter roting . .

Discharge Q

Cose 4

Cose 5

->-," 'C

<:

":::'

N

>.

'"'"co "

'"

~

'"

Disc horg e 0

'" ~ '"

407

'C

<:

":::' '"

"'" " (;, Discharge Q

FIG. 15-10. Classification of tailwater conditions for the design of scour-protection works.

to show the relation between the sequent depthY2 and the discharge Q; Because of the difference in the relative positions of the two rating curves, Leliavsky (43] has suggested that the design may be considered according to five different classes of condi tions (shown respectively by fLVe cases in . Fig. 15-10): Class 1 represents all ideal conditions in which the two rating curves always coincide. This means that case 1 in Fig. 15-9 exists at all times and that a jump will form at the desired place on a protective apron at all discharges. . Conditions of thisolass are mrely encountered naturally. Class 2 represents the conditions in whic.h the jump rating curve is always at a higher stagetha,n the taiIwater rating curve. ThiS means that case 2 in Fig. 15-9 exists at all times (i.e., the tail water depth is lower than the jump sequent depth) and that the jump will form at a certain place far downstream. An effective method of ensuring that the i

I

~.

I

·1

"

SENERGY DISSIPATOR

EniDRA.ULIC . JUMP AND ITS USE A

408

RAP'LDLY VARIED .FLOW

greater than the sequent depth is advisable to make certain that the jump will stay on Hm apron. 6. When the Froude number is than 10, a stilling basin using the jump a dissipator may no longer be the most economical dissipatiun device. In this case, the difference between the initial and sequent depths is great, and) generally spe&king, 8, very dj'lep basin with high basin may .not be retaining wall is required. The cost of the commensurate with the results obtained. A bucket of dissipator 1 may give comparable results at lower cost. 15-9. Control of Jump by Sills. The hydraulic jump can be controlled or effected by sills of various designs, such as sharp-crested weir, brcadcrested weir, and ~,brupt rise and drop in channel floor. The function of the sill is to ensure the formation of a jump and to control its position under all probable operating conditions. . Interesting experiments [46J have shown that the forces acting on the sill in a jump decrease rapidly to a minimum as the downstream end of the jump is moved upstream to a position approximately over the silL The force then increases slowly to a constunt ,;alue f.s the jump is moved farther upstream. This change in force on the sill is probably due to a change in the velocity distribut.ion from one end of the jump to the other, since llonuniform distribution of velocity is· a characteristic of such rapidly varied flow. As a result, the momentum in the nonuniformdistribution section is greatly increased. Theoretically: speaking, the control of hydraulic jump by s~lls can be analyzed by t.he momentum theory; Because of lack of accurate of the velocity distribu-" tim!., however, the theoretical analysis cannot predIct the quantitative result very closely. For useful design information one has to rely upon e..~perimental studies. . Dimensional a::mlysis shows that the relations the Froude. number Fl or F of the approachingflo\~) the height 12 of the approaching depth Vl, the depth Y2 upstream from the aill, the distance X from the toe aLthe jump to the sill, and the downstream depth 1/3 may be expressed as

as

:l

=,

~ ( F, ~, ~:)

This function can be determined quantitatively by model studies.

an

The

1 This is \Ipturned bucket provided a.t the toe of !I. sp'illway to deflect the ov·erflow up :through the tailw!l.ter. If tlle ta.i1water is high enough t:o submerge the bl~cket, :Ii. roller wiU form downstrelj.m from the bucket and tend to mo~e bed materials toward the dam. thus preventing Eel'io~!s scour at the toe of the dam:. Onthe other h!lD.d,if ~he ta.ilwater is low, the overflow will be thrown up and alit so th!lt it will strike a solid-rock channel, if any, at a safe distance below the dam; See [44), [45]1 p.nd Arts; 1<1-10 and 1 4 - 1 1 , · . .

409 ,

. ..,. . ·scont~olled by the sill, however, cannot be exact pOSItIOn of the Jump. a d l t d this positio.n can be repre. d 1 f cally In the rno e s u YI . t . cletermme ana Y .1 ,. X .d· The ratio is taken as constan sented by the ra:10 between. an "uY2~iellt to ~nsure a complete jump. in each tes~1 hav!llg a m~gn~tud~till~ basin should be made at lea~t In the deSIgn, the lengt .0 a ~ gh' er· the length of the basm .' X.'. }i'
9

. , F hIli !lnd Xly~ for FIG. 15-11. Experimental relal.lons among, t, (A.jler Fm'Bte1' "'Id Bhinc/e [23J.1 •

.

II.

".",,_~. re"C~Ll

weir.

'd d th t the high bottom velocities

T

~:!~~~~ ~:~i:S~:~:~:!~:;;:I v:lue ;ollsidered safe for the down- . .

mtatyh' be a ee stream channel condit.ion.

.

.

. . of ex )el'imental data

Control by Sharp-c1'e$ted Wetr. On the baSIS dI- 1 d ·d'a. .., d Sk 'nde [231 have eve ope a h an d th'Coretical analYSIS ' l' orst.er . an . n (1) Froude numb ei'F 0 f , (Fin- 15-11) showing the relatIOns D.mong· .' d h A

o· , b tween the well' heIght h an t e the appr~achln~ ~ow, (2~ ~;) ~~:l~ati~ between the distance X fram approachmg dept Yl, an. d th dep! tIl y upstream from the Well', h' to the well: an e 2 " toe 0 f t e Jump. 1 . f the effec. of a (TivBJ;l weir for known This pe~mrts an ana !~IS 0 ,."h t the norm.:'l.l tailwater anproach and tallwater condItions, pIovlded t a. 'st.t that iil pro• . ff t th discharge over t he, Well' Cle} <, ~~X:~ ~o;: +O~.;5~~ Or~inarilY, ul1su'bmerged conditions pteya~l ,over

glf.l.Dl

(15-3)

"

t?e

:33

. 1 This condition was first observed by Bazin (see reference [12] of Chap. 14) find . later demon:;trated theoreticn.l1.y by Bakhmetcfi [27]. , .

) 410

411

RAPIDlJYVARIED FLOW

HYDRAULIC JUMP AND ITS USE AS ENERGY DISSIPATOR

a wide l;allge of Thus, v.lvl in Eq. (15-3) can be dropped. If submergence occurs, the jump is forced upstream with possible drowning as a .result, In the diagram, any point is represented by a pair of coordinat~s (F,h/Vl)' If the point lies within t,he curves, hydniulic jump will occur, with its relative position indicated by the corresponding interpolated value of X/Y2. Points lying above and to the left of an interpolated curve represent the conditions under which the weir is t,oo nigh, so that the jump will be .forced upstream and possibly drowned at the sonrce. Points lying to the right,of the curve repre'sent the c011ditiol1s under which the weir is too low, so that the jump will be forced dO'wllstreal1l and possibly washed out. Y{hen the weir is as low as this, it ma~r be crossed only by a single undular surface rise, forming a standing swell (Fig. 15-6). For design purposes, it, is proposed that the curve X/Y2 = 5' in the diagram be used. Laboratory expedence has shown that the highest required weir does not necessarily occur under conditiollSOf maximum discharge [20]. Consequent.!}', the highest required weir should have the largest required value of h within the expected l'3.l1ge of as determined from the diagram, B. Control by Broad-crested Weir. For a br'Jad-crested weir, ift11e downstream depth is lower than the critical depth OIl wp of the weir, that is, if Ys < (2Y2 + h)/3, the tailwater iyillnot affect appreci~~bly the rela-· tion between the head water elevation and the Thus, .the discharge over a unit width of the weir can be written'

mental data, this curve may be usecL as. a guide in .proportioning broad-c;rested weir as. jump control, provided stilling basin Yl < (2Y2 + h)/3. A broad-cre.~ted weir has certain advantages in comparison with some other types of control. It has greater structural stability than a sharpcrested weir and usually requires lower cost of exc[1.vation than an abrupt rise.

q

= V IY1, H can be reduced to

·Since q

0.433

= Yz

2.667:F 2 (1 .

(

+ )

+ ya! }tIy2.) YI

.F .2 .S1 1:; ~ 2r---~+-----1-----~~~~--~-····~············---r----~-----1

6

(15-4)

8

9

F'IG. 15-12. Analytical relations between F and

"-hit

for a broad-cre~ted weir.

(A/tel'

Forslar and Skdr.de [23J.)

the above equation C .. ControZ by Abrupt Rise. From the experimental data, Forster and Skrinde (23J have developed a diagram (Fig, 15-13) showing the reh.ltions among F, Ya/Yl, and h/Yi of an abrupt rise for X 5(11 Ya). The conusing siste!lcy of the relations was verified by a theoretical the momentum theory, sImilar to that made for a broad-crested weir (Example 3-2). This diagram the prediction of the performance of a given abrupf, rise when Vi, illl Yz, YI, and 11. are known. In the diagram, a point (F,Y3/Yl) lying above the line Va "" Yz represents the condition of Y3 > Yz in which the abrupt rise would serve only to increa.se the dl'owllirlg effect, Fora point lying wit,hin the .experimental range between the lines for Ya Y2 and Y3 y., the position of· the point relative to the. corresponding h/y, curve indicates the effe6t of the abrupt rise on the flow pattern. Thus, if the point lies on the corresponding hllh curve, a jump will form with X = 5(h + Va). Irthe point lies 011 the left and above the curve, the rise. is too high, and the

+

(15-5)

(15-6)

This equation giv.es the relation between h/Yl and F and can be plotted . as the curve shown in Fig. 15-12. Forster :and Skrinde (23J have found that this curve coinddes ·with the: experimental data for an abrupt rise with y~ = Yo for X = 5(h + Ya)., Despite the lack of furtherexperiThis is

7

Values of F,

When a hydmulic jump is effecte.d by the weir, YZiYl can be related to F (15-5) becon1es through Eq. (3-21). Then,

l

4

, i

~.

~ 'H:;' . Yz h

II, and F

(l.

(3-17) except that 1" is replaced by 11"

J

.oj

..

\

412

HYDRAULIC JUMP AND ITS

RAPIDLY VARIED FLOW

tH3

'USE' AS ENERGY DISSIPATOR Jump troyels upslreom -

jump will be forced upstream and may finally be drowned. If the point is at the right of the curve, the rise is too low, and the jump will be forced downsti'eam toward the' abrupt rise and mayfinnJly be wil.shed out. If a point (F,Y3/Yl) lies in the diagram below the line Y3 = 1'" then the normal downstream flow is sl.lpercritical. A jump is followed by a critical section cre..'Lted over the crest of the abrupt rise, so that the rise acts as a weir, and the design diagram (Fig. 15-11) for the sharp-Cl'ested weir may be . used.

1

I

V1-

v,

Y, ----+-

1

, ~.

l \

16~

Volue s of F

FIG. 15-13. Experimental relations among F, YJ/V I, and 7t/y, for an abrupt rise. Forster and Shhule [23].)

(AJter

For design purposes, the diagram (Fig. 15-13) can be used to determine the necessary length and depth of stilling basin when V 1, Yr, and Ya are known. It is proposed that a point (F,1/3/Yl) be first defii,led for conditions at or near maximum discharge and tha,t the corresponding value of hjyl be determined by interpolation .. By repeating this procedure for other discharges within the expected range of discharge, a largest required value of h enD be obtained. This value should be used for the highest required rise. A minimum height. of rise necessary to prevent the jump from being washed out can also be thus determined. 15-10. Control of Jump by Abrupt Drop. The control of hydraulic jump by sills is useful if the downstream depth is smaller than the sequent depth for a normal jump. If the downstremu depth is btrger than the sequent depth for a normal jump, a drop in the channel flooi' must be used in order to ensme a jump. This condition occurs generaUy at the end of the expansion of R supel'critical flow. For Il. given appro
.

i4n~~-l 12

~"'--+---+-----t-.--

,;::- 10 f-.----+---j--.-

;,. '"

i :1--~qL___;~..w~-~

-f

----+----t-------1

~"i-3-~.:--

.' OO~~'~"--~--~2~-~~3

4L----::---;:s 5

F=vl/fol , •

1

FlO 15-14. Experin1antal and analytIcal Il:Dr~pt drop. . (After E. Y. Hsu [471.)

•

Ie

7

8

9

10

. I f I f ns among F Y';YL, and lI'Yl or an a10' .

I i

I

RAPIDLY VARIED FLOW.

414

will begin to· tr:'L\'ci do "'llS cream , Evidently, the drop does not control the jump in these two regions. The jum.p is stable and the drop is effective for its desired purpose only in regions 2 and 4. The intermediate region 3 represents an undular state of flow without a breaking front. By applying the continuity and momentum equations in an analysis similar totlutt made for the broad-crested weir (Example 3-2), HSll [47J has shown that, for region 2, (15-7) and, for region 4, .

,. i

, i·

j

(15-8)

These equations have been verified by experimen lcs. 1 The rebtions among F, Y./Yl, and h/Yl are shown in Fig. 15-14. In Fig. 15-14, each curve for a given h/YI has two relatively straight limbs connected by a short straight portion near the middle. The leftside limb represents the condition corresponding t.o rl~gion 2 and the right~ side limb represents the condition corresponding to region 4. This diagram may be used for design purposes to determine the relative height of drop required to stabilize a jump for any given combination of discharge, upstream depth, and downstream depth. 15-11. StillingBasins of Generalized Design. In important works or works that involve a lflrge number of stilling basins, generalize.d desigll:> for the basins are often necessary fol: economy a.nd to meet specific requirements. 'These designs can be developed through years of experience and observations on existing structures, or by model investigatiolls, or from both. The basins thus designed are usually provided with special appurtenances, including chute-blocks, sills, and. baffle piers. The chute blocks ~Lre used to form a serrated device at the entrr.nce to the stilling basin. Their function is to furrow the incoming jet and. lift a portion of it fr0111 the fioor, producing 11 shorter length of i ump thail would be possible without them. These blocks also tend to stabilize the jump and thus to improve its performance. The sill , either dentated 2 or solid, is usually provided at the end of. the stilling basin. Its function is to reduce further the length of the Jump and to control seoul'. For large basins that are designed for high inconl1

I\

F2 = liYa/Y1 [( ~ + 1)2 _ (J0.)2l 7't 1 - yJivl \ Y l 1 l 1 J

A simplified analysis and further experiments were also·made later by Moore and

Morgan [481. , The dentnted sill or serrated baffle is also known as the· Rehbock sm; because it was first patented by Prof. Theodor Rehbock after it was developed from experiments made during 1924 to 1927 at the Technical University of KarLsruhe, Germany.

HYDRAULIC JUjI,lP AND ITS USE AS ENERGY DISSIPATOR

415

ing velocities, the sill is usually dent::Ltcd to perform the additional function of diffusing the residual portion of the high-velocity jet that may reach the end of the basin. Baffle piers are blocks pla,ced in intermediate positions across the basin· floor. Their function is to dissipate energy mostly by impact action. The baffle piers are very useful in small structures with low incoming velocities. They are unsuitable,· however, where high velocities make c!witation possibie. In certaiIl circumstances, they must be designed to withstand impact from ice or floating debris. There are many generalized designs of stilling basin that use a hydraulic jump as the means of energy dissipation.! Three typical designs will be described in the subsequent artides: 1. The SAF basin. This is reCl)mmended for use on snutll scructUl'es such as small spillways, outlet works and small canal structures whBre FJ = 1.7 to 17. The reduction in ba~in length achieved through the use of appurtenances designed for the basin is about 80% (70 to 90%). 2. The USBR basin II. This is recommended for use on large structures such a,s large spillways, la.rge canal stnlCtures, etc., for FJ > 4.5. The jumwand-basin length is reduced about 33 % with the us~ of appurtenances. 3. The USBR basin IV. This is recommended for use with jumps of FI = 2.5 to 4.5, which usually OCCll!' on canal structures and diversion dams. This design reduces excessive waves created in imperfect jumps. It should be noted that these designs are only typical examples and applying them to stilling basins under t.hat caution should be used entirely different design conditions. The principle of the stilling basin applies also t.o the design of a canal drop (.01' canal fall), which is a sti'ucture built to secure the lowering of the water surface of a canal and the safe destruction of the energy so liberated. The canal drop is sometimes designed with a contracted width like the Parshall flume. Such a drop is known as a fiumed drop, which can economically be built together witha crossing bridge and used as a meter or a regulator as well [50,60-63]. 15-12. The SAF Stilling Basin. This basin (Fig. 15-15; SAF denol;es "Saint Anthony FalLs") W2.S developed at the St. Anthony Falls Hydraulic Laboratory, University of l\iinnesota, for use on sma.lJ ,drainage structures such as those built by the U.S, Soil Conservation Service. The design rules summ[Lrized by the investigator Blaisdell [22,64] are as follows: L The length LB of the stilling basin for Froude nuinbers between FI = 1.7 and FI = 17 is determined by LB' = 4.5ydF1G.76.

in

1 For mOl'e info!'.mation see [9], [25J, [34], [35], [43J, and [49J to [,58J. developed by the U.S. Army Corps of Engineers, see 1591.

For designs

.) j I

\

I (

1

·416

. RAPIDLY .vARIED FLOW

HYDRAULIC JUMP AND I'rs· USE AS ENERGY PrSSIPA'J,'OR

RECTANGuLAR STILLING Bil.SIN HALF-PLAN

,

\

I

j, I

l.

CENTERLINe:

TRAPEZOIDA;"

SECTION

RECTAN GULA~ STiLLING BASIN

i

it

DOWNSTREAM ELEVATION

Fw.·15-15. Proportions of he SAF ba.sin.

(U.S. Soil

'

C~senlation Service [64J.)

417

2. The height of the chute blocks ::md floor blocks is y 11 and the width . and spachlg are approximately 0.731/1. 3. The distance from the upstream end of the stilling basin to the floor blocks is L11/3. . 4. No floor block should· be placed closer to the side v;all than 3y/.8. 5. The floor blocks should, be pbced downstream from the VjJ"~""'''50 between the chute blocks. 6: The floor blocks should occupy between 4{J and 55$ of the stillinghasin width. . 7; The widths and of the floor blocks fOl'divel'ging stilling basins should be increased in proportiOll to the increase in stilling-basin width at the floor-block location. 8. The height of end sill is by c. O.07y 2, 'tV here y 2 is the theoretical sequent depth corresponding t,Q 1f1. g~ The depth of tailwater abov6 the stilling-basin floor is given by yl = (LlO - F 1 2/120)Y2, for 1.7 to 5.5; by YI' = O.85Y2, for F1 = 5.5 to 11; and by 'Y2' = (LOO FN800)Y2, for F1 = 11 t'o 17. 10. The height of the side wall 1lbove the maximum tailwater depth to be expected during the life of the structure is given by z = Y2/3. II. Wing walls should be equfll in height to the stilling-basin side walls, The top of the wing wall should have i1 slope of 1 on 1. 12. The. wing wall should be placed at an angle of 45° to the outlet center li.~1e. i3. 'l1.e stilling-basin side wail~ may be parallel {as in 1} rectangular stilling basin) 01' they may diverge as an extension of the transition sidl;) walls (as in a trapezoid~l stilling basin). 14. A cutoff wall of nominal depth should be used at the end of the stilling basin. . 15. The effect Qf entirained air should be neglected in the design of the stilling basin. 15-13. USER Stilling Basin II. From intensive studies of many existing structures and laboratory illvestigations, various tYPElS of generalized design of stilling basin have been developed by theD.S, Bureau of Reclamation [34,35}. L"SBR basin I is the basin· created by a· jump occurring on a flat floor with no appurtenances. This can be designed easily by following the principles descdbed in the' early D,l'ticles of this chapter. However; such a basin is usually not very pructic!).! because of its expeI)sive length and its lack of control. USER basin IIi is designed for a pu~pose similar to that of the SAF basill, but it has a higher factor of safety, which is neceSS!1ry for Bureau use. The pel'forrhance of this basin in~icates that the jump-and..:basilllength can be reduced about 60% . with the appurtenances, as compared .vith 80 % for .the SAF basin. Therefol'e, the SAF basin is shorter and more economical but, in cons&-

II


i

RAPIDLl VARIED FJ.,OW

'118

-I

.5

.b.·.D2

-\

VI F "--

I~ (c

I

1 (a]

28 26

I

24

V

i

2220

.

I ~.

--

(J, +8F}-; L",

' -:;c-=--

I

I /

i

18

I

."'-:;;'

I

14

.£

/

1 OJ/

/

/

'0/ . 0'/

I

I?~

1'/ /[

0

i

.j

/

I

/,/

I

1/

17 . /Sj/

I

;} /

! I I ii-r-

i

1<9;. I

I

\// i///

I

~6

I'

!

j; ",i

A.7Q~

1--..

I

///

I :'/.1 '< Mi~fml,lm -W depth /

2

11# .

12

, 1/;, ~?~ '/

10

/

4

6

'S

F,'

10

:0

8 6

4

(dl

/_M

~

\~

o

,-

..

0

..•

!

)

I

·2

4

1 i

f-~

..

;.........-

"/

2r-#"

16

/,1

~' /

I

14

!

10/'/

II

12

Vj ./gD,

.l

t----.

i 6

iO

(3

-F 1

~

12

14

16

18

20

V, .,.tgi), ~~

(b)

FIG. 15-16. Design curves and proportions of USBR ba.sin II. (U.S. Bureau of Reda.;nation [34].) fa} Definition of 5j·mbols; (b) minimum. tailwatet depths; (c) length .of hydraulic jump; (c) n.pprotimate water surface and pressure profiles (conjugate depth = sequent depth); (e) recommended proportions.

i

FlO. 16--16 (Connn!(.ed). 419

I

I' I

,

420

RAPIDLY VAlUED FLOW

quene!!',. has ~ lower safety factor. USER basin V is us6d where struc~ tU~'al er,onOlll,te8 dl~ti1.~e th6 lise .of a sloping a.Pl'Oll, usually on high~dam s~dlways. ~ he ?nnclple of deSign for hydraulic on sloping apron

:Vlll ~e descl'lbed 111 Art.. ,lS-16. USBR basins II and IV will be described III tillS and the next articles; respectively. . USBR basin II was developed foJ' stilling basins in common use for hlg~~dam a,~1(t eUl'th-dam spillways and for large canal structures. The . ba,sln contall1S {:hute blocks at the upstream end and a dentated sillnel1r the do:vl1stl'eo.m end. No baffle piers are used because the relatively high velo?ltles the jUl,:\lP might canSe cavitation on· piel;". The detailed and the data for COn1FJUtutiollS al'e shown in Fig. 1.5-16. The rules recommended for the design are as follows: 1. .s~t apron eleva.tio~l to utilize full sequent tailwater depth, plus ~n acl~led mctol' of sdety 1f needed. The dashed lines in Fig. 15-16b are gmdes dn1wl: for vtl,dous ratios of actUttl tailwaterdepth .to sequent depth. ,::itud1es of exis~il1g in:dicate that most of the b,usins wer<, de~lgn~(l for sequent. tallwatel' depth or 1,,8s. HowBver, there is a. lilI:cit, WhlCh.l~ ~overned by the curve labeled "Minin~um TW depth." TillS curve IndICates the point at which the front oithe jump mo\'cS [l:\~Tay from the chute blocks. In other words, any additional lowering of the tail. water depth would cause the ju:np to leave the 'basin; that would pro~uce u "sweep-out." For design purposes, the b!l.Sin should not be deSIgned for less than sequentclepth. For additional safety, in fact, the Bureau recommends that f, minimum safety margin of 5 % of be added to f,he sequent depth. .

2. Eosin II may be effec.tive dow::l to u Froude number of 4, but the lowcr values should not be. t.aken for granted. For lower values, W~:we supp!'esSlOll are recommended. 3. The length of basin can be obt~'l.ined from the length-of.-jump curve III 15-16.::.. . . . 4. T~e height o~ chute blocks is equ!!.l to the depth Dl of flow ent.ering the bl1S111. !he Width and spacing should,be approximately equal to D ; l however, tillS may be varied to eliminate the need for fractional blocks. . A· s'pac~ eqll~l to O.5D l is prefemble along; eMh wall to reduce spray and mamtam desJl'able pressul'e..'!. . . sill is equal to O.ZD~, and the rilaximum .5. The heighc of the wI~th and spacing recommended is approximately O.15D 2 • In this deslgn a block is recommended adjn.cent to each side wall (Fig. 16-16e). The slOPe of the continuous portion of the end sill is 2; 1. In the case of 1111rrow basi~" whic!.l would involve only a few dentates according to the a.b?v~ rule,' It IS adVisable to reduce the width and th.e spacing; provided thiS IS done pr'oportionally. Reducing the . .vidth and actually

ilYDRAULlC JU¥P AND ITS USE AS ENERGY DISSIPATOR

improves the peiorruance in narrow b!l.Sins; thus, the minimum width and spacing of the dentates is governed only by structural conside~·ations. 6. It is not lleCeSS::l.rv to the chute blocks and the sill dentates. In fact this practice is ~s1:ltl.llY inadvisable from' a construction standpoint. 7. The verification tests 011 basin II indicated no perceptible change in the stilling-basin action with respect to the s!ope of the chute preceding the basin. The slope of chute ":I1l'ied from 0.6: 1 to 2; 1 in these tests. Actually, the slope Of the chute does have 9.n on the hydraulic jump in sorile cases. l It is recommended tqat the sharp intersection between chute and basin apron be replaced with a cltrve of reasonable i'adius (R ~ '1,0,) when th'l slope of the chute. is 1; 1 or . Chute blocks can be incorporated on the curved face as readily as on the plane surface". On steep chutes the length of top surface on the chute blocks should be made sufficiently loug to deflect the Jet. The above rules ,,,ill result in a safe, cOl1servaUve stilLing basin for with fall up to ZOO ft and. for flows up M 500 cfs per foot of basin width, provided the jet entering the basin is I'easonably uniform both in velocity and in depth. For g,Tet\ter falls; larger unit discharges, 01' possible asymmetry, a model study of the specific design is recommended. 8. The approximate water-surface and pressure profiles of a jump in the basin are shown mI5-1M. Example 15-2. Pruportion a USBR llr.illing basin II for th.e overflow spillway designed in Example 14-1. The tailwater elevation is at El. 920.0. Solt,lion. Entering Fig. 14-15 with a h<;ad of 17.7 ft over the crest and !l. tot-al fall of 120 ft. the velocity of fiow at the toe of the spillwa.y is 79 fps. Thus, the depth of flow is 7;,000/(250 X 79) = 3.8 ft., !J.nd the Froude nuraber is 79/ = 7.13. Entering Fig. Hi-HliJ with f\ = 7.13, the solid line gives the ratio of '['IV depth to Dl as 9.7 .. As TW depth and sequent ta.i1wawl' d~pth D. a!e identical in this case, D.= 9.7 X 3.8 = 36.9 ft. The miniImnu tll.ihvater line for basin II on Fig. 15-11lb shows that 2. margin 'of safety of about 4 % can be' expec~ed for the I).bove Fronde number. The elevation of th~ basin floor ia placed at EI. 8~3.1. j~hus, the tobl fall will be 116.9 ft, but this will not change the flow velocity o.ppredr..bly. Should it be desired to. provide a IDru:gin of safety of 80/0, the following procedure ma.y be fol!owed. Consulting Fig. 15-16b, th6 li~8 for minimum TW depth for basin II gives TW depth/D, = 9.2 for F, = 7.13. The tnilwater d~pth a.t which sweep-out is incipient is TW •• = 9.2 X 3.8 = 35.0 ft. Adding 8% to this figuro, the stilling 37.S ft 'or basin should be positioned again for a tail water depth of 35.0 X LOS l.OZ4D.. The ba.sin floor will be placed at EJ. 882.2.

v3.8v

1 The slope of the chute has little effect on the jump as long as the velocity distribution and depth of tiowa.re reasonably uniform on entering the jump.. If the chute is long and fiat, the velocity maybe concentrated in one part of the flow section, resulting in an asymmetric8.1 jump with strollg side eddies. The operation of suell (l. .jump is expensive and should be a.voided in ail stilling basins. Also, when the angle of divergence of the chute is too la.rge for water tp follow properly, the jump will become rough and uncertain its position.

of

i

j

I

422

RAPIDLY VARIED FLOW

HYDRAULIC JUMP AND ITS USE AS ENERGY DISSIPATon

The length of basin can be obtained by entering the curve in Fig. 15-16c with F, = 7.13. Tllll~, LID, = 4.16, or L = Lll = 4.Ui X 36.9 = 154 ft. The height" wid th, and spacing of the chute blocks as recommended are D I; thus the dimension can be 3 £t 10 ill. The Might of the dentated sill is O.2D" or 7 ft 5 in., and th~ wicl~h anc1 spacing of the dent,ates can be O.ISD" or 5 ft 6 in.

15-15. The Straight Drop Spillway. The aerated free-fallillg nappe in a stralght drop spillway (Fig. 15-18) will reverse its curvature and turn smoot.hly into snperclitical flow on the apron. '. Consequently, a hydmulic jump may be formed downstream. Based on his own experimentai data

423

15-14. USER Stilling Basin IV. When F 1 =2.5 to 4.5, an, oscillating jump WIll be produced in ,the stilling basin, generating a wave that is difficult to dampen (Art. 15-3). USBR. basin IV (Fig. i5-H) is designed to combat. this problem by eliminating the w~ve at its SOUl'ce. 1 This is Imlmllllll//TIIlI//IlIII/I?/dIlIII/llwm ,..j---;---- L

-..j

Fro.I5-IS. Flow geometry of a straight drop spillway.

and those of Moore [4.0J and Bakhmeteff and Feodoroff [65J, Rand [6l1J found t.hat the flow geometry at straight drop spillways can, be described by functions of the drop numL'61', which is defined as

o Fractional

D=L gh

space'

3

-

VI = mox.tooth 'Hid Ih OJ

; - 5 p oce,: 2.5w

Top

o

~d

li'IG. 15-17. Pro,portions of USBR basin IV.

(U.S. B1}.realt of Recla.ma.tr:on [34].)

achieved bv intensifying the roller, which a.ppears in the upper portion of the jump "(Fig. 15-2), with directional jecs deflected from large chute 'blocks. The number of chute blocks shown iil Fig. 15-17 is the minimum required to serve the purpose. For better hydraulic performance, it is desil'fl.ble to construct the blocks narrower than indicated, preferably O.75D 1 , and t.o set the tailwatel' depth 5 to 10% greater than the sequent depth of the jump. The length of the basin is made equal to the length of the jump in a horizontal stilling basin without appurtenances and, chus can be determined from the curve in Fig. 15-4 (i.e., also equal to the l~ngth of USBR basin I or L 1 )., Basin IV is applicable to rectangular cross sections only. liThe Bureau has also developed alt!lrnative designs to substitute for basin IV, such ns the drop energy dissipator (Art.' 1&-15), wave suppressors, and the impact-type energy dissipator [34,35].

(15-9)

where q is the discharge per unit width of the crest of overfall, 9 i's the acceleration of gravity, and h is the height of the drop. Tho functions are

'surface on 5° slope

4.30Do.27

(15-10)

Vv h = 1 _. OODc·.22

(15-11)

~!l

(15-12)

=

=

0.54Do.m

~2 = 1.66Do.27

.

~

! \

I I

I I

I

."

1.,

J

(15-13)

where La is the drop length, that is, the distance from the drop waU to the position of 'the depth YI; Y'P is the pool depth under the napp'e; VI is the depth at the toe of the nappe or the beginning of the hydraulic jump; and Y2 is the tailwater depth sequent to Yl. The position of the depth Vl can be approximately determined by the straight line ABC which joins th.e point A on the apron at the position of Yl, the point B on the axis of the nappe at the height of pool depth, and the point C on the axis of the nappe lIt'the crest of the fall. The fact that these three points lie on a straight line was also verified by experiment. For a given height h and discharge q per unit width of the fall crest, ' the sequent depth Y2 and the drop length La can be computed by Eqs. (15-10) and (15-13). On the one hand, if the tailwater depth is less than

-1

,1

424

I)

r)

(J \

) ;

!)

r '

If

I)

UAPIDLY VARIED FIJOW

,HYDRAULIC J;UM:P AND ITS USE AS

Y2, the hydraulic jump will recede downstream. On the other hand, if the tailwater depth is greater thanY2, the jump will be submerged. As the trtilwater level rises, the spillway crest may be finally submerged. The spillway will still be effective if the submergence does not reach the control depth Oil the spillway crest. The upper surface of the submerged nappe lllay be assumed as a straight line tangent ito the, upper surfn.ce of the free nappe at the point w~lere the nappe plunges into the tailwater. The upper surface of the free nappe may be represented by the general equation given in Art. 14-l.

In the above discussiun it is assumed that the length of the spillway {)l"eBt is the same as the width of the approach channel. If '~he crest length is less than the width of the approach challll81, the {)ontl'ltction at the ends of tlle Spillway notch will be so great that the ends of the nappe may lane! beyond the stilljng~basin sidewalls and the eoncentration of high velocities at the center of the outlet may cause additional scour in the clownstrBam chnnnei. It is, therefore, important to design the npproach end properly by shaping the approach channel to reduce the effect of end contractions. The straight drop spillway is commonly installed in small drainage structlll'es by the U.S. Soil Conservation Service. The simplest Ioi'm of such a structure, known as the box inlet drop spillu1ay, is simply a rectlmgular box open at the top !end at the downstream end [67-69]. Storm runoff is directed to, the box by dikes and head \Valls, enters over the' upstream end and two sides, and leaves through .the open downstream end , lBading to t.. channel outlet. A generalized design has also been deve:oped by the.Service as a result of tests and anu.lyses at the St. Anthony lealls Hydraulic Laboratory [70,71]. , By placing a gridiron or grate on top of the straight drop spillway, the overfalling jet can be .separated into a number of long thin sheets of water which fall nearly vertically into the channel below., Thus the energy in the jet can be dissipated without ~'esort to t.he Use of hydraulic jump, n.nd. hence wave action can be l'educed if Fl = 2.5 to 4.5. This scheme hai:l been adopted by t.he U.S. Buren.u of Reclamation [34,35Jfor developing a so-c!1.lled drop energy dissipator as a substitute design for USBR basin IV (Art. 15-14). In this desi5n, the grate may be composed of a 'series of beams, such as steel ,rails, channel irons, or timbers, which form slots pamllel to the direction of flow. The width of the slots is equal'to two-thirds the width of the beams. If the mils' are tilted dow·nward ~t nn angle of 30 or more, the grate is self-cleaning. On the other hand,: if the grate is tilted upward, it can check the upstr~am water level but IIfay pose a cleaning problem. The length ,of the grate slots can be compllted by , L a-

4.1Q WN V2gYl

(15-14)

~NERGY

D1SSIPATOR .

where Q is the total: dischnrge in cfs, W is th,e width

~f

425

a space !n ft,

N is the number of spaces, g is the graV'itatiollnl: acceleratIOn, and YlIS the

;

i I

-I I

!

depth of flow in th(~ canal upstream. . ' . . 15-16. Jump in Sloping Channels. In the ~nalYSls of ~y:l1'auhc ]~,mps in sloping channels or channels having ap?recla~le slope, It IS essentld.l to consider the weight of water in the jump; III honzontnl chann~ls ~he effect of this weight is negligible. Thus, the~momentum form:llas lor Jlll~pS.O~ horizontal floor cmmut be applied straIghtforwardly to Jumps on ~'P~l!lb floor As will be shoi,vn in this article, however, the momentum P~l!lCIP.\~ can be used todel'ive a~l equation analogous to E~. (3-21), :VlllCh WI expenmentally. . . 'ical functio'l that has to be cleterDlll1ed COIl t am an e m p l l " h I de Enrly studies on hydraulic jumps in sloping c anne ~ wel'~ ~~a by Riegal and Beebe [9J and by Ellms [72,73], Later I1lVestlg~'LlOn~ were m:'de by Bakhmeteff and Matzke [74] nnd also by YameIl an Kindsvnt.er [75J. . '. f IIydril.ulic jump in sloping ch11l1ne!s may occur I~l ::anous orms, ~s . F' Case 1 is a typical form , but It IS not common III shown III jig. I'"J 10 -. • . , ~se" ' 2. to , 4 kn. own as drowned-out Jumps, pI.ac t'IC,al pI'oblems , " C'" . . t are common forms anci usually appear simply as jets of w~ter plunglllg m. ,o.a d'ownstream pool below the steep slop~. For pra;,tI~al PU~'poses, It~: believed that the solutions for the typIcal form 01 c_ase 1 a.nd fo: t drownecl-ollt jumps nre mlltually, applicable. Case b shows the Jump on' an adverse slope. This is a rare type of jump, and no adequate experimental data are ava.ilable !!.t the present moment. ' . For the analysis of the jlimp of case 1, a rectangulm~ channel of umt 'wi.dth is assuH),ed. Considering all effective forces paraLel to the channel bottom, the mOll'lentulll equation lllay be written

9!!!. ce 2Y2 g

- (3llll) = P l

-

P2

+ lYsin e -

Fj'

(3-14)

,

wh~re

Q = lI1d 1 Tt2 = V 1d l /d 2, P l = O.5'Wd I 2 cos 0, P2 = O,5wd 2 2 cos (j, F(, ·IS neg'I'IgI'bl e, 'a.n d (3 .\ and,~ (3 may , be taken 2 s unity. . ,If the .. . surface profile of the jump is a straight line, the weIght of w.ater I~l the Jump can be computed. The discrepancy between the straIght-lme and actual profiles and the effect of slope may be cOl'l'ect,ecl by a factor Ie Thus,. (15-15) Substituting Eq. (b-15)'ill Eq., (3-14), le,tting Fl

=

Y,/vgch; and.

1 The workBtal'tedi~ 1936 by David L. Yarnell at the Iow!1Institute bf IIydralllip n!;se:1rch, Iowa City, Iowa, was interrupted by his death in 1937,. ~he Y~rne~l datfl. were lent to the Tennessee Valley Aut,hority in 1939 for all extenSIV
426 simplifying,

where

(~y

_

427

HYDRAULIC JUMP .AND ITS USE AS ENF.RGY DlSSIP.:o\:rOR

R,A.PlDLY VARIED FLOW

Since d 1 = 111 cosO and d~ .= Yz cos fJ, Eq, (15-18) may also be written

(ZGZ

+ 1) ~ + 2G2= 0

(15-10)

~

Yl

(15-17)

G

.

! (

=

HC vi-+ 8G2 - 1) .'

(15-19)

The above two equations ai'e analogous to Eq. (3-21). Since G = f(F1,f)), these equations indicate that ddd~ l:md are functions of FI I,nd fJ •

'-I

I

TI 'e is a general belief [70J th,l,t J( and L/(dz - d 1 ) vary primarily with leI d h tl at G is a function of F I and 0, or. G = f(F 1,8). . F I o.n l ellee, h. 'Nith Eq. (3-20) for level-floor Jump" a If Eq. (1.5-16) IS .' r'd t Following the solution . '1 'ty between the two equatlollS :8 e\ 1 en . , SImI an ' ( - 16) . . .. ltlv for Eq. (3-20), the solution of 10- ) IS appa.eI •

H(V1 + 8CP -

(15-18)

1)

Co se I

\6

17

Ie

'9, 20

FIG. 15-20. Experiment!!.l relll.tions I::ietween F, lind lIz/Y, or d,/d J for jumps in sloping cha,nnels. '

case :}

Case 2.

Cose 4

The depth ratio y2/Yl or d2/d 1 oan be shown as a fUllCt~Ol1 of FI and 00 (Le. sin 8) by t...~e chart in Fig. 15-20, which is ba.sed .on the experimental data of Hickox [77], Kindsvater [751, Bakhmeteff and Matzke [74J, and the U.S. Bureau Reclamation [34,35}. Similarly, the relative length of jump L/ih may also be shown as a lunction of Fl aU;:l 00 and represented by curves based on the experimental data .of the Bureau, of ,Reqiamatioll (Fig, 15-21). The dashed lines indic:Lte the parts where the curves are not well defined by the available data. The diagrams in Figs. 15-20 and 15-21, are based on limited' experimental data ;with considerable' interpolation; nevertheless they provide useful il1io,rmation for practical purposes. The following rules for designing a stillip.g basin with sloping apron

or

.

co ••

5

FIG,: 15-19. HydT2.ulic jumps in sloping channels.

j

"1 '-

.J I

\ I

f

i)

428 )

I

:

)

429

RAPIDLY VARIED FL>OW

HYDRA.UL>IC JUMP AND l'.rS USE A.S ];}NERGY DISSIPA.TOR

(USER basin V) nee extracted fromrecommendntionsmade by the IT.S. Bureau of Recla.l1u1tion [34,35J:

6. The slc;rpe of the chute upstre:1mJrom a stilling basin has Uttleeffect on the jump u.s long as the distribution of velocity and depth of flow are reasonably uniform on entering the jump. 7. A snlall solid triangular sill with a sloping upstream surface, placed at the end of the apron, isthe only appurtenanGe needed. This serves to lift the flow as it leaves the apron and thus acts to conUol scour. Its dimensioas are not critical; the most effective height i" between 0.05 and 0.10 of the vertical distance Qf the sequent tailwatel' elevation above the bottom of&he toe of the jump, and the surface slope Chll be 3: 1 to 2: 1:

1. Determine an n.pron arra.ngement that will give the greatest , economy for the mtiximum dischn.rge condition. This is the governing factor and the only justifica.tion for using a sloping apron.

6

5

\ I

.

' 2l-+-I-;!'+-I-1)

Section A'A

'0 .

;:

:3

4

5

6

FlG. 15-21. Length in terms Bw'el:l1< of Redwnalion [34J.)

1

f

or

e

9

10

Ij

17

18

19

20

F, • 1r,/:I9d,

seq'.le~t

depth of jumps ill eloping

chllnll~ls.

(U.S,

2. Position the apron 80 that the front of the jump will form at the end of the slope for the maximum discharge a.nd tnihva~el' condition. 3. The portion of the jump to be confined on the stilliilg basin is a uecision for the uesigner,. depending on the qUlllit.y of the rnatel'in.l in the river bed and other conditions. The average ovel'-o.ll apron is about 130 % of the length of jump for the U1:1xi::num discharge condition. 4. With the apron designed properly for the maximum discharge COll: dition, the next 8t~p is to be certain that the tailwater depth [1.ud basin : length available for energy dissipation are sufficient for, say, one-quarter, one-half, and three-quarters of capacity. . 5. A hCll'izontnl apron will perform on par with the sloping apron for high Proude numbers, if propel' tailwater is provided. upstre~llll

\,

7

.,

FIG. 15.,22. Oblique hydf/mJic jtlmp .

I

15-17. The Oblique Jump. When a supercriticn,l flow is deflected inward to the course of the flow by a verticalbounchtry (Fig. 15-22), the depth of flow '111 will increase abruptly to a depth Yz along a wavefront CD which extends out from the point of boundary discontinuity at a wave angle {3 that depends in magnitude on the angle of deflection (J of the boundary. This phenomenon resembles a normal hydraulic jump but with the ohange in depth occurring along an oblique front; hence it may be called an obl'iq(~1i hydra'l.#c j1tmp.l When 8 = 0., oblique jump becomes the familiar nydraulic jump in which I;he wavefront is normal to the direction of ftow, or (3 90.°, Referring to the relationship of velocity vectors before the .iump in Fig. 15-22, the velocity normal to the wavefront if:1 V"I VI sill fJ w11ere V lis the velocity of flow before the jump. The Froude number normal to

I I"

I II

i

! I

.

1 For original infonnlltion see [781 and [79J. Tho oblique hydr!Lulic ju'mp or oblique. sta.nding wave is also known as the shock wave, by analogy to the C:1.Se in :mpersoniv flow of gases, The basic development of thi:5 subject was aacDmplisherl by Rouse and

White .[SDI.

t.

'-1

I

HYDRAULIC JUMP AND ITS USE AS ENERGY DISSIPATOR

RAPIDLY VARIED FLOW

431

the wD,vefront before the jump is, therefore,

F nl

=

V"l --= ~gYl

=

Y 1 sin (3 ~gYI

=

F

. (3

-113111

(15-20)

-

Considering a section A-A nOl'lnal tb the wavefront, it is seen that a normal hydraulic jump occurs in this section and that Eq. (3··21) call be applied. Substituting Eq, (15-20) for Fl in Eq. (3-21), the ratio of the sequent to initial depth is 1:!~ =

Y,

7Z( Vl+8F 1 2 sin~ -

1)

( 15-21)

This is the eqllation tlmt represents the condition for ~\:l oblique hydraulic jump to take place. Iteferring tQ Fig. 15-22, the ta,ngential velocit.ies before and after the jump are VII = TT "i/t::m (3 and V 12 = V,,2/tan «(3 - 0). Since no Jl1ornentum chfl,l1ge takes place patallel to the wavefront, these two velocities shou1d be equal, or V"I tan-f3 (15-22) 'V~; = tan ({3 - 0) By the condition of continuity 'VI V"I = Y! Y,,2, 'the above equation 08,n be written _ Y2 tan f3 (15-23) Y; = tB.n «(3 - 8)

I i

I I

,I

I

II

-\

'-

~.

Elimino,tingudUl from Eqs. (15-21) and (15-2:3), a relationship involving F 1; 0, and .13 is obtained:

I \

tan f3 (~1 + SF l ? t;;in (3 ....: 3) tfl,n 0 ~-~~==~~~~~~ - 2tan 2 {3 VI 8F l 2 sin 2 f3 -- 1 2

+

+

This equation should produce the value of t3 if F 1 and. e are g-iven. However, a direct solution of t.his equation for {3 in terms of Fl and e is practically impossible. Ippen [78J has prepared a four-quadrant graph _(Fig. 1:'5-23) showing all relationships expressed by Eqs. (15-21), (15-23), and (15-24). This gl'aph is 8elf-explanatory and can be used for the solut,ion of an oblique hydraulic jump. Since the oblique jump is a normal jump across the section A-A, the energy 10:0:0 in the oblique j'ump can be computed by Eq. (3-24). In practical problems involving an oblique jump, Y2!YI' is usually small. Thus, the head loss may often be neglected in design. Like~. normal jump, if vdvi < 2 (or F"I < 1.7, according to Art. 15-3), the oblique jump becomes undular. This' fact has been confirmed both experimentally and theoretically by It1pen [79J. It should be noted that an oblique jump will rarely oectll' alone in an

1 I

FIG. 15-23. Genera.! relations among Fl"e, /3, y,iy" CLnd F, for oblique hydraulic jumps. (Allel' A. T. lppen ~781.) _ _

ordimtrychannel, because multiple oblique jumps due to reflections on opposite ,-valls and intersections with other wavefronts will develop. Further discussion on this subject will be given in the sub5equent chapters.

15-1. Show that the {oilowing equation lIl[Lybe written for a hydraulic jump taking place in a. horizontal trapezoidilJ channel (some of the notation is given in Table 2-1):

il

r'

+ (2.51 + l)r' + (1.5t + l)(t + 1)1" + [D.W + (I

-

3F l ')(1

+ I)]r + 1)2

- 3F,'(t

where ,.

L ...

.- i

1 PROBLEMS

1

"':'/:":;'"

.-1

=

y,/y"

1 = b/ZYl, and Fl

=V,;VOy,.

= 0

(15-25)

432

15-2. Sh()w I,hat t.heequation written

,

L

,,

I I /.

(

i

I!

:

!

!

!

I

fOl"

. . , HYDRAULIC JUMP AND ITS USE AS ENERGY DISSIPATOR

now

a hydraulic jump. in a pa.rabulic channel mo.y be (15-2G)

Ydv'ilih. where r = Vo/lll and F', 16-3. Verify Eqs. (15-0 and (15-2). . 16-4. In Bidon~'s experiment it was found that, for V, = 5.59 fps and !II = 0.208 ft m a rectnngularchannel J the sequent depth y, was 0.613 fi" Determine' (a:) ~he alternate .depth, (b) the tbeoretic~.l sequent depthJ (I;) the energy ioss in the jump, .(d) t'he relative loss, '(e) ·the efficiency of the jump, (f) the relr,tive height. of the i limp, (g) the lengt.h of the jump, and (il) the type of jUlnp; . ' . lIi~5. In a !'ec~augular channel with b = 20 'ft, n '= 0.03, B,nd So = 0.04, the depth ?I un.lform flow IS 3 ft. A low dam which keeps 7 it of water immediately behiud it IS built at thee downstream end of the chanuel. If thtc backwater surface behind the do.m is assumed horizontal, find the pussibility of developing a hydraulic jump in the channel. ,If ~he jump will OCCllr, determine «(,:) the discharge over the spillway of the dam, (b) ,lie !ump heIght, (.G) the energy loss m the jlllnp, Cd) the efficiency of jump, and (e) the ·dlstali~.e of the Jump from the dam. 16-6. Locate the hY,d.raulic jump in Example 15-1 if the ch9.nnel has a free-overfall 01itlet.50Q it do\vnstream from the Y~na contracta. ·15-7.'\Vater flowing over a low spillway of bl'oael width passes on toa level concrete apron at 12 £t below the spillw"y crest, The head over the SpillW!e}, crest is 8 ft, and the tailwn.ter depth is 10 ft. Determine the location of thc jump. Use' Eq. 0'1--9) with C = 3.61 for computing the discharge n,nd Eq. (14-19) for computing tbe depth of Ho\\' at the spillway toe with a velocity coeffici
1~ 1

I I !

RAPIDLY VARIED

l;

. .

l' -----~

FIG. 15-24. A canal falL

I

~oss of 5 ~t by hydraulic jump. Determine the position of the hydraulic jump assuromg negligible channel friction and ignoring the effect of cha-nnei sLope on the jump. (HINT: Use the result obtained ill the preceding problem.)

433

16-12. A canal fall 40 ft long and c3:i'rying 240 cis is designed to dissipate a head loss of 4 ft .. The channei of the fall has a.slope ofl on 5 and diverges fr.om a width of 8 ft at the upstream end to 24 ft at th~ downstream end. Determine the loeation of the hydraulic jump, assuming a frictionless channel and ignoring the slope effect on the jump. (HIN'l': Divide the channel into, say, four reaches, n.nd COlll'pute El and E, at each section fol' f'>E = 4 ft.) . 15-13. Water flowing under a sluicc ga.te dischal'ges into a rectangular plain stilling basin h.oNing the same wideh ~3 the gat.a. After t.he contra~tion of the jet the flow has an average velocity of 8.0 ips and !1. depth of 6 ft. Determine (a) the s~quellt tailwater d'lpth, (b) the length of the b~"sin required to confine the jump, (e) the effectiveness of .the basin to dissipate energy (i.e., efficiency.of th'e jump), and (d) the type of iump to be expectetl. 15-14. Design the stilling basin for EX'ample 15-2, using jump control by (n) sharp-crested weir, (b) broa.d-crested weir, and (e) abrupt l'ise. 15-15. Design the st.illing basin for Exu.mple 15-2, using a jump control by abrupt drop. 15-16. Proportion an SAF stilling basin for Example 15-2. 15-17. Construot the jump rating curve at the 'toe of the spillwo.y designed in Example 14-1. . . Hi-i8. Discuss the development of an energy-dissipatiDn structure at the toe of the IOpillway under consider'lt·wn in the preceding problem, providecl tile tll.illl'ater rating curve can be r~presented' by (al Q = 40y', (b) Q = 100y', (c) Q = 2y', Dad (d) Q = 9tJOy"', 15-19.. Design the energy-dissipation structure, if needed, for the different tailwater conditions givcn in the preceding problem.

a

FIG. 15-25. A ladder of caacf,des . 15-20. Poggi [81,82] has proposed ll. ladder of cascades; as shown in Fig. 15-25. Design thi.s structure for the given datu.. 'fhe profile of the spillway crest is assuTIled of ,"YES shape. . . 15-21. Derive an equation of the upper sLlrface of the free n!~PJle over n. straight drop spillway. (H1N'l': See Art. 14-1, and note tr.at h. . . J11 = ~.3.) lli~22. A straight drop spillway is used io. a ditch to cffcet a drop of 6 ft in t.he ditch grade!. The dit~h is 14 it wide ami carries a nniform flow of 330 cfs. The spillway ha.s a crest length of 12.5 it and takes a hcad of 4 ft for the given discharge. The normal depth of flow in the downstream ditcn is 4 ft, but uncler flood conditio!lll backwater from a stream may mise the tailwater so that its level is 1.5 ~t above the spillway crestl Proportion the struchlre aSiSuming [l.ny type of stilling b~sin. The upstrenm a.pproach to the ~pillway is properly desiglled, and the velocity Head on the spillway ; crest' may be i g n o r e d . ' 15-23. Solve Prob. 15-13 ifthe floor slope of the stilling basin:is 0.10. ·15-24. Solve Prob. 15-11 if the sloping effect or the channel is'zonsidel'ed. 15-25. Design the stilling basinfol" Example 15-2 'on an apron with a slope of 0.15.

I

t

434

RAPIDLY VARIED FLOW

HYDRAULIC JUMP AND ITS USE AS ENERGYDISSIPATOR

15-26. In a flow 23'08 fps in velocity and 0.70 ft. deep, an oblique hydraulic jump is produc.ed by [l. deflecting vertical wall making a 15° angle with the direct.ion of the flow. Detet'mine (u) tha wave angle, (b) the sequent depth, and (0) the energy loss .

r

REFERENCES

I

L Giorgio Bidone: Obs.ervatiorl6 sur Ie hauteur du Iessaut hydralllique en 1818

2.

3.

4.

5.

13.

7.

8. 9.

10. 11.

12.

13.

(Observations on t·he he,ght of the hydraulic jump in i81S), a report presented at the Dec. 12, 1819, meeting of the Royal Academy oi Science of Turin and later incorporated as. a part qf [2], pp. 21-80. Giorgio Bidone: Experiences sur Ie remous et la propagation des ondes (Experiments on backwater and th'e propagation of w!1ves), illemorie deUa Realp-· Accarle1711:a delle Scienze di Torino, Turin, vol. 25, pp. 21-112, 1820. J. B. Belanger; "Essai sur Ill. solution r.mnerique de quelques problemes relatifs au mouvemcnt permanent des eallX COli l'a.ntes " ("Essay on the Numericai Solution of Some Prohlems Relati';e to the Steady Flow of Water"), Ce.rilian-Goeury, Paris, 1828. J. A. Ch. Bresse: "COIU'S de mecanique appliquee," .2e partie, HydrauiiquB (" Course in Applied Mechanics," pt. 2,' Hydraulics), rl'fa.llet-Bachelier, P"':'is, 1860. ' . H. Dtlrcy and H. Bazin: Re~herche$ exp~ri!llentale's relatives aux remous et a la propn.gatiop. des ondes (Experiment.al research 0v; l)ackwater and wave propa.gat.ion), in vol. II. of "Recherches hydrauliques" ("Hydraulic Researches"), A.c.ademie des Sciences, Paris, 1865. rvlansfield l'vlcrriman: II A Treatise on Hydraulics," John Wiley & Sons, New York, and Clul.pman'and Hall, London, 4th cd., 18\)4; 10tlr ed., 1916. Also see R. Ferriday; The hydraulic jUn:lP, ETI(!in.cer'ing New8-Record, vol. 34, no. 2, p. 28, July 11, 1895. . A. H. Gibson: The formation of stB.nding waves in an open strea.m; paper '4D81, Minules of Proceedings of the In3iitlttionof Civil Engineers, London, vol. 197, pt.' lII,pp. 233-242, 1913-14. . E. R. Kenn'ison: The hydraulic jump in open channel flow at high velocitles, Tmnsactions, Ameri::an Society of Civil Engl:neers, vol. 80, pp. 338-353, 19i5. Sberman M. Woodward: Theory of thehydl'[lulic jump and backwater curves; and Ross i\L Riegel and John C. Be.ebe: The hyifraulic jump as a means of dissipating energy, Iffimni Conser1Jan.c!' Distdel; Technical Repor!, pt. [] L, Dayton, Ohio, 1917, pp. 63-118. ' A. ICochand M. Carstanjen: "Von der Bewegung des ,"Vassers und den dabei auftretenden.KriHten" (" On Flow of Water anrl the Associated Forces"), SpringerVerlag, Berlin, 1926. E. G. W. Lindquist: Anordningal' ftir effektiv energieomvand1ing vid foten av iiverfalls{\:1mmal' (Arrangements for effective energy dissipation at the toes of dams), "Anniversary Volume," Royal Te.chnical University, Stockholm, Sweden, . . 1927. I{urt Safranez; Wechselsprung und die Energievernichtung des Wassel's (Hydrau- . lie jump !tnd energy dissipation of water), Del' BcJ.1tingenieur, Berlin, vol. 8, no. 49" pp. '898-905; no. 50, p. !J26, 1927. \ J.Einwachter: Berechnung der in der Wehrbreite gemessenen Liingenausdehnung von Dechialzen (Computation' of expansion of rollers with respect t.o width of weir) Wa.sserkraft und Wasserw}rlschafl, Munich, vol. 27, No. 14, pp. 157-159, alld no. 21, pp. 245-249, 1932; vol. 23, no. 17, pp. 200-202, Hi33.

I

II 1

i

I

I :r 'I

i

I I

!I •

435

14. J. Einwachter: Wassersprung- und Deckwltlzenlange (The length of the hydraulic jump and of the surface roller), Wa38erkrafl wui Wasserwirtschaft, Munich, vol. 30, no. 8,pp. 85-88, Apr. 17, 1935. . 15. J. Smetana: ExperimentaIni st.udie vodniho skoku (Experimental study' of hydraulic ,jump), C. 2-143, 1933; and Experimentalni studie vodllfho skoku vzcluteho (Experimental study of drowned hydraulic jump), ZPl'ri.uy lTe;'ejn~ Slttzby Technick~, Czechoslovakia, 193";'. " 16. J. Smetana: Model'll typ~s of movable dams: Great.est. dimensions attained in each t~'pe. Principles, of design of the fixed and movable parts of these dams and principles to be followed in their working, in order to reduce undeunining to a minimum, paper 51, 16lh Tnlemalionll! COnllre$3 of Nwigation, Brussels, pt. III, 1935. 17. Bcris A. Bakhmeteff and Arthur K l\.Iatzke: The hydra,ulic jump in terms of dynamic similr.,rity, T?'(I.nsact'ions, American Society of Cill.:l Enginears, vol. 101, pp. 630-647, 1936, ' , . 13. L. Escancle; Etude theorique et experime~tale de I'ecoulement. par vanne de fond (Theoretical and experimental study offlow through sluice gates), Revue g~n~ral.~ de l'hydmldique, Paris, vol. 4, no. 19, pp. 25-29, no. 20, pp. 72-79, and no. 21, pp: 120-128,1938; vol. 5, no. 25, pp. 21-34, no. 26, pp. 65-77, and 110. ·28, pp,131-139, 1930. / . " 19. D. Citrin~; II saito di Bidone (The hydraulic jump), UEnergia eletlrica, Milano, vol. 16, no. 6, pp. 441-465, June, and no. 7, pp. 517-527, July; 1939. Conr,ains a resume of work done up to Hl39. 20. Guido Nebbia: S11i dissipatod a saitO di Bidone: Ricerca 5perimentale (On dissipation by hydraulic jump: Experimental researches), L' Energ£a elellrica, 11/ilano, vol. 17, no. 6, pp. 325-356, June, 1940. . 21. CM] E. ICindsva.ter: 'rhe hydra.ulic jump in sioping channels, Tran$acliolls. American Society of Civil Engineers, vol. 109, pp. 1107-1120, 1944. 22. Fl:e~ W. ~:Iaisdel1; D.evelopmen~ and hydra.ulic design, Saint Anthony Fall~ stIll!llg baSin, Tra.nsacltons, AllImcan Soc!el-y of Civil Enllineer8, vol. '113, pp. 483520, 1948. . .' ' 23. John VIT, !orstel' an~ Raymond A. Skrinde; Control of the hydraulic jump by sills, Transacttons, Amer1cun. Society of Civil Engineers, vol. 115, pp. 973-\)87, 1950. 24. Hunt.er Rouse, T. T. Sia.o, and S. Nagaratnam: Turbulence characteristics of thll hydraulic jump, paper 1528, Proceedings, American SoC::ety of Civil Engineers J01l1'1lal, H.ydl·,w.lics Divi$ion., vol. 84, 00. HY1, pt. I, pp. 1-30, February, 1958.' 25. Til: st~Il.dmg wave or hydraulic jump, GO:Jernmen1. oj Tndia. Central Boal'd of hrtgai:.on and Power, Publication 7, Simla, India, 2d ed., Aug. 15, 1950. , 26.J. O'IDe Mello Flores: Le re.ssaut (The hydraulic jump), La Houille blanch~, Grenoole, 9th yr., 1'.0. 6, pp. 811-822, December, 1954. 27. Boris A. Bakhmcte/I; "Hydraulics of Open CI1[mnels," McGraw-Hili Book Compo.ny, Inc., New York, 1932. 28. A. G .. Levy and J. vi. Ellms: The hydraulic jump as a mixing device, Journal, Amencan TValenuork.$ Association, no. 1, pp. 1-23, January, 1927. , 29.~. ~. Kali.nske and James M. Robert,son: Closed conduit flow, in Entrainment of all' In flowmg water: A symposium, Traltsactions, American Sociely of Civil Engi'neers, vol. 108, pp. 1435-1447, 1943. . 30. G. H. Hie-ltox: Graphical solution for hydraulb jump, Civil Engineering, vol. 4, no. 5, p. 270, May, 1934. ' , 31. C . .T. Posey and P. S. Hs{ng: Hydraulic jump in trapezoidal cha.nnel!';, En[line(!'l'ing News-Record, vol. 121, no. 25, pp. 797-798, Dp.c. 22, 1938. . 32. E. "y. Lane and C. E. Kindsvater: Hydraulic jump hi. enclosed conduits, Engineenng NeTvs-RecQrd, vol. 121, no. 26, pp. 815-817, Dec. 29,1938.. .

-I,

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436

RAPIDLY VARIED FLO'W

33. J. C. SteHns: 'rhe hydraulic jump in stanMrd conduits. Civil En",,'ne' I .0 n '7 0 .., v' annll, vo . 3 , JlO. 1 , Pl'. ;)1.15-56, etober, 1933, 34. Research studics on s~illillg basin,q, energy dissipatol's, and associated appurterUHH:es, U,~, B'((rea1t of R£ciamatiorl, Hydral£lic L(tuor(t/,ory Reprn-t No, Hl'd-399, June I, 195.:., . . . 35, J, N. Bradley a~d A, J, Peterka: The Hydraulic design of stilling basins: Hydraulic Jumps On a hOI'IZOlltal nprQn (Basin I), paper 1401; High d!lIliS, em'th and hll'ge can~d S~l'llctures (Basin II), paper 1402; Short stilling basins for sLn~ctul'es, SIlUll! outlet works, aml sInnll spillw!l.Ys (Basih III), paper g03; b'SI~ and wO-ve SUppl'BSSIJI'S for canal structures, outlet wO!'ks, o.nd diversiOIl dams ~Basln IV!: ?404; Stilling basin with sloping apron (Basin V), paper H05; ,011l!l1l bl1sl!lS pipe or o:Jen channel ou~lets -no tailwatel' required (Easin VI) pap~r, 1406, ProCttcdi'r;gs, A'llLel'ican Scciety of Cicil Engill.ee"s, JOUI'fl,(l', H'ldr(wUc; 36, .i?1V~s·'~n,. ~Ol;~, no. HY5, pp. 1-2?, 1-14, 1-:2, 1-20, 1-32, llncll-11, 00tub~r, 1957. I,ll! C l.Oaha~ez. Untersuchltngeu uh":r de;) Wechselsprung (Researches relating tQ the l~y?rauhc JUlllp}, ,Del' BcmingenielL1', Berlil", yoL 10, no, 31, pp,64.9-651, 1£)20. A brter SlI1llll1,ru:y l~ given 1:' D~n?ld p, Bames: Length of hydraltlic jump investig~ted at Berim, CWll Enguzeenng, vol. 4, no .. 5, pp. 262-263, M~',y, 11)34, 37. l\.urt 1linge des Wassersprunges (Length of hydl'!l.ulic jUnto) Wasserk~'(!Jt una IV~~e~;cirschaft, Munich, vol. 28, no, 2,]" pp, 271-282, 1933: ' 38, 1\., Wo~e'~I(I. asserspnmg: Deckwalze und Ausfluss untel' eiuer Schi!tze (The hydraulic Jump: Its top roll and discharge through a. sluice gn.te) Warsaw 1931. 39, Y, I. Anwtn: Opredelenie dliny gidravlit;heskogo pl'yzhk!lo (The cl8termina'Liolt of th~ l~ngth of the h}'dr~ulic jump), I zvestiia 1'S€$O'hlZnl)!io N aHchno- Iss/edov"lel.lrogo ~nstt:1i'a G'l;arotekhn,',kt (Tra~8IJctions, AIl-Un:iotl Scie'atiji,c ReMarch Insl'itl,te of .IyrllUl.
v,

Go~energ()lzdat, Moscol". and Lellingrad, 11154, p, 337. . , 42, JulIUS \Vels!Jach: "Die Experimentalhydraulilc" (" Experimental Hvdraulics "', ' . . " Fl'eiberg, 1855, p. 52.

43, Serge Ldiavsky: "Il'l'igatiOll and Hydraulio " Chapman & Hall, Ltd., Lolld,:m, 1955, vol. 1. . 44. J. B, 'l'iffunj': Labol'atol'Y reseal'c.h applied to the hydraulic design of large dams U.S. }Vatcl'way$ EXp€1'im.er,t Btati01!, Bulletin 32, 1948. ' 45. 1\1. ~' 1'1'!.cPhel'son and. M. E, IC~rr: A s,tudy of bucket-type energy dissipat.or :hElI ~~.~n$tlcs, Pmceel.luI.9s, Amencan SOC'J.ety of Civil Enyilleers, J oumal, Hydrtm.ItCS Dwnton, paper 1266, vol. 83, no, 1-12, June, 1957, and correcti(:>ns, paper 110. llY4, pp. 57~i14.. August, . 46. Ralph 1\1. Dis311ssion on Control of the hydraulic jump by silla by Johu W. ~'orstel' and Raymond A, Skrinde, Tro.rw).c{iom, A,merican Societ~ Ci~il En(!~lteeJ's, voL llS, pp, 1003-1006, 1950, : 47, En-Yull' Hsu: Discussion on Control of the hydraulic jump by siils, by John '.V, Forster Il.lld Raymond A. Skrinde,Ttu:rIsactions, American SoCiel'l/ of Ci'vi! En!!ivol. 115, pp, 988-991, '1950, . ·18. Walter·L, Moore a~d Carl W., Morgan: The hydr.tulic jump at o.ll. abrupt. drop, pap:r,14<19, PrOCeed11lg8, Amencan Society of Civil Engineers, J 01<1''11(11, Hydraulics Dunmon, voL 83, no, RY6} pt. I, pp. 1-21, December 1957 49. William .. P, Cl'(l'\ger, Joel D. Justin, and Julian Hinds': "En~ineering for Dams," John Wiley & Sons, Inc., New York, 1945, vol. I, pp. 73-89.

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I I

HYDRA.ULIC JUMP A.ND ITS USE A.S EN:E[tGY DISSIPATOR

437

.50, Calvin Victor Davis (editor-in-chie£): "Handbonk of Applied Hydraulics," 2d eeL, l\'IGGn'w-HiIl Book Company, Inc" New Yol'l" 1~52, pp, 281-288 !,ud 813-846, 51. Civil and strllctural design, vol. 1 in "Design of TVA Projects," 'l'ennes888 Valley "lutho1'it1}, Technical Report 24, 11)52, pp, 41-60, 52. Armin Schoklitsch: "Hydraulic StrUiJtlire.>," tr!tnsla.t~'d frolll the German by Samuel Shulits, American Society of Me0hunico.\ Engineers, New' York, 1937, . vol. 2, pp. \)13-926. 53. Armin Schoklitsch: "H::mdbuch des WaSSel'balleS" ("Handbook of HydraulirJ 2, pp, 816-825. Erl(:;ineeriug"), Springer-Verlag, Vienna, 1952, 54. C, Maxwell Stanley: Study of stil!ing··basin Tmnsacl·ions, i!mer'icm. Socie/y of Civil E'lyf,neers, vol. 99, pp. 490-523, 1934. . 55. JacDhE. Wal'lloclc: Spillways and energy dissipl1tol's, PmceBdings of J!yelrauh.cs Confer'ence, 'Uni(leTsity of lawa, Studies in Eng~'nlle1'infl, Bulle/in 20, 1940, pp. 142-15(). 56. John R. FreeIr.an (editor): "HydrauU" Laboratory Practice," American Society of Mechanical Engineers, Ne\v York, 1929. 57. Ahmed. Shukry.: The efficacy of fiool"siils uncleI' drowned hydraulic jumps,paper 1260, P1'oceed.ings, Ammcnn Society of CiLil Engii1.~eta, Journal, }J.r;elrtmlics D'ivision, vol. 83, no. HY3,. pp. 1-18, Jnne', 1957, . 58. L. E,scande: L'Ewde sur modele l'Muit des OIIVl'!l.ges de rupture de charge (The stlldr of en!Orgy dissipators with the aid of a,small-sca.le modell, Le Gdnio civil, vol. 115, no. 25, pp, 429-433, Dec. 16, H139. 59. R. H. Berryhill: Stilling basin experiences of the Corpa of Engineers, .paper 121H, Proceedings, A,merimn Soc'iety of CiliU Engi,,'.e.ers, Jo!mu,l, Hydmtlhcs Dirrisiolt, vol. 83, no. HY3, pp, I-Sa; June, 1957. 60, B. S, Talwani a.nd S. T, Ghotauk..1.l': Design of canal [ails, I""iqalio'n and P01uer, J o1
SCS.'l'P-79, May, 1949, 65. Boris '.'1.., Bakhmeteff and N. V. FeodofofI: Discussion ali Energy los.~ at the b2.ge of free QI'erfail, by Walter L, 'Moore, Tranzar,t·ions, Amp.rican Society of Civil Enginecr$, vol. 108, pp. 1364-l373, 1943. G6, Wa.tt~r Rand: Flow geometry at stmight drop spillwa.ys, paper 7tH, Proceedings, American Bociety of Civil Engtneel8, vol. 81, pp. 1-13, September, 1U55. 57. Charles A. Donnelly: Desi;;:;n of an outlet fOI' hox' inlet drop spillwa.ys, U,S, Comet'jjet/·ion Service, Report SCS-TP-63, November, 19-17. 138. Fred W. BIltisdell and Charles A. Donnelly; Capacity of box inlet drop spillwn.ys uuder free and submerged ./low conditions, Uni1lersity of J1fin1!.j:sota, SI. AntllOny FaUs Hyd7'alilic Laboratory, Technical Paper 7, Janul,1.l'Y, 1951. Gg. Fred W. Blaisdell and Chltrles A, DDnnelly: Hydra.ulic design of the bOl( inlet dmp spillway, Uni;;er,sity of lI1inrtesola, SI. Anthony Falls Hydraulic Laoon:uo'''Y,Ted!n':cal Paper 8, Jl1.nua.ry, 11)51; also U.S. Soil Con~e""Mian Service, Report BCS·TP106, July, 1951. ' 10, Charle. A. Donnelly and Fred W. Blaisdell: Straight drop spillway sLilling bo.sin,

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438

RAPIDLY VARIED FLOW

Unive'l'sil!l of Minnesota, St. Anthony PaJJ..~ HydrauUc Laborcti.o1'lJ, Techniw[ Paper 15, Sel·. B, November, 1954. ' 71. Fred W. Blaisdell and Charles A. Dovneily: The box inlet drop spillway !lnd its outlet, Tnl1lsaclions, Amel'ican. SocictyoJ Civil Engilleel'S, vol. 121, pp. 955-986, 1\)56. 72, Il" W. ElIms: Computa.tion of tail-water depth of the hydraulic jump in sloping flumes, paper Hyd. 50-5. Tl'anMcHon.s, A merict1n Society of 111 echa!,ic"l EngilLce7'll, vol. 50, no, 3G, p,p, 1-6, S"ptember-Decembel', 1928. "

73. R. W. Ellms: Hydraulic jump in sloping and borizontallbmes,paper Hyd, 54-6, Tmn~actiQns, klnerican Soddy of ilIechanlcru E'?lgineers, 1r OI. 54, nb. 22, PI1. 113n9, Nov. 30, 1932.' , ' 74, B. A. Bakhmeteff and A. E. Mat~kB: The hydraulic jump in sloped channels, po,per Hyd. 60-1, Tm1tSa.clions, Amel'iccm Sociely of M echanicai Engineer", vol. 60, DO. 2, pp. 111-118, February, 1938. 75. Cf1.l'l 11:. KinJsvo.t.er: The hydraulic jump in sloping channels, Transactions, A'II!eric(U~ 8ode/y of Cl:vil Enginee.l·s, vol. 109, pp. 1107-1120, 1944. 76. J. C, S~evens: Discussi.:.n on The hydl'3.ulic jump in sloping cha.nnels, by Ca!'l E. I{indsv:tter, Tl·ClllSa..;Uons,' A.llterican Society Civil Engineers, ~ol. 109, pp. 1125-1135, HH4. . . 77. G. It. Hickox: Discussion on The h}'clraulic jump in sloping channels, by Cad E. Kindsvater, T)'a1t8r.L~ti{}n.s, Am,el'ican Society of Cioil Engineers, vol. 109, pp. 1141-1145, 1044. . . 1ft Arthur T. Ippen: Mechanics of 5upercl'iticII! flow, lSt paper of High-velocity flow in open channels: A symposiulll, Transactions, American Society of Civil Engineers, vol. 116, pp. 268-2Q5, H/Sl. . 79. Arthur T. lppcn and Donald R. F. Harleman: 'Verificatio!l of theory for oblique st.anding waves,-:l"·af1.Sac{.·ions, Ameriwn Society of CivU Engineers, I'D\. 121, pp, r,78-094, 195(i. ~ , 80. Hunter Rouse: "Fluid Mechanics for Hydraulic Engineers," McGra.w-HHl Book Company, Inc.; New York, 1938.' , ' , . . 81. Bruno Poggi: Sopra gli sco.ricatori a sca.la di stramazzi (On the flow in a ladder of cascades), L' Ene1"gia elettric(l, Mila.no, val. 26, ao. 10, pp. 600-604, Od,ober, 1949. 82, Bruno Poggi: Lo scaricatori a scala di stramazzi (FiGW in a ladder of cascadtt,)' L' EnergiCt eleUrica,ilfi/a.no, vol. 33, no. 1., pp. a3-40, January, 1956. This pa.per describes the laboratory tests.

II

'I

,.j

CHAPTER

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16

FLOW IN CHANNELS OF NONLIN]!:AR ALIGNMENT

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>1 16-1, Nature of the Flow. The presence of curveI'; or bends in aligi1Hlent is unavoidable in the design of open channels. Difficulties in design often ariee because of the complexit,y of the flow around a curved path. The streamlines of the flow are not only curvilinear but also interwoven, resulting in spiral cmrellts and cross waves. Furthermore, the cent::-ifugal force acting on the flow around a bend produces, a unique feature known as su.perelevation, that is,a rise ill the water surface at the outer bank with an accompanying lowering at the inner bank. Also, the velocity distribution in the channel sec~ions in the bend is very irregular and the coefficients a and fJ are usuu.lly far greater than unity. In ch!1nnels of nonlinear alignment" flows behave differently according to the state .of the flow. Generally spe:l.king, subcritical flow shows smooth water surface and slight superelevation, ",haren,s supercriticr.l fio,w exhibits characteristic cross-wave-disturbance patterns on the surface and thus exaggerates the superelevation. In the study of subcritica1 flow, the spiral currents are of primary inter- . est. These currents are essentiaUy a friction phenomenon; hence their analysis requires use of the Reynolds number as a pars,meter. , In the study of sl1percriticD.i flow, the formation of cross waves is of major COMel'll. These waves represent the gravity effect of the free surface upon the flow; hence their analysis willi'ely on the.use of Fronde number as a basic parameter. 16-2. Spiral Flow, Spiral flo'w l'efersto movement of water particles along a helical path in the general direction of the flow. Thus, in addit.ion to the major velocity component normal to the channel cross section, . there are transverBe velocity components on the cross section. These trarisverse components will create so-called secondary flow in the plane of the croas section. Spiral fiow in curved channels was first observed by Thomson [IJ in 1876. Since then many studies have been .made [2-7J. It is believed that. this phenomenon is due mainly to (1) friction on the channel walls, which causes higher filarnental velocities near the center of 'the channel th.3.!l near the walls; (2) centrifugal force, which deflects the particles of water from straight-line motion; and (3) a vertical velocity distribution 439

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which in the 'appl'or,ch chan~lel and thus initiates spiral motion in the flow, The centrifugal force is responsible also for tl~e superelevl1tiOl1 in the flow surface, . It is generally knowr, that, looking downstream, a chM1Uel curve to the right causes a counterclockwise spiral, \Yherei),~.a cht:l,llnel curve to the left causes. a clockwise spir~1L. If the curve is followed by a long tnngent, the spiral flow developed 1\1 the curve will for somE: distance dowllstream. The actual pa.ttern of a spiral flow is cornplicatedand three-dimensionaL In order to record th:o actual flow pattel'll, it is desirable to use a specially designed instrument that can measure dil'cctlv the velocity :lUch [is the pit~t. used components in different coordinate by Shulery [2]. Spiral flow exists in straight channels (Art. 2-4) £.3 :welt IlS in curved channels. In a curved channel, ho\vever, the spi!'f:.l flow lnduced by the centrifugal force is "Vary pronounced L1.nd irregulaF the bend. The co::nplicate~ pattern of flow is caused by thB interference of the spiral flow ongmated m the strR.ightapproach channel with that generated in the curve. The strongest lateral currents usually o.ppC\UI' close to the outer· 1'[,\11 f1.t the mid-section of the cm"'le, where they h,w'3 n. general direction toward the outside of the curve. Their direction rend position cha.nge· gradually through the second half of the curve until the exi.t is reached, . where they approach the inside wall with upward inclination. In ordpl' to delilleate the magnitude and effect of the spilil\l flow in different curves under v(1.rying conditions of flow,Shukry [2] hn~ used a term known as the slre:nglh a sp'iral flow. This term is defined as th.~ pe\'~ent!l.ge ratio of the mean kinetic energy of theJateral motion to the total ldn8tic onergy of flow at a given cross section. It is to be noted that, the kinetic energy of flow depends Oll the sqtmre of the velocity. Re.fel'rin o· to the channel cross section represented the xy plane in Fig. 2-6, th~ strength'of the spiral flow at this section is (H;-I)

where V,yis the men.n-vEllocity vector projected 011 the xy plD.neand F is the mean velocity in the section. ; Thus, for a flow with all streamlines O. parallel to the axis of the channel, SZY The pitot sphere designed by ShukrYi has'five bl'llSS tubes encased ill a ·~asing stem. has a small br!Lss to.pping n.t the top enn to pel'lilit a rUbber-tuhingconnecttrlll Wit" !l pl'e:~$ul'e nlanOl11ete~'" At t!le bottcm end the tube is bent inside a sphere (\nd then termmates a.t a.aertam hole lI1' the shell of the sphere, which thus ()ontRin~ five l:oles.: The openings of the holes a:re strategically located on the'stlria.ce of the sphere so tha.t, by rotating the sphere and with c!Llibration, the dil'ecti~n II.nd ma.gnitude of 3. yeloclty vector of flow ca.ll be determined, 1

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From the experimental results obtained by'Shukry for subcl'itic:11 flow l1round ~ liend in a rectanglllal' flome, ~he following are noted: • 1. S*u is compa~:(\tively. rtt low R of the approach flow, but clecreases considenl.bly by illCl'ensing R. . 2. S~y decreases gradually with the increase of the radius-width ratio rJb mid attains practically its minimum l11n.gnitude (I.e" the curve effect [I,ppro!Lches the least amount) .a.t 7 c /b =: 3.0. 3. S%y decreases as the depth-width ratio. ulb increases. ,1,. increases as the de\,iation (j of the curve becames large. For the range of 8/180 0 from 0.0 (straight channel) to 0.5, the inerease in is ne::wly twice that for the ro,nge from 0.5 to-LO. . . . 5. The kinetic energy of the btel~al currents ill a curve is reln.tively 8m.all comfH1.red with the ellergy in the. longitudinal current.s ~1.ncl, consequently, ptays only a minor pM'~ in t he energy lo.ss du~ to. te~1Cl.resistu.nce. Spirnl flow occurs in no,tural rivers as well as \11 artifiCIal channels [5]. Howi?ver, it mfl.)' not exist at all ill a curved channel if th~ strength of the spiral motion is so weak that its. effect i~ pr[l.ctically elimll1utedby the r,h::mnel friction. This is the case in mo.ny l1:1tuml rivers whel'e the ratio or depth to width is small (henc~, is llmall) and where the forces teJ;l.cling t.o produce spiralHovv Me overcome by the complicated forces . reflUlting from bed and btl.nk l'oughne'3ses [6,7J. 16-3, Energy Loss. AccQrding to. IV!lHler [81, the energy line and flow profile in a uniform cUl'v'ed chan.nel may be show'l1 as ill 16-1. Ca.se 1 illustrates the Imbc;'itical flow in a curve bet\veen two tnngentch::mriels. Without the curve. fio.w would occur :J.t the· nOl'nml depth y" corresponding to the pal'ticuiar discharge. With the Clll've, the energy line at the beginning A or the curve is raised by [1.11 amount hr. A major p[~rt. of this energy is dissipated ovel' the length of the curve. The l'eml),\I1\llg part is can-jed over a distance L' in the down"tret\l11 channel BB', which is required for the How to revert to norma.\. The slope or the energy line between A [md B' is gre~btel' than the bottom slope S~, aud the energy line meets the energy line for the normal flo';'{ it B'. In oreler to I:aise. the. energy line at the point A a.bove the Ilormal line, wuter must be ba.cked up in the chal'in.'el upstream from A. Fi'om the specific-energy curve of the flow in the Channel, it can be shown that a r1se of hJ in the energy line requU'es a corresponding rise in water surface by an amount /::'y which is greater than hj. This rise in water surfa::e indicates that the presence of a. bend in a channel of subcriticn.l flow has abackwater effect ~imilar to that of a weir or dam. For the computation of the backW!l,te~' profile, the start,ipg point m.:t.y be set at A ~vith a depth eqtU11 to y" -t; /::,y. The bD-ckwater profile is of the 1111 type, Wflich extends upstream frorn A and is asymptoti'e to the norma.l-depth line. . Case 2 illustrates 8upercriticaL flow in. the curved channel. It ca'n be

442

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FLOW IN CHANNELS OF NONLINEAIL ALIGl'It.1EN'f

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L-~,__~~~b~,_~.__~o~~J'__~

023456780 4 Reynclds' number R (mult:plied by 10 )

0,25 0.50 0.75 Rotio of ongle of curval"r.

1.00

1.20

'\

:

1,10

1---

1.00 ..:! 0.90

.,

! 1

I

!,

~l~"~

,

I, I

'1/

l-J

Case :0

FIG. 16-1. Energy line and flow pl'ofile around a curve.

II

water slll'f3.ce l'is~s above the critical-depth line, a hydraulic jump will be

(

~~~.

.

g 0.80 ~

~~

~

0.70

,

I

'E 0.60

to

~;:; a

u

0.50 0,40

0.30 0,20

J

0,10

I

.

Case 3 illustrates 8upercritical flow when the normal depth is only slightly below the critical depth. The wavy surface in the upsti'eam channel is due tothe fact that the water surlace is raised above the critical depth, so-that an unClular jump is produced. The total enel'gyloss due to curve resistance can be expressed in terms

\

3 4 5 6 7 8 0.50 ReynOlds' numbe, R (multiplied by 104 )

2.

3.00

it

0.70 0.90 110 Depth-brealh rolio

FIG. 113-:2. Experimental parametric functions of the coefficient (Aft~r A. Shukry [2J.)

t

1.30

of curve resist.a.nce.

445

FLOW IN CHANNELS OF NONLINEAR ALIGNMENT

444

HAPIDLY VARIED FLOW

of the velocity head:

V2

(16-2)

fc 2g

where l' is the mean velocity in the section and Ie is the coeffiCient of Clll've resistance. The coefficient fe varies cOl1sider(tbly with each of the parameters R of the approach flow, ?'e/b, y/b, and 8/180°. Fumilies of curves for these parameters based all the experimellt,~ made by Shuki'Y [21 (see [9] (tnd [lOJ for other studies) are shown in Fig. 16-2. The curves CM be used to determine approxim[,tely the value of to in sm.ooth curved channels. For allY given c(tse, fe is obtnined by first fixing its v(tlue with respect to two variables and then adju~tillg with respect to the third and' fourth vm·j(tbles.

:)

:I :.'!

Example 16-1. Given R = 55,500, "c/b = 1.30, y/b = 0.8, and 0/180° = 0.556. Determine j" Sohltion. First, keeping y/b = 1.00 and 0/180° = 0.50, it is found that, for R = 55,500 and ",/b = 1.30,f,'~0.200 (Fig, 16-2c). Then, keeping l'clb = 1.00 and 8/180 0 = 0.50, it is found that, for R = 55,500 and y/b "" 1.00, j, = 0.230 and ~hnt, for R = 55,500 and y/b 0.80, j, = 0,275 (Fig. 16, 2b), Adjusting the condit,ion of y/b = l.00 to that of u/b = Q,80, the conected coefficient = 0.200 X 0,275/0.230 = 0.239.' Now, keeping y/b LOO and relb = 1.00, it is found that, for R = 55,500 and 8/180· = 0.556, j, = 0.245 (Fig, 16-2a), Similariy, adjus~ing t,lle condition of 8/180· 0.50 to that of 8/180° = 0.556, the finally' corrected coefficient = 0.239 X 0.2'15/0.230 0.::\55, say, 0,26, The above procedure of interpolation is shown in Table 16-1. TABLE

16-1.

DETER~f1NA'rloN 01.' THE 'COEFFICIEN'r Ol' CURVE REsrSTANCE BY L~TER['()LA'I'lON

Step

y/b

II/180°

R

r,/b

(1) (2) (3) (4)

1. 00

0.50 . 0.50 0.50 0.50 0.556 0,556

55,500 55,500 55,500 55,500

1.30 1.00 1.00' 1,2·0 1,00 1. 30

(5)

1.00 0.80 0.80 1.00

(6)

0.80

55,500: 55,500

i i

I

j,

a

~n

~

0/

~~-1Z----z=::J=

if.!I R~o,,~,; I Ii ("'s$'O{) • \ I 1

:~

I\

~

,/~/

\

~\'\, ~

I

9 10

'

60--1-

,1

5

_~ _ ':I{\j-',!<:! a 7~

(b)

FORWARD VELOCITIES

_~_Cer,terhr
I s

---PQ/:I':~~~~~~l~

,~

20 ,07'0==?O-!~--___t:\oO~

-----7'J~

. -_ ______ :-__ _

,f-~:"L-1_- -~-~ep 1_ -==--_-~~

'"

"

/":. "-~ "'0 I"~;).')

1

0,200 : By Fig, 16-2c 0,230 By Fig. 16-2b 0.275 By Fig, 16-21> 0.239 :j3y interp"O!ation: (1)[(3)/(2)J 0,245 By Fig, 16-Za 0.255 i By interpolatioll: (4)[(5)/(2)1

~

701

ij<~/ .. ~~~-=-:-~-- ---=-~=:q. ~'iJO-f-... L ~--

~I

Remarks

16-4. Superelevation. Studies on the supen;.hiVl1.tion in water surface; arO'lllld a curved chuimel are many [11-14J. From the experimental results obtained by Shukry (Fig, 16-3), it oan be seen thn.tthe path of the thread of maximum velocity in ourved channel deviates fr0111 it-snormal course a section opstream from the bend. At point el, the puth almost touches the inside wall of the chunl1el and the ,,'ater surface attains its minimum level. Beyond point el, the path gradnally moves outward until it orosses the center line of the flume at the section that passes

at,

tal SURfACE LEVELS

i

)

0".: , --;;::'::::"_-:'::-,_

..... .

.

\

. I' J d forward velo0ities in flow around Contour lin~~ ~~, e'lual [~Y1aC;tU~:~: ~~els measured in cm (= 0,3937 ill.) a 180 be:l~. . (Ajt6'l' .... ::1L " y ~', ,: /b = 1.00; YA/b =; }.OO; and, with TT A = and velOCItIes III em/sec (-R 0.~3;8315~S\"I'e~.e subscript ..1 refers to conditions at the 77.8 em/sec (= 3,55 fps), A - I , ", inlet-approach section of the bend,

FIG. 16-3. 0

throngh the separation poiut 3, For different bends ,under vm'~~us C~ll" d't" .. of flow the- point of maximum surface depresslOn cl and th~ pcmt I lOn::; .' 1 t d (s given in To,ble 16-2, The positlOllS of of separatlOll s ,v ere oc(t e , \ . . 1 9 1 IJ the two points n,l'C given in tenils of the ratios of theIr ang es , d (tl1C d" . d f th inlet of the bend to the total angle fJ of· the ben . measure rom e ' 1 1 ff t 1 b a i n g the These positiolls were found to be only slig It y a ec ec y v r.Y h . , t 'U/ b Hence fol' pmctieal pmposes, Table 16-2, ",hlO gIves par~~e e l l ' f . /1. ~ 1 00 can also be applied to other values of y/b. pOSItIOn va uea or y v - . , .

446

RAPIDLY vARIED FLOW

FLOW IN CHANNELl:! OF NONLINEAR ALIGNMENT

In using the table for interpolation, the procedure i.'> similar to that for f. values (ExlUllple 16-1). 16-2. LOCATCONS OF PorN~'S OF MAXUlUiI
The average forward velocity is

V.

TADLE

(POINT

R

=

Parameter 9d/9

,i~ =

lD,500 I

I

(J,/6

.i 8d/ O [

t~

= 31,500

I ed/B I 0,/8

e,/e

(A) 0/180°

r./b 0.500 1.00 2.00 3.00

21,000

=

so/a !8,/e

ILOOO I I

[I. i I

1·8

d

/8

I 0;/&

0.111 0.111 0.1'18 0.051l

0,222 0,250 0.333 0.778

= 1.00,

I

a

1

cons~ant

I

0.333 D.IS\) 0.185 0.111 I

0.667 0.3:131 0.6'37 o.400 I O. 7.lD 0.556 O. 8S9 0,500 0.250 0.611 0.306 0.889 0.6137 1.000 0.445 0.296 0.556 0 .. 371 I 0.926 0.4'i.5 1. 000 0.778 o 167 0.805 0.278 I 0. 833 1 0.33311.000 ,

Ti

--1

T,

C

~---ln-­ To( ri

~·o -

(16-5)

. {" V dT

" (E -

CZ/2g1· 2 ) dl'

=~-~- = :C.'.'~__~~_ _ __ 'l'o -

ri

T-:;J -

-)

1';.

C2

(16-6)

In the above equations 7'. and J'i are, ret;pectively, the outer and inner radii of the curve. NO'N, the discharge i5

-I

(16-7)

If Q, 7'." !'" and E are given, the r::onstant C can be determined from Eq. (16-7). The velocity pond depth at any radius l' are then obtained by Eqs. (16-3) and (16-4). Thus, the superelevation t.h of the water surface can be shown to be

-1'

1

.. Alter A. Shukry [Zi. t No separation.

'The forward-velocity distribution and the water-surface profile at the section of ma.xinllun surface depression may be estimated by the assumption of a theor.etical free-vortex distribution of velocity. Th.is assumption holds as long as the flo'w is ,mbrr'itical. By the law oj free-vortex' motion, I the following expression can be written: v. -

=

c

Y

.

=

E

I'

(16-8)

I

For the practical application of the above equations, the position of point d is first deGermir.ed with th~ aid of Table 16-2. Thespecific energy can be c.omputed at any section A in the approach challliel by (16-9)

where CIA is the energy coefnciept and II A. is the mean forward velocity. The specific energy at, the radial section passing through point d is

(16-3)

-

l'

where v. is the forward ftlo.mentf.l velocity in the curve at a ra.dial di'stanc.e l' from the center of curvature, and C is the so-called circ'ltlat1:on constant in a free-vortex motion. Let E be the speGiEiq energy at a.ny section and y the depth of flow at a distance l' from the center of curvature; then

1

Ym

2(JT ,1';

.. ..

(B) I'c/b

~~---

_1~o -

= E - ---

1

1

f' (C/r) dT =

and the average depth of flow is

= 73,500

I 000 1. DOD 11. 000 I 1. 000 1.000 1. 000 000 1.000 1. DOD 11. 0.111 0.250 0 180 0500 0250 1061110.30'1'880 0 ,,7 1. 000 0.033 t 0.067 t' 0,100 t 0.133 t 10.333 t 0.000 t iO.OOO t 0 . 044. t 0.056 J t 1 0 . 167 t

6/180°

0.50 0.75 1.00

1

IR

0.50, a constant

_____ 1 _ _ ....

0.Z5

I~ = ~2,500

447

!J 2

_..2-

2g

(16-4)

For an elaborate ·ma.thematica.l ,analysis by the l.aW' of free-vortex motion see 1121.

where L is the channel length between sectiCYl1 A and the section COll~aining point. d; Sf is the friction slope, which may be either determined experimentally or computed by the M~nni:ng formula; and h f is the. energy loss due to curve resistance, which may be determined a.'> showni in Example .16-1. The coefficient 0.4 was found to be practically constant for any curve. .11le above m~thod was found to be. reasonably accurate as long as the angle of the c1.irve was greater than 90° .. For smaller angles, Sl~ukry assumed that C varies lineatly 'with e from rV.. at e = 0 to its full v::dl1e at. 0 = 90°. Therefore, for an.y angle e less than 90°, the Clrcu-

-\

,I I

448 (

i

RAPIDLY VARIED FLOW

FLOW IN CHANNELS OF NONLINEAR ALIGN?tENT

lation constant can .be multiplied by a correction factor equal to

o+

(1

where iT.. is the xnean fonval'd velocity in [l, straight channal. Supereievation in CUl'vcd channels nuty also be detcl'l"nined less aCClU'ate but simpler formul!t8 which lire bilsed on the application of Newton's second bw of motion to the. <::6ntrifug:11 action in the Cl1l've. Assuming that all filamental velocities in the bend are equal to the mean velocity V, and tllat. all streamlines have a radius of curvature rc, the transverse water surface can be shown to be a straight line, and a simple formula for sllperelevation can be obtained: V z 2b

(16-11 )

where b is the width of the channeL Applying Newton's second law to each streamline and then integrating the whole channel sectioll, Gras.~of [131 W(l.S able to show that the trailS" verse surface profile is a logarithmic curve and that the supel'elevation is ,

TT 2

2.SD -~-' 10 0 ' ~ g

';'

1';

(16 .. 12)

Woodward )4,151 l.l.ssumed th,tt the velocity is zero at the banks and has a 11l[Lximmu value at the center, varying in between according to . a parabolic curve .. Using Newton's seconcpaw, he obtained the fol1o\l'.ing formula for sllperelevation:

Of the above three simple superelevl1tion formulas, it has been found that Eq. (16.. 13) gives the best re!;lults, but Ilone of them is more accurate than the free-vortex formula (16-8). 16.. 5. Cross Waves. Cross waves· Ine usually found in supercriticaJ flow ill channels of lIonlinear aEgnment and channels with nonpi'ismatic .

I I

I Cross waves ill supercritical flow iJJ.wa~er are' aualogolls to the shock waves In supersonic ·t!01V of gase.~. This analogy was first noticed by Prandtl [16J, Hiab~ucbinsky [17], and von Karman 1181 and then investigated experimeatally by PreisIVerlc [19J and others. These scientists, however, were iIiterestecl ill'imrtrHy ·in applications of this ~tinciple to sttpersonic flow of gases. La,ter, comprehensive $tudies of .the subject! in hydraulic applications ware made by Knapp, Ippen, ilnd othel's [20-26). To facilitate the analysis of shocld wave in gases, B\lSemann [27J developed !l. gra.phical method known !\s the melhod of charactedstics, which was lall"r applied to hydraulic ;problems by Preiswerk, Ippen, and Knapp. For detfliled de$criptioll5 of this method, the reader should refer to [23J, [2·11, and [27].

449

;;ection (Chap. 17), These wa ~res, fonniilg a disturbance pattern tha~ . can persist for a .consideral:>le distance downstream, are ci),u:;ed by the turning effect of the cUl'ved walls, which does not act eqmtlly on all stremnlines in the channel section. In a curved channel, the outer wall, which turns inward to t.he flow', will produce all. oblique h}'dmujc jump rmd a corresponding pnsitive disturbance lirie or positive wavefront l . (Art. 15-17). The inner vIall, which turns away from the flo\v, will develop a oblique expansion - t { .............,"-waue and a disturbance line or negative wavefront l (Example {aj 16-2). The disturbance lines thus produced by both outer and inner walls wii! be reflected back and forth between walls and will interfere with each other, resulting in a dis~'~l'b::Ll1ce pattern of cross waves. Similarly, in any channel of nOIllinear alignment, cross waves may be {bl formed in various pattel'lIs, such as those shown in Fig. 16-4. It may be nuted that the cross waves in Fig. 16..4b can be canceled, as shown in Fig. 16-4c, making t.he line Art coincident with the first positive wavefront AA'. For large deflec- FlO. 16-4. Rectangular channels of 1l01ltion angles of channel walls, the plism~tic alignment. positive wavefront may be determined by the rnet.hiJd developed in Art. 15-17 fl,nd the negative wavefront by the method to be described later in Example 16-2, For small deflection angles, such as the infinitesiiru?l increment of the deflection angle in curved chn.nnels, the Cl'osswaves mo.y be determined by the method described below, In studying the de\~elopment of these methods it should be noted that, for sm::dl deflection angles a·lld for large deflection that create negative wavefl:on'ts, the specific energy across the wavefront can be ass limed constant since little energy dissipation is actually involved. For hrge deftectioil angles that create positive w~wefrOl~tsl however, the ,

.

I The disturbance line OJ' wave angle created by an oblique bYif W!lvefront d\IC to nn oblique expansion ,i-ave. The negative disturbJl,nce Iille Ilctjlally only marks the beginning of a. distuxbance region; it is not a. distinctly, defined Iiuc like the positive one. i

. -- 1

, 45,0

RAPIDLY VARIED FLOW

specific energy cannot be assumed COl).s.tant because the wavefront is built up to a :mbstantial height and so the oblique hydraulic jump across. the front will consume a.n appreciable amount of . Considering the supercritical flow in a curved channel of constant wi~th 'b and radius 7', (Fig. 16-5), the first small disturbance caused by the

FLO,\, IN CHANNE(,S OF:NONLINEAR ALIGNMENT

451

. ary line AB AI, the< flow is unaffected by.the curve and thus continues to move in its original direction; that is, parallel to the tipstre!un tangent. Beyond point B, the two wA.vefronts AB and AlB affect each othel' and are 110 propagated in straight lines but in the curved paths ED and BC, TheOllt,cr COll.cave wall A C tends to deflect the flow, .yhich would otherwise follow the tangential direction. Consequently, the water surface is raised higher and higher around the outer wall up to a maximum at C. After G the effect of the inner wall, which is to lower the water surface, begins to operate., Thus, the water surface along the outer wall starts to drop. On Uie imler convex side, the flow tends t~ depart from the wall, and the water surfa.ce is depressed lower rmd lower around AID until the point D is reached. After D the effect of the outer wall comes into pl!),y and the surface' begins to rise again. The reflection oC disturbance waves from the outer B.nd inner walls will not come j,o equilibrium 01' .stop when they meet near the center of the channel. They will continlle to be reflected back and forth' across' the Chal111el, crmsing lhe surface profiles the walls t,o h£1 ve a Beries of maxima, and minima of surface elevation, approximately at f~ngles 0, 30, 58, . , . , from the beginning of the curve. The e marks half the wavelength of the disturbance pattern. For practical purposes, . it may be assumed that the points of maximum and minimum for each (J occur on the same radial line; such as DC for the first maxiphase ~um 'and minim um. 1 By geometry (Fig. 16-5), the cen tnd angle to the first maximum can be shown to be 8

.. tal1-' (21",

2b

+ 51 tan f3

-1

I

I

i

·1

j

I I

(16-14)

FIG. 16-5. Cross-w~ve patt;mdor supercritico.l flow ina. llllrved channeL

where the wave ~\ngle f3 is approximately sin- 1 (V/gU/l') , as will be Sh0W11 below by Eq. (16-15). In order to simplify the computation of the water-surface el"vation 2 the following major assumptions al'eto bfl made: (1) two-dimensional flow, (2) constant velocity across the cross section, (3) horizontal channel, (4) frictioilleas flow, and (5) vertical channel walls. Assumptions 3 and 4 do not exclude the application of the results to sloping channels if the slope compensates for the friction, For the curved outer wa.lI, the changes in the angle of deflection and in the depth are gradual and small and, thus, can be represented' byde and

curvature of the outer wa.ll starts at the beginning point. A and is propagated along the line AB, which an initial wave angle /3 with the tangent extended beyond point A; In the meantime, the initial distu~bance produced by the i~'ll1er wall is propagated along the line A'B. The . two p'ropagation fronts meet at point B. . Upstream . from the bOWld-

1 Actuo.lly th~ locations of the first maximum and minimljm do not oceUl' exactly on the ra,diaf line OC but slightly on the left side nnd right side of DC, respectively (Fig, 16-5). 'However, the error involved in the l15surnption is negligible. This assumption applies also to the locations of the subsequent maxima. lLnd minimn.. ~ A det&iled study of the complete surface contours is possible by means of the method of ch!l.t!l,cteristios (23]. .

a

!

.- ~I

-\

\

.. ·1

.\

I

.~

. 452

r' (.

r

RAPIDLY VARIED i"LO'"

~.

ely, l:espectively. Assuming YI = 1/2 = y lor an obliq\.\e hydraulic jump of infinitesimal height dy, Eq. (15-21) bccomes l . ' .

I

[ I i

I

1 V Fl = F = -.-- = - . sm ,6 . vfiiY Substituting y for YI Rnd y

ely for Y2, Eq. (15"~3) may be reduced to

+

I ay

(16-1.5)

. Ii sec~ {f tan e = . tan f3 .:- tan e ~--~----­

I

For a small angle, tr,ri e may be replaeed by de, and ,tilll (; in the denominator is negligible compar'ed wit.h tan (f. Thus, Eq. (l6-16) becomes 11

(16-17)

By eombining Eqs. (16~15) tind (16-17), the following is obta.ined· dy

V~

=

---

g

tan (3 de

(16-18)

. According to assumption 4. above, the specific energy may be COIl-' fHdered constant. Since E = y + V2/2g, Y = v/2g(E ;- V). Substituting this. expression for V in Eqs. (16-15) and (16-18) and then eliminating fj by Eq. (16-15), the following is obtained: ely

de =

I

I

'2(E - y)

_ ,/<.,' -I 8 -. -v u tan

Sillce 2E' '

.,. ':<

..

. (16-19)

The exact solution of Eq. (16~19) for 9 gives

I

I

.,

vY

v2E - 3y

=

I - -3y- - - bn- I - -1- 31Y , \j 2E - 3y '. ''';3 '\ 2E - 3y

y(2

8 = '\/

+

;, Q

+

t

(16-20)*

tuu- 1 - - - " ' - = + const VF2 - 1 .

(16-21)

cons

Fi) (why?), Eq. (16-20) may be written

tan- I

',/3· --== .y F~ - 1

1

This equation can be used to cll.lculilte the change in depth along th . ~ walls at the begiuning of the curve. The integration constant Clll1 be determined by the condition that, for e = 0, the depth y is the initial , This equation ill believed to be tl'Ue only for supercritical flow in wide rectan"ular channels ... For bro~d ~ppli~atians, Erigelund and Munch-Petersen [28] have devel~ped . Ii. generahzed equatIOn, which h!l5_been found in good agreement with experimental d~ta covering considerable variation of the Froude number (even for F' eqtla\ to or shghtly less than unity) uud of the clepth-,viclth ratio of the channel. • 1'ile matliematicai derivation of the equation of this form was first made by von Klirman [18]. ' .

453

depth Y I. Equ[1.tion (16-21) is, however, involved and inconvenient for prar.:tical usage, even with the aid of a graphical c.hal't such fl·S that develope.d by Ippell (23]. According to Knapp and Ippen [21-241, adequate results may be, obtained by .the .much simpler equation . V2 sin l g

y= -

(

,6

8' +-) 2

(16-22)

This cqu[1.tion was developed as a result of actwd flow mensurements. 1 In using Eqs. (16-21) and (16-22), the angle eis positive for depths along . the outer' wall and negative for depths a.long the inner Wl~IL The depth at the first maximum heights of the cross-wave disturbance may be obtained by using the value of 8 computed by Eq. (Hi-H). In cross section GD (Fig. 1 (}-5), where the first maximum height of the cross wuve occurs at the outer wall, the line ala represents the theoretical position of t·he wat~r surfacein the channel cross section if the channel were straight, line b' 6 the theoretical positioll of the water surface 011 the curVE: if the flo';" were subcritical, and line e'e the actual water surfacc on the curve when the flow is supercriticaL It is evident that b is higher than a and b' is lower than a' by ail. amount equal to half the supe.rdevntion, or 6.h/2.By Eq. (16-11), this amount is Y?b/2TcY. According to t.he experimental investigations made by Ippen and Kn[1.PP [21,231, e is higher than- a by an amount eqlml to_about 6.h, or highcr than b by about (',.h/2. The positioll el is lower than b' by ['.bout t::.h/2. Similar conditions exist in other cro.3S sections where maximum height of water stll'face occurs at the outer w8.1l. In cros.'3 sectio:l FG, where the minimum cro~s-w(we height OCCll1:S at the outer w<.111, thfj actual water 'surface is identical with 11.' a because the . effect of the disturbance .cross wave is prnctically offset by ,the superelevation. Simibrly, in other cross sections where the wave height is minimum at the outer wall, t.he water 13urfILce aSSUlIles (\, position as if the flQw were in a straight channel. The distance .fiG along the wall represents a half wavelength, which sustains a central angle 8.. This length may be approximated by AG'

(16-16)

ely = -.-~,-- de . 8m f3 eos fj

FLOW IN CHANNELS 'OF NONLINEAH ALIGNMENT

i

!r

I· t

1 On the basis of !l.Ssumption 4 and, hence, of the conservation o{ energy, the velocity in any stl'eamline must chani!;e as the depth changes, since E='y + V'/2g must be con,~tant. The flow E1,l'Ound the outer wall, boing the deepest, should be the slowest. Actual measurements, however, indicate on the contrary tl\r.t the velocity around,the autc,. w(Lllli~lllains constant or even increases slightly, whereas f\long the inner W:L1l the velocitj"'decreases. It is believed th:.t this dispD.rity is due to varyhig effects of the channel irictian, which actua.lly render ass~ll1ption 4 invalid. Consequently, it is rell.soned that a constant velocity may be assumed. Equ:.tion (16-22) was, therefore, derived from the assumption of constant velocity, which rep!E1,ces assumption 4. Since the author cannot discern here the rigor of the origirull mr.thematical derivatioll, he would rather consider Eq. (16-22) to be empirical. .

454

455

RAFID1JY VARIED FLOW

FLOW IN 'CHANNELS 'OF' NONLINEAR ALIGNMENT

or b/tan f3 by that the angle AG' A' = (3. Thus, the wavelength is 2b/tull fl. Frain the above discussion, it may be concluded that the disturbance wave p.l1ttel'll, which osoillntes about the plane represented by has a wl:welength of2b/tl1n.6 and an amplitude of 112b/2TcU. This can be used to cstimn.te roughly the s~lpercritica[-flow profile ill simple curved

where FI a.nd F. are, respectively, the initi",ll1.nd fina.l Froude numbers of the flow. The rehtion among F t , F,; and th.e tot!1-l deflectioll angle ~, can be represented by Eq. (16-21), In that equation, e '" 0 when F = Fl and 0 = 8, when F.= F.. Thus, if F, aud 8, are give!l, F, can be determined. The angular mnge in which the depth change takes place may be determined by (11)-23). Since the depth cha,nge does not involVe 'appreciable energy depths before and after the change ca.n be rela.ted by 'assuming a constant er,ergy, 'Thus, the mtio of the final to ,the tnitia.! u(~pth can be shown to be

chn.nlle[s. For disturbance continuing into thedowl1stream channel, the primal':\, wavelength is sti!! 2b/tan p. AI>, the curvature suddenly changes to straight alignment, a r,ew disturbl1l1ce pattern, starting with a m
W2N6

·,·1 (

In til" disturba,nce .region, the decreaS
in

16-6.' Design Considerations for Subcritical Flow.

In general, curves

are unde3irable in open channels, because they will increase frictional

I

.

Mm?q#:;:J;;77/7:/7//?//t::::Lm7///.7//~7/A; ,

i

Seclloo A'-A

Fw. 16-6. Oblique

expa.n~ion

I

wave,

i

i! Sol-uliQn. The oblique expa.nsion wave occurs when the channel wllll is turned oute ward from the flow at all o.ngle {Jc (Fig. 16·6). The depth of flow decreases in the fanshaped region of the wave disturbo.nce delineated by wave angles (1, and {:J" measured with'l'eference to tbe initial and final flow dirac.tians, respectively. By Eq. (16-15), and

(15-23)

I

I

I t

I.,

loss and letta to dllnger of serious locn.l erosion due to spiml flow. Reduction of spira; flow is the mlljor conce.rn in the design of curved ch:J.n~lels for subcritical flow, For a proper proportioning of the curve, a ratio of To/V = 3 is recolnmended, because it will give the least, radius at whichthe . effect clUB to spired flow is minimi.zed (Art. 16-2). In erodible ch!l.l1!lsls, the action of spiral flow will develop a tion in the bed. The size of the cOllfiguration is minimal if 'l'r/b is 3.0 OF larger. When bl111k protection is necess::I.ry, it is needed most on the outer bank at the do\vnstream end of t.he curve and to a lesser degree on the inner bank at the beginning of the CUl've. This is suggested in accordance with the behavior of spiral flow. In an alluvial bend it seems that the configuration of the channel cross sectioll is defined more or less in accordance with certain l1a,turallaws. This subject has been by many scientists and engiheers of river hydraulics since [29]. According to the studies by Ripley [30], the configuration at an. alluvial river bend (Fig. 16-7) may be represented the following empirical equation:

I x~ - 0.433) ( 1 Y = 6.35D ( ,\/0.437 - T2

Xl() + r;

(16-25)

where y is the ordiril1teot depth in ft, :1: is the abscissa in ft, Dis the hydraulic dept,h in ft, T is the top width in ft, r" is the mdius of curvature on the concave side of't.he channel in ft, and J( is a coefficient equal to

I

456

I I

RAPIDLY VARIED FLOW

:FLOW. IN CHANNELS OF NONLINEAR ALIGNMENT

.17.52. The crigin of the cool'dillates for this equation is on the water l>'Urface' at n pOlut equidistant. from the banks~ ~l1using :this equation the foil.oviring I'emarks shoulcl' be noted: . 1. For],. less than 40 tiInes the square root of t,he water area, no further deepening of the channel seems ttl result from the increased curvature; hence, in such cases, the ivalue of riJ used in t.he equation should be 40 VA. Consequently, 'bends are const,!'llctive and stahle when r. is . greater them 40 wher~!l.S sharper bends are destructive, te,nding to shift the channel. 2. For r. gren.ter thn.ll about 110 VA, the equation bacome.s il~ valid.

457

the radius r of the curve and the velocity Y; that is,

. VZ gr

(16-26)

In order~o avoid abrupt chailges in flow condition, banking should be introduced gradually from zero to its full amount, starting at both ends of the curve. rt should be noted that the slope thus computed is good only fqr the velo('.it,y given. When the veloeity chiLnges, the computed banking will be less effective under new flow condition:,. Hence, banking; is most suitable in channels which ordiM,rily opern.te nt or neo,1'

I

I .r-

Center of Cur .... o1vre

FtG. 10-7. Empirical channel

cro~s

section at river bend.

o FIG. 16-8. Ph\ll of sill insta:llativn in a curved ch3.nnel.

\

i

3. The equation ma.y be npp!i¢d to cU'rved channels not occupying the entire width of the wntenvay or to those at the river entrance created by a . single curved jetty. In such cases, K 26.28 and the'value 1)f y thus compltted shouldbe in<.:reMecl 14%. 4. On a crossover bar when the c:hannel is neitheI' on a curve nor in IIstrnight .reach, the maximum depth is iI.bout 14.5 % less than the computed value. 5. The eqtlatioll generally gives a 'width of channel at hydraulic depth about 20 % greater thfUl the actlJf.1 width. 6. The equation is very approximate j nevertheless, in the absence of a, better method for determining shap~ ofcrbss section at a river beild or at a bend in a dredged c[mai, it will give satisfactory answers to many practiGal problems. : . . 16-1. Design Considerations for Supercritical Flow. the major issue in the ,design of curved channels for supel'critical flow is to elimil11'te or reduce the superelevatioll and cross-wave disturbance pattern. Knapp [24J h~s suggested the following:methods to achieve this objective: A. Eanking. By banking is jrneant use of a. bottom cl'OSS slope that supply a l£\teral force to counteract the centrifuglll;:lCt~on of the flow. The required cross slopeS, caA be computed by equati,ng the gravity component [Llong the cross siop~ to the centrifugal force determined by

,vill

i I.

(A.jlel' R. T. Knapp [24].)

the designed flQW coridition. Fac~ors against lnmking inc-lude co;;tly . excavation and possible siiting or erosion ~;'long the inner w[ll1 dUl'in~ low flows. . B. Multiple Curved 11 anes. The !:;upel'elevation and disturbance pa,ttern can be reduced by concentric vc.nes which divide the ehallllel width into a series of ml.l'l'OW curved channels. This method is IVJt generally practicfl.l, O-\ld it becomes impossible in channels ctl.lTYing debris of sizes larger than the sllbdivided width. O. Ea8e'TIumt Cv.T1)e~. ,The disturbance ir~ a simple cUl'ved oh"nne1 may be reduced by employing a compound curve. The best design is a SImple curve of radius To preceded nnd followed by a section of ::motlle!' simple curve whose lentics ... D. Diagonct! .StUs. Diagonal sills installed on the bottom of the channel near the ends of the curve will prodLloe all effect correspondi'ng to that of im easement . curve. An effective layout of the sills, shown III Fig. '16-8, has been develQped expet:imentally. The optimum sill angie ,

,

'>

I

458 O!

is 30°.

I

RAPIDLY VARIED FLOW

I

The distance Lus may be estimated by

,I

I

(16-27)

I I

where K is 11 ooefficient;; FOt, noi'mo,} cases, the value of J( lies between 0.9 and 1.15. In accordance with the the length LU1 is

0.30L'

+

b

! i

,,I

(16-28)

l

i

where L' = b/tan p, that "is the half wftvelength of a channel disturb:?nce. This method can be used as a remedial measure in channels which have been designed as simple circular curves or other un:=mtisfactory fOl'ms. The major disadvantages of this method are mail1tenal1.;e cost, pronounced distUl'bitnCe at low flows, and possible cavitation at extreme high-velocity flows. The pronounced disturbll.nce l1t low flows may cause uneasiness and distrust in an unexperienced observer liS he may not be ea.'5ily convinced that the distul'bance will be greatly reduced at high d~sigl1 flows. .

PROBLEMS 1~-1. A 120 bend with Ii 10 In. and 1', = 15 in. is designed to c.onnect two etra.lght channele of the same width, ca.rrying 1.3 cis at a normal depth of i in. Determine the bend loss. 16-2. Det~rm~ne the specific energy at the section containing, maximum surrace depreSSIOn In the bend given in the preceding problem. Assume 0: = 1.0 alld n 0.01. . , 16-S. Compute the WH.IA'.['-sr:r profile in the cross section containing maximum surface depression, as in the problem, using the law of free YortE'x.' What is the snpel'elevation? 16-4. Det.ermine the superelevation in the preceding problem by (a) Eq. (16-11}, (b) ;Eq. (1.5-12), and (e) Eg. (16-13). . 16-5: Verify Eq .. (16-14). 16~6. DeLermine the approximate flow profile ill the curved channel given in Prob. 16-1 If the approach channel carries a supercriticl11 fio\I' at a depth of 0,5 in. .16-7. Design the curvad channel for. a rectangular flume to turn,lj.n angle of 50° wIth t1 radms of 250.it.. The flume is'12 ft wide and built of smooth concrete. The de.sign discharge is 350 cfs at a slope of 1 %. ' . 1~-8 .. Using Eq. (16-25), compute the channel cross 8ec~ion at a bend in the MissisSIPPI Rl;erj where A. = 148,010 ft 2, T '"' 2,340 it, and 1. 18/300 ft. result WIth the actuat cross section, which is as follows: 0

,

o

x

y

1,170 1,000 800 600 400

0.00 36.00 88.00' 108.00 111.00

y

x

ZOO

°

-200 -400

97.00 74,00 65.00 55.00

1I

y '-660 -800 -1,000 -1,170

44.00 38.00 24.00 0.00

I

.{

! \ l

FLOW IN CHANNELS OF rWNJJINEAR ALIGNMEN'f

459

REFERENCES

l· Jallles Tholnson: Oil the origin a.nd winding of rivers in allu,rial I,laius, with rema.rks on the flow around bends in pipes, Proceedings, Royal Society of L01!rf,on, vol. 25, pp. 5-8, Mar, 4, 1876. 2. Ahmed Shukl'Y: Flow arour:d bends in an open flume, T1·ansa.dionli, Ame1'ican Society oj Civil Engineer.s, voL 115, pp. 751-779, 1950. 3. A. Hinderks:' NebenstrBmungen in gekrilmmten Kaniilen (Secondary flow in Gun'eel can!l.ls), Zeitschdfl. des rre1·~in.~ deldscher I ngenieuTe, Berlin, vol. 71, no. 5J, pp. iTi9-1783, Dec. 17, 1927. 4. Bmllo Poggi: Col'l'enti veloci nei (lana-Ii in cm·va.(8wift flow in curved channels), L'Ell,ergia eleUrit;a, Mila/Ito, vol. 33, no. 5, pp. 465-480, May, !f)5G. 5. F. L. Blue, Jr., J. K. Herbert, a.nd R. I,. Lancefield: FibII' around a river bend inveRtigated, Cl:"n En!j1:neenng, vol. 4, no. 5, pp. 258-260, May, 1934. G. Herbert D. Vogd Ilnd Paul W. Thompson: Flow in ril'erbends, Civil Engineerinll, vol. 3, 110. 5, pp. 2613-263, May, 1933. . 7. Herbert D. Vogel and Paul W, Thompson: Existence of helicoidal flow, Ci1iil EnfJ1.needll(J, vol. 4, no. 7, pp. 370-371, July, 1\J34. 8. Robert Muller: Theoretische Grundl:J.gen del' Fluss- und Wildbachvcrbauungun (Theoretical principles for mgulatioll of rivers and t{)rrent.s), Eixlgeniissi,che Teclmi~che HClchsclntle; Z1ldch, jJ[itieii1,t~{le" de,. Ye.rsuchsanslalt fiLl' Was.,erbau 1.nd El'dbau, No.4, 1943. 9. SaHjiva. Puttu llaj\l.: Versuche i.iber den SCl'Omungswicierstll.nd gekrummter offencl' Kaniile (St'.lcly on the flow resistance ill CllT\'cd open channels), lI{Uleibngen des Hyd1'f.mi::schlm [nstitu.ts del' Tec/',nischen H ochscknle Miinchin, no. 6, pp. 45-60, Munich, 1933. English tTll.nslaticn by Cla.rence E. Bal'cMey, R.",is~allcc to fiow in curved open dmnnels, A1I!,~,':c(m ,society of Civil Engineers, vol. 63, pt. 2, p. 49 after p. 1834, November, 1937. 10. C. H. Yen and J. W. Howe: Efferti'; of channel shape on.losse5 in a canal bend, Civ';: EnrJineering, vol. 12, no. 1/ pp. 2.8-20/ 19':12, 11. H. Wittma,!l and P. B6ss: "Wasser und in gckrihnmten FIuss-strecken" (" Vi' a.ter and Eed-load Movement, in Curved River Reaches "), Springer'''erlr.g, Bfll'lin, 1938. 12. C. E. Mockmol'e: Flo\l' around hends ill stable channels, TnlllSact·':oTl.!, Amel'ican Society of Civil Engineers, vol. 109, Tip. 5£13-618, 1944. 13. Armin' Schoklitsch: ".:Hydraulic Stmctul'es," trallslated from the German by Samuel Shntitz, American SGciety of M'lchftnical Engine~rs, NEllV' York, 1937, . vol. I, p. 151. . 14; Shenna.n ))..1. Woodward: Hydraulics of the Mi!l.rni Rood control project, ilIia.1ll,i Cons6"unncy Distr':ct, Tel'hn·ica.l Repi,,·t, Pt. V Il, Dayton, Ohio, 1920. 15. Sherman M. Woodwa.rd and Charles J. PORCY, "Hydraulics of Steady Flow in Open Channels," Jobn Wiley & Sons, Inc., Nt'w York, 1941, p. H2. 16. L. Pral!·::hl: Abriss de,. StrQmungslehre ("Outline of the Theory of Flow"), VieIVeg-Verlag, Brunswkk, Germany, 1031. 17. D. Riabouchinsky: Sur l'analogie hydraulique des mouvements d'un ffuidp. compressible (On hydralllic anl.Llogy of flaws of' a compressible Jiuid), Com.p/.es renaw, de I' Acad.!?nie des Scie1!ce~, vol. 195, pp. 998-999, 1932; vol. H)9, pp. 632-634, 1934. 18. Theodor yon Karman: Eill;; i)rai
-1

460

RAPI~LY: VAlUED FLOW

19. El'Ils~ l'reiswerJc: Anwendung gllSdynamischer Methoclen auf Wassel~t\... . f . 01 . , omUll"en m.. t reler Jcrllli.ehe .(Application of the methods of ga.s dynamics to water fl"ow E,dgenBssischB iechnische H ochsch~de dwith Ifree . surface), .. . 'Z~'rich , 'nf,t1e,'1 ,. ,ungell aus ~?1\. nsttb~t ~~L1' Ae"odllnam'ik, No.7, 1938. Euglish translation by S. Rei~s as l'iuttOnal Advtso,'y CommtUe~for Aeronmtlics, 'I'echnicaI1l1emoranda Nos 934 .d 935, M' arch, 19'1D. . ' • ar. 0

20. Arthur T. Ippen and Robert T. Knapp; A study of high velocity flow in curved channels of rectangular cross section, 1"ransaclions, American Geophl sical U '. vol. 17, pp. 516-521, 1936. Y mon,

21. Robert T. Kna~~alld Arthur T. Ipj:>en: Ol:rvilineur fiow of liquids with free sur~.\l.ces at veloCitIes abov: that of wave propagation, Proceedings of the 6th Interna/tonal (')on9 res s of Apphed Mecillznics, Cambridge, Mass., John Wile' & Sons Inc., New York, pp. 531-536, H)38. . Y , 22. Arthur T. ~ppen: GM-wave Rnruogies in ope:l~challnel flow, Proceedings of the 2d Hydrm'/tcs Conference, June 1-1, 1 19./,2 St(tie [/niU'Ts't'· of I~w'a J S'Lolluozes ,. '!"L . '" -r: ~:y E'ngtnee''',g, B1tl/etm 21, no, 400, pp. 248·-265, 1943. 23. :~r~hur T. 1ppen: Mechanic.s of sllp~reritioal flow, 1st paper in High-Yelo~ity flow HI open channels: A symposIUm, T"unsar;iions, A.1nerican Society of Civil E'lUI~'nee-rs vol. 1] 6, pp. 268-295, 1951. • , 24. R~bert T .. KnapJ): I?esign of channel Cu\,ves for slipercril-iC!l.1 flow, 2d paper in oS

I

....

,

.J

'-'

HIg~l-veloCity. flow In open channels: A ~ymposium, Tmn~'actions, A.mer.jean SOCIety of 0.0.1 Eng·ineers, vol. 116, pp. 2B6-325, 1951. . 25. ~rth~lr T. Ip~en and J?hn H. Dawso'l: Design of channel contractions, 3d paper In ~IJgh-veloclty Ro.w m op~n chunnels: A symposium, T,-a·)tsaclions, American So.;teiy of Cw,l Engtneers, vol. 116, pp. 326-346, 1951. , Z6. Hllnte~' R.ouse, l!. V; Bhoota, and En-Yun HSll: Design of c]lannel expallsior.s, 4th paper 11111Jgh-ve.lo.clty f[~w in open channels: A symposium, Tranaactions, Amer,can SocIety of C1Vl! E'Il{j!1lo,,'s, vol. J 16, pp. 347-3G3. 1051. 27. A. B:I~el~~nn: "?asd~r:.~,mik" ("
J:

sur

?

CHAPTER

17

FLOW THROUGH NONPRISMATIC 'CHANNEL SECTIONS

Ru.pidly varied flow through nonprismatic sections! is found frequently in v
.Qw (P3Y. - .elV l ) g

=

PI - P 2

-

Pi - F/

By the continuity equation! Q = V\b1YI

=

Vsb,Ya.

Fol' convenience

1 Gradu'ally varied flow through nonprismatic channel sections has been covered in Arts. 3-5, 3-8, and 11-,1 to 11-7. . • The following discussion is based 011 a.tre!Ltmeut given by Jaeger [J-3]. 461

1

J FLOW THROUGH NONPRISMATIC CHANNEL SE<"'TIONS

RAPIDLY YARIED FLOW

462

Region 1. Region 2. su bcritical. Region 3. Region 4. supercrit.icaL

in the theoret.ieal discussion, it may be assumed t11at Ff = 0, lh = P3 = 1, and Y2 = Va • . Under these conditions, Eq. (17-1) may be reduced to F

(yJ/Vl)[ (VJ/Yl)2 - 1J 2[(Ya/Yl) - 1/(b a/b 1

2 _

1

)J

-

(17 -2)

. where F 12 V 12 /gYl' This equation call be plotted as. shown in Fig. 17-3, using bl/b l as a parameter. The family of CUl'yes thus plotted are

Horizontal

contrQction

463

Flow is supercritical throughout the transition. Flow through the tra,nsition passes. from supercritical to Flow is subcrit.ical thr"Otighout the transition. Flow through. t.he tra,l1sition passes from subcritical to

!

Horizontal

expor.sion

Vertical expansion

.FIG. 17-1. Sudden tra:'lsitions.

Fw. 17-2. Anl11ysis of a horizontal contraction.

I \

similar hyperbolas of a higher order, having·the following cbaracteri:otics: 1. The curves are considered only for positive values of F 1 and Ya/YI' 2. All hyperbolas pass through the points (Fl ;;= 0, Ya/YI = 0) and (Fl = 0, Ya/VI 1) and are asymptotic to the yerticr.lline Ya/VI = bdh 3. The special case of ba/b 1 =:= 1, indicai,ing the hydraulic jump in a prismatic channel, is represented by the curve . Fl2

=

0.5 (~) i(Ya) + 111.1 Yl L Yt

(Why?). . 4. The upstream flow is supercdtical ill the regionaboye the horizontal line F 12 = 1 or F I = 1 and sub critical below t.his line... The downstream 2 flow is subcritical in the region below the curve Fl~ = (!Ja!Yl)3(ba/ b1) and supercritical above it. (Why?) Consequently, four regions in the figure, representing four regimes, are created:

I

1

I

FIG. 17-3. Plot of Fli aga.inst 1)./1/1, using D.lol as a parameter fQ; the ana.!ysi;; of sudden horizontal contractions (shown by the full curves) and expa.nsions (shown by the dashed curves).

5. Theoretically spea.king, certain parts of the curves represent flOWE thatc!1lmot actually occur, because they necessita.te an increase of 61\ergy, 01' a negative energy loss, which is contra.dictory to the fact that the flow always loses energy in passing through a traUl)ition. The difference between the energies before a.nd after the transition is

I

IlE

I

I

ar

IlE Yl

+ -V2(1 - y. -V3 2g 1 + F12 - r~ + 2 1..l!1 2(Y3hh)2(b /b

Yl

1

2

2

-_.!L_-J a

1)2

(17-3)

(17-4)

464

RAP IDLY VARIED FLOW

l<'LOW THllOtlGHNONPRISMA.TIC CHANNEL SEC'l'IONS

By applying this eq uatio!l to a certain part of the curves, the flow can be found to be impossible if the computed value of /::,.E is negative.. It should be noted, however, that this discussion is intended to present a theoretical t'malysisof the phen6menon and to develop a classification of the flow through sudden transitions. In real problems, the theoretically impossible flow may become actdally possible becanse tbe iJ.ssumptions made in the ab~ve derivation may not be true under actual circumstances.!

, I. I

405

. by Formica ' . [4], . !LS shOwn in Fig, 17-4. .Tl1e t YdP11cn( . I measured' were made _ '._ ) I (col1~rn.ctlon) expansIOn flow profiles and energy 1·'Illes f· or . hall 'fi encrn-y 1"'(-5. " Th" enel'''Y speCI ill c e the b . h are sown m '" lines represent . . t e11k ~\'hel'e the tl'(1nsltlOll t, 'es pc, Y TI CI Vtj'/. ""V' Near the section . 35 30

f-'-~ ,

25 20 em

15

+._._---~t ' 'I...-., ~-. .

®

10 5

.

I

1

-0

E. ~=·-·4

a

®

35

~ 4

30

FIG: 17-4. VariolIs designs of sudden transitiolls -[01' (Afler G. [i'cwmir,a [4.).) ChELnnel ;vidti! 355 mm for wider sections an,1 mm for r.a.rro\ver sections. l"or' COli tractions, channel slopl} 0.00023 at wider sections and 0.00096 fit narrower sectio11S; fo!' expansions; 0.00104 at narrower sections' and 0.00073 at wider sections.

By i1 similar analysis, the following equation can be written for Yl): zontal expansion, (assuming Y2

11

25 em

hori-

(17-5)

This eqtmtion is l'epresentei:! by the dashl"d curves in 'Fig. 17-3. Similar analyses also be made for vertical contractions and expansions. investigations on flow through sudden transitions will be in the ncx~ three articles, 17-2. Sub critical Flow through Sudden Transitions. For sub critical flow through sudden transitions, e)cperiments' on varklUS designs 1 Under actual cil'cum~taneell, fh and ,e,.are not exact~y equa.l to 1.0, a.nd y~ could be YI > :liz > !ii. Thus, the. lJegative energy loss would ~ecome positive, and the thEm.! reticallyimpoSSible flow: wouhl become ,actually possible. The loss !J.E is !L very small amollnt and callJ:e~dily be changed from negative to positive by I!. slight change in the items ill Eq. (17~4).

.

-1

o ~udderl tr!)nsition~.

Il-nd energy lines througll

FlO. 17·5, Typica.l measured flow (/lftet G, Formica [41.) In the 1 em = 0.3937 in.

1 Is-1

6.03532 cf., 1 m ~ 3.2Bl it, .and

velocity heu.d cannot be measured easily because of tdhea tUI1'~hule~t tcon~ r ' . e vel". .ICll., " . strop ' IJr exten e. dition: of flow; hence, the energy I'mes are h l'lltel'Cept betweell the extended upstream and downstream h~els flIt t e . . transitional section represents~ t he energy 1oss flE'. The• astens .' c S lOwn . fo~ the sudden expansion iildidates the position of the energy lme assum-

\7-6

.

,

co~B?-cient

lng a == shows the computed values of the energy different section."3 of the ~hannel for various designs of transItIOns.

at The

0:

rI I

467

RAPIDLY VARIED FLO)V

FLOW THROUGH NON PRISMATIC CHANNEL SEC"l'IONS

vhlues are apparently very close to unity immediately after the sudden contractions hut are generally higher than unity after the .sudden expansions. This indicates that the flow in a sudden expansion is irregularly diffused.

channel, as in designs II to IV. The differences among these three designs are evidently insignificant. Again, the energy loss in a sudden expansion can be reduced by gradua.lly enlarging the channel section or decreasing the angle of divergence

466

I

I

! 1.1 0

r---r--r----'-~

-+-~

1.08

1.l0

.-

-

__ .,..r--

!.08

--o--~

a

1.06

~~

'1.04

l'>.'

1.00

!.to

--

1.08

a

1.04

~,

1.00 1.10

,,

.

1.06

1.04

.......... ---~

1.08

a

,

~

Il

1.06 f--- 1--

,

C'" 41

1.04 1.02 ~ 1.00 1.10

....... --:---=-

F t--

I.oe !.os

n,

.0'" 49

1.04 1.02

1.00 1

~,~

""'-----

0, A

91- i-®-···-,

Bf--- -$---0 ----1.:.L_

...

/'

E -~

!

~r ,

6'

2

'"

0-25.2

\\

f"

~ '-~,

V

I

.--

0

I.

;-

taf~, ~

;':1~"

,-.=,~

---<

I

I ,,/ {g;/

<

-~

~

V

f--

'--

---r

:

I

3

:-..

tOO

,'I

4

-".

-

-4} ... -.<~~.L-..

"'}-- - --¥..:....- V

/

.,..-:"'

.1.02

Q

....

"lV'

-

o:~jt1irttJt~1

'::l~*lll-ttB

B 1.003

Sec1iqn

>-~-


1.08

0" 25

(06

1.02

,

'y-

'"

.' I-Q.J--'~

ts- I

Q = 15.6

---iJ---

"1',

,J:''';F--

1.02

LtO

--

1

.:j

5 0

0

bed

•

I

9

'~~-~ ~

71c

---

.

1

+~ ....c::.

(After G.

I

o

10

The head losses for various designs of trallsitions at different discharges are shown iIi Fig. 17-7. It can be seen that in general the sudden contractions have higher losses ,than the sudden expansions. In a sudden contraction the flow is first cOlltraoted and then expanded. A process of' conversion from potential to kinetic energy is followedimmedia,tely by a process of reconvBrsion from kinetic to potential energy. As a result, much less energy than in a 'sudden expansion is recovered. However, the' energyloss in a sudden contraction of design I can be greatly reduced by modifying the sharp-edged corners of the entrance of the reduced

-',

=

I~

L----

,......

20

~-

2S

....

~

-

1

·1

r r"

W-~ ~d

~: ~~ V-w-

30 35 q [1s"1

f,t

/vp, .IJf

[-

J--

•

I

I

.~:====== FIG. 17-6. Variation of the energy coefficient near sudden transitions. Fo,mica [4J.) 113- 1 = 0.03532 cIs.

1:=· ---"1-f-

""""".-~"'"
Sf---

h

iI

40

45·

50

.

55

FIG. 17-7. Head losses in'sudden transitions., (After G. Fc>rmica [4J.) ers and 1 mm = 0.03937 in.

113-;1

== 0,03532

(8 in Fig. 17-4), but this advantage may be nullified by such modifications as those in designs 6 to 8. The lengt.\i of the gradual enlargement of the expansion has a limit beyond which the. gain in efficiency becomes insignificant. . In closed conduits [5], the energy loss in a sudden contractio~ may be

I I

(

.'\

-"l

f.

468

,",r'

RAPIDLY VARIED FLOW

expressed by

. V3~' E=K-' 29 and in a sudden expansion by

(17-G)

\ t

,

FLOW THROUGH NONPRISMATlC CHANNEL SECTIONS

4G9

[6] h!1Ve found that straight contracticinsal'e always ~etter ~hancurv~~ contractions of equal length of contraction from the s~andpomt~f m~:I­ mum height. Accordingly, they ha.ve pl'op?sed a procedure of des~on for straight contractions, which will be descnb~d below. For t.he deSign

(17-7)

E=

wherel( and € are coefficients, and VI and V3 are the upstream and downstren.m velocities, respectively. By applying these equations to open

'VI

\

c'

AJ

tOl

A

FIG. 17-,8. Cross wave,; in

it

curved tranaitil;m.

(Cou?'iesy of A. T. Jppen.)

channels, Formica obtained the following average values of expansions:

€

for sudden

~~~~Of,~e~igrL':"", .10,821 :,871 ~.681 :',~I :'27l~.291 :A51 :.4~Accorcling to the experimental data obtained by Formica, the' values of

K for sudden contractions seem to vary in a wide range, generally increasing with the discharge. The approximate median value of J( for design I is 0,10 and, for designs II to IV, 0.06. 17 -3. Contractions in Supercritical Flow. When supercritical flow is introduced thro\lgh a contraction with symmetrical converging wa.lls, cross waves similar to those developed in a curved channel (Art. 16-5) will appear. The cross waves in a contraction, however, are symmetrical with respect to the centerline of the' channel (Fig. 17-8). Theanalytical study of the problem can be performed by the principles of the mechanics of supercritical flow described in the preceding chapter. From an experimental and analytical investigation, Ippeu and Dawson

Along wall

AI on 9 ce n Ie,.'.;,;Ii;,;."e=--_r-_ _ __ • h hz

r-~"----l

F------------,;

1

t

I"

tel

FlO. 17-9. Designs of skajght-waU contractions. (After Ip~cn and Da,"'son ..[6].) . ~~) Generai dis~ul'bance patterns; (b) minimum downstream distUl'bance, (e) .,chema Ie profile.

experi'. . of curved contractions, the cross-wave patt ern ID!'.}'b'.e determil-ed mentally by model test or !'.na.1yt.ically by the method of charactenstlcs (see first footnote to Art. 1 6 - 5 ) . ' .' In supel'critical flow through a straight contraction (Flg, 17-ga) ,symmetrical shock wl1ves are developed at points A and At at .the entrance. These waves extend across the channel at. wave angles {h, mter~ect ~t B on the center li~le of t.he channel, and, finally, af~er some· modlfic~tlO~, reach the opposite walls at C and Ci. In the regIOns A!'W and A BG , t,he flow proceeds through a new field characterized by the Froude number F~. Note that, ot the end of the contraction, negative disturbances are

r

I

";

'1

\ 470

RAPIDLY YARI:EJD :FLOW

created at points D and D" These disturbances will result in more complicated disturbances dOIVnstream. In a good contraction design, it is possible to minimize the downstream distu:rballces. This can be done by directing the shock waves to the opposite walls at D and D', thus car.celhig off theoretically the newly created negative disturbances. As 11 result, the flow will be calm in the downstream channeL This is the situation shown in Fig. 17-9b. From the geomeky of this situation, the letlgth of the contraction can be shown to be L

~

=

471

I

1 \

I I

velocity from a closed condBit, sluice gl1te, spillway, or steep chute. If such an expansion is ll}ade to diverge too rapidly, the major part of the flow fail to follow the boundaries. Studies by Hom:-ma and Shima . [7] indicate that separation of flow like that shown in 17-10 may occur. The separll.tion surfaces shown by the dashed lines act as solid boundaries within which the flow hail the characteristics to those in a channel of decre!1sing width. If the divergence of the expansion is

,,,,ill

!

(17-8)

By cOlltitluity of the flow, b1y1V\ = baY~V!

h

FLOW THROUGH NONPRXSMATlC CHANNEL SECTIONS

1

I

I

(~)%(Fa) Yl,

FI ,

I

Q,or

~

(17-9)

The above equations and either Fig. 15-23 or Eq. (15-24) can be use.d to design a strn.ight contraction with minimum downstream disturbances. Generally speaking, high values of FI and low values of Ya!Yl will give a long contra.ction. To reduce the length of the contraction, the advisable va.lue of Y3/YL seems to be bGtween 2 and 3, provided that F J stays well above the critical value. . Example 1'1-1. Design a straight contraction con!1ecting two recta.ngular channels 12 ft and 6 ft wide. The discharge through the contra.ction is 200 cI3. The depth of the ll,pproach Bow is 0.70 ft. . Sa/uliano Since Q 200 crs and A = 12 X 0.70 - 8,4 fP, the velocity of the approach flow jg VI 200/8.4 = Z3.8 ips. The Froude number Pi = 5.0l. Assume y,/y, 2: and take b, = 12, b, = 6, and FI 5.01j Eq. (17-9) F3 = 3.54. This vI111.1.e should not be less than 1.0 or even too close to 1. Now, assume a volue of 9, say, 15', and take Fl = 5.01; the diagram in Fig. 15-23 gives y';V' = 2.60 and F. 2.S. . A second determination using the same e = 15° and repla.cing F, by Fi 2.8 pro· duces values of y,/y" ... 1.80 and F; 1.77 from Fig. 15-23 (corresponding to vliv, and F" respectively, in the dia.gram). Howe.rer, these values do not neces3arily represent the actual flow condition in the required design, since,the flow condition downstream ffia,y be complica.ted by the negative disturbances origina.ting from the points D and D'. . Multiply 11./1/1 by 1/./y%; the first trial va,lu(l of y';y, is equal to 2.50 X 1.80 4.68. Since this does not a.gree with the Il.Ssumed value (that is, 2.00), the procedure should be repea.ted with lJ, new value of " until agreement is reached between the assm;ned y,/y, a.nd the value obtained by the trial. . ' , Alter several trials, the correct angle 9:is found to be 5°. With 9 = 5° and'F, 5.01, the diagram in Fig. 15-23 gives II1./Yl= 1.50 and y,/y, = 1.35. The 'value of 1/3/1/1 = 1.50 X L35 = 2.03, which is close to the assumed value, By Eq. (17-8), the length of the contraction is found to be 34.3 ft.

17-4. Expansions in Supercritical Flow.. Channel expansion in supercritical flow occurs frequently at places where flow, emerges at high

I

Eddies and circ.ulo1ion

C --- --,

'

_ ....,) LSePOrOliOn --.,

surfoce ---

1

,I

i

.J FIG. 17-10. Separation of How in an expa.nsion of rapid, divergence.

I 1

L

too gradual, waste of. structural material will result. Furthermore, if local disturbances of gre'at wave height are produced by improper boundary geometry, either at the expansion or farther downstream, the walls may fail to confine thc flow, Satisfactory desigl1 for the expansion is, therefore,of practical importance. . The best \\ray to study a particular channel expansion i:6 by cut-andtry investigation through model tests. Using an analytical approacb, strealb.lines and water-surface contours for any expansion can be con'structed directly by the method of chameteristics first footnote, Art. 16·5), provided that (1) the channel walls are vertical and the floor is horizontal, (2) the energy loss due to boundary resistance is negligihle, and (3) the pl'essure is hydrostatically distributed. Owing to these limitations, the analytical method ca.n give only approximate results. From both experimental and analytical studies, Rouse, Bhoota, and Hsu [8J have obtained the following results, which may be found useful in the preliminary design of ch!l.nnel expansions in supercritical flow:

J I

J

!

h 472

;FLOW THROUGH NONPIUSMATLC' CHANNEr., SECTIONS

. 1. The generalization of experimen1;al data for channel expansions may be expl'e:::sed by the following relationship:.

value of b1iYl and for various valuesofF l l1re shown in Fig. 17-12. This form of expansion boundary corresponds approximately to the shape of the streamlines that" corifine about 90 % of the flow. N cite that the begi;ming of this expansion is sufficiently gl'O.dual to reduce the effect of nonhydl'Ostatic pressure distribution to a minirriun1, so that the factor bdYI is no longer an essential vl1riri.ble. In thisiol'm, the general iliCrel1Se

y

. 1h

I

=

X

z)

(b1F.' b;,

f

(17-10)

where y is the depth of flow, Y1 is the depth of the approach flow F 1 is the Froude number of the approach flow, b1 is the channel width, x'is the 1.5

1iIf.~0.90 00.801>0.40 "0.7000.30 eO.60 1iI0.20 "'0.50 ()O. 1 0

i' I

.

473

RAPIDLY VARIED FLOW

----

1 "

-----

\1 \, 2.0 )

"

~rG. 17-11. Genel'!\lized surface contours 81OU.

"~

r I

")

I,

I:

x

VOiues of D,F,"

)

2.0

fOI"

(A/leI' ROllse, Bhoota, and Hsu [8).)

FIG; 17-12. GeneraJized" expansion .. (tifler Rouse,

superc.itica.l flow through a1rupt expan-

longitudinal coordinate measured from the outlet section and z is the lateral coordinate measured from the center line of the ch~nlleL ". 2. For abrup~ ex?ansions, a dimensionless diagram (Fig. 17-11) was devel?~ed. Th~s dl~gram, gen~ralizing the experiment!!:L data in the fllnctl~nal relatlOnshlp of Eq. (17-10)] may be used for the prelimina.ry analyslS of an abrupt expansion. " . 3 .. From the .experiments, the most satisfactory boundary form for an efficlen~ expansIOn was found to be ." ".

z

. b;.

I( x

=

~ \b 1F 1

)~'

+ 72

For" expansions designed fol' thiiS' form, the surface

(17-11)

contou~Q

for a mean

!I I

"I

i

I

()

sllTfac;~ BI~oola,

contours for suphrcdtical flow t.hrough gra.t\utli and FIs"1< (8J.)

in the boundary angle is sufficient to eliminate the format.ion of cross waves, yet not so great as to cause an undue change in depth across any norm:]'l 3ection. " . The boundary represented by Eq.(17-ll), howev~l', will diverge indefinitely. For practical purposes, the divergent wal1s of the ~xpan­ sion will usuallY be followed by parallei wn.lls with either an abrupt or a gradual transition, As a result, positive disturbances dQwnstream may be produced. When practical circumstl1nces permit, such disturbances" can be eliminated by a hydmulic jU1"l.lP M or near the end of the ~xpan­ sioH. To avoid dangerous asymmetry oIthe flow at ~he end of the ¢xpansion, the jump may be stabilized by a drop on the, channel fiool' (Art. 15-10).

474

FLOW THROUGH NONPIUSMATIC CIL-\.N}\/EL SECTIONS

RAPIDLY Y ARIED FLOW

4. The disturbance in the downstream chaimel may also be eliminated if the transition mentioned r,bove is designed with a well-pl:oportioned reversal of the wall curvature. In snch a design, the positive and negative disturbances developed. by the reversid of the curvature may offset each other so that the flow is restored to complete uniformity at . the end of the transition. Figure 17-13 shows the boundary curves of such a desigrl, derived by the method of characteristics. Although these

~--"--~----'---~~~~~~--~~-4---££=3t'__~_~~5 I .

~:

= 25

b,

I

.

..

1.1

---I' I

I----+-----+-a~b~+--+-~ = 2--+---/--b,

•

-1...---1---.-1 I

I

_I 2

:3

·4

5

6

7

8

-=-_--L-_J 9

10

11

12

Values of _x_ b, F,

F':ra: 17-13. Generalized boundary ~urve3 for channel expansion method of chamcteristics. (Aft~r Rouse, Bhoola; and Hsu [8].) channel width ..

de~ermined by the b, = dowl!stream

curves merely represent generalized conditions, they may be. used as guides in preliminary design in order to giye as little angularity of the final flow as is consistent with the practical requirements of t.ransition length. The U.S, Soil Conservation Service [9J has i,)jSO Hlade tests on sunercritical flow t.hrough expansions in conjunction with the SAF stiiling basin (Art. 15-12). The tests were made on transitions with straight fla.ring side walls. and 1 % channel slope: The major findings of the tests are: 1. A length of approach channel preceding the transition equal to 5YI is satisfactory, where YI is the approach depth of flow. 2.. rhe maximum permissible side-wall divergence is .1 in SF I if cross waves of excessiv.e heigp.t are to be avoided. . . 3. The use of the expQ..Dsion ahead of a SAF stilling basin is economicl'Ll. 4. Design charts providing information on relative depth contours, and flow. conditions within the transition were ,developed.

.;

i

! ,

, 'I..

475

17 -5. Constrictions. A constriction in an open channel cOl1stitutes a , reach of sudden reduction in t,he channcl tross section. The effect of a constriction on the 110w depends mainly oil theboundnry geometry, the discharge, and the state of flow.' The phenomeilOll is USUltlly so complicated that thc resulting fluw P;),tConstric!ion tern is not readily subject to any analytical solution. A pradi(',al E" >' E~ solution is possible, however, through systeml1tic experimental investigation. The flow through a constriction m:1Y be subcritical or sllpercriticaL When the flow is subcritic!I,I, the constriction will induce a . pronounced backw8.ter effect t.hr.t extends a long disjjance· upstream (Fig. 17-14a and b), When the (b) flow is supercritical, the constriction will disturb only the ,vater surface that is adj!1cent to the up_· stream side of t.he constriction and will not extend the effect farther upstream (Fig. 17-14c). If the [c) npstre!l,m water surface is dammed up to a. depth greater than the critical depth, the surface tyill form. !LIl En < E~ 81 profile, extending upstream oni y for a short distance and then ending with a hydraulic jump (Fig. 17-14d). . (d) . A critical contl'Ol sec.tkm mayor FIG. 17-14. Constriction in uniform-Holv may not exist at, the constriction, chlLn~el (a, b) in subcrlt,ical flow; (e, d) depending O~l the magnitude of the in sllpercritlc~,l flow. energy E" of the nonmJ.! flmv in com. parison with the energy E/ of the oritical flow at the constriction. For a short constriction, this situation is shown in Fig. 17-14. In the CaBe of a long constriction, the constriction itself constitutes a channeL The 'control section" when it exist,s, may be at either the upstream or the downstream end of the constriction, depending on whether the slope of the constricted channel is steep or mild. The entrance !1.ud ou~let of the constriction then act as a contraction and an expansion, respectively. The critical-flow flume (Art. 4-6) presents a unique case of channel constriction. The flow through the flume may be either free from or drowned

-I I

1

--I! I

-I

i

.I

.~

[

. -.{

476

j.

by the tailwater. A critical section exists at or near the neck of the flume only when the flow is free. Illvestigations on the subject of flow through constrictions are many, but most of them have dealt with subcritical flow. The contracted-opening method, I v, method fOT determining flood discharges which has been very popular with Americn,n engineers since 1918 or earlier, is based on the application of the energy and continuity equations to the· fiow through a constriction in the waterw[1.Y. The first laboratory inv·estigation in the United States is believed to have been m[1.de by Lane [ll}. This investigation dealt with simple constrictions of flows having Froude numbers slightly higher than those uSlmlly fOllnd in lll'.1;ural channels. In recent years, :.L ne.w development in the study of constrictions in subcritic

FLOW THROUGH NONP1USlIIATlC CHANNEL SECTIONS

RAPIDLY VARIED FLOW

Section (OJ _ _ _

B

T -~

(1) b (2) (3) Ccb

I

________________

I

'Ln"-- -~~

I

(4)

~

It

Live slreom

bo~ndory

Eddy zone

~__~ ___________-L~__~--I--------------~ (0)

Constriction

Note: horizontal si;cle distorted

Oalum·

(bl . Backwater profile Normal profile

ho'hj hOn'hln

j /.

" Channel boftom

Datum

HOlik [lOJ credited S. M: Woodward with the ~evelopnlent of this method for use

by the Mi"mi Consen'ancy District, The othel· equn.lly important method reported by Houk is tile slope-a.rea. method (Art. 6-9). , In Fig. i7-15, it is'shown that the flow through the constriction is subcritical, e.s in Fig. 17-14a.. This is t.ile case usually encountered in practical problems, If th~ water surface drops below the· critical-depth line M the constriction, supercritical flow will occtu·, as shown ill Fig. 17-14b. To reestablish subcritical flow downstream, a hydraulic' jum{will be developed near the downstream side of the constriction. The method described here, however, is not applicable to this case.

.UC~'VW"

I

(e)

FIG,

[14],) loss.

17-15. Definitioll sketch of flow through CDnatl·iction. (After l"}'(fey cmd Carter ((~) PIau; (b) elevation; (c) elevlltioIl, ad'lpted to ltsstunptioll of zero friction

478

RAPiDLY VARIED FLOW

FLOW TJ;mOUGH NONPRISMATIC CHANNEL SECTIONS

"'[tter. the water passes through the contraction, the contracted f>tremn reMhes a minimum width at section 2, which corresponds to the vena contracta in an orifice flow. After the vena contracta, the live stre&111 begins to expand until it reaches downstrea.m section 4, where the uniform-flow is reestablished in the Jull-width channel. Between sections 3. and 4, the flow is gntdually varied, Over the whole reach from sections 0 to 4: encomp:1ssed by the backwater "ffect of the C012stricti on, the total energy loss .is the same as that for uniform flow. FOl' a simple analysis of the flow characteristics, the following conditions m!1-y be assumed: L The channel .ftOOl' is horizontal or nearly so. . 2. Laboratory observatiol1s have shown that the level of the water at the vena contracta can be closely approximated by the leve! at sect.ion 3, which is at the downstream face of the constrictioll. H611Ce, the depth Y2 can be taken as Ya. . . . . 3., A coefficient of dischr.rge is intl'ciduced in the a.naIJ~sis to include the over-all effect of contraction, eddy loss, nonuniform distribution of velocity, and nonh)rdrostatic distribution of pressure. The related notatiOll to be used in the analysis is defined as follo,,'s: hd = eddy loss due to turbulence el1gend,~red in the upstream separation tones. It assumed that this loss can be exuri8SE;ea in terms of the velocity head at section 3,01'

is

h. = k.(Y:NZg) 0<1

aI'

and 0'3 alld a..'

=

=

where /'" is a coefficient. , energy coefficients at sections 1 and 3, respectively: I pressm:e coeffici"~nts at 13ecl;ions 1 and 3, reSpec.tlVe y. It is assumed that the deviation from hydrostatic distribu tioil of pressure can be in teni1s -of the velocity head at section 3, Thus, OI/Y'

=

kl'(y~2/2g)

+ Yl

where k 1> is a coefficient responsible for the nonhydro- . static pressure distribution. Cc = coefficient of contraction at section Applying the energy equation to sections 1 and 3 17-15c), the following can be writte~1:

3:

1112 0'1 -

2g

+

, al

VI

11 3 2

Oil

-2 g

+

;, 0'3 Yl

+

h,

+ hi

where hi is the frictional loss. Then, assuming a\' /c,,(vN2g) + YJ, and h, k.(T'N2g) ,

1, aaty,

=

(17-13)1

479

The discharge through section 3 is (17-14) .

Q = Co;4JTT, where ila is the water for the disclu,rge,

~l.l~ea

at section 3.

Soh'lng Eqs. (17-13) and (17-14)

I---·----------·~~;"

Q

CA. ,\/2f) .

where t.h equn.l to

= Til

'I

(6.1& \

hi .

+

(17-15)

al

.j

Ya and where C is all over-all coeificient of discharge, (17-1G)

This is a t-he(lretical expression. For prac,tical uses, the value of C ma.y be expressed by a dime.nsionless flJI1ction and evaluated experimentally, By a dimellllioiutl ahalysis oi the essenthl,t geometric and hydraulic factol's that govel'u the flow, it can be shown that C is expressible £l.'3 a function of, the or

C

=

f[m, Fl> rIb, r'V/b, 6,
+ Yb)/2b, x/b,

i':,

t/(y'J

+ 6.h),j, Ljb] (17-11)

The hydraulic factors contained in this fUllction represent the following various physical eff~cts: 1. The effect of channel cont-ruction is represented by the contraction ral,7:o ni. This ratio is eq lIal to 1 - Kb! K Il , in which [(IJ is the conveyance of the tUlcontracted ariproach section r at normal dischal'ge and [(b isthe conveyance of the contracted section: 3 which has the SHme normal depth and roughneils chn.l'acteristics as the approach :section. For a recta,ngular approach section of width B and a recta.ngular contracted section of width b, this ratio is?n = 1 - biB, AC01tV8yanc.7J 1'al'io (J may be defined as 1 -?n. Thus ~ 1 or 7n=O when there is no constriction. , 2. The effect of Froude n~mber is rep.res~nted by the Froude number at section 3, or F3 Q/(As V"iYa). It is necessary to assume a di.scharge for the initial computation of the value F 3. The final value of F 1 can be determined by successive approximations. If the value of F 3 is f{reater than 0.8, nearly critical or supercritical velocities may occur in section 3; . then the method described here is ina,pplicable. . 3. The effect of entmnce rounding of theabutmellt is represented by rib, where r is the radius Of rounding of the entrallC~ corner of abutments for vertical-faced constrictions. 4. The effect of chamfers of the abutment is represented by TV/b and tl , where W is th~ length of wing wall measured ill a direction normal to section 3 and where' 8 is the acute angle between a wing wall and the plane of constriction. .

i r

-I

'.

I

I

Il r,

!

I

.1

i

480

Jt

RAPIDLY VA.RIED FLOW

_

I J

.,'"'" .J::. ",-

.~ ~

"'0

-~ o

-'" c~

.~.g uc

=-10 ~-

.,~

"

u

" 'u

Ca)

o

-

~

:3

-:

o

~

o

;0

:.::

(b) 1.2

r

tt b

io

(0 -~

tD

E

j

tl)

~'---"--r-, --'"'.~~

:T

~!' "0,

f' I

"" c:: " "0o.

\

I

Li

n

II

~~~ 11 , )

1

I,

I J

!

1

"II

I-

,',

N d

f---'+-'\--"l-

'0

l-----+--+-!f-.--,

~ ."


e:0

';; 5!

v,

3!
e:0

~~j'il 1 '"

'0

i

[ -~ 3:1

,

I

o

(e)

FIG. 17-16. Disclul.rge coefficient for constriction of type I opening, vertical embankments (E = 0) and vertical abutments. (U.S. Geol(J!lfe~1 Survey [12J.) (a) Base . ' curve for Coefficient of discharge; (b) vl1l'ia.tion of discharge coefficient with Fl'oude nll.mbru·j (cJ "Il;liation of discharge coefficient with entrance rounding.

481

•

482

RAPIDLY VARIED FLOW

'5. The eff8()t of angularity of cOIlst~iction is represented. by
+

483

FLOW THROUGH NONPRISMATIC CHANNEL SECTIONS

" D

~ o.ao j-----'-----'----+---+---=--..;...:...,=c-=?---=+-==l---l,...,i Stone-erd conditions

:6

X=-+Yb ~'O.~O

~

o

'0 0.70

.!!

~

_t_,o

u 0.60 ·u 0

+

I

j ;0

~~ +LI'

IT

~-.


FJ ' 0.2 1o 0.7 • '1.00

I

10

3D

20

qo

50

60

70

90

100

m;lPer c:-t'nt af channel cont"roction

(a)

+

1 ·.1

tOO

<1>'0' 10'

(17-18)

where C' is the standard value of the coefficient of discharge correspond..:" ing to a standard condition of all effects mentioned above; and where the 1{'s are coefficients that can be used to adj ust the v:alue of C;' to a given nonstandard condition of the Fronde number, entrance rounding, chamfE?r, angularity, side depths, side slope, bridge submergence, and bridge piles and piers, respectively. From the laboratory in,restigationby the U.S. Geological Survey [12], a set of curves we~e developed for the determination of these coefficients for four different types of constriction (Figs. 17-16 to 17-23). . . When a waterway is coritmcted by bridge piers or piles alone, it may be al3sumed that C' = I, and C is a.djusted by Ki only. To determine

I I

-25'-

--_._----

0.90

_2Q.·_

Ko/>

\i~

0.80

--,-

0

(e)

.i

i {

J

-I

41"-~---~i

d.70

10

20

40

30 m: Per cel'lt

of

50

so

c.honnel controclion

F~G. 17-17. Discha.l'ge coefficient. for co;\strictiou of ty pe II opening, ~mbankmellts WIth 1: l ~lope (E = l). and vertical abutl'lent..,. (U.S. Geological SW'vey [12].) (a.) Base curve for coeffiCIent of discharge; (b) variation of dischnrge coefficient with (y. + Yb)!2b ratio; (c) variation of discharge coefficient with angularity.

I

484

},'LOW THROUGH NONPIUSIIUTIC CHANNEL SECTIONS

RAPIDLY Y ARiED FLOW

485

1 I

/TI =

P·H cetM of c.!-.on/lle.l cant:oeliol1

(a) m=Per cen1 of chonl'\el contraction

(((.)

0.05

K,~0.90

0.800:--"---'10--.--'20

30

50

60

70

90

100

m" Per cant of c.l';onnt:1 c.onftoelion

(b)

'T~~~~ ---+-'--+----+-=.4 ----1----1----,

0.90 K¢

(b)

-

-~Q.'- ----- I-----\~--

0.80

40

80

rn= Per cerri ot channel Conln::lcflon

(e)

F~G. 17-18. Discha.rge coefficient fOl" constriotion of type II opening, embankment.~ With 2:1 slope (E = 2) and vertical abutments. (U.S. G?oiogiealSlI1'vey [12].) (a) Base curve for coefficient of dischl1.rge; (6) ·vl1.riation 'of c!ischaro'e coefficient with (y. -I- y,)/2b rafio; (el vMia.tioll of discharge coefficient wit.h angLI1:1rity.

J

1

,--."..,,_ _---'--c-_-'--_ 0.06 0.10 0.12 0.14

{=Rotio

or

x tcwidth of opening

(e)

FIG. 17-19. Discha.rge coefficient for constriction of type III opening, embankmerrts and obutments with 1: 1 slope (E = 1). (U.S. Geolorrica1 Survey [12J.) (a) Base . curve for coefficient of discharge; (b) variation of discharge coefficient with angula.rity; (I) variation of discharge coefficient with x/b ratio.

)-

486

nAP~DLY

FLOW THIWUGH NONPRISMATIC CHANNEL SECTION'S

VARIED FLOW

487

~

~

'0

>

"0

l; ~

~ 0.90 I----'~"

. ~

~

0.80

I--_~_-I-.

___-'-_~"",,-!..::::,.,,,,,,_

'il

lO.70 ~

" ..'"~

50 rn£P~r

cf)'nt

~l

60

70

60

90

ioe

m" Per eel'l

channel tontrct:tion

at chon,,1

con1r~ction

(a)

(a)

11'\'

Per ctJn1 ;,t chonr.et con1rClcfion

m ~ Per cent af chollne:1 conlrocfion

(bJ

(b)

1.15

1.10

I

i

Kw

,

1.05 I

;. ,r

\

\.00

0 ~:r Rafio of x 10 wi_oth 01 opening

(c) FIG. 17-20. Discharge coefficient for constriction of type III opening, embankments and abutments with 2:1 slope (Iff = 2). (U.S. Geological Surull11 [12J.) (a) Base curve for coefficient of discharge; (b) variation of dIscharge coefficient with angularity; (el variation of discharge coefficient with x/b raj,io.

I I

I

! !

B:rAI"'f~ Ie of wing Wl)1I

(c)

,:IG. 17:21. Discharge coefficient for cOllstridion of type IV opening, embankments .nth L L. ~lope (Iff ~ 1) and vertical abutments with wing wa!ls. (U.S. Geological Sw-ve:y. [12J.). (a) Base curve for coefficient of dillchslge' (b) variation of dlschargo ~oeffiClalit wIth ll.IJ.gularity; (c) variation of coefficient \.yith wing-wall angle.

l

.!

FLOW THROUGH NONPRISMA'rrc CHANNBLS.ECTIONS

489

1.00_=~=c

f f

! t

I

!

K

,

---r[---11

,.00\-----:;0",--..

osoo

0:,

0.2

0.;

0.4 .

(a)

(b)

• (a) t.OO'

I PI~E:S

0/>- O'

--'10'0·-----

t:";WdZl

0.90

___

o

0.70

~rr

4"i---~-

. ~l

·10

j!.~

0

A3

OI'OO~~~~' g~ ~<-'--l .i! 0,90

J

',DO

0.90

",-

~---

0

0

~

K~

O.BO

C 0

Q

--20'------

~O

.

60

30

70 .

60

m:l. Plllr cent of ~hanne:1 contraction

(bj

m=Pcr cent of c:honnel con/toction

(c) P'ERS

= =

~

j·o

1.00

l-----,~-.,..-'--'-..,---r===+===F=33;:::;1 I I 0.05 I '1 I '

~lo.H5:::::i::::I:o:.lo:+±I-::-:-:-::::::±I===:J~=-_.-L-r i so 20

30

4Q

50

m~PI!"

60

70

.80

100

cenl of channel confraction

(d) KO 1.• 10

FIG.

I.003).,O~.-::::;J35'"'.,..---,4-:f;O"·:---.-:4f;5;;;·---;;5';;;O'·--,;'5"5.--7 • 60 , e·Ao~l.

01 wing ... 11

(d)

FIG. 17-22. Dischtlrge coefficient for constrict.ion of type IV opening, embankments wiUc 2: 1 slope (E' = 2) and vertical abutments with wing. walls. (U.S. Geological Sw'vey [121J (a) Base CUl've for coefficient of discharge; (b) variation of discharge coefficient wIth angLIl.arity; (e) variation of,discharge eoelficiellt with' Froude number; (d) variation of discIlarge coefficient with 'wing-wall angle.' . 488

17-~3. IC, K"

and K,. curves for constrictions of types I to IV openings. (U.S. Geolov'iGu! St,rvey [12J.) (a) Varia~ion of discharge coeff.cient with eccentrici~y; (b) . variation of discharge coefficient with degree of submergence of bridge; (e) variation of discharge coefficient with area of bridge piles; (d) variation of discharge coefflCieut with area of bridge piers.

K; from the diagraro(Fig. l7-23c), enter the hOl'izontuj scale at the propel' value of m, then move verticidly to the vaiue of Lib, horizontully'to the line marked j = 0.10, vertically to the vulue of j, and horizontally to the value of f{j. If j happens to be greater than O.lCl, an approximate computation may be made ~y taking C = C' for the given m and omitting Kj •

490

RAPIDLY VARIED FLO!\"

FLOW THROUGH NO:!lfPRISiHATlC CHANN'BL SECT£ONS .

It is that ~rtaiIi combinations of the empirical coefficients applied to C' may appear to yield a value of C greater than 1.0. In sllch cases, howBvei·, a value of C 1.0 should be used. . Tile friction-loss term h, in the dif;lcharge equation, Eq. (17-15), represent.s the total loss of head due to .friction between sections 1 and 3. This loss is equal to the sum of the loss in the approach reach of lellgth La from section 1 to the upstream side of the contracted opening and the lo~s ill the contracted reach of length L. The total head-friction' loss .may be cpmptited by.

L"(VI~d(3Y +L(I~3Y

stage .to the backwater stage at section I iskrlOwn as the backwater of lhe c(Jnslriction. The'distance Lilt is the difference in water-surface elevation between secti.ons 1 and 3. From Eq. (17-15) the value of tJ.h can be. shown to be V

2

.

~l 2~ + hI

(17-21)

The rat.io h*/ tAh is called the backwater l'atio, which is known to be a function of the cluH1l1el roughness, percentage of chamiel contraction,

(17-19)

where [(land J(. are the total conveyances of sections 1 and 3, respec. (17-15) the above expression for hI, and tively. Substitut!'l Q/ A! for l'l; the may ·be expressed as Q

= 8.02CA a (17-20)

This equation can be llsed for computing the required discharge. ExaI~ple 17-2. Compute the flood disch~rge through a. highwa.y bridge opening in a practically ·sl.l'aight and uniform reach of a river. An appioar.h section 300 ft upst.ream from the bridge ha.s the characteristics described in E:,ampl~ 6-1. At t~e ccin~ra~ted section, the bridge spans over ·the main channel and the slde channel 15 entirel)' blocked up by the highway embankment. The constriction thus created is 30 it long and 180 ft wide and has a type J opening (Fig. 17-16). The a\"erage water surrace at the contracted section wa.s found to be 5· ft below that at the approach section. It is assumed that both the main and ~ide channel sections are approxim(t.~ely rectlJ.ngu1~I'. Thus, the dimensions are 180 ft wide by 29.8 ft deep for the m(t.in ch:;,n!leI and 390.4 It wide by 14.6 it deep fcr the side channel. Solution. From the datil in Exa.mpl( 2-1.8 = 4,460 ft', p, = 180-+ 2 X 24.8 =' 229.G ft, n, = 1;)..5 ft, 11. == 0_035, and thus J(, = 1.38 X 10". Other required da.tB are C'l = 1.29, At = 11,070 fV, Xl = 3.14 X 10', t!.h;= 5 ft, L = 30 ft, and "" 300 it. By Eq. (17-20) 1 Q = 60,000 cfs. The above computation is based on the sta,tPard condition that F, = 0.5.: Now, F, = 0.470, so it is to :repeat the computation for e.djustment oC F,. By successIve l!.pproximations, final Q is found to be. 59,500 cfs. In the case.of typ.e II and nI openings, C is not a significa.nt f~tncti6n of F l ; hence, no successive a.pproxlInations a.re necessa.ry 'ft'r the determination of ,Q.

17-7. Backwater Effect due to Constri4tion. hl* (designated

In Fig. 17-15 the increase also by h*) in: water surface from the normal

0.040 Manning's un"

0.050

0060

0.070

FIG. 17-24. The effect Qf cha.nue! roughness on the backwat,er ratio for basic-type constrictiolls. (;iji8" Tracy and CarieI' [141.)

and constriction geometry. By means of the continuity, energy, and momentum principles, it is possible to obtain an approximate solntiOI1 for the backwater ratio 011 the assumption that normal boundary fdction losses are zero, .. A more practical solution of the problem, however, may. be rea,chad by laboratory investigation. . TracY and Carter [14J hn.ve made a laboratory investigation at the Georgia Institute oi Technology and the University Illinois 011 the backwater effect due to. vertical-faced constrictions with squnrs-edged ahutments. The experime!ltaJ data can be pl6tted.as shown in Fig. 17-24 . to indicate the relationship among backwater ratio, Mannii1g's 'n, and contractiol1 ratio '17'., It can be seen that the channel roughness is relatively' unimpol'tn,nt us a factor iIi determining the backwater ratio. In .fact, the limit of change jn the backwater ratio due to roucrliness is ·practically reached at an n of about 0.050. The laborat,ory i~vestiga­ tion also reveals that the influence of cross-sectional shape on backwater ·ratio is included in the c6ntraction ratio.

of

.J (

I

492

FLOW THROUGH NONPRISMATIC CHANNEL SEC'l'IONS

RAPIDLY VARIED l"LOW

The backw[\,t(-~r I':1tio in Fig. 17-24 is for constriction of basic type., that is fol' a vertica.l-raced cor,striction ,v-ith square abutments. The backw~lter mtio for other types of constriction may be obtained by multiplying the backwater ratio by an adjustment factor lea. This factor has been found to be a function of the contraction ratio m and the ratio CIC"",;" Cb "" ~lncl C are, respectively, the discharge coefficients for the

Taking an approach sect-ion 350 ft ~lpstream from the constriction, the friction slope section and the contracted section ma.y be estimated by the Manning equation S, =Q~n2j2.22rl'Rj,~. Now Q = 50,000 cf~, n = 0.035, A = 400 X 20 = 8;000 ft', and-R = 18.2 ft,; hence, Sf = 0.000-!48. The frictional loss h! = O.000-!48 X 350 = 0.16. AsBuming h~ = 0.20, e:.h = 0.20/0.26 = 0.77. \Vith As = 350 X 19.'13 = 6,800 ·ft' and C = 0.87, V a'j2gC' = 1.1l. By Eq. (17-21), uh = 1.11 - 0.59 + 0,16 = 0.68. This vaiue is less than the t.ri:J.i value 0.77. The computation is to be repeated until agreemeat is reached. Before this' is done, however, the value of C should be adjust~d lor F,. Now Fa = 50,000/(6,800 X --v'32.2 X 19.'13) = 0.29. Using Fig. 17-16b, C = 0.87 X 0_955 = 0_83, Thus, Va2/2gC' = 1.22 and e:./i = 0.79. This is close enough to 0.77; hence,a r6peat.ed cumputation for successive approximation is not required. between~his

;.

1

..I 0

\~~_I

\ "I . . . . . ~

.

V73--~--'--'---~I---'

'il73

1--,.... V73

0.9

o.s

0.6

FIG. 17-25. The effe(:t of constriction geometry on the backwater ratio. Tracy and Ca"leT [14]-)

(After

b1Lsic type and for other types of constriction that CM be determined by the method described in the preceding article. The value of Cb.", can be Clbtained directly from Fig. 17-16a and b. Based on experimental data, the relationship among lea, m, and CIC b .", is shown in Fig. 17-25. Example 17-3. A stream ,discharging 50,000 cfs at flood stage flows through a symmetrical constriction having a type I opening which reduces the normal width of the w~terway from 400 to 350 ft. If the average -normal depth of flow in the uncoiltracted sectiun is 20 ft, what will be the height of the backwater caused by the con. strictLoll? The opening of the constriction is 30 ft long. The coefficient of roughness of the channel is n = 0.035 and "'1 = l.10. So/-ution. T.he contraction ratio 1)1 = 5%00 =0.125. With n = 0.035; the chart in Fig; 17-24 gives h*,/e:.h = 0.26. With L/b = 3%50 = 0.086 and F, = 0.5, the chart in. Fig. 17-16a gives C' = C = 0.85.

493.

i

I l

J:

~. ~.

17 -8. Flow through Culverts. A culvert is a unique type of constriction, and its entrance is a speciallcirid of contraction. The culven a'cts ",
RAPiDLY. VARIED FLO,y

494

"''"

r,-rr-rr-'---"'--~---'

to the contrac.tion of Row in the form of a jet under a sluice gate. Thii': high-veloc.il,y let will continue through the barrel length, but its velocil;Y will be reduced slowly as head is gradually lost by friction. If the culvert. is not sufficiently long to allow the expanding depth of flow below the contraction to rise and fill the barrel, the culvel·t will never flow fulL

+-11 II t: !

., 0

.; 0.05

~

a. 0

.,

".,

+-t--+-j-L I

0.04

/' v

11

"

,

I 0.01

I

f

/

:II/,.V

0 0

\

I

1\'\ \ \ "'- I',",~)J)\

\

'"~~ l\\~ ~~~'" ""r::: t::;::

t!

1

~

{'

~(j'c'

\ I

i

~

'"o

'"od

00

i

if>

o c5

d

-t+-

I

.I

2 _, ~t- 1--1 (lon9_) I ~Type

Vi~ L...-\--i

10

I

20

\

0.01

f--

o

"

f-.--

....!o

f---

;::J 0

oj

:::.:::....l--Ul

o

o

'" I

0 .'

...Jjo

30

L

d

I\:

\

~ \

l

~

J\

...

..c:

.~

!,

>

r-< ;::J
o

'"

bJ)

0;

..s

J

"d Q

.~b

\

oj

!

\ +-1 1- -.-

\

pt~o~\O Jg~\ r

"'0

,

\ I

\

f--+--

FIG. 17-26. Criteria for hydraulically short and long box ane! pipe culv~rts with concrete barrels :J.nd squ&re, rounded, or beveled flush entrances from a vertical headwall, either with or without wing walls. (U.S. Geological 81J.rvey 1271·)

Snch a culvert is considered hydra1Glically short. Other\vise, the culvert, is long, for it will flow full like a pipe. ' . 'Whether a cu\vert is hydraulically short or long cannot be determined by the length of the barrel alone. It depends on other ch~racteristics, such as slope, size, entrance geometry, headwater, entmnce and outlet conditionA, etc. A culvert may become hydraulically short, that is, it may flow part.ly full, even wheli ttJ.e headwtLter is greater than its cr)tical value. For this situation, Carter [27] has prepared charts (Figs; 17-26 and 17.:.27) which may be used to distinguish roughly bel',ween a hydrau-

1

'"

...,

.~~

I

..c:cD

:;:: -3

d

,

L'J

,.....'"

.0

1

0.02

~

F§ r-=: ~

'"0-

q

.,00

T

D

~i(b~~"~ \.\ ,

",-" ~~

~"

hydrau~ican!l

f::: t::=

S!

r--::: I--

d

j

o

~

o

o

od

'C/)

'I

OJ

q


lD

g

..'"

C,J

I

I Ul ·0

,s... :~

'"

;

00

'" g

o

o

if>

\

I ~

495

t

.• '!



8

!

Type). V

I ,/ )

I%i

::::;

~Io

1

q

I

\ I '.(Shorl)

I

0.02 I--

,

1

I '17

I

0.03 I---- I--

o.o~_ f--

I

II I -'

O)

I----

Ii····I--

'" 0

\

I----

"" ~


~

..JIO

0.06

.::l

~

f--

o

l


.~

'"

I

~

I

-,:-." -.

~.~

..

',.

_. -',_.

-- - - -".

->.:':."

RAPIDLY VARIED FLOW FLOW THROUGH NONPRISMATIC CHANNEL SECTIONS

lically short and 11. hydraulically long culvert;. Under suitable conditions, n hydrttulica!ly short culvert with submerged entran~e may prime itself automatically ,md thus flow full. AGcordi:1g to the laboratory tiotls by Li and Patterson [28], this self-priming ,~ctiol1 is due to a rise of the water up to t.he t.op of the culyert caused in most c\\ses by a Jump, the backwater eff0ct of the outlet, 01' a st,lndilig surface Wiwe developed i.nside the barreL . For Pl'!lctiooal plll'po:>es, cuivert flow 111£\)' he clf\,ssified irlto six as shown in Fig. 17-28. The ic1enti5cfltion of en~h type may be explained accQrding to the followi'ng outline. A. Ontlct submerged ....

Type (1) Outlet submerged H >d 1/,

>d

Full

ftOlV

(2) Outlet uIlBubmerged

H>H"

y; < d

.

l?ull now

. , .... ,.,. Type 1

B. Outlet' unsubmcl'ged 1. Headwater greater than the critical value

a. Culvert hydraulically long .............. . ....... . Type b. Culvert hydraulicdly short ,. , .. " .Type 2. Hcud\vater less than the critical value a. Tuilwater higher than the cl'itici11 dept.h, . , ... , , .....Type b. Tailwater lower than t.he critical depth i, Slope subcritical. , .. , ............ , ....... ~ .... Type ii. Slope sllpel'cl'itimll. ...... : . . . . . .. . .......... Type

2

(;1) OuClet uns~bmeq~ed If> H· y,.< d Pa~tly fun

3

4 5 6

If the outlot. is the cllivert wi1lflow full iike 8. pipe, and the flow will be of type 1. . If the outlet is not submerged, the headwrtter may be either 01' less tlmn the cl'itic:11 vn,!ue': When the headwater ii; greater than the critical the cuh
(4) Outlet unllubmerged If < H· y, > 1h Subcl'itic£ll Bow

(5) Outlet unsubmei'g;erl

H
< y,

.

SuiJcl'itical flo,,' Control at outlet

j

(6) Outlet ,'nsllbmeq;ed

H 'Yr

< U' < Ye.

Supercritica,[ flow Control at entra.nce

FIG. 17-28. Types of mtl'rert flow.

i t

i

I "

497

j 498

II

RAPIDLY VARIED FLOW

..

FLOW THROUGH NONPRISMA'l'IC CHANNEL SECTIONs

may be obtained by means of the charts in Figs. 17-29 and 17-30 for box l culverts and circular culverts, respectively. Both charts supply information only fOl' (l1llvert.s having square-edged entrance.s. For culverts having rOllnded entrances under average conditions, the value of H/d may be roughly estimated the following expressions, in whIch Hid refers to the l'!1,tio of headwater to barrel height for culvert with squareedged entra.nce:

'"

'u

'0

'"

I HId,

Type

X

2

'+ 5

:3

10

20

30 4050

Circutar, .. Box, .. " ..

'

.. ,

.5 ..;

:

. h d t.e on 00" 'ulver~~ with square-edgGd enefstlll'rna~lBng de:.,.. \d~I,a.'of U.S.AB:reat, ~f Public Roads [29].) U. I. aJe ,

100 r--t-'-Jj,..-

50 f--+--lrl-t-+-t-;--y:

J.

J 0 I----t---'H...,..

N<

00 t")

':t

\i')

Discna rg e io cis, 0

Fl 17-?O Chart for estimating headwater on circuln.r cu~vert,sl )Wlth square-edged G., v'f l' ' owmg pa"ly " full . (Ba~ed . on data of F. T. Mav.s [17 . entrl).nces.

the outlet.

Type G How is supercritio~l and, hence, the control seotion

~ntrfanche. flows na YSlS 0 t ese

is aAtth1e

may follow the p' rocedure described in Art,S. '" " d t '1 d 11:1 to 11-3. The U.S. Geolo:gical S].1rvey, [27J has deve~oped a e al e. 'd that may be used for the hydraulic computatIon of a culvert proce . teIsotU"IOn 0 f the problem design ureFor' practical purposes, an apprOXlma

,~."

..

10

< Hid <

1.5

0.87 H/d 0;36 + 0.64l!ld

Hid> 1.5 1.09 0.02

+ O.WHlll + OA6H fa

17 -9. Obstructions. An obstruction in open-channel flow present:) a phenomenon very similar to that of a COllstrictioll, since both have the effect .of contracting the crOSS-Sectional area of the flow. Howe'ver, the constriction reduces the craSi! section to a single opening, whereas the obstraction cJ:l'lates at leapt ·two opeilings. The Df contraction offered by [I constriction is usually higher than that by an obstruction. ' The types 'obstruction commonly encountered in engineering problems include piers, pile trestles, tra.sh racks, piers and abutments on top of ovedlow spillways, etc. 111vestioO'ations of these 17-31. Definition sketch showing problems areuumerous in hydraulic, Fla. How through all oblltru;:;~ion. literature. ~

I

of

g 40f--t--lrl-t-+-t-r-iTT 'i;

< 1.0

O.87Hld 1.00Illd

....

OischoJge in cis per II oj width.a/b

, FlO. 17-29, ~harG folr tl'llnces, flowmg p!l.r~ y

499

The flow through an obskuctionmay be sub critical or. sllpercritical. The energy equation fOl; the l'eacll between the contracted section 2 and section 3 below th.e contraction (Fig.1i-31) is VZ2' . ' E ( V2 + = Y!1 (17-22) (/ , or ,

~Y2(2

2)

+ F21)

= YJ(2

+ F32)

(17-23)

data for box culverts were prepared by analogy from experience with circUlar pipes and are beli~ved to be conservative; tht\t is, they will give h!l!\.d as high as is likely to occur uncier adverse conditions. 1 The

For important'referen<;es on the subject or' flo\v through bridge pi
~restjes.

I

£i00

RAPWLY V AnIED. FLOW

TRROUCH NONPRISMATIC CHANNEL SECTIONS

ivhen: F2 = V 2 /Vgyz, Fa c= V3/";{jY~, and e the percentage of energy recovery, sinceel1ergy loss will occur between the sections. By. the continuity of flow,

. V2b~Y2

·.F22U2Y2~

or

• when')

(f

bdB3.

= Y1Bay. Fa 2 Y3 J:.

Eliminating 1/2 and:!i3 from Eqs. (17i23) and (17-25), 2· _

U

-

e1Fa2(2 --1•

FzZ(2

F~2)3

(17-26)

+ Fa2)l

,;\7h6n the flow at section 2 is critical, F,

,

(17-25) .

=

1.

The v1111.1e of F 3 that

1

[ I

I \

501

ingly, the flo'll is subcriticnl iUhe value of Fa faUs on the left side of the plotted curve 9£ an itSsumed energy loss andslipercritical if F J f
FIG. 17...aZ. Criterion for subcritical and mpercrit'cai flows through an iJbstructiQll,

satisfies this condition is called the limiting value and is designated by F aL, Thus, for I, the above equation becomes . u~ ~

.

=

2 27E 1F,L (2 + Fn 2)3

(17-27)

For: a given IT, therefol'e, the flow through/ the obstru~tiol1 is critical if Fa T F aL . Examination of Eq. (17-26) indicates thatithe flow through thei obstruction is 8ubcritical if < FaL and supel'critical if Fa > F aL • Eqt\ation (17-27) is plotted ~s shown ill 17-32 fori E 1 (no energy losS), € = 0:95 (5 % energy lo~s), and E OJ) (10% energy loss). Accord-

where Ie" is a coefficient depending on the degrae of channel contraction and on the characteristics of the obstruction, Ii is .m adjustment factor intended to reduce the depth y. to Y2, and (J is a coefficient wl1ich corrects for the velocity of approach. The notation used in the formul!\ is shown of in Fig. 17-31. The backwater is represented by h'l or h". The f) varies, approaching zero a~ the contraction effect approaches zero and becoming high whe11 the· fiow is shooting and turbulent, but it can: Ol'dinai'ily be tn.ken as 0.3. The value of f3 varies with the cOll\;eYlll1ce ratio, as showp in Fig.17-33.. .. ~ In deriving the d'Aubnisson formula, the energy equation is t.pplied to the approach channel section 1 alld to the cOlltraded section 2 (Fig. . ' . ~ The "'Yei~ba.ch formuLa. is based UPOll t.he ~ssumpti6n that the flow ente:ring the contracted ing thrDugh I.l. . this assumption [3'2J also have

mll.)' b.e ca.lculated as the Burn o( two portions, the lower p\,-rt, flow· orifice a.nd Lite upper part flowin~ over a weir •. Ap~arelltly, and has no theoretical basis. Houk [10] and Bhbeucley it unWll.rril.L1ted and cOlltrary toestnblished hydra.ulk t!teory.

II

I I

I

RAPIDLY VARIED FLOW

502

17-31) or, 1wing the l)Ota.tion in the figure,

,

V 22

+ YI = CJ!z 2i + Y2 +

h I

(17-29)

503

FTJOV.r THROUGH NONPRfSMATIC CHANNEL SECTIONS.

experimental data, Yarnell derived the coefficients for these formulas. Based on this information, the following values of [(N alid Ie] are recommended for practical use"

+

Conveyance

where Yl = Y2 10 2, l,he velocity-distribution coefficients ell and a~ are assumed 'equal to 1.0, and the friction loss hI is Msrimed zero. Taking

Type of pier

0.9

0.8

.0.7

ra~io u

0.6

0,5

I---;--I·······:······:---l-,-~,--l---:--'-

"1-

..

f......-.. -I------l~"--.+-.-I-­

::>

~ 1.4 ___...!i--........:.-_+_-+ 1.21----1----+

00

02

0.3

0.4

0.5

0.6

Caoveyonce rotio

FIG. 17-33. Va.lues of f3 in ~he N.:lgler bridge-pier formula.

is a coefficient of contraction, and simplifying, (17-30)

Q

el' Aubuisson made no distinction between the depths Yz and Yz, but assumed Yz

= Y3.

Accordingly, the d'Aubuisson formula is Q

-1

SqU&re noses and tails .......... . Semicircular noses and tails .... . 90" triangular noses and tails. Twin-r,yiinder with or without ,.'. O.

1.8

"' 1.6

"'-

K,l b2Ya "/2gji~

+ VIi.

(17-31)

where K.{ depends mainly on the degree of channel cOl'ltraction and OIl. the shape and orientation of the obstl'l1c&ion. The assumption Y2 Ya implies that the backwater h* = ha = hz in Fig. 17-31. Accordingly, the backwater ratio has a va:ueof 1.0. This),,; of course, not always true, although in many cases there is very little difference between Y2 and Yl' For this reason, the d'Aubuisson formula is genel'ally recognized to be empirical and approximate. The following' are extracted from the results of Yarnell's investigation on bridge. piers of variolls shapes and sizes: L 'The height of the back~vl),ter due to bridge piers varies directly with the depth of the unobstructed channeL . 2. The Nagler, d' Aubuisson, and Rehbock formulas give' approximately correct results for ordinary velocities when the proper coefficients al'e used, but they do not. hold for extremely high velocities. From his

3, For flow of low yeladty and least turbulence, the more efficient shapes arfJ len.'S-shaped nose and '"ail, lens-sh~"ped nose and sell1icircular tail, semicirClilar nose. and lens-shaped tail, co'llyex no'se and tail, and semicircuhll" nose and, taiL 1 However, the data are 110t sufficient to differentiate RlnOllg these shapes for high of contraction. 4. Twin-cylinder piers either with or without connecting diaphragms, piers with 90" triangular noses and tails, a.nd piers with rec;::ssed webs are less efficient hydraulically them those just mentioned, and piers with sqnare noses and t?ils are least efficient, 5. Application of batter to the ends of slightly increases their hydraulic efficiency, that is, raises the KN and values. 6. the length of a pier from 4 times the widt.h to 13 time;:: . the width has compal'atively little effect on its 'hydraulic efficiency, II! some cases the efficiency .is t.hus increased anti in other case.~ decreased. The optimunl length-width ratio probably varies with velocity a.ndis generaUybetwe:en 4 and 7. On the average, the values of KN and ](A will increase about 3 to 5 % for the increase of the ratio from 4 to 13. 7. Placing the piers at an angle with the current has an insignificant effect on the alUount of backwater if~the angle is than 10°. Placing 0 the piers at 20 or more m
about 7

at 20°.

In order' to utilize the results of Lhe

and to fa~iIitate .

lens-shaped nDse or tail is fDl'lned by two convex C~ll'ves tangent to the si;:lcs of pier l),nd of radills twice the pier width. Th", convex nose 01' I,ail is ionned:by two curves ('illlgent t-o the sides of pier find circum$criberl on i'\n cquila.tenl triangle.

\

I

504

505

FLOW THROUGH NONPRISMATIC CHANNEL SECTIONS

RAPIDLY ·VARIED FLOW

,2.4

hydraulic compu1",t;ion for practical applications, Yarnell has developed empirical fotmul:.l.s and graphical solutions for the 'bridge-pier problem. The chr..rt in Fig. 17-3<1, designed by Yarnell but slightly modified for , the preSeJlt purpose, supplies a quick solution for subcritical flow through bridge piers. With the appropriate values of u and V 3 } the value of x , can be obtained from the left part of the chart. Then enter this value and the proper values of F,2 and shape factor 1{ on the right part of the chatt, and read the backwater h.... The shapes shown, on the chart

1'-

i

2.2

I

2.0 r--- r--

, 1.8 r--- r-- 1.6

I

-'Vi

I

I

II

/ vi r- -

I

Yo

I lYi[ll I ',ti't1 '-0==0 I I. ~

I

I

i

-

I

J4f", L -~. ~ f11J o k. ':'=

I

e==)-

--_. --+----

I

0.6

0.4

.

C=>"-l

0.8

1-I'
l!

-

, .... - f-

1.0

Vf;-

c:::::J~

I

I

h" 1.4

1.2 -

I

._-

0,2

I

0.6

r-

---

4

3

5

,

,

FIG. 17-35. Chart for determining baclC\\,ater of supercritical fiow through bridge piers. (After D. L. Yarnell 142].) . F:IG. 17-34. Chart for determining backwater of subcritical flow through bridge pier~.~

(Aile)' D. L. Yarnell [42].)

are square noses and tails, 90" triangular noses and tails, twin-cylinder piers with and without connecting diaphragm, and semicircular noses and tails. If the backwater is given am). it is required to determine the discharge, it will be necessary to as:mme a value of V 3 and to carry out the computation by trial and error until a aolution ia obta.ined. The chart in Fig. 17-35 gives a graphical solution for supercritlcal flow through bridge piers. With ,the propel' va.lue of (1', determine the limiting value F3L from Eq. (17-27) or Fig. 17-32. Then enter Fa/FoL . . on the chart and read h*/y~ for the given pier shape. Tliis value multiplied by Ya gives the backwater h *. If Yl, v!, bz, B 3 , and the pier shape are known and it is required to determine ;the dischar:ge, h* /Ya may be computed from h*/V! = (YI ~ Va)/Y3; Fa/F3L may betaken from the chart; u = bdB3;'and F3L may be obtained from Fig. 17-2701' Fig. 17~32. The vaiue of F3 can then be calculated, and from it V3 and then Q, Yarnell's graphical solutions apply to piers with length-width ratio

1.2

I .'

l/f<==)

Length -width rolio =13

- - c..._

---

. ----

,- /

y-

/-'

'\7"-

-'

;,. :7' /"

~

--- ---e:"L. -.... - -- -- -f/

'··"l-

FIG, 17-36. Effect of increase in l~ngth of bridge pier.

I

-.-

-

--' --

c::::::J

Length-wIdth

F./F 3L [421.)

I

.

-'--

---

----

2

(Based'Qn data of D. L. Yarnell

,

506

RAPIDLY VARIED FLOW

'

507

FLOW THROUGH NONPRISMA'l.'IC CHANNEL SECTIONS

equal to 4. For ratios of, 7 and 13, the effect of increasing length on backwater is I:Ihown in Fig; 17-36, which is plotted \vith Yarnell's data. It seems that tM bZ,ckwater caused by the long piers is greater when the pier ends are semicircular than when they are square. It is pl'obahle that an, abrupt entrance in the case of square pier ends tends to'decrease friction losses for (t short distance downstream because of its effect on the velocity distribution, since the velocity along the walls is reduced. The effect of bridge piers present ill a constriction has been considel;'ed in Alt. 17-6 and in Fig. 17-23d. 17-11. Flow through Pile Trestles. Yarnell's investigation [41] indica.tes that the N{tgler formula may be suitably applied to subcrltical flo w passing a pile trestle a,net the d' Au buissoll formula to superc.ritical flow passing a pile trestle. The following coefficients are recommen.ded for use in these formulas:

where {3 is a coefficient having the :values listed below: Form of rack beLt'

Value of (3

Square nose and tail, Lis = 5. _., ....... , ' ..... '" Squ!1rtl nose and semicircul
2.42 1.83 1.67 1,79

-1

I

0,76

Spangler [55] has extended t.he experiment and determined the value of f3 for IX = 30 0 , ,45 0 , and 60 0 , Ac.cording to F'ellenius [54], an empirical formula for c can be given as follows: C =

. {j + Ii )2 sm=

' S Ji. ( S

(17 -3L~)

.j

Type of t"e811e Single-track 5-pile trestle ben t Parallel to curren t. . . .. . .. . ............. , At to" angle with current ......... '.' ..... . At 20° angle with current .............. , .. At 30° angle with current ................ . Double-track JO-pile trestle bent ............ . Two sillgle-track 5-pile trestle b'~nts: ..... , . , .

where the coefficient 0.90 0.90 0.89 0.87 0.82 0.79

0.96

,

V2

= c2g-

T'alue of /J. Square nose and tail With sharp corners, Lis ~10 ...... , .................... 7.1 With sharp corners, Lis = 12 .......... , ...... _....... 6.2 ,\Vith slightly rounded corners, Lis = 8 to 11. .......... 6.1 Semioircular nose and tail, Lis = 7 ......•........•.. .... 5. I)

0.88 0.86

(17-33)

I I.

I -- .-:<

V(tlue "fx

1.0 1.0

LO

i.5

In general, ~c = 1.0 for bars having sharp 01' slightly rounded corners and x = 1.5 for ba!'s having rounded corners. For cross-connected and clipped rack bars, the value of Ji. should be increa.'led by about 22.5 %. Scimemi [56] and Kozeny [50] have provided values of c, /3, and Ji., and other data for racks iustalled in several hydropower plants . 17 -13. Unqerfiow Gates. Certain control gates ir, cana.ls may be called unde7'ftow gates 1 frcim the f act that. water passes underneath the struct ure. Common examples are the sluice gate, Tain~er gate (or radial gate) I and rolling gate (Fig. 17-37). In d.esigning such gates the hydraulic engineer is most interested in two major .features: the 'head-discharge relationship and the preSSUl'e distribution over the gate surfaces for various positions of the gate ~nd forms of the gate lip. The form of the lip will .not only affect the velol:ity and pressure distributions and the energy loss in flow through the gate opening, but may also develop very disturbing vibrations that should be avoided during gate operation. As

(17-32)

where V is the velocity of approach ahead of the rack and ,1vhere c is a coefficient depending on the cross-sectional form, thickness IS, length L of the l'ack bEll', clear distance b between bars, angie /J of il1cliuation of the bar from the horizontal, and angle a between the direction of flow and the length of the bal'. ;On the basis of the experimental data for rack bars of various forms a~d with a = 0, Kirschmel' :[52 ,53] has set up the following formula for c: ,

,,c = {3 (S)H Jj sin 0

and exponent x have the followil1g values:

Form of rack bar

The amount of channel cont.raction is to, be t.aken as the average diameter cif the piles plus the thickness of t,he sway bracing, disregarding the angle at -which the bent is set with the current. ' 'The effect of trestle piles present in a constriction has been considered in Art. 17-5 and in Fig. 17-23c. 17 -12. Flow th~ough Trash Racks. For flow through trash racks, the .designer is primurily concel;ned with the amollnt of head loss due to the resistance of the rack. This may be expressed in terms of the velocity head of the approach flow, or h,

Ji.

.1

1 In contr~t ~o the underfk..r gate is the ,ovet'flow gate through which the water flows over the stl1fctur~. The drum gate (Art., 14-9) is an exa'mple of au overflow gate_ , Hydraulically speaking, the overflow gate ,acts like! a weir as much as the underflow gate act~ like an orifice. There are also designs for which the water flows above ,and below the struct·ure at the same time (Fig. 17-37).

I

FLOW THROUGH NONPRISMATIC CHANNl!lL SECTIONS .

RAPIDLY VARIEh FLOW

508

the design of the lip varies considerably, independent investigation of the . lip for a. particul[1r design is usually necessary. I By the energy equation, it can be shown that the discharge through an . underflow gate may be expressed as

Q= where

aLh~-IJ-:(-Y1-+-a-';~'7. ~-;:::-)

(17-35)

a is the coefficient of discharge, L is the length of the gate, h is

For the vertical sluice gate, l experimentally determined curves representing the value of a have been prepared by Hflnry [68], aFl Shown in •. Fig. 17-38; The dashed line. A represents the result obtained by Eq. (17.:.35) on the basis of t11e energy principle; the dashed line B is obtained through the use of the. momentum principle. The value of :IfQ is the Froude number of the flow through the gate· opening. . Similat· curves fo.r the Tainter gate (Fig. 17-39) were prepared by Toch [69] on the basis of an experimental study" induding the pivot height as a

+Qt.

O.4l--I-+'<:""'1I---.J,c...,.61~';""'F-~F-;~I=--l

c Sluice gate with combined oyer- and underflow

Sluice gate

0.1

.'1, ~

Z / Tointer gote

FIG. 17-37. Underflow gates.

the height of the gate opening, Yl is tbe upstream· depth of flow, and ex. V 12/ 2g is the velocity head of the approach flow. The outflow of the gate may be either free or submerged, depending on the tailwater depth. For submerged flow, Yl in the above equation should be replaced by the effective head, or the difference between the upstream and downstream depths. . . . For the .purpose of experimental.studies, the ~elocity-h~ad term m Eq. (17-35) may be omitted, and its effect may be mcluded m ~hecoefficient a. Thus, (17-36) Q = aLh, V'2i1h

04

..

where a is ~ coefficient depending the geometry of the str~cture and on the upstream and downstream de~ths. The form of this e~uation is . the same for both free and submerg~d flows;

.'

1

For

!--f----jJ-l--ll----jF:=.:I-/!='"'

studi~s on ~uice-gate lips of variods designs, see [66J .a.nd [67] .

FIG. 17-38. Discharge coeffil.llentJor vertical sluice gate.

(_4/Iel' H. R. Henry [68].)

:variable i~addition. to o~her variables that ,:vere used by Metzler [70] m an earl~er analYSIS. 'Ihe U.S. Army Engmeera Waterways Experiment ~tatlOn [74] has ~lso prepared design charts for use in the design of Tamter gates, partlcularly for those installed on spillway. crests of .WES-standard shapes (Art. 14-3). . . . In the case of a sluice gate with combined overflow and underflow (Fig. 17-37), the overflow discharge Qu is a function of II only, but the underflow discharge QL is a function of yo + AYQ as well as Yl. If AYo 0 and if the lower (under~ow) jet is free, the total discharge Q = Qu + QL is independent of the ~ailwater depth Y2. IfAYo is greater than zero the lo:ver jet is drowned. The problem ·may be ~olved by the continuit; equatIOn and the momentum equation betweenicross sections 0 and 2. I

For other studies, see [39J and [71]. For a theoretical:analysis of the flowthrough a sharp-edged sluice gate, see [72] and [73]. ~. . 1

510

RAPIDLY VARIED FLOW

FLOW' THROUGH NONPRISMA'l'W CHANNEL SEiCT'IONS.

Under norma.l conditions, y~ is a function of Q. Both yo + "b.Yo and Q£ are unknown but must satisfy an energy equa.tion similar to Eq. (17-35). The use of the momentum equation will automatically include any friction losses. The pair of equations may be solved by trial and error or graphically, The result thus obtained has been found to agree with an experimental finding derived by Escande [2,711.

511

(a)

"

I I

(hl

-----'--~--------4~

.I . .FlO. 17-39. DischArge coefficient Ior Tainter gate.

(After A. Tock [69).)

The pressure on the surface of the gate can! be determined accuratel~ either from flow-net analysis or oy actual measurement on model or prototype. 17-40a shows the nO,fmal pressure ol1,th~ s1..u:face o.f a; Tainter gate (a simil?-r pattern of hOrlZontal-pressure dlst.rlbutlOn eXIsts on a sluice gate), the1horizontal pressure on the gate openmg and on the upstream and downstream cross section.s, and the vertical pressure on th~ channel bed. The normal pressure on the gate surface may be repre-: sented by its horizontal a.nd vertical components PH andF v, as shown in Fig. 17-40b. The magnitude and positions of these forces may be

'

-

•

(el

FIG. 17-40. Determination of pressures for Tainter gate. ,

,

o?tained by graphical determina.tion from the pressure-distribution diagram .. A less accurate but. simpler method of determining these pressur~s IS to assume that the h~rizol1tal pressure on the gate surface is hy~rostatically distributed and tpat the vertical pressure is equal to the ;ve.lght of the water replaced by the gate structure above its surface t,FIg. 17-40c). The horizontal force may also be determined by the mo~.entum relationship. However, this determination will not give the pOSItIon of the force. Sin\:e all water pressures are normal to the gate

512

RAPIDLY Y ARIED FLOW

513

FLOW THROUGH NONPRISMATIC CHANNEL SECTIONS

I J

I

1

I

surface, the res1lltant of the 'pressures must pass through the trunnion' axis O. The outflow from the gate uswiJly constitutes 11 high-velocity jet, capable of eroding the downstream channel bed. l The erosion thus developed may be avoided by means of a stilling basin (Art. HHl). If no preventive measures are provided, the erosion D:1ay result in a deep seoUl' hole, the formation of which depend' on the type of the jet. According to the studies of Escande [71] on fiow through a. channel drop downstream from [l, sluioe gate/ four types' of jet are possible: (1) submerged How with rising j Elt, ,(2) submerged flow with diving jet, (3) free flow with rising jet, and (4) free flow with diving jet. Generally speaking, the diving jet will induce less pressure on the vertical surface of the drop than the rising i et and will usually develop a scour hole with its deepest point closer to the drop than the rising jet. In submerged Bow, the depth of scour ,hole depends on the tall water depth, whereas in free flow the downstream conditions have little or no effeot upon the depth of scouring. 17 -14. Channel Junctions. Flow ,through a channel junction is a phenomenon that involves numerous variables, such as the number of the adjoining channels, the angles of intersection, the shape and, slope of the channels, the'directions and discharges of flow, the rounding of the corners at the junction, etc. The problem IS so complicated that only· a. few simple and specific cases have been studied. The conclusions of such studies indicatE;. that gene1'8.l1zation of the problem is not possible or even desirable. When the application of hydraulic theory t.o the problem encounters limitations, !L model study will give the best solution fOl' the How characteristics involved. For stcbc1"itical flow passing through the channels at a junction, Ta.ylor [80] has investigated the specifiC cases sholvn in Fig. 17-41. The channels . are horizontal and of equal width.' In the case 9f combining flow, the following assumptions were made: (1) the flow is from channels 1 and 2 into channel 3; (2) channels 1 and 3 lie in a straight line; (3) the flow is parallel to the chn.nnel w
will

1 The erosion of channel bed due to jet is a subject beyond the scope of this book. For detailed informatiDn see [1] to [3] and [75] to [77]. 2 Other examples of jets below hydraulic stnictures in channels have been found and studied by Camichel [78]. The erosion due to iet through a bridge opening may cause a deep scour hole generruly known as the bl1le hole. An experimental study of such a problem has been reported by Hickenlooper, Guillou, and Chow [79J.

these assumptions and by the application of the momentum ,equation to the junction in the direction of 1 to 3, Taylor was able to obtain the following equation: l . '

i

r

.k2

i

n.~(ni

=

4nll ![2n. - nl(l

- 1)

+ cos 0) + ny -

1]

(17-37)

=:

where k2 = ~22/2gY2' nq Q2/Q3, ny = Ya/Yb, V 2 is the velocit,y in ~han~el 2, Y: IS the depth 0, flow in channel 2, Ya is the depth above the, JunctIOn, Yb IS the depth below the junction, and 8 is the angle between the. merghig chann~ls. Taking n. as. a parameter, k2 may be plotted agamst nv for any gIven 8.

(oj

(b]

FIG. 17-41. Simple channel junctions (a) combining flow; (b) dividing flow.

Equation (17-37) was verified by experiments on junctions with () = 45° and 135°. It was found that the agreement between theory and experiment was good, for 0 = 45°. ,There was no agreement' however for () = ,135°. It ii? believed that this was because the veloci~y distrib~tiol1 below the junction was distorted and the flow did not remain parallel t? the chann~l wal~s. The expedmental data showed clearly that assumptlOn 5 above 1s valId, regardless of the angle of intersection of the channels. In t~e case Of d2~viding flow the problem c::mnot easily be analyzed theoretlC~Il~'. The application of the momentum principle is difficult be.cause. It 1.nvolves some unknown quantities, and assumptions which might slmphf! :he determination of these quantities, such as assumption 5 for the, combll1lng flow, are not available. Basically, the division of the flow "1111 depend upon the backwater effects of the two branch channels and ~he dynamic conditions existing at the junction. If the divided flow IS to be combined again at a certain point downstream a solution ' of the problem described in Art. 11-9 may be applied. 1

In applyirig the momentum equation to the problem the hydrostatic pressure

exer.~ed by the flow in chann.el 2 is ~ol1nt.erac.ted by the pr~ssure on the opposite wall, prov lded the water surface m the JunctIOn IS esser,tially flat. Thus, the only force the flow from channel 2 can contribute to the flow in the straight channel· 1-3 is its mom~ntum component in the direction of 1 to 3.

514

RAPIDLY VARIED FLOW 1.35

I /

1.30 1.25 ~ 1.20 E .<::

Q. US

I

'"

;7

Cl

1.10 1.05

2/

1.00

!Lo

t10

,--

Vi

rn

AiT

~W~cP

1.00

the division of flow of a given discharge QJ. First assume QI; then Q2 = Qa - QI. The corresponding depths YI and Y2 can be obtained from the rating curves of channels 1 and 2. For Yl/Y2, the ratio Ya/Y2 can be determined from Fig. 17-42.. By assuming other values of QII the correspoildirlg ratios YZ/Y2 c<,n be obtained. Thus, Qz/Qa is plotted against Ya/Y2, as shown by the curve. A in Fig. 17-43. The intersection C)f this curve with the ka curves now gives all possible combinatior.s of the 0.20 B variables, among which one value of . k3 ·would correspond to the flow of, I say, 500 cfs entering the junction:: ! 0.16 1----. Tbe llext step is to plot the ill terSected k3 values against the corresponding Ya. For the example shown by the .., 0.12 line A in Fig. 17-43, ka = 0.10, o ~ . Ya/Y2 ~ 1.067! and Q2/Qa = 0.677 or ~ " !I Q2 = 0.677 QJ. From the ratiag curve ~ 0.08 of channel 2, the V2 corresponding 'to , V3z~ :-the ·Q2 just computed can be· found ~kl=2Y I 9 l and, hence,. Ya= l.067Y2. The plot 0.04 II of k, ~,gainst lis can be constructed, as shmvl1 by the curve B in Fig. o 17-44. The proper value of k. must 4 6 8 10 satisfy not only curve B but also the V?IUes of YJ solid. curve showing the relationship FIG. 17-44. Factor k, D.S a funccion of ka = V 3~ /2gY3~ The ill tersection of Ya. (Afler E. If. Taylm' [80).) the two curves gives the required values of lea and Y3. With this le" the corresponding value of Q2/ Q3 may be determined from Fig. 17-43, and the divided discharges Ql and Q2 can . be determined accordihgly. For 8upercritical flow passing through the channels at a jUllcLiQll, the flow condition is more complicated .. Studies of a few cases of combining flow of the general arrangement :~hown in Fig. I7- n have been made by thel;.T.S. Soil Conservation Service [81]. In such cases, a hydraulic jump m~.y forin in one or both of the inlet channels, depending lIpon the junction design, the discharges, velocities, and related flow conditions intheadjoinillg channels. Thus, a large increase in the height of the side walls in the vicinity of the j unction may be necessary. When the prihclple of momentum is applied to sllch a problem, it is necessary to know the position of the jump or the depth of ~ow at the uj:oEtl'eam edge of the junction in order to compute the momentum contributed by the. inlet· flows. If the flow passing through the junction is supercriticaI. without the formation of any hydraulic jump upstream, the flow condi-

/

. 'I

~r~

1.30

1.20

Depth rolio, ~

.,

FIG. 17-42. RelaLionship between depths in a 90" flow division. [80J.)

-

(Afler E. H. TaylO1'

"i~ 1.20!1--~-l---t-~

.§

1.1 51----l------I---j-----

.t:

.Ii ., o

ti Oll---'--l----+---t 1.05

il

)

1 I 1

.1

lh '\

M

Yz

. Ii'!

&;15

FLOW THROUGH NONPRISMATlC CHANNEL SECTIONS

---

E

Quantity rolio, ~ 2 3

FIG. 17·43. Characteristics 9{ a. 90· flo~v division.

(Afl~r E. H. TaY'loT [80J.)

For the specific junction showll in Fig. 17-41b, Taylor has made an experimental approach to the problem as follows: . _. For any given value of 8, it is possible to correlate the (hmenSlOnle~s ratios Q2/Qa, Y3/Y~, yt/Y2, and ka = V.;2/2gYa, all derived from the expel'lmentally determined data. For g:= 900, such correlation curves ~re shown in Figs. 17-42 and 17-43. These curves can be used to determme

t

,J

1

., (

. 1516

F,LOW THROUGH NONPRISlIfATIC CHANNEL SEC'1'IONS

)

tiol1 will be coniplicnted by the development of cross waves similar to those discussi:d in Art. 1~-5. These wayes will propagate for aconsider. able distullce downstream before being damped by the channel.friction. Hence, side wt1lls higher than those normally provided are requ"ired not only nM:' the junction but also for ll. considerable distance downstream.

)

PROBLEMS 1'1-1. Verify Eq. (17-5), and interpl'et the dashed curves in Fig. 17-3 that represent this equation. ' . . 17-Z.. Formica's data. for sudden expansions, plot a Cllrve showing the relac tiollship betwe(ln •.aucl the angle of divergenc-e fI, and then aiscuss this rela.tionship. 17-3. COnllmte. the wave angles fJ, anG fl. of th? contruction designe.t in Example 17-1, and s\;:e~(·h the position of the shock wf,vefronts. 17 -4. a straight 'contmction to redu~e the width of \I. rectanguiar channel fl'Olll Q. to 3 The channel carrips a discharge of 4;33 ds with an approach velocity of 7.2 fl)S. ASSUl1l11 11;/11i 3.00. 17-5. Compute: the wiivenng!es of the design inProb. 17-4, and sketch the posi'tlQIl Q! Ill\) sll(ld: wiwefront..'. 17-8. ))c>lign the c-ontracl.ion in EXIIlJ1ple 17-1, !l.Ssllmbg 'Ya/Y, = 3.00. 17-7. A c.hnnllcl 6 ftwide (;afries a discharge of 100 cfs at a velocity of 20 fpS. COIl~tl'Uct th.l surface contour of the !low through a sq3 ropresented by Eq. (l7-ll), constl'uct the slIrface contour of the

,l' (

How ill the (·1:jllHltiiou.

17 -9. !II 11 ke n preliminary design 'of a curved e.xpansjO!l for the channel in Prob. 17-7 with Iho !lid of the ,Ihnellsion\ess boundary c·urves detetmiHed by the method of !,hnral'tcri~ti('l! (Fij!;. 17-13). AsslIm() hi/II = 3. ' . 17-10..Solve EXam1)le 17-.2 fol' tlw following addition"t'conditions: (a) the center lille. Of-thfJ bridge and highwn.y emballkmellt makes an angle of 70" with the directioll of flow, (I,) the ~i1t.m!lCe is rounded fol' rib = 0.10, [lnd (e) the bridge has a pier 30 ft loug nnd 10 rt wide at I,hl) middle of the waterway opening. 17-11. Dctermine the flood disdlllrge through 11 bridge conBtrintion having a type III opel1illg with (;mb~lIlm1ont Ilnd abutment slope 2:1. The COl1strietion 30 rt long r(!.dtl(·t~ t'le nonnal wid th of w!lterWt1)' from'
I

J

517

17~le. the preceding problem if the headwater is 13 ft above the inv'ert of the entrance jl.nd the barml is: rough, having n = 0.025. '. . 17-17. A llighway bridge 350 ftloIlg h!iS six piers with semicircular nos~s and tails, ea.ch 8 It wide and equl'.lly spaced. At the peak of a flood, the backwat~r was found to l:!e 0.30 it for the seven openings. The average depth of flow immediately down:stream from the opening was 8 ft, Compute the flood discharge, (a) by the Na.gler formula, (b) by the d'Aubuiss.oIl formula, and (Il) by the Yarnell charts. 17-1.8. A. bridge has four piers wit4 semicircular noses and ta.ils, each 40 ft long by 10 ft During a flood peak of 45,000 cfs, the total width the strealIl was 390 It, and the depth at a. downstream section was 19.4 ft. Estimate the backwater. ' . . 17-1.9. Soive the preceding probleIll by the USGS method d~8Cribed in Art. 17-7. The chaonel roughness coefficient is ,It 0.03. Unavailable .data.are to be reasoIl. abl)' a.s5umed. 17-20. If the bridge in Prob. 17-17 ill exposed to a flood discharge of 35,000 C[6, <:5timate the backvrater. The rating cur"" of the chanIlel gives a normal depth of . 8.5 it for this UllScmLrg'e. 17-21. A pile-trestle I',dlroad bridge 840 ft long is supported by bents, each 2 ft wide and spaced 11 n center to center. At the peak of a flood, the ayerage ha.ckwat,er V{as found to be 0.15 ft. The aver1:1ge depth of flow immediately downstream from t.he trestle W3S 6 ft. Estimate the bnc.kwater. 1'l~22. Verify Eq. (17-37).

of

REFERENCES 1. Charles Jaeger: Der Mischungsvorgang bei plt'tzlichem QuerschnittsUbergang (The mixing process in sudden transitions), lVasserkTaft 'l.!nd Was'lenoirtsckafl, Mllnich,voi. 31, no: 24, pp. gOo-SOD, 193B. Z. Charles J a.eger: "Er,gineering Fluid M¢chanics," translated from the German. by P. O. Wolf, Blackie & Son, Ltd" London and Glasgow, HI5S, l>p. 157-169. 3. Charles Jaeger: De I'impulsion totnJe et de ses rapports avec l'enel'gie tot.ale d'un courant liqui<:le A. sUr:lace libre (The total :impulse and it..~ relations with the total energy of a. iree-surface liquid flow), Relme ytnJral.e de ~;hydratlh'q!le, Paris, vol. 13, no. 39, pp. 143-151, 1947. 4. Gianni Formica: Esperienze preliminari sulle perdite di carico Dei can ali, dovute a cambiamenti di sezione (Preliminary test on head losses in channels due to C!OSSsectional changes), L'Energia eleltricll,iIfila.no, vol. 32, no. 7, pp. 554-568, July, 1955; reprinted as lstitu.tQ di I dr{Lulica e Coslruzioni I dmulichc, Milano, :nie'lr!crie e ~t'!.ldi No. 124, 19'55. 5. R. L. Daugherty and A. C. Ingeraoll: "Fluid Mechanics," McGraw-Hill Book Company, Inc., New York, 1954, pp.19O-HI3. . 6. Arthtlr T. Ippen and John H. Dawson:: Design of channel contractions, 3d paper in High-velocity flow in open channels: A symposium, Transactions, American Sociel.y·of Ciflil Engineers, vol. 116, pp. 326-346, 1951. . 7. Mtl.S.a!lhl;Hom-ma and Sukeyuki Shima: On the flolV in a gradu!illy dIverged open channel,: The Japa.n Scienc~ Series yoL 2, no. 3, pp. 253-·260, 1952. 8. Hlmter ;Rouse, B. V. Bhoota., Design of channel' expansions, 4th pap~r in High-velocity flow in op:en channels: A symposillm, Transactions, Amt
I

518 10. 11.

12. 13.

14.

15. 16. . 17. '18.

19.

!

RAPIDLY VARIED FLOW

critical,-elocities, U.S. l!oil COnse~1Jation Se71rice, ProgreS$ Rept SCS-TP-76, April, 1949. Ivan-E. lluuk: Calculation of flow in open channels, Miami COnSe,.vlmcy District, Techniml Report, Pt. [V, Dayton, Ohio, 1918. " E. W. Lane: Experiments On the flow of water through cont,ro.etions In an open channel, Transaction$, American Sociel!! of Civil Engineers, voL 83, pp. 1l49-1208~ 1919-1920. , Carl E.IGndsvo.~er, Rolland W. Carter, and H. J. Tracy: Computation of peak discharge at contractions, U.S. Geological ,'lurvey, Circulol' No. 284, 1953. C~J,fl E. I'l:illdsvater and'Rolland W. Carter: Tranquil Jlow through open-channel aonstric-tion", Transactions, American SQci~ly of Civil Engineers, Vl)[. 120, pp. 955-980, 1955. Hubert J. Tracy and Rolland W. Carter: Backwater effects of open-channel constrictions, ,['ransad,ions, American Society of Ciuil Engineers, vol. 120, pp. 993-1006, 1955. Tate Dalrymple: Measuring floods, in Floorls, val. III of Symposia Darcy, IntM'national Association of Scien(ific Hydralou!!, Pttblicatio1! No. 42, 1!J56, pp. 38(1-<1.04. David L. Yarnell, Floyd,A. Na.gler, l1ll,d Sherman M. Woodward: Flow of w[j,~el' ~hrough culvel'~s, University of [o1Oa, StudifJ~ in Engineering, Bulletin 1, 1926. , l!~. T. Mavis: The hydraulics of culverts/ The Pennsyl,'ania Stale College, ElIgineer'£ng Experim6,tl Station, Bulletin 56, Oct. 1, 1942. LorenZ' G. Straub and Heruy M, Morris: Hydraulic data comparison of concrete and corrugated metal culYert' pipes, Unillil1'sily of Minnesota, St. Anthony Falls Hydraulics Laboratory, Technical Paper 3, SeTie~ B, July, 1950. • . Lorenz G. Straub 'and Henry Morris: Hydnl.l.lhc tests ou concrete culver. pipes, University of Minnesota, St. Anthony Falls Hydraulics Laboratory, Technical Paper ND. 4, Seri,w B, July, 1 9 5 0 . ' "

20. Lorenz G. Stra.ub and Henry Morris: Hydra1.llic tests on corrugated metal culvert pipes, Univer~ily of }/in'Lesota, St. Anthony Falls Hydra'.J.lics Laboratory, Technical ' . Pa.per No.5, SeriBs B, July, 1950. 21. L0renz G. Straub, Al...-in G. AndBrson, :md Chades E. Bowers: Effect of

Inl~t

design on capacity of culverts on steep slopes, U nive"sity of M innesoLa, St. A n{ hony Falls Hydraulics Laboratory, Project Report No. 37, August, 1953. 22. Lorenz G. S~ra.ub, Alvin G. Anderson, an.d Charles E. Bowers: Importance of inlet design on culvert capacity I in CuI-vert hydraulics, Highway Research Boa:rd, . Research. Reporll;i-B, Washington, D.C., 1953, pp. 53-67. . 23; Roy H. Shoemaker, Jr., and Leslie A. Clayton: Model studms of ta.perEd Inlets for box culverts, in Culvert hydraulics, iIighway Resecrch Bilard, Research Rep!lTt' 15-B, Washington, D.C., 1953, pp. 1,-45; , 24. John L. French: Fi~st progress report on r.ydraulics of short pipes: Hydr(l.ulic characteristics of commonly lLSe,d pipe entnmces, U.S. National Bweau of Stand(J;rds, Report 4444, Dec. 28, 1955. ' ' . , 25. John L. French: Second progress report on hydraulics of culverts: Pressure and r~sistance characteiilStics of a model pipe culvert, U.S. National Bureau of Standai'ds, Report 4911, Oct. 29, 1956. ' . 26. John L ..French: Third progress report on hydraulics of culvert.s:'Effect of approach channel characteristics on model pipe culvert operation', U.S. National Bureau Of Standards, R.cport 5306, June 3, 1957. " ' 27. R. W. Ca.rter: Coinputa~ion of peak discha.rge at culverts, U.S. Gealogical Survey, Circular 376, 1957." . ' ., '. 28. Wen-Hsiullg Li and Calvin Patterson: Free outlets and self-pnmlng action of

a.

FLOW THROUGH NONP1USMATIO CHANNEL SECTIONS

519

culverts, paper 1009, Proceedings, Ame1'icfm Society of Civil Engineers,'Joll.rnal, Hyd"cmlics Division, no. HY3, pp. 1-22, J~me, 19M. 29. Hydraulb cha.rt, No. 1043, U.S. Bureau of Plwlic Roads, Hydmlllic Chcr.rts, March, 1951." . ' . 30. J. F. d'Aubuisson de Voisins: "Trait.! d'hydraulique" (" Trea.tise on Hydraulics "), 2d ed.,.Pitois, Lenaut & Cie, Paris, 1840. 31. Juli.1I3 Wei~bach: "Die ExperimentE.lhydraltiik H ("Experimental Hydraulics"), Frelburg, Germa.ny, 1855. . 32. ,T. F. Bubendey.: Stau an Briicken un.d Durchlassen (B!J.ckwatel' of bridges and culverts), art. 25, chap. 3, pt. 3, yol. 1, in "Handbuch der Ingenieurwissenschll.ften " 4th ed., W. Engelmann, Leipdg, 1911, pp. 707-773. ' 33. FiQyd A, Nagler: Obstruction of bridge piers to the flow of water, 1'1'an,sactions, American S,'cietyaf Civil Engineers, vol. 82, pp. 334-395, 1918. 34. H .. D. l{rey: Berechntlug des Statle5 infolge von Quenschnittverengungen (CalculatIOn of backwater due to cross-sectional contraction), Zentralblat{der Bauverwa/tung, Berlin, vol. 39, no. 79, pp. 47-2-475, Sept. 27, H)19. 35. '~h. Rehbodc·. Zur Frage des BrUckenstaue8 {On the problem or b!'idgecous~ric­ tIon), Zeniralbla.t! der Bauvenoa.Utmg, Derlin, vol. 3g, no, 37, pp. 197-200, 1919. 3,!t Th. Rehbock: Verfa,hren zur Bestimmung des Briickenstaues bei rein stromenden Wasserdurchfluss (A method of determining the backwater at bridges for distinctly streaming flow), a volume published to celebrate the dedication of the new building of ~he DIvision of Engineering Structures of the Technical Institute of Ka:rlsruhe, Ge;mll.ny, Nov. 26, 1921, pp. 7-13; a.lso if' Der BauingenieuT, Berlin, vol. 2, no. 22, pp. 603-609, 1921. 37. Th; Rehbock: BrUckenstau und Walzenbildung (Bad(water and eddies Iltbddges) Der BattingQnieur, Berlin, vol. 2, no. 13, pp. 3<11--347, 1921. ' 38. Paul BOss:" Berechnung der Wasserspiegellage bairn Wechsel des Fliessz\l~tandes" (" C?mputa.tion of W~t.er Surface with Ohanges in Flow Type "), Springer-Verlag, Berhn, 1919, Il,nd VDI-Verlag, Berlin, 1927. 39. Alexa.nder Koeh and Ma.'{ Carstanjen: tt Von del' Btnnlgung des Wo..."SBl'S und den da.bei auftreten,den Kriiften" (" Moysment of Water and Associated Force Phenonlena"), Springer-Verlag, Berlin, 1926', pp. 179-185. 40. Giulio De Marchi, "Idraulica." ("HydrILulics"), Ulrico Hoepli, Milan, 1930, vol. I, pp. 401-404. 41. David, L. Yarnell: Pile trestles Mchannel obstructions, U.S. Departmenl ol ' Agricult-u.Te, Technical Bulletin No. 429, July, 1934. 42. David L. Yarnell: Bridge piers as channel obstwctions [T.S. Depr:lrtmenl of AgriC'ltUur 6, Technical BuUetin No. 442, November, 1934. ' 43. L: Escande al;ld G., Saba.~he: Sur l'emploi des prolils aerodynamiqtles pour les pIles de ba;rages deversolrs, de barrages mobiles et les piles des ponts (Use of ~erOdyn~llC profiles for mov",blp. dams aod bridge piers), Re!lU.~ ¢nemle ds I hydrauhqne, Paris, vol. 2, no. 10, pp. 546-5li5, 1936. ' 44. Leopold Escande: Recherches sur l'ecoulement de l'eau entre les piles des ponts (Researches on the flow of water through the piers of bridges), pt. A, and Remll.rque sur Ie cll.icul du remous provoqlle par un pont (Remark on the calculation of the backwater caused·by a bridge), pt. B, Le. G~nie civil, vol. U5, no. 6, pp. 113-117,' A~g. 5; no. 7, pp. 138-140, Aug. 12; and llO; 13, Pl" 259-260, Sept. 13, 1939. 45. Leopold Esnande: ~xperience.s sur l'ecoulement entre piles de ponts (Experihtents on flow through piers of bridges), Compte.; renduil de l' Acad.emie. des Sciences vol. 209, pp: 14-16, July 3 1939. ' , ·16. Leopold Escande: Sur l'e~oulement entre piles de ponts (On the flow through

.oj I

J 520

t,

RAPIDLY "VA.RIED FLOW

FLOW THROUGH NONPRISMATlC CHANNEl, SECl'IONS

piers of bridges), Com.ptell rCIu:i.ItS d.e l' Acadpmie des Sciences, vol. 108, pp. 1970197:/;, JIlne 19, l ! l 3 ! J . . . 47. Leopold Esco.nde: Recherches sur l'ecoulement de l'eau entre .pll~. de ponts (Resellrcl.es on the How of w:J.ter through piers of bridges), I;e G~nte C'tmt, voL 115, '. no. 5, pp. 113-117, Aug. 5; and no,. 7, p~. 1~8-1.40, Aug. 12. 1;39. 48. Etienne Crausse: Sur un phenomene d oscl\latlOn du plarr d eau provo que pa. I\~coulement autour d'obstllcles en form de piies de ponts (On the phenomenon of oscillation of watel' surface caused by flow a.round obstacles in the form ot bridge piers), CDmp!es rendus £I;: l' Academie des.8ciences, voL 209, pp. 197-199,

63.

64. 65.

Julv 24,. 1939. . 49. Otto St~eck: "Gmnd- und Wa.sserbau" (il Foundation lLnd Hydraulic Engmeering") Spriuger-Verlag. Berlin, 1950, vol. 2, pp. 420,,434. 50. J o~ef ' Ko~eny: "HydrauHk j' (" Hydraulics"), Springer-Verlag, Vienna, 195jl,

66.

Pll. 546-549. 51, l)hili-pp Forchheimer: "Hydrt'!.'ulik" ("Hydraulics"), 3d ed., Teubner yerlagsg;~sellsch!lft, Leipzig and Berlin, Hl30, pp. 519-522 on backwater a~ bru:!g-e piers and pp. 522-:524 on ioss of head at trash. ri1cks. . 52. Armin Schoklitsch: "Handbuch des Wasaerbaues" ("Handbook of Hydrauhc Engineering"), Springer-Verlag, Vienna, 1950, vol. 1, pp. 122-124 on backwater oJ bridge piers and pp. 125-127 on loss of head :J.t trash racks.· . 53. Otto Kirschmer: Unters\\chungen liber den Gefallsverlust an Rechen (Studles on the head loss through a rack), Mitteilungen des hydraulischen I nst-Uuts der technisc/llm HocMclmle JltIilnchen, no. 1, pp. 21--41, Mllliic~l, 1926.. . 54. W. ~'ellellius: Undel'sokinglll' betroffande faUf5riuster 1 skyddsgrmder vld vattenkraftanlOggninp;er (Studies of heu.d·loss through racks in power pla.nts), Tran$aclirms, Hyr/"alliic Institute of the Royal Technical University of Stockholm, no. 5, 1927. . . 55 .. J. 'Spangler: Untersuchungen tiber den Vel'lust all Rechen beischr!i.g~ Z~stromullg (Studies on the head tollS through a ra.l!k inclined to stream flow). r.bttel{un~en d~s hydr~lllichen I nstituts 4er technischen H ochschttle M tincher., no. 2, pp. 46-60,

, I

j

Munich, 1928. . .. • . 56. E. Scimemi: Jlilievi sperimentali sui funxionan:.ento idraulico del grandl. Impla~tl industriali ('Experimental studies on the hydmulic function of large mdustrHl.l plants, L' Energia ele/lrica, MilanI), vel. 10, no. 9, pp.7Q5-723, Sep~ernber; no. 11, pp. B97.-924, November, 1933. 57. Armin Schoklit.sch: "Hydraulic Structures," translated from the German by Samuel Shulits, American Society of Mechanical Engineers, New York, 1937, . . . . . vol. II, pp. 891-892. . 58. Leopold Escande: Expression de Ie. perte de charge a. la traversee des gnlles (E:<:pressionof the head loss in water through grills), Com:ptu rendlls de I' AcCld~mie des Sciences va!. 218, pp. 179-181, Jan. 31,1944. 59. Leopold Escande;' 1!:turle experirnentalii de 'Ia perte de charge Ji. In. .tllra)verCsee des grilles (Experiment!!.1 studies of head lo~s m water through gn s, ,ompws rer,dlls de l'Acad~",ie des Sciences, vol. 218"pp. 266-268, Feb. 14, 1944. 60. Leopold EsCl1nde: Etude theoriq,ue, et experimentl1le la perte de charge de l'eal.l i Ie. traversee d'une grille (Theoretical and expemnentl1l study of tho head loss in waLer through 0.. grill), Le G~rlie civil, vol. 122, no.. Z3, pp. 188-190, Dec. 1; and flO. 24, pp. ·197-HJ8, Dec. 15, 1945. 61. David H. Kent: l\lodt"ls of hydra.'alic stnlctures, pt. I, Water Power, London, vol. 7, no. S, pp. 301-307, August, 1955.. Gives head losses through tailrace fish

?e.

'11 d screens. 62. FI~yd A. Nagl~r and Albion Davis:.Experiments on discharge over Spl ways an

6;.

69. 70. 71.

models, Keokuk Da.m, Transactions, American' Society of Civil Engtneers, vol. 94, pp.77F820, 1930. Louis G. Puis: Spillway discharge capacity of Wilson Dam, Transactions, American Society of Civil Engineers, vol. 95, pp. 316-329, 1931. E. Camichel, L. Escande, and G. SalJl;.th~~ Sur la similitude des barrages a COD.tractions latel'~les (On the similitude of dams with lateral contractions), Compte. rendus de l' AcaMrnie deJl Sciences, voL 194, pp. 807:-809, Mar. 7, 1932. E. Camichel, L. Escande, and P. Dupin: RemarqUB!l sur certains phenom~nes de contractions laterales dans Ire barrages (Not~son cert.ain phenomena of late.ral contractions ill 'dams), 90mples rendus de l' Acad6mie de.t Sciences, vol. 197, pp. 722-725, Oct. 9, 1933. O. Mueller: Schwingungsuntersuchungen an unher:;trornten Weh,en (Vihration studies on underflow weirs), Mitteilungen aer preussischen Vel'suchsanstall fur WM3erbau und Schiffba.u, no. 13, Berlin, 1933.' J. B. Tiffany: Laboratory research applied to the hydraulic design of large d&.ms, U.S.A.rmy Enflin~r8 Waterways Ezperi-m;enl Station, 'Bulletin 32, 194!f Harold R. Henry: Discussion of Diffusion of submerged jets, by M. L. Albel·tson, Y. B. Dai , R. A. Jensen, and Hunter Rouse, Tra1l.8(JctiQ:/I.I!, American Society of . Civil Enaineers, vol. 115, pp. 63i-69-i. 1950. Arthur Tach; Discharge chara.cteristics of Tainter gates, Transactions, A'lL61'ican Society of Civil Engineers. yoL 120, pp. 290-300, 1955. . Hunter Rouse (editor): "Engineering Hydraulics," John Wiley &; Sons, Inc., New York, 1950, pp. 540-541-. Leopold Escande:lttude theoriq,ue et experimentale de l'e~o\llement pal' vanne de fond (Theoretical and experimental study of fiow through sluice gates), Revue g~n~rale de l'hydraulique. Paris, vol. 4, No. 19, I)P. 2.5-29, no. 20, pp. 72-79, and 00.21, pp. 120-128, 1938; vol. .5, no. 25, pp. 21-34, no, 26, pp. 65-77 and 131-139, 1~~

I

I !

521

..

72. Georg Pager: 'tlber'den Strijm~ngsvorga.ng an einer unterstromten scharfkantigen PlanschUtzll (The flow characteristics at an underflow sluice gate), Zeitschrift fur angewtl.ndtil Mathema.#k und Jiechanik, Berlin, vol. 17, no. 5, pp. 259-269, October. 1937. . 73. T. Brook Benja.min: On the flow in channels whf'n rigid obstacles are. placed in Lhe stream, Jlm'mal of Fluid Mechanics, L
I I

l I.

522

RAPIDLY VARIED FLOW

to fI~vi!l.1 hydraulics); Revue generllle de l'hydral!lique, Pll.ris, vol. 1, !lO. 5, 'pp. 235242, and no. fI, pp. 293-2g9, 1935. . . 79. Irby J. Hiclccnlooper, John C. Guillou, and Ven Te Chow: HydraulIc stu~les of ~ highway bridge, University of lilinoi.s, Civil Engineerinfl Stl.die~, H ydraHhc Engtneering Be·ties No.4, June, 1957. . ' . 80. Edward H. Taylor: Flow characteristics at rectangular open-channel Junctions, Transactions American 8od~!y of Civil ,Engineers, voL 109, pp. 893-003, 1014. 81. Charles E. Bowers: Studies of open-channel junctions, pt. V of Hydraulic mode! studies for Whiting Field Nl1vll.l Air Station, lI>lilton, Florida, Univ6rlli!y OJ Minnesota, Sl. Anth~my Falls Hydraulic Laboratory, Technical Paper No. 6j Serie3 B, Januaty, 1950.

PART V

UNSTEADY FLOW

}

I

i

!

I

!

i

I J

The unsteady-flow problem most commol1Iy encountered in open channels deals with translatory waves. The translatory wave is a gravity wave that propagates in an open chal1l1el and results in appre<::iable. displacement of the water particles in a direction parallel to the flow. l In actual open-chi.mne] flow, the water particleslllay also oscillate. However, the effect of oscillation is insigllificant in the problems to be discussed in this book. . . For purposes of analytical discussion, unsteady flow is classified into two types, namely, gradually varied ;md rapidly varied ullsteady flow. In the first type the curvature of the wave profile is mild, and the change in depth is graduaL The vertical component of the acceleration of the water particles is negligible in comparison wich the tota! acceieratiol1, whereas the effect of channel friction is usually appreciable and should be taken Into account in &\1 accurate 'analysis. In the second type the curvature of the wa.ve profile is very large and so the surface of the profile may become virtually discontinuous. 'rhe vertical acceleratioll component, therefore, plays an important part in the phenOInenOI1, whereas the effect of channel friction is practically negligible in comparison with the dynamic effect of the flow, Common examples of gradually varied unsteady flow are Hood waves and waves due to slow operation of COlltrolling,structures, such as .the and sluices in ship locks. Exa.mples of rapidly varied unsteady flow are surges of various kinds caused by quick operation of controlling structures. Gradually varied unsteady flow will be discussed in Chaps. 18 and 20, and rapidly varied unsteady fiow in Chap. 19. 2 For extensive iniorml1tion on unsteudy flow in open channels, see [5J to [20J. 1 Another basic type of gra.vity wave is the oscillatory wave, in which the W8.ter particles oscillate in an orbit Il.bout a mean position but do not display appreciable displac~ent In the direction of wa.ve propagation. 2 Other kinds of waves !lod tides in open ohannels a.re beyond the scope of the present work. For comprehensive inform!l.tion or-water wa.ves, see [IJ to [3!. For references on tidal hydraulics, the reader should refer to the extensive bibliographies prepared by t!;.e Committee on Tidal Hydraulics, U.S. Army Corps of Engineers [4J.

523

. ...

\

CHAPTER

18

GRADUALLY VARIED UNSTEADY FLOW

1 18-1. Continu;ity of Unsteady Flew. The law of continuity, for unstea,.dy flow may be established by considering the conservation of m~ss in an infinitesimal space betw~en tWQohannel sections (Fig. 18-1). In unsteady flow, the discharge changes with dista.nce at a rate fJQ/ax,

loilio1 wafer surface

FIG. 18-1. Concimdty of 'Uustea.dy flow.

and the depth changes with time at a rate ay/at. r~e change in discharge through space in the time dt is (aQ/ax) d$ dt. The corresponding change in ch['.nnel storage in space is T dx (ay/iJt) dt = dx, (aA/at) dt. Since water is incompressiblel-~~!?Il!::!~lus thech@J~'l in stomge should be zero; that is,

(~~) Liz dt + T dx (~~) dt = (!~) dx dt + dx (~1) dt

= 0

SImplifying,

+ 7' ay = at aQ + aA =

fJQ

ax

or

At a given section, Q

ax

at

0

(I8-1)

0

(18-2) , ../

J

V A; thus Eq. (lS-I) becomes

fJ{VA)

ax, 525

+ T ay = 0 at

(18-3) ."",.-

j inJ

j

I

~6P ~c

'c/'d'v:mt-rt9 ·

!

526

·1

UNS'rEADY FLOW

527

GRADUAr,LY VARIl!JD UNSTEADY FLOW

or

+ V ~~ + T ay= a

A aV

ax

ax

Since the hydraulic -depth D may be written

=

at

(1&-»'/

A/T and aA =. T ay, the above equa.tion . (18-5) ,

small, that ~he ac~e].eration is in the direction of x,' and that its vertical c?mponel1~ IS neghglble, Thus, the work done by this force through a distance between .the· two channei· seotions shown in the figure is (w/g)(a V/at) dx. ThIS amount of work is equal to the energy los, due t? acceleration. Dividing by W,' the loss iil head is expressed b;

II

d:

I /'

i

(1/ghav/3t) dx.

, The over-all challge in head in the infinitesimal leilgth dx may bA treated exactly as in steady flow, except, that an additional loss dUe t~

The above equations are all forms of the cont~'nuity equation j07' unsteady flow in open channels. For. f1 rectl1ugubr channel of infinite width, Eq. (18-I) may be written iJq • ay _ 0 (18-6) ax T at -

~~d' _ _-->I ,Acc!!!lerot1on Irne.s.Jope :$0

1-' -r""~ ===--=.. - -

where q is the- discharge per unit width. This expression Wfl,S first, introduced by Saint-Venant. [21). When the channel is to feed laterally with ::t. supplementary di.,chnrg.;) of q' pel' unit length, for instance, into an area that is being flooded over a dike, Eq. (18-2) may be written ' aQ ao;

+ aA + q' CJt

= 0

ax

+ aA + aA' at·.

at

29

.:l-

-::"....

[n~y~:::::::::-==-~ e

J1 oPe

,--':-z-I

I

.r )'

I

0

(18-8)

where A' is the water area of the side' section. This equation also applies to the case of a ohannel containing groins; the water oircllIates between the groins to guide the fiow in the main channel but not to /L,M"'Rrr...~SH1tribute discharge. ~ 18-2. Dynamic Equation for Unsteady Flow. For simplicity,! :(f;<() .+l1tJ/y' unsteady flow will be treated like two~dill1ensional steady flow except - - - - - that an additional variable for the time element will be used. This time variable takes into account the variation in velocity of flow and accordingly brings to the fore the aoceleration, ,'thich produces force and causes additional enel~gy losses in the flow. 2 With reference to Fig. 18-2, the force due to acceleration J V! at acting on a unit weight w of water is equal to (wig) avjt1t; that is, force = roass X acceleration. , It is assumed that the slope of the channel is i For rigorous analyses~of unsteady flow, see [5J, [6], [9J, [111, !l5] and [20J. The acceleration is positive for an increasing velocity of flow and negative for a ' ' decreasing velocity of flo'Y. 1

2

.. -.-,":

"

~

-

I
h,'S,dx

'

i . ,¥:+d(af-qZ) ed. Slope,s

"" •

1 1

~.

Channel b

....L.,_ - - '_'" 1 L L' __ y~u~~o':.:. __ ~"--.l z

y+dy

-i

---l

a

I

=

hiha,qaT dX

Sf/'---'-

WOle,

~ISodl

(18-'1)

'If the channel consists of a deep main section and an extensive, shallow side section, it l1HW be assumed that., whereas the discharge in the main section is relatively very high, the side section cOli tributes only to storage but not to discharge. Thus, Eq. (18-2) may be written aQ

I

Q::!..~

~

i

z+dz

.

FIG. 18-2. Simplified represent,ation of energy in unsteady flow. J,-----"'....

acceleration must be included. The total loss inhead will.consist of two J parts: the loss due to friction hI = Sf &..and the loss due to accel r t' ~ h -(lIr)(ri2tTrrl.1'+)d":" 1'h . . ealOn -~__._Lf!~~ e Ime mdicating the loss due to acceleratim'C' shown as. th~ accelemNon line; its slope is Sa = (I/ql (;IV iat):="J3., t~S energy pnllelple, the foHowing may be writt~~-'_._ e 2

,

+

y

D'V2 + 2i =z+

Simplifying, d

!he left side of

dz .

+y+

dy

(+ +. 2{j-

V +~ + 2

2g

D'V2)

.2

V

=

-...'I,

(

d D'V2 2g

)

+!:. a v

+S

.

d. gat x ~ ----. (18-9)

1 aV'" .

d:I; -

gat dx

d.

.J

(18-10)

-I

thi~ equation represents the change in total head. The

":vo terms o~ the l'lght. are the head losses due to friction and acceleration, respectively. I

ThIS equation states that the change in tot-'ll head

Sin~e there are two inde end

t

.:.,

'

.

velocity head are' each comp~sed:~ ~::I;a~:ss. ~h:~~.\th:ChaIlges in depth an~in and d(",P/2g) = [aCo: V '12g)/ax] dx + [a(o:V;/2 )/ ~]' y (ay/ax) tirc + (By/a,) dl does not vary with tim. az/a~ _ d-/d d g at dt. As the channel bottom ., ~ - ~ x an az/at = 0 Fo th k' f ' 1" these mathematical details 'tt d f . r· e sa e 0 simp IClty, . are oml e rom the present analysis.

,I

I

528

I-

i

529

. UNSTJ':ADT FLOW

GRA.DUALLTVAIUED UNSTEA.DY FLOW

in gra.dually varied unsteady flow depellds Oil the effects of friction and acceleration. In gradually varied sten.dy flow, the chn.nge in head depends entirely on friction, provided eddy loss is negligible. Dividing through by dx in Eq. (18-10) and utilizing parti<'ll diffel'entials,

Q2. The depth of the wavefront is gradually varied from the upstreani section to the downstream section. During a time interval t, the wavefront moves forward a di~tallce equal to V",t. The wave velo'city is greater than either of th~ mean velocities V 1 and Vi' in the uniform-flow regions. When the wavefront passell ~ver the flow in the channel, it takes in a steady discharge (Y", - V 1) A i at the front. Since the "rave configuration is of stable form and coilstant volume, the front leaves behind an equal st.eady discharge Qo = (V,. - V 2)A 2 to the

+ Sf a +.i. (0:2g. V2) + .!g aatv = 0 x av + --1 &Y Jz + o:TT -::+ -~ + Sf. = (I

-'--:-=

or

g

a.1:

IL8L-- 0;1-

(18-11.) (18-12)

ThL'S is the geneml dynamic (J(pwtivn 101' gr'adnally tlaried unsteady flow. The friction slope in the equation can be evaluated by the Manning formula, the Chezy formula, or any other suitable uniform-flow furmula. For prisnmtic channels, i.e., for -dZ/CJX = So, Eq. (18-12) may be written (18-13)

I,

I

The continuity and dynamic equations for gradually varied unsteady flow were first published by Saint-Yel1:1.ut [21,22J. The validity of these equations has been verified by. many observations and experiments. _-~How~.!~12-?wing to their !!l.!~th~m!l:~~cal ~<:lln:pJ,:,.xity., e:.Xfl:.<;~ ~ntegra!ioll of the equatio!!!L-i:::1!ra!lt..icalll':.. jIIJ.RQ~ibJe~ ~~~,-~ct~I:!~Ll!Pm!Qatiolls, ~ ~~utiOl:"?~ th'U~.ID.l~ti.Q!!E!J!!~Y.~.Ql)t!tiiled by approximate sten IIJ.ethod§ or by methods based on siIIJ.plifying assuIIJ.ptions . . 18-3. Monoclinal Rlsing Wave. A special caSe of uilsteady now which is nearly possible in prismatic Ch2..1111els is uniformly pTogrei1~ve This type of flow has a stable ,"?ove profile that will not change in as .it moves dowl1 the channel: According to this definition, uniformly progressive flow has the following notable features: (1) the successive positions of the wavefront at different times are parallel; (2) the velocity of the wavefront, or celerity, is greater than the mean water velocity at any section of the wave; and (3) the wave configuration travels downstream with a constantvelocl}ty, but the mean water velocity in the cross section may vary from section to section as the hydraulic radius and surface slope change. Of va,dons forms of wave configuration in -r }ih~.m01!f'clinal riS'in~will pc taken as a the following discussion, because such a wave can be approximated to most flood mwes in ria.tural channeLs il.nd because it is subject to simple mathematical treatment. ! The 111onoclillal rising wave (Fig. 18-3) is il. translatory wave of stable .form, that is, a uniformly progressive wave, that travelsdowl1 the channel at constant velocity V w from an upstream region of uniform flow having 'lhI V 1, and Ql to a downstream region of uniform flow having YZl Va, and

, FIG. 18-3, Monoclinal rising wave.

upstream side. The steady dischlLrge Qo which flows through the wavefront in' the upstream clirE.'ction is called the OV/3T7"un. Equating the values of the steady discharge, Q.

= (Vw

Solving for V w,

( 18-H:) (18-15)

Since in uniform-flow may be written (18-16)

The above equations can be used for the computation of the velocity of the monoclinal wave. Equation (18-15) shows that, if there is no initial flow, that if Vl 0 and = 0, then V",= V z• If there is initial flow, then V w is always than either TTl or V 2. Thiil is true because the wave must move more rapidly than the water particles ; ------?>-if it is to make up the volume of the wave configuration at any time. ~ In natural channels, the velocity of an assumed mOlloclinal rising flood wa.ve may be d~termined by the so~called Kleitz-Seddan principle. . Equation (18-16) indicates that the velocity of a monoclinalhsing wave \ is a function of the water .area' and the discharge relationship for the 1 channel. Figure 18-4 shows a curve representing such a relationship. For ordinary channel sections in which. the velocity increases as the water

530

GRADUALLY vAln~D UNSTEADY FLOW

UNSTEADY FLOW

area increases, this curve is concave upward. In the figure, tan lh = QdAl = 17 1 , tall O2 = QdAz = V 2 , and tan Ow = (Ql - Q2)/(A 1 A 2) = 17",= the slope of the line P 1P 2 • Since the curVe is concave upward, it can be seen that VUI must be. greater than either V lor V 2. For a maximum V,., the slope of the line P1P z must be a maximum. This occurs when QI = Q2 or when the point P 2 approaches the point PI and Q

0,

--------

PI

I,

,./ I

/17 I

/1 'j I

I

./ /1 L-t-Um,ting position of ~ Pz // I !. I / I ! I / I ! 1 / I I I _L_ ! , ...,. I Pz tori (max VJ "

I

I

I

')A

FIG. 18-4.. Graphical interpretation of water area, discharge, allti velocity relations in IDonodinal rising wave. . .

the slope of the secant P 1P 2 approaches the slope of the tangent at point PI as a limit. Therefore, dQ

(V"')m~~ = dA

(18-17)

maximllmwave velocity.and the water velucity. in a prismatic channel. On the basis of the Manning formula, the raLio can be shown to be 1.67, 1.33, and 1.44, for wide rectangular, triangular, and wide parabolic cha.nnels, respectively. On the basis of the Chez;y formula, the corresponding ratios are 1.50, 1.25, and 1.33. Equations (18-17) and (18-18) apply only to small rises in flood stage, where the wavefront has such slight slope and rise that the term dQ/dy can appropriately be taken from the rating curve for uniform flow. The principle o£ the above method was originally developeu, mathematically by Kleitz [27], but was discovered from actual observations by Seddon [28] in a strIdy of gage heights on the Mississippi and Missouri Ri'!ers. Wilkinson [29J has described the application of the method to the Clinch· and Lower Tennessee Rivers. He conoluded that the wave velocity in such natural channels can be well approximated by this method. ~ He found that mid-points of rise or fall stages were best suited for determining the velocity of an observed wave. 18-4. Dynamic Equation.for Uniformly Progressive Flow. In a, time interval dt, the w!),vefront of a uniformly progressive flow (Fig. 18-3) will travel a distance V", dt. It can be shown that,as the wave advallces, the velocity of flow decreases with distance at a rate -ov/ax. Therefore, the change in velocity in dt is dV = - V,. dt (aV lax) . . Using partial differentials, av/at = -VwcaV/CIx). . An equ:atiOll similar to Eq. (18-14) can be l.vritten for the general case of uniformly progressive flow; that. is, Q. = (V", - Tf)A. Differentiating thiR equation with respect to x, av/ax =' CQo/AD)(ay/Clx). Substituting the above expressions for ov/at and all'/ih:, respectively, in Eq; (18-13), assuming a = 1, and simplifying, By/aJ: IO!' the flow in a prismatic channel is

ay = dX

Since dA = T dy, the above equation becomes (18-18)

where dQ/dy is the slope of the rating curve of the channel. If the top width of the channel section is known, the maximum velocity can be computed by this equation. Likewise, if the maximum velocity is known, the average width of the channel can be determined. The term d!)Jdy at a given stage rna;: be computed from the mean slope of the ..!-ating_curves at gaging stations in the reach under col1sicieratiojl. The maximum velocity 17,. can. be determined from the distance and time of the wave travel between th!:.l stations. - . • Equation(18-18) may also be used to determine the ratio bet'iveen the

~

.531

1 -

8 0 - Sf Q.Z/gA2D

(18-19)

Since Sf = Q2/K2 = YZA2/K2 = (V",A - Qo)2/K2, Eq. (18-19) becomes By 8 0 . - (VtDA - Q.P/J(2 (18-20) . ax= l-QNgA2D

!

i

!

1I i

This is the dynamic eq·ttation for uniformly progn;ssive ftowin a p.1'ilimalic channel. Since the flow is unsteadY;-Uiepositfon of the wave • prOftl€ischanging constantly, but the shape of the profile at ariy instant remains the same. Equation (18-20), therefore, describes a "snapshot ':' view of the profile at any instant. For wide rectangularchannels J the equation can he further simplified and becomes mathematically integrable. It should be noted that ay/ax represents the insta.ntaneous surface slope of the wave profile. At a. given instant of time, this is

I

-I

\

·1

i

(

I j

I

532

UNSTEADY FLOW

In the equation, the normal discharge Q" can be determined from the rating cui've at the station, and other quantities can be measured in the field. A similar. equation has been developed empirically by Jones [30J and is used widely for correcting the normal discharge to obtain the true discharge in a river when the stage is changing. In the Jones formula, however, the wave velocity .in Eq. (18-24) is l;eplaced by the surface velocity, .which is much easier to determine but lower than the hue wave velocity. A more exact equation than Eq, (18-24) may be obtained from Eq, (18-21) by letting dy/dx -j/V" and solving for Q. However, the derivation and the resulting form of the equation are too complicated for practi.!l1l purposes. 18-5. Wave Profile of Uniformly Progressive Flow. The wave profile of uniformly flow can be easily der~ved for wide rect~1ng1.llar channels by using the Chezy formula. For the unit width of a wide rectangular channel, A R D = y. By the Chezy formula, ](2 = 02y3. Substituting these expressions in Eq. (18-20), the slope at any point on the wave profile at a given instant of time may be expressed as

~f the velocity i~ very low, so that the second te~ of the denominator

v:hlCh covers the velocity-head. effect, can be neglected, the above eaua~ . tlOn may be written • dy . (18-22)

Q = Q"

(18-23)

, L~t j represen~ the ratf'l of rise in the water surfJ).ce at a given channel ,sectIOn,: 1?en, J = V,,:dy/dx, or dy/dx -JJT1... Substitutin this expressIon 111 Eq. (18-23), . g

:Q = Q", ,

'"

(18-24) 1

This equation can be used t~ compute approximatblY the true discharge of .an unsteady flow at 13. given station, provided the velocity is low.

(V",y - Q.)2/0as o

= So 1" y

(18-21)

dx

533

GRADUALLY YA.RdllD UNSTEADY FLew

the same as the permanent slope dy / dx and hence: may also be expressed by the latter. . There is an lLpparent similarity betwee~ (18' -'10) . . .' ., and th e 'd' yna.mIC . e~ll~tr~n for steLl,dy varied flow [say, (9-16) with a = I}. This slllilanty can, be exp:ained as follows. Imagine that an observer on the ~ha.unel ban1l: ,IS l'~nnm.glljter the uniformly progressive flow of Fig. 18-3 In the same dn'ectwn and at the same velocity, V., as the "mvefront. H~ will see 11 picture of a stead v flow like that ill Fig. 18-5; the "water surfa~e will appeilr statioD.ary .to him,and the flow will. show a discharge Qc and a velocity ,Vw Y nt every point along the channel. Therefore, if Qo is FIG. 18-5. View appearil\g to the observar .regarded as a steady dischaq,~e, who follows the wavefront of a uniformly the steady-v!l.ried-flow equation progreJ;slve flow, can be applied directly 'to this . . uniformly progressive flow except t.~at the dIscharge used for computmg the frictional slope should be set equal to .VwA ~ Qo. This discharge is responsible for the frictional loss. The dlscha~ge [~t any section of a uniformly progressive flow is Q = VlOA - Q.. ~ hl1s, Q. Vwi! - Q, ll.nd Eq. (18~20) beco:Q:les

i

(18-25)

y3 - QNg

Whendy/dx O,theaboveequationgivesy 3 - (V",y - Q.)2/02S 0 0 This is a cubic equation in y; the nature orits three roots may be determined by a discriminant ~.l The equation ',~ill give three different real roots if ~ > 0; three real roots, two of them alike, if ~ = 0; and one real root and two imaginary roots if ~ < O. It can be shown that t.he third case will produce no possible wave profiles and that the second is a special case of the first. For A > OJ the cubic equation may be written (y -: Yl)(Y y~)(y - Va) = 0, where the two positive roots Yl and Y2 are, respectively, the final and initial ncrm.aldepths to which the wave profiles are asymptotic~ The third root can be found to be v. = Q~2/ 02S oYlY 'l.. Since Y3 is. not asymptotic to any real branch of the wave profile, it has no physical significance. When dyjdz = r;o, Eq. (18-25) gives y = vQ,,2/u = !Ie. This is called the OI.lBn'Un critical depth, or the critical depth corresponding to a steady discharge equal to the overrun. Therefore, the wave profile. is vertical !at the overrun-critical-depth line. . Substituting y.3for Q,zjg and (y.- Yl)(Y - Y2)(y '- Ya) for the numerator in Eq. (18-25) and solving for x, ,

I

X '7

1

1 So

f

,

;

,;3 - y 3

": < dy (y - Yt)(Y - yz)(y - Ya) .

.

The equa.tion can be written in. the form y'

11 = 18c'f'c.

4Cl"C,

I

+ C)IC~I

-

4c 2' '-'27c ....

,

+ 0 1 !:

(18-26)

!

+ cIY~ + elY +' Ila == O.

'l:'hen,

534

UNSTEADY FLOW

GRADUALLY VARIED UNSTEADY FL01V

where C l is an integration constant. With the given values of V w , Qo, So, a.nd C, thi$ equation represents a number of possible wave pro1iles. For a monoclinal rising wave having an initial stageYl and a. final stage Y2, the possible profiles are shown ill Fig. 18-6 for Yc < Y2 and Yc > 1/2. Theoretically, in the case Yc < Y2, the wavefront extends an infinite distance downstream, since the wave profile is asymptotic to the down-: stream stage. This may seem unreasonable, because the wave actually starts at a finite time and tra.vels with a finite velocity. The explanation is that the wave profile descdhed above will become stable only after

voir into a dry 'wide rectangular channel. Use I,he Manning formula for expressing the velocity of uniform :t!o\v. , Solution. In this case the wavefront moves in a steeply inclined wall of water whose profile is apparently uncilanging as long 'as 'the channel conditions remain fixed and the source .of supply is constant. Therefore, this can be considered a special case of uniformly progressive flow, known specifics-lly as t.he roll wave in which A! 0 Vz = 0, Q" = 0, and V .. "" v', c -,/ii~s;. Subs~itutin:g the ~:alues of Qo alld of V wand K = CA "/11 ill Eq. (18-20), dy

Ja: == S.

(

535

(18-21)

1

LetYl ;, Y" which is the norm!!.l depth at the crest of the W/l,ve, where the flow is pr:l.ctically uniform. Solve Eq. (18-27) for dx; then 1

1 ) V/Y.

d

Y

(18-28)

, . Choose the tip of the wavefront, whe.·e x = 0 and y = 0, as the origin; and integrate . the above equation; the length of the wave profile from the wavefront to the s~cti()n of de;>th '!/ is

x = FIG. 18-6. Profiles of uniformly progressive Row (11:) when y, , 1(b) Ivhen Y. > Y2 or V", > Va + V OYz.

< Yz or 17.. < 172 + V01hi

.!.

r (1 _ =---,1'---,-~)'" dy 1 V/'.}.

S. }O

1/

(18-29) .

Integrating, x =

the wave attain,s equilibrium. The first traces of the wave are transmitted downstream by dynamic propaga.tion at a velocity V 2 + v'YiJ'lJ which is much greater than the fin1ll wave velocity V". By this actiol1 the downstream end of an actual wave will approach the theoretic:),} asymptotic form vel'y rapidly. If the velocity. V,. is greater than V 2 Vfiiiz, it is evidently impossible for the wavefront to extend an infinite distance downstream. This is the case 'when Yc > Y2. The downstream end of the wave is terminated with an abrupt front which in theory should cross the overrun-critical-depth line vertically .. The front is essentially a rapidly varied ullstea,dy flo,;', known specifically as the hydraulic bOr'e (Art. 19-2).

y.,

S;!z + In

(l

z)]

(IS-30)

Rhine :River at the German-Swiss frontier. By a finite-difference graphical calculation of the char>'Lcturislic equations (Art. 20-2) developed by Craya (33], Re was able to compute the fiov.' t.o be expeded from the destruction of 11 do.m for one specific ~ and of Chezy's resistance fa,rtQr. -rul~lud;:r the p;es;;nce of some water initially below the dam, cau~ing the formation of a bore in his problem. In 1952, Levin [341 described I!. solu~ion of the problem by a graphical method~ In the S!l.rtle year, Dressler [35] presented a solution in which the resistance effect was evaluat€d by the Chezy formula.. This solution gives good agreement with observed data exceptin the region near the tip of the wavefront. Accordingly, it is believed that the Chezy formula may not be adequate for application to bighly unsteady flow or to t,he condition of turblJlence in the tip region. In 1953, Dressler and Whitham [36J improved the solution by a more rigorous analysis of the, flQW at the tip region. Dressler J36) &lso developed a method of analyzing the flow during the very small time interval after motion begins. , . Experimental data of the darn-breaking problem are few, notably the early Ilxpcriment,s made by SSholditsh [37] and Egui:3.zaroff [SS] and the later coues by the U.S. National Bureau of Standa.rds [36). . , A problem similar ~o that of dam breaking deals with the flow due to sudden lifting of Ii'. sluice gate. A theoretical analysis of the wave profiles t)lUS developed was made by Pohle [39) in 1950, assuming a. velocity potential flow in Lagrangian coordinates. Experiments on this problem aiso were made at the National Bureau of Standards. A compB,risOD. of the theories and experiments on both problems WllS preSented by I?ressler [36J ill 1954. A comprehensive treatment of the dam-breaking problem along hnes worked out by Pohle was giver. by Stoker [5, pp. 513-522j in 1951.

+

Example 18-1. Determine the wave profile of the nood generated by the failure of a dam. 1 The failUre causes a sudden release of the. impOunding water from i!.s reserI This is the so-called dam-break problem, which is well-known in hydra,ulics. In 1892, Ritter [31) offered· the first solu~loll of the problem, using thea.pproximate

Saint-Venimt equation but ignoring the effects of the frictional and turbulence resist· ance to the unsteady flow. Consequently, this unrealistic classical solution was not 8upportedby the experimental findings and more practics.l analyses had to be sought. In 19B, Forchheimel" [5, pp. 267-271\ presented a ~ummary of most of the previous work on this problem, including attem pts to include the'resistance effects. Toward the end of World War [32) was instructed by the Allied Command to study the problem in anticipation of possible destructiQn of the large da.ms on the

n, Re

I.

-I

.1r

536

UNSTEADY FLOW

where z = V/V". 10 failure of the

ron

Example 18~3. Determine the wa.ve velocity and prome of a uniformly progressive flow in a. wide open channel if YI = 25 ft, Y. = 10 ft, Qhezy's C = 100, and So = 0.0004. Solu.tion. By the Chazy formula a.nd Eq. (18-15), V., = 12.45 fps. By Eq. (18-14), Q, = 61.25 cis per unit width. Since the discriminant e. is positive, the numerator of Eq. (18-25) has three rea! roots, which are Vi = 25, y. = 10, and y, = 61.25 i /(100·X 0.0004 X 25 X 10) = 3.76. Tht; overrun critical depth y" = ..y61.25~/32.2 = 4.89 ft. Since y. < Y" no hydraulic bore will be formed. Substituting all known values in Eq. (18-26), .

This. equation Fepresents the required profile of the wave due If the upstream supply of flow is limited, the profile 'of the wl1ve prnpB.gating ujlstream will be deseribed in Example 19-5, Example 18-2. Determine the profile of the monoclinal rising wave in a wide recta.ngular channel. The wave is lI.S~umed to move with a constant velocity with the nl!l.ximum stage unchanged over a long' period of time. . So/tllion; J'or unit width of the wide rectangular channel, Al = 111 and A, ~ y •. Using the Chezy formula, V, = C Vy 1S0 !Lnd V, = C vY;S~. Substituting these expressiCins in Eq, (18-15) and simplifying, d;lIll.1

a

Vw = )T,G G = 1 - (Yl/y.)~2 1 - yJ/y.

where

(18-31) :1)=

(18-32)

x =

A,V,--AV Ax - A

. i I

i

(18-34)

dy dx = So - Sf

Since V

= C vYS; and V.

= C

S,

=

Vy,S" ~

C'y

=

(18-35)

S,z (G _ g -z !)'

(18-36)

or

=

G - 1)' -z

So ( .

So - ~- G -

dx=Y2{I+ (GZ,-G+l)" So (% ..,.. 1)[.' - \G' - l)z (G _ 1)'J

+

08-37)

}d.

(18-38)

U

By integration,

_ v, { I ln z+3_za

x-S;

I'

I

I

_ (G- 1)'[(2G'

+ G,

(l-z)

+ I)(G + 1) -4J In 2z + I)' - 4 2.

2(3 - 20) V(G

+ 1)

(G -- 1)'(2G 2(3-2G)

. -In{.'-(G'-I)z+(G--,I)·j

- (G' - 1) - (G - 1) V(G - (G2 - 1) + (G - i) v(G

+ 1)" + 1)'

- 4} - ;I (18-39)"

where z = vlv" 11> is the depth at high stage, V is the depth at any section along the wave profile, G i~ defined .by Eq. (18·32), Yl is the depth at low stage, and C1 is a constant tha.t call be det.ernuned from the condition that x = 0 when Y = (YI y.)/2,.

y' - 116.7 d (y _ 25)(y _ lO)(y _ 3.76) . Y

+ C1

(18-40)

f (1 .

48.68 _ 9.437 _ 0.4~9'1) 25 - Y y -10 Y - 3.76

dy

+ C1

(18-41)

= 0.0004

[ Y

+ In (y

(25 - y) ••.•• ] _ 10)9.431(y _ 3.76)',<19'

' T

CI

(18-42)

Taking the origin at x= 0, y = 24, Eq. (18-42) gives C1 = 5,840, by t.hi.s equat,ion is plott.ed as shown in Fig. 18-6a.

ThB wave profile

18-6. Wave Propagation. Before further discussions of unsteady flow, it seems pertinent to discuss the propagation oj gravity., wa~ ~___--",,-;,-Channe}s. For this purpose, a single form of gravity wave, known as the' solitary wave, will be deacribed. . The solitarywav8 was first observed a.!!QjpvesiiK~ted experimentally by Russell [41].1 It has a simple form (Fig. 18-7), consisting wholly of . an elevation without any associated trough or depression what.ever. The wave lies entirely abOVe the normal water SlU'face and moves smoothly and quietly without; turbulence at any place along its profile. In a ~_>lrictionless ~ann~.!~e wave can travel an infinite distance without change of shape or velocity, but in an actual channel the height of the wave is gradually reduced by the effects of friction, Such a wave can be produced in E. laboratory by a sudden horizontal displacement of the gate in the channel (Fig. 18-7) or by other means .•,In natu!:.,e, such waves .have been generated by earthquakes and have been obserye~ to tra;:;l across oceens.~=q Consider a solitary wave, traveling to the right in a rectangular chatrilel, with celerity.c (Fig. 18-7a). An observer on the shore running along with the wave crest at a velocity equal to the (',elerity will see a

+

1

x

\

".

Eq. (18-34) givRs

Substituting this equntion fGr Sf in Eq. (18-35), dy dx

1

i

where Z = y/y" The wave profile under consideration is long and flat; hence, the mean velocity V will change very slowly with respect to both time t and distance x, Thus, aT';ax = 0 and DV IB/ = O. From Eq. (18-13), using ciifferentials,

f

By integration,

!

Eliminating V w from Eqs. (18-31) and (18 e33) and solving for V,

1)

_1_

0.0001

(18-33)

Gz - - V. V= ( G - -

1 0.0004

Expmssed in partial fractions,

By Eq, (18-15), replacing Al and V l by A and V, respectively, for any given section V" =

537

GRADUALLY VARIED UNS'l'EADY FLOW

c'

I It was fu'st discovered by Russell in 1834. The mathemn.t,ical analysis of the solitary wave was originally wo,kec! out jndependently by Boussinesq [42J and Rayleigh [43]. A complete account of the analysis can be found in Lamb [3, pp. 423·126].

For a. refined· solution of this problem, see [36J. of this equation was originally given by Moots [40].

* Derivation

I

I

--I 538

UNSTEADY FLOW

GRADUALLY. VAlUED UNSTEADY FLOW

pictul'e of steady flow (Fig. 18-7b) in which the wave appears to stand still while the flow moves at a velocity equal to c in magnitude. Neglecting friction and assuming a small slope and a = 1 at all sections, the energy equation between the tlormal section of the flow and the section

539

the celerity of a solitary wave in a rectangular channel is C

=Vg(y

+ h)

(18-48)

According to a Gompleteanalysis described byLamb [3, pp. 423-426J, a more accurate equation for gravity waves in general, but stlll assuming

small heights, is c ~

y

~.

foX t

\j21r an

h 21ry

T

(18-49)

j ..l

1

where A is the wavelength from crest to crest. This equ.,tion is generally known as the c.elerity equation in honor of its originator [47j. In

>;7////9///)#/9/////)/// (b)

FIG. 18-7. Generation of asotitary WaNe. (a) Unsteadj- flow; (b) the flow that appears steady to un observer following the wave crest.

at the wave crest may be written

c2

Y+2g=y+h+

CZ (

Y )2 y+h

(18-43)

Solving for r,

c where h is the wltve height above the normal water surface. of moderate height, Eq. (18-44) may be approximfl,ted by c=

~gy (1 + ~:) = vgY(l + :~)

(18-44) For waves

(18-45)

This equation is commonly known as the Saint- Venant celerity equation in honor of its originator [44J. For waves of small height, h is negligible. TIlliS, c = Vgy This is the equation for the propaga.tion of small waves in rectangular channels. It is commonly known as the Lagrange celeriiy equation after Lagrange [45J, who first derived it.· Similarly, it can be shown that the celerity of small waves in llonrectangular cha1111els is·. (18-47) c where D is the hydraulic depth and is equal to A/T. .In the above analysis, neither the centrifugal-force effect of the wave curvature nor the vertical component of the acceleration of the water particles is considered. According to field observ:ations mMe by Russell [41] and experiments made by Bazin [46\, a more suitable equn.tioll fa!'

FlG. 18·8. Wave patterns createq. by disturbances, (a) Still water, V = 0; (b) sub·c.ritical flow, V < c; (t;) critical flow; V = c; (d) Bupercritical flow, V > c.

deep water, where y is large comp!u'ed with A,the above equation becomes c. -./g;.,/'Er. For small-wave heights, Ais large compared with hand tanh (21rY/A) may be· replaced by 21rY/)... Thus, Eq. (18-49) becomes Eq. (lS-46). The equation for celerity, either Eq. (18-46) or (18-47), may be used. for studying the propagation of gravity waves. If a pebble is dropped into still the wave pattern can be represented by the concentric circles shown in Fig. 18~Sa. The waves travel away from the source of the disturbance in all directions with a velocity or celerity equal to c. If the water is fio.w-ing, the wave pattern produced by a disturbance will be displn.cedin the direction of flow. When the velocity V of the water is less than the celerity, the wave pattern is as shown in Fig. 18-8b. Since the velocity of flow is less than the celerity, it is possible for the wave to t.ravel upstream at a velocity equal to V tD

"'"

C -.

V

The wave traveling downstream is in the direction of ·fiow. is ~ncreased to 17.. c 17

+

(18-50)

Its velocity (IS-51)·

",

1

540

Note that thb celerity represented byEq. (18-47) is identical with the critical velocity of the flo;~. (1-10)].; The fiow under consider-\Ltion is, therefore, subcl'iticaL '¥) . . When the velocity of the water is equal to the celerity, the wavefronts . in the l+pstream direction ILre stationary, or V", = 0, and' those in the downstream direction have a velocity equal to V w' = 2c. This wa.ve pattern is showl{ in Fig. lS~8c. The flow is, app~relltlYl criticaL. When the velocity of the water'is greater than the cel<;1rity, the waves will travel downstream only. This wave pattern is show11111 Fig. 18-8d. The flo"r is, apparently, supercriticaL The lines tangent to the wa.vefronts lie at an angle to the original direction of flow. The angle fJ is the (16-15), 01' . wave angle, and its magnitude is given by Sill

fJ

=

~_

v~ ~ ~

These Yectol'S are, as 8. general rule, parallel to the channel axis; so Eq. 18~53 mny be reduced to a simple algebraic sum =

y±c

ods suggested by Thomas [15] will be discussed. briefly.

They are th~

. method oJ finite incre"!-e.h~ (called the complete method by Thomas) and '-themethOd o{tl'ial and e;;O'I'. . '

By the method of finite increments, a reotangular channel is diyided finite length L:.:t, The time interval under consideration into reaches

of

(18-52)

where Fist.he Fl'oude number. The celel'ity c must be olearlydistinguished from the ab80lute waye velocityY",. The celerity is the velocity of a wave relatiYe to the yelocity of flow. When the wave is propagated tln'ough still water, the celerity is identical with the absolute velocitJr.. In open ch!>l1nels, the absolute wave velocity is the Velocity of the wave relative to a certiLin fixed section of the channel. Mathemntically, a general equation may be written, expressing the absolute wa ve'velodty as the vectorial sum of the celerity and the undisturbed velocity 11 of the water through which the Wl\ve is propagated; this i.s (18-53) v'" = V + c

v...

541

GRADUALLY YARIEDUNSTEADY IFLOW

UNSTEA.DY FLOW

FIG. 18-9. Definition sketch for the method of finite increments, The subscript 1 is for upstree,m section at initial state; 2 il> fOI' upstream I!e()ti~o oiter At; 3 is for downstream section at initW state; and 4 is for downstream section !lfter tJ.t,

For a given reach, various elements are designated in Fig. IS-9. The follo\"ting notation, is used to adapt the unsteady-flow equations to the use of finite increm~nts:

is L:.t.

A =

A4 = -;r 2:A

'~~--=-~_,::" ..C--_:

T = Tl

+ Tz +4 Ts +. T4

R = Rl

+ Rz + R. + R4

=

V = VI

+ V + VI + V\

=2:V

(1S-54) ,

where the vekJCities are considered positive in the downstream direction and negative in the upstream direction. The initial flow ·of water in the' channel is assumed to be ill the downstream direction. In discllssing the direction of progression of wayes in channels, three principal case~ may be recognized: (1); positively prO{JT6s8ive flow: if the disturbance progresses dOIVllstream, (2) negatively progressive flow if the disturbance progresses upstream, and (3) mixed progressive flow if the disturbance or disturbances progress both! upstream and downstl'ea6. and will meet and colllbine. • I 18-7. Solutii:ln of the Unsteady-flow Ejquations. Owing to complexity of flow conditi~nsJ the general unstes.dy-flow equations defy exact mathe. matical solution. For demol1stI'atiYe p1u'poses, two approximate meth- .

= ~T 4

4

4

2

.

2:R

4

4

Expressing the partial differentials by finite iucrements,

ay

ax

.!. (us 2

L:.x

VI

+ Yt -

L:.x

Y2) = /

-

av =! (Va -Vl + ~=-~t) ~

2

~

~

i!

fJQ = aCAl') = (AlVa - A 1 Y 1 fJx ax 2 L:.x

A1V I

+A

2V 2

-

YI

+ Y2 VI

+ Vz -

Va -~

2~

+ A4V4 -. AZV2) , L:.x

AiVs - A,V, .

2L:.x

- Ya - V4 .

-~&E~-

/

542

UNSTEADY FLOW

uy = !2 (Y2 iU

aV

Yl -I-

2

- VI 6.t

GBADUALLY YARIED UNSTEADY FLOW

=

t:d.

f:,.t

=! (172

y~)

Y4 -

+V

4 -

173)

fit

Substituting these expressions in Eq. (18-1) and solying for 17 4,

Y4

llxT.T(YI- Y2

=,

Substituting Sj and simplifying,

So

;:=:

+ Y3 -

Y4)/46.t

A,

+ A1V1 +

A 2'V z - AaVJ (18-55) . .

= V2/C 2R and the above expressions in

(1:17)2 1h + y~ - y~ - Y4 _ 4C2 T.R 2llx

(18-13)

a:::Z~~'-..'..~~__,,-__.:':

(18-56) In a given proble~, tIle quantities Yl, A 1,Vl , Yl, A. 2 , 172 , Y3, AJ,.and Vz are known from the initial conditions and from the compuk'~lOns on previous reaches. The unknown quantities Y4 and 17<\ can be o.btained by solving Eqs. (18-55) and. (18-56) simultaneously. Repeatmg :he computation for subsequent reaches, the complete prohle and veloc1ty variation of the flow can be determined. It is clear, however, that the solution by the method of finite increments is extraordinarily ~aborious. In the trial-aud-error method, the wave profile through the given reach is first sketched arbitrarily and then revised if the sketched profile f~ils to satisfy all necessary conditions. The procedure for the computatlOn of a positively progressive flow is as follows: .... . . L For a given reach, draw the wave profile at the 11lltialmstant and sketeh the wave profile at the final instant by extending the known wave proS.l,,,: fromtlle upstl'eam reach. The extended profile is shown by the dashed line in Fig. 18-9. 2. Measure the depths Yl, Y2, YJ, and Y4 and the surfa,ce siopes Sl, S2, S anc1 S at the sections and instants under consideration. 3, • 3. Compute the water areas A l , A z, A~, and A4! the hy dra~ l'lC ra d" 11 R 1, R 2, R.! and R" the velocities V l ! 17 2, 17" and 17 4 , fl,nd the discharges Ql, Q2> Q" al'ld Q4. . .• +' . 4. Substitute the above quantities il?-to the contInUIty equaulOn. Using finite increments, Eq. (18-2) gives 0

CQI

+

Q2 -

Qa - Q4) III

= (-.4.1 +

A2 - A.

+

A4) llx

(18-57)

If this equation is not satisfied by the given vldues, revise.the ske~che.d . profile and repeat the work until a satisfactor}' check in thiS equatIOn 1S obtained. • . 5. Continue the procedure from reach to reach, proceeding downstream,

543

It should be noted that the above procedure" does not include the velocity-head and acceleration-head terms. If these terms were to be included, the procedure woald become so tedious as to be almost prohibitive for practical purposes. One of the most important problems in gradually varied unsteady flow is the routing of a flood wave through a channel. Many practical solutions of this problem have been developed, Because of its unique nature, the problem with its pr
or

= -

~ (V 817 + ~ (JQ) + (S~ -

So - SI = ay ax

+.! g

!

Sf) ox

(v iW + Av aQ + a~) ox

at

all. ilx

(18-58)

(18-59)

This is the reqniredequatioll. However, a solution of this equation is extremely difficult even by finite-increment approximations, One important practical problem dealing with spatiully varied unsteady surfaoe flow is the determination of the unsteady runoff of the overland flow reaulting from r.ainfall. t The riischarge of the runoff with the time since the rainfall began; thel'efofe, the flow is unsteady. If the rate of rainfall is kept constant, a. time of equilibrium will eventually be reached. At this time, the discharge is equa.l to the rate of rainfall and. the runoff becomes 8~eady. When rainfall ceases, the runoff recedes and the discharge tapers off accordingly. In the recession the runoff becomes unsteady again. For practical purposes two empirical methods have been developed, one for turbulent flow and the other for laminar flow. The empirical relations used in these methods were obtained from experiments co";'ering a wide ra:nge of varintion in conditions. Hence, they give only an a,pproximate answer to the problem. A. Laminar Flow. The variation of the runoff discharge with time is u8uallyrepl'esented by a hydro(lraph, which is Il. curve. by plotting the ,discharge against the time. 2 From H011lyses of the' hydrographs reSUlting from simulated rainfall 0f constant rate, Izzard 152J fouIld that the form of the rising hydrograph can be represented .by a sJngle diment For advanced studies of this problem, the reader should refer to the works of Iwagaki [48-51j. .

• A hydrograph may also be a durve of the stage or velocity of the flow plotted

agairjst the time (Art. 20-2).

·f

1

uNs-rE .... Dy FLOW

544 1.0 0.9

I

~~+

/'"

V

0.8

0.6

L-

,-

'-

0.5

/ J

0.4 I---

~

I

0.3

---

,/

0.2 0.1

..,/ 0.1.

V

0.2

/

I

I

I 0.4

0.5

~

v.6

01

0.8

fJ 0.9

=

to -.

FIG. 18-10. Dimensioniess hydrograph of overland flow.

(After C. F. Izzard [52].)

r sionless curve, as shown in Fig. 18-10.

D.

ddention in fV, that is, the volume of water in surface flow on a strip of unit width at the time t since the rainfall began detention in f1;3 at equilibrium It was found empirically that this could be expressed in general by (I 8-(2) D.= KLq.~3 Actually, the exponent was found to vary from about 0.2 for very smooth pavement to nearly 0.4 for turf. The value of J( depends on the rainfall intensity i, the slope of i3urfnce 8, and a roughness facto!: c; that is,

tn.

I

=

(I8-61 )

60q.

'---+~

_. 0.3

D

-

=

J( ='

O.OOO7i+ c . S~'i

(18-63)

This equation was developed for slope not steeper than about 0.04. The roughness factor c was evaluated as follows:

The notation involved is: Type of surface

discharge of surface flow, in cfs per ft of width, at time t since the rainfall began q. = discharge of surface flow, in cfs pel' ft of width, at equilibrium At the equilibrium condition, the rate of supply of the rainfall is equal to the outflow discharge. If i is the rainfall intensity in in./hr and L is the length of surface flow in ft, then q

545

Therefore, the equi-

2D.

t.

/

~

o

equilibrium (the area below the curve). librium time t. is expressed by

~

'---

.~

I

CT

o

-

V

0.7

0' .....

GRADUALLY VARIED UNSTEADY FLOW

=

iL q. = 43,200

(18-60)

It should be noted that bhe equilibrium condition is reached asymptotically. t = time in min since the rainfall ~egali t, = t.ime of equilibriUlil, in min Since the equilibrium condition is approached asymptotically, the time t. must be det8rmined arbitrarily. In Fig. 18-10, t. must be determined arbitrarily. In Fig. 18-10, t. is defined as the time when q reaches 0.97q,; that is,ql.q. = 0.97. It ";as - found empirically that the volume of water, represented by D. in W, in the surface flow on a strip of unit width at equilibrium (the area above the curve) is substantially equal to the volume of water that has been discharged in the time required to reach

Value of c

Very smooth !'sphalt pavement .............. Tar and aand pevement. . . . . . . . .. .......... Crushed-slate roofing p!'per. . . . . . . . . . . . . . . . .. Concrete pavement, normal condition. . . . .. .. Tar-and-gravel pavement. ... ; . . . .. .......... Closely clipped sod. .. . . . . . . . . . . . . .. ........ Dense bluegrass turf. . . . . . . . . . . . . . . . . . . . . . ..

0.0070 0.0075 0.0082 0.0120 0.0170 0.0460 0.0600

. When rainfall C8ases, the runoff decreases. The time t, from the beginning of the recession hydrograph to the point where qlq. = r is

t

= T

DoF'(r) 50q.

(18-64)

where Do is the detention corresponding to D. after the cease of rainfall, ,\'I'hich is the detention when i = 0; and where F(r) = 0.5(r% - 1)

(18-65)

Equation (18-64) is derived mathematically from the finding that detention on th'e recession curve is proportional to the one-third power of the discharge, Le.,

!!... = (~)~ Do q.

= r14

545

UNSTEADY FLOW

GRADUALLY VARIED UNSTEADY FLOW

Using the dimensionless hydrograph and the above equations, it is possible to construct II hydrogmph for surface runoff due to a rainfall of given intensity and duration: It is understood that the experiments conducted for deriving the dimensionless hydrograph and the above empirical equations were made under the condition that the flow was lamiilar at all times. Thel'efol'e,the method is most suitable for laminar flow and, according to Izzard, should be limited to cases where the product of the rainfall intensity in inches per hour and the length of surface flow in feet. is Jess than 500 .. . B. Tt~rbu.lent Flow. With reference to the type of flow ranging from fully turbulent to laminar, Horton [53J has developed' the following equation for the discharge of a spatioJly varied tlDsteady surface flow due to a uniform rate of rainfall excess or supply rate: . q

=

.

utanh"'

lmm+1

t ] (uK) lim 60

are recommended: Type of

i I!'

u tanh 2

r' (cL)0.50. 8°·25 ] 0.~22t

Ij

./ • t

I • t

~ I.

'IF'<:ro,, ,/>:in/hr

(18-67)

• t

FIG. 18-11. Derivation of hydrograph for a rainfall of tinite duration.

overland flow due to 11 uniform rate of rainfall lasting indefinitely. However, the hydrograph for rainfall of finite 9,uration t can be easily derived . In Fig. 18-11, the top figure shows t,he hydrograph for rainfall of infinite '. duration, and the middle figure shows an offset hydrograph for which the same rainfall starts at a. time of t later. The difference between these two hydrographs will, evidently, produce the bottom figure, which is the hydrograph for rainfall of 'duration t. .

(18~68)

In connection with the use of this j:)quatioil, the following values of c

;/

. J

. According to Horton, Eq. (18-66) is strictly rational for 75.% turbulent flow, or m = 2.00. The equation has been used in drainage design of airfields [54:1. For average airfield surfaces, 75% turbulence may be assumed, and the Horton equation may be written =

0.02 0.10 0.20 0.40 0.80

/ConJinuous rainfall excess or i in/hr

tr

(18-66)

where S is the surface slope in the direction of flow j I is the factor of turbulence, that is, 0.75(3.0 - m); c is the roughness factor;. and L is the length of the eleniental'Y strip of t'lIrfed, bare, or paved surface, in ft.

.q

i

Value of G

The Horton equatioll is most suitable for turbulent flow with high discharge, The use of this equo'tion will produce a hydrograph for the

=

- 1,020 VS K .IcL

~1!rfaGe

Smooth pavement ........................ ;........ Bare, packed soil, free of stone ..... ,. . ................... Poor grass covar, or mt,dera.tely rough bare surface......... . Average grass cover. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dense grn.ss cover. . . . . . . . . . . . . . . . . ... ....................

discharge at .the lower end of an elementary strip, in cfs/acre or in./hr . Ij = rate of rainfall excess or supply rate, in in./hr. The rainfall excess is the net amount of rainfall that becomes direct rUDoff, since a portion of the total rainfall is lost by evaporatiOn, infiltration, etc . .m = an exponent depending on the state of flow; m = 3.00 for laminar flow and % for fully turbulent flow. t = time from the beginning of supply, in min K = a constant depending on characteristics of the drainage surface; it is expressed by

where q .

547

;

I

E:umple 18-4. A smooth aspbalt highway pavement is 72 ft long and has a slope of 0.005. Determine the hydrograph at the downstream end of the pavement when a minfall of 1.8!l in./hI' is applied for 10 miD..

UNSTEADY FLOW '

increru;e in discharge, aiter which the curve drops rapidly. This inorease in discharge is due to the fact that during rainfall the amount of'detention on the; payement is grea ter than the amount required to cause }he snme discharge after rainfall has ceased. For i = OJ Eq. (18-6.3) gives K == 0.041 and Eq, (18-62) gives D~ = 0.434 ft·. The excess detention is, therefore, 0.515 0.434 = 0.081 ft·, discharging at'a rate equal to or greater than q, ~ 0.00315 da. The length of time ~ required to discharge the ;,xcess is, therefore, 0.081/0:00315 26 aee or 0.43 min. Accordingly, the beginning point of the actual ourve is 10 + 0.43 = 10.43 min; at this time /] 0.00316 cfs and the detention 0,434, ft'. The recession time Of the hydrograph (Fig. 18-12) caJl eMily bl) computed by Eq. (18-64).

TABLE 18·1. Com'UTA't'ION OF TR.lll HYDROGRAI'H OF A.N OvERLAND FLOW

1 2'.00 ' 2.50 3.00 4.00· 5,00 5.45 6.00 7.00 8.00 9.00 10.00 10.43

I >,

0,367 0,458 0.550 0.733 0.917 ,1.000

0,0002 0,0010 0,0015 0.0021 0.0027 0.0030 0.0031 0.0032 0.0032 0.0032 0.0032 0.0032 0.0032 0.0019 0,0013 0.0009 0.0006 0,0003

0,30 0,48 0.65 CQ5

°0.97

,[)4

1.00 1.00 1.00 1.00 LOO 1.00 0.60 0.40 0.30 020 0.10

1O.9l

11 40 11 ,86 12,64

14.57

0.21

i 0.48

0.42

0.97 1.43

0.02 0.£'0 1.80

2,21 4.14

PROBLEMS 18-1. Assuming that the flow condition does not change with time, derive from Eqs. (I8-1) and (18-13) the continuity and dynamic equations for gradually yaried steady flow in prismatic channels.· , 18-2. Show that the overrun of a mon:lclinal rising wave may be e)(pres~ed 118

Q.

=

lOmm.

,

and

'

0.004

....

..

., Q,E ., ,.

..

-u

0.003

:v

I

0"-

.E'O 'II

0.002

II

~~

0'0 .c ,u,]t

~~ .~

0.001

Dr:;

'" 0

'L 0

II I

VI

I

,

\.

I I "-

!

6

6

.10

!

ii

I

I

!

i

4

I

'I

I

I

V: i 2:

1

12

I

Nt

FtCl. 18-12. Hydrograph of ,an overland floW.

4

16

.

(18-7Q)

When the minbll ee:J.ses at 10 min, it will be noted that there is a. momenta.ry ,

'- r:;

(18-69)

18-3. Show that the overrun Md velocity of a Inonocliual rising Waye iu a wide rectangular challnel of uDit width may be expressed, respectively, 8.8

Solulion. The given dat,a. are L = 72 ft, S = 0.005, i 1.89 in.,/hr, and the total 10 lnin. Since 1.89 X 72 < 500, the Izzard method I\,pplies. For smooth asphalt pavement, C = 0.007. ,By tbe equations oHhe Izzard method, th" constants are c0l!'puted as g, = 0.00315, K = 0.Oi8:', D, = 0.515, and t; = 5.'15. The comput.ati()ll of the rising curve pf the hydrograph is shown ill Table 18-1 and in ]"1g. 18-12, The value of qlg. for the corresponding value of tit, is obtained from the dimensionless hycirograph (Fig. 18-10). At.t = i, = 5.45 min, the discharge q= 0.97'1. = 0.0031 cf~ per unit width of the pavenlen~. IIen"eforth, the discharge is assumed to reach the full value of q. up to t

549

GRADUALLY VAIU);;P UNSTEADY FLOW

,

I ! I

I

(18-71)

where 'C is the Chezy resistance factor. 18-4. On the 333-mile reach of the lVlisBouri River below Kansas City, Seddon found that an average rating curve and ve!.ocity in miles per day can be expressed as Q. = 0.15(1.17y + 19)' and V" = 70 + 3.27y, respectively. Show tha.t the average top width of the channel in this reach can be elCprCSlled as T = 5.62(y + 16.3PI ,(y + 21.4), and construct the' average shape of .the channel section. ' 18-5. On the basis of the Manning formula., determine the ratio of the maximum wave velocity to the water velocity of a mOMclin!l.1 rising wave in (a) a wi~e rectangular channels (b) a triangular channel, and (e) a wide parabolic channel. ' ;18-6. Solve the above problem on the ba.sis of the Ch';zy formula. 18-7. 'Using the Manning formula, derive an expression for the wave profile of the Hood due to failure of a d a m . , ' . 18-8. Given So ~ 0.0004 and Yn ~ 25 ft, plot the profile of the roll w~ve derived jn Example 18-1. ' 18-9. Solve the above problem, using the'iestllt of Prob. 18-7 for thCl profile, and comp",re the result with that obtained above. . 18-10. Flot :the wave profile derived in Ex~mple 1$-2, when Vi = 10 ft, it. "'" 25 ft, and S. "" O.OOp4.·. , . i 18-11. SolVl! Example 18-3 for Y. - 0, 2, 5, aod 15 ft, and shoW that the surface' slope of the vni.ve configuration decreases wi~h the decrease of the initial stage Y2. la-l~. Solve Example 18-3 if S. = 0.004. : 18-13. Solve ExAmple 18-3 if S. = 0.06. i

55,0

UNSTEADY FLOW

GRADUALLY VARIED UNSTEADY FLOW

IS-14. The initinl and final depths of a. monoclinal flood wave are .10 ft and 25 ft, respectively. Given C = 100 and S. ='0.0004, determine the percentage inel'eases in the true'dischHrgc over the normal discharge at depths of 15 and 20 ft. 18-15. Compute the Reyn<:>lds number of the discharge from the pavement described in EXllmple 18-4, aud show that the 1I.0\V laminar. The discharge immediately a.fter the rainfnll ceasE'S is assumed to increase momentarfly about 10 %. 18-1S. Solve Example 18-4 if the rainfall is 3.78 in./hr. 1B-17. Solve E:
of

REFERENCES 1. J. J. Stoker: "Water Waves," vol. IV of "Pure and Applied' M:athematics," Interscience Publishers, Inc., New York, 1957. 2. G, H, Keulegan; Wave motioil, chap. XI of "Engineering Hydraulics/' edited by Hunter Rouse, John Wiley & Sons, Inc., New York, 1850, pp. 711-768. 3. Sir Horace Lamb: "Hydrodynamics," 6th ed., Dever Publications, New York, 1932. For tidal waves, see chap. VIII, pp, 250-3(}Z; for surface waves, see chap. IX, pp. 3(}i3-475, , 4. U.S. Army Corps of Engineers: Bibliography on tidul hydraulics, CommiJ,lee on. Tic/.{).! 'IIlIdmulics, RepQrCNo. 2, Februa.ry, 19M; Su.pplement No.1, June, 1955; Supple.men/. No.2, May, 1957j and Supplement No.3, May, 1959. 5. Philipp ForchheiUler: "Hydraulil, " (" Hydr!l.ulics "), 3d cd., Teubner Verlagsp;esell. Bchaft; Leipzig; and Berlin, 1930, o. Charles Jaeger: "Er.gineering Fluid Mechanics," translated fr?1l1 the Germ&n by P. O. Wolf, Blackie & Son, Ltd., London and Glasgow, 1956, 7. Takeo Kinosita: Hydrodynamical study on the ftood flow, in Floods, vol. III of Symposia Darcy, Intemal{onal Association of Scientific Hydrology, Publica~ion No,

42, 1956, pp. 56-63. . , ' . , 8. N. J, Dahl: On non-permanent flow in open canals, Proceedings of the 61h General Meeting, Ini.ern.alional ASli/ociatiiJnjor Hydniltlic Researc'h, The Hague 1855ivoL 4, pp. D19-1 to DI9-1B, 1955, • 9. Fr!l.no;ois Serre: Contribution a I'et·o.de des econlements perm~"nents etvarial:lles dans les ca.na.ux (Contribtltion to the study of permanent and nonpermanent flows in channels), La Houille blanche, Grenoble, 8th yr., no. 3, pp, 374-38B, June-July; a.nd no. 6, pp. 830-872, December, 1953. 10. J. C. SchOnfeld: Distortion of long waves. Equilibrium a.nd stability, Assemolee gtnera!e de Bnae!!es. 1951, Inte1'1l.11J.ional AssociaHon of Scientific Hydrology, Publication No. 30, 1951, vol. 4, pp. 140,:,157. . ' 11. Bruc.e R. Gilcrest: Flood routing, ·chap. X in "Engineering Hydr,a.ulics," edited by Hunter Rouse, John Wiley &. Sons,Inc., New York, 1950, pp. :635-710. 12. Giulio, Supino: SUf I'amortissement:des iI!tumescences dans les canaux: (On the damping of translatory waves in channels), R;wue ge-Mrole de I'h:ydraulique, vol. 18, no, 57; pp. 144.,.147, 1950.

I

I I I

551

13. D. N. Dietz: A new method for calculating the conduct of translation w>tves in prismaticcannls, PhI/sica, vol. 8, no. 2, pp. '177--1\)5, February, 1941. 14. Giulio Supino: Sur la propagation des ondes dans les Cll.naUl( (Or,. the propaga.tion of waves in channels), Revua !len~"'ale de l'hydra:uliquB, Paris, vol. 5, no. 29, pp. 260-262, 1939. ' , . 15. Ha.rold A. Thomas: The hydraulics of flood movements in rivers, Carnegie Inslitute of Tecknology, Engineering B1Lllelin, 1934; reprinted in 1936 and 1937. 16, L. Cagniard: Hydrodynamiqt1e fiuviale: Regimes variables (Fluvial hydrQdynaIllic~: Variable regimes), ReVILe gtnerale de I 'hlldrG'U.iiql1e, Paris, vol. 3, no. 15, May-June, 1937, 17, M:;.sse: D~ {ntumescenl~es dans lea torrents (Tra.nslatory W!l.ves in torrents), Revue iJdnerale de l'hydrGu1iqlle, Puris, vol. 3, no. 18, pp. 305-305, 1937. 18. Ph. Deymi{j: Propa.gaLion d'une intumescence allongee (Propagation of a grn.duaUy varied translatory wave), RevM genb:ale iie l'hydrauliq!Le, Pal:.is, vol. 1-2, no. 3, pp. 138-142,1935-1936, 19. Pierre Masse: L'amortisMment de." intumescences (The damping of tmnsla.torr waves), Revue gen~T!ll.e de l'hydrauliq!L6, Paris, voL 1-2, no. 6, pp: 300-308, 19351936. 20. Josef Frank: "Nichtstationare Vo;·ga.itge in den Zuleitungs- und Ableitungska.n1i.len von WIl.Bserkraftwerken" (".unst.eady Flow in Heo.draces ",od Tailral;es of Hydropower Plants"), Springer-Verlag, Berlin, 1957. 21. A. J. C, Barre de Sa.int-Venant; "E:tudes th.eoriques et roent'des eaux courantes" ("Theoretical a.nd Practical Paris, 1848. 22. A. J, C. Barre de Saint-Venont: Theorie du mouvlmlent non perma.nent des eaux, avec application aUx cruea des rivieres et A I'introduction des marees dans leur !its . (Theory of the nonpermanent movement of waters with o.pplication to the fl~ods of rivers and to the introduction of the tides within their beds), Comple.i rend1LS des seanc(lS de I' Acad6mie des Sciences, vol. 73, pp. 147-154 and 237-240, 1871. 23. H; Reineke: Die Berechnung der Tidewelle im Tideflusse (Compmation of tida.i wave in a tidal river), Besondcrc MifJ.e~7:ungen, J ah.,.buch Jur die GelOtisserklmde Narddeutschlands, Berlin, vol. 3, no. 4, 1921. 24. J. Frank and J. SchUller: "Scliwingungen in den Zuleitungl'- und Abldtung.i- ' kaniilen von Wa.sserkm.fta.nlagen" (" Oscilla.tions in Her.draces and Tailra.ces of Hydropower Pla.nts"), Sprir-ger-Verlag, Berlin, 1 9 3 8 . ' ' 25. H. Favre: "lttude theorique et experimentale des ondes de translation' dans.les canauJ( deco\lverts" ("Theoretical and Experimental S!.udy of Translatorv Wa.ves in Open Cbannels"), Dunod, Paris, 1\J35. . • . 26, Carlo Drioli: Esperienoe sui mota perturbato nei canali industriali (Experiment on in industrial canals), U ElI erg ia. eleitrica, }I;[ilano, voL 14,110.4, pp. 285305, appeng,ice, pp. 306-311,Apdl, 1937. 27. M. IGeitz: Note sur la tluiorie du rnOUvement non permanent des liquides et sur applicati?u a la propagation des crues des rivieres (Note on the theory of unste~dy How of liquids and on application tC' flood propaga.tion in rivers), Anna.les des 'fonts at chauss.es, ser. 5, vol. 16, 2e semestre, pp. 133-196, 1877. 28. James A., Seddon: River hydraulica, Transactions, A meriron Sociely oj Civil E111linecrs, vol. 43, pp. 179-229, 1900, American. 29. J. H. Wilkinspn: Translatory waves in natural channels, Socidy oj Civil EnginCer8, voL 110, pp. 1203-1225, 1945. ' 30. Benjamin E. Jones: A method for COITBctio.g river discharge for !J. changing stage U.S. Geological Survey, Wat;:r Supply Paper 375 (e), 1916, pp. 117-130. "

"1

'f .1

,I I

-I ·1

~'II

./

11 552

jJ

l. ")

)

I UNSTEADY FLOW

31. A. Ritter: Die FortpfianzlIng der Wa.sserwellen (Propagation?f waves), Zeitschrift clc.~ Verm:ms, d,'u/,scher Ingenieure, vol. 36, no. 33, pp. 947-954, Aug. 13, 1892. 32. R. Re: Etu~le du lacher instantane d'une ret.enue d'eau dans un canal 'par la met'hocle graphique (Study of the instantaneous relea,se of water in a reservoir to 1\ canai by the graphical method), La HOl,ille blanche, Grenobl~, 1st yr., no. 3, pp.181-~187, NIay, 1946. ' . 83. A Craya: Calcul graphique des regimes variables dans les canaux (Graphical calculation of variable flow:-regimes in open channels), La Houille blanche, Gre'noble, 1st yr., no. 1, pp. 1\)-38, November, 1945; no. 2, pp. 117-130" March, 1946. 34. Leon Le;"in: Mouvement non permanent sur les COUrs d'eau a la suite de rupture de bllrrage (Unsteadyfic.w in channels following, the rupture of dam), Revlle glne.rale de l'hydrmdiqlle, Paris, vol. 18, no. 72, pp. 297-315, November-December, 1952. 35. Robert F. Dressler: Bydraulic resistance effect upon th~ dam-break functions, paper 2356, Jotmwl of Research, U.S. National BW'ea,t: of Slandard.~, vol. 49, DO. 3, pp. 217-225, September, 1952. 36. Robert F. Dressler: Comparison of theories a.nd' experiments for the hydl"Jl.ulic d~m-hrep_k wave, Asse11lbl~e g~n~raIB de Rome, 1954, International A.~_qociat(on (If Scientific Hydrology, Plibliroii011. No~ 38, voL 3, pp. 319-328, 1954. 37. A. Schoklitsch,: Uber Dambruchwellen (On ,"aves produced by broken dams), SI{:Z1mgsberu:hte, l1f athematisch-nat'lrwissenschaftliche Klasse, Akademie der J,Vissenschajlen'in Wien, vol. 126, pt. !la, pp. 1489-1514, Vienna, H1l7. 38. 1. B. Eguiazaroff: Regulation of the water level in the reaches of canalized rivers and regulation of the flow b.elow the last lock dam according to whether the water power is or is not lIsed, Report no, 8, in Inland Navigation, 2d question, sec. 1, 16th 1IltemMionai Cong1'ess of NaviGation, Brussels, 1935. 39. Frederick V, Pohle,: Motion of wave due t,Q breaking of E< dam and related problems, paper No.8 in Symposium on gravity waves, U.S. National Bureau of SIa:lldal'di;, Circular 521, 1952, pp. 47-53 .. 40. EllIler E. Moots: A study in flood waves, University of Iowa, Studies in Engineering, Bulletin 14, 1938. ' 41. J. Scott Russell: Heoort on waves, R~ort of the British Association for the Advallce. meni of Science, 1844, liP: 311-3g0. 42. J. V. Boussinesq: Sur Ie mouvement permanent va.rie de.l'eau dans les. tuyaux ~e conduite et dans les canb.UX decouverts (On the steady varied flow of water In conduits and open channels), Comptes rendlls des s~ances de I' AcadMnie des Sciences, vol. 73, pp. 101-105, 1871. 43. Lord Rayleigh: On waves, The London, Edinburgh and Dublin Ph.:losophical M[!~adne and Jot/rnai of SC1:ence, ser. 5, vol. 1, pp. 257-279, April, 1876. 44. A. J. C. Barre de Saint-Venant: D~mon5traliori elementaire de Ill. formule de propagation d'une onde ou d'une intumescence dans un cana.l prismatique; e~ remarques sur les propagatiollS du son et ,de la lumiere, sur les ressauts,: ainsi que SlIT Ill. distinctiori des rivieres et d~s torrents (Elementary demonstratIOn of the propagation formula for a wave or a translat.ory wave iii. a prismatic channel; and remarks on the propagation of sound and light, on hydraulic jumps, and also on the'distinction between rivers ,and torrents), Comptes rendus des sB(lnCes de l' Acad~mic d~1i Scienceli, vol. 71, pp. 186-195, July 18, 1870. 45. Joseph L. de Lagrange: "Mecanique analytique" (" A~alytical Mechanics"), Paris, 1788, pt. 2, sec. II, art. 2, p. 192. , ' 46. H. BiJ.zin: Experiences sur la'propagation des ondes Ie long d'un cours d'eau torrentueux, et confirmation par ces experiences des formules donnees par M.

1

GRADUALLY YARIED UNSTEADY .FLOW

!!

Boussinesq, dans sa tiu\orie du mouvement gj'aduellement varie des Iluic\es (Experiments on wave,propagation in torrential flow and their confirmation or the BOllssinesq equations for gradually varied flow), Comptes rendus des s~ances de I' Academie des Sciences, vol. 100,pp. 1492-1494., June 15, 1885. 47. Sir George Biddle Airy: Tides and waves, "Encyclopaedia Metropolitana," London, 1845, pp_ 24F396. 48. Yuichi Iwagaki: Theory of flow on road surface, Memoirs of Ihe Faculty of Engineering, '[(yoto University, Japan, vol. 13, no. 3, pp. 13g-147, July, 1951. 49. Y. Iwagaki and T. Sueishi: Approximate method for culoula.tion of llost.eac\y flo\\" in open channels with lateral flow, Proceedings, 4th Japan National Congressfo'}" Applied ivlechanics, pp. 235-240, March, 1954. 50. Yuicni hvagaki and Tomit,aro Sueishi: On the unsteady flow in open channels with uniforIn lateral inflow (in Japanese), Proceedings, fapan Society of Cillil Engineers, Tokyo, vol. 39, no. 11, pp. 575-583, November, 1954. . 5J. Y,uichi Iwagaki and Takuma TakasM: On the effects of rainfall and drainage basin charactel'iBtics on runoff .fela~ion (in Japanese), Proceedings, 5th Anniversary of the Establishment of the Disaster Prevention ResearcH InsW"te, EYoto University, I{yoto,Japan, pp. 1,91-200, November, 1956. 52. Carl F. Izzard: Hy.draulics of runoff from developed surfaces, Proceedings of the iJ6th Annllal Meeting of the Highway Research Board, vol. 26, pp. 129-146, December, 1946. 53. Robert E. Horton: The interpretation and application of runoff plot experiments with reference to soil erosion problems, Proceedings, Soil Science Society of Am.erica, vol. 3, pp. 840-349, 1938. 54. Gail A. Hathaway: Design of drainage facilities, in Military airfields: A sympo. sium, Transactions, American Socie.ty of Civil Engineer!!, vol. HO, pp.' 697-733 1945.

(,

553

RA.PIDLY V.ARIED UNSTEADY FLOW

555

Solying Eq. (18-15) for V 2, 172 = V I :"':l CHAPTER

+ V"u:.t12 -

V",A 1

(19-4)

A2

19

Substituting the above expression fot V 2 in Eq. (19-3) and reducing,

RAPIDLY VARIED UNSTEADY FLOW

vr ( v .. -

or

. Flow. If th 19-1. Uniformly ProgressIve . e front of a :rnonocliniil h O~ ~n abrupt change iil curvature or a sudden change. 111 wave ''''' '"' Th' ff t be p"O . depth the flow in the front is rapidly varied, . . IS ~ ec m~y h • ~ duced' for example by a sudden increase in gate opelllng at tee anne , ' 111 "F"19. 19-1, The velocity of the mass of W
(19-5)

V",

(19-6)

G . ,

f

t

•

~GO" II U

or

This is a genera! equation expressing the absolute w,ave velocity of the flow as shown in Fig. 19-1. Mathematically speaking, the sign i;l frOll t of the square-root term in the n,bove equations may also be However, since the w"we is mo,ring downstream in the directioll of the initial flow, its velocity must be greater than the velocity of the !llitilw flow. In othel' words, V", - V j should be positive. Therefore, only the plus sign is considere9. practical. . If the initial velocity VI = 0, that is, if the WlI.ve travels in still water, then the square-root term in the above equations 18 equal to the absolute VI, velocity of the wave. In allY case, this term, being equal to V represents the velocity of the wave with respect to the velocity of the initial flow. It is, therefore, t,he celerity; that .

1.,. _~_. . .". *-_- ",=-_

'-2_-+1T"

....

--WAi'h

.

W/////m)W#407»;;;:

11}

FIG. 19-1. Rapidly .varied uniformly progressive flow.

between the gate and the ...vavefl'Ont is increased from 1'"1 to V z, an~,the . . accord'lllgIy. By J.Newton's second law of"'t motion. mOPlentum is increased t' . the 'unbalanced force required to change the momentum p~ ~m Hl1e is the product of the mass and the ill velocity per um une, or F

~

(V",

V 2 )A 21V(V 2

(J

-

V 1)

c=

wA1Yl

where fj: and iiI are the centroida.l depths of the areas. above values of F and simplifying, ' (V.. - V 2 )(V 2

-

. Al _ ) V 1)= ( ih - A2 Yl g 55~

(19-8)

(19-1)

c

= wA~y~

-

For rectangular chanuels, fit Thus, Eq. (19-8) becomes

where 10 is the unit weight of water. 'l'hi'l Ullbalanced force is equal to the difference between the hydrostatic pressures on the areas A2 and Al at sections 2 and 1,reapectively; that is, '

F

(19-7)

(19·9)

It ca.n be shown that, fOl"\VaVes of moderate height, Eq. (19-9) becomes Eq. (i8-45), FQr very small waves, Eq, (19-9) becomes Eq. (18-46), For very small waves in llourectangulal' channels, Eq,' (1£)-8) beconles Eq. (18-47). In a.ll cases, (19-7) may be written

(19-2)

Equating the . i J

(19-10)

(19-3)

I

J

which is identical with Eq. (18~51). Theoretically, there are four typ~s of rapidly varied ullsteady flow

·~,l

I

556

UNSTEADY FLOW ,

(Fig. 19-2): type A, having an advancing wavefront moving downstream; type B, having nil advancing l"avefro!lt moving upstream; type C, having a retreating wavefront moving' downstream; and type D, having a retreating wavefront moving upstream. Type A has just been described. 1"01' type B, it can be shovtn that

v'" = I"I

(19-11)

c - Y1

which is identical with Jiiq. (18-50). It can also be shown that Eq.. (19-10) applies to type C and Eq. (19-11) to type D. It is theoretically

~ I 'I,_Y, Y, --Vt r

I

I

;;"?'~ml////;;/I';"-//h-:--"';:r-;'-/,/,-

Type A-Advl)l"lci"2 dcwfIsttifam !P(U:!IVl1I 1utQe}

557

RANDLY VARIEDUNS'l'EA.DY FLOW

,

'l"ype B-Ad'lol():.citlQ up!\'lr'l:lJn (p:1sHive sun;~e:

r-=F:::[ Y, -V, I

J»/;;7///74//~r/h'l/}"

T:tpe C-Rerr.cnJlQ dawit- l'ype O-fL!1t!II:;HJ:"q u?,siteD .." (!\egoli,,: shearn :,,e201iy~ slug2) SUf08:

top waves will retreat faster than the bottom ones. The result is that , the wavefront becomes sloping and eventul111y flattens out. Owing to the presence of channel friction, however, the wave profile of art advancing front n.ctmilly changes. This. effect may not be noticeable in relatively short artificial chQ.unels of small friction effect, but it is significant ill long reaches of natural channels, where ~riction is considerable. • . 19-2. The Moving Hydraulic Jump. When a rapidly vfl.l'ied unsteady How involves a change in stage from subcl'itical to supercritical, a so-called moving hydl'(J,1tlic jump will result. An observer on the bank. running n.long with the wave crest. at the same velocity as the. wave will see a stationary hydraulic jump. This is illustrated by the figures at t.he bottom in Fig. 19-2 for four types of flow. All equations derived in the preceding article apply also to these flows. However, another method of . deriving the equation may be described below_. For a stationary hydraulic jump, (3-21) can be reduced to (19-12) ,

Tipe B

7,!pe C

Typ~

D

,

The right side of the equation is equr,l to the celerity expressed by Eq. (19-9). Applying this equation tothe steady hydraulic jump of type A in Fig. 19-2, as seen by the observer, V 1 should be replaced by V.. - 'f\; that

FIG. 19-2. li'ol11' tYlJes of rapidly varied uiliformly progressive flow. (Top) Unsteady flows; (bottom)' the corresponding flows that' appear stea.dy to an observer following the wavefront~

possible for the velo\}ities V 1 and V 2 to be opposite in direction. In that co,se, a velocity is considered negative if its direction is opposite to th£\L shown in 19-2. The absolute velocity ll", of the wave may alsob9 expressed by Eq. (18-15). Since thi8 equation is derived on the basis of the principle of continuity, it applies to both graduilly and mpidly varied monoclinal rising waves. It should be noted that the wave ;profile is stable for s.n advancing front and uristable for a retreating front. The wavefront can be assumed to be.made bp of a large number of very small waves placed one on top of the othel:. Since the wave on top has greater depth thl1uthe one below it, it has grelltel' velocity, according to (18-46), abd moves faste~·. III the case of the advancing£ront, the top waves will, therefore, overtake th~ bottom waves in the forward direction. The teildency is for the wav!:ls to combine and to form eventually a single large wavefront which :i8 steep o,nd stable. In the case of the retreating front, the

(19-13)

or

(19-14)

V .. =

This is identical with Eq. (l9-1O). type Bilow, gY2 (Yl

2Yl

+ 1;2)

Applying the same procedure to

- VI

= C- V1

(19-15)

This equation is identical with (19-11). It can be seen that type C and type Dflows are, not actually possible, because fL stable jump cannot be formed if the flow is from high stage .to low stage. For this reason, neither type C .nor type D can have a stable steep front like th!:l.t of a hydraulic jump. The term "moving hydraulic jump" is usually used synonymously with "surge" and "hydraulic bore." However," surge" more often refers to a moving hydraulic jump due to abru~t decrease or increase in flow, such as. that caused by the sudden closirig or opening of a gate.

558

UNSTEADY .FLOW

RAPIDLY VARIED UNSTEADY FLOW

"Hydraulic bore" usually refers to a moving hydraulic jump due to tidal effects; it possesses a sharp and. steep advancing front.l When the moving hydraulic jump is specifically noted as a surge, it is furthe.rclassified into two kinds, namely, the positive surge and the negative surge. 2 The positive surge elevo.tes the water surface, advancing either upstre2im or downstream with a stable front (Fig. 19-2, types A and B). The negative surge depresses the water surface, retreating

.

at a speed of 50 mph or more. 200,000 ds.

559

The discharge was estimated at about

:Type B usually occurs in tidal rivers. Typical examples' are the Hangchow bore at Raining on the Chien Tang River, China (Fig. 19-3); the bore on the Seyern River near Gloucester,' England; anel the 'bores 011 rivers connecting the Bay of Fundy, Nova Scotia. In' power caI'mls serving hydraulic turbines, type B flow, known as rejection surge, will . oecm' as a result of suddell dzcrease in power output. .

- r

I'.

~

..,

.,

,

.~

'-ocus of Wove front

.

~ ~ \\;:-"r";:-----....--... ---l . ---l. . --. ~~ Bup~ -=

~ \~ , /// ,.//~V1 --..

l

1 j

.

~

~-

j

~

either upstreo.m or downstream with an unstable front (Fig. 19-2, types C and D ) . ' . For the types of moving hydraulic jump illustmted in Fig. 19~2, typical examples can be quoted. For type A, examples are the fo.ffious J ohnstown flood of 1889, caused by the failure of an eart,h dam at the South Fork Reservoir in Pennsylvo.niu [6-8]; the Heppner flood, due to a cloudburst storm on Willpw Creek, Oregon, in 1903 [9]; and the flood due to the failure of St. Francis Dam near Los Angeles, Catif., in 1928 [10]. In the J ohnstowll c~ei. the flood was led bya bore which was initially 125 to 150 fthigh at the da.m and which was reduced to 30 to 40 ft as it rushed down the 15, miles' of the narrow, windi.ng va.lley to Johnstown For a theoretical an~ experimental investigation' of the bore, see [lj.and [2J. • See [3] to [51 for aniIytical and experimental studies.

'1

I ! I

I

-

(

Locus of wave frant

w

..

~

~

.//~~ Vz~

.

I

adVancing

--..... V ...... -

~

-r~-==

.

",

Type

~

~

FIG. 19·3. Hangchow bore at Haining em the Chie?- Tang Rinr;. China.. The wave· tront was about 16 it h,igh, traveling at high velocIty, Seven mm after It could first be distinguished on the horizon the wave had passed. The water rellched a final height. of about 28 ft within 30 min. The width of the river at the place of obser'mtion was about 1 mile. (Caurle.~y of Mead and Hunt, I n c . ) . .

~ '///-7.:;:

.

Type A - Qdvancing downstream

\ 1

.

~

/~

1 ;j

1 !

V-

.

/

"h~-07-~

Type C-retreating downstream

FIG. 19-4. Rapidly va.ried flow

'

i~

'

v2 ---....

--

.

Vw ___~ __ ~_

~up'stre~ .

;;,-..

~

'/

~~

.

.~

Type D-retrealing

inclined channels.

(Afte,· M. D. Chedousov [11].)

Type C generally results irom .a sudden decrease in supply of flow. upstream, such as that I;aused by the closing of a head gate in a canal or at the tailrace of a hydro plant. . Type. D, kl:ow'~ as demand S"..trge, Occurs in power canals servi ng ~ydmuhc tllrbmes If the demand at the lower end of the canal is suddenly

Increased.

.

In Fig: 19-2, tl;Ie initial. and final stages of a po.ssing surge ~tre assumed to ?~ ~lllform .. Such surges OCcur frequently io channels of sma.ll slope. . In lIl,chned channels, the four types of sur"e will Occur as shown in Fig 19-4 [111. '" .

I

J

.19-3. Positive Surges. Positive surges .have an advancing front With t?e profile of a moving hydra.ulic jump. When the height of the surge ~s sn:al~, the surge 'appears undular like an unduhi.r jump. When the hel~ht IS Increasing, the unClubtion will eventually disappe.'l.r and the surge Will have a sharp and steep front.

..-. _. -..·_ . . ·_···. _···--.. -- . 1 !

I

(

I

560

\

Consider the positive surge of type A (Fig, 19-2); the absolute wave velocity can be expressed byEq. (18-15). For a rectangular channel, the velocity is

I

YI

\

,

Y2

)

2

Also, from Eq. (19-18), V

+ Y2)g 2YIV;--

(Yi

(19-17)

This equation represents the relationship among the initial and final velqcities and depths of the surge .. Similarly; it can be shown that' Eq. (19-17) applies also to surge of type B (lfig. 19-2). Note that, if Eq. (H)~17) is used to determine Yi or Y2, it must be solved by trial ::tnd error. MUltiplying Eq. (19-17) by the square of Eq. (19-9) lma simplifying, it CMl be shown that (Hl-IS)

where h =Y2 - Yl) or the height of surge, and where c is the celerity. On the right side of the equa.tioll, the positive applies to type B surge a.nd the negative sign to type A surge. When the height of surge is smn,ll compared with the depth of flow, 111 "'" Y2. Thus, Eq.(19-18) may be written

± c

I i

I I

h

(19-20)

21/1

+ Y2

~g

(19-23)

h=~V

(19-24)

l'

1 and

I

c(

{/ ,Yi

) V

(19-22)

If the height of the surge is small relative to the depth of flow, Eqs. (19-21) and (19-22) may be approximated, . respectlvely, by =

c

g

B. 111eeting of Two Surges. When two surges meet that are opposite in direction, . the result is the generation of two new surges tra.veling in the reversed dil'ections. This is illustrated in 19-5 with all notation indicated. Applying Eq .. (19-17) to the left-side new surge with l' ~ V, Y2 ;", V, aIid VI replaced by V 1,

+

(YI - yP (VI y)g (19-25) 2Y1Y

(19-1\))

In order to avoid the l.:o.nfusiol1 of the sign convention, the use of Eqs. (19-18) und (19-10) may be simplified by ignoring the negative sign 011 their righL sides and remembering that. hand c are always assumed to . be positive quantitie13. Accordingly, VI - 172 should always be positive. If V 2 < 1711 then 'VI V 2 must be replaced by V 2 - l T i • U' VI and V 2 are in opposite directions, then their sum V 1 + V 2 must be used illstead of their difference. . .Several typical ct1.Ses of the al11Llysis of positive surges are given !l.S follows: A. Surge due to Sudden Stoppage of Flow. If wate~ flowing in a channel with a velocity V is checked instantaneously, a surge of type B will be produced .. In this case, V 1 = V and 1'2 = 0; 50(19-17) gives

(19-21)

ThllS, the height of the surge is

(19-15)

Y1

Eliminating 1'", from Eqs. (19-14) and (19-16) and simplifying, ,I

561 .

RAPIDLY YARIED UNSTEADY FLOW

UNSTEADY FLOW

(01

(P)

Fw.19-5. Meeting of two surges. (al Before thE' meeting; (b) after the meeting.

Applying Eq. (Hl-17) to the right-side V, V 1 = - VI', Y2 = y, ILnd YI

new surge with V 2

( ll 1'+ V)2

=

(y' 1

y)~ (y{

+

= Y/,

y)g 2y{v

(19-2G)

. Solving the above two equations simultaneously, the unknowns l' and y . can be determined. The absolute wave velocities of the new surges can be determined from Eq. (19-16). Thus," V,. =

and

V,.'

11

Vl~Yl/

Y-

VI

+ Vy yr'

(19-27) (19-28)

When the height of the surges is small relative to the. depth of flow,

Eq. (19-19) may be applied.

which is similar in form to Eq. (19-30).

Thus, (19-29)

,)//7/7777//7/tl

l

(19-33)

Applying

=

. . ) (Ut + Y2)g (Yt- Y2 2"':'=--::::2-'-=-"':"'::" ,

YtY2

-[

Solving the above four equationssiwultaneously, the unknowns V 2 , 1/z, V 21 , and Y2' can be determined, The wave velocities can, therefore, be determined by Eq. (19-16). When the height of surges 1" small relative to the depth gf flow, (19-19) applies. Thus, Eqs. (19-30) and (19-31) may be written (19-34)

(19-35)

and

where c and c' are the celerities of the right-side and len-side suI'ges, respectl vely.

#

_ (Vi -'- V 2 P

By geometric continuity,

By hydraulic continuity,

The positive sign applies to the right-side surge and the negative sign to the left-side surge: C. Surge Cro:>:>ing a Step. In surge analysis [. step on the channel' floor may be real or fictitious. The fictiFricUorl; tious step is, usually assumed to simulate borrier the effect of channel slope.. The lliethod I: Pro ..'d ed for of analysis, however, is the same in both {rlc'ion effect ! cases. .......--_......:\l.,I-,t~ In previous discussions, the slQpe of T! cha.nnels has been assumed small. For Y, ~VI I V;_ y; channels of large slope, the ana.lytlis of surge moyement will be illustrated by l:-3'//?/)'T.r7I'/7/ Example 19-2. In simplified calculations, Ne9ligible . {oj difference in the effect of slope ma.y be treated by conveloeily head j.J verting the slope into a series of steps. Vw~ In this way the c,lmnnel length is divided t-;~ J il.· ,,'t ~. into a number of reache.3. In ea.ch reach, I VZ_Y.ZI'y'-.+-V Y' the fioor is considered horiwntal, and the ~' - VI ! .( v; __ ( actual drop in floor elevation due to slope .:h7?777d771d i I effect. is represented by a step (Fig. 19-6) f!);'7//7T/?/hzhw: placed either in the middle or at one end IO! of the reach. The height F of'the step is FIG. 10-6. Arrival of surge at a equal to the slope of the' channel multistep rmd friction barrier. (a) Before the arriva.l; (b 1 after the plied by the length of the reach. This simplified calculation is usually used in arrivaL. ' analyses in which the effect of channel friction is also considered (Art. 19-5). When a surge arrives :J.t a step, the result is to produce two new surges, one traveling upstream and the at-her downstream. Figure 19-6 illustrates the an'ivaI of a surge at the step and the condition immediately after arrival. Applying Eq. (19-17) to the left-side Ilew surge,

I

563

RAPIDLY Y ARIEl) . UNSTEA.DY FLOW

i

UNSTEADY FLOW

562

Example HI-1. A steady flow s.t 6.80 fps and 42.90 It deep is suddenly stopped at the downstream end of the 9,700 ft reach oJ a. frictionless rectangular channel. Determine the surge thus produced. Solulion. A reflected suq;e advancing upstrea.m is produced. The velocity at the downstream end evidently cha.nges from V I = 6.80 fps to V. = O. By. Eq. (19-17) or (19-20), with Yl = 42.90 ft, the depth or the oovallcing surge y" CllJl be computed as 51.09 f t . · ' , Note that Eq. (W-I7) or (19-20) must be solved by trialllnd error. A simplified procedure is to use Eq. (18-46) together with Eq. (19-24). Thus, Eq. (I8-4cG) g1\'e;; c = V32.2 X 42.g 37.2 ips, and then Eq. (19-24) gives h = :17.2 X G.80/32.2 = 7.85 it, which makes 1b = 42.90 + 7.85 = 50.75 it. This value may be used as the best estimate /Dr the trial-a.nd-error solution oJ Eq. (l9-17) or (19-20). By Eq. (19-10), the wave velocity V..; 35.6 fps. The time requir~d for tb.e wavefront to arrive at the upstrea.m end of the reach is 9,700/35.6 272 sep_ The celerity is 35.6 + 6.8 ... 42.4 Ips. Example 19-2. A gradually va.ried steady flow is decreased a.t the lower end of an inclined rectangular channel by the sudden parti'a.l closing of a gate. Describe!!. method for determining the. surge thus produced (also. see Art. 19-5)_ " Solulian. Referring to Fig. 19-7, the channel has a slope S. and carries a steady discharge Q. The depth y and velocity V of the steady flow at any. section, say nn, can be computed. . For the determination of the surge due to sudden decrease 01 flow at section '00, a step computation may be eI)1ployed.' The channel is divided in~· 9.- number. of reaches, each ha.ving a length equal to t.:r.. The computation may start at section 00 and. then proceed by steps from reach to rea.ch. Consider the step computation from section n'n' to section nn. The flow condition at section 11,'.,.' is assumed to ha ve been determined, and it is now requifed to determine the l3urge height h. and velocity at section nn '

(19-30)

(19-17) to the right-side new surge, . (19-31)

.v...

I i

! I;

)

r

1

'~

1-

554

r

I

I (1~37j

where 11 is the depth iUld l' is the velocity of flow at section nn. If the surge height is re\ll.Uvely small, say h.;fi, a.I!I, the above eqtnltion may be replaced by , : (19-38)

The time required for the smgc to move from section n'1l."to nn. mILY be <:lxpressed as (I(l-39)

where

(19~40)

) \

Let W represent the sh£.,ded !1.re~1 (Fig. 19-7), \vhich is

a. measure of the increase in

~~~~ZXZ'<~~lola' I I

I

,I

FiG. 19-7, Positive surge moving ,tpstream in an inclined channel.

Thus,

, W '" t;Q AI

(19-4.1)

where t.Q is th~ reduction i'[j discha.rge clue to partial closing of the grlte. El i~linating 1'.. and' At the abo"e thre~ equations, the velocity of surge may be written '

1:1 ~he CQmp~ta.tion, first aSSume h. ,assnm\ug

!I.

2

A~ Ax

_ l',/

(19-42)

he

I

I, II i

t

of

(7.00

By Ee:. (19-32), lh' (19-31) gives

= y,

V:)2

=

+ 2.3&.

(41.65

")' (41.65 + !{,Jg y83.30yz

Thus, for the surge traveling downstream, ,Eq.

(48 71 )2 (53.47 + Yl)g (0 - Y , ')'- y", 102.l8(YJ + 2.38)

Ys!l~ =

I

I

chanl\el storage due to advance of the S1lrge in time Ilt,

I

Then, compute the values of Yv by Eqs. (19-31) and (19-42), respectjve\y. If the two values of V .. so computed do not agree,assume another 11. and repeat the computation, until the two values of V", agree ~nd hence the correct values Ii. and V'" Me obtained. To start Lhe step computation at section 00, Eq,. (19-36) may be used to replace Eq. (19-42), since in Eq.(19-36} the va.lue of AQn at section 00 is equal to IlQ. Example 19-3. A step with F= 2.38 It is located at the upstrea.m end of the reach considered in Example 19-1. Determine the flow condition immediately after the: sru.ge arrives at the barrier. The flow upstream from the reach llUder consideration has fl, sk;ady velocity of 7.00 fps'in the downstream direction shd a depth of 41.65 ft. Solt,tUm. At the step the sur'ge produced in Example 19-1 is spli't into two com: ponent surges, one tra.veling fa,rther upstream and. the other, II- fe-fleeted surge, travel,ing dowllstream. For the surge tra.veling upstream, Eq. (19-30) gi\'es

(19-44)

By continuity of fiOlY, Eq. (Hi-33) gives

~

V,,_

565

ILIi.PIDLY VARIED UNSTEADY FLOW

The dischn:rge t\lrotlgh the surge front is

-y

1

. i

"

UNSTllJADy FLOW

wllere b is the width of the channel. By Eq. (Hl-15), using Eq. (18-45) for c,

)

'l,

Th~s, W may determined by geometry, horu;ontal water surface in the channel after the passing' of the surge.

V,'(Yt

+ 2.38)

(19-45)

Equtl.tions (19-43) to (19-45} may be solved simultaneously by trial and error. Assumin[ 'V, = 0.52 fps, Eq. (19-43) gives Y2 '.19.34 ft a.nd Eq. (l9-44) gives V,' =: 0.50 'ips. Equation (19-45) serves as 3. check. Thus, 0.52 X 411.34 "" 0.50 X 51.72. If Eq. (19-45) is not satisfied, other vruues of V, should be tried and the computation repeated until the condition of flow continuity a.t the step is lIVe!l" established. It $holl1d be noted that, in this ex"mple, the water-surface elevations on the two sides of :the step are not equa.lllnder steady-flow conditio,ns; the difference is 41.65 + 2.38 42.90 1.13 ft. This difference is due to the friction effect and can be treated in combinatio!l with the slope effect in surge analyses (see Art. 19-5). The effect of friction in a rllILch may be assumed to be concentrated at a. fictitious element knowu as the friction bGJ'1'ier. This element ma.y be positioned conveniently a.t the section where a. ~tep is provided for the slope effect. On both sides of the barrier the water surfa.ces are kept horizonta.l, but the depths of flow are not equa.\ j eflch represents the avera.ge dep~h of s~ady flow in its reach. The depth of steady flow may be determined by a lIow-profilecomputatioll on the basis of,a. given channel roughnlll!s. When the surge arrives at the barrier, thefailter water on one side of the barrier will bump into the slower water on the other side. The result is to shoot up a. little water spout, which immediately separates tho flo,,, into two new surges, traveling in reversed directions. At, the barrier there rema.ins ;. little standing wave. rrhi.~ accounts for the diffe'reI)ce between the two velocity, heads that now exist on either side of the barrier. In, general, this difference is so small that it ca.n be' neglected. ' Example 1'\1-4. Derive a.n showing the approximate a~nount of energy of 0. surge. SolutlilJn. Consider an element of surge h~Ying length in the direction of the channel'equal to unity; the potential of the surge is equal to the work done by Efting lip the mass of water wkT to a height h/2, or

a

(19-46)

666

KE.

=

wV 'YT' •2 g

height~

(19-47)

where y is the depth of water snd V is the velodty of flow. By Eq. (1,9-23) or Eq. (HI-19), !liS the case may be, the shove equation, may be reduced to '

l{.E. = 'Ulh'gi(£ 2c' Assuming

R

(19-48)

surge of small height,

of y. By the momentum principle, the corresponding change in hydrostatic pressure should be equal to the force required to change the momentum of the vertical element between y and y + dy. Considering a unit width of the channel and assuming PI = /32 = 1, W

Z y2

W

2'

-

W

(y

.

+ dy)2 = g (y + }f dy) (V + V ...l dV .

dy

. V =

The total energy of surge per unitJength is, therefore, =

P.E,

+ ILE.

=

wh'T

(19-50)

19-4:. Negative Surges. Negative surges are. not stable in form, because the upper portions of the wa.ve travel faster than the lower portions (Art. 19-1). If the initial profile of, the surge is assumed to have a

r Y2

Assumed profile ofter 2.1 sec -

-:,,---

Assumed profile one: t sec

~,-

.

Assump.d surge profile immed,otely

dy 1 V2

~ [

' ;

Y

l:!

VW1

-<--'

I I...

, FIG.

-,

VI-

YI

)//.1,/'//'yhhw/7/,//?//mJ/?,.;;:v/.?7

19-9, Analysis of a negfl.t,ive surge.

one on top of the other. The velocity of the small wave a.t tho point under consideration may be expressed ar'.()ording to Eq. (19-11) as Vw FIG,

19-8. Propagation of

II.

llegative surge due tr, sudden lift of a sluice gate.

=

vgy -

V

(19-53)

Similarly, the velocity at the wave crest is

steep front, it will soon flatten out as the surge moves through the channel (Fig. 19-8). If the height of the surge is modemte or small compared with the depth of flow, the equations derived for a positive surge can be applied to determine approximatdy the propagation of the negative surge. If the height of the surge is relatively large, a more elaborate analysis is necessary as follows: Figure'19-9 shows a type D surge (Fig. 19-2) of relatively large height, retreating in an upstl'eam direction. The surge is caused by the sudden lifting of a sluice gate. The wave velocity of the surge actually varies from point to point. For example, V .. is the wave velocity at a point on the surface of the wave where the depth is y and the velocity of Jlow through the section is V. During a time interval dt, the change in y is dy. The valu~ of dy is positive for an in,crease of y and negative for a decrease

...... !'--: .. ; .. -.

I

. I

.1 w

. ,?X?i/,-//?7-0?V/.?7/,T/.//.//?

ofter creation

-- -=- - - --

~

.

",

j

t

f

, f

___ ~w~_

---

I

- r

(19-52)

. g

As described previously (Art.· Hl-l), the whole wavefront ~all be assumed to be made up of a large number of very small waves placed -=-

J

V

~~--=-.dV

-

and the above equation becomes (19-49)

(19-51)

Simplifying the above equation and neglecting the differeritial terms of higher order,

c=-../iY

E

567

RAPIDLY VARIJ
UNSTEADY FLOW

where w is the UlliL weight of water, T is the top width, and h is the surge The kinetic energy of the element is evidently equal to '

V .. 2

=

...}gY2 - Vz

(19-54)

and, at the wave trough, (19-55)

When the surge is not ,too high, a straight-line relation between V"'l and V",2 may be assumed. Thus, the mean velocity of the wave may be considered to be . (19-56) Now, eliminating V", between Eqs. (19-52) and (19-53),

!!JL --/y

dV,

- ...}Y

(19-57)

569 RAPIDLY VAlUED UNSTEADY FLOW

568 .

. a ower canal This information is result of iL sudden rejection of load l~ f P stabli6hi~1g the height of wall the design of the cana or e . d'n requrre 1 nece6So,ry to prevent overflow. d" f teady flow hl a power canaL nputed, When the load Figure 19-11a shoWS ,th~ C?n,ltlOn a ~ The flow profile and the fnctlOll loss can e COl

UNSTEADY FLOW

Integrating this equaiion from Y2 to Y and from V 2 to V, and solving . for V, v (19-58) From Eq. (19-53), tr", = 3 -0iY - 2 vgy; -V2 (19-59)

Velocity ----2.r .heod

Thus, the wave velocity at the trough of the w:w.e is

",

V",t

i I

= :;

.ygy-; -

2 'VgY2 - V 2

\

A

I.

=

(Vw2 - V~'I)t

V

Profile of the origiMI steod Y flow

/

. -

--...

.

I

I ;

~

Iplont

- .../~

,

tal

.

---~-~--h-l-e-

v.+

-------jd

I

)(

E:tample 19-5. Show that the equation of the wa.ve profile, reSUlting from the failure of. a drun is in the form of

(

x = 2t

\

v'iiii. -

31

[bj

I

~cenler line

I, )

I I

/

I

,

/

1 I

/.

tel

of

'I ::.:-

I

.

,Rolf wove ...,. V.... i =2.,..I9Y2,

·

'19-5. Surge in Power Canals. Engineers are interested in the determination of the maximum stage of water that could be developed. 8J.I a.

_--.=-....I

---=-=- --:-1l ' I

I

I

l'

Maximum .tage 01 th'l rejection

\1·U,Q9

!

~~ ~ (;;""

19-10. Wa.ve profile due to dam failure.

Equation (19·(i2) represents a. par~bQla with:vertical axis and vertex on the' cha.nnel bottom, as shown in Fig. 19-10. At the site of the dam, x = 0 and the depthy. 4y./9, Qwing to the channel friction, the actual profile takes the form iudiCl!.ted by the dashed line. . This profile has a rounded ,front at the downstream end, forming a bore (see Example 18-1). At the upstream end, the theoretical profile thus developed has been checked satisfMtorlly with experiments by Schoklitsch[121. .

r--------~

--

~

l"XG.

V=0-+1 /

Surge profile when tne w'!ve front reoches the ,.servo,.:!

~-l.v

'it:,

I

Ihe dam

L.-!I ":..

Jm

Vmax--""'"

J

.....

Yz

Sur,)" prl)f lie wnen the wo'!e front reaches the reservolf

I

-vgy

v.=o

I

-vIiii .

where::t: is the distance measured from the da.m site, y is the depth of the wa.ve profile, lit is the depth of the impounding wa.ter, and t is the time a.fter the dam broke. Solu/ian. SiuGe the impoundiug wa.ter velocity, or V. = 0,. the. wave velocity by .Eq. (19-59) is Y", = 3 2 Since V 18 is in the negative direction of z, x = - V"I, which gives Eq. (19-62).

I

.

;C~:.=lW/&~~

---..

of

To power

.......Jr,..

~;

(19-61) ,

analysis oan be applied sin:ilarly to a ne2:atlV'e

The . type C.

I

Reservoir.

(19-60)

Let l be the time elapsed since the surge was created or, in thif! easel since the sluice gat.e 'was opened. At t = 0, the wavelength A O. After t sec, the wavelength is equal to

, \

.

\ !

FIG.

19-11'. Development of rejection

~urge in friction!!.I. channel.

. . . ' on sur' e advancing upstream is produced, : is suddenly thrown off, a reJectl d' g t sual observations, the water as shown in Fig. 19~1lb. Aceol' 0 +~s approximately level. Thus, surface ab downstream from the wave ro~" the water surface throughout when the wavefront reaches the re(s;:volf ). However, a steadily 11C the entire canal becomes level Ig.

m1.

i9_

570

UNSTEADY JrLOW

571

RAPIDLY VARIED UNSTEADY FLOW

increasing volume of water must pass through the wavefront to fill the space corresponding to abc. Because of this action and the effect of friction, the wavefront is being greatly reduced in height and velocity on its way to the reservoir. Furthermore, as the surge reaches the reservoir, the velocity of ft.ow in the canal must show a progressive increase from a maximum at the reservoir to zero at the downstream end where there is no outlet for relief. As a result, water will be built up to a maximum stage at the downstream end, as shown in Fig. 19-11d, sometime soon after the arrival of the surge front at the resenroir. For the determination of the maximum stage produced by the rejection surge,l Johnson [14,15J has applied an analytical method 2 to the design of the supply canal of the Queenston-Chippewa plant in the vicinity of Niagara Falls . . In this example (Fig, 19-12a), the entire length of the canal is divided into four rea.ches each 9,700 ft long. At the very beginning, t = 0, the flow is steady, and the flow proftle can be computed. The drop in water-surface elevation in each reach is indicated at the division section between reaches, for exampte, 1.13 ft for reach 4in }I'ig. 19-12b (see Example 19-3). The water surface in each reach is assumed leveL Note that the friction in each reach at the steadyflow condition is considered in the computation of the flow profile. The slope effect of the channel is represented by a drop on the channel floor at the division sec Lion between the reaches, for example, 2.38 ft for reach 4 (Art. 19-30). The water depths ill feet are represented by the figures wriHen vertically. The velocities in feet per second are r.epresented by the figures written horizontally. In reach 4, for example,. the depth is 42.90 ft and the velocity is 6.80 fps. When the load is suddenly thrown off, a surge is developed at the dowllstream end Qf the canal (Fig. 19-12c). The computation of this initial surge has been shown in Example 19-1. After 272 sec have passed since the surge was produced, the surge arrives at the barrier at the upstream end of reach 4 (}I'ig. 19-12d). The water depth throughout reach 4 is 51.09 ft; and the velocity of flow is zero. At the barrier, the surge is soon split int.o two component surges, one traveling upstream and the. other downstream. The computation for these surges has been given in Example 19-3. The component surge traveling downstream will soon reach the downstream end of the canaL After it reaches the end, it will be reflected. The reflected surge will travel upstream again. The component surge traveling upstream will soon re:ach another friction barrier farther upstream. At the barrier it will split into two components, one traveling farther upstream and one traveling down-

stream. When one surge traveling tlpstream meets another surge traveling downstream,. two new surges will emerge. The, computl).tion for the new surges has heen described in Art. 19-3B. The surges will continue to tra vel, split, and emerge. The computation

"'"

-

-

......

-1'>1I

~.

~7);/J) /777.1»)17) 77))..'7"I/?;P>?/./ 1I'l//7777//Ml/mmr/.J//?7 I f.-- 9,700' 9,700'-----4---- 9,700'---+J..-:.- 9,700'-----1 (al I", ·~8.aoo' I

.1.

ci'"

1'1

~

I""l

;-,--fr~-~-'--IF----~-· --ir=-----~-~r . ~I

1 I,

(b J

-.

I"'l

~I

~I

'

' ';':.;;''

I

V~7.4~ __ '::~_0_....:N"",!~_7_.0_0_ _N:,;.,,;..,._ _6_.8_0_ _ -l

'"

I

::.11

';1

'" oon

I

"'I "!,

...,

;;;

0

<:$

l-

; ,.., '"

a

ao

'"ci


".

-

~!,

ci

~I

N 7.00 '!l 0 1 --~--'''''~I~.---~~-~--~,--~~--~

7.20

"'I. .n

27<' sec

.,.

.oj

.....

C

"' ci

ci

.-

., '",..; "! ,.., .,. 0

N 0

"'

ci

Velocify

10

oJ

_....,

.D

_.

recoIJe ry

/1~ .1. -1

f

'"IX! "!'" '" 0 <1> ...'" '"ci r-. .; on, N .,.,..; "j "': " on " '" "' ,.., ., I .f--I.. "!! ..j... ,,!I I -1..... "11 1 '" I~) \...-,1 I ":1." ,I, "'I. ,I. "J......::!i.-----=---->i 4.SZ45,087 5.11.0 4.899 5,{24 4.545 4.350 4.3710.<40· 0.230 u1

oJ

oJ

on

S·~

0

~

N

.

.

7. 65.mox,. rrH I,B51

sec

III

FIG. 19-12. Analysis ";f reject.iQ:J.surge by the Johnson method~

will {ollow' accordingly until th8 maximum stage is obtained l1.t the downstream end of the canaL In the example given by Johnson, the maximum stage is found to be 7.65 ft above the reservoir pool level (Fig .. 19-12e), This will be reached 1,851 sec after the load is rejected. It is apparent that the Johnson method is very laborious. The proc.edute becomes very involved as the computation conHnues, for numerOWl surges will be produced an.d propagated. The surge in a canal can

See [131 for another method. % ·lohnson gave credit to P, Wahlman for this method,

1

I

1 \

-

\

572

l I

U~STEApy

FLOW

RAPIDLY YARlED UNSTEADY FLOW

also be studied by hydraulic models' and electronic computers, which are sometimes found much eltsier though more expensive. The analysi8 of. surge is important also in the design of ventilated taill'ace tunnels for underground hydropower stations. In E'Uch cases, demand surge is usually developed as n result of asuddell increase in the lond output [16,17]. Rejection surges ~n powel: c!111f1.ls can be controlled by two commonly used deviee::;; longitudinal weirs along the canal and tiiting gates on top of the spillway, either aut.omatic or turhine-opemted; The theory for longitudinal side weirs hM been developed by Drioli [18], Schmidt [191, and Citrini [20,21J. Citril1i's theory is based on the method of characteristics and hM been verified satisfactorily by GBlltilini [22], Pel1~ti [23], and De M(1.\,chi De Marchi hm;·!\.lso reported experimeuts on the use of the tilting gn.te as a control device. For a comprehensive discllssion of un~teady flow, the reader should refe!' to the trenti~e by Frank [20J. 19-6. Surge .1n Navigation Canals. Surges may occur in navigation canals as a· result of lock operations, tmffic disturbrt.nces, tidal a.ctions, or other causes. Under severe conditions they will affect the canal tl·n.ffic critically either by depressing the wll.ter level and thereby reducing the effective channel depth 01' by raising the water level and thus encroaching upon the freeboard. They will also impose sudden impact load upon vessels, cD.lml structures, and mn.chinery operating miter gates} In 1110st lHIVign.tiol1 canals, ,')urges are small and insigIiifical1t. Neverthele~s, some consideration of the surge problem is desirable in can..l desigll, particularly if the traffic is to be heavy and if .the canal is . long and controlled high head In this article, surges due to Surges due to traffic disturbances lock operatio!ls are mainly are e~~cluded, fol' they are significant only for the consideration of vessel power requirements. due to tidal actions a.re beyond the scope of this book, since they cOllstitute an important topic il1 tidal hydraulics. Surges may (1.180 OCCIII' in lock chambe~'s; this, however, is usually .regarded as a special problem in lock design [15, pp. 1I:JO-213J. There are several methods of controlling surges in navigation canals among them the following:· . , 1. Increasing equalization time. surge height is, in gmiel'al, propOl,tional to its discharge rate 01' inversely proportional to the time of lock: equalization. Thus, the surge height can be reduced to any desired degree by nppropri,tte increase in ~qualization time. In all existing cnnfll, this cp.n be done by throttling the inlet ports, and, in a new design, :

:

I

l For !tn analysi:; of surges deve]opedby partially or fully opening a !:niter gate, see [27]. .

573

by providing inlet ports or low capacity. This method is simple' and inexpensive but hss'the disadvantage of delaying the traffic. , 2. Increasing channel dimensions. . An increase in channel depth WIll iricretl.se the celerity and reduce the surge height. An increase in channel width will decrease the surge height without change in celerity. In either case, the. velucity of flow and hence the collision ha,7iard are reduced. 3. Lateral reservoirs. These reservoirs are connected to the lock as well as to the canal with control valves, and are installed at the upstream as well as the downstream lateral sides of the lock. In filling. the lock, water is taken' in from the upstream reservoir, which is later to be filled tip ,'gnin to its original level by ta.king in wa.ter slowly from the upstream cll.nat In emptying the lock, water is released to the downstream reservoir, which is then to be emptied down to its original level by releasing water gradually to the d')WJ1strcam canal. In essence, the effect of the hiaral reservoirs is to reduce the time of lock equalization without developing severe surges in Lila canal. . . 4. Channel expansions. Sudden. expansions are provided in the callal 011 the ,upstream &.ud downstream sides of the lock. Such exp,1.l1siollS will produce negative reflections,which can cancel the traveling Burges partially. , The method of analysis of surges in navigatioq canals can best be illustrated by a numerical example} In navigation canals: surges of large height cannot be tolerated nor can the occurrence of high velocities. Thus the analysis can be greatly simplified 1n many pI';Jblems by assuming possible surges of height aua flow of negligible velocity effects. Since the generation, propagation, and reflect. ion of surges are essentially dynllmic and gravit!1.tion111 phenomena, the liiurges can also be studied by model tests based on the Froude In.w. . . Example 19-6. A surge is prod~ced in a navigation canal by em{ltying the lock chamber. The discharge out of the chamber is shown 8.5 iJ. function of time in Fig. 19-13£1. The equaliza.tion time is 11) min. The canal has a l·~tangular section 200 It wide and'a mean wa.ter depth of 15 ft. The slope of tbe canal is assumed horizontal. Determine the surge profile. . . Solution. The computation for the determination of the Burge profile is shown in Table 19-1. In the analysis, t.he discharge is assumed to be reieased at inter'/a\s of time. In each interval a surge of small height is produced and will travel downstream at a, speed computed by.the formulas derived in t,he previous articles. Thus, the resulting surge configuration is formed by acoumulating the fronts of the incremental surges. The columna in the table lire explained as follows: . CoL 1. Time in see since the discharge sta.rted. The interva.l of time is determined arbitrarily. accurate determination, a small interval should be used. , CoL 2. Discharge in cis per ft-of channel width, equal to the total discharge show,n by the hydrograph (Fig; 19-130) at the corresponding time divided by 2QO It "

! '

,

,

i

1 Simllar methods with further approximations are used by tIle U.S. Army Corps 'fJf Engineem [28]. : .

574 Col. 3.

Initial velocity of flow in fps before the incremen.tal surge arrives, equal to

V. (col. 4) of the previous step Col. 4. Final v(~l()city of flow in fps after tbe incremental passes,
maximum height of the surge is found to be about 2 ft, which is'generally required in
<

TABLE

19-1.

COMPUTATION OF SURGE

PROFILE

L

o

o 0 a 15.00 12.5 0 0.83 15.00 25.0 0.83 1.(10 15.57 37.5 1.50 2.33 16.11 50.0 2<33 3.01 16.63 46.9 3.01 2.74 17.12 43.8 2.74 2.5916.92 <40.6 1 2.592.4216.81 37.52.422.2416.69 34.4 2.24 2.08 16.56 31.3 2.08 1.90 16.45 28.1 1.90 1.72 16.32 25.0 1.72 J .55 16_19 18.8 1.55 1.17 16.07 12<5 L17 0.79 15.80 6.3079 0.41 15.53 0.411015<17 0

____~ 1 __~__~.....

Col. 5. Col. 6.

_~

o 600 0 15.00 22.0 22.0 0_57 I 570 0.57 1I:L57 22.4 22.4 0.54 15.11 22<8 23.6 1.11 540 0 52 16<63 23.2 ~4.8 1.63 510 2.12 480 0.4917.12 23<5 25.8 -0.20 16.92 23_3 26.3 1. 92 450 --0.11 16.8123.326,0 1. 81 420 -0.12 Hi,fi9 23_225.8 1.69 390 -0.1315.5623.1 1 25.5 1. 56 360 -0.11 16.45 23.0,'25.2 1.45 330 -0.13 16.02 2:L'9 25.0 1. 32 300 -0.13 16
__~_ _ _~

113,200 12,800 12,700 12,500 12,400 11 ,800 10,900 10,100 9,200 8,300 7,500 6,700 5;900 4,400

.L-__~__

'"' .,

8.000

!r

6.000

~~

(12)

(11)

12.000

II

10,000

IN A NAVIGA'l'ION CANAL

.l:'!

30 60 90 120 150 180 210 240 270 300 330 360 420 480 540 600

575

RAPIDLY VARIED UNSTEADY FLOW

UNSTEADY FLOW

'5U>

0

I . ,

J

4POO

·1

i 4

6 5 Time i., min

{a)

.!: 2

I' - -

I

I

-- -.J...--

o

V o ~~ o

I

I

2,800 1,400

Initial depth in ft, equal to the final depth y, of the previous step Height of the incrementa.! surge in ft, computed by Eq. (19-19) or

j

i

-r-~

!

<-<

5,000

......

;---

I

------

-.:;;..;;-

,

-T

j

\

Ii

}.

.la, 00 0

D1stom:e in ft

j <'

lb)

(19-63) As an approxima.tion, tlie value c (col. 8} of the previous step is used in this equation. Aiter l = 150 sec, V, becomes less tha.n V 2 ; tha.·t 15, the nega.tive incremen.tal surge occurs. Thus, the computed his TI".''''.I''''_ Col. 7. Final depth of flow in ft, equal to the iniLi"l depth in col. 5 plus the height of the incrementa! surge in col. 6 Co\. 8. Celerity in Ips, co}nputed by Eq. (19-9) Col. 9. Absolute velocity of surge in fps, computed by Eq. (19-16) Col. 10. Cumulative. va.lue of the height of in.cremimtal5urge (coL 5) Col. 11. Travel time in sec until the discharge stops, equal to the equa.li~ation time (600 sec) minus the time in col. 1 since the discharge started Col. 12. Trovel in ft for ea.ch increment.aLsurge until the discharge stops, equal to the product of the time in col. 11 and the absolute velocity in col. 9 Plotting the travel distance against the cumulative height of the incremental the whole configuration of the surge is shown in Fig. 19-13b. This is in the downstrea.m caDal after a period equal to the equalizatiou time is

FIG. HI-IS. Analysis of surge in. a navigation eanllJ. profile.

(.. ) Hydrogra.ph; (b) surge

19-7. Surge through Channel Transitions. When a surge arrives at a channel transition, it will usually split into two component surges, Oile traveling forward and the other backwB.rd. A typical case of a positive surge passillg through a channel contraction has been described in Art. Hl-3C. In this articie, however, more general but simplified cases are presented. Figure 19-14 shows four cases in which the positive and negative surges are passing through channel contractions and expansions. In each case an incoming surge of height hI and celerity Cl is split into !l. forward surg~ of height h2 and celerity C2 and a backward or refiecte~ surge of height hs and celerity Ca. For simplicity, the sur-gas are assumed of small

.(

j I

i! 576

I

I

UNSTEA.DY FLOW

T Ih1Cl = T 2h 2 c2

+ T 3haca

(19-64)

where Til T z, and Ta are top widths of the surges. '

I

Solving Eqs. (19-66) and (19-68) for h2 and

h", (19-69) (19-70)

and

The ratio of the reflected height to the initial height of a surge is kn.own a~ the reflection coefficient (Jr_ Letting T2 = nT 1 and using Eq. (19-71)

1I I

,

As a further approximation, 1:1

I ?WT/T/ffffO?7il , ( oj

7-1"77. .7;;'-:7,///;;.»??/~

Ii>l

fa?WJ;·//.i7//h//Y"';;'l ~Q...:''''a'7?/rr:/.''""z'''''/;;'''''/$,

'

'rhe relationship between

n n n

<1

11.

>1

By the principle of conservation of energy, the total amount of energy rema.ins the same after the split of the incoming surge. With the aid of ' (19-50), the relationship of energy among the surges during a common time interval may be written as follows: (19-65)

For the case of a positive surge passing through a contraction (Fig. Cl "'" Ca and TI = Ta. Thus, Eqs. (19-64) and (19-65) become

19-14a),

-

hS)Ci = T 2h2c2 h3 2)Cl = T2hz 2C2

(19-66) (19-67)

From Eqs. (19-66) and (Hl-57) or by geometry, (19-68)

11

and C is shown below:

Reflection coefficient

Reflection

Cr 1 Positive Partial positive Or > 0 None Or - 0 Partial negative Or < 0 -1 Negative Or

0

1

n= «>

FIG. 19-14. Surge through channel transitions. (a) surge through n. ~on­ tro:ction; (0) positive surge through all enlll.rgemllnt; (0) negative surge through n contraction; (d) negative surge through an enlargement.

T1(h i TI(h 12

c.; so (19-72)

(el

r

""

"

Width rfl.tio

J

577

RA.PIDLY VARIED UNSTEADY FLOW

height, so th
Chaullel condition A dead-end barrier Contraction No tmnsition Expansion Entrance to a lake or sea

In the case of positive reflection, a positive surge is reflected as, a positive surge and a negative surge as e. negative surge. In the case of negative reflection, a positive surge is reflected as a negative surge, and vice versa. Example 19-7. III a navigation canal, forebays are const:ucted betw'een the lO,ck and thecanala in order to provide adequate, Il.Ct.'OtnIDodo.hon, for the traffic (FIg., 19-15). The depth of wat.er in the downstream forebay and canal is 9 ft. The widt~a of the iorebay and caDal are 300 ft and 150 ft, res:pectively. Determine the a.pproxlmate flow condition when a surge, produced by a sudden relense of 1,800 cis of water from the lock, arriveB 'a.t, the transition between the for-sbay and the ,ca.nal. Solu.tion. 'I'he initial surge has a celerity approxima.tely equal to c. 17.0 fps and a height hI 0= ilQ,/c,T, = 1,800/(17.0 X 300) .'"' 0.353 ft. 'I'?e. c~lerity in .the canal is-approximately,equal to, t., since the depth 13 the same as It IS ill the forebllY. By Eq. (19-69), .

....;gg-

h2

'""

L,

""~~'---"

ThuB, h. h. h. - O.11B ft. By Eq. (19-71), Or 0.333.

=

.2 X 300 X ?OO + 150

0.353 "

= 0.471 it

The reflectioll coefficient is 0.118/0.353 .: 0.334.

578

UNST'EADY FLOW

Forebay

579

RAPIDLY VARIED UNSTEADY FLOW

The disc}targe entering the canal is I:1Q, =c,T.h. = 17.0 X 150 X 0.471 = 1;200 ds . . The reflected discharg;: is D.Q, = 1,800 - 1,200 = 600 ds. .--_F_o_re_b_o-'-y_ _ _ _ _ _ __

-.-~--~---+I----~~--..:~S-j~:-n~-c~ FIG. 19-15. Analysis of surge through the trunsitio!l bet·ween forebay and canal. J. Kozeny [31J.)

the two adjoining channels 'with celerities (:2 and /;3, respectively. Assuming that surge heights are small compared with the depth of water ill the chaunels, thB celerities can be computed by Eq. (18-'16). In the meantime, a reflected surge will travel along channel I at a celerity hpproximately equal to Cl. Figure 19-16b shows the condition at the junction immediately after the arrival of the incoming surge.

(Afler

_I II

v31r-c,

h

Now, if the tmnsition between the callul and the lock is assumed gndual, as shown in Fig. 19-15, the computation can be mude by letting h. = 11., dh, and c.T, = cIT, d(c,T,). Accordingly, Eq. (19-69) may be written

+

+

Ch·~nnel

II

V4

-f=

Woter orea=A2

·

V3=O· h3=O

m

Channel Weier arHo=A,

CI

(10-73)

hi

= const,

Vc';;r.)

= canst

(19-75)

From the above two equations, (19-76)

(b)

FIG. 19-16. SUrge at channel junction. ·(a) Before arrival of surge; (b) ufter B.i:rival. of surge.

Since the surge heights are assumed small compared with the depths of flow in the channelS, Eqs.(19-1 9) and (19-2'1) can be applied. Thus, the following equations ca.n be written for channels I, II, and III, respectively:

With T,lT. = 2, the above equation gives hol"l = 1.414_ . Thus. h, = 1.41411., = 0.499 It and h, = 0.146 ft. The corresponding C, = 0.413, I:1Q, = 1,~70 cis, nnd t.Q, = 530 cis. This p.xample illustrates t.he application of the theory of surges to the hydraulic st.udy of ship lock and forebay. Such applications have been described by many hydrnulicians, among t,hem Dantscher [29,30] and !{o~eny [31J.

19-8. Surge at Channel Junctions. When a surge arrives at a channel junction, it will split into· several surges, each entering· a connecting channeL A simple case, shown in Fig. 19-16, may be used to explain the phenomenon. . . Figure 19-16a shows that a surge in channel I is approaching the junction of three rectangular channels. The surge has a height hi and celerity CI.· Channel I has a water area A I and depth of fiowYI. Since there are ~o surges in channels II and III, h2 = hs = O. The initial velocities in the channels are taken as zero merely for the sake of simplicity, although any velocities could be assumed. When the incoming surge reaches. the· junction, its height is reduced because of the increased surface area presented to the surge by the two additional channels. Surges of reduced height h will then travel through

Chonnel I Woler area = Al

(a)

(19-74)

Similarly, it can be shoi>l'n that log (h,

~

Channel [ Woler orec=A,

Int,egrating and simplifying the above equation, log (h, vC;:r;.)

-1

For channel I

hi - h = ~ (17 4 - V 1)

(19-77)

.~

V1

(19-78)

h = ~ V2

(19-79)

h ..:.. -

(19-80)

g

. For channel I

hi

=

9

For channel II

9

For channel III

-C3

V3

9

By the law of continuity of flow, it is required that the .quantity of water fioYving into the junction be equal to that flowing out of the junction, or A,V 4

=

A.V•. + A3V3

(19-81J*

Solving the above equations simultaneously .for h, h

=

.

where

~!lAi let.

I = A1 C1

'" V. has a direction opposite to c,.

(19-82)

+ A2 + A3 C2

(:3

(19-83)

-.J

580

UNSTEADY ,FLOW ,

RAPIDLY VARlED UNSTEADY 'FLOW

.

If ::\11 incoming surge of height h2 nppr:oaches the junction from channel II, then it can heshown that (19-84)

Similarly, if fin incoming surge of height h3 approaches the junction from chn.nnel III, then h

(19-85)

By the above principle, it is possible tp analyze surges in a complicated channel network,' Such it methtld has been developed by Swain [32,33] for all approximate det.ermina.tion of surge flowa in an interconnected system of estuarine channels in response to tidal variations. In this method, the friction barrier is assumed to he located at the midc,ile of a chanl1el reach. The surges are assumed to have small heights compared with the depth of flow, so that simplified equations can be applied .. To further expedite the computation, it is desirable to have coincidental arrival of surges at a barrier or junction at some given instant. This is made possible by the use of a procedure that iuvolves a modification of the true length of the channel reaches. The analytic method developed by Swain was a.ctually applied to a problem for the determination of flows in intercollnected eatuo.rine channels in the Delta area of California ea.st of Suisun Bay. The same problem was studied by the U,S. Bureau of Reciamation using ;an elaborate hydraulic model [34-36] and an electronic anriJog model [37J. The results of the SWj:1.in method were checked i satisfactorily .. 1.9-9. Pulsating Flow. "When the slope of a channel becomes very steep, supercritical flow of uniform depth in the channel will brel1k into a train of. traveling wo.ves or pulses, known as pulsating flow. Such an unsteady flow occurs in various situations and in each situation it has specific significance. When it occurs in inclined flumes and spillw:1Ys, the increased height of t.he flow requires additional freeboard to prevent. spillage, and ,the concentrated mo.ss of the wavefrol1ts calls for additional structural safety factors against transient pressures and stresses. In hydraulic models, pr~sence of the flow often interferes with similarity conditions. On farm, lands and unstabilized toads, the high scouring: capacity of the flow causes surface erosions. In chemical processes, the effect of the flow will increase the mass-transfer rate between gas and liquid diffusion reactiqns. ' I. . • There have beeuni.llllerou8 'studies on the phenomenon of pulsating; flow. Most of them/ ihowever, were made printarily in connection with I the investigation of the mechanism of instabilit~ of flow (Art. 8-8). An· analytical and experimental study of pulsati~g flow which has long

581

intere,sted hydraulic engineers Was made by Thomas [38]. In this study pUlsating flow was described as a flow consisting of two parts, a roughly tumbling head and a smooth' tail sectiOll. The results of the study indicate that, for puLSating fio'" to occur in a wide rectangular channel, the channel slope must be more than 4' times the critical slope or the velocity more than twice the critical velocity. Another investigation was by Mayer [39J, 'Who found that pulsating flow can be classified into two distinct types, namely, r~ll waves arid slug flow. According to Mayer, rpll waves are char!1cterized by transverse ridge
582 velocity, that or

UNSTEADY FLOW

when L5V ~ V Vic ~ 2

+c

RAPIDLY VARIED UNSTEADY FLOW

(19-86) n9-87)

Since Vic is equal to the Froude number and since roll waves can be formed only on supercritical slopes) the range of Froude numbei' for the formation of roll waves is 1 & F & 2. Solving (6-43) for S, it can be 8how'n that 8 = 3F2/R. Thus, the range of slope for the formation of roll waves is 3/R :::2 S ~ 12/R. Slug flow can be·formed when the surface velocity is great.er than the wave velocity. Theoretically, the formation of slug flow will occur when F > 2 and S > 12/R. PROBLEMS 19-1. Prove (19-11). 19-2. Prove Eg. (19-15). . 19-3. Assuming the energy loss in thp. moving hydraulic jump t.o be negligible, Koch an'd l27J have derived the following equation ior the wave velooity or.a. surge; (19-88)

where the positive sign is for type.A and C surges and the negative sign is for type B and D SUrges (Fig. 19-2). Verify this equation. . 19-!!. Assuming atl initial depth of 20 ft, an initia.l velocity of 15 fps, a.nd a rectangttlar channel cross section, compute the velocity and overrun of thE'. bore shown in . Fig: 19·3 and estimate the distance trs.vel>ad by the bore in 7 min. 19-5. Compute the height", depths, and wave velocities of the two component surges produced in Example 19-3. . 19-6. A positive surge 0.63 fthigh and 51.72 ft deep is moving ill still water 5l.0a. ft deep with a wave velocity of 41.70 fps toward the daRd end of a channeL Deeermine the height and wave velocity of the reflected S'Irge after the original surge hits the dead end. 19-'1'. Solve Example 19-1 if the original steady flow has a velocity of 10 fps and a depth of SO ft . . 19-8. Solve Example 19-3, if the step has a height of F = 3 ft. 19-9. Two positive surges 3 and 2 ft in height, rc~pe~tively, move in opposite directions toward each other in a frictionless borizontal channel where the water is initially stationary at a depth of 30 ft .. The high surge has a wave velocity of 30 fps and the low surge a wave velocity of 20 fps. Determine the flow condition after the surges meet. . 19-10. Solve Example 19-1 if the chanuel has a slope of 0.01. The initial steady flow is assumed uniform. 19-11. Show that the celerity Iorrn4la in the form of Eq. (18-45) applies only to positive surges and that the corresponding celerity formula. for negative surges is (19-89)

583

.,;gy;,

lS-12. Referring to shotv that the crest velocity V ... = that the trough velocity Vw! thl3.t the velocity of flow through the !lam site is % and that the through the dam site is %7 yg !l,li. 19-13. Contmuet.he oi the example illustrated in Fig. 19-12, Bnd determine the flow conditions a.t 420, 504,552, 743, and 767 sec, \€spectively, after the load was suddenly thrown off. 19-14. Soive Example 19-6 if the equllJi2'ation time is (a) S min and (b) 12 min .. The dil!charge from the lock increase.• linearl)"from 0 to 10,000 cfs for the first 20% of the time and then decrease$ linearly to 0 for the remaining part of the· time. Study the effect due to cbange in eqllalizati'm time. 19-16. Solve EX!l.J11ple 19-7 if the discharge is 1,000 cis. . 19-16. Three horizontal frictionlE'lis channels '}, 6, and S ft wide meet 'at a jnnction. The W!l.ter in the channels has an initicl statiana.ry depth of 10 ft. When an incoming surge 0.4 ft bigh approaches the junction fro~ the narroWI..'St ci]!l.nnel ddermine the flovi condition after the surge enters thl:! junction. Compute the relle~ticn lloeffi. cient of the junction.

vYY.,

REFERENCES

1. J. C. Schonfeld: Theoretioal considerations on an experimental bore, Proceedings of the 8lh General Meeling, Intematicmal Association for Hydm11lic Resear,it, The Hague 1955, vol. I, pp. AlS-1 to AlS-12, H155. 2. T. B. BenjalOin o,lld M. J. Light-hi!!; On .cnoidal waves and bores, P1'oceedillgs, Royal Society of 224, no. 1159, pp. 448-460, July 22, 1954. 3. A. M. mnllie and J. C. Orkney; Experiments on flow of water from !l. reservoir through an open horizontal channel: II, The formation of hydraulic jump, Pl'OcOO!24, No,'ember, 1957. S. Robert E. Horton: Channel waves subject chiefly to mOmenhlfl1 crmtnl, U.S. 80il COn;;er~Gtion. Service, SCS-TP-16, May, 193&. . 6. The JohnstQwn disaster, E1?gineen·ng News, yol. 21, no. 23, pp. 517-518, J,me 8, 1889. 7. A. L. A. Himmehvright: The Johnstown flood, Harper':; vol. lU7, pp. 443-4,55, September, 1933. 8. Richard O'Connor: "Johnstown; The. Day the Dam Broke," J. B, Lippincott Company, Phila.deJphia., 1957. 9. The Heppner dis[l.ster, Enr;{neerin.g News, voL 50, ilo. 3, pp. July 1I'J, 1903. 10, Commission finds failure of St. Francis Dam due to defedive foundaiion, Engi-' "eering NP.1IJR-Record, voL 100, no. 14, pp. 553-S55, Apr. 5, 11)28 . .11. M. D. Chertousov: "Cidmvlika: SpetsiulnYl ICurs" ("Hydraulics: Special Course"), Gosenergoizdat, Moscow, 1957, pp. 437-453. 12. A. Schoklitsch: Uber DambnHlhweUen. (On waves produced by broken dams), Sitzunllsbe"ichte, J1{att.ematisch-naI1ITwisseIl3chajtliche Klasse, Akade1nie der Wissenschaflen in. Wien, v·o1. 126, pt. lIn, pp. 1489-1514, Vienna, lI1l7. 13. E. T .. Haws: Surges and waVes in open channels, }Vater Power, vol. 6, n~?- 11, pp; 419-422, November, 1954. . 14. R. D. Johnson: The correlatiQn.of momentum and energy changes in steady flow with varying velocity 8.lld the applicat.ion of the former to problems of unstea.dy

l l'

584

15. 16. 17.

18.

1 \

19.

ZOo

Zl.

22.

'23,

24..

25.

UNSTEADY FLOW

flow or surges in open channels, Hydro-Electlic Conferen'ce, 1922, Engineers and Engineering, The Engineers Club of Philadelphia, pp .. 234-240, July, 1922. George n. llidl: "Hydraulic Trnnsients," McGraw-Hill Book Company, Inc., New York, 1951, pp, 217-224, Charles .Jaeger: "Engineering Fluid Mechanics," translated from the German by p, O. Wolf, Blackie & Son, Ltd., Lon'don and Glasgow, 1956, pp. 381-392. G. Bll.tn: Utilisation des valeurs rMllites dans Ie dOlnaine des regimes'tran.sitpires des Cil.naux ouvcrts et leur application dans Ie cas des gllieries utilisees en reservoirs d'eclusees (Use bf reduced values in the translatory regions of open channels and their application in the case of tunnels used as storage re3ervoi~s), Proceedings of the 6th {ieneral Meel·j.ng, Intemational AssQciation for Hydraulic Resp-arch, The Hagl
!:

t. I

iI, II

I

I 1

RAPIDLY VARIED UNSTEADY FLOW

585.

26. Josef Frank: "Nichtstationiire Vorgange in den Zuleitungs- und Ableitungskaniiien 'von Wasserkraftwerken" ("Unsteady Flow in Headraces and Taiimces of Hydropower Plants"), Springer-Verlag, Bcrlin, June, 1957. 27. Alexander Koch and Max CllrstanjeLl: "Von der Bewegung des Wassers und dabei auftretenden I{riiften" (" Movement of Water and Associated Forces"), SpringerVerlag, Berlin, 1926, pp. 132-150. 28. U.S. Army Corps of Engineers: Hydraulic design: Surges in canals, Civil Works Construction, Engineering j{amtal, March, 1949, pt. CXVI, chap. 6, 13 pp. 29. Kll.Sper Dantscher: Wfillderwellen in Schiffahrtskan1ilen (Tra,·eling waves in navigation cp_naIR), Wasserkraft 1md Wasserwirtschafl, Munich, voL 35, no. 7, pp. 145-147, July 15, 1940. . 3Q. Kasper Dantscher: Die Wanderwelle in Schiffahrtskanal (The traveling wave in a navigation canal), Wasserkraft und lVasserwi?·tschaft, Munich, vol. 35, no. IO, pp. 226-229, Oct. 15, 19'10. . ., 31, Josef Koteny, "Hydraulik" ("Hydraulics"), Springer-Verlag, Vienna, 1953, pp. 263-265. . 32. Francis E, Swain: Determinat.ion of flolYs in interconnected estullrine (:hannels, U.S. Bureau of Reclamation, 2'echnical lYIemora71dt"'~ 640, February, 1951. . 33. Francis E. 'Swain: DctcrmintLtion of flows in intercoIlnected estuarine channels produc~d by the wmbined effects of tidal fluctuatioM and gravity flows, Transactions, Anw'ican Ge1tphysi~al Union, voL 32, no.' 5, pp. G53-672, Oct.ober, 1951. 34. D. J. Hebert and F. C. Lowe:. Progress report on model studies of the Sr.cramentoSan Joaquin Ddta, Central Valley Project, California, U.S. B"reau of Reclamation, HvdralLlic Laboratory Report No. Hyd-142, Apr. 10, H)44. 35. D. J. Hebert and J. E. "Warnock: Skeleton outline of a plan for developing t.he Delta-Mendota Irrigation Water Supply and Salt-Water Repulsion In the Sacmmento-San Joaquin Delta Region, 'Central VaHey Project, U.S. B1LrealL of Reclamation, Hyd-rUliliee Laboratory Report No. Hyd-145, July 10, 1944. 36. Cherry Creek Dam arid Reservoir: Report or'model Gtudies, spillway and stillin::o; bn.sin, Hydraulic LaboratDry lteport No. Hyd-146, prepared by U.S. Bureau bf Reclamation for U.S. Army Cofps of Engineers, July, 1944. 37. R. E. Glover, D. J. Hebert, and C. R. D9.um: Application of an hydraulic problem, in Electrical am.logics D.nd electronic computers: A symposium, T'ran8act'ions, American,Society oj Civil Engineers, 'vbL, UB, pp. JOIO-·101G, 1953. 38. Harold A. - Thomas: The propagation of waves iIi steep prismatic conduits, Proceedings of Hydraulic Conference, State University of IOlVa; St1tdies in Engineering, Blllletin 20, March, 1£1'10, pp. 214-229. 39. Paui G. H. Mayer: A study of roll waves and sIng flows in inclined open channels, doctor"l thesis, Cornell University, Ithaca, N.Y.,' S~ptember, 11)57.

."

FLOOD ROUTING

CHAPTER

20

FLOOD ROUTING

2£1-1. Routing of Flood.. It may be asaumed that the configuration of a flood wave moving through a shortregu1ar channel reach where the. cha,nnel resistance is relatively low will remain unchanged. The flood movement can. be treated simply as.a uniformly progressive flow, as discussed in Chap. 18. If, however, the channel is irregular and the resistance is high, the wave configuration will.be modified appreciably as it moves through the reach. The qetermination of this modification of a flood flow is knO'\'lD as flood routing. AsdesClibed in Art. 18-7, Thomas has two hydraulic methods methods can be used for the solution of unsteady-flow equations. for the purpose of flood routing. However, they are too laborious for actual applications. More practical methods on· hydraulic principles will be described in the following two articles.! In engineering hydrology, flood routing is au important technique necessary for the complete solution of a flood-control problem and for the s3,tisfactory operg,tion of a flood-p.rediction service. For such purpose!,>, flood routing is recognized as a pl'ocedure required in order to determille the hydl'ograph at oue point on a stream from the known hydrogrllph at an upstream point. Modern E,lectronir. machines can easily rOllLe a flood directly from its sources. Accordingly, the meaning of flood routing has been extended to include the routing of the movement of water from ra,iilfall to runoff. The routing technique applies both to channel reaches and to reservoirs. In the latter case, it is called reservoir routing. When it is used to determine the combined flood at a dO'i\'nstream point due to floods in several upstream tributaries and in the main stream, the technique is known as flood synthesis. The hydraulic method of flood routing is distinguished from the hydrologic method by the fact that the hydraulic method is based on the solution of the basic differential equations for unsteady flow in open channels whereas the hydrologic method makes no direct use of these equationil but a.pproximl1tes in some sense to their solutions. The hydrologic method is in general simpler but fails to give entirely satisfactory results I

For other hydra.ulie methods of flood routing) see [I] to [3j.

586

.

I

f

\

f I

I

ill

' /

587

in problems other than that of deterrilining t.he progress of a flood down ·a long .river.· For example, when a flood comes through a junction, backwater IS usually produced. When a flood is regulated by a dam, surges are generally evolved. The backwater effect and the effect· of surges in these ,Pl'Oblems can be a.ccurately evaluated only by the basic hydraulic equatIons employed in the hydraulic method, but not by the hydrologic method. 20-2. Method of Characteristics. A strict hydraulic method of flood routing has been shown to be extremeJy complicated and difficult. However, various simplified methods developed for practical purposes. Nhny of these methods belong to the general method of characterist'ic8, which is based on the solution of a set of characteristic equations of unsteady flow. Outstallding contributions to the development of such methods h::we been made by Massau [4,5], Henry [6J, Bergeron [7,8], Khristiftnovich [9}. Levin [10], Cl'aya [Ill, Arkhangelski! [12], }~olsters [13)141, StokeI' [15-17], Putman [181, Lamoen [19], Dmitriev [20], Es?offier [21!, ~ansford [22), Lin [23], Uchida [24], Nougf\fo [25~28L Iwagakl. and Suelshl [29-31}, Isaacson, Stoker, and Troesch [32,33J, nnd ma~lY others. The method developed by Lin will be described in this artICle, Acc,ording to Mas;s[1.u, the following equatiolls for unstea.dy flow may be wntten: _ .

J)fh,J~~ 4~J? ------7' IF

ay

+

a;r;

C £j.;}1:-~ D aTf O'n iC-r)I.UJ

a.1;

!

F alT

+ ~ af!

+ Tf

+

9

a.);

!I

ay d.-r; ike

9

at

0

+. i!..!L d- t d at!J

aT'd'"- ...L aT! dt • iI!

In the above equations)

So

dY

S,

(20-1) (20-2) (20-3) (20-4)

ay/ax is the slope of water surfllce, ay/at is the

chan:~e of .depth of flo,v with respect to time, aT' lax is the cho,nge of velOCity wl~h respe?t to distance, avfat is the of velocity with respect to tllne, SO IS the cfll:l.l1l1el slope, .91 is the friction slope, dy is the totlll change of depth, and dV is the total change of velocity. Note that Eq. (20-1) is a dynamic 'equation repl'e'3entiug assuming a: = 1 and that Eq. (20-2) is the continuity equation identical with ~q. (18-5). lfsyation (20-3) indicates thllt the totnl change ill depth

equal to the sum, of the partial changes in depth due to distance a.tid ~hne) re~pe~tively. Similarly) Eq. (20-4) indicates that the total change 10 velOCIty IS equal to the sum of the partial changes in velocity due to distance and time, respectively. IS

..

( I

588

UNSTEAD-r FLOW

-D(So - S)

+ Dd~ _

J g dt ------~~~--~

g dt

I'

£"dy +..!:dydx gdt

.~ (dX)2 _ 2 V dx + V g ,dt

2

_

(J

dt dt (20-5)

D

g

It may be assumed that a flood wa.ve is composed Of a great number of infinitesimal surges. Thepl'opagation of ~he flood wave may be treated as the propagation of the surges. These surges are formed as a result of disturbances caused by the flood, and each surge has IJ. discontinuous surface profile. At the point of discontinuity, the water surface breaks and the slope By/a;!; has two values. Since the two surface slopes do not bear any definitc reln.tionship to each otIler, the vaiue ofay/ax must be' indeterminate; or, mathematically, ay/ax = %. When the denominator of Eq. (20-5) is set t.o zero,

d:t

=

eV

± c) dl

where c = ...jgD. For wide channels, c = v' gy. is set to zero and Eq. (20-6) is used, d(V

± 2c)

deY

+ 2c)

(20-7)

dx dt

(20-8)

C

=

g(So - Sf) dt

=

V - c

(20-9) (20-10)

dev - 2c) = g(Sa - Sf) dt I

(20-11)

It CD.n be seen that Eqs. (20-8) and. (20-10) D.l'e identical with Eq. (18-54), expressiI1g the velocity of wave ·prcipagation. These equations can be represellted graphically on an ;rt plane, as sh01vn iIi Fig. 20-1. . For a finite time increment ilt, the point p represents the position of the channel section; under: consideration at timet ilt, and the pOInts u and d represeJilt, respectively, the posi~ions of c~rtain upstrean\. and downstream ;sections at time t. The. velocity of wave propagatio.n can be

+

+

When the numerator

=g(So - Sf) dt

+

I

represented by the slopes of lines constructed on. the xt plane. When the' flow is subcriticaJ,1 as it is in most streams, that is, when V < c, the slope of the line ·up, a positive quantity, represe~lts V cof Eq. (20-8), 8.nd the slope of the line dp, a negative quantIty, represents V - c of Eq. (20-10). These lines up anddp al'e hereafter re:e.rred to as r.har~cter­ ..1.S
(20-6)

The above equations are known as the eqttations of chamcteristics. The method for deriving these equations was first given by Massau [4J. Later, IMassD.l1 [5J developed atrial-und-enor procedure for 9.pplying these equations to problems of unsteady flow. Owing to the laboriousness of the procedure, the method had not been very popular until later investigators, among them Lin, simplified the procedure through gmphical and other approaches. }'he equations of characteristic§) !.lliJ,Y.be wrjt~ clx _ TT dt - r

589

FLOOD ROUTING

Solving the above four equations ilimultaneously for ay/ax,

(aJ

[b]

FIG. 20·l. Graphical: representation of, :vav~ prop~gation for sub critical flow. Flow condition at inflow; (b) Row condItIOn In channel. .

(a)

section p after the time interval 6t. Similarly, point d represents the position of the downstrea~ section fwm which a surge once developed . ' will arrive at section p after At. The flow conditions at the upstream and downstream sectIOns are to be indicated by the subscripts 'U and d, l'es~ectively. In~egrating Eq. (20-9) from time t to t ilt, which interval is the time reqUired f~r the :'[l.ve to travel from the upstre9iID section to the section ull;del' conSIderatIOn, .

+

V

+ 2c =

V"

+ 2c + gSa M u

(1+-'>.1

- g it

S, dt

(20,12

)

For a smail value of t:.t

J

I+A1

I

th~

S, dt

=

(S,,,,

+2 Sf) Atap~lica.blc: su~ercritical

1 Theoretica.lly, method of cha.ract.eristics is similarly to flow .. However, the possibility of disturba.nce due to formatIOn of standmg waves would have to be taken· into acrioun~ [23].

For units .in miles and hours, the equation becomes

So Eq. (20-12) may be writt,en V where

+ 2c = G + K (J", = V + 2c" + K" Ie {J(So-2Sf") Ilt-

(20-13) (20-14)

u

u

(20-15)

V

= . /39,5008 0 Ilt -

.\1

(20-16)

(20-25)

Equation (20-24) or (20-25) contains three variables c, V, and IC; So and For this equation, two auxiliary plots may be constmcted: (1) a plot of c against TT using f{ as a parameter, which is called "the

Similarly, the following equation may be obtained by integrating Eq: (20-11) over the interval required for the wave to travel from the dm"l11strea.m sectioll to the section under oonsideration:.

x in ini

I '.

(20-17)

where

(20~18)

and

(20-10) J(

]( cH

1,"t1O,000n 2 tlt

C:..t are known.

and

Eliminating

591

. F.LOOD ROUTING

UN,sl'lllilY FLOW

590

'200 A

I.

c'P

'/

I .I I

from Eqs. (20-13) and (20-17) and solving far c,

.

\ i

~O.D I ~~5'

i

jhrL~q

(20-20)

1'<::--.,J:-'-7p",-+..:.,w.'-b-.~l--+.---'j'

-i

16.6 mi .......

J

This equation can be used for computing c at the time t + Ilt if the upstream !l.ud downstream flow cOllditions at the time t are given. Eliminating c from Eqs, (20-13) and (20-17) and solving for l' V _ J(

=

or, from Eq. (20-20),

Y-K=

Gu

+ G~

(20-21)

+ 2c

(20-22)

2

Using the Manning formula, Sf = n!V2/2.22R~~. expression for 31 ill Eq. (20-16) and solving for V, o Ilt -- Ie R"~ v '\I'flG.IS 7.25n~ .

/16.1So III - J( 1> 742nz Ilt '. c

\j

1 .I

Substituting this

(20-23)

From Eqs. (20-22) and (20-23J, the values of Y and IC can be determined. , The solution for c and V, however, can be simplified by a graphical procedure. Taking the wide channel as an example, R = y. c~!g. Thus, Eq. (20-23) becomes

v

I

(20-24)

The units in t-his equation are expressed in terms of feet ane. seconds.

Fw

2(~:la..

Routill[l; by the method of

chamctcristic.~:

the ;ri plane. j

plot of c vs. V for K" (Fig. 20-2e), and (2) ~L plot ofT! - J{ against V using c as a parameter, which is called "the plot of T1 ]( ·vs. V for. c" (Fig. 20-2j) , The use of these two plots simplify the problem, . which is to solve for c and Yat time t + Ilt if the flow conditions c and V at the upstream and downstream sections at time t are given. In illustrating the application of the gmphical procedure, it is assunred (1) that the channel infinitely long and very wide, (2) that t.he initial flow is uniform and steady, IJ,.nd (3) that the time interval Ilt is COlllltant. The data required for the solution include the initial flow condition, the

I

I.

"vill

is

-1

592

FLOOD ROUTING

UNS'l>EADY FLOW

593

as

inflow hydrograph I with the stage y a function of time t the channel roughness, the cha.nnel slope, Ulld the length of the channel to be. routed. The whole procedure is to be developed by steps at equal time 4

x in rnl

Fl.G. 20~2c. Routing by the method of characteristics: the Vx pla.ne.

::: .5

.

. ~

•

l

I

I·

I

t- +3200 I

x in rni

FrG. 20-2b> Routing by thifmethod of chara.cteristics: the ex plane.

intervals.

For any step, the procedure is as follows:

A. Preliminary Com1mtation

,

1. Compute the initial velocity V" and depth y" by the Mannillg formula, and the initial celerity by c,. 2. Using Eq. (20~24) or (20-25) construct plot of C VB. V for J( (Fig. 20-2(3) and the plot of V - K VB. V for c 30-2f). 3. Construct the referen.ce lines OB and AB 20-1 and Fig. 20-2a). , Fo~ oth~ types of hydrogra~h, for example, in the form of the discharge q ~ f(t), a modification of the procedure IS neCeS5ll.ry. For a detcdled disC\I~~iQnl :;;ee [23].

J(1n mi FIG.

20-2d. Routing by the method of chara.cteristics: the 'Ii:!' plane.

The slopes of these lines are equal to V .. + e" and y" c.., respectively. Point 0 may be placed at any position on the z .axis. 4. Construct the inflow hydrograph with c plotted Ilgo.inst t (Fig. 20-2g). Since the given hydrograph is in the form of y J(t) , it is necessary to convertyto c by the relation c = vgy. 5. Obtain the values of c for various t from the hydrograph constructed

594

595

UNSTEADY FLOW

FLOOD ROU'l'lNG

ill the preceding step, and plot them 011 the ex plane at x = 0 (Fig. 20-2b). ALso, plot the vahle of V,. on the Vx plane at x = 0 (Fig. 20-2c).

through the time I (Fig. 20-'10.), the values of Cd and Vd are equal to the values of c and V 2,t time t. 3. From theploi.J of C VB. V for J( (Fig .. 20-2e), find the rca corresponding to Va and Cd, al1d then compute Gd by Eq. (20-18) . . 4. Compute Y I{ by Eq. (20-22) with the value of c obtained froin the inflow hydrograph (Fig. 20-2(1).

B. Computation JOT Flow Condition at the Inflow 1. Draw a trial characteristic p'd' (Fig. 20-10.) with a stope in tb~ genera.l . direction of AB.· From the inflow hydrograph (Fig. 20-2g) , it can be

i I I I

I

·-1

j'

~

I

,I

I

I

I I ~

2.5

3

v in FIG.

.. .

····--'4·

3.5

4.5

5

mph

20-2f. Routing by the method of characteristic.s: the 'V - J( vs. l' fore plot.

3.5 Vir mph

FIG. 20-2e. Routing by the metllod of (;haracteristics: the

C VB.

V for K plot.

seen whether the value of c is increasing 01; decreasing at a given time. Accordingly, the slope of the trial characteristic should be either :Ratter 01' steeper tha.n that of the previous step. .2. The values of c and V at time t are assumed to have been determined and plotted against X, as shO~Vll in Fig. 20-2b andc. Accordingly, at the point where the trial characteristic intersects the horizontal line passing

-"~

..,', -: "- -

---,--~'"-,~

..

"-,"

.",,;.'

5. Determine t' from the plot of V - J( VB. V for c (Fig. 20-2f). 6. Since the values of c and II are known, check the slope of the tri~tl characteristic. The slope of this characteristic should be equal to the average of the slopes at its ends, or [(TT - c) (Vd - cd)]/2. If necessary, repeat the' procedure with a new trial characteristic until a. satis- . factory check is obtained.

+

C. Computation oj Flow Condil.ion in the Channel

1. Construct two trial characteristics up and pd at point p and il). the general directions of DB and AB, respectively (Fig. 20-1b). To start

I

59B

UNSTEADY FLOW

22 -

9. Extend the procedure step by step to the desired lengths of time .and channel. ' 10. From the ex plane, a hydrogl'aph at any section of the channel CUll tle constructed. Constructing a vertical line through the' given x~ the line will intersect the curves of the e;t; plane, giving values of e at different t. The value of e may then be converted to y by the relatidn y ::: e2 /g.

! '--

/""

2I

I

Givert inflow,,-

'i

2 01----

E

f

8----,- l -

.!;

...

/

I

16

A. Preliminarycomputalion

=

\

/ I

f.

I-

-

\ 1

9

.J: 0.

-- I---~--

c,

/

---

: I

/. 14

1£V

o

10

\ -L \

c

\ -~, 30

40

50 60 in hr

70

80

90

100

20-2g. Routing by the method of charactE:ristics: thehydrograpns.

3. Determine Ku and Kd from the plot of C VI:I. V for K (Figs. 20-2e). 4. Compute all. andGd by Eqs. (20-14) and (20-18), respectively. 5. Compute C byEq. (20-20). , 6.' Compute V - K by Eq. (20'::21). r 7. 'Determine V frotn the plot of V - K vs. V for e(Fig. 20-2f). 8. With the flow donditioris determined at a sufficient number of points along the chan~el, construct the plots of·!V and e vs. x at the time t + !:It on the exand Vx planes (Figs. 20-2b and ~0-2e). C

=

Vgy

= -V0!21.5 =

---,

j

l"IG.

By this equal.ion, the auxiliftry plots (Fig. 20-2e and fJ are computed ,as shown in Table 20-1, p. 600. 3. On the xl plane (Fig. 20-2a) construct the reference lines OB and AB, the slopes of which are 2.60 + 14.0 = 16.6 mph nnd 2.60 - 14.0 = -H.4 mph, respectively. 4. ,construct the inflow hydrogr'1ph (Fig. 20-2g) by the foJlowi~lg equation:

-

I-----

' (20-20)

- --

1\

/ 20

(20-25) gives

,

,f--1:~ompvted outflow

=

1. By the Manning formula, 'Un = 13.1 ft. Thus, Vn 50/13.1 3.82 fps = 2.60 mph, and c. = v13.1V = 20:~ fps = 14.0 mph. 2. Using the mile-hour units, At '= 3 hr, 11 = 0.03, and So = 1/5,280. Equation

1 15

Example 20-1. An infinitely long and wide channel carri~s an initial uniform flow of 50 cfs per foot of width.' If a hydrogrnph l'epresented by y = 21.5 - 8.4 cos "(1l't/48) ft is imposed on the upstream end of the channel, route the flood through the first 50 miles for a pp.riod of 24 hr. Given 11 = 0.03 and So = 1/5,280. Sohdion. The computation follows the procedure described above.

-

\

597

FLOOD ROUTING

the computation, the point p may be so located that the pointu is at x = 0 on the horizontal line passing through the time t. 2. From the 0: and Vx planes of the previous computations (Figs. 20-2b Il.nd 20-2c), determine eu and V" at x" and Cd and Vcl at Xd (Fig. 20-1b).

r

I

I· I

I

I

I

- 8.4 cos (~'1/48)1 fps . 3.87 v21.5.- 8.4 cos (1r1/48) mph

(20-27)

5. By Eg. (20.27), compute the values oi C Il.t the upstream curl of the channel (at x = 0) for ,t = 0,3, . '. _ ,24 hr, and plot them in Fig, 20·2b. Also, plot the value of V = 2.60 mph for x = 0 and t = 0 in Fig. 20-2c. . . R. ComputatioT! Jar flow condiliQn at inflo·w. The step of computation from t = 12 to t = 18 hr ~vill be t.n.ken for illustration, assuming that the values of c and V along the channel at t = 12 hr have been determined in a previous step. The following' procedure rlescribes the determination of c and V at the inflow at t = 18 hr. 1. Draw a trb.l characteristic at point 100 in thp. generlLl dirp.ction of AB (Fig. 20-Za). The slope of this characteristin should be somewhat flatter than 'that of the previou." step, that is, flatter th~n the slope of the chllructeristic!at 12a, because the inflo\~ hydrograph CFig. 20-2g) indicntes that c is increasing during the interval from t = 12 to 15 hr. The downstream end of this 6ho.racteristic is at t: = 12 hr and Xd = 36 miles. . 2. With Xd '"' 36 miles and t = 12 hr, find Cd = 14.38 and V~ = 2.72 from Figs. .20-2Q and c. Theae value" are ent~red in eols. 9 !lnd 8, respectively, of Table 20-2. 3. IWEth Cd = 14.38 and V. = 2:72, ,find Kd = -0.04 from Fig. 20-2e, and enter the value in col. 10 of Table 20-2. By Eq. (20·18), Gd = 2.72 - 2 X 14.38 - 0.64, = -26:68, as shown in col. 12 of Table 20-2. ' 1

This example is adapted from [231.

598

4. From tbe inflow bydrograph (Fig. 2()..2g), c= 15.84 for t = 15. By'Eq. (20-22), V - K, = -26.68 + :.! X 15.84= 5.00, a.~ shown in col. 13 of Table 20-2. 5. With V - K = 5.00 ande = 15.84, find V = 3.19 from Fig. 20-2f and enter tbe value in col. 2 of Tobie 20-2. 6.Tbe slope of the charoct,eristic should be (2.72 - 14.38 + 3.19 ~ 15.84)/2 = -12.16, which gives Xd = 12.16 X 3 = 3.6.48 miles. This value is dose to x~ = 36 miles for the trial characteristic. If f,he agreement is not dose, a new trial characteristic must be drown and the computation repeated.

file do not change. In other words, the flow mfl.y be considered \)niformly progressive aiterthe flow conditions at points a and.b are determined .. This may be explained in Fig. 20-3 as below. Oil thE; ex plane, draw a hor'izolltalline passing through the / = 9 hr curve at x = O. This line intersects the t = 12 hr curve at x = 13 miles, Similarly, a horizontal Jine

)

I "j

,

A

7

C. Computation for jInw condition in the channel

1. Draw two trial cha.racteristics at 15& in the general directions of OB and AB. Note that tile characteristic in the direction of OB starts from the point 12a. Thus, x. = 0, x = 52 miles, and x~ = 86 miles. 2. From Figs. 20-2& and 0, find c. = 15.22 and V. = 2.96 nt x" = 0, and Cd = 14.02 and V ~ = 2.59 at x" = 86 miles. Enter thes"l values in cols. 5 to .3 of Table 20-2. 3. From Fig. 20-2e, find Ku = -1.08, as shown in col. 7 of Table 20-2, for t:u = 15.22 and Vu = 2.96; and also [{~ = 0, as shown in col. 10, for Cd = 14.02 and 1'd = 2.59. 4. By Eq. (20-14), G. = 2.95 2 X 15.22 - 1.08 = 32.32, as shown in col. 11 of 'Table 20-2. By Eq. (20-18), Gd = 2.59 - 2 X 14.02 = -25.45, as shown in col. 12 of Table 20-2. 5. By Eq. (20-20), c = (32.32 + 25.45)/4 = 14.44, as 'shown in col. 3 01 Table 20-2. 6. By E<\.. (20-21), l' - K = (32.32 - 25.45)/2 = 3.44, as shown in col. i3 of Table 20-2. 7. From Fig. 20-21, find V = 2.75, as shown in col. 2 of Table 20-2, for c = 14A4 and V - K = 3.44. 8. Use the values' of c and V to check the slopes of the trial characteristi~s. Thus, x = (2.75 + 14.44) X 3 = 51.57 miles and Xd = -(2.75 - 14.44) X 3 + 51.5'7 = 86.64 miles. The trial slopes are satisfactory. Otherwise, new charactc:ristics should be drawn and the computation repeated. 9. Proceed with the computation step by step, as shown in Table '20-2. 10. Draw a vertical line x = 50 miles on the ex plane (Fig. 20-2b). The linll intersects the curves, giving yalues of c at various times. Plotting these values of c against time, II. hYdrograph at x = 50 miles for a period of 24 hr is obtained, as shown in Fig. 20-2tp. This hydro graph call be converted to y = J(t) by means of the relation

+

599

FLOOD ROUT1NG

UNSTEADY FLOW

! . \

°

·1

x

at

y = c'/g.

.

It may be noted that, after the Bow conditions at points (l and b for each time.interval are' determined, part of the curves on the ex plane and Vx plane for ~he first 50 miles or so can be tmced in approximately and then revised or refined by further determination of more points, such ·as c and d, in the channel. It can be seen that the graphical procedure offers a practical solution of Massa.u's equations; Nevertheless, the computation involved is stilt complicated and requires practice for routine applications. For accura.te solutions, the time interval .c.l should be reduced, all graphs should be prepared on large scales, and the computation should be carried out to sufficient significant figures. For Table 20-2, the computation was done on a slide rille, since the example was given only for illustrative purposes. The values in cols. 7 and 10 were roughly estimated from the curves; they are not accurate even tD the decimal places showD. For accurate computation of K, Eqs. (20-15) and (20-19) should h~ve been used. ; In this particular example of a long uniform channel with init.ially uniform flow, it may be assumed that the values of c ~nd V ata particuli1r section of the wave pro-

-':".:

, FIG. 20-3. Simplified procedure for Example 20.,1. the Vx plane is drawn through the t =9 hr curve ttt x = O.This lineinterseds the t = 12 hr curve also at x = 13 miles. It is, therefore, evident. that values of c and V at the points U = 9 hr, x = 0) and (/ = 12 hr, x = 13 miles) rue equaL This evidence may simplifY the further s'Leps of the routing computation. The dashed characteristics and ce" at 15c may be dmwn parallel to the chamcteristks b&' and bb", respectively, at 12&. The upstrettm poiIlt c' of the'cha,racteristic ee' is placed at x = 13 miles ort the 12-hr line. Thus, the values of c and V at 15c are equal to those !:.t 1Zh. Similarly, the vaiues of e and V at points 9b, 12c, and 15d are all equal and cttn be determined in the same way, As a result, the computation of the ';fllues of c Oil

ee'

1 l.

) .600 .

\

UNSTEADY FLOW

1J.~1:l ~ is need~ only at point (J, at the inflow a.nd at point 11 in the channel for each tIme mt~rvaL The va.lues of C IUld.V a.t ot~er points in the channel can be detcrmin.;d by the SimplIfied procedure shown by the dashed linea in Fig. 20-2a.

i

20-1.

TABLE

I)

I K

= . .,..7

'_~

____ 1.0,1

0.01

I\

-0.5'1'

L' )

-1.0

-~.51'

I' r)

-~.O -2.5

I,

-3.01

) 1

0.0750 0.0166 0.0776 0.078,1 0.0192 0o.'OO~O~~g u

00816

TABLE

1

20-2 ..

V [: -

c •

~

i

[

)

= 47.2

Iv - K

V F -

c~;

=

c~~

54.3

F

=

61.6

K

;-;;-3j2.0;13.54 Z.54 3.093.09 3.13 3.63 3.16 4.16 3.20 4.70 \3.Z3 5.23 '3.26j 5.76 3 29 6.29

3.62! 3.66 3.70 3.74 3.78 3.82 13.851

1(, ---1----

3.62 4.16

4.70

5.83 &.38

5.24 5.80 5.78 4.34 6.34 6.32 14 .39 6.89 6.85\4.4317.43 5.0S I

K

V .

I .'

1

i

C.

'K

I

y

~~I~!~

IT d

(8)

Cd

6.93 7 All 8.03

Kd

I 1

Gu

Gd

I.... .I. ... '. i" ... 1"..

I'

30'64'1'--24'4?

.

I' .....

)

"I' ....

)

1

)

') ) )

1

) )

.~1

i

V

K

~~Ii!.!l_~, __--,--

~b ~.611406-0.1026\)[14.31-0.48~.5814.00

)

I

V

cH

-

.6114.07 -0.11' .... '". ,,' ...... 2 5814 00 O ! '? .69IH.31-0,48 .... :5914:02 0 1..... 1: ; : . : : .6014.02 -0.012.6J l,r.07 -0. 58H.aO: 0 \}(1 2,8614.71 -0.59," ", ... , ....... 26214.05 -0391 .. : .. ":'25:8;1 0 130.83-25.42 12a 1~.9615.22-1.081 .... 1 .......... 2.67114,16 -0 75j 26 12b 2.68H. 251-0.402.86iI4.71-0.692.581,'LOO. O' l5a 3.19115.841-1. 81] .... 2 .1214 .38 -0.64 , 681 1511 2.7514.44 -0.69 . 2.9615.22 -l.08,2.S9!14.02[ 0 f<:t2'S'Z _; . _ 18a33S'1651-191 ,) 1"" I ~5.4() 2773 . : . ! ' j...... )2.831<1,68- 19°1· 18b 2.~2[14. 72! -0.833.1915,85 -1. 8112 ,(;,2',14.08 -2578 1 21a3.,,\) 17.211-2.341;' ... ..,.... . 2 . 9315.02 -1.40 ..... -28: 51 21b2.9815.231·-l.06\.'I.SS',.16.511-1.912.68114.22 -'0.6513449 -26 2'.1a )3:7917: .90 -Z.49 · ....... 3.08115.55 -1.50\ .. : .. -21l 24b 30815551-1. 493. 59117 -2.3412.7014.32 -0.6035. 67 -:26:54,

)

1

xl

= 40,3

--

STEP COMl'UTATIC>l'i'SOF FLOOD ROUTING :BY THE METHOD

3tl 6a Bb

)

\.

\

.2O-i

OF CHARACTERISTICS

) \

di

2.53, 1.53 2.58 2.58 1 262,3.12 2.64,3.64 j2.61' 4.17 2.701'4.70 \2.73 5.23 2.75 5.75 1

j)

,

THE AUXILlI\RY PLOTS IN ExAMPLE

=~31~!-'-'[-C-=-1-6,-,-I-c-=-1-8,-,I-c----2-0-,--~""-2-2-,

C

\

3810

])

FOR

COMPUTATION

ai'59 "T?5':~

I

1::::~6'

-0:;433:0811

1

FLOOD ROU1!ING

{)01

Ohio; (2) movement of the 1947 flood through the confluence of the Ohio and Mi;ssissippi rivers at Cairo, Ill.; and (3) movement of the 1948 and 1950 flood waves in the 184-mile long and na.rrow Kentucky Reservoir on the Lower Teunessee River. The second problem is the same as that described in Art. 11-10 and Fig. 11-] 5 except that the flow is treated as unsteady in the present problem. .The computations ·for floW profiles were thus made for times t = 0, 2.5,4, 10, and <:tJ hI', after the beginning of the flood 50 miles up the Ohio River (see Fig. 11-15). The flow profiles for t = 00 are identical with those computed for steady flow. Solutioll of practical problems by the numerico.l method is generally very tedious. In order to speed up the computation work, the Corps has employed an electronic digito.l computer, such as the RemingtonRand UNIVAC I (Universal Automatic Computer I). For small quantities of work, however, the Corps has found tha.t a po,rt of the computn.tion can be performed less expensively and more efficiently by 3. portable computer, such as the Burroughs EIOl desk-size computer; the use of larger machines is justifiable only for very involved problems. 20-3. Method of Diffu.sion Analogy. An approximate hydraulic approach to the problem of flood routing in natural channels has been deyeloped us-illg the ciD.ssicn.l stc.tisticl).l theory of flow diffusion [35]. According to this theory, IL differential equation may be written for the diffusion of an unsteady flow of particles as follows: (20-28)

2.72 17

9.

2.61

;:;~ 4.04

3.08 5,00 3.44 5.29 93 53.65 4 .'0 4

'~~l ::;~

Similar numericahnethods for the solut'ion of the differential equations of unsteady flow [Eqs. (20-1) to (20-4)J were developed by Stoker [16 171 fLnd . by ~saacs~n, Stoker, and Troesch [32,33]. The methods ~ere applIed wlth satlsfa.ctmy results to three actual.problems presented by the U.S. A:my Corps of Engi~eers. [34): (1) movement of the 1945 flood in the 375-mll~ reach of the Ohio River from Wheeling ' W .Va . t'II . . , to C'mcmna

where Nis the number of p:1l'ticles, t is the tim9, x is the dista.nce, and K is a coefficient known l),S ditrw~·b}ity. When the pl1rtides are flOWi11g in a direction along the X axis, this equation gives the particle distributi()U in the dii'ection of flow as a function of time and position. This theory is commonly applied to problems of heat tl'il.l1sier r3B]; there Eq.(20-28) represents Fourier's ge-ne'rallGw of heat conduction, ih which N desigllates temperature and K is known as thermal d~tJusiuity [37]. . In natural streams, the disturbances of flow caused by local channel irregularities have definite magnitUde at any time and position. They are mixed, dissipated, and diffused as the flow moves along the channel. In ll.pplying the theory of flow diffusion to the flow of water, it may be assumed that the diffusion of the dist,urbanc~s is analogous to the diffusion of the particles. If the over-all effect of the disturbances on flow is represented by the variation in the flow depth y, Eq, (20-28) may be written (20-29)

"

602

J:.,.j-...:...........

603

UNSTEADY FLOW

FLOOD ROUTING

In natural stres,ms the local irregularities provide irregular storage, and the above equation reflects the rate of change in channel storage due to irregularities. Including this item in Eq. (18-1) for continuity of flow in p~ismatic channels, the continuity equation for flow in natural channels may be written

The propagation of this unit flood may be determined by Eq. (20-33). As a numerical example, using V_,,=70 em/sec; J( = 10 7 cm 2 /sec; tit = 5 hr; and x ,= 2.2, 14, 21, and. 32km, Hayamihas computed the propagation of the unit ffood, as shown in Fig. 20-'1. It can be seen that,

:..!

(20-30)

It is assumed further that the channel is relatively wide and that the flow on the average is uniform and steady. Thus', the discharge per unit width of channel may be represented by the Ch~zy iormula1 q = C . ./8 0 Vi, Substi~uting

(20-31)

this equs.tion for q in Eq. (20-30) and simplifying, ---.~ ay t,. r-/ayH -.:. at +(~~e . :-

K

'0.\

a2y

ax':

(20-32)

,

This is the basic differential eqt;¢{ion for flood }1:ow in nat7);ral streams. It can be seen that the coefficient of ayi~/ax in this equation depends on the ~ l'esistance and slopt and that the coeffIcient of a2 y/ax 2 depends on the channel iW'Igularities . .:.. __ This equation was orig'inally developed by Hayami [381 through an r 'I elaborate mathematical derivation.. A similar method of routing based on diffusion analogy was also presented by Appleby [39]. The value of diffusivity K in ordinary streams was estima.ted in the range from 10 6 to 10 7 cm 2 /sec. In large rivers, such as the Mississippi River: in the United States and the Yangtse River in China., the value of K would be of the order of lOS cm 2/sec. . A solution of Eq. (20-32) by Hayami results in the following equation for the propagation of a flood wave: . ~y - Yn Yo

,/

= 1 _~' r,./2VJ[i rV",~; _ X2 _ -y'; Jo exp l2K

(VwX)'2] d' K 4KX.i

\

"-j

" 1

(20-33)

where y is the depth at a point a distance x from the upstream end of the reach under consideration, y~ is the normal depth of flow at the same point before the flood arrives, yo is the depth at the upstream end, t is the time, J( is the diffusiv:ity, V", = 1.5V, V is the mean velocity, and X is the variable. A fictitious wave having a constant depth and lasting a llilit time interval tit may be assumed(Fig. 20-4). This wave is called a unit flood.

Elcpsed time I in hr

Elapsed lime I in hr

(a)

(b)

FIG. 20-4.. Propl>gl>tion of Cal a solitary unit lluud Md. Cb) two successive unit Boods. (After S. Hal/ami [38].)

as the wave propagates dowllstream, its shape gri\.dual1y becomes asymmetrical and fi:J;ttens out. In routing an actual fiood, the hydrograph of the fiood may be divided into a number of unit flood hydrognt.phs (Fig. 20-5). All hydrographs are plotted with the time against the sta.ge. The propagation of each unit flood can be computed by Eq. (20-33). By the principle of super-

1 The Manning formuIa may also be used. In this case, the second term on the left side of Eq. (20-32) would be written (1.49 v'Sc/n)ay~s/ax.

"'-~'

I, ..

'~'

..

.. '.

'.

",",'

..

) 604

I \

UN:;lTEADY FLOW

FLOOD

position, the propaga.ted height of the given flood:is equal to the sum of the propa.gntod heights of all the unit :floods. This method has been used to compute the propagation of a unit fioodartificiaUy produced in the Yedo Rivet· ill Japan; the theoretical computations were found ill good agree,mellt with the observations. In order to simplify and expedite the I'outing pr9cedure by this method, electronic' analog com!ai puters have been developed on the basis of (20-32) and applied to several rivers in Japan with satisfactory results [40-44J.

ROUT~NG

605

where ilS/ilUs the change in storage during a period t::.t, I is the !lver~~ge inflow during tit, and 0 is the lwerage o1itilow during t::.t. The value of AS!t::.t is positive when the storage is increasing and nega.tive when the storage is decreasing. , This equation constitutes the basis. for a ~ydro­ logic procedure of routing in which ilt is known as the nmhng PBNQd. The rate of storage can be plotted against time, as shown in 2O-6b, from which it can be seen that the, storage iB increasing before the time '(;l,t which the inflow equals theoutflovr and decreasing after that time. The cumulative 'area below the curve (shaded :l1rea in the figure) represents the volume of storage at !l. time t after the beginning of the flood. By plotting this volume against the time, a storage-volume / curve can be obtained, as shown in Fig. 20-6c. This curve hilS a pev.k representing the maximum volume of storage that occurs at the time when the inflow equal:> the outflow. If the storage is plotted against the outflow discharge, the resulting curve will generally take the form of a loop, such as that illustrated in F'ig. 20-7. Siorage As shown ill this figure, the stornge for FIG. 20-7. The storage-outflow reb.a giv~n discharge on the rising (or tionship. falling) part of the flood, wave will be , ' greater than (or less than) the sl;orage c'orrespollding to the condition of steady fiow. The storage-outflow relationship for the condition of steady flow is represented by the dashed curve, which is approxinlately !l.t the average position of the two limbs ot: the loop. ' The'storage in a channel reach for ullstendy flow depends prim!j.l'ily on the inflow and outflow discharges and on th~ geometric and hydrauli.:; , characteristics of the channel and its control features. It can be assumed that the upstl'eam and downstream end sections of the reach have the same mean discharge.and storage relationships with respect to the depth: of flowy. Then the following equations may be written:

1

(G)

li.1'

lime

Time

Fro. 20-5. Dividing hydrograph into unit-' flood h'ydrographs. ,

I I

I I

I

I

I

"'!

I

I

t

[e,

, "

ll~

f I' ,

t

20-4. Principle of Hydrologic Routing. When a :flood wave passes throngh a channel reach, the inflow and outflow hydrographs o Time ... at the upstream aJ:).d downstream , ends of the reach, respectively, are }7'IG. 20-6. Relationships among inflow, outflow, and storage in a. channel reach as shown ill 20-6a. ASS1.lrn- due to a passing flood. ' ing it. negiigible amount of loss 01' gain of \:V~1.ter.in the course of flbw through the reach,the total areas: under the hyclrogruphs are equal,. since the volume of the flood :water is unchanged. In: natUl'al the channel resistance und storage capacity are high; ,consequently the flood-wave characteristics will be considetably modified. As shown ill Fig, 20-6a, the floodpell,k is attenuated and: delayed. The difference between the ord' s of ,the inflow and outflow hydrogr!l.phs, represented by the rsh ' , areas in the figure, is ~qual to the rat.e of storage 'of water in the re~ch i thn.t is, i '

ilS ilt

=

I

0

:t

and

I =

fly'!!

(20-35)

0

ay"

(20-36) (20-37)

Si d:. by,:, bym S.

(20·38)

where a a.nd n~ express the depth-discharge characteristics of the s~ctions,. band m express the mean depth-storage characteristics of the reaph, and 3, and are the storages referring to the depths at the upstre(l.m and downstream end sections respectively. Eliminating y from Eqs. (20-35)

606 and

UNSTEADY FLOW

(20~37)

FLOOD ROGTING

and from Eqs. (20-36) and (20-38),

a(i)""n

(20-39)

Eo b(~)""n

(20-40)

E,= and

Let X be a dimensionless factor ~hatdefines the relative weights given to inflow and outflow in the determination of the storage volume within the reach. Then, the storage at any given time may be expressed as' (20-41) S = XE. +(1 When the stages in a reach are determined by the control at its downstream eud, for example, at the spillwa.y of a level-pool reservoir, the storage is a 80113 function of the outflow; therefore, X = O. If the stora,ge due to backwater effect at the upstream end of the reservoir is significant., X will be greater than zero. In uniform channels, equal weight is given to illflo\,. and outflow, and X 0.5. Substituting Eqs. (20-39) and (20~40) for E, and S.; respectively, in . Eq. (20-41) and simplifying, S

K[X I~

+ (1

- X)O-']

(20-42)

where K bla"'''' alid x min. In prismatic rectangular channels, discharge varies with the five-thirds power of the dept.h on the basis of the Manning formula, and storage varies with the first power. Since n = % and m I, the exponent x == 0.6. In natural channeJs, m may be considerably greater than unity and hence :t; is larger than 0.6. Mady, hydrologic routing procedures have been developed on the general basis of Eq. (20-42). For simplicity and for practical purposes, x is commonly assumed to be unity. The hydrologic approach to the problem of flood routing is based 011 the storage-discharge relationship described above. It is assumed that. dynarniceffects of flow are negligible and that storage is a single-\'alued function of discharge. This assumption implies that the flow is changing slowly with time. Effects of 'abnormal surface slope in modifying discharge and in changing channel storage are, therefore, neglected. This procedure is approximately correct for ordinary st.reams with small slopes. .When storage is plotted against discharge, the resulting loop is usually narro\v and an average curve may be fitted ill to represent .the storage a.s a single-valued function of the discharge. If the loop is wide, it can be reduced to a single-line relationship by an adjustIi:lent such as tha.t emploY,cd in the Muskingum method [45]. In this method the adjustment is made possible by varying the values of K and X in Eq. (20-42). .

607

. In streams having steep slopes, the . dynamic effects of flow are pro~ nounced and call1lOt be neglected. Consequently, for such streams the hydrologic method of flood routing maybe found unsatisfactory. 20-5.' Methods of Hydrologic Routing. Numerous hydrologic methods of flood routing h.we been developed. Descriptions of these method4J can be found in the literature on engineering hydrology [45-50]. In this article, therefore, the important methods will be mentioned only by name and references will he provided for furt.her study. A simple hydrologic method, however, "rill be described in the next article. In general, hydrologic methods of, flood routing may be classified into h'TO groups: t.he analytical audthe instrumental. There are many analytical methods of routing. For routing through reservoirs, the well:.known Rippl ma,ss curve [51] is widely llsed [52-54]. Other graphical m~thods of routing have also been developed, sllch as the method of Sorensen (55 i 561. For routing through rivers, the storagedischarge relationship is simplified in the methods developed by Meyer [57j, PuIs [47,58,59J, Wilson [60], Oheng [61], Johnstone and Cross [54), Knappell, Stratton, and Davis [56J, and Chow {52]. Semigraphical procedures include the methods of Goodrich fo3J, Rutter, Graves, and Snyder [64J, Wisler and Brater [65J, and Steinberg [66J. Other simplified meth~ ods use a nomograph, as suggested by Linsley [67], a str~ght. slide rule as suggested by Posey [68,69], and circular computers as suggested by Shepley and Walton [70J. A simplified method of successive average was developed by Tatum [48,71]. One rather popular and satisfactory method, known as the Mt~I;!";in{}um method was developed by :!I.1:cCarthy [46,72]. In this method, Eq. (20-42) is used as the working equatioll, assuming x L; that is, S = [([Xl

+

(1 _. X)OI

(20-43)

where K and X are to be determined from the channel chamctel'istics under study. In flood forecasting or control and operation of multiple-purpose river projects, the stage of flood is a major concern and a procedure for :;~a(Je routing is needed; For this purpose, a method involving the use of multiple-line charts was pl'QPosed by Lane [73]. An improved procedure ",vas later developed by Kohler [74], which requires on6 chart'for de~erminiug the normal relationship bet, ween gages in the main river and auxiliary charts, one for each tributary. These charts can be used to determine corrections to be applied to the predicted normal stage.. The .U.S. Mississippi River Commission [75] has used the Puis method of stage routing. For accurate prediction of stages on very flat rivers, a method of stage routing was suggested by RA.y and l'vlondschein [76]. In connection with the design of levees for flood control, the routing

r

608 1

UNSTEADY FLOW

techniqu~

shoulci be extended to solve the problem of so-called ftow-Unl: This problem is to estimate the maximum elevation of the water surface at all points along the leveed channel during the passage of a design floou. Once this elevation is determined, the height of the levees is given by adding a suitable freeboard. For such purposes, a simple method of calculation was developed by: W. M. Mulholland (46]. computation.

(

ROUTING

609

The U.S. Weathe.r Bureau [79-81,48] has developed an electronic analog, shown in Fig. 20-8. This machine automatically produces the outflow hydrograph while the operator traces the inflow hydrograph with a stylUS. Resistances in the electric circuit. can be adjusted to simulate the conditions for each reach of the river, as determined from past floods. The model illustrated ill the figure has two inflow positions,

I

!

i FIG. 20·8. Eleatronic f1ood·ronting a.nalog.

FiG. ~U-!l. uEDA tiood computer. Engineers.)

(Courtesy of E. E. AbboU, U.S. Army Corps of

Bureau.)

r \

In order to speed up routing operations, instrumental routing becomes necessary. For this purpose, two kinds of routing machines have been developed: the mechanical and the elec.tronic. Mechanical routing 'machines are usually designed for reservoir routing. The U.S. A~'my Corps of Engineers has constructed two types: an integrating machine designed by Tarpley (77J and a rolling-type flood router by Harkness [78; 45, pp. 674':"676]. r Electronic routing machines are electronic analog and digital com- . puters. The principle of the electronic analog is to utilize the analogy between the flow of current in an electric circuit and the flow of water in a channel or river system. In other words, an electri.c circuit can be constructed with its circuit equation analogous to the. routing equation.

allowing processing of two sources of inflow affecting the final outflow downstream .. More positions can be added, if necessary. In other words, the machine also does flood synthesis. The use of such a m[~chille has a decided advan:tage over analytical methods in that it solves the routing equation in differential rather than incremental form. Incrementa) form is often unavoidable in analytical methods. . Furthermore, the entire hydrograph can be routed more rapidly than by analytical computatioll. . Similarly, the U.S. Army Corps of uses n GEDA (Goodyear Electronic Dif;Ierential Analyzer, 20-9) in preliminary studies of flood projects and in touting flood through reservoirs. 20-6. A Simple Hydrologic Method of Routing. The method to be described here is approximate but simple and suitable for practical

610

UNSTEADY FLOW

I1 and 0 1

instantaneous inflow and outflow, respectively, at the end of the first routing period or at the beginning of the second touting period . 8 1 . = storage in the reach at the beginning of the second routing period S 2 = storage in the reach at the end of the first routing period or at the begi~ning of the second routing period At = routing period

Then

=

I

=

o=

~ + [2) = 2

(0 1

~S = 8 2 -

.

average inflow during l:!.t

+ O = average outflow· during l:!.t 2. 2 )

81

= change iin storage during

611

FLOOD ROUTING

purposes. Likp. most hydroiogic methods of routing, this method is based on the following assumptions: L The channel is divided into a number of reaches. Each reach is relatively short and has practically constant physical characteristics. The flood is then routed successively from reach to reach. In general, the shortest practical reaoh is the section between the two nea.rest gaging stations. 2. The discharge data are given at equal time intervals or routing periods. Within this period the increase or decrease of inflow and outflow is assumed to vary linearly. A short routing period is preferable . but will multiply the labor of computation. 3. The inflow and outflow are both taken as a measure of storage in the reach. This assumption is almost true if a: flood is being routed through a le\Tel-pool reservoir where the variation in storage between the falling and rising stages of the flood wave is not appreciable, In the case of a stream, the length of a reach for routing must not be too long or these variations will be exaggerated. .Theoreticaliy, the length of the reach should not exceed the product of the routing period and the average velocity of the flow in the reach (why?), although it has been found in many cases that reaches considerably longer are permissible [54,pp. 173-1'75J. 4. The flow in the reach, local accretions from ungaged tributary flows, ground water, rainfall or any form of precipitation, and local decrements due to evaporation or seepage are ignored if the amounts are small. If the amounts are large, they are either added to or deducted from the inflow as the case may be. In routing a flood from the first routing period to the second routing period, let

At

.

Substituting these expressions in Eq. (20-34) and simplifying,

. 8t

2 tJ.t - 0 1

+I +I J

2

S2

= 2 l:!.t

+a

2

.1 (20-44) J

If I I, I 2, O~, and O2 are expressed in cis, 8 1 and 8 2 in acre-ft, and tJ.t in

-,1

tt

UlUl

+ + -0

FIG. 20-10. Characteristic cun'es for flood routing.

days, noting that 1 cfs Eq. (20-44) beco:rnes

=

-\

-s

S-<>-

FIG. 20-11. Const.ruction of the average S curve from a wide S-curve loop.

2 afd (acl'e-feet per day) approximat.ely, then

(20-45) When tJ.t

=

1 d.ay, as it is in most cascs, the above equation becomes (20-46)

. This equation constitutes the basis of the present method. If tit is not equal to 1 day, then the values of Sri tJ.t and 82/l:!.t should be used in place of Sl and S2, respectively, for the computation and construction of the curves in the procedure to be described a..') follows: A. Construction of Characteristic Curves. These curves are constructed on a plane of S vs. 0 + S (Fig. 20-10). The abscissa represents the storage in acre-ft, and t.he ordinate represents the sum of the outflow in cfs and the storage in acre-ft. The curves can be constructed given any· two of thesetbree items: informat.ion on inflow, information on,out~ flow, and storage data of a historical flood in the reach under consideration. The third item can be calculated from the two given items by Eq. (20-34).

). 612

UNSTEADY FLOW!

l

which corresponds to point A in the previous 0Yllle. Successive values of outflow can be obtained, and the outflow hydrograph can be constructed. The above procedure for d~terroining the outflow can be proved as follows: Extend CD and EF horizontally toward the left to meet the . ordinate axis at H ,and G, respectively. Then, from Fig. 20-10,

. 1. Draw the 450" liM, starting at the origin at a slope of 45°.

+

,2. Plot S agaillst,O S, obtaining the S curve. . As described previously. (Alt. 20~4), the plot of st,orage against out-

\ I

flow will form ., loop, and so will the plot of S against 0 + S. If the loop of the S curve is not wide, an eye-fitted aVemge curve may be drawn to represcnt [l. line::ll; relationship between S o.nd 0 + S. If the width of the loop is fairly wide, the average S curVe ml'.y be constructed as:shown in Fig. 20-11. In this figure, a curve ofS against 1 + S also in t,he fonn of a loop is plotted on the left side of the ordinate axis and the 8 curve in loop fo!'m i~ plotted on the right side of the axis. On both sides' of the ordinate axis, vertical lines are constructed at thc equ.al ~bsci8sas' corresponding to. a certain storage 8'. The lines intersect the loops at poiuts A, B, A', and B'. The points Aand A' are on the limbs of ~he rising stage of the loops, and B rmd B' are on the limbs of falling stage. Draw stl'o.ightlines AA' and BE', which intersect at C. Draw a horizonto.llille irOlil C to the right to lTIf::et t.he verticul interc(~pt AB at point C', C'is the r.equil'cd point on the nyerage S curve corresponding to the given stomge 3'. It can be shown that the ordinate of C; repl'esen~s the sum of S'and a certain discha.rge Q whose value is the average for the rising and fulling of the flood stage. 3. Construct the image curve (1 curve in Fig. 20-10) whose abscissa is equal to 8 t 0 1• Tlus is an image of the 45° line reflected horizontully on the left side of the curve. The curve may be constructed easily by making the hOl'izontal intercept between the curve and the's curve equal to the horizonto.lintercept be~\v~en the 8 cur~e and the 45° line. . B. Determination of the Outflow. This is based on the char:l.cteristic curvtjs (Fig. 20-10) and the inflow hydrogl'nph of the flood to be routed. The routing period is taken as 1 day. 1. The initial outflow at the beginning of the first routing period must be known or assumed. Not.e that, if the value is assumed, the error involved in assuming th~ vruue will not be magnified enough to produce appreciable effect on the result. 2. Locate [l. horizonto.lintercept AB equal to the initial outflow .between the 45 0 line and the S curve. 3. Extend AB towo.I'd the left to meet the 1 curve at C. 4. Extend AB to,vard the" right to point D, making CD = 1 1 + Iz. 5. Draw the vertioalline DE from D upward, to meet the 45° line at E. 6. Draw the horizontnlline EF from E to the left to meet the S curve ~~

. .

.7. Measure the length of EF, which is eqU9l1 to the outflow at the end of the routIng period, or Oz. . ' ; 8. Continue the cycle of the above steps :by sta.rting from point F,

613

FIIOOD ROUTING

I i

HD

and Therefore,

. HA -- CA + CD"", 8 1 - 0 1 + II GE = GF + FE ,= 8 2 + FE HD =GE S~ - 0 1 + 11 + I .. = 8 2 + FE

+

12

(20-47) .

CoropfLring this equation with Eq. (20-46), it is evident that FE must be equai to O2• Example 2a-~. The infiow and out.flow'data of a historic!!.l flood for a channel rench are given in cols. 1, 2, lind 5 or Table 20-3. Determine the !>utflow hydrograph of a flood whose inflow dlJ.t/l. are given ill cols. 1 !!.nd 'e: of Table 20-4. TABLE

20-3.

COMP1:.'TJ\.~ION 01' CaAB.ACTEFtIS't'IC CURVES

"(All quantities ill thou3ands)

(4)

(2)

(1)

I, I, !I.-larch 23

24 25

25 27 28 29

30 31 i

i

I

I

April

1

22,8 23.6 57.4 59.5 161.1. 158.5· 279.2 269.6 277.7 .268.2 195.9 189,2 133.9 129.4 92.8 96.1 71.9 i'L5 58.9

56.s1,

5

45.1 -37.3 32.8 29.6

6

26.6

2

3 4

24.1

l I1.

I

I

(6)

(}utfiow

Inflow Adj, ds

(5)

aid

Given, cis

I, +;,

o.

Ay,

80.2 215.9 428.1 53i.:S 457.4 318,6 222.2 164.1 128.7 101.9 82.4 70.1 62.'4 56.:2 50.~

(8)

(7)

Storage

Av,

Ch!mge>:

Accum,

aid

!lfd

!I.e-it

8, S,

Ot

18.5 29.5 94.5 211.2 270.8i 229.3 168.5! 119.7 9E;'.8

64.5 50.2 40.0 34.0 30.4 28.0 26.2 ,512.1

0 1 +0, S. - S, 48.0 124,0 305.7 482.0 500.1 397.8 288.2 216.5 16l.3 114.7 90.2 74.0 {i4.4

58.4 54.2

(9)

27.5 59.7 151.6 274.0 329.8 287.1 207.9 141.9 90.1 57.5 44.7

32,2

9l.9 122.4 55.8 -42.7 -79.2' -66.0 -51.8 -32.6, -12.8 -7.8 -3.9 -2.0 -2.2 -3.5

36.9 i

33.0 31.0

46.0 89.2

246.1 485:2 600.6 516.4 376A 261.6 1813.9 122.0 94.9 76.9 67.0

28.8

61.4 56.8

25.3

51.5

614

-"I

U-\'lSTElADY FLOW

FLOOD ROUTING

Solution. The computation for ·characteristic curves is 8hl?wn in Table 20-3. In col. 2 the inflow Gt\tn ttre made up.of three parts: the daily inflow at the upstream end of th;' reB.ch, either gaged or routed from the upstream reach i the daily inflow from gaged or routed tributaries entering the reach; and the loca.J. inflow contributed by ungaged tributa.ries and la.nd nreas draining directly into the stream. The local inBow Inay be evaluated directly from the distribution of rainfal! of the storm over the area draining directly into the reach. As the total outflow of a flood ispresuumbly equal to the total inflow, the sums of the values in cols. 2 and 5 should agree with TARLE

20-4.

COMPU,!?,TION OF TilE OUTFLOW: HYDROGRAPf!:.

(Ali quantities in thousands)

600

(2)

(3) Inflow

Date (day)

A'll, dd

I,

+ I,

.. ..


,I

(5)

Given, ds

Adj,

0, 0,

01 01

2S1.0 111.S

20.3 65.0 129.0 126.0 70.0

72.5

40.0

1st 2d

20.0 113.2

3(1 4th

180.0 71.0

Sth 6th 7th

40.S 32.(} 2S.0

8th

22,5 20.0

47.S

19.0 17.5

39.0

9th 10th lith Total ...

.1

Outflow

Given, cfs

II I,

~;

400

.t 300

(4)

~,

500

0

(1)

615

~

133.2 293.2

. 07.0

42.5 36.5

29.0 24.0 21.0 19.5 18.9 562.7

cfs

20.3 134.8 128.8 125.8

S

FIG. 20-12. Characteristic curves of Examplc·20-2.

59.8· i

39.8 ~ ~8.8

23.8 2{;L8

19.3 18.7 560.7

each other. As is usually the case, however, the sums here are not equal to each other. Assuming that the outRoy, values are closer to the tr)le values tha,n the inflow values, the inflow is adjusted by multiplying the values in col. Ii by the ratio 1,512.1/1,566.3 0.9654. The adiusted inflow values are listed in col. 3. The a.verage inflow and outflow in afd a.re equal to I j + I z a.nd 0, + 6" respectively, where I" I" 01, and O. are in efs. Value~ of I I + I, and 0, + 02 are given in cola. 4 and 6, respectively. The' difference between t,he values in eols. 4 and 6 is equal to the change in storage listed in ~ol. 7, liinee (II + I.J (0, + 0,) = S, - S, = AS. A positive value in col. 7 indicates that the storage is increasing; a negative value indicates that the storage is decreasing. Column 8 gives the cumula.tive stora.ge oomputed from the va.lues ill col. 7.' The first entry in col. 8, i·.e., 27.5, rep:resents the storege in the reach on March 23 just before the flood began, which should be given or estimated. The sum of thevlLlues in cols. 5 a.nd 8, or 0, + S" is ent.ered in col. 9. From the values in ools. 8: and 9, the cha.racheristic curveS are constructed, a.a shown in Fig. 20-12.

Dole

FIG, 2~-13. Inflow and outflow bydrographs for Example 20-2.

The computat,ion of the outflow from the inflow of Il. given flood is shown in Table 20-4. In cols. 1 and 2 are inflow data. Column 3 gives the average inHow inafd, or II + I 2• In coL 4, the initial outtlow is gi ven as 20.3. Other values of outflow are obtained from the characteristic curves in accordance with the procedure described previously. The outflow values, except for the given initial outflow, are ad;usted and entered io col. 5 so that the total inflow agrees with the total outRow. The

"~

i !

t

616

i

UNSTEADY FLOW

inflow .and out.f1ow hydrographs of the given flood are constructed as shown in Fig. 20-13 from the vtl.lues of cois. I, 2, and 3.

!

'FLOOD ROUTING

617

change in stage at Sta. B of the flood under investigation as a. result of the proposed pla.u, Assuming that the discha.rge is a single-valued f unction of the ataga.

(

"REFERENCES

PROBLEMS ZO~l. Extend the routing of the flood in Example 20-1 to a period of (a) 36 hr !l.nii (b) 120 hr, 20-2. Route the following hypothetical solitary flood through the channel reach described,in Example 20-1:

Ti'me since flo.od began, hr

Celerity, cfs

o

14.0 15.0 15,,j 15.9 '14.0

3 6 9 12

The flow conditions before and after the flood wave Me uniform/having a celerity . ' of 14.0 cfs. 20-3. Verify the un'it-flood computation sho\vn in Fig, 20-4. 20-i. In a level-pool reservoir in which discharge is not controlled, the pl!ak outflow must occur where the' outflo,w hydrograph intercepts the inflow hydrograph; that is, P' in Fig. 28-6 must be at L Why? ' 20-li. In the development of a flood-control project for e. river bEtsin, the largest flood on recol'd is investiga.ted. The discho.rges of this flood at an upstrea.m station A llnd a downst.rc!t1U station B were observed, as shown in the accornpnnying table.

-

Inflow at Stl', A from Date

Main, cfs

-------" ;[llly 29

13,600

30 20,100 31 106,000" Aug. 1 :2 3 4

5 6 7

1, (

I

8

,

92,800 49,600 22,700 11,000 8,050 13,800 19,000 14,500

I 'Iributo.ries,

Local,

cis

cia

1,900 46,300

3,000 63,800 72,lQO 21,100

50,100 I5,900 0,200 2,900 2,200 2,10.0 3,200 5,300 3;200

8,900 6,200 4,100 3,100 5,200 8,100 3,800

Outflow at Sta. B Discha.rge, ds 20,800 33,000 80,800 110,000 112,000 lIZ,OOO '102,000 68,200 26,100 20,800 21,700

Discharges held "out at Sta. A, Gage ht, due to proposed ft reservoirs, cf. 14.4 19.6 35.1 43.0 43.5 43,4 41.1 31.5

16.8 14.4 14,8

0 60,000 149,000 96,400 46.100 17,900 2,000 2,100 3,200 4,800 2,200

The total inflow at Sta.. Ais equal to the sum of discha.rges from riIn.in stream, tributaries, and local areas a.djacent to the reach. The storage in the reach between StaG. A and B" at to,; beginning of the flood )5 estimated at 50,000 acre-ft., For the purpose of flood control, a system of reservoirs is proposed on the tributaries above Sta. A. It is found that the effect of the reservoirs is to hold out discharges from the flow at 8ta. A, ~ shown in the 11l.5t column of the table. Predict the

1. N. J. Dahl; On non-permanent flow ill open cana.ls, Proceedings of the 6th Gene,-al Meeling, International Asso,ia.lionJor Hydraulic Resea.rch, The Ha.gue 1956; vol. 4, pp. D19-1 to D19·16, 1955.

2. T8.keo IGnosita: Hydrodynamicn.i study on the flood flow, in Floods, voJ. III of SYTnposia Darcy, International A~sQciaHon o.J Scientific Hyd"ology, PubliCa/ian No. 42, 1956, pp. &5-63. " 3. ,D. N. Dietz; A new method for calculating the conduct of translation waves in prismatic canals, Physica, ~·ol. 8, no. 2, pp. 177-195, Febru/l.Yy, 1941. 4. Junius Massau: Appendice au M:emoire sur l'integration graphique (Appendix: to Memoir on graphical integration), Annales de I' Association des Ingd'lli~u1's sDrti~ des Eco.les Speciales de Ganc, voL 12, pp. 185-444, Ghent, BalgiuIll, 1889. ' 5. Junius l'Ihssau: Memoire sur l'integration graph,ique des equations aux derivees partie lies (Graphical integration of partial differential equations "lith special applications to unstelLdy Bow in open channels), Annale8- de l' Association .des [-ngJnieurs sortis des Ecoles Speciale;; de Grmd; vol. 23, pp. 95-214, Ghent, Belgium, 1900. 6. MlJ.rc Hemy: Propagation des intumescences dans un canal rectangulaire'-( Propagation of, translatory waves in a rectangular channei), RellUe g~n()I'ale de l'hydrallZi'lue, Paris, vol. 4, no. 19, pp. 17-24; no. 20, pp. 65-71, 1938. 7: Louis Bergeron: Methode graphique generaie de calcul des propagatiolls d'ondes ·planes (General graphical method of computation of the propagations of plane waves), Memc)1:res, Societ~ dell Ingenieurs Civiis de la Fra.nce, pp. 407-497, JulyAugust, 1937, , 8. Louis Bergeron: Methode graphiqllIJ pour te calcul dell ondes de tro.ns~ation· (Graphical method for the complttation of translatory waves), Sw;~te Fran.~aise des Mecaniciens, Bulletin No, 7, Paris, 1953. 9. S. A. Kh1'istianovich: Neustanovh'sheiesia dvizhenie v kanalakh i rekokh (Unsteady "motion in channels a.nd rivers), in "Nekotoryie Voprosy Mekhani\d SploahnoI Srady" ("Several Questions on the Mecha.nics of ContinuouH' Media"), Academy of Sciences, U.S.S.R., 1938, pp. 13-154. 10, J..eon Levin: Methode graphique de calcul du mouven1ent Iion permanent dans' lescanailJ( a ecoulement libra (Grapbic method of computation of unsLeady Bow in open channels), Le Gtnie civil, vol. 119, no. 11-12, pp. 109-113, Ma.rch 12-14, 1942. 11. A. Craya: Ca,lcul graphique des regimes variables dans les canaux (Graphical computa.tion of varia.ble regimes of flow in channels), La Hou.ille blall,che, Grenoble, new series, lilt yr., no. I, pp. 19-38, Novembel', 1945, and no, 2, pp. 111-130, March, 1946. ' I 12. V. A. Arli:hangels!dl: "Raschety NcustaDovivshegosia Dvizheniia. y Otkrytykh Vodotokakh" ("Cnlculation of Unsteady Flow in Open Channels"), Academy of Sciences, U.S.S.R., 1947. . 13, H. Holsters: Le Ca.lcul du Illouvement non perma.nent dans les rivieres par Is. methode dite des "Iignes cl'influence" (The computation of unsLeady fiow in rivers by the so-called "influence-lines" method), Revu.e generale de l'hydra!tl'i2tIB, Paris, vol. 13, no. 37, pp. 36-39, no. 38, pp. 93-94, no. 39, pp. 121-130, no. 40, pp. 20Z-206, and [l0. 41, pp.,237-245, 1947. .

618

UNSTEADY FLOW

14.. H. Holsters: Le Calcul du mouvement non permanent dans les rivie .."s par la methode dite des "Iignes d'influence" (Calculation of nonpermanent flow in rivers by t.he method kflown as "influence lines"), La Houill~ blanche, Grenoble, 8th yr., no. 4., pp. 49fi-5U9, August-September,1~.53. . 15. J. J. StokE>r: The formation of breaker,,; and bores, New York Univer.lily, Comnmnications on Applied ltfathemalics, vo!. I, no. I, pp. 1-87, January, 1948. 16. J .•r. Stoker: Nunierical solution of flood prediction and river regulation problems, I: Derivativ'n of ·hasic theory and formulation of numeriea! methods of attack, New York University, Institute or Mathematical Sciences, Report No. IMM-200,1953 . . 17. J. J. Stoker: II ';Vater waves," vol. IV of "Pure and Applied Mathematics," Interscience Publishers, Inc., New York,1957. . . 18. Henri J. Putman: Unst,ead:r flow ill open channels, Transactionll, Amer·ican Geophysical Union, vol. 29, no. 2, pp. 227--232, April, 1948. Discussion by Pin-Nam Lin, vol. 30, no. 2, pp. 302-306, April, 1949. 19. J. Lamoen: Tides and current velocities in.1 s%-lev~1 canal, E71!linceri:I(I, vol. 168, no. 4.357, pp. 97-99, July 29, 1949. 20. G. J. Dmitriev: Vychislenie kharakt~ristik cst[l.noviTfshegosia pJavllo izmeniaiushchegosia dvizheniio. v prizmaticheskikh ruslnkh (Computat,i'Jil of characterist'cs of a steady gradually varying movement in prismatic channels), Comptes rend". (Dokludy) de l'AC(1d~mie de~ Scienceli de l'U.R.S.S., Akademiia NatLk S.S.S.R., Leningrad, vol. 68, no. 5, pp. 825-827, 194.9. 21. Francis F. Escoffier: A graphical method for investigating the stability of flow in op~n cbannels or in closed conduits flowing part.ly full, Transactions, America11 Geophysical Union, vol. 31, no. 4, pp. 583-586, August, 1950. 22. G. D. Ransford: Contribution to first order theory of translation waves, La Houille blanclw, Grenoble, 6th yr., no. 6, pp. 761-763, September-October, 1951.; . 23. Pin-Nam Lin: Numerical analysis of continuousunsteady flow in open channels, 'l'ransCtctions, Americ.an Geophysical Union, vol. 33, no. 2, pp. 227-234., April, 1952. Discussions by J. C. Schonfeld and Fin.. Nam Lin, vol. 34, no. 5, pp. 7nnl5, October, 1953. 2'1. Shigeo Uchida: On the analysis of flood wave in a reservoir by the method of c1111.racteristic"s, Proceedings oj the Ed Japan National Congress for Applied lIfechan~ ics, pp. 271-276, 1952. 25. J. N ougaro: Recherches experimenta.les sur les iritumescenoes dans les canaux decouverts (Experimental researches on translatory wave" in open channels) SociJt~ Fm1<~aise des M~caniCiens, Bulldin·No. 9, Paris, pp. 23-35, 1953. 26. J. Nougaro: 'Theoretical and experimental studies o.f the propagation of the transI~"tion waves in open channels, Proceedings. oj.the Minnesota International Hydra11r lics Convention, Joint 'Meeting of International Associationfor Hydraulic Reliearch and Hydraulic,s Diuision oj American Society of Civil Engineers, pp. 555-559, August, 1953. 27. Jean NotlgarD: Etude theorique et experimentale de la propagation des intumescences dans les canaux decouvert.<; (Theoretir.al and experimenta.l study of the propaga.Uon of translation waves ill open chaunels), PtJblications scientifique.~ et techniques d1J Ministere de PAir, France, No. 284, 1953. 28. J. Nougaro: Methode graphique pour Ie caJcul de 1iJ, propagation des intumescences dans les canaux cle~()uverts (Graphical method for the computation of the propagation of translatory waves in open channels), Proceedings of the 6th General Meeting, International Association jor Hydraulic Research, The HaglLe 1965, vol. 4, pp. D5-1 to D5-15, 1955.

FLOOD ROUTING

l

r

G19

29. Yu,ichi Iwagaki and Tomitaro Sueishi: On the unsteady.flow in open cha.nnels with uniform lateral inflow (in' Japanese), P1'Occcdings, Japan Soci'-tll oj Civil Engineers" vol. 29, no. H, Tokyo, November, 1954. 30. Yui,~hi Iwagaki: Fund~.mental studies on the runoff analysis by charaderistics, Kyoto University; Disaster Prevention Research 1 nstilule, Bulletin No. 10, Kyoto, Japan, December, 1955. . ./ 4 31- Tomitaro Sueishi: On the run-orr analyds by the method of characterist.ics (in Ja.panese), Transactions. Japan Society of C7:"il Engirwers, no. 29, pp. 74-87, Tokyo, December, 1055. 32. E. J. Isaacson, J. J. Stoker, and B. A. 'Troesch: Numeric.al ~oIution of flood prediction and river regula Lion problems: Report 2, Numerical solution of flood problems in simplified model" of the Ohio River and the junction of the Ohio and Mississippi Rivers, New Y orlc Uni,'er.su1l, I nslitule of lVIathcIII.atica/ SC'iences, Report No. I11JjI-I-205, 1954. as. E. J. Isaa:cson, J. J. Stoker, acd B. A. Troesch: NUIIlerical solution .of flood prediction and river regulation problems: Report 3, Restllts of thr, numericll.J prediction of the 1945 and 194.8 floods in the Ohio River, of the 194.7 flood tin'ough tIle junction of the OIJio 2"nd Mississippi Rivers, and of the floods of 1950 and 194.8 through l(eutuGky Reservoir, IV ew Y orlc University, Inst,:tute of Mathell1(!tica.L Science~, Report No.IMM-NYU-235, 1956. 34. Edward A. Lawler and Fra.lLlr V. Druml: Hydraulic 'problem solution on electronic computers, paper 1515, Proceeding", American Society of Civil Engineers, J o!!rnal, l-Faterways and Harbors Division, vol. 8~, no. \VW1, pp. 1-38, January, 1958. 35. Georg Joos: "Theoreticai Physics," 2d ed., Hafner Publishing Company, N~w York, 1950, pp. 590-594. 36. Shib-I Pai: "Viscous Flow Theorl',"yol. II, "Turbulent Flow," D. Van Nostrand Company, Iuc., P,inGetcll, N.J., 1%7, pp. 1/9'::183 and 186-187. 37. Alfred Schack: "Industrial Heat TransJer," translaLed from the 'German by Hans Goldscilmidt and Everett P. Partridge, John Wiley & Sons, Inc., New York, 1933, p. 29. 38. Shoitiro Hayami: On the propagation of flood WD.ves, Kyoio Unive.,·sity, Disa.~ter P)'evention Research lnsl/}ute, Bulletin 1, Kyoto, Japan, Deeember, 1951. 39. F. V. Appleby: Runoff dyno"mics: A hee,t conduction 2"na!ogue of stora.ge flow in channel networks, A.s.scmbUe G~n~rale de Rom~, 1954, InternationaL Association of 8cientifi·c Hyd1'oloflY, Publicat·ion No. 38, yol. 3, pp.338-3<18, 1954. 40. 'fojiro Ishihara, Shoitiro Hayami, and Shigenori Hayarni: On the electronic analog computer [or flood routing, Proceedings of the Japan A.cadem!l, ·/O!. 30, no. 9, pp. 891-895, Tokyo, 1954. 41. Tojiro Ishihara and Yasuo Ishihar!i: On the elect,ronic analog computer for !lood routing (in Japanese), Transactions, Japan Society of Civil Engineers, no. 24, pp.
J "1

·1

'1

1

620

! I,

I

I I

I·

I

,

UNSTlj:ADY FLOW

FLOOD ROUTIN,G

45. B. R: Gilcrest: Flood routing, chap. X of "Engineering Hydraulics," edited by Hunter Ronse, John Wil.,y & Sons, Inc., New York, 1950, pp. 635-710. 46. Flood rou'lillg, chap. Vof "Flood Control," The Engineer School, FaIt Belvoir, Virginia, H)40, pp. 127-177. ' 47. Flood routing, cbap. 6.10 of pt. 6, Flood Hydrology, vol. IV, Water Studies, ' U.S. Blcrea1' of Reda?/tillion Manual, Dec. 30, 1947: 48. Ray K: LinskW, Jr., Max A.Kohler, Joseph L. H. Pa;ulhus: II Applied Hydrology," , McGraw-Hill Dook Company, Inc., New York, 1940, chap. HI, Stream routing, ; pp. 485-541. 49.: Ven Te Chow: Hydrologic studies of floods in the United States, in Floods, voL In ' , 'of Symposia Darcy, Inter1lational Assaciation of Scientific Hydrology, Pllblical.ion No. 42, HJ56, pp. 134-170. 50. Flood routing, art. 3.17 of sec. 4, Hydrology, U.S. Boil Conseroalion Service, Enginee,ing Handbook, Supplement A, 1957, pp. 1-28. 51. W. Rippl: The capacity of stora.ge-reservoir for water-supply, Mim.iell of Proceedinf/.s, Instilution of Civil Eng'ineers, London, vol. 71; pp. 270-2';8, 1883. ·52. Armin Schoklitsch: H Hydxaulic Structures," translated from the Gernlan by Samuel Shulit.s, American .society of Mechanical Engineers, NeW' YOl'k, 1937, vol. 1, pp. 65-67. , 53. H. Ii:. Barrows: "\Vater Power Engineering," 3d ed., McGraw-Hill Book Company, Inc., New York, 1943, pp. 199-201. 54. Don'Johnstone and William P. Cross: "Elements of Applied Hydrology," The Ronald Press Company, New York, 1949, pp. 1Ii3-167. 55. Kenneth E. Sorensen: Graphical solution of hydraulic problem:!, TraTMIactions, American Society of Ct'vil Engineers, vol. 118, pp. 61-77, H)53, 56. Calvin Victor Davis (editor-in-chief): "Handbook of Applied Hydr!lulic~," McGnnv-Hill Book CompallY, Inc., New York, 2d ed., 1952.. Sec. 1, IUvcr :regulation by reservoirs, by Theodore T. Kna.ppen, James H. Stratton, and Calvill V. Davis, pp. 1-21; and appendix B, Graphical aids to hydraulic computations: by Kenneth E. Soren!?en, pp: 1229-1248. 57. Otto H. Meyer: Simplified flood routing, Civil Enginecn'ng, vol. 11, no, 5, pp, ·306-307, May, 1941. 58. Louis O. Puis; Flood regulation of the Tennessee River, House DOCU1l2eni, !lo. 185, 70th CongresS, 1Gt Session, U.S. Government Printing Office, Washington, D.C., 1928, pt. 2, appendix B: . , 59. Stanley S. Butler: "Engineering Hydrology," Prentice-Hall Inc., Englewood Cliffs, N.J., 1957, pp. 208-212. 60. Walter T. Wilson: A gr!l.phical flood-routing method, Transac:l.ilYnf, American Geophysical Un'ion, V!;ll. 22, pt. III; pp. 893-897, Hi41. 61. l\. M. Cheng: A graphical solution fOI' flood rGuhing problems, Civil Engineering, vol. 16, 110. 3, pp. 126-128, March, 1946. 62. Ven'fe Chow: A practical procedure of fio(ld routing, Civil Engineering and Pt,bLic Works Review, London, vol. 4~, no. 542, pp. 586-588, A\lgllst; 1951; reprinted as Universit1! of Illinois, Civil Enr/ineering Studies, Hydra.ulic Engineering Beries No.1, Urbana, Ill" Nov. 1, 1951. 63. R 'D. Goodrich: Rapid calculation of reservoir discharge, Civil Engineering, voL i, no: 5, pp, 417-418, February, 1931.

6(1. L B. Steinl;>erg! A method of flood routillg, Civil Enyineering, vol. 8; no. 7! pp. 476-477, July, T938. ... . 67. R. K. Linsley: Use of nomogl'l!.r5i1s in solving strea.mflow routing problems, Oiilil " Engilteerin{f, vol. 14, no. 5, pp. 201}-210, l\1aYi 1944. 68; C. J. PQsey: SIicle rule forl'olltillg floods throug!l stomge reservoirs or In.kes, Engineering News-Rec(wd, vol. Hi, pp. 580-581, Apr, 25, 1935. 69. Sherlll11.n .11£. Woodwa.rd ami Chesley J. Posey: "Hydraulics .of Steady Flow in Open Cha.n:neis" John 'Wiley & Sons, Inc" New York, 1941, pp. 133.,.145. 70. J. M, Shepley ~nd C. B. "Val ton: Solving reservoir problems with cirel,l!!.r pointby-point COlllput~r, Civil E'I!lineel'ing, vol. lZ, no. 3, Pl'. 154-155, IVlarch, 1942: 71. F. E.Tatum: A SiUlplificd method of !Outing flood flows through natum! va.lley stora.ge, unpublished m~mo"mdum, U.S. Engineer's .office, Rock lsland, IlL, 1IIa.y 29, 1940, . 72. O. T .. MoCr,rthy: The unit hydrogrrtph and flood routing, unpublished manllscript, presented at a conferCHlce of the North AtlAntic Division, U.s. Army, Corps of Enginee!'s, June 24, 1938. . 73. E. W. Lane: Predicting stages for the Lower Mis.sissippi, Civil Engineering, vol. 7; .no. 2, pp. 122-125, February, 1937. ' '. :'. H. I'vb;..: A. Kohler: A forecu.:;ting technique for routing m\d cOll\hlllll\g flow'ln t.eflHS of stage, Tra.nsactions, American Geophysical Union, voL 25, pt, VI, pp. 1030-1o:J5,

, 64.

,

~~m~'i!~t~:~ia~' o~'cio;ti~di:e~~~, ~;OI~·I:'y::r ~7~~;9~, r~~~~,

65. C,

0;

Transactions,

Wisler and E. F, Brater: Aidirect method of flood routing,. TransactiDns, AmeHcan Society of Civil Engineer~, voL 107, pp. 1519-1529, 1942.

H!44.

621

.

75. Ralph E. King: Stage predictions for flood contra! opero.tions, T,'Msactiolts, American Society a/Civil Engineers, \'01. 117, pp. 600-GIl8, H)52. 76. William E. Ray and Herman F. Mondllchein: A method of fnrecn.stillg stage.~ on Jl~t rivers, Tran..sa.cli01l.s, Ame1'ica.1! GeophyS'ical Union, vol. 38, no. 5, pp. 698-707, . . October, 1957. 77. J. F. TllXpleY, Jr.: A new integrilting machine, MiWary E71ginee,·, voL 32, no. 181, pp. 39-43,1 \)39. .. : . . • 78. Frank B. Harkness: HarkMss flood rout~l':, SpecificatIOns of construction and operat.ion, Pa.t~nt File No, 2,550,692, U.S: Patent Ot1i;:e, Washington, D.C., May 1, i951. . . . 7\l. R. ic Linsley, L, W. Foskett, and M. fl.. Kohler: Electronic device speeds flood forecasting, Ellyintering News-Record, vol; 141, no. 2?, pp. 64-66.< Dec. 23, 194~, 80. R K Linsley, L. W. Foskett, and 1--,1. A. Kohler: Use "i electronical ant!.loe;:r lfl flood wave analysis, Comptes rerUius et rapports, de. l'AssembMe G~ne'rale d'Oslo, 19-28 aiJid 194.8, International Association of Scientific Hydro/agy, Publication No 29, 11)48, vol. I, pp. 221-227. 81. M. A. I{o,hler,: Application of electronic flow routing 1l1l;l.log, Transactions, American Society 01 C~'vil Engineers, vol. 118, pp. 1028-/045, 1953.

I l

APPENDIXES

'; j I I

1

I

I

,I

1

[

1I

:j

I

,I .1

I

;1

;1.

j

I

r,

',"j .

ApPElND~X:

A. GJilOMETRIC ELEMENTS FOR CmCULAR CilANNEL Sli:CTlONS dialJleter R hydulUlic radius do y depth of How T "'. top width A = wuter area D "" hydraulic clepth p wetter perimeter Z = A ~. = section factor for critio:d-flow comp u ta.tion

A d o1

.Y.. d. \

I I

r

d,'"

0.0066 0.0132 0.01!)7

O.OO6~

0.0134 0.0202 o 0268 0.0;;36

0.0001 0.000-1

0.0326

0.1990 0.2800 0.3412 0.::'!)1\) 0.4359

0.05

0.0105 o 0147

0.06 0.07 0.08 0.09 0.10

0.0192 0.024:3 0.0294 0.0350 0.0409

O.494::J 0.5355 0.5735 0.6094 0.0435

0.0389 0.0451 0.0513 0.OS74 0.0635

0.4.750 o 5103 0.542(; 0.5724 0.6000

0.0·106 O.(H74 O.01i42 0.0612 0.0682

0.0039 0.0053 0.0069 0.0087 0.0107

0.0470 0.0534 0.0<300 0.OGG8 0.0739

0.6761 0.7075 0.7377 0.7670 0.79M

0.0695 0.075-1 0.081a 0.0871 0.0929

0.6258 0.6499. 0.6n6 0.61;)40 0.7141

0.0752 0.0822 0.0892· 0.1034

0.0129 0.0153 O.Oli\) 0,0217 0.0238

0.0811 0.0961 0.1039 0.1118

0.8230 0.8500 0.8763 0.9020 0.9273

0.0985 0.1042 0.1097 0.1152 0.1206

0.7332 0.7513 0.7GS-t 0.7846 0.8000

0.1106 0.1178 0.1252 0.1324 0.1398

o 0270 0.030* 0.0;339 0.0378 0.0418

0.0220 0.0247 0.0273

0.9521 0.976 .. 1.0003 1.0239 .1.0472

0.1259 0.1:312 0.1364 o~ 1416 0.14136

0.8146 0.8285

o 8-H7

0.25

0.Wl9 0.1281 0.1365 0.1449 0, HiSS

0.8542 0.86(10

0.1472 0.154[\ 0.1622 0.1096 0.1774

0.0460 . 0 0503 0.0549 0.0597 0.0646

0.0301 0.0333 0.0.359 o 0394 0.0427

0.2() 0.27 0.28 O.2D 0.30

0.1623 0.1711 0.1800 0.1890 0.1982

1. 0701 1.0928 1.1152 1.1373 L 1593

0.15\6 0.1566 0.1614 Q.1662 0.1709

0.8773 0.8879 0.8980 0.9075 0.9165

0.1850 0.1925 0.2004 0.2084 0.2162

0.0697 0.0751 0.0805 0.0862 0.09Z1

0.0464 0.0"97 0.0536 o 0571 00610

0.21 0.22 0.23

,

Z

d,

O.O'!;

o 24 \

D

d.

02838 0.3
0.15 0.17 0.18 0.10 0.20

\

T

cIo

00013 0.0037

o.n

I-

R

001 0.02 0.03

0.12 0.13 0.14 0.15

\

P d.

O.OOLIO

i

O.OSB5

O.

0;OZ5Z

i

625

O.O
O.OUIO

0.0017 0.0027

0.0002 0.0005 0.OU09 0.0015

0.0022 O.OO;H 0.00·10

0.1]052 0.00()5 0.0079 0,001)5

·o.oua 0.0131 0.0152 0.0173 O.OHJ()

~I 626

l

APPENDIXES

627

APPENDLX A ApPENDIX A. GEOMETllIC ELEMENTS FOR CIRCULA.R

A.

AI'I.'EN.DrX

CHANNEL SECTlO~S (C01Itinued)

GEOMETRIC EI,EMENTS FOR CmCULAll

CHA.NNEL SECTIONS

P

A do'

.'!!.. d,

R da

.do

D d.

T d.

--0:31 0.32 0.33 0.34 0.35

0.2074 0.2167 0.2260 0.2355 0.2450

0.36 0.37 0.38 0.39 0.40 0.41 0.42 0.43 0.44 . 0.45 0.46 0.47 0..48 0.49 0.50

I

1.2025 1.2239 1.2451 1.2661

0.1755 0.1801 0.1848 O. ]891 0.1935

0.9250 0.9330 0.9404 0.9474 0.9539

0.2546 0.2642 0.2739 0,2836 0.2934

1.2870 1.3078 1.3284 1.3490 1.3694

0.1978 0.2020 0.2061 0.2102 Q.2142

0.9600 0.9656 0.9708 0.9755 0.9798

0.3032 0.3132 0.3229 0.3328 0.3428

1.3898 1.4303 1.4505 1.4706

0.2181 0.2220 0.2257 0.22(H 0.2331

0.3527 0.3627 0.3727 0.3827 0.3927

1. t907 1.5108 1.5308 1.5508 1.5708

O. '1027

0.51 0.52 0..53 0.54 0.55

0.4127 0.4227 0.4327 0.4426

0.56 0.57 0.58 0.59 0.60

0.4526 0.4625 0.-1723 0.4822 0.4920

0.61 0.62 0.63 0.64 0.65

0.5018 0.5115 0.52t2 0.5308 0.5404

:i d.

0.2242 I) :232:l O.240.f 0.2486 0.2568

0.0981 0.1044 0.1107 0.1112 0.1241

0.2652 0.273(3 0.2822 0.2908 0.2994

0.1310 (),1381 0,1>153

0.1528 0.11303

0,1050

0.9837 0.9871 0.9902 0.9928 0.9950

0.3082 0.3172 9·3262 0.3352 0.3446

0.1682 0.1761 0.1844 0.1927 0.2011

0.1100 0: 1147 .0.1196 0.1245 0.1298

0.2366 0.2400 0.243'1 0.2467 0.2500

0.9968 0.9982 0.9992 0.9998 1.0000

0.3538 0.3634 0.3730 0.3828 0.3928

0.2098 0.1348 0.2186 0.1401 0.1452 0.2275 0.2366 . 0.1505 0.2459 0.1558

1. 5908 1.6108 1.6308 1.6509 1. 6710

0.2531 0.2561 0.2591 0.2620 0,2649

0.9998 0.9992 0:9982 0.9968 0.9950

I 0.4028 0.4130

1.6911 1.7113

0.2676 0.2703 0.2728 0.2753 0.2776

1.7926 0.2797 1.8132 0.2818 1.8338 a 2839 1.8546 0.2860 I .1.8755 0.2881

1.1810

I I

I

1. 4101

I

A.R~~

Z

do" s

1.73~5

1. 7518

1.7722

l

I

I

0.0650

0.0864 0.0909 0.0955

0.71

I

0.4448

0.2553 0.2650 0.2748 0.2848 O.2!H9

0.1610 0.1664 0.1715 0,1772 0.1825

0.9928 0.9902 o 9871 0.9837 0.9798

0.4558 0.4670 0.4786 0.4902 0.5022

0.3051 0.3158 0.3263 0.3373 0.3484

0.1878 0.1933 0.1987 0.204'1 0.2092

0.9755 0.9708 0.95513 0.9600 0.9539

0.5144 0.5270 0.5398 0.5530 0.5666

0.3560 0.3710 0.3830 0.3945 . 0,4p66

0.2146 0.2199 0.2252 0.2302 0.2358

0.4234 0.4340

A do'

i

0.0736 0.0776 0.0820

0.1020

i

I

0.66 ! 0.67 J 0.68 0.69 0.70 !

o.OMn

j

I

I!

!

0.72 0.73 .0.74 0.75 0.76 0.77 0.78 0.79 0.80

I \

I

(conlinued)

z

T D R d~ d. ---'--I·---___~I·-~-~·· - ; - - - - 1 - - - - - -

P

d.

0.4188 0.4309 0.4437 0.45136 o -lG94

0.5804 0.5948

0.54()9 0.5594 0.5687 0.5780 o 5872

1.8965 1.9177 1. 9391 1.9606 1.9823

0.289£1 0.2917 0.2935 0.2950 0.2962

0.9174 0.9404 0.9330 o 9250 0:9165

.0.6096

0.5964 0.6054 0.6143 0.6231 0.6318

2.0042 2.0254 2.0438 2.07H 2.01)44

0.21)73 0,2£184 0.2095 0.3006 0.3017

0.9075 0.8980 0.8870 '0..87i3 o 86GO

0.6572 0.6742 0.6918 0.7104 0.7296

o 531l:!.

0.6404

2.1176 2.1412 2 1652 2.1895 2.2143

I)

3025 0.3032 0.3037 0.3040 0..3042

0.8542 0.8417 0.8285 0.8146 08000

o 1498 o 7710

o..'5G!)5

0.7934 0.8170 08420

0.3044 0.3043 0.3041 03038 0.3033

0.7846 0,7684 0.7513 0.7332 0.7141

0.8686 0.8970 0.9276 0.9606 0.9964

0.6489

0.6573 0.6655 0.6736

.'

0.6250 0.6408

!

0.2407 0.2460 0,2.510

o 2560 o 2DU8

0.4831 0.·jJJG4 0.5100 05248

O.2IJ.5:j 0.2702 0.2751 0.27H4 0.2840

0.5540

0,5850 0.6011 0.G177

0.2888 0.2930 o 2060 0.3008 0.3045

0.G347 0.6524 0.6707 0.6897 0.7098

0.3082 O.aJl8 0.3151 0.3182 0.3212

0.81 0.82 0.83 0.34 0.85

0.6893 0.6969 0.7043 0.7115

2.2395 2.2653 2.29l() 2.3186 2.3462

0.86 0.87 0.38 0.89 0.90

0.7186 0.7254 0.7320 0.7380 0.7445

2.3746 2 4038 2.4341 2.4655 2.4981

0.3026 0.3017 0.3008 0.2996 0.2980

0.6940 0.6726 0.ti49\l 0.15258 O.GOOO

1.0354 1.0784 1. 12fH 1.1800 1 2408

0.7:307 0.7528 0.7754. 0.8016 0.8285

0.3240 0.3204 0.3286 0.3307 0.3324

0.91 0.92 0.93 0.9:1 0.95

0.7504 0.7560 0.7612 0.7662 0.7707

2.5322 2.5681 2.6061 :2 .6467 26906

0.2963 0.21)44 0.2922 0.28913 O.28(l4

0.5724 0.5426 0.5103 0.4750 0.4359

L3UO 1.3932 1.4918 1.6130 1.7682

0.8586 0.8917 0.9202 0.9725 1,0242

0.3336 0,:334.5 0.3350 0.3353 0.3349

0.96 0.97 0.98 0.99 1.00

0.7749 0.7785 0.78t6 0.7841 0.7854

2.7389 2.7934 2.8578 2 9412 3,1416

0.2830 0.2787 0.2735 0.2665 0.2500

0.391\)

0.3412 0.2800 0.1990 0.0000

1.9770 2.2820 2.7916 3.9400

1.0888 1.1752 1.3050 1.5554

'"

""

0.3.'34:0 0.3322 0.3291 0.3248 0.3117

0.68~5

I

I

- 1

-I

I i

II

!'

,i "

l

Apl'ENDL"

II

B.

GEOMETRIC

ELE~'[ENTS FOR TRAPEZOIDAL, TRUNGULAR, AND

PA1lAl:IOLIC CHANN.I!a, SllCT40NS

(The lise of the chari,s" l'<:quil'es no e:1:plnlk'ltion.)

I

I I I

r

* Repl'odnced with the pet'mis~ion Qf the U.S, Agriculturai Research Service, through the ~,ourtesy of Mr, W~ 0, Ree, from" Handbook of Cha.nnel Design for Soil Mld 'Vater COli.s<:l'vation," prepal'ed by StillwMer Outdoor Hyclraulic Lnboratory, n.S. Soil Conservntion Service, Sc...ct.TP-Gl, March, 1047, Md revised June, 1954.

1 II

!

! I,

I \

'I

1

I

1 ,

.

I r

y;

629

~/~

..

0: ::!

'-6

'"

'"""o ", ~

FIG. B-l. Geometric elements of trupc"oidlll dlr>rmels with

1; 1

side slopes.

:t: 0:

'"

" 'is l?

'" ....

C»

" ~

E

"0

>-

:L

4

S

S 7 8910

0.2

0.3

A,;!Q. A, liz

FIG. B-2. Hydraulic eJemc,ots

or tJ'tLpezoidllJ channel:; with n If

1 side slopes.

...

'.

, ~

...-L-..

-

.... 0::

.2

."

.,.

2

to;)

'-'"

Areo, A,

1,2

FIG. B-3. Hydraulic elements oftrapozoidal channels with 2:1 side slopo!!.

0::

"".w w

.; " '0 l2' .2 "3 0

.f5 "., J::

Areo,A, tlZ FlO. B-4. Hydraulic element.s of t.rapezoidal channels wit.h 2~i: 1 side slopes.

t-

l!w:",--+--+.--J-j-+-i

.!!

O.')~~--!-,-+-1---1--1-4-

:c

0.7

g -g,

0.8

r--r--+-+-++-b I

---\.---i--.".j"~";::

0.6

0.5· §----i-'7"'g-,,4--+-7"i"-

a

~

10

. 15

Area, A • 0 2

FIG. B-5. Hydraulic elements of trapezoidal channels with 3: 1 side slopes.

;:::

Ii.

.;

"

15

B

~

Ql

.;1

:;
'"J:

>.

. FIG. B-6. Hydraulic elements of tr!l.pezoidal channels with 4: 1 side slopes,

(-'~-

(

-'--

~(--

FIG. B-7. Hydraulic etemehts.of trapezoidal channels with 5:1 sid" siol?es.

,.

....

rl .;

a

-0

~

c.. e.->

-1

(.3

30

t :r:

..____

.---,~u.w

15

Afea,A, HZ

FIG. B-3, Hydranlic elements of trapezoidal cbannels with 6:1 side slopes.

'!

""'"''!''''!''''l'''

2.5i=---,...·+-----,-;---~+_H_+-··--!-

-..i~1--r-::7F7""+__::.:r'9-::l_-

:1'

" 1--

a:

'"::>

'i5 0> <;;:J

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e

.!'!

:; 0

.;;;>..

::c

:';

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., '.'J

FIG. B-O. Hydmulic elements of trillnguli1r chltnnels.

B-IO. Hydraulic elements of parabolic ch.annels.

i, I:

I

I

......... Oai'l!' ..., 1.0

»:;1(;> m:::;,,",

.s:::: .......

T

, iTi

ApPENDIX

4)-

>=

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.,.

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g; ~ ~ i3 88

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o' dd dd

r

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(0£. ~ ~ ~

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ro

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u

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rrp rT""T"Tn-r'll;
cici

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it l'i n I _II> "'

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0

d

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,

,T

f:l

>-oJ

<1l

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0

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So.

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. a

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P<

<=l

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.~~;

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iii

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z

•

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:

i:iA

~.2m

rri .~.,3

c: 0'0

0

d

I

d

'"

Q ::l 0._

"ro >';:: ~ ~

....°A_ ~8

u. t..O:';l'JIJOw

t:

..0> .....-. ~-

~ ..,

~

0

Z

~

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0

:::

_

rl

~ t..?

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~

d

-"00

0

E::

:s

'" 0

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gad ~ §....

CJ

..:::0:21

i""I"'lN'T'T'TITm=J"'1'~'rT'l'l'l'ITI'"'1""r'"":rrITlTT'JT"""F!"'T'l

:>

.

en

0-::;:-

1

-< :;E

11"'

~

~;;:: .., _ 0~ , .a

:&sj

.-

0

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"CO

It·

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a. • ...

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.. . .g.., ..

::I

::20

J.

I \

640

UN

AND F(iJ.,N)_s.

IoU

TABLE D-l. THE VARIE\)-FLOW FUN(!TION FOR POSITtv,E SLO>'ES, F{1t,N)

e

~.,.

I

0

z

:t·

2.2

2A

2..6

2.8

0.000 0.020 0.040 0.060 0.080

0,000 0.020 0.0,10 0.D60 0.080

3.0

3.2

3.4

3.6

4.0

:;:; ... - t ......

r=.,

Z

....

~ -"" §

I

..:I b

IoU 1

;0..,>-:>

~ i:! :::> .. "0

.;:

I

TABLE OF THE VARIED-FLO'\',' FUNCT40NS·

.0.-0

~"'V~ ..,;t 0'-

1

D.

F(u,N) =

gOO::;

0.000 0.000 0.020 0.020 0.0·1.0 0.040 0.060 U.03O 0.080 0.080 0.100 0,100 0.120 0.120. 0.140 0.1·10 0.161 0.161 0.181 0.181 0.202 0.201 0.223 0.222 0.244 0.243 0.26.'5 0.263 0.286 0.284 O.SO 0.307 0.305 0.32 0.329 0.326 0.34 0.351 0.:148 0.3? 0.372 0.369 0.395 0.2·93 0.38 0.40 0.418 0.414 0 .. 42 0.442 0.431 0.44 0.465. 0.460 0 0.489 0.'183 0.514 0.507 0 0.539 0.531 0,565 0.557 0.592 0.582 0.619 0.608 0,648 0.635 0,676 0.663 10.691 0.678 0.7061 0.692 0.722. 0.707 0.738 0.722 0.754 0.737 0.65 0.66 0.771 0.753 0.67. 0.787 0.769 0.68 804 0.785 o 69 0.822 0.804[

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 u.20 ,p.22 0.2-1 0.26 0.28

I

°

0.000 0.000 0.000 0.000 0.020 0.020 0.020 o 020 O.OrlO 0.040 0.040 0.040 0.060 0.060 0.060 0.060 0.080 0.080 0.080 0.080 o. 100! 0.100 0.100. 0.100 0.100 0.100 0.120 0.120 0.120 • 0.120 0.120 0.120 0.140 i 0.140 0.140 0.140 0.140 0.140 O.UlD 0.160 0.160 0.160 0.160 O.HIO 0.181 0.180 0.180 o 180 o 180 0.180 0.201 0.201 0.200 0.200 0.200 0.200 0.221 0.221 0.221 0.220 0.220 0.220 0.242 • 0.241 0.241 o 241 0.2·10 0.240 0.2tl2 0.262 0.261 0.261 0.261 0.260 0.283 0.28:& .0.282' 0.281 0.281 0.281 0.304 0.303 i 0.302 0.302 0.301 0.301 0.325 0.324 0.323 0.322 0.322 0.321 0.34.6 0.344 0.343 0;343 0.342 0.342 0.367 () .3613 0.364 0.363 0.3133 0.362 0.38g 0.387 0.385 0.3&<1 ·0.383 0.383 0.411 0.40810.407 00405 0 0.403 0.433 0.430 0.428 OA26 0 10.424 0.456 a 452! 0,450 0.448 o A:16 • 0.445' 0,479 0.475 I OA72 0.470 0.468 i 0.466 0.502 0.497 I 0.494 0.492 0.489 GA88 0.525 0.52110.517 0.514 0.511 0.509 0.550 0.54410.540 0.536 0.5:34 0.531 0.574 0.568 0.563 0.559 0.558 0.554 0.599 0.593 [ 0.587 0.583 0.579 0,576 0.626 0.618[ 0.612 0 . 607 o 603 o 599 I 0.653 0.644 0.637 0.631 0.627 0.623 0.667 0.657 0.650 0.6H 0.1339 0.835 0.680 0.671 0.663 0.657 0.651 0.847 0.694 0.68'1 0.6713 0.609 0.1364 0.659 0.709 0.698, 0.690 0.683 0.677 0.67'2 0.724 0.712 0.703 0.696 0.689 0.684 0.738 0.727 0.717 0.709 0.703 0.697 0.754 0.742.0.731 0.723 0.716 0.710 0.769] 0.75710.746 0.737 a 729 0.723 0.785 0.772 0.761 0.751 0.743 0.731 1

0.000 0.000 0.020 0.020 0.0,10 0.0;10 0.060 0.060 0.080 0.080 0.100 0.100 0.120 0.120 0.140 0.140 0.160 0.160 0.1800.180 0.20010.200 0.220' 0.220 0.240 0.240 0.260 0.260 0.280 0.280 0.301 0.300 0.321 0.321 0.341 0.341 0.362 0.361 0.382 0.382 0.403 00402 0.423 0.423 0.444 i 0.443 OA650·H14 o .486 0 .485 0.508 0.506 0.529 0.528 0.551 0.550 0.574 0.572 0.596 0.594 0.620 0.617 0.631 0.628 0.643'0.640 0.655 0,652 0.667 0.664 0.680 0.676 0.692 0.688 0,70,,) 0.701 0.718 0:71.3 0.731 0.726

!

* The table of the varied-flow fUlletion fOI' i)ositive slopes F(II,N) is 1'\}pl'oduced from Ven Te Cho\v, Inteljratin' g the equntion of gradually n.ded fiow, PToceedin(lo, A JIIerican Society of Cil,ii I~!li"eej's, vol. 81, paper no. 838, pp. 1-32, Novembel', 1955. Tpe ta.ble of the varied-flow [tinction for negative slOpes F{1L,N) -8, isrepl'oduced from the author's closing discussion of this pU.per in roceedings, voL 83, JOli1'llut of Hyd1'auUC8 Di.vision, no. HYl, paper no. 1177, pp. 9-22, February, 1957.

641

643

APPENDIX D

642

APPENDIXES TABLE

TABLE

D-l. T.Hf:

VARIED-F!,OW FUNCTION FQ!l. POSITIVE SLOPES,

2.4

2.6 1 2.8

0.819 0.802 0.836 0.819 o.85S 0.B3r} 0.874 0.854 Oy892 I 0.868

0.787 0.804 0.820 0.837 0.854

3.0 0.775 0.791 0.807 0.823 0.840

3.4

3.2

F(u,N) (continued)'

3.8 I

3.6

0.766 0.757 0.7810.772 0.796 0.786 0.811 0.802 0.827 0.817

0.750 0.764 0.779 0.793 0.808

0.744 0.758 0.772 0.786 o 800

1

0.890 0.872 0.909 0.890 0.930 0,909 (I.gSO,0.929 0.971 0.949

0.857 0.874 0.892 0.911 0.930

0.844· 0.861 0.B78 0.896 0.914

0.833 0.849 O.SGG 0.883 0.901

0.823 0.839 0.855 0.872 0.889

0.815 J 0,808 0.830 0.823 0.846 0.838 0.863 0.854 0.879 0.870

0.994 1.017 1.041 1.067 1.094 1.121; 1.153 1.182 1.228 1.255

0.970 0.992 1.015 1.039 1.064

0.950 0.971 0.g93 1. OHi 1.040

0.934 0.954 0 .. 974 0.9£16 l.OW

0.919 0.938 0.958 0:979 1.001

0.907 0.925 0.945 0.965 0.985

0.895 0.914 0.932 0.952 0.9:12

0.887 0.904 0.922 0940 0.960

1.043 .1,024 1.048 1,095 1.074 1.124 1.10[ 1 155 1.131

1.007 1.031 1.055 1.081 1.110

993 1.015 1.039 1.064 1.091

0.980 1.002 1.025 1.049 1.075

1.140 1.173 1.210 1.251 1.297

0.8 5 0.8 6 0.87 0.8 8 0.8 9

1,201 1.238 1.272 1.314 1.357

1.157 1.192 1. 223 1.262 1.302

1. 091 1.119 1.149 1.181 1. 216

1.065 1.092 1,120 1.151 1.183

0.9 0 0.9 1 0.9 2 0.9 3 0.9 4

1.401 1.452 1.505 1.564 1. 615

1.343 1.294 1.253 1.389 1.338 1.29'1 1.'138 1.351 1.340 1 . '1\l3 1. 435 11.391 1.5G8 1. 501 1.4.49

1.218 1.257 1.300 1.348 1.403

1.189 1 225 1.266 1.311 1.363

1.163 1.197 1.236 1.279 1.328

1.120 1.152 1.187 1.22G 1.270

1,103 1.133 1. 166 1.204 1.246

1.582 1.665 i 1. 780 1.853 1.946 i 2.056! 2.212 2:478 3.097

1.467 1.,515 1.644 1.707 1.783

1.423 1.497 1.590 1.6'19 1.720

1.385 1.352 1.322 1.45411.41711.385 1.543 1.501 1.461 1.598 1.554 1.514 1eGGG 1.617 1.575

1.296 1.355 1.431 1.47\1 1.536

11.737 1. 833

1.652 1. 741 0.9 70· 1.969 1.866 1.945 75 2.055 09 0.9 80 2.164 2.045 0.9 85 2.294 2.165 0.9 90 2.477 2.333 0.9 95 2.792 2.621 0.9 99 3.523 3.292 LO 00 " I ~ LO 01 3.317 I 2.931 LO05 2.587 2.266 1.0 10 2.273 1.977 1.0 15 2.090 1.807 LO 20 1.961 1.711 1.779 1.531 1.0 3 1.651 1.410 1.0 4 1.552 1.334 1.0 5 LO 6 1:472 1 250 LO 7 1.404 1.195

0.9

O.g60 (](J

1.346 1.139 1.08 1.295 1.089 1.09 1.1 0 • 1.250 1.050 1.1 1 · 1.209 1.014 0.981 1.1 2 • 1.172

1.518 1.601 707 1.773 L855 1.959 2.106 2.355 2.931

11.

1.0G8

1.880 1.812 2.017 UNO 2.250 2.159 2.788 2.6133

°

!

0:>

'"

2.640 2.022 1.757 1.602 ·1:493

2.399 1.818 1.572 1.428 1.327

2.184 1.649 1.419 1.286 1.191

2.008 1.50[l 1.291

1.340 1.232 1.150 1.082 1.026

1.186 1;086 1.010 0.948, 0.896

1.060 3.967 0.896 0.838 0.790

0.955 0.86610.79010.725 0.668 0.868 0.78510.714 0.653 0.600 0.802 0.723 0.656 0.598 0.548 O.HS 0:672 0.608 0.553 0.50G 0.703 0.630 0.569 0.516 0.471

0.978 0.935 0.897 0.864 0.833

0.851 0.812 0.777 0·741; 0.718

0.749 0.713 0.681 0.652 0.626

0.565 0.631 0.601 0.575 0.551

1.16G

1.078

1.14 1.15 l.It' 1.17 1.13 1.19 1.20 1.22 1.24

1.26 1.23 1.30 1.32 1.34

0.485 0.457 0.433 0.411 0.392

2.4 ,

2.6

I 2.8

1!~3.2

'.3.4 ...

3.6

0.602 0.5290,468 11.10710.9210.7800 (l69 0.5810.50910.450 1.078 0.892 0.756/0.64710.561 1),490 0.432 1.052 0.870 0.734 0.627 0.54.2 0.473 i 0.417 1.027 0.850 0.713 '0.608 0.525 0.458 ' OA02 0.825 0.810 0.787 0.755 0725

0.694 0.676 0.659 0.628 0.600

0,591 0,574 0.559 0.531 0.505

OA17 OAOO 0.384 0.369 0.356

(3.8

4.0

0 .'374'

1

0.358 0.343 0.329 0.317

0.509 0,494 0.480 0.454 OA31

0.443 0.388 0.343 0.4.29 0.37510.331 0.'1016 1 0.363 0.320 .0.392' 0.341 0.299 0.371 0.322 ,O.28[

0.305 0 O. O. O. .0219

1,0.85S 0.692 0.1574 0.482 0.410 0.827 0.666 0.551 0.461 0.391 0.80°10.644 0.530 . 0.44'20.373 0.775 0.625 0.510 0.424 0.35i 1 0.752,0.6050.4920.4080.342

0.351 0.304 0.2135 0.334 0.28810.250 0.3180.274 0.2:37 0.3lH' 0.260,0.225 0.290 0.24810.214

0.23310.205 0.219\ O. HJ3 0.207 0.11l1 0.191i 0.171 0.185 0.162

1.003 0.981 0.950 0.922 0.1387

I

I

1.46 1:48 l.50 1.55 L60

0.64210.5050.4050.3300.273'0.2270.19: 0.1()2 0.139 0119 0627,0.4930.3940.3200.2630.2190.1840.1560.1330.113 0.613 0.480 0.383 0.310 0.2.";5 0.211 0.177! 0.149 0.127 0.108 0.580 0.451 0.358 0.288 0.2.15 0 194 0.1(31 0.135 0.114 0.097 0.5510.425 0.335 0.269 0.218 0.1790.148 0.123 0.103 0.087

1.65 1.70 1.75 1.80 L85 1.90 1.95 2.00 210 2.20

0.52.5 0.402 0.3160.2510.2030.165 0.136 0.1130.0940.079 0.501 0.381 0.298 0.236 0.189 0.153 0.125 0.103 0.086 0.072 0.480 0.362 0.282 0.222 0 177 0.H3 Q.1l6 0.095 0.079 0.06,5 0.400 0.34Q 0.267 0.209 a. lOG I 0.13:3 ° 103 0.088 O. 072 0.050 0.442 0.332 0.254 0.198 0.15610.125 0.100 0.082 0'.057 O.OSS 0.425,0.;3150.2420.1880.1470.1170.094 0.07G 0.002 0.050 0.40? 0.304 0.231 0178 0.139 °.110 0.088 0.070 0.057 0.046 0.3D5 0.292 0.2210.[090.1320.10·10.0820.01)80.0530.043 0.3!Hi 0.27.'3 a 202 0.154 0.119 1i 0.092 (J.on 0.058 0.0~6 0.037 0.34B 0.253 0.1813 0.141 0.107! 0.083 0.0511 OOSI 0.040 0,032

2.3 0.325 0.235 0.173 0.120 0.09810.075 2.4 0.308 0.220 0100 0.1:1\10.0890.058 2.5 0.2920.2070.1500.110 0.082 0.062 2.6 10.277 0.197 0 140 0.102 0.07G 0.0.57 2.70.264 0.188 01310.005 0.0700.05'3

11.

Q.535 0.506 0.480 0.457 0.436

i

F(11,N) (conlin'l!ed)

0.731 0.588 0.475 0.393 0.329 0.278 0.237,0.20'1 0.176 0.153 0.711 0.567 0.459 0.378 0.31(\ 0.266 0.2ZG 0.19,1 0.167 0.145 1.'10,0.6920.5480.4440.3650.304 O.25G 0.21701850.1590.138 1.42 0.674 0.533 0.431 0353 0.293 0.246 0.208. 0.177 0.1521.°.131 1.44. 0.658 0.517 0.417 0.341 0.282 ·0.23(j 01\l9 0.169 '0.145 0.125

i

0.595 0.563 0.536 U.511 0.488

2.2

1.36 1.33

1.752 1. 699 1. 652 LIn 0 1.873 1. 814! 1. 761 1.714 2 . 07!) 2 008 11. 945 1. 889 2. 554 1 2 .4.57,2.370 2 293 co "" '" 1 '" I 856' 1. 725 1.610. 1 1.S08 1.384 1. 279 i 1. 188 1.107 1. 182 1. 089 007 0.936 1.065 0.978 0.902 0.836 0.982 . 0.900 ! 0.828 0.766

00

00

THE VARIED-FLOW FUNCTION Foa POSITIVE SLOPES,

-T.13 -11.138 [0:950 I 0.805/ 0.692

0.769 0.752 0.766 0.780 o 794

0.8 1 0.8 2 0.8 3 0.8 4

0.80

~",1

4.0

0.940 0.913 0,961,0933 0.98510.954 1.007 0.976 1.031 0.998 LOli6 1.022 1.083 1.046 l.UO 1.072 1.139 1.099 1.171 1.129

0.7 5 0.7 6 0.7 7 0.7 8 0.7 9

D-l.

0,441 0.415 0.392 0.372 0.354

0.0.t5 0.035 0.028 0.040 0.031 0.024 0.035 0,028 0.022 0.03:~ 0.025 0.019 0.029 0.022 0.017

2.8 2.9 3.0 3.5 4.0

0.2520.1760.124 0.08(10.01350.0-180.0360.0270.0200.015 [0.241 O.WG 0.11.7 0.083 (J.050 0.044.0.03:3 0.024 0.018 0.014 ;,0.230 0.159 0.1l0,1 0.078 0.056 0.0.J.1' 0.0::\0 0.022 0,{)17 0.012 0.1900.1260.085'1°.0590.0410.(21) 0.0210.015 0.01l 0.008 0.1610.104 0.0690.0460.0310.022 0.015 0.010 0.007 0.005

4.5 5.0 6.0 7.0 8.0

0.1390.0870.0571°.037.0025 0.122' 0.076 0.048 0.03110.020 0.098 O. aGO 0.0313 0 022 0 . 014. 0.081'0,048 0.028,0.017 0.010 .0.OGg 0.040 0.022 0.013 0.008

9.0 10.0 20.0

.J.

00.58 0.052 0.0.,!7 0.043 0.030

O.Oli 0.013 0.009 0.006 0.005

0.01110.008 O.oun 0.006 O. naG IJ. 004 0.004 O. 002 0.003.0.002

0.005 O.OOt,· o. 002 0.002 0'001 '

0.004 0003 0.002 0.001 0.001

10.060 0.034 0.01910.01110.006 0..004 0.002 0.001 0.001 0.000 10.0530028 0.016 O.OO!) 0.005 0.003 0.002 0.001 0.001 0.000 0.0'23 0,018,0.011 i 0.0(10 0.002 0.00110.0011/1.000 0.000 1 0.000 'Ii

.~

), ,

'\

I

I • 1

I

i

(J44 TAlILlll

D-1.

T,H!l YARH!n-nOW FONCTIOH 1'0& POSITfVE SLOPES,

4.2

,

I

0.00 0.02 O. [1·1 0.00 0.08 0.10 0.12 O.H 0 III 0.18 \).2f) 0.2'2 0.2-l

o.'ili 0.28

0.30 0,32 0.34

0.31; 0.38 U,4ll 0.42 0.4-1

Vi

0.000 0.000 O.O:W 0.020 O.O.fU 0.040 O. OM O.OUO 0.080 0,080 0.100 0.120 0 1 . ..10 1 0.160 0.180 . 0.200 0.220 0.240 0.260 0.280

5.0 a 000 0.020 0.040 O.OGO 0.080

6.2 0.000 0.020 0.040 0.000 0,080

(I,son

0300 0321 0.320 0.341' 0.3-l0 O.3lil! 0.301 U.:j81 0.381 OA02.0AOI o , 422 ' 0.421 o H31 0.442 0..163 I tL -!()2 o , ~ 84 . U. 483

D .300 0.320 0.340 0.360 i 0 .. 381

TADLE

F(lt,N) (contintled)

G.B

0.000 0.000 0.000 0.020 0.020 0.020 0.040 0.0,10 0.040 0:000 0.060 (i.OGO 0.080 0.U80 0.080

0.100 0.100 0.100 0.100 0.120 0,120 0.120 0.120 0.140 0.140 0.140 O.UO O. HiD 0.160, O. 160 O. HlO 0.180 0.lS0 0 180 0.180 0.200 0.:)1)0 0.200 0.200 U.220 0.220 0.220 0.2.20 0.240 o 'HO 0.2'10 0.240 o 260 o.30d O.2liO O.2no 0.280 0.21,;0 0.280 o ::80

0.000 0.020 o 040 O.OGO 0.080

0.000 0.020 0.040 O.OGO 0.080

0.000 fJ.020 0.040 0.0.60 0.080

·0.80 0.81 0.82 0.83 0.84

0.878 10.865 !0.854 0.895\ 0.881.0.869 0.913 0.897 0.885 0.931.0.914\ 0.901 0.949 0.932 0.918 0,969 0.990 1,012 1.035 1.060

0.100 0.120 0.1'10 O. HlO 0.180

0.100 0.120 0.140 0.160 0.180

0.100 0.120 0.140 O,lliO 0,180

0.100 O. J20 0.140 O.WO o 180

0.100 0.120 0.1'·!0 O.WO 0.180

0.85 0.86 0.87 0.88 .0.89

0.950 0.970 0.990 1.012 1.035

0.935 0.95-1 0.973 0.99-4 1.015

0.200 0.220 0,240 0.360 0.280

0.200 0.220 0,2!O 0.260 0.280

0.200 0.220 0.2-10 0.260 0.230

0.200 0.220 0.2'10 0.260 0.280

0.200 0.220 0.240

0.90, 1. 08711.060 1.116 1.088 0.91 '0.92 .1.148 1.117 1.184 1.151 '0.93 1.225 1 188 0.9t 0.950 1.272 1.232 0.060 . 1.329 1.285 o .970 \1. 402\1. 351 0.975 1, 447 1 1 ,393 0.980 . 1.502 1.443

1.039 1.064 1.092 1.123 1.158'

0,2~0

0.380

0.300 0.300 0,300 0.300 0.300 0.300 0.300'

0.3-10 0.340 0.340 0,340 0.340 0.340 0.340 0.360 10.360 0.360 0.3GO 0.3GO 0,3130 D.3nO 0.380' 0.380 0.380 0.380 0380 0.380 0,380

OAOI 0.400 0.400 b 400 a.lOa 0.·100 OAOO 0.-100 0 .421 0,421 0.-120 0 ... 20 0,.,120 0.-120 0,'120 0.420

·0.985 0.990 0.995 0.999 1.000

I . -,

i

I

0.711 0.71

o.n

o.n 0.74

0.75 0.7il

r

U.77

0.78 0.71)

0.735 0.7'18 0.761 0.774 0.788 0.802 o 817 O.8:n 0.847 10.8\l2

0.7'21 O.HO 0.752 0.765 0.779

0.722 0.7'34 0.7-1u 0.759 0.711

0.702 0.784 0.778 0.7!Jl. O.soo 0.820 0.811 0.80-1\ 0.834 0.825 0.817 0.849 0 a-39 0.831

10.798

0.710 0.708 0,706 I. 0.721 0.719 0.717 0.732 0.730 0.728 0.743 0.741 0.'/39 0.75-4.1°·752 0.750 o 773 0.769 O.'((lB 1 0.763 i 0.761 0.786 0.782 o. Fa I 0.775 • 0.773 0,798 0.794 01.\J~10'Z87jO.784 0.811 0.806 o . 80_ 0 ' 199 0 ,796 0.824 0.819 0.a1sI0.811 0.808.

0.71710.714 0.729 I o.nu 0.741 0.737 0.753 0.749 0.7GG 0.761

THE VARIED-FLOW FUNCTION FpR POSITfVE SLOPES,

5.0

0.320 0.320 0.320 0.320 0,320 0.320 0.320

g:~~~1 g:~~~

D-l.

7.8

OA41 0.441 O,Hl 0.4'11 0.440 o 4-10 0.440 ' 0.4'10 0,402 OAill OAU1Io.401 0.460 O,-!liO O,·lGO ·0.4()O O. 482 0.481 0.481 0.-181 oA80 0.480 0.480 0.480 0,'18 0.50 . 0.505 \ 0,504 0.503 0,502 0.501. 0.501 0.501 0.500 0,500 0,500 0")05')'\ 0.522 0.522: 0.521 0.521 0.521 0.520 0.520 0.52 .,,_1,. 0 . 5')5' ~ 0.5·1 o 5!8· 0.5·16 a.5·!.! 0.5'1;; 0.5-12j 0.542 0.541 O.SH 0.541 0 541 0.50 0.570 0.56·1 0.5G;; 0.562 0.562- 0.5Gl o .5ul C5GI 0.58 0.5U:l o 585 0,583 0.583 0.582 0.582 0.581 0.581 0.(114 0.60 o . till 0 ..!lOS O.GOG o . (l05 a. (;04 0,1303 0.G02 O.GO:'! 0.601 U. (21) 0, uz:'! U.OW 0.617 O.tH5 0.1114 0.fi13 0.IH2 0.612Io.till 0.01 1 O.GZ 0.637 0.63:1 0 mw O.G28 0.G26 0,625 0,(\24 U.623 O.G22 0 022 0,G4U U,6-14 O.!Hl 0.038 o 133G 10.635 0,G34 0.633 0.G32 0.(',32 O.llS 0.64 o.Uti! 0.U5G 0.G52 o 649 0.G·l7 0:{l46 0.645 0.644 0643 10.642 ().65 0.673 O.OG7 O. till3 O. llt30 0,6580.656 0.655 0.654 0.()5S o .65:~ 0.G6 0.G85 0.67\) 1 0.G75 0.(l72 0.66!! I 0.6U7 0.GOU.0.665 O.(l().j, 0,663 0.G()7 o .Gll! 0: 08U 0.683 0.680 10.678 0.G76: 0.675 0.674 Q.6n. 0.G7 (j .u3 0.70n O.7113· O,60S 0.694 0.6\)1 G.GSl) 0.Hil7 :0.686 0.68i) 0,6!H O.ti9 0.122 o 715\ 0.710 0.70G 0.703 0.700 0.698 0.m.l6 0.695 O.G\)'1 OAl)

645

APPENDIX· D

APPENDIXES

0.712 0.723 0.73-1 0.H6 ·0.757

0.759 0.771 0.782 O. 71B 0.805

I

1

,)

0.845 0.8.38 0.860 0.852 0.875 \ 0.866 0.890. 0.S81 0.906.0.897 1 0.923 0.912 0.940 \ 0.030 0.95!) (1.9-17 0.97810.966 0.999 0 986

G.G

6.2

5.8

0.893 0.913 0.929 0.1146 0.964

1.045 1.072 i 1.1?11 1. li;14

1.029 1.054 1. 081 l:1l3

1.016 1.039 1.065 1.095

",

1.033 0.737 0.610 0.537 0.'186 I

..,

""

0.951 0.870 0.669 0.612 0.551 0.502 0.483'0."1-4.0 0.436 0.394

L03 1.04 1.05 1.06 1.07

O.filS 0.535 0,469 0.415 0 370 0.333 O.&M 0.477 i 0.415 1 0.365 0.324\ 0.290 05(H 0.432 0.374 10.328,0.239 0.259 0.164 0,395 0.342 0.298' 0.262 0.233 0.431 0,3GO 0.315 0.273 0.239 i 0.212

1.13. 1.14 1.15 1.16 1.17

0.305 0.291 0'.278 0,26D 0.255

1.18' 1.19. 1.20 1.22 1. 24

0,244 0.235 0.226 0.209 0.195

I

0.3-11 0.319 0.299 0.282 0.267

Q,';!g2 0.272 0.254 0.239 0.225

0.252' 0.234 0.218 0.204 0.192

0.253! 0.240 0.229 0.218 0.208 0.1!)\} 0,1111 0.183 0.168 0.155

0.212 0.201 0.191 0.181 0.173

0.18'1 0.170 0.161 0.153 0.145

0.165 0.157 0.150 0.138 0.127

0.887 0.88~ 0.901 0.896 0.91() 0.911 0.932 0.927 0.949 0.943

0.891 0.90(l 0.922 0.938 0.956

o £lUO 0.986 1 0.979 1.006 0.999 1.029 1.021 I 1.054 1.0H

1. 021 11.007 0.994 0.984 0.9711 0.967

1.417 1. 036 0,873 0.778 0.111

0.4.03 0 379 . 1 0.357 0.338 0.321

1.264 1.138 0.915 0.817 0.766.0.681 Q.68010,602 o li20 0.54.6 ,

.7.8

0.905 0.921 0.937 0.955 0.974

1.172 1 1.148 1.128 1.21711.188 1 1(;7 1.275 i 1.246 1.319 1. 31111. 280 ' 1.250 1.354 I 1.339 1.2881 1.573 1.508! 1.454 1.409 I 1.372 1.337 1.6711.593\1.537 1.487 1 4'14 1.404 1.838 1.751 1.678 1.617 1,565 1.519 1.917 1. 8,15 1.780

I

7.4

i

0.828 0.823 0.820 0.818 0.841 0.836 0.833 0.830 0.854 0.850 0.846 0.842 0.8138 0.8133 0.859\ 0.85.5 0.882. 0.877 0.872 0.8118

1.199 1.248 1.310 1.34.8 1.395

,.!," "!,02 '2.~02

7.0'

o . 832 . 0.84(l 0.8GO 0.874 0,889

1.001 1.00f! 1.010 L015 1.02

1.08 1.09 L 10 1.11 1.12

0.705 0.716 0.727 0.737 o 748

.5.4

F(It,N) (cohtimled)

I

o . 220 O. 194 0.204 0.179 i 0.18910.165. O. 17G 1 0,] 5,1 0.IG5·0.143

1.003 1,027 1.050 1.080

0.905 1.016 1.040 1.066

l.Ul 1.149 1.197 1.227 1.262

1.09.7 1.084 1.133 1,119 1.179 1.162 1.207 1.190 1.241 1.221

1.309 1.313 1.479 1,725

1.2B4 1.3<1& 1.451 1.678

L073 1.106 1 148 1.173

".201

1.23S 1.243 1.319 1.297 1.416 l.3SS 1.635 1 596

eo

00

'"

"

0.803 0.553 0.459 0.S\19 0.358

0'.746 0.526 0,422 0.366 Q.327

0.697 0.481 0.389 0.33() 0.300

0.651 0.447 0.360 0.310 0.271)

0.300 I 0. 272 1 0.249 0.262 0.236 0.2140.231 0.:2080.189 0.209 0.187 0,170 O.H}! 0.168 0.151

0,228 0.105 0.1740.154. 0.136

0.172 0.158 0.146 0.135 0.125

0 153 0.140 0.129 0.119 0.1l0

0.137 0.125 ' 0.114 0.105 0.097

0,123 0.112 0.102 0.094 0.086

0.155 0.H6 0.137 0.130 i 0,123

0.135 0.126 O.l1S 0.111 0.105

0.111' 0.109 0.102 0.096 0,090

0.102' 0.095' 0.089 0.084 0.078

0.090 0.084 0.078 0.072 0.068

0,080 0,074 0,068 0.064 0,060

0.1113 0.110 0.W5 0.095 0.08ti

0.099 0.094 0.OB8 0.080 0.072

0.085 0.080 0.076 0.M8 0.060

0.073 0.068 0.064 0.057 0

0,063 0.059 0.056 0.049 0.044

D.OSS 0.051 0.048 0.042 0.038

TABLE TABLE

D-L

,~Z I

THE V,\It1lm-FLOW FUNCTION FOR POSITfVE SLOPES,

4.2

1

4.6

I 5.0 I'

5.45.~

6.2

0.1l71~.095

~O;

6.6

7.0'

7.4

I

7.8

1.36 1.38 1.40 ] .42 1.44

0.134 0.127 0.120 0.114 O.lOS

0.103 0.097 0.092 0.OS7 0.082

0.081 0.076 0.071 0.067 0.063

0.064 0 060 0.056 0.052 0049

0.052 0.048 0.044 0.041 0.038

0.042 0.038 0.036 0.033 0.030

0.034.0.028 0.032 0.026 0.028 0.023 0.026 0:021 0.024 0.019

fl.023 0.021 0.0]9 0.017 0.016

1.46 1.48 1.50 1.55 1.60

0.103 0.098 0.093 0.083 0.074

0.077 0.073 0.069 0.061 0.054

0.059 0.046 0.056 0.043 0.05310.040 0.046 0.035 0.040 0.030

0.036 0.033 0.031 0.026 0.023

0.028 0.026 0.024 0.020 0.017

0.022 0 ..021 1).020 0.016 0.013

0.018 0.017 0.015 0.012 0.010

0.014 I 0.012 0.01310.010 0.012 0.009 O.OlD 0.008 0.008 0.006

1.65 1.70 1.75 1.80 1.85

0.067 0.060 0.054 0.049 O. 045

0.048 0.0·13 0.038 0.034 0.031

0.03,5 0.031 0.027 0.024 0.022

0.026 0.023 0.020 0.017 0.015

0.019 0.016 0.014 0.012 0.011

0.012 0.010 0.009 O. 008

0.011 0.009 0,008 1 0.007 a .00B

O.OOS 0.007 0.006 0.005 O. 004

0.006 0.00.5 0.004 0.004 0.003

1.90 1. 95 2.00 2.10 2.20

0.041 0.02S 0.0200.014 O. 038 O. 026 a . a 18 O. a 12 0.035 0.023 0.016 0.011 0.0300.0190.0130.009 0.025 O. Oll) 0.011 o. 007

0 . 010 a .008 0.007 0.006 0.00.5

0.007 0.006 0.005 0.004 0.004

0.005 O. 004 0.004 0.003 O. 002

0.004 O. 003 0.003 0.002 a .001

0.003 0.002 0.002 0.002 0.002 0.001 0.0010.001 0.001 0.001

2.3 2.4 2.5 2.6 2.7

0.022 0.019 0.017 0.015 0.Ot3

0 . 003 0.002 0.002 0.001 0.001

0.002 0.001 0.001 0.001 0.001

0.001 0.001 0.001 0.001 o.oeo

0.001 0.001 0.000 0.000 0.000

0.014 0.012 0.010 0.009 0.008

I 0.014

0 009 0.0060.004'1 0.008 0.005 0.00:3 0.006 0.004 o.ooa 0.005 0.003 0.002 0.005.0.003 0.002 1

0.019 0.017 10.016 0.014 0.013 1

I

0.005 a 004 0.003 0.003 0.002

0.001 0.001 0.000 0.000 0.000

2.8 10.012'0.0070.00410.002 0.001 0.001 o.oor 0.0000.000 0.000 2.9 0.0100.0060.00410.0020.0010.0010.0000.000 0.0000.000 3.0 10.009, 0.005 0.003 0.002 0.001 0.001 0.000 0.000 0.000 0.000 3.50.006 0.003 0.002 0.001 0.001 0.000 0.000 0.000 0.000 0.000 4~0 ,0.004 0.002 0.001 0.000 0.000 0.000 0.000 0.000 0.000 0.000

'il.0 10.0 20.0

0.003 0.001 0.001 0.000 0.0020.0010.000 0.000 0.0010.0000.0000.000 0.001 0.000 0.000 0.000 o. dO~ 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000

I

0.000 0:000 0.000 0.000 0.000

D-L

THE VARIED-FLOW FUNC'TION Fon POSITIVE SLOfES,F(u,N)

F(u.,N) (continued)

1.26 -'0.182 ,0.145 0.079 0.0550.0460.03910.033 1 1.28 0.170,0.1350.1080.0880.072 0.060 0.050 o.on 0.03510.030 1.30 .0.160 0.126 0.100 0.081 0.066 0.0~4 0.045 0.037 0:031 0.026 1. 32 0.150 0.118 0.093 0.075 0.061 (). O:JO 0.041 0.034 0.028 0.024 1.340.142 0.110 0.087 0.069 10.056 0.045 0.037 0.030 0.025 0.021

4.5 5.0 6.0 7.0 8.0

647

APPENDIX: D

646

0.000 0:000 0.000 0.000 o. 000

0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000

a . 000 0.000 0.000 0.000 0 . 000 0 .000 o. 000 I 0.000 o. 000 0: 000 0.000 0.000 0.000 0.000 0.000 0.0000.000) 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0 .. 000 1

~

8.2

0.00 0.02 0.04 0.06 0.08

0.000 0.·)20 0.040 0.060 0.080

0.10 [l.12 o 14 0.16 O.lS 0.20 0.22

9.0

S.6

9.8

9.4

----

----1----

0.000 0.020 0.040 o 060 0.080

0.000 0.000 0.020 0.020 0.040 0.040 O.OGO .. 0.060 0.080 0.080

0.100 0.120 0.140 0.160 0.180

O.lDO 0.120 O.UO 0.160 0.180

0.100 0.120 0.140 O.HiO 0.180

0.200 0.220 0.240 0.260 0.280

0.200 0.220 0.240 0.260 0.280

0.300 0.320 0.340 0.360 0.380

I

0.000 o 020 0.040 0.060 0.080

0.100 0.120. 0.140 0.160 0.180

0.100 0.120 0.140 0.160 0.180

0.200 0.'220 0.240 0.260 0.280

0.200 0.220 0.2'10 0.260 0.280

0.200 0.220 0.240 0.260 0.280

0.300 0.320 0.340 0.360 0.380

0.300 0.320 0.340 0.360 0.380

0.300 0.320 0.340 0.360 0.380

II 0.300 0.320

OA8

0.400 0.420 0.440 0.460 0.480

0.400 O. '120 0.440 0.400 0.480

0.400 0.420 0.44.0 0.460 0,480

0.400 0.420 0.440 0.460 0.4CiO

1 DADO 0.420 0.440 0.460 0,480

0 . .50 0.52 0.54 0.56 0.58

'0.500 0.520 0.540 0.561 0.581

0.500 0.520 0.540 o ..'iB0 0.581

0.500 0.520 0.540 0.560 0.580

0.500 0.520 0 . .')40 O.5uO 0.580

0.60 0.61 0.62 0.63 0.64

0.601 0.611 0.621 0.632 0.6'12

0.1301 0.611 0.621 0.631 0.641

O.GOI 0.611 0.621 0.631 0.641

o 500 0.520 0.540 0.560 1 0.580 0.600 0.6J1 . 0.621 0.631 0.641

0.69

0.652 0.662 0.673 0.683 0.694

0.652 0.662 0.672 0.683 0.693

0.651 0.662 0.672 0.682 0.692

0.651 0.661 0.672 0.682 0.092

0.651 0.661 0.671 0.681 0.692.

0.70 0.71 0.72 0.73 0.74

0.704 0.715 0.726 0.736 0.747

0.704 o 714 0.725 0.735 0.746

0.703 0.713 0.724 0.734 0.745

0.702 0.il3 0.723 0.734 0.744.

0.702 0.712 0.723 0.733 0.744

0.75 0.76 0.77 0.78 0.79

0.758 0.769 0.780' 0.792 0.804

0.75i 0.768 0 .. 779 0.790 0.802

0.756 0.767 0.778 0.789 0.800

0.755 0.766 0.777 0.788 0.799

0.754 0.765 0.776 0.787 0.798

O.U 0.26 0.28 0.30 0.3~

0.34 0.36

0.38 0.40 0 .. 42 0.44 0.46

0.65 0.66 0.67

0.68

(continued)

---

I

I

I

I

t 0-

.l l

0.340 0.360 0.380

I

.!

0.600 O.GlD 0.621 0.631 0.611

r

1 I

;

/ 1

.,I

-~-~

~.-~

-.I-~-.--~-

i

648

APPENDIXES

APPENDIX D

r F(>II ,N) (COl1iin.l~ell)

I

TABLE

D-l.

THE VAlUED-now 'FO'NCTION FOR POSITIVE SLOPES,

0.8

,

0.80 0.81 0.S2 083 0.84 0.85

:'

J

j

II

I

0,80

0,37 0.88 o.S!)

o,no

a 815

0.827 0.8:l\) 0 852

O.SUS

0.878 0.SU2 rUJ07 a 921

0,937 0 (,)54

{UJI

0, !r72·

O.!)2 () II:!

O.lllll

O.H·I

I.U:Hi

,I

°0.\)50

I

'(J,Hi5 (I. \l80 0.\i85 0.\)\11)

1.130

!

!J!i!)

0,\);0

n,D!).)

() I'

t, ,

I/.

O,ntH! 1.O!H)

1.001 1.005 1. U10 :1.(jJ5 I.U:20 1.03

J 0-1 1,05

I

!

'j

I J

!

I

I

I 2:2-1 1 275 1,3133 1.5UO

'"

0(\14 0 -1:.'0 0.::137 0.:'>8U 0,257 0,2U O.ln

1. 13

n.07l

1.][

1. 15'

I

I 157

I 187

1.12

1.07 108 1.1J!1 1.10

1.11

,,'

1.002 1.0117

0 158 0.140 0.J:.l3 0.111 0, 1111 0. Uti:.' 0.084 li.077

LOU

!

1,012.

1.16 1.17 1.18

j,

1.19

"j

1 ,).) 1.24

1.20

O,OUS 0 001 Q,OS!i 0.052 0.048 O.!H5

O.OH 0,037 0.032

0:813 0.825 0 8;j7 0.S4\) 0.80:.! 0.875 a 389

0,\)U3

0,918 0,\)33 0.\)4\) 0 (.)()7 0 980 1.()OU 1,0211 105:; 1.085 1.1:£.1 I 147 1.175 1.210 1 :WO L:H2

1.530

""

0.&77 o .3ll1 0.313

O.:W!J O. :.>:;17 0.1~5

0, !li5 0: l-l3 0.]'17 0.11:'> 0. J(j I O.O!)1

0.08;3 0.U75 O.OUU 0.OU:3 0,058. O.Oi>4 0.050

a.u·w

0.042

0,0;~9

0.037 0.032 O.0::!8

0.811 0.823 0.8:35

O.S4/

0.8LiO

O. O. O,!)OO

0.0i4 O,('):l~)

0.9·14 0.801 .0.USO G,I);)\)

1.02:2 I>OH 1.074 i 112 1 1:3-1 1 lUO 1.

1 1

0.810 0.8:,\2 0.8:l3 0.8·15 0.858 0.870 0,883

0, sn:' 0.011 0.925 0.0·10 0.057 0,1),15 0 Ufi·l

LOlli 1 . 0·10 1.0Li3

1.100

1. 122 1.150 1. 1

1.3U2

1.470

1. co

o,S·W O.:lliS

0.2!14 0 O.

0.1 O. U.

o.:ns o 2:37

O::?UU

O. liD

0,14:3 0.12-1:

0, IlL!

(J.IOU

0.10:.'

0.0\1·1 O. 0 0 Oli7

0,0()3 0.08:2 O,UU

0,()()7

00li2 0.051)

0,05:.' 0.0·18 0.0.J5 0.0-11 0,037 o 0;;4 0,032 0.0:'>8 0.024

O.OliO

O,USS 0.050 O,O·H)

0,043 0,040 0,1)30 0,0;33 O,O:jO

U,023 0,024 0.0:.n

I.ZG

0.820

1.28

0.8:31 0.8H

l.30 1.32

u.85iJ

0,S08 0,881 0.894

1.3-1 1.36 1.38

O.ll:.!:.!

1 42

lAO

O,DOS

1,4,4

0.93;"

1.'16 1.48·

%3

°0,070

1.50 1.55

O. ~'Sli

1 >UlO LO:>::!

1. GO

1 65 1.70

1.053

1.087 1. 108 1.132 1.165 1,208 I 280 IA47

""

0.519 0,350

8.2

---

0.808

1.7.5

1.80 1.85 1. 90 1.95 2,00 2.10 2,;20 2,3 :2 4 2,5

'"

0.494 0,:,3] 0.262 (I

:223

2,1\ 2.7

0 190

O.tSu O,(:H 0.lI5

2.8 2.9 3.0 3.5 4.0 4.5 5,0 6.0 7.0 8.0 IJ.O 10.0 20.0

0.OD8

0.086 0,077 O.OLiv 0.OW2 0.055 0050 0,045 0.041 0.0:38 0,035 0.0;;2 0.029 0.027 0.025 0,021 0 018

f r, ,..\

649

0.028 0.025 0.022 0.020. 0,018 0.016 0,014 0.013 0.011 0.\)10 O.OO!) 0.009 0.008 O.OOG .0.005 0.004 0003 0.002 0.002 0.002 O,OOI O.OOl 0.001 0.001 0.000 0,000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

o.obO

. 0.000 U.OOO 0,000 O.OOU

8.6 -------~-

0.02-1 0.021 O.OH)

0 .. 017 0.015 0.013 0,012 0.011 0.009 0.008 0.008 0,007

O.OUO

0.005 0.004 0.003 0,002 0,002 0.001 0.001 0.001 0.001 O.OOl 0.000 0.000 0.000 0.000 0,000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0,000 0.000 0.000 0,000 0.000 0.000 0.000 0.000

I.

9.0

F{1!,N) (continued) .

9.4

9,8

0.018 0.016 0,014 0.012 0.010 0.009 0,008 0.007 0,006 0.00(3 0.005 0.004 0.00", 0.003 0.002 0.002 0.001 0,001 0,001 0,001 0,001 0.000 0.000 0,000 0,000 0,000 0.000 0:000 0.000 0.000 0.000 0.000 0,000 0,000 0.000 0.000 U.OOO 0.000 0.000 0,000 0.000 0,000 0.000

0.016 O.OH 0.012 0.010 0.009 0.008 0.007 O.OOG 0.005 0.005 O.OOt 0.004 0.003 0.003 0.002 0.001 0.001 0,001 0,001 0,001 0.000 0,000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0,000 0,000 0.000 0.000

---

0.021 0.018 0.016 0.014 0.012 0.011 0.010

D.OO?

0.008 0.007 0.00(3 0.005 0.005 0.004 0.003 0.002 0.002 0.002 0.00] I 0.00] ! 0.001 0.00] 0.000 0.000 0.000 0.000 0.000 0,000 0.000 0.000 0,000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 . 0.000 0:000 0.000 0,000 0.000

'.

II

650

'I

~N

''''1 APPENDIXES

D-2.

TAlJr.E

I

Tlll'l

V A.RIED~Ft,OW

FUNCTION FOR, NEGATIVE Sr.OPES, F(u.,N)_8~

1

.-~-

~~l~~--=- ~:~I~I~I~~I~I~!~~J~

:1

1:'J i,

0.00 0,02 0.04 0.06 0.08

1

!

I

I

0 0(\0,0.000 I 0.000 0.000 0 .GOO : 0.000 0.000 0.000 0.000 I 0.000 0,020 0.020 0.020 0.020 0.0:l0, 0.020 0.020 10.020 0.020 0.020 10.040 1I 0.040! 0.040 0.040 0 01010.040 0.040' 0.040 0.040 0.040 0.060 0.060 0.060 0.060 0.060,0 01'10 0.01')0 0.060 0.060 0.060 O. OBO 0.080 0.080 I 0.080 0.080 0 080 0.080 0.080 0.080 0,080 o.mm! 0.100 10.100 II 0, 100 0,100 I (J,100 0 100 0 00 0 00 0,119 10,11910.120 1,°,120 0,12°10120 0,1::1910,139 0.140 i 0 140 0,140 ,) 1'10 0.140 0,1400',140 0 140 0,158'0,159 0.15910,160 0,160,0.160 0,160 0,1(0)°,1601,0.160 0.1780,1790.1790,1800.1801°.1800,1800.1800.18010,180 01197 0, HIB 0,199 0.199 0.200' 0.200 0.200 0.200 0.200 0.200 0.216 0,217,0.218 0.219 0,219! 0.220 0.220 0.2201°.220 0.220 0.234 0.236 '0,237 0.238 0.239 0.240 0240 0240 0240 0240 0.253 0.255110.256 0.257 0.258 0.259' . . . 0.259 0.260 10,260 0.260 0,272,0.274'( 0 275 0.276 0,277 0.278 0.278 0.279 0.280 0.280

!

I

I

i

0:i20io:~2010:~~lO

!' '1'·

,

r

8:igg

I

:

,

0.30 0,32 0.34 0.36 0.38

0.29110.2930.29410.295 0.2[)5IO,297 0.308,0.311. 0.313 J 0.31410.31610.317 .0.326 0.329 10.331 0.333 0.33510.337 0.344 ,O.347! 0.350 0.36210.354 0.356 1 0.362,0.355 0.3GS 0.371 0.373 0.374

0,40 0.42 0,44 0.46 0.48

0.38010,384 0.387,0.3901°.392 0.397 '°.401.0.405,0.407 0.409 0.414 10,419 0.423; 0,426,0.429 0.431,0.437 0.440 0.44410.447 0.447,0.453 0.458 OAfH 0.464

0.50

0..163 j' OA70 0.475 0.479 0.432 0.485 0.437

I

0.52 0.54 0.56 0.58 .

0 479 0.494 0.509 0.524

0.485 0.501 .0.5H 0.533

0.60 0.81 0.62 0.63 0.64 0.65 0.66 0,67 0.68 0.69

0.540 0.547 0 .•554 0.562 0.569 0.576 0.583 0.590 0,597 0.603 O.GW 0.617 0.624 0.630 0.637

0.548 1,0.555 0 556 10 55S 0.553 0.571 0.571 0.679 0.579 0.586 0.585 0:592 0.593 0.600 0.599 0.607 0.607 0.e15 0.613 0.621 0.620 0.629 0.627 0,636 0.634 0.1l43 0.641 0.650 0,648 0.657

0.70 0,71 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79

0.491 0.507 0.523\ 0,539

0.494,0.499 0.512\ 0,516 0.523 0.533 0.54.5' 0.550

0.298,0.299 0.318! 0,319 0.3381' 0,339 0.357 0.358 0.376,0.377

0.339 0.358 0.377

0,298 0,318 0.333 0.357 0.375

0.394 0.39510.3'J6 0.412'1'0.41310A14 0 432,0.433 \ 0.434 0,451' 0.452 0.453 00467 0.469 0.471 0.472

O.39G 0.4U 0.435 0.154 0,47:,

0.393 0./J,11 0.430 0.449 0.502 0.520 0.537 0.554

0.48910,491 0.5OS 0.507 0.509 0,522 0.525 0.527 0.54.0 10.543 0.545 0.558 0.561 0.563

0.5Gl 0 .'569 0,513 0.585 0.592 0.599, 0,607. 0,614 0.622 0.629

0.566 0.571 0.575 0 575 0 579 0 583 o 583 0,578 0.51B 0,590 0,595 0,599 0.598 0.602 0,607 0.605 0.610 0.615 0.613 0:618 0.622 0.621 0.626 0,631 0.628 0.634 0,639 0.635 0,641 0.646

0.537 0.544 0.651 0.659 0,665

0.644 o 6.51 0.658 0.665 0.672

0.649 0.657 0.664 0.672 0.679

0.5M 0.661 0.669 0.677 0.684.

0,578,0.581 0 58710 58<:1 0:595 0.598 0.603 0.607 O.61I 0.615 0.519' 0.623 o 626 0.630 0.635 0.039 0.643 0.647 0.651 0.655 0,659 0.653 0.666 0.671 0,674 0,679 0.682 o 687 0.689 0.694

I

0.299 0,3Hl

0.402 0.511 0.529 D.5,n 0.567 0.583

- 00.6059g 0.60, 0.618 0,626 0.63 0.643 0.651 0.659

0.667 0.674 0.682 0.691 0.698

0.643 0.655 0.664 0.671 0.679 0.686 0.691 I O.G96 0.701 0,705 0.649 0.656 0.662 0.fi68

0.661 0.667 0,673 0.680

0.670 0.677 0.683 0.689

0.679 0.G85 0.G92 0.698

0,687 0.693 0.700 0.705

0.693 0.700 0.707 0.713

651

APPENDIX D

0.G09 0.705 0.713 0.719

0.704 0.711 0.718 0.724

0.709 0.715 0.723 0.729

0.713 0.719 0.727 0.733

TABLE D-2.,THE VAl:'l.IED-FLOW.FuNCTION YOH Nn:ClATlvESLOrE~,ji'(tt,N)_gQ (continued)

2.2

2,4

2:6

3.8

0.685 0.H91 ° 698 o 703 0.709

0.095 .701 0.707 0.713 o . 719

0.703 0.710 0.717 0.722 ! 0.729

0.741 0.749 0.755 0.762 0.769

,.

\

I

!

0.80 0.8! 0.82 0.83 O.S-!

0,074 0.680 0.686 0.692 0698 0.704 0.710 0.715 0.721 0.727 0.732 0,738 0.743 0.7,19 0.n4

j

0.715 0.721 0.727 .0.733 0.B8 0.739 0.89 0.744 0.90 0.750 0.91 0.754 0.92 0.761 0.93 0.767 0.94 0.950 0.759 C),H2 O.gGO o. 7ii41 0,777 0,970 0.770 0.782 0,975 0,772,0.785 0.980 0,775 0 , i87 0,985 0.777! 0.790 0.990 0.780 0.793 0.995 0.782 0.7!l5 1.000 0.785 0.797 1.005 0.788 0.799

0.754j 0.i6,t 0.7£10 0.770 0.766 0.776 0.77210 782 o 777 0.787 0.783 0,793 0.788 0.798 0.793 0.803 0.796 0.805 0.79S 0.803 0.801 0,811 0.804 0.814 0.806 0.816 0.808 0.818 0.810 0.820

1.010 1.015 1.020 1.03 1.04

0.812 0.815 0.818 0.822 0.829

0.85 0.81\ 0.87

!

I

LOS

1.00 1.07 1.08 1.09 1.10 1.11 1.12 1.13 1.14 1.15 1 16 1 17 1.18 1.19 1.20 ·1.22 1.24 1.26 1.28

I

0,790 O.80~ 0.793,0.80", 0,795 0,807 0,800 0.811 O.80li 10,816

I

0.72510.735 0.731 0.741 0.7381'0:747 0.74310.753 0,749Io.i58

0.822 0.824 0.828 0.832 0.837

0.810 0.821 0.831 0.841 0.815 0.825 0.837 0.846 0.S19 0.831 0.841 0.851 0.824 0.836 0.846 0.856 0.828 10.840 0.851 0.860 0.883 0,845 10 855 0:865 0.837 10 84.9 ° 8GO 0.870 o. 8!~ ! 0.8541 0.864 0.873 o . 8 _Ii , 0, 858 0.868 0.878 0.851 0.861,0.872 0,,881 0,855 o 866 i o.876 0.886 0.859 0.870 0.880 0.890 0.804 0.8740.884.0.893 0.868 0.8781 Q.888 0.897 0.872 0.882 0.892 0.901 0.876 0.886 0.896 0,.904, 0.880 0.891. 0.900 0.909 0.888 0.898 0.908 0.917 0,.900 0910 0.919 0.927 0.908 0.91i 0.920 0,.934

I

0.765 0,744 0.752 0.771 0.750 0.758 0.777 O.7oG 0.764 0.783 0.762 0.770 0.789 0.7671o.77G 0.781 0.789 0.795 0.787 0.795 0.801 0.7!)3 0,800 0.807 0.709 0 807 0.812 0.804 \ 0.813 0.818 0.801 0.80li 0,819 0.823 0.807 0.815 0.824 0.829 0.812 0.820! 0,82(j 0,834 0.814 0.822,0,828 0,836 0.81810.825 0,830 0.839 0.820 O.827i08~3IO,841 ° 822 0 830 0 8.~7 0.844 0.82408:)210.84010.847 0.826 0.834 0.842\ 0.849 0.829 0.837 0.845 0.862 0,831 0,840 0.847 0,855 0,833 0.843 0,850,0.808 0.837 0.84.; 0.8~3IO.8~0 0.841 o . 850 0.8.)7 i O. 8/)4 0.845 0.855 0:8621°,870

0.851 o 859 0.86710.874 0.855 0.860 0.805 0.870 0.874 0.878 0.882 0.886 0,8gb

I) .864

0.871,0.879 0.869! 0.8761 0.88:) 0.873 0.880 i 0.887 0.877 0.885 io .892

I

I

0.895 0,899 0,902 0.906 0.910

0.881 0.890 0.897 O. 886 0,894, (UIOO o , 891 U. 897 i 0 904 0.895 0.9021°,908 O. 899 I O. 905 0.912 0, 903 1 0 ,910,0.916 0.90710.91410.920 0.911 0.917,0.923 0.915. 0.92110.927 0,918 0.925 0.931

0.913 0.917 0.925 0.9;:!') 0.945

0.921 0,929 0.935 0.942 0 .. 948

0.928 0,932 0.940 0.948 0.954

0.9,14

o.o:>s

0.945 0.954 0.%0

0.770 0.775 0.777 0.782 0,783 i 0.788 0.789! 0,7~J4 0, 7'..l51 O.BOO 0.8010.807 0.807 0.812 0.81310.818 0.818 0.823 ° 824 0.829 0.829 0.835 0,835 0.841 0.840 o 84G 0.84:J (),848 0,845 0,851 0.847 0,853 0.850,0.856 0.75:) j 0,859 0.85(; 0.852 0.858 i 0.864 0.861 . 0.867 0.80 4 10.870 0,8GG i 0.872 0,871 0.877 0877 0.883 0,881 0.887 0.88.5 0.8!B o .S89 0.896 0.893 0,900 0.898 0.904 0,903 0.908 0.907 a.BI2 0,910 0.916 0.n4. 0.919 0.918 0.923 0, 0.928' 0 0.931 b. 0,9:H O. 0.939 0 0.942 0.940 0.945 0.944 0.949 0.950 0.. 955 0,900 0.9fl4 0.965 0.970

I I

I

II .... :1

,

i

-

J

\

652 TABLE

APPENDLXES

D-2.' .Till':

VARIED-FLOW FUNCTlOT, . FOR

2.4

l.H lAG LIS

L75

:5.1) ;l.5 4.0 4.5 5.0

!:i.0 7.0 8,0 \J.O

10.0

:

5.0

5.5

r

0.000 0.020 0.0'10 0.060 0.080

0.000 0.020 0.040 0.060 0.080

O. 97D 0.984 i O. 08D , 0 AlD31 0 097 0 084 0.1)89' O.!)!)5' 0.9\)8 1.001 o. Hno \unlS 1. 000 1.003 1. OOll 0.995 1,000 1.00-' 100-11 01U 0.900 1.005 1.008 1.011 LOl·1

1. 000 1.004 1. OO!) 1. 0) 1:1.0Ui

0.10 0,12 0:14

0,100 0.120 0.140 0.160 0.180

0.100 0.120 0.140 0.160 0.180

0.100 0.120 0.1;10 O.HIO

o 100 0,120 0.140 0.160 0.180

0.100 0.120 0.140 0.160 0.180 .

0,200 0.220 .0.240 0.260 0.280

0.200 0.220 0.2':10 0.260 0.280

0.200 0.220 '0.240 0.260 0.280·

0.300 0.320 0.340 0.3130 0.380

0.300 . 0.320 0.340 0,360 0.380

0.300 0.320 0.340 0.360 0.380

I

' 0.073' 0.979 0,984 0989, 0.9\H

I

I

1.001' 1.005 1.000 LOI2 1.012' 1.0IG 1.020 1.022 1.024)' 1.027 1.030,1.032 1.035 1.037 LO'''' 1 _0'1 ,,
0.066 () UTi 0 :978 0.,983 O.US9,

• ., ( . ,

L05i 1.0U5 1.071 107£' 1.085

1.015.1.017 1.0241' 1.020 1.0:H 1.035 1 " a41 /1 . (042 1.049 1.049

a

1.019 1.028 1.035 1. 042 1.0-18

1.0r.fj 1 • A·" v iJu 1 . 0-5 <:I I. 053 L064 1 00') 11 ono ' 058 1.0li9 . .} •. 1.077 Lg~! i i:2~~ i:g~~ 1.081 1.0791 1.075 Lon 1.08211'.073111,075

i: g~ U~g U~~ U8& ~:?g~ Lgg~ U~g, ~ .g~~' i

1

Ll!JO 1.175 1.162 l.ISt} 1.140 1.1;~I

LIS0 1.141 113:1: 1.124 1.117 1.110 1 103 1.1O:l 1.152 L142' 1.133 1.124 L11G 1.109

1. Oil'; 1.0~0 1101 l.OO'!

1.121

1.113 1,105 1.098

1.228 1.208 1.189 1.173 1'158 Ll.:J.li 1.132 1 1,2:;0 1.210 1.10G 1:178 1.162 1..150 1.. 137

1.1'2:2 1 112 1.103 1.125 1.115 1,106

1

4.5

I O.!l'i' j 0.950 O. 90ll ,0.957 0.OG5 0.072 G. Iln4 I 0.072 0, l!7!) ,0.tlm','O.!!77 0.983 0.07710.98310.989 0'\)0 90 0093 .!Hlt L 02 1.007 1.012 1.017 1.0'20 1.020 ,1.029 1.032 1.03011.042 1.0-14

:Ui 2.6 ~.8

4.2

0.000 0,020 0.040 0.060 r 0.080

I. Hll L17li

2 9

;

0.000 0.020 0.040 0.060 0.080

0.9Gl 0.\)(\7 U. \J7:~ U.iJ79 0.979 0.985

2.~

'l.7

4.0 0.000 0.0"20 0.040 0.060 0:080

0.055 0,\)61 0.907 ° Ilia

1.107 1.103 1.100 1.096 '1.0031090 l.085

:L:1

~ 0.00 0.02 0.04· 0.06 0.08

U~~ ~: ~;~

L.W

3.8 0.1175 0 0: ~~6 0.\)91 0.996

1.056 1,.005 1.075 1.074 1.074 1.073 I'.072 L08G 1.085' 1.08-1' 1.082 LOllI. 1.0n7 1.0!)5 i 1.092 1.090 1.087

L85 LDO l.U5 2,00 2.10

VARIEn~FLOW FUNcnON ,'OR N li:OATIYE SLOPES, F(1I,N) _s, (continued)

0.981 \J7G 0: 0.987 0.993

105211.05311054 1055 • L O(J14 1 . AU""I1.0li4 1.064

LilO

THE

° \J8~

°O·98~

1.55 1.60 l.(;5 1. 70

D-2.

0.9·18 0.\).55 0.9(52 0.90B O.D74

I

1.50

TABLE

0.941 0.948 O. \130 O. OJ!) i {j. 9JS O. U55 0.9:17 0.9461°.954 0·9Ul 10.IN.! 0.952, O.ouo I. 0.967,

0.\)15

1.~0

NEG'\'TIVESLOI'E~ F(11 N)'-SCI (ci' .1) -. ;;1'", 'VtJ..ln'lleI.L,

2.6 :

o.\J:n

L~2

/353

APPENDIX D

r~?j i'}~~ i'~~~ U~~ idt~ Ln~ i:i~g

1.2·j!) 1.224 1.203 1.200 1.232 L32U 1.286 1.251 1.352 1308 1.270 1.37.. 1,325128:1

1.184 1, 206 1. 22:) L:2:l5 1,245

1 HiS L15-1 1 185 LllJ7 1 11>8 L i7U I.?O~ 11.18;3 1.212 1, 188

1,.,/001.3-121202 ~ . ~3~ 11. ~G~ I· LlO::l ...·1 .,,7:, 1.313 L·iG! laS4 1.:119 1-171 1.3!),! 1.324

1,252 I. 200 1,261\ 1269 1.27:2

i.221 L195 . 1. 225 1. 1gO 1 22[) 1.201 1 2:n 1.20;~ 1.233 1.203

L 2!J:l

1.140 l.151 1.158 LHi2 I.HlG

U~g

Ugg

1.128, L117 l.138 I I 125 1. 142 1.129 l.HGl.131 L 149 1.1:14

1.107 1.113 1 117 1.119 1.121

1.136 L l:s0 1.137 1.137 1.137

1.1:12 1.122 1.122 IT'? 1:122

i'Ub

O,HI 0.18. 0.20 0.200 0.200 0.220 0.220 0.22 I 0.2-i I' 0.240 0.240 0.260 0.26 ! 0.260 0.28 , 0.280 0.280 0,30 0.300 0.300 0.32 0.320 0.320 0.339 0.340 0.34 0.36' 0.359 0.360 0,378 0.38 0.379 0.397 0.398 0.40 0,42 0.417 0.418 0.436 0.437 0.44 0,46 0.455 0.'156 0.48 0.474. 0 .. 475 0.50 OA03 0.494 0,512 0.52 0:513 0.54 0'.531 ' 0.532 0.549 0.550 0.56 0.58 0.567 0.569 0.60 0.585 0.587 0.61 0.594 0.596 0,605 0.62 0.603 , 0.IH2. 0.63 0.615 0.620 0.6·1 0.623 0.65 0.629 0.632 0.66 0.637 0.640 . 0.67 0.646 0.649 . 0.65.4 0.66 0.657 0.69. 0.662 0.665 0.70 0.670 0,673 0.71 0.678 0.681 0.72 0.686 0.689 0.73 0.694 0.698 0.74 0.702 0,705 0,75 0,.709 0.712 0.70 0.717 0.720 0,724 0.77 0.727 Q.78 0;.731 0.735 0.79 0,742 °i,738

! I

I

II

o.180~·

0.398 0.418 0.437 0.457 0,476

0.400 ·0.420 0.440 0.459 o 479

0.400 0,419 0.439 0.458 . 0.478

0.495 0.515 0.S33 0.552 0.570

0.497 ' 0.517 0536 0.555 0.574

0.498 0.518 0.537 0.558 0,576

0.589 0.598 0.607 0.61G 0.G25

0.593 0,602 0.611 0.020 0,629

0.595 0,604 0;013 0.622 0.631

0.634 0.643 0.652 0.660 0.668

0.638 0.647 0.656 0.665 0,674

0.640 0.650 0.659 0.668 0.677

0.677 0.685 0.694 0.702 0.710

0.682 - 0.690 0,G99 0.707 0,716

0.686 0.fi94 0.703 0.712 0.720

0.717 0.725 0.733 0.14,0 0.748

0.724 .0.731 0.739 0.747 0.754

0.728 0.7;36 0.744 0.752 0.760

, .

.~ 654 TABLE

APPENDIXES

D-2. THE VAnJE1)-FLOW

~~I 0.80 0.81 0.82 0.83 0,84,

0,85 0,86 0.87 0.88 0,89 0,90 0.91 0.92 0,1l3 0.94 0,950, 0.960 0,970 0.975 0.980 0.985 0,990 0.995 1.000 1.005 1.010 1.015 1.020 1.03 1.04 1.05 1.0G·

1.07 1.08 1.09 . 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 l.18

4.0

FUNC'TJON FOil NEGATIVE SLOPES, F(IL,N)_s. (continued)

4.2

I

4.5

5.0

5.5

0.762 0.770 0.777 0,784 U.791 0.798 0.804 U.811 0.818 0.825 0.831 0.837 0.. 844 0,850 0.856 O.8Cl 0.8137 0,1)72 0.875 0,878

0.768 0.770 0,78:3 o 79Q 0.798 0.805 0.812 0.819 0,826 U.8;{2 0.839 0.845 0.851 . 0.857 0.8G4 0.8!}!) 0.875 0.881 0.883 0.886 0.8S9 0.891 0,894 0.897 0.899 0.90'2 0.904 0.907 0.911 0.916 o 920 0.924 0,928 0,932

TABLE

0.746 0.753 0.760 0,766 . 0,773 0.780 0,78G 0,793 0,799 0.805 0.811 0,817 0.823 0.829 0.835 0.840 0,846 0.851 0.854 [J,8S7 0.859 0,861 0.86,1 .0.SIl7 0.870. 0,873 0.875 0.877 0.882 0.888 a.892 0.896 0.901 0.905 0.909 0.913 0.917 0.921 0,925 0.928 0,932 0.936 0.~3g

.1. 1f)

0,943 0,947

- 1. 20 1. 22 1.24 i,2G 1. 28

0.950 0,95G 0,952 0,9!l8 0,974

0.750 0.757 0,764 0.771 0,778 0.784 0.791 0,797 0.803 0,810 0,816 0,821 0,828 0,833 0.8.,10 0.845 0.8G1 0.8G6 0.859 0.861 U.863 0.867 0.8G9 0.873 0,874 0.878 0.880 0.883 0.887 0.893 0.897 0.. 901 0.906 0,910 0.914 0.918 0,921 0.926 0.929 0'.933 o 936 0.041 0.944 0.947 0.950 0.953 0.057 0.91l2 0.971 0,9,77

0.755 0.702 0.709 O.i7G 0.783 0.790 0,797 0.803 0.810 0.816 0,822 0.828 0.834 0.8.J.0 0.84G 0.852 . 0.857 0.863 0.86G 0.8G8 0.870 0,873 0.876 0.879 0,881 o 884 0.886 0.8S9 0.893 0.898 0.903 0.907 0.911 0.91G 0.920 0.923 0,927 0,931 0.935 0.938 0.942 0.945 0,948 0.951 0.954 0,958 0,964 0,970 0.975 O,I)Sl

0.880 0.883 0.885 0.887 0,890 O.SfH 0.89G 0.898 0.902 0.907 0.911 0,915 0.919 0,923 0,927

~

0.931 0,935 0.939 0.943 0,947 0,950 0.953 C.957 0.%0 6 9G3 ·O.96G 0.972 0.077 U.ll82 0.987

0.936

0.94.0 0.941 0.948 0.951 0.954 0.957 0.9GO. 0.903 0.965 0.968 '0.970 0.1)76 0,981 0.986 O.Y\JO

D-2. THE.y ARlED-·FLOW

~-I

.~

I

I

655

. APPENDIX D

,\

I

.1

I i

I

j

J

1

I, I

i

1.30 1.32 1.34 1.36 1.38 1.40 1.42 1.44 1.46 1.48 1.50 1.55 l. 60 1.65 1.70. 1. 75 1.80 1.85 1.90 1. 95 2.00 2.10 2,20 2,3 2.4 2.5 2.6 2,7 2.8 2.9 3.U 3.5 4.U 4.5 5,0 (} 0 7 0 8,0 9,0 10,0

4.0

FU;XCTlON FOn NEGATIVE SLOPES, F(H,N)~.), (rOllil:nrwl)

I

4.2

.Uj

I- -5.0- - - -5.5- -

0.979 I 0.978 U.nS5 I U.991 U.985 I 0.98G U,OllO i 0.995 0.9\12 . 0,\195 U.09U U.999 0.9\)4 0.!1\)6 U ()!)9 1.002 ./).998 1.00U ! 1 ,00:3 ,1,00C 1.001 I 1,004 1.000 1.006 1.005 I 1,008 1.010 I.U12 I,U09 1 J ,013 1,014 I.U16 1, U18 1.014 1.016 1.017 1,01\) 1,020 1.016 1.020 . J .022 1,020 1.U22 1. U21 1,02!) J ,020 I,U28 1.029 [,0:34 1. U:35 1. U:35 J .032 1.041 J.04U 1.039 I I.U:)6 1,04<'; 1, U43 I 1.0:30 1047 1.0S2 1. 051 1.047 1.042 1.057 1.05G I.U51 1.045 1.061 1 U,59 1,054 i .1l·17 1. UOS 1. 0(j0 ' 1.U57 ! 1.0 ..llJ 1051 . 1.008 1 UG4 1.05\1 1,071 1.0(12 I 1.053 1,068 1.07G 1, U71 I.U5G 1.065 1,058 1. DSa I . U7:3 1.0G8 1,084 1,079 LU71 I.U60 1.087 1.081 1,073 1,001 1.)00 1.0831.075 1.062 1.U92 1.085 1 OG:3 1.076 1. Ug-! 1,087 1,063 1.077 1. DOG 1.0SS 1.078 I.UG'l 1, U!)8 1 08f) 1.U65 1. Ui\) I,OY9 1.U!)O 1.0GS. 1.080 1.10:3 1. UIl:3 1.082 1.06(1 1.10G 1 Ulli 1.067 1.U84 1.108 1.0DS 1,085 1,067 1,110 J ,oun 1.068 1.085 1,1U 1.100 1.085 1.068 1.111 1.100 1,086 1.068 1.086 . 1.111 1.100 1 OG8 1.111 1 100 1.086 I.OG8 I. III 1,100 1.068 1.U86

I

I

I

I

I I

0.994 0.997 I,U01 1. 005 1.008 J .Ull 1.014 1,0IG 1.018 1,020 1.022 . 1.028 1, (J:3U I,U:H I.U:37 1.0:39 I.U41 1.04:3 .1.045 1.U46 1.(]47 i.O-!!l 1.050 1.051 1,0.";2 1.053 1.054 1. U5-! I,U5! 1.U55 1.055 1,055 I.05!l J, USG 1 05G

--)

J i I

!

.1

-t

-\

..l •_J)

)\

,(J5G ,()51i

05li

J 05(; I.U5!i

._' !

, \

I

I ApPENDIX

E.

TAllLE OF THE VARIED-FLOW FUNCTIO"lS :FOR CIRCULAR SEOTION,,*

* Prepared and suppli<>d for publication through the courtcsy of Mr. C. J. Keif()l' and Mr. H. H. Chu 'of the Department of Public Works, City of Chicago. The step 'lines in the' table show the location of n,ormal depth. They should not be crossed by interpolation.

I

657

TABl.E

E-1.

X

V"aIED-FLOW FUNC'rlON

it.'

.~

0:59 0.58 O.!17 0.56 . 0.55 0.(\.1 0.53 O.5~

0.51 0.50' OAD O.~g

..

O.H 0.~6

0.~5

0.44 0.43 0.~2

OAI. 0.40 0,39 0.38 0.37 0.36 0.35 0.3. 0.33 0.32 0.31

<:n <:rt

<0

~.30

0.20 0.28 0,27 O.2G O.lUi 0.g4

0.23

0, 2~ . 0.21

0.20 0.19 O.IH 0.17 O.li 0.15 O.I~

0,13

0.12 Q,l1 °

(

'.

°';1421°'5516'060191068791°'825211 375110 3181Io.2119;O IIi(H,O.I2S0'O'I01~0.09031'O.0179 0.0712,006541°.06351° 061S;0 OW1t053910.01S5,O.0309;0.033fO.0287iO 0248 1 0.53040 57311° 5328,0 7213,0 g9g010 510G 0,25M 0.1829 0.1410,0.11450 09700.08170 07080.00480 0593,0 0580 0.05\)4 0 :l551 10.O'192.0.0H4 0.0307,0,0308 02iW 00220

°

°

°

6~00,O 1809 Lom 3514'0 2110'0, \6931°.125.1 {) 10220.0875;0 0139 0 06420.0593,0 0542,0,0529'0.0514'0 050-11°,0450 0406'0 033. O. 1rn,IO• 0510 0.5641 o.al54jo 0902 0 1>1961--;0 2781;°,185110 13920.1113 0 0916 a am Q ,006S 0,058.lI0 0535,0 0492 °.048010.0468,0.0459;0.0410,0, om'0,o308 O. 0223'0 0193 1 1 0.583010.0388 0.7246 n.31lo,ruo 4438 0.2260,D.l593 0 1225,0.009010,08t9 0.0711 l0 0804 05281°,04811°.0441'°.04370,°"" 0.OH710.0374,O 0339'0,0581 0,0204'0 0177 0.6023 o.f!I:lHIO 76401l.oof!l:l 0.31 H jlU8U5 0 1380 0.1080 {) 08820,01350,0640'0,0545 o.o·m 0 0,1420.010010.03911°.03871'° Ol80i,l.OlI0 0.0~09'0,0251 O.0l1l7 l\O 0163 0,02271069260.81-1£ 1.825 0,2·1,,7,0 Hl!l1 O.12113 0.09(,-1 1007860.0657 00512'004920.04330.0400 0.0369 0,0360 0.03510.0345 0309 O. 0.0171 0.0148 1 .01800 01500.0136 0,6446,0 7~U:O.S817'03ll(iO:O, ID71!O.131S 10.1051 0,0844 °.0701 0.0589 0.051"10 04 H 0 03911°.03611°.0334 03270 0318\°.031310.02811°. 0.66S~.0 702:!,~ 0 2112 °.1641,0. H8710 092310 01480 00340,05260 04580,04000,03530.0321 0 03030.02950,02880.0283 0025.0, .01630,01421°.0123 1 1 1 o 6947,0 80891 10 21060.1335 0 102010.081\ 0 06630.05570.0412 0 0.10 ° 03590.03181°.02960.0214 0 02670.0060 025110,02300.02 0149001290.0112 O. 12~~,O 8717 0 3J2u,O.1707!0, lIS 110 089~IO,0714 1),05871°,04960 042°1°,0310,0 0:,23 0.0230 0.D2M {) 02470.024°1°.02350 023110,0203 0.01 .013·1 O.Olli 0,0102 0.7591,0.97670 2380!0. HHIO.101410.07nllo.OS25 0 05210.04420.0377 0.03311° 0200 ° 025710 02390.02210.0217 0.021110 020&10 0\87 I) 01 ,OI2~ (} OHIO 0 0002 o S0l51"i9S6-i;o IS3I,O.1136,O.OSHiO 06830.05540,04010 03910.0337002960 02590 02301 02130 0200 O.OI~2 0 01900 0I860.0I68Io.0IM,0, .0100 0 00~5Io.ooa3 o 6118'1°.2033 o. 1115 10060 015°10.05970 013710 OlOB 0.0347 (l 03001°.0205 (I 02320 02000 01920.0[78'0 0113 0 017Q 0.0 01510 01380 .00991° 0085 (1,00'15 1 1 0,95150.2069;0 12150 085810 06.5;6.0522.0 04~910 03610 0308 00261 (1,0235 0 020710 OlaM 0171 Q 016010.015.<10,01521° 0 .01351° 012110 00S9 (I 007710.00~7 .01210,01!00 00800 00u9 o.0061 \3 l67110.159~IO.I014 0 o73510.(!5fii}io.045110 0317003190,0273002371°,021000,84001511001520 DHH1.0139 0, 0136 (10 1 1 0 2650[V 1215;0 os,; 0.0332.0 04~lfO'O 100 °,03:120 0281 0.02400 0210 ° 01860 01640 01460 01350 0127 0 OI~3 0.01211° 0 ,01OS 0 0071 0 0062 005.. 1O.I~17 0 1042 O.U7~' 0 05UO .iH3G Q.0150 10.0201 0 02~7 0.0212 ° 01S60 01640 01450.01300 0121 0 0 .00OB 0 .00050 0014 0 006:! 0 00550 0048 o. 13981 036~,O. .o,68,a.0'l7'ljo.o30510 02550.02170 0lS1 0.011\.! Q,QIH a.olnla 01150 01070.0 .0095 o. OOgSIO' 00650 0050 0 004 0 0043 o H07,O.Gna.o 04(l·/'O 0324,0,02660,02230,01011° 016510 OIH 0 012H 01131°.01010.0004 O. 0 00840 00750 ,OMS 0 C.Q50 000.3 0.0038 0.O&l5;O.06!}l;" .0;,8:0.0281;0 02310.01950.01670,01,130 01261°,01110.00990 00890.0082 0.007S O. 0.00740 0067 O. 0 00010.00" 0 003& a 0033 0.07330 Gro90 0299;00243;0.0"01,0 0170 O.OHSlI.OI26 0 01100.00970,0087 0.00780.0072 00680 0.00650 ,00580. 000151° 00390,003{1) D029 1 00580.00030.00000 0068000510. O.0205:~ 04JO:0. ,02,To.nwo.O!iIIO,01i70,OI26IO,0IlOO 00%0,00850,00160 .00510. 0.OU400 00340 OQ2910 0026 10 0503 " Ol63IQ,027S,o.i)22110 0131!0 015°10 01280 OIlO lJ il(l96 0 0084 0.007.1 0.00661°.0059 0,0055 0 .00500 00500, OOH O,OOH 0 0035000300.00260.0023 1 1 0.OH9.0 0307,O.023Z·V.O!89jO 0155,0.01300.0110 9 00950,0032 0,0012 0.00640 00570.00520.00470, 0014 0,00·13 O. 00380.003510 0020 0 00260 00220.0020 0.O~5!!,O.025O;Q.o:ml G.016! 0 OI3?,0.0I1! 0.0094 0 !l!!S21t1.0U1110 0062 a.oosso 00500.004500041 \I .00360.00310 MS.J 0003°10 Uil26 il 00220,00190.0017 o.,L9" 0 Q:).'fi,O 0170,0 0131,o.omlo.00!l. O,OOSl 0 0010 fi. 006! 0 005.10.0043 0 oo.~ 0.0038 0 003610 0023 a 0032 O. .0029 0.00'18 O.OO~210,OOl~ 0 0017 0.0015 ,00:1;( 0 0021\0 00250.0022 ij,OOWIO 00\110 DOli \) 0013 :0.0214 0.0183 1°.01<4;0 QI16,0,0096,O 0981 () ~'O69 0.0000 0 00520.0046 0.0041 0 0036jO,0033 0.003110 'O.020:l;1/ O[&!lo.om;O.0098 0.0082'10,006910. 00511 0 0051 0 00.(50,00390.003510,Q031 O. 0025 O. .00231°.00230.0021 0 0019 (),OOl6 0: OOIl 0 OiH~ o· OOll .0033 0.OU39 0 002Hl .0020 0.00100.00180 00160 0014 0 0012'0 OOW 0 0009 0.Ol59,O Ol~3,O 0102;0.0083'1°.0068,0.005810 00,9 Q 00'3,i!. 0.0139,0 OI071Q.OO,~5 0 00690 0051 10.0049'\0.00.!! 1fi.OOS6'O.OO: C'Wl 0.00251° G!l220 .0017 0.00160,0015 0 00l~ O.OOI~'O.OOIOlo 0009 0:0008 o OilS Q.OOSOIO,OOll 0.005710 0048 IO.00·n 0.0035 0.0000 0 0.1 00230 00310,001910. .0014 0 00Hlo.0012 0 0011 0.001010 00080.00070.000(\ 1 ODIM \0.000400373 Q.0058,O,0048 a 004010 003-10. 00250.VO 00190 00!610 0012'0.00120.00120.001000010 0 IJ008 0.0007000050,0005 0.0077,0.0060 0 OO~910 0039\0.0032,0 0028 O. .00210 00 OOloIO.OOHjO, 0013 O. OOH)j;:;.OOIO 0,0010 0 00090.00080.00070 OU05'0 00050 0004 10.OOfi2,~ 00.8 00391°.00320.00271°.0023.0 001711.,,3 001300012 n oml 0. .OOOE 0 @08\0 00081°.00070000.0.0008 o.OOiJ510 000400004 Q.0050,0.00391O 00320 00261\).0022 o.OO!SI Q .('214 0 OOlZIJ 0011 {I 0010 00009 O. 0(1)11° 000610.00080 0006 O. .00051°.0005 (l 0004 0 00030.0003 o 0040,0,00310 00250,00210.0017 O.OO;S 0 .DUll 0 OOW 000090.0008:0 000'/0 .00060.00050.00050 0005 O. ,000.0.00040,00030 OOuS o.a002 o 00321 00250 002010 00181~.OOH 0 0012 v 0.:>19 000081) 0001 (I 00050 .0005,,°.0004 000110 OOU1[o 0004 00030.000310 000> 000020 0002 0.00250 0019000160001300011 0 000!l 10 ifJilS 0 OOO~ 0 ilOOSjo 0005 D. 0904 0 OOO~ 000030.0003000030,00030.00030.00020.0002 U 00020,0001 10.6472 0.5935

°

°

°

°

°

°

°

°

°

1

°

I

°

° I ° ° °

°

°

°

1

0.0019!0,001510.0012jO 0010,0. <) 000110 OOOB (} 0005 0005'1°.00040 00030 00151\),0012,0 00091° 00080 0005 G0050 000. 0 OOO~ O. ,00030 1 !TOOtlo 00030 o OOll,O ooo9jO 00010 OIloa!o .00040 9003 ,00020 1 1 0,0008 oooa 0 00050 00010 00010 1 OOO{'O 0004 j Q,tlOO3 0 0003,0.0002 10 {.Q02.0 0000° <> 000510 0001 0 0003.0. 00020 0002IQ.OOOllO 0001 0001000011°.00011° 1 o 0004 00030 0003 1O. .0002 10.0002 .000110 0001 0001100001 O.DOC! 000010 0001 1 0001 0001\0 0001,0 0001\ 0.00030.0002 (I 00020 .0001 1{L0001 00010 00010 1 1 0.00021 0002 0 000110 0001 .0001 °.0001'0.000 1 0,0001'1° 0001 1 1 !l 0001 °.00010 0001000010.0001 I 1 I ' 1 1 o 10 0 IQ ;0 10 10 10

° °

°

~,000\g.oooirUOII0

°.

1

i

.00030.000310.00031000021° 00020.0002 (} 00021° 00020 0002'0 00020.00020,0002 ° 00021°.0001 0 0.000210.000zl 0001 °,0001 °,0001 0.0001 0 00010 I) 0001 0 0001000010.0001 O.OOO! ° 0001,0 000] 0 1 0 00011°,00011° 0001 00001\0.000100001[0,0001 0.0001 0 000110 0001 0 0001

,0

0002 0001 0001 0001

°

\0

'0

I

I'

10

10

i

1

1

'I

10

10

\

'I 0

00020000100001 00010 0001 0.0001 00010 0001 0 0001 00010 0001

10

10 ."----'---'---

\_--',

]0

TAELE

~'_~I~i l.OO 0.99 0.98 0.97 0.96 0,95 O,g, 0.93 0.92 0. ~l . 0.90 0. 89 0.1l8 0.81 0.86 0.85 0.8l '" 0.83 '" . 0.82' o O.BI O. B!l 0.79 0. 78 0.77 0.76 0.75 0.74 0.73 C.72 0; n· O. 70 0.69 0.68

0.~7

0.68 0,65 0.64 0.63 0.62 0.61 0.60 0.59

0.58 0.S7 0.56 0.55

0.54 0.53 0.52· 0;51 0.50 O.iS 0,48 047 04.

o.j,

0.41 0.43 0.42 0.41 0,40 0.39 0.38 0.37 0.36 0.35 0.31 0.33 0.32 0.31 0.30 0'.29 0.28 0.27 O.Z6 0.25 0.21 0.23 0.22 .0.21 O.W 0.19 0.19 0.17 ~. 0• .16",. 0.15 0.1I 0.13 0.12 0.11 0.10 0.05 0,.00

50 0.

1 0.5.,

I

0;60

E-2. V .... RTED-FLO~I;

1.~I~·7°.I~I~'I~~I.~1

'100

0.0'

FUNC'1'ION'

Y

I~ .Qa~ J.071~J 1.091~11.15 ~:~I:.:~J~:1~:~j~.

;--.' O.ll2~50.46i!OO.30i!Ol(j.2IQ2;O.13530.11280.01l51:0.06i!OO.05520.0482

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0

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.) NAME INDEX Abbot, H. I,., 94n., 126 Ackers, P. i 32.9, 352 Aridison,.H., 85 Agroskin, 1. 1., 401n., 436. Ahm:lcl, N., 521 Airy, G. B., 553 Albertson, .M. L., 204, 213, 521 Allen,J, 17, 217n., 246 Allen, J. W., 329, 352 l\.mbTGSe, H. H., 212 Anderson, A. G., 493, 518 Appleby, F. V.,.602, 619 Atavin, V. I., 398n., 436 Arghyropoulos, P. A., lSi} ArkhangelskiI, V. A., XlI., 587, 617 Auroy, F.,g91 Averianov, S. F., 36

Betnmlili, D., 40, 21711.. BerryhiH, R. H., 437 Bettes, P., 17, 217n., 246 Bhoot.a, B. V., 460, 471-474,517 Biclone, G., 393, 434 Binnie, A. M., .583 Biot, M. A., 248 Bla,i~l(lell, F. \V., 361, 390, 393, 415, 435, 437-438, 517 Blasius, H., 17 mench, T., xn., 12G, 190 Blue, F. L., Jr., 4.59 Borland, W. M., 190 Boss; P., 42n., !ll, 459, 519 Boudin, M., 227, 247 BQ1Jssinesq, J., 27, 37, 5in., ill, 217n., 358, ::\62n., 300, 455, 460, 537n., 552-553 Bovard, M., 352 . Bowers, C. E., 493, 518, 522 Bowlus, F. D., 78, 85 Boyer" M. C., 207, 213 Brad!~y, J. N., 36S-3G9, 381-382, 391392,436 . . Brf.thms, A., 94, 125, 1G8".. Braine, C. D. C:, 82, 'S5 Bmter, E. P., 17, 37, G1, 85, 127, 156, 390, G07, 6Z0 Bresse, J. A. C., 54n~, GOn., 61, 217n., 246, 253, 258, 294, 357n.. 393, 434 Bre~t.ing, A. E., 213 . Brudenell, ROo N., 3G9, 391 Bubendey, J. F., 501n., 519 Buchler, B., 36\), 391 . Buscmann, A., H8n., 460' Butler, S. S., 620

Babbitt, H. R, 36, 295 Bakhmetelf,·R A., vii, x, 1311.., 18, 37, 61, 20G, 212-213, 246, 252-255, 294, 297, 306, 324"':325, 393, 398-400, 409""., 423,425, 427, 435, 437-438 BalloJfet, A., 74n., 85 .Banks, R. B., 136, 155 Barbarossa., H. L., 208, 213 Bardsley, C. E., 17, 459 Barillon, R G., 358-359 Barker, C. L., 61 Barrows, H. Ie, 620 B~ta, G., 584' Baticle, E., 253, 294 Baller, W. J., 198-199,212,383,391 BaUIl1;LnD, E. R., 295 . Bazill, H. E., 17,61,94-95,99,126,213, 353-364, 390, 393, 409, 43-1, 538, 552 Beebe, J. C., 393, 425, 434 Ca.gniard; L., 551 Beij, K. H., 351 en-michel, C. M., 512, 521 Belanger, J. B., 30n., 37, 49n., 59n., 61':'62, 217n., 246, 393, 434 Co.michel, E., 521 Ca.mp, T. R, 106, 127, 135, 327, 334n., Belok(ln, P. N., 138n., 156 Benjamin, T. B., 521, 583 351 Cfl.mpbell, F. B., 359 Bennett, J., 16 Bergeron, L., 587, 617 Cariiio, 1. D., 189 ClLrison, E. J., 190-191 Bermel, K. ,T., 85 663

..

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664

I

I.

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NAME INDEX

Carscanjen, M., 3D3, 434, 519, 582, 585 De Marchi, G., xi, 14n., 84, 231 247 Co.rter, A. C., 191 327-328,. 341, 351,390, 519, 572, 58l Carter, R. W., :)!l(l, 476-477. 491-482 De Mello Flores, J. 0., 435 494, 518 " . de Prany, G., 93n. Chabert, J., 296 Deymie, Ph., 551 Chardonnet, 353 Dietz, D. N., 551, 611 Charnomsldl, J., 26311., 295 Dmitriev, G. 436 581 618 Doddiah', D., ., " Chll.telain,P., 295 Cheng, H. M., 601, 620· Doeringsfeld, H. A., 61 Chertousov, M. D., vii·n.] 295, 559, 583 Doland, J. J., 208, 213 CMzy, A., 93 Doll, E. H., 191 Chien, 180 Donnelly, C. A", 437-438 xr;., xii, 83, 126-l21, 155, Chow, V. Douma, J. H., 38, 391 208 295, 352, 521n., 522, (J07, Dressler, R. F., 214, 5:351t., 552 620: Drioli, C., 551, 572, f,I3'l Chu, H. H., 253, 261-262, 2\J5, li57 Druml, F. V., 619 ChugMY, R. R., xi, BOn., 83, 131n., 1:~3, du Boys, P., 168"., 190 Dupin, P., 521 ' 155 Chun, Y. D., 295 Dupuit, A. J. E. J., 253, 294 Citrini, D., 14n., 8'1, 327, 351, 393, 435, . Durand, W. F., 17 572, 584 Clayton, L. A;, 49:3, 518 . Coleman, G. S., ~J28, a51 Eddy, H. P., 36 Ehrenberger, R, 38, 253 . Collinge, V. K, 329, 353 Conard, R. G., 1.91 Einstein, H. A., 136, 155, 166n., 190 208 21;3 ' , COile, V. M., 75n., 85 . Einwachter, 393, 434-435 Contessini, F., 74n., 84 Eiseniohr, W. 37 COl'bett, D.1V1., 8.1 Coriolie, G., 21, :37 Eisner, F., 11, Cornish, 210, 213 Elevatorski, E. A., 391, -J04n. Cotton, J. 85 EUms, J. W., 435 Cowan, W. Ellms,R. W., 438 Cox, G. N., Engel, F. Y.A. 1271., 18, 74n., 84 Engels, 328, Coyne, A., 384, 391 Engelund, 452n., 4130 CrElusse, :E., viin., 62,248, 520 Craya, A., :no, 214, 53511., 552. 587, 617 EquiazaroIf, B., 535n., 552 Creager, W. P., 361, 374, 389-::WO, 436 Escnnde, L., 3DO, 393, 435, 437, 510, 512 519-521 ' Cross, W. P., 007, ()20 01'0111', F. 85 Escaffier, F. F., 214, 231. 2-10 242 248 295, 587, 618 .", Crump, E. 74n., 84 Cunningham, A. J. C., 991 •. , 126 Etcheverry, B. A., 160, 113, 189, 1\10 Euler, L., 40 Ezra, A. A" 272,'295 Dahl, N. 550, 617 Dai, Y. B., Dalrymple, T., 518 Fan, C. 170n.-171n., 191 Dimel, P., 385 Favre, H., 329, 350, 352, 551 Fawer C Dalltscher, 578, 585 Darcy, H., 811., Hi, lISn., lZiJ, 213 30;;\ 434 FeJlenlus;\V., .d' Aubui~soil de Voisins, J. F.: i'in.; 16, Feodorofi, N. V., Ferriday, R., 393, 501, 519 . Daugherty, R. L., 517 Florey, Q. L., W\), 191 Damn, C. It, 585 Forchheimer, Ph., ,125, 110n., 191, 294, Davis, A., 520 3;'-7-328,351,358,520, 534n.) 550 . Formica, G., 2,1,7,464-468, 516-511 Davis, C. V., 3~O, 431, 601, (120 Dawson. J. H., 460,. 4(38--!6!), 511 Forster, J. IN., 393, 409-41'2, 435-435 Fortier, S., 166, 173, 190 Delleur, J. W., 212

665

NAME INDEX

Foskett, L. W., 621 H. A., 359 J., 295, 353, 551, 512, 585 Frazer, 'Y., 329, 34G, 352 Fredenhagen, V. B., 191 Freeman, J: R., 37, 431 Prencb, J. L., 493, 518

Gang1,lillet, E., 94, 125

M. le, 391 Garot, r., 352 Gauckler, Ph., 99n., 126 Gentiiilii, B., 329,351, 572, 584 . Gh.:ll'!l.l'delli, L., 247 Ghotankll.f, S. T.,. 431 Gibb, H. M., 36 Gibson, A. H., 37, 393, 434 Gilcrest, B. R., 550, 620 Glover, R. E" 191, 585 Goldschmidt, H., 619 Goodrich, R. D., 607, 620 Gotaas, H. B., 85 Gl1lshof, F., 253, 294, 448 Graves, Q. B., 607, 620 Grimm, C. I., 296 Groyer, N, C., 31 'Qrzywienski, A., 390 GUil101.1, J. C., 51'211., 522 Gumensky, D .. B., 391 Gunder, D. F., 225, 247 G!1n~uli,

Hagen, G. H. L., 93n., 9911., 125 HaU, L. S., 38, 30111.., 326 Ha.n·,a, F. R, 204, 212 Harkness, F. B., 608, 621 Harleman, D. R. F., 438 Harms, F., IS Harrington, A. 'iV., 37 Ho.l'I·$, C. W., 379, 391 Harrison, A.S., 295 Harrold, J. C., 3Ul . HMumi, M., 3211. Hathaway, G. A., 553 Haws, E. T., 583 Hayarni, Shigel1ari, 619 f,[ayami, Sboitiro, 602-603, 619 Hebert, D. J., 585 Hem)', H. R., 509, 521 Henry, M., 587, 611 Herbert, J. IL, 459 Hering, E., 125 Herschel, C., 9310., 125 L J., 51211., 522 31,363,390,427,435,438

Hirnmelwright, A. L. A., 583 Hinderks, A., 459 Hinds, J., 312-314., 318, 320-327, 342, 350, 361, 389, 436 Holsters, H., 587, 617-618 Hom-mo., M., 16, 18, 237, 248, 295, 471, . 517 ISO, 156 Hopf, HOI.ton, E., 5371.., 61, 108, 127, 136, 150, 155-156,392,546, 553, 583 Houk, 1. E., 125, 156, 189, 437, 47011., 50171.., 518 Howe, J. "V., 459 P. 8., 435 fuu, Y., 359, 413-414, 436, 460, 471414, 517 Humphreys, A.. A., 9471.., 120 Husted, A. G., 28311., 295 Ince, S., 61 Ingersoll, A. C., 511

Inglis, O. C., 74n., 84 ·Ippen, A. T., xi, 361, 390,431,438,
C., :xi, 5571.., 61-62, 83, 231, 246358,461, 517, 550, 584

A. H., 1 -!on., 84 Jansen, B., 246, 391 Jeffrey", Ii., 150, 156, 214 Jegorow, S. A., 16, IS' Jensen, R. A., 521 John$o!l, C. F" lOOn., 127 Johnson, J. 'N., 2H, 31 Johnson, RD., 510-571, 583 Johnstone, D., 607, 620 Jones, B. E., 533, 551 Jones, L. E., 296 G.,61i) 3. D., 361, 374, 389-390, 4.36 Kalinske, A. ·A., 435 Ko.ndaswa.rny, P. IL, :)HO Klrr, M. H., 436 I{eifer, O. J., I(eIl.Iledy, R. G., Kennison, K. R, 393,

666

NA.ME.INDEX

Kent, D. H., 520 ICestin, J., 212 ICeulegan, G. le, 9, 17, 98, ;':'..1;6-;<,'..10. 213-214, 34"r-:;'l8, 353, 550 Khllfagi, A., 7411.., 84 Kholodovsky, Professor, 255n. Khristianovich, S. A., 587, 617 Kindsvater, C. E., 18, 390, 393, '425, 427, 435; 438, 476, 518 King, H. W., 17, 37, 53n., Gl, 85, 108n., , 127, 156, 390, 621 Kinosita, 1'.; 550, 617 Kil'pich, P. Z., 131n., 155 Klrschiner, 0., xi, 10, 12, 17, 506, 520 IUeitz, M., 531, 551 KMPP, R. 1'., 4'lBn., 4053, '156-457, 460 lfuappen, T. T., 607, 620 Koch, A" 393, 434, 582, 585 Koch, P., 247, 519 Kohler, M. A., 607-60'S, 620-621 Koloseus, H. J.,,392 ICo]upaila, 8., xi-xii, '28, 37, ,295 Koieny, J., 11, 18, 253,294, 358, 507, 520, 578, 585 . f{rey, H. D., 519 ' Kuntzmann, J., 352 Kutter, W. R., 94, 125 l{viatl{ovskii, V ~ 8., ?,8n. Lacey, G., 166n., 190 l,agrange, J. L. de, 538, ,552 Lamb, H., 537n., 539, 550 Lamoen, J., 537, 618 ' Lallcenelcl, R. L., 459 Lane, E. W., lOS, 127, 17On., 174, 190-· 191, 359, 3~)0, 435, 476, 517,607,621 Langbein, W. B., 213 . Lansford, W. M., 9, 17,326 Lauffer, H" 38 Lawler, E. A., 619 Lazard, A., 238n., 248 Leach, H. R., 156, 271, 295 Lee, M., 253-254, 295 Leighly, J. R, 169, Hll LeJiavsky, 8., xn., 190, 406n., 436 Leopold, L. B., 16611.., 190 Levi, 1. 1., 255n., 295 Levin, L., 53~n;, 552, 587, 617 Levy, A. G., 4.15 Li, W. H., 327, 334-336, 351, 496, 518 Lighthill, M. J., 583 Lin, C. C., 214 Lin, P.N' f 587-588, 618 Lindley, E. 8., 190 . Lindquist, E. G. W., 28n., 37, lOOn., 125, 213, 393, 434

Linford, A., 74"., 84 Linsley, R. re, 607, 620-621 Lotter, G. K, 136-137, 155 Lowe, F. C., 585 LudWig, .J. H., 85 Ludwig, R. G., 85 McCar~hy, G. 1'., 507; :321 McHenry, D., 191 McNown, J. 8., 359 J:l'fcPhel'son, M. B., '.4313 Maddook, T., Jr., 166n., 190 Maitre, R, 391 Manning, R, 98, 99n., 126 Marichu.l, A., 61, 390 Masoni, U., 253, 293 MllSSl!.U, J., 587-588, 617 Ma.sse, P., 237-238, 247, 551 Masse, R., 551 . Matzke, A. E., 246, 259n., 295, 393, 398400, 425, 427, 43.'>, 438 Mavis, F. 1'., 80, 85,493, 49S, 518 Mayer, P. G. R, 581, 585 Merriman, M., 393, 434 Mertell, A" 22 7 , 22711", 247 Metcalf, L., 36 Metzler, D. E., 508 Meyer, O. H., 607, 620 Meyer-Feter, E., 350 Meyers, J. S., lOS, 127 l;I1eynardi, G., 353 Miller, C. N., 349, 351 Miller, C. R., 190 Mitchell,W. D., 326. !\'IockmoI'B, C. K, 459 Molesworth, G. L., 189 Mondsch'lin, H. F., 007,621 MOllonobe, N., 253, 295 Montagn, A. M.R., 437 , Moore, W. L., 204, 398-399, 41411.., 423, 436-437 . Moots, E. E., 53611.., 552 Morgan, O. W., 4l4n., 438 Monis, H. iii., Jr., 12, 18, 196, 197n., 212, 193, 518 MostkolV, l\LA., 61, 32\l, 337, 33911..,352 Mouret, G., 225, 227, 238n., 247 Mueller, 0., 521 Muhlhofer, L., 136, 155 M\llholland, W. M., 608 MiUler, R., 441, 459 Munch-Petersen, J., 452-n., 4()O

Na.garatn~, S., 393, 435 Nagler, F. A., 493, 501, 518-520·

NAME INDEX

Nebbia, Neliclo v, I. Nelson, H. C., NeWnl(lll, 21)5 Newmark, M., ::i Nikuradse, J., 9, 17, 201n., 202, 212, 214 Nimmo, W. H. R, :3:2\), 352 NDsedl.l., G., 339,.., 352 No ugar'J , J., 618 Obolensky, S., 391 O'llri.en, III. P., 2$), 37, 213 O'Connor, 583 elser" O ..T., 101 Qrklley, J,. C., Orth, J., 35.3 Owen, W. M., 17 Pa,ger, G., 521 Pai, S. I., 619

667

RI1.n1ser, O. E., 115, 12i

n,ltnd, W" 204, 212, 423, 437 Raus{ord, G. D., 587, 618 Ray, W. E., 1307, G21 R.ayleigh, Lord, 537n., 552 Raytchine, N., 2v5 Re, K, 5:34,11..-53511.,55'1 R.ee, W. 0.;8.), 180, 1\)1,-13211,640 .Renboek, Th., 13n., 18, 2!Jr.., 37, 362, 3I)O,.4'14n., 501, 51\) Reineke, R., 5.51 Rtliss, S., 400 Riabouchinsky, 44.871., 459 Rich, G. 584 Riegel, R. 39:3, 425, 434 Rip Ie:;', H. 0., 4SG, 400 Rippl, \V., G20 552 Ritter, A" xi, 14, 17-18, 435 Robertson, J. 213 Robinson, A. R., RohwAr, C., 85 ROllse, H., xi, 14,..15, 18, Gl, 24u, 302, 380, 30S, '42B, 4:l5, 4:38, 460;. 471517,521,550 391 Roy, S. RUhlman, Russell, J. 8., Rut.ter, E. J., 607,

Palmer, H. K., 78, 85 Palmer, V. J., If)! Pannell, J. R., 16 Pr,rshall, R L., is, 8,5 Part,ridge, E.P., 6ID Patterson, O. C., .,H)(J, 518 Patterson, G. lV., 214 Pall 1hll!;, J. L. H., lI2!] Sabatlu), G., 519, 521 Pavlovskil, N. N., viin., lOOn., J26, Snfrallell, IL, 393, 398, 434, 436 laIn., 155,15(i, 281 , Saint:'ifeno,llt, A.. J. C, l~fI.rr'; de, 210, Penati,8" 217,.., 52l', 551-552 Perrollet, J. Sandover, J. A., ' Pest,r~cov, Dr., Sao, T. T" 393, 'i3S ,PeterkA., A. J., -1,36 Saugey, 30411. Pickels, G. W., 126 Schn;;k, A., G19 Pikafov, F. I., ·1::16 Sc!mffernak, Y., 253 Poggi, R, 43:3, 438, 459 Schlichting, H., 195, 212 Pohle, F. V., 5:3511" 5.'52 S(;hmiclt" M., "iilt., 329, 346,352,572,584 Poinctl,re, H" 237, 247 Schneckenberg, E. C., 02, 125, 127 Poncelet,J, V., 217-n. Sci1nepper, D., 127 Posey, O. J., vii, 1il"., 18, '2413, 2gB, 4::15, Schoder, E. 'Y., 28n., 37, 390 45!J, (JU7, ll2I . Schoklisch, A., 253, 295, 327, a51, 437, Powell, R. W., 95, 126, 204, 213-21'1 459, 520-521, 5:35n:,· 552, 568, 583, PrandtJ, L., 0,12, Ii, 200-201, 212, 448n., 620 ' 459 ' Schonfeld, J. 550, 583, 613 Prnsil, F., 358-359 Schuller, J., Preiswerk, 4-1,811.., 460 Schultz, E. A., 105,127 Price, W. II., Schnemi, E., 3(Hn., 390, 507, 520 PuIs, L. G., 521, 607, 620 Scobey, F. C., lOa, 115, 126-127, 16G, Pllt.man, H. J., 587, G18 , 173, 190, 213, 3'J6 Seddon, J. A., 5~1, 551 Serre, F., 358, 550 Rafter, (J. W., 53n., 390 Simny, M. B., 3n Raju, S.P" 17, ..IS!) Rakhlllanoff, A. N., 131n., 155, Z81n., 296 Shl1rn1a, K. R., 437

:

~

,'l

I J \

I

I

668

NAME INDEX

Shchapov, N. M., 37 Shepley, J. 607, 621 Shima, 8., '171, 517 Jr., 493, 518 37, 431, 440-441, 45!J Shulit.s, Samuel, 437, 459, 520, 620 Silber, R, ·"iin., Sil1:icrm:'..!l, E., 17 It., 295 Skdnde, A. J 393, 409-412, 435-436 : Smetana, J., 390,.393, 435 Smith, D., 328, 351 Snyder, F. F., 607, 620 Sorcrtllen, K E., 607, 620 Spangler, .1., 501, 520 Stanley, C. M., 437 Stall~on,

T. E., Hi

Troesch, B. A., 587, 600; 619 Tulta, H., 352 Turner, K, B., 28n., 37 ','

uGhlda, S., 587,618 Vanoni, V. A., 2'12 Van Vliet, R., 156 Verlernlkov, V. V., 210, 2'14 Villemon te, J. R, 85 Vl2.di~:avljevitch, Z., 125 Vogel,H. D., 459 von I(d,rrrdn, Th., 9, 17,37, 201n., 212, ' 248, 448n., 4.',)2n., 459 Von Seggern, 1\1. E., 253-254" 295

Stain, M. F.,35l Steinberg, L E., 2SOn., '296, 607, 621 lVah!man, P., 570n. Stevens, J. C., 13n., 18, 78, 82-Sa, 85, 'WaltOll, C. B., 607, 621 .436, 438 Warnock, J. K, 437, 585 Stokei, J. J., xn., 325, 53511.., 550,587, Weaver, R. M., 436 ' 600, 618-019 ' Wcisbach, J' I Sn., 16,40, 401n.,436, 501, StmtWll, J. H., 607, 620 519 Stmub, L. G., 17, 4!}3, 518 , ,;ells, E. A" Jr., 85 Streck, 0., 520 , Werner, P. W., 346, 353 St-reeter, V. L., 189, 212 White, M. 246, 429 Strickler, A., 99n., 126, 206, 213 ,:llhitham, O. 535n. Sueishi, T., 587, 619 Wien, \V'I 18 Supino, G., 550-551 Wilcox, E. 10671.., 127 Swain, F. E., 5aO, 585 Wilkinson, J. 531,551 Wilm, H. G., Wilson, W: T., 607, 620 ,TD.k!lS~o, T., 553 Wisler, C. 0., 607, 620 Talwani, B. S., 437 Wittman, R., 459 T>\fpley, J. F., Jr., 608, 1)21 Wolf, P.O., 61, 83, 246,.358, 517, 550, Tatum, F. E., 607, 621 , 584 ,Taylor, E. H., 512-515, 522 Woodburn, J. G" 61 Terrell, P. \V., 190 Woodw9.rd, S. M., vii, Won., 121, 246, Thomas, H. A., 214, 541, 551', 581, 585 2ge, 393, '134, 448, 459, 493, 518, 621 Thomas, R, 521 ' , W6ycioki, K, 3gSn., 436 'l'holllp~on, P.W., 459 Thomsoll, J., 439, 459 1:iffany, J. B., 436, 521 Yarnell, D. L., 106n., 127, 425, 493, 501- ' Tison, L. J., xi, 61, 247 i 50.6, 518-519 , Toch, A., 509-510, 521 Yassin, A. M" 155 Toebl!!l, C., 92, 125 Yen, C. H., 459 Tolkmitt, G., 253, 294 Yih, C. S., 214, 359 Tracy, H, J., 53n., 61, 476-477, 491-492, 51B • Trautwine, J. C.,i Jr., 61, 126; 390 Zie~kiewicz, O. C., 583

SUBJECT INDEX A profiles, 230 Abrupt drop for jump control, 412-414 Abrilpt riEe for jump control, 411'-412 Accaleratioll line, .S27 .. };.dverse slope, 223, 259-260 Aeration of nappe, 352-363 Air entrainment, :-13-36 Airy celerity equation, 539 }Jl-American Canal, 188-189 Alluvial riVer bends, 455-456 Alternate depth, 41, 551', Angle of repose, 171-172 Artificial roughness; 202n. Atterberg soil test, 173n. Backwater, of bridge piers, 501-506 of cOllstriction, 490-493 of dam, 319. effect of, 72 endpoint, 319 . Backwater curve, 71, 222n. Backwater envelope carve, 319 Backwater mtio, 491 Baffle piers, 415 Banking, 4513 Bazin da.ta, for broad-crestcd weir, 53n. for channel roughness, 11, 95, 99n. for overflow spillWays, 363-36'4 for rough channels, 204 for sharp-crested weirs, :;163, 382, 390 for wavy surfaces, 204 Bazin formula, 95 Bazin pro file, 363 , Ba~in'f! rr., 95 Bed load, 106 Bed-Iond fUnction, 166n. Bends in alluvia.I rivers, 455-456 Bernoulli Emergy equation, ,40 Best hydraulic section, 160:"162 : Blllaius equation, 8 Blasius-Prandtl-voll Kd.rman curve, 10 Blue hole, 5121l.. Bottom ro.ck, flow through,: 337-340 Bounda.ry layer, computation, 198-200 development, 192-193

Bounda.ry layer, laminar, 193 on overflow spillwll.Ys, 199-200 rougluless effect on, 200 turbulent, 193 of uniform :flow, 89 Boussinesq coefficient, 27 Boussinesq number, 13n. Boussinesq theory, 358 Box inlet drop spillway, 424 Bresse met,hod, 258 Bresse theory, 357 Bridge piers, 482, 489, 501-50G Bridge piles, 482, 489 Bridges, submergence of, 482, 489 Broad-crested weir!;, 52-53, 80 for jump control, 410-H1 tests, BalJill, 53n. Cornell, !lSn. Michigan, 53n. MinnesQta and Wa.shington, 53n.

C profiles, 230 Canal drop, 415 Canal faU, 415, 432

19 delivery of, 297-306 navig!l.tion, 572-S75 outlet. a.nd entrance, 30'1-306 Ca.pillary 'I'I'aVI!, 12 Cascaoos, ladder of, 433 Celerity, 13, 538-540, 582 Channel li:'lignment;lOii'(106-109 Channel contrachiO!\, 41-49, 258; 468410,479, Channel expansion, 57-59 Channel geometry, 20-24 principle of, 16611.. Cha-anel irregUlarities, 103 Channel junctions, 321-323, 512-516 Cha.nnel sections, 20 best hydro.tllic, 160-162 circular (see Circular channels) of constant: critical flow, 82 of eonstant'hydraulic udius, 152 geometric elementa, 22-24

669

)

)

uoo

NAME.INDEX

, Linford, A~, 74n., 84 Lmsley, R K, 607, 620-521 Keulega.n, G. 17, '98, 202-205 L-otwr, G. Ie., 136-137 155 213-214, 353, 550 ' Lowe, F. C., 585 ' Ludwig, J. H., 85 Kl;aJagl, A.. , 74n., 8", Ludwig, R. G., 85 Kho.lo~ovsk!, Professor, 2oon. , K~nstl!LnoVlch, S. A., 587, 1317 Kmdsvntel', C. E. 18 390 ~9q . 427, 435, 438, 476, lilS " , u, 425, McCart.hy, G. T., 607, 621 McHenry, D., Ull mng, H. iN., 17, 37, 53n. 61 85 108 . 1~7, 156, 300, 621 ' , , n., McNowD, J. S.,359 McPhersoIl, M. E., 436 Kllloslta, T., 550 617 Ma~di)ck, T., Jr., 156n. 190 19,rpich, P. Z'J 13171., 155 MlUtre, R.,' 391 I I\'lr~chtner, 0" xi, 10, 12 17 60(\ 520 Cleltm, M., 5in, 551 ' , I Mf1n.ning, R., 98, 09n., 126 Manchal, A., 61 390 ~napp, R. T., 448",.,453,456-457 460 Masoni, U., 253; 293 ' T nappeu, T. T., 607, 620 ' Massau, J., 587-588 617 KOch, A., 393, 434, 582 585 Ma.!;Se,P., 237-:t:~$,'247 551 Koch, p" 247 519 ' Ma.sse, R., 551 ' Kohler, M. A~, 607-608, 520-621 Kolcseus, H. J., 392 ' Mat~koeo' A. E., 246, 25911., ::W5 393 398. • .425, 427, 435, 438 ' , I{ol~paila; S., xi-xii, 28, 37, 295 ·1{ozeuy, J., 11, 18, 258 294 358 ~O~I, MMav1s, F~ T., 80, 85, 493, 498 518 520, 578, 585 ," ,u a.y~r, P. G. H., 581,585' . I{rey, R. D., 51\) Mernman, M., 393, 434 ' li:untzmann, J" 352 Ivie!:ten, A" 227, 237n.., 247 Metcalf, L., 36 l{u~t;el', W. R., 94, 125 Metzler, D. E., 509 Kv!atkovskii; V. S.,28n. Meyer, O. R, 607, 620 Meyer-Peter, E., 3'50 Laoey, G., 166fl., 190 Meyers, J. 8., 105 127 Meynardi, G., 353 J. L. de, 538, 552 537n., 539, 550 Miller, C. N., 349, 351 L a.moen, 587 618 Miller, O. R. 190 Laucetield, L'., 459 ' Mitcheli, W. 'D.; 326 ,Lane, E. W., 105, 127, 17011.; 174; 19(}-: ?1ockmore, C. E., 459 L 10~1 359, 390, 435, 476'517607 6zi ::'vloleswol'th, G. L., 189 II>ngbelO, IV. B., 21;) , , , . ~1Qnc\snhein, H. F., G07, 621 Lansford, W. M., 9, 17 326 Mononobe, N., 253, 295 ' Lauffer, H., 38 IVIQntll.gu, A. M. R., 437 La.wler, E. A' J 619 Moore, W. L., 204, 398-399 414n ""3 436-437 . . , -= , La.zard, A., 238n., 248 ;)'1oots, E. 536n., 552 Leaoh, H. R, 156, 271, 295 Le e, M' J 253-254 295 Morg.a.n, C. 4141'1.., 436 Lei~hly, J. B., 16:9, 191 ' Mor~~2,~93~5'1t' 12, 18, 196, 197n." Lelia. vsky, S., An., 190 406n.. Leo~old, L. B., 166n., 'l!lO .. Mostkow, M. (ll 329 337 339 M t G '" n. 352 oure,., 227 238n 2:17 ' LeVI, I. 1., 255n., 295 Mueller, 0.) ,'" ' Levin, L., 535n., 552, 587 617 Levy, A. G., 435 " MUhlhofer, L., 13G, 155 :Mulho\!and, W. M., 508 L~, W..R, 327, 334-336, 351 4.96 518 MtiUer, R., 441, 45\) Lighthill, M. J., 583 " Lin, C. C., 214 M~mch-Petersen, J., 452n., 460 L~n, P. N., 587-588, 618 t;ndle y , E. S." 190 Nll.ga.ra.tn~m, S.,'393, 435 mdquisc, E. G. W., 28n. 37 lOOn 125 213, 393, 434 ",'" Nagler, F. A., 493, SOl, 518-520 Kent, ·D. H., 520

Kestitl, J., 2'12

~,

I

I

436

NAME INDEX

Nebbin, G., 74n., 84,393,435 Nelidov, t. M., 246 Nelson, H. 0., 17-18 Newman, A., 295 Newmark, N. J\'1., xi Nikma.dse, J., 9, 17, 201n., 202, 212, 214 NImmo, W. H. R, :129,352 Nosed!!., G., 329, 339n., 352 Nougnro, J., 587, 618 ' Qbolensky, 8., 391 O'Brien, M. P., 29, 37, 213 O'Connor, n.., 583 Olsen, O. J., 169, lin Orkney, J. C., 583 Orlh; J., 353 Ow,;;n, W. M.; 17

667

C. E., 115, 127 Ra.nd, W:, 204, 212,423,437 Ra.nsford, G. D., 587,618 Ray, W. E., 607, ()21 Ra.yleigh, Lord, 537n., 552 Rayt.chine, N., 295 Re, It., 534n:-535n., 552 Ree, W. 0.,' 85, 180, 191, 629, MO Rellbock, Th., 13n., 18, 2911.., 37, 362, 390, 414n., 50J. 519 Reineke, H., 551 Reiss, 8., 460 Rinbouchinsky) 44an., 459 Ric11, G. R., 584 Riegel, R. M., 393, 425, 434 Ripley, H, C., 455, 460 Rippl, W., 6'20 Ritter, A" 53471.., 552 , Rcbertsoll, J. M" xi, 14, 17-18, 435 Robinson, A. R., 204, 213 Rohwer, C" 85 Rouse, H., Ai, 14-15, 18, 61, 246, 3132, 390, 393, 429, 435, 4.38,,460, 47i474,517,521,550 Roy, S. K., 391 R\ihlmlln, M., 253, 294 Russell, J. S., 537-538, 552 Rutter, E . .I., 607, 620

RIl.IUBer,

F'ager, G., 521 rai, S. {HI) , P!1.Imer, Ie., 78, 85 P",lmer, J., 191 Pannell, J. 'R., t6 Farshall, R. L., 75, 85 Part.ridge, E. P., Gi\) p,lttersOll, C. C., 496, S18 I'ltttel'SOll, O. W., 214 Paulhus, J. L, fl., 520 P~vlovski\, N. N., viit<, lOOn., 12(\, 13h;., 136-138, 155,. 156,281 Panati, 8" 572, 584 Perronllt,' J. R., 93n. Pest.recov, Dr., 255n. .Peterka, A. J., 436 Pickels, G. W., 126 Pikalov, F. I., 436 P;jggi, B., 433, 438, 459 Poble, F. V., 535n., 552 Poincare, H., 237, 247 poncelet, J. V., 21711,. ,Pose)', C J, vii, 13n., 18, 246, 296, 435,' 459, 607, fi21 Powell, R. W., 95, 126, 204,213- 2 14' PrandLl, L., 9, 12, 17,200-201,212, 448n., 459 Prasil, F:., 358-359 Preiswerk, E" 448n., 460 Price, W. H., 359 ./ P;.
Sa.bathe, G., 519, 521 Sn.frn.nez, K, 31)3, 398, 434, 436 Saint-Veuant, A. J. C. Burre de, 210, 2171>., 525, 528; 551-;552 Sandover, J. A., 583 : Sao, T. T., 393, 435 Sn.ugey, 39411. Scha.ck, A., 619 Schaiiernak, F., 253 , Schlichting, H., 195, 212 Schmidt, .11>1.; viin., 329, 346, 352, 512, 584 Schr.eckcnberg, E. C., 92, 125, 127 Schnepper, D., 127 Schoder, E.W., 28n., 37, 390 Schoklisch, A.) 2-53, 205, 327, 35], 437, 459, 520-521, 53Sn., 552, 568, 583, 620 SohOnfeld, J. C., 550, 58q, 618 SchuHer, J., 551 : Schultz, E. A., 105, 127 : Scimemi, E., 364,~., 390, 507, 520 Scobey, F; C., 103, 115, 126-127, 166, 173, 190, 213, 326 Seddon, J. A., 531, 551 Serre, F., 358, 550 Sho.ny, M. n., 392 Sha.rm:J., K. R., 437

,\

'';-.

670

SUBJECT INDEX

Channel section.s, hydrostatic catenary, Circular channels, best'hydraulic section . 22, 35, 161 161 . ' for irrigation c&nr.IR, 175n. flow measurement in, 82-83 lill tearia, 22 . geometric elemeuts, 21,23 625-627 parabolic (sec l'ambolic channels) hydra.ulic exponent ivl, 67' rectangula.r (see Rectangular channels) hydraulic exponentN) 132 semi-elliptical, of high order, :ZOn. Manning's n in, 106, 134-135 tr~.pezoidal (see Tra.pezoidal channels) roughness vari!ltion in, 106, 135 tria.ngular (see Triangular channels) values, of AR'~/do~~, 130 yelocity distribution in, 24.-26 Qf Z/d ou , ti5 . -vertical, 20 (See also Circular conduits) Channel si~e Rnd shape, 104, 106-109 Circular cocduits, depth, of maximum Channel slope (see. Slope) 134-135, 152 Channels, of adverse slope, 223; 25\)-260 of maximum velocity, 1:34-135, 152 oi compcllud sections, 138-140 flow (IT-(JS, lZftn., curved, energy loss in, 441 132-136 erodible,IM-119 flow profiles in, 230··231 types, ix r<:nlghnllSB coefficient in, 106, 135 wHh grn.dually closing crown, 67-6!1, varied-fiow fUnction, 202 131-135,230-231, 261-252 Circalatioll (1onsto.nt, 44G . grassed, 179-188 Coefficient, Be llssinesq, 27 h.orizontnl (see HQrizontal chanilels) of contraction, 374-377 ice-covered, 137-138, 153 CorioIis, 27 of luge slope, 33 of curve resistance, -1+1 lining, 157-158, 160 of discharge, for bottom rucks, 339 mild,223 through constrictions, 478-492 of negative slope (~ee adV8l1le, above) for overtio\\' spillway", 366 .open, IS, 26-27, 258 for sharp-crested weirs, 302, 3GB-369 prismatic and nonprimnatic, 20 for submerged overflow spillwo.ys, steep, 223 386 with vadnnle hydraulic exponents, of drag, 94n. 260-261 of energy (see En~rgy coefficient) of variable slope, 238 of energy loss, 468 (See also Channel·sect.ions; Circular of frictioa, 171 . channels; NOJ;lerodible channel:;; of pl'es;;;lI'e distribution, 32, 50 Nonprismatic channels; Pnrabolic of retarciance, 179 chanuels; Rect.ll.ngula.r channels; of roughness) 92 Rough channels; Slope; TrapeBIl.'liin's 111, 95 ze,id"J ch,1nnels; Trill.ugular chanin circular cOlldtJits) 106, 135 nels) Horton table for, 108 Charo.cteristic depth, 238n, Kutter!s n, 04 . Characteristic length, for Froude uumManning's n (see Manning's n) bert 13 . Cohesive material, pijl'missible tracth'e for Reynolds numher, 7-8 fome for, 165, 174 for turbnlent mixing, 200 perlllissible velocity for, 165 Characteristics, 589 Comple.t€ method for lillstclLdy flow, M1 Chez}' formda, 93-94 Corr;plete turbulence, 12 Ch6zy's C, 93-9& Composite roughness, l:JG-140 formulas for, BfLzin, 95 Conc!Lve flow, 30-31 G; Ie, 94 " Confluence of rivers, 321-323 Mannillg, 100 Conjugate depth, 418 Pavlovski( 100 Conjugate normal depth) 231 Powell,95 ConsolidlLted-shear test, ~ 73n. for smooth and rough cha.nnels, 204 .Constant critical flow, channel of, 82 Chezy's resistance fn.cto·r (soe CMzy's C') Const!).nt hydratilic radius ehannel, 152 Chute, 19, 324 Constrictions, 475-476 Chute blocks, 414 backwa.ter due to, 490-493

SUBJECT INDEX

Continuity equation, 5 of unsteady flow, 525-526 Continuous flow, Ci" Contracted--opening method, V6, 476 Con traction, coeffll:iellt Of, 374-377 Contraction distance, 258 Contractiun ratio, 479· Contractions, -17-49, 468-470 Control, of !low; 70-74, 237· of hydnmlic jump, 408-414 Control scction, 70, 234-237 in laterru spillway chl>nnels, 342 Conversion lcss, 310 Convex flow, 30-31 Conveya;lCe, 128 Conyeyancfl ratio, 479 Co,iolis coefficient, 27 Cowo.n's method for Manning's n, 106109 Creager prums) 364n. modified, 36411.. Critic!!.l depth,41 in circular sec tion, 82 curves for computation, 65 overrun, 533 in rectangular sections, 81-82 Critical discharge, 64 . through rectangular sections, 66 through sections of various shapes, 81 Criticru flow, ~pplicl1.tion of energy principle,4'2n. . appliaution of theorem of momentum, 54n, computlltion (RM CriticpJ·flow computat.ion) criterifL, 42-43, 55, 59-60, 63 definition, '13, 43, 55 Critical-flow oomputation, 63-70 hydraulic exponent. for, 66'--68 section fa"tor of, 64 Critical-flow flnme, 74 for closed conduits, 82-83 San Dim!l.S, 78 Critical-flow profiles, 23&-241 Critical roughnes;;, 195 Critical sectio n, 63 Critieal slope, 53 . at ·given normal depth, 142, 154 Critical-slope cur"e, 243 Critical state of flow, 13 Critical tractive force, 172 Critical velocity, 13 Cross waves, 448, 468 Cui verts, 20 . flow through, 493-499 flow proJiles in, 265-266 flow types in, 496-499

671

Culverts, hydraulically long, 494 hydraulicnJly short, 494 Curve resistance, coefficient of) 444 Curved vanes, 457 Curvllillear flow, 31-34 .

Dam-break problem, 534-536, 568 Darcy's friction factor, 211 Darcy-Weishach formula, 8, 124 d' AubuissOll formula., 502 Deliyery of canal, 297-306 for constant discharge, 300-302 for constant downstream dellth, 299300 for constant upstream depth, 297~299 for super critical flow, 302-303 Delh-ery curve, 297 Demand surge, 559 De Marchi profile, 354n. Denver tests, 364n . Depth, alternate, 41, 5511.. characteristic, 238n, conj uga.te, 418 conjugate normal, 231 critic!>l (see Critical depth) of flow, 22 of flow lIec tion, 22 initial, 46 normal, 91, 129 in circUlar conduits, 230-231 determination of, 140-142 sequent, 46, 5511.., 307 transitional, 23711.., 23B, 242-244 Diffusivity, 501 Direct-integration method, 252-262 Direct jurop, 45, 397 Direct st,epmetn.od, 262-265 Discharge, coefficient of (see Coefficient of discharge) through cOllStrictions, 476-490 critica.l, 64; 68, 81 through culverts, 493-499 through gates, 508 in·ice-covered channels, 137-138 'maximum, in canals, 299 normal, 129 for 1-ft fa.ll, 281 through piel's) 501.,.506 through pile trestles, 5D6 service, 303 transitiona.l, 242 through trash rll.cks, 506-507 Discharge rating curve, 70 for inflow to canals, 805 Discontinuo\lB flow, 5 Displacemeat thickness, 193

"

, .'

672.

SUBJECT INDEX

Diatribution (see Pressure distribution) Disturbance lines, 449, 454-455 Douma formulu" 36 Drag, coeffieit)ut of, 94n. Drag force, 168n. . . Drain tiles, 1'0ughness variation in, 135 Drawdown curve, 71 Drop, 20 Drop energy dissipator, 42Zn., 424-425 Diop number, 423 . Drop spillways, 423-425 Drowned-out hydraulic jumps, 425 Drum gates, 380-382, 507n. Dynamic equatioo, of gradually varied flow, 218-222 for spatially varied flow, 332 for uniformly progressive flow; 531 lor unsteady flow, 526-528 Dynamic viScosity, 8

Ezra method, 272-274 for natural channels, 284:-292 foR relationship, 8-13 for roughchannelll, 11-13 for !IIIlooth channels, 9-10 Factor, of flow resistance, 91

;;hape,210 Fall increaser, 394n. F&.w<)r tbeory, 358 Fish skimmer, 337 Flip b.ucket, overflow spillways, 85-36 Flood discharge, cllmputation, 146-148, 476-490

Flood plain, roughness of, 104, 113 Flood routing, 586 Flood synthesis, 586 Flo,", concave, 30-31 continuous, 5 convex, 30-31

Easement curve, 457

Eddy loss, 267 Efficiency of hydraulic jump, 896 Einstein's bed-load function, 161m. End point of backwater, 319 . Energy, minimum, tbeorem of, 42 in nonprisnllLtic channels, 40-49 in open-channel fiow, 39-40 specific, 41 in surges, 565-(;66 Energy coefficient, 21 in spatially varied flow, 345 at sudden transitions, 465-466 Energy dissipators, bucket-type, 408 drop-type, 42211.., 42·4-425 hydraulic jump for, 404-408 impaot-tYpe, 422n,. Energy equation, B!;'rnoulli, 40 Energy grade line, 3 Energy gradient, slope of, 40 Energy line, 3 Energy loss, coefficient of, 46.8 in hydraulic jump, 60

(See also Losses) Energy reoovelY through obstructions, 50U Entrance to cano.ls, 304-305 Equations of characteristics, 588 Eqttivo.lent n value, 136 Erodible channels, 164-179

types, ix Erosion below gates, 512 Escailde profile, 364n. • Expansion in chanllel, 57-59 Expansion wave, oblique, 449,
critical (see CritiellJ flow) curvilinear, 31 depth,2Z discontinuous, 5

gradually varied, 6-7 grl\dually varied unsteady, 7 instability oi, 204 isolated-roughness, 12n., 196 lamirlll.r,7 opeu-chanuel (see Open-channel liow) overland,544 parallel, 30, 33 passing islands, 320-321 pipe, 4 progressive, 540, 555-557 pulsating, 580 quasi-normal, 23811.. quasi-smootb, 12n., 196

rapid, 13 rapidly varied, 6 rapidly. varied unsteady, 7 regimes of, 14:-16 rough-surfMe, 196-198 sec?ndary, 12, 439 shooting, 13 skimn:>lng, 196 slug, 581 spatially varied (see Spatio.lIy varied flow) .spiral, 25-26, 439-440 state of, 7-14 steady, 5 steady uniform, 5 streaming, 13 . subcritical, t3 supercritical, 13

673

SUBJECT INDEX

Flow, torrential, 13 tranquil, 13. turbulent (see Turbulent flow) uniform (see Uniform flow) Ullstcac\y (see Unsteady Bow)

unsteady varied, 6, 7, 523 varied,6 wake-interference, 12n.) 196 Flow control, 70-74, 237 Flow-line computll.tion, 608 Fl'),;,{ measurement, 72--81 }I'low-net analysis, 358, 374, 384 Flow pro liles, analysis, 232-237 in channels; of adverse slope, 259-260 with gradualli closing crown, 230231, 261-262 of variable hydraulic exponents, 261 or variab1e slope, 238 cla::sification of, 226-232 .in closed oonduits, 230-231 critical, 238-241 dillcontin'Jity in, 224 in Idctionless rectanguial' cha.nnels, 292 . in horizontal channels, 259-260 in nonprisIDll.tic channels, 235-237, 306-309 point of inflection in, 224-227, 292 along side weir, 340-341 o(spa.tially varied fiow, 235-237, 333-331 transitional, 238 types, 241 nodal,

sp.ddle, 241 spiral, 241 vortex, 241 typical examples, 229 in wide horizontal channels, 292 Flow~resistll.Llce fa.ctor, 91 Flumed drop, 415 Flumes, 19 Parshall, 72--81 Force, drag, 168n. plu:! momentum, 54/1.. shear, 168n.. specific, 53-56 of stream, 5<1n. tracth-e (.ee Tractive force) Force coefficient, 50 Fort Collins test,g, 364n. Foriier-Scobey table of permissible velocities, 165-166 Fourier's law, 601 Free overfall, 44 Free surface, 3 Free vortex, 446

Freeboard, 159-160, 311

Friction·, coefficient of, 171 Friction barrier, 562, 565

Friction factor,

8~ 13

..

Friction loss in transitions, 319 Friction velocity, 195, 201 Frictionless rectangUlar chaJlnels, 292

Froude number, 13, 43 Function, bed-load, 166n. varied-flow, 254

G. K. formula, 94 graphical solution, 95 Ganguillet and Kutter formula (see G. K. formuia) Gates, drum, 380-382, 507n. overflow,S07n.

radial (Tainter) I 507-5ill rolling, 507-508 . sluice ($8e Sluice gates) Tainter (rndial), 507-512 underflow, 507-512

Gauckler formula, 99\'1. GEDA fiood computer, 609 Geometric el1~ment8, ·01 channel section, 22 . of circular section, 21, 23, 625-627 curves of, 270-271 . of parabolic channels, 639 . of paraboli<: sections, 21 of rectangular sections, 21 rounded-cornered, 21 .• of trapezoidal section, 21 of triangular section, 21 round-bottom, ·21 Gra.dient., energy, 40 velocity, 200

Gradual hydre.u!ic jump, 45 Gradually varied

now, (}

basic assumptions, 217-218 dynamic equatIon of, 218-222 for wide rectangular channeLs, 222 Gra.du!l.lly-varied~Row equation, 219-222 in tapered· channels, 246 Gradually varied unsteady flow, 7 Gra.phical-integration method, 249-252 Grassed channels, 179-188 centipede, 180 . design, examples, 186-188 for maximum capa.city, 181-188 for stability, 186-187 design procedure, 184:-188 n-VR curves, 182-183 permillsfule velocity, 184-185 retardance, cJ.a;ssmca tiOD, 181 selection, 184

I

!I'

I

, 6'74

SUBJEC'l' INDEX

Grassed chllJlnels, se'leei;ion of graSS' 184 " Gravity effect on flow,]3 Gravity wave, 13 ' Grimm method, 280n.

H pro files, 230 H.llgen formulo.,' 99n. Hangchow bore, 558 Heppner flood, 558 High stage, 41 , Highway gutter, flow in, 151 Horizontal channels, flow profiles in, 259-

260

[,

r i

I (

'

varied floW in, 333-337 slope, 223 Hor~on equation, 546 Horton'table for roughness coefficient, 108 bore, 534, 557-558 depth, 13, 23 arop, 43-44 gradual,45 Hydraulic exponent lYI, 66-68 , graphical determination, 68 Hydraulic e:~p()nent N, 131 g;raphical determination, 133 variation with depth, 132-134 ,Hydraulic exponents, of circular conduits, 67, 132 for critical-ilow computation (see: , Hydraulic exponcll.t M) for uniform-flow computation (se~ Hydraulic exponent N) Hydraulic gr!l.de line 3 Hydraulic jump, 45-'46 applications, 393-394 characteristi~s, 396-398 control, 408-414 abrupt drop for, 412-414 a.brupt rise for, 411-412 direct, 45, 391 drowned-01.\t, 425 efficiency, 396 elimination of, in canals, 305-306, at transitions, 57-59, 3l4. a,s dissipatvI, 404--408 energy in, 60 gradual,45 height, 396 initial depth, 45 length, 398-399 loc~tion, 390-404 moving, 557 ob)ique, 429-431 oscjllating, 395 in pllZe.bollc chiJ.llnels, 432

SUBJECT INDEx

Hydraulic jump, relatIve loss, 396 sequent depth, 46 in sloping cllann.els, 425-429 steady, 3f/5-396 strong) 395-396 submerged below sluice, 60 surface profiles, 399 in trapezoiual channels 431 types, 395-396, 407-408 undular, 45, 395

weak,39S

:

Rydro.ulic ra.dius, 23 Hydra.ulic routing, 586 Hydrogra.ph, 543, .592 Hydrologic routing, 586, 804 Hydrostatic catena.q, 22, 35, 161 Ioe-covered channels, 137-138, 153 Impact-type energy dissipator, 422n. Inflow-discharge-rating curve 305' Initial depth, 46 ' relative, 396 Inlet loss, 311 Instability of flow, 204, 210 Inverted sipholl, 317-319 Irrigation canals. 175n. Islands in rivers; 320-321 lsoclinal method, 346-347

Isola.ted-roughneilll flow, 12n., 196 Jaeger theorem, 5571.: ' Johnson method, 570-572 Johnstown flood, ,558 Jonfl5 formula, 533 Junction problems, 321-323, 512-516 for surges, 578-580 I(o.nsll.S River, flow-prof.ie' computntJon, 276-292 ' junction problem, 321 Kennedy fornnlla, 16671.:' :Kinematic vLqcosity, 7-8 Kinetic.fiow factor, 13n. Kineticity, l3n. Kir~chmer data for rough channels, 11 KIeltz-Seddon principle, 529 Kutter's'l1" 94 ' dimensions oi, 9871.. Lacey's regime theory)166n.' Ladder of cascades, 43'3 Lagrange celerity equation, 538 Laminar boundary layer, 193

Lamiu!Lr flow, 7 criterion, 7-8, 150 unst,eady, spatially vaded, 543 Lnminar suble.yer, 194 Laminar surf!!.Ce flow, spatially varied, 348:"349, 543 ' uniform, 149-150 Laue-Davis profile, 364n. Later8.1 spillway channels, 342-346 dis,~harge in, 350 Lateral tra.ctive force, 176 Leach diagram, 271-272 Length, cn!>racteristic, 7-8, 13 mixing, 200 Limit, liquid, 173n. {lla~tic, 17371., Limit slope, 142,243-244 , Lining of channels, 151-158, 160 Lintearia, 22 Liquid limit, 173n. LocD-l phenomenon, 6, 43-'Hi Losses, energy; in channels, 21871.., due, to conversion, 310 to eddy, 267 at inlet, 311 at outlet, 311' in pipes, 8 o.t transitions, 310-311 Low stage, 41 Lowel' normal depth, 231 Lucite, 10811.., 110

M profiles, 228 M value (see Hydraulic exponent At) Maddock-Leopold's principle of nhannel geometry, l66n. Manning; formula, 93 exponent for, 99n.--l00n. . for friction factor I, 124 international adoption, lOOn., nomog;ra'Pruc solution, 840 lVhnning's n, 99 in circular conduits, 106, 134-135 composite roughness, 135-·140 determinatipn oC, 101-123, 206-210 Cowan'll method for, 106,.,109 dimensions, 9an.-!l9n. of drain tiles, 135 equivalent value, 136 faciors afieqting, 101-1m~ on flood plains, 104; for friction factor I. 124 for ice-covered channels, 137-138 for Panama Canal, 105-106 photographs of channels for, 114-123 relation to roughness height, 206

675

Manning's 11-, for sewers, 106, i35 table ior, 108-114 . theoretical interpretation, 205 Maximunl discharge, in c8.nals,299 in circula.r conduits, 134-136 theorem, 59n. Maximum permissible velocity, 157, 165167 ' Maximum velocity, 24-25 Meandering, 106-109: Method, of characteristics, 44811., 587 of diffusion analogy, 601 of dired integration, 252-262 of finite increments, 541 of numerical integration, 261-262, 3 111-

34G of permissible velocity} 1[14, 167-168 of rela.Xll.tion, 358, 37411.., 384 of singula.r 'Point, 237-242 of tr!l.ctive force, 164, 175 of tria.l and error for unsteD-dy flow,

5i1 Mild channels, 223 Mild slope, 63 Minimum-energy th!:!ocem, 42 l\'linimum penniEsible velocity, 158 Mississippi River, Ba.~in'll1'1, 97 Chezy's C, 97 . diffusi vi ty, 602 gagings,94 junction problem, 322-323 Rutter's n, 97 Ma.nning',s n, 105 roughness datil., 206-207 Missouri River, ftQw~prQlile computa.tion, 27&:-292 junction problem, 321 Mixed progressive flow, 540 Mixing lengtll, 200 Momentum of o!len.channel no,,', 49-53 Momentum coefficient, 27 Momentum flux of stream, 54?t. Momentum principle, 49; 56-59 Monaclin::.! rising. wave, 528 Moving hydraulic jump, 557 Muskingum method, 60(J-(\07

N value ($ee Bydnl.lilic exponent N) n-VR curves, 180, 182-183 . ~a.gler formula, 501 ~appe, over spillways, 370-37?, 387 : over weirs, 361 ' Na.ppe aera.tion, 362-363 Naviga.tion canals, 572-575 Negative slope, 223. Negative surges, 566-568

676

SUBJECT INDEX

Negatively progressive flow, 540 NeutceJizing reach, 306 Nikuradse sand roughness, 202 Nikuradse's dfl,!.tt for smooth and rough pipes, 20l~204 Nodal flow pro.fi1es, 241 N oncohesi ve material, angle of'repose, 171-172 tractive force, 173-174 Nonerodible channels, 157 design, by best hydraulic section, 161 by width-depth r!ltio, 162-163 Nonerodible velocity, 165 Nopprisroa.tic chll.l:mels, 20 application of momentum principle, 5&-59 energy in, 46-49 flow profiles in, 235-237, 306-309 gradually-varied-flow equation, 246 Nonsilting and noneroding velocity, 166 Nonsilting velocity, 158 Nonsustaining slope, 223, 259 Normsl depth, 91, 129 in circular conduits, 230-231 coill ugate, 231 for comput,a.tion, 130 aeterml,oatao·n of, 140-1'42 lower, 231 upper, 231 Normal discharge, 129 Normal slop'l, 142 Number, Boussinesq, 13n. Froude, 13, 43 Reynokls,7 Vedernikov, 210 Weber, ix Oblique expansion wave, 449; 45,1, Oblique hydraulic jump, 429-431 Obstructions, 104, 106-109, 499--501 Ohio River junction problem, 322-323 Open channel, 19 wide, 26-27 Open-channeHiow,3 energy lu, 39-4{) momentulD in, 49-53 state, 7-14. t.ypes, 4-7 Open-flow tunnel, 20 Oscilla.ting jump,.396 Oscillating wave, 523n, Outlet and entrance of canals} 304-306 Outlet loss, Overliow 507n, O"erf!ow spillwa.ys (see Spillwa.ys) Overland £low, 14, 149, 544

Overrun, 529 Overrun critica.l depth, 533 Panama. Canal, for, 105 Parabolic channels, 188 best hydraulic section, Hll geometric elements, 63£1 of high order, ZOn. hydraulic jump in, 432 Parall"l flow, 30 pressure distriou tio!! in, 33 Parsha.ll flume, 72-81 . PavlC'vskii forlllula., 100, Permissible maXimu.lTI veloc.itie:o, 1657166' Permissible traetiv!,\ force, 165, 172~174 (SeE, also Tractiva force) Permissible velpcities, ror cohesive ma- ' terial, 165--167 corrections for dElpth and sinuosity, 167 ' in grassed channels, 184-185 maximum, 157, 165-167 method of, 167-168 minimu..-u, 158 U.S.S.R. data, 166-167 Piers, baffle, 415 bridge, 501-1506 in gated spillways, 370-380 Pile trestles, 506 ' Pipe flow, 3 PitO& sphere, 440 Plastic limit, 173n. Plasticity index, 173n. Positive slope, 223 Posi',ive surge, 559--566 Positively progressive flo,;\', 540 Powell formula, 95 Powell's f,.98 Power canals, 568-572 Prandtl-von Karman curve, 12 PrandtI-von IC&rmAn equation, 9 ' Prandtl-von Ka.rman universal-velocity- ':' distribution la.w, 201 ~ Pressure on gates, 510-512 Pressure coefficient, 32 Pressure distribution, 30-34 coefficieuts, 32, 50 in curvilinear fiow, 34 effect of slope on, 32-34 hydrostatic 1e.w of, 30'-32 in parallel How, 33 Pressure-dlstribu tion ooefficien ts, 32, 50 Pressure-head correction, 31 Prismatic channel, 20 Progressive flow, 540, 554-557 Puisa.ting flow, 580 '

, ".

SUBJECT INDEX'

Q-constant curve, 300 . Quasi-normal flow, 238n. Quasi-smooth flow, 121>., 196 'Radial (Tainter) ga,te, 507-512 Raft chute, 324 Rapid flow, 13 Ra.pidly varied flow, 6 ch,\r!Lcteristics, 357 Rapidly varied unsteady How, 7 Recovary of velocity head, 311 RectAngular chJlnnels, best hydra,ulic section, 61 frictionless, flow profile in, 2!l2 geometric el~ments, 21 hydra,ulic exponents, 66-67, 132 round--corneri:d, 20-21 values, or. AR;;/dD'~, 130 of Z/rlG u , 65 wide, '25-27 Reflection ooefficient, 577 Regime theory, 165'n.. Regimes of flow, 14-16 Rebbock 41411. Rehhock formula, 352 Rejection 559 Relative of hydraulic jump, 396 Rclative loss of hydraulic jump, 396 Relative roughness, 195 Reserroir rou ting, 586 Resistanc<;l modulus, 28h, Retarciv.ncc, coefficient of, 179 degree of, 181 Reynolds number, 7 critical, 8-9 Rippl mass curve, 607 River hydraulics, lS Road surface. flow on, 348-349 Roll waves, 210, 535, 581 discovery, 210 Rolling gates, 507--508 Rough cb&nnels, Bazht's data, 11, 204 j-R relationship, 11 Kirschmer's data, 11 Vanvick's data., 11-12 velocity distribution in, 202 Rough pipes, Nikuradse's data., 202n. R(JUgh-surface flow, 196-198 Roughness, channel, artificial, 202n, Bll.zin data fot, 11, 95, ggn. composite, 136-140 critica.l} 195 aue, to alignment, 103, 106-109 to bed load, 106 to irregularities, 103 "',0 size a.nd shape, 104, 106-109

677

Roughness, chlJ.Ilnel, of i~e-()Qvered channels, 137-138 isolated, 196 . relative, 195 size measure, 11 surface (see Smface roughness) varia.tion in sewers, 135 ' 'Roughness coefficient (see Coefficien t, of roughness) , Roughness factor for overland flow, 545 Roughness height, 195-196 related to Manning's n, 206 Routing, hydraulic, 586' hydrologic, 586, 604 Routing period, 605 Rugosity coefficient, 92n.

r {

B pro Jiles, 228-·230 S~ddle How'proJiles, 241 SAF stilling basin, 4) 5-417 Saint-Venant celerity equation, 5S8 San Dimas flume, 78 . Scale ratio, model, 16 Scimemi profile, 364n. Scouring, 103-104, 406-407,512 Secondary flow, 12, 439 Section factor, for critical-flow computation, 23, 64 , f'Jr uniform-flow computa.tion, 23, 128 Separation of flow, 471 Sequent depth, 46, 55n. relative, 397 Sen-ice die charge in eana.ls, 303 Sewers, roughnBSs vll.riation in, 135 Shape factor, 210 Sharp-orested weirs, 360-362 Bazin da.ta. ior, 36B, 382, 3QO for jump control, 40Q-410 Shear foree, 168n. Sheaf velocity, 201 Sheet flo,;\', 14, 148 Shock wave, 4Z9n., 448n. Shooting flow, 13 Side-channel spillways, 328 Side slopes of channel, 158-159 Side weirs, 340-341 Sills, in curved channels, 457 I(lr jump control, 408-409, 414 Silting, 103-104 ' , Singular point, 238 Ski-jump spillways, 384--385 Skimming flow, 196 SIQpe, cha.nnel, adverse, 223 change in, 232-234 critical, 63, 142, 154 definition, 4011.., 94n.

I \

,I

.)

)

[

"

II

678

SUBJECT INDE.'4

Slope, channel, effect on pressure distributiou J 32-34 horizontal, 223 ' large, 33 limit, 142, 243-244 mild, 63 negative, 223 non sustaining, 223, 259 norm(J.l, 142 positive, 223 side, 158--159 sIllall, 33 subcritical, 63 Sli percri tico.i, 63 sustaining; 223 variable, 238 zero, 223 of channel bottom, 4() for 'energy gradient, 40 of ellergy line, 40 ill uniform ilow, 40 of water surface, 40 Slope-area method 146-148 Slug flow, 581' " ,Sluice gates, 507-510 contraction dista.oce after, 258 How profile 24.0 257 submerged jump afte-~, 60 vena contracta I:lelow, 401 Smetana profile, 364n. Smooth pipes, Nikuradse data for 20ln.

204

'

Solitary wave, 537-538 Spatially varied flow, 5 afiillysis of flow profiles, 235-237 through bottom rack 337-340 with dscreasing discharge, 328-329, 332-333, 346 dynamic equation for 332 in horizontal cha.nnel~, 333-337 with increasing .discharge, 327-332, 341-:-142 method of numerical integration 341345 .. ' through side weirs, 340-341 surfllCe flow, 347-349 types of, 334 unsteady surfaoe, ,543 Spatially-varied-flow eqlu~tion 332 Spncific energy, 41 ' Specific-energy curve 41-42 Specific force, 53-56 ' . Specific-force curve, 54 Spillway channel, 306-309 342-346 Spillways, box inlet '424 straight, ' , overflow, Bn.zin data fOf, 363-364

I

Spillways, overflow, boundary layer on 199-200· , . bu.cket a.t toe, 36, 384 crest shapes, 363-365 design ot1l9ction, 367-358 design head, 368--369 discharge of, 365-370 flip bucket, 35-36 ' flow at toe, 382-384 Fort Collins teats, 364n. gated, 370-374 na.ppe profiles on, 370-373,,387 ,preiJaure on, 374, 378-380, <387 rating, 3G8-S70 . submerged, 385-388 sidc-cha.nnel, 328 ski-jump, 384-385 Spiral flow, 25-26, 439, strength, 440 2-11 Spiral flow Stahle hydraulic section, 1'76-179 Stage, 22 high and low, 41 Stage-fa.ll-discharge methOd, 280-284 Stage-verslls-Q/Wcurve, 281 Stage routing, 607 Stand of grass, 180 Standard step method, 2tl5-268 for natural ch9Jlneis 271-280 Standing swell, '402 ' Standing \1'8."6, 393n. Stanton diagram, 8 St!1.te of flow, laminr..r, 1 sub critical. 13 llupercritic'9-1. 13 transitional,' 7 tlIrbulcnt, 7 Steady flow, 5 Steady jump, 395-396 Steady uniform flow, 5 Steep cbanneld, 223 StolP method, direct, 262-265 stlludard, 265-268, 274-280 St,ilIing basin, 401 , generalized design, 414-411) SAF, 415-417 with sloping apron, 427-429 USBR,415, 417-422, 427-429 Streaming flow, '13 ' Strength of spiral flow 440 Strickler formula, 99n.', 206 Strickler's. constant, 20(3, Strong jump, 3g5-396 Subcritical slope, 63 Subcriticll.l sta.te of flow, 13 Sublltyel', laminM 194 8lldClen tni.nsition~, 461-468

SimJECT INDlllX

Sllpercritjcru slope, 63 , Supercritieal state of flow, 13 : Superelevlltion, 439 ' , Surface curve, 222 Surface flow, .spatially varied, 347-349 spatially varied unsteady, 543 uniform, 148-150 ,. Surface irr.eg\ll!1rities, 106-109' Surface roughness, 101-102 ,~oncept, 104 explanatio1l; 194 hydraulically rongh, 1-95 nydraulic!l.lly smOGtll, 195 wa.vy, 195 Surface tensiou, 7, Surge. 5S7 demand,559 energy in, 565-566 at junctions, 578-580' in navigation caT.lnls, 572-575 negative, 566-568 positive, 559-566 in power ca.nals, 563-572 rejection, 559 t.hrough transitions, 575-578 Suspended load, 106 Sustaining slope,223

I

I L I

t

I f·

I,

Tables, geometric elements of circular conduits, 625-627 permissible velocities, 165, 1S5 roughness coefficient n, 109-i13 varied-flnw function!!, 255,259, 641661 Tainter gate, 507-1H2 Theorem, of maximum discharge, 5%. of minimunl energy, 42 Thermal diffusivity, 601 Tidal hydraulics, ii, 5231'1.< TOIl- width" 22

To,:nintill.l flow, 13 Total force nf stream, 54n. Tractive force, 168 , critical, 172 distribul;ion, 169 : lateral, 176 . method of design for, 175 p~rmissible, 165, 172-174 unit, 168, }(i9, 173 Tr~ctive-forc"l ratio, 170-171 Tra.nqui! flow, 13 . ' Tra.nsition loss, 3LO-311 Tlranaitional depth, 2;!7n., 238, 2'!2-244' Transitional discharge, 242 Trllnsil;ional pro.file, 238 Transitionailitate of flow, 7

679

Transitions, gradual, 47-49, 51-59,310319 ' sudden, 461-468 surges through, 575-578 Transitory zone of uniform flow, 90 Tl'!l.nsla.tory l'rave, 523 . 'rrll.p3~oidli.1 channels, best hydraulic ~ilctioll, 161-162 geometric elements, '21, G2D-G~i7 hydraulic expDnent M, 66-67 hydraulic exponent N, 131-132 hydraulic jumps in, 431 values, of AR'Sldo'i, 130 of Z/dou, 65 TrllSh racks, 506-507 Trin.ngular channels, 20 best hydraulic seotion, 161 geometric elements, 21, 639 l'ollnded:bottom, 20-'21 Turbulence, cDmp!ete, 12 Turbulent boundary layer, 193 Turbulent flow, 7 criterioll, 7-8, 150 mllan velocity, 20J-205 unsteady spatially varied, 546 velooity distribution, 200-202 Turbulent surface flow, spatia!!y varied, 348 uniform, 150 tntrarapid velocit.y, 89 Underflow gates, 507-512 Undular jump, 45, 395 Uniform flow, 5 computation, 128-150 establishment, 88-91 que.lificnticins, 88 instabili ty, 210 steady, 5 theoretical equation for, 202-205 transito ry oon e, 90 unstea.dy,5 Uniform-flow equations, theoretical, 202-205 ' Uniform-flow forrhula.s, 91 Uniform surfaoe flow, 148-150 criterion, 150 Uuiformly progressive ftolt, dyna.mic equation, 531 wave pro£ile, 533-537 . Uniformly progressive WlIve, 528,,·· Unit flood, 602 Unlt tractive force, 168 distribution, 169 maximum, 169 permissible, 173

,

'.~

j

.! ,

;~

680

SUllJECT INDEX

UNIVAC, 601 . Univemal-velocity-distribution lq,w, 201 Unsteady How, 5-6, 523 . continuity, 525-526 dynamic equation, 526-528 rll.pidly varied, 1 Unstendy uniform flow, 5 Unsteady varied flow, 6, 1, 523 Upper normal depth, 2:31 U8BR stilling basins, 415, 411-422,427429 Varjerl flow, 1) gradually I 6' gradua.lly unsteady, 7, 523 rapidly, 6 rapidly unstell.dy, 7, 523 spatially, 5, 327 Varied-flow fUllction, 254 for adverse slopes, 259 far circula.r conduits, 262 tables, 255, 259, 641-561 Varwick'a data for rough channe18, 1112 Vedemikov number, 210 Vegetll.l retardance, 171l, 184 Vegeta.tion roughness, 102-103, 106-109 Velocity, critical, 13 friction, 195, 201 measurement, 21 nonerodible, 165 nonsilting. 158 and non eroding, 166 shear, 201 . . uLtmrnpid, 89 Velooity distribution, 24-26 t,urbulent.flow, 200-202 Velocity-distribution coefficients, 21':"30 for compound sections, 139-140 equations lor, 211 Velocity-distribution law, 201 Velocity gradient, 200 Velocity-head ratio, .l3n. Vebcity-head recovery, 311 Vena contracta below sluiee, 401 Venturi flUme, 14 . Vrilemonte weir sill, 80

Viscosity, dynamic, 8 effect on flow, 7 kinematic, 7-8 of water, 8 Vortex flow profiles, 241 Wake-interference flow, 12n., 196 Wash-water troughs, 34.9 Water area, 22 . Wat.!r seal, 317 Wave angle, 429 Wave pro Olea, uniformly progressive flml', 533-537 Wave suppressors, 422n. Wa.ves, capillary, 12 . graVity, 13 . monoclinal rising, 528 . oblique expansion, 449, 454. oscillatory, 523n. propagation, 13, 537 roll, 210, 535, 581· . shock, 42\1n., 448n. solitary, 531-538 translatory, 523 uniformly progressive, 528 Weak jump, 395 Weber nu.mber, Ix Weirs, broad-crested (see Broad-<:rested weirs) sharp-<:rested, 360-362 Villemonte sill for, 80 Weisbach formula, 501 WEB Ge~al Spillway Tests-CW 801, ~64, 37011.. , 374 WES st~ndard spillway shapes, 364'365 iQietteo-perimeter, 22 Wide horizonta,l channels, 292 Wide open cbapnel, 2f1-27 . critical a.nd normal depths in, 258 Width-depth ratio, 162-163 Ya.ngtze River, diffusivity, 602

Zero :slope, 223

,1

,