Gas Dynamics by Ethirajan Rathakrishnan

Fourth Edition

GAS DYNAMICS Ethirajan Rathakrishnan

` 425.00 GAS DYNAMICS, Fourth Edition Ethirajan Rathakrishnan

© 2012 by PHI Learning Private Limited, New Delhi. All rights reserved. No part of this book may be reproduced in any form, by mimeograph or any other means, without permission in writing from the publisher. ISBN-978-81-203-4605-5 The export rights of this book are vested solely with the publisher. Eleventh Printing (Fourth Edition)

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June, 2012

Published by Asoke K. Ghosh, PHI Learning Private Limited, M-97, Connaught Circus, New Delhi-110001 and Printed by Rajkamal Electric Press, Plot No. 2, Phase IV, HSIDC, Kundli-131028, Sonepat, Haryana.

Contents Preface ................................................................................................ xi Preface to the Third Edition ................................................................ xiii Preface to the Second Edition ............................................................... xv Preface to the First Edition ................................................................ xvii

1. Some Preliminary Thoughts ................................................... 1–17 1.1 1.2 1.3 1.4 1.5 1.6 1.7

Gas Dynamics—A Brief History .............................................. 1 Compressibility ........................................................................... 2 Supersonic Flow—What Is It? ................................................. 5 Speed of Sound .......................................................................... 6 Temperature Rise ..................................................................... 10 Mach Angle .............................................................................. 12 Summary ................................................................................... 15

2. Basic Equations of Compressible Flow ................................. 18–42 2.1 Thermodynamics of Fluid Flow ............................................. 18 2.2 First Law of Thermodynamics (Energy Equation) .............. 19 2.3 The Second Law of Thermodynamics (Entropy Equation).... 23 2.4 Thermal and Calorical Properties .......................................... 24 2.5 The Perfect Gas ...................................................................... 26 2.6 Summary ................................................................................... 35 Problems ...................................................................................... 39

3. Wave Propagation ................................................................ 43–46 3.1 3.2 3.3 3.4 3.5

Introduction .............................................................................. Wave Propagation .................................................................... Velocity of Sound .................................................................... Subsonic and Supersonic Flows .............................................. Summary ...................................................................................

43 43 44 44 45

4. Steady One-Dimensional Flow .............................................47–105 4.1 4.2 4.3 4.4

Introduction .............................................................................. The Fundamental Equations ................................................... Discharge from a Reservoir .................................................... Streamtube Area–Velocity Relation ....................................... vii

47 47 51 61

viii

Contents

4.5 De Laval Nozzle ...................................................................... 64 4.6 Supersonic Flow Generation ................................................... 72 4.7 Diffusers .................................................................................... 82 4.8 Dynamic Head Measurement in Compressible Flow ............ 86 4.9 Pressure Coefficient ................................................................. 91 4.10 Summary ................................................................................... 93 Problems ...................................................................................... 96

5. Normal Shock Waves ......................................................... 106–145 5.1 Introduction ............................................................................. 106 5.2 Equations of Motion for a Normal Shock Wave ................. 107 5.3 The Normal Shock Relations for a Perfect Gas ................. 108 5.4 Change of Stagnation or Total Pressure across the Shock ... 112 5.5 Hugoniot Equation .................................................................. 115 5.6 The Propagating Shock Wave ............................................... 118 5.7 Reflected Shock Wave ............................................................ 124 5.8 Centred Expansion Wave ....................................................... 129 5.9 Shock Tube .............................................................................. 132 5.10 Summary .................................................................................. 137 Problems .................................................................................... 141

6. Oblique Shock and Expansion Waves ................................ 146–212 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11

Introduction ............................................................................. 146 Oblique Shock Relations ........................................................ 147 Relation between b and q ........................................................... 150 Shock Polar ............................................................................. 153 Supersonic Flow over a Wedge ............................................. 155 Weak Oblique Shocks ............................................................. 158 Supersonic Compression ......................................................... 160 Supersonic Expansion by Turning ......................................... 161 The Prandtl–Meyer Expansion .............................................. 162 Simple and Nonsimple Regions ............................................. 170 Reflection and Intersection of Shocks and Expansion Waves .................................................................... 170 6.12 Detached Shocks ..................................................................... 186 6.13 Mach Reflection ...................................................................... 187 6.14 Shock-Expansion Theory ........................................................ 191 6.15 Thin Aerofoil Theory ............................................................. 196 6.16 Summary .................................................................................. 203 Problems .................................................................................... 205

7. Potential Equation for Compressible Flow ........................ 213–231 7.1 7.2 7.3 7.4

Introduction ............................................................................. 213 Crocco’s Theorem ................................................................... 213 The General Potential Equation for Three-Dimensional Flow ......................................................... 217 Linearization of the Potential Equation ............................... 219

Contents

ix

7.5 Potential Equation for Bodies of Revolution ...................... 222 7.6 Boundary Conditions .............................................................. 224 7.7 Pressure Coefficient ................................................................ 227 7.8 Summary .................................................................................. 228 Problems .................................................................................... 231

8. Similarity Rule .................................................................. 232–269 8.1 8.2

Introduction ............................................................................. 232 Two-Dimensional Flow: The Prandtl–Glauert Rule for Subsonic Flow ................................................................... 232 8.3 Prandtl–Glauert Rule for Supersonic Flow: Versions I and II .................................................................... 240 8.4 The von Karman Rule for Transonic Flow ......................... 243 8.5 Hypersonic Similarity ............................................................. 246 8.6 Three-Dimensional Flow: The Gothert Rule ........................ 249 8.7 Critical Mach Number ........................................................... 259 8.8 Summary .................................................................................. 264 Problems .................................................................................... 268

9. Two-Dimensional Compressible Flows .............................. 270–282 9.1 Introduction ............................................................................. 270 9.2 General Linear Solution for Supersonic Flow ...................... 271 9.3 Flow along a Wave-Shaped Wall .......................................... 276 9.4 Summary .................................................................................. 280 Problems .................................................................................... 281

10. Prandtl–Meyer Flow ......................................................... 283–289 10.1 Introduction ............................................................................. 283 10.2 Thermodynamic Considerations ............................................. 284 10.3 Prandtl–Meyer Expansion Fan .............................................. 284 10.4 Reflections ............................................................................... 287 10.5 Summary .................................................................................. 288 Problems .................................................................................... 288

11. Flow with Friction and Heat Transfer ............................... 290–317 11.1 Introduction ............................................................................. 290 11.2 Flow in Constant-Area Duct with Friction ......................... 290 11.3 Adiabatic, Constant-Area Flow of a Perfect Gas ............... 292 11.4 Flow with Heating or Cooling in Ducts .............................. 302 11.5 Summary .................................................................................. 309 Problems .................................................................................... 313

12. Method of Characteristics ................................................. 318–342 12.1 12.2 12.3 12.4 12.5

Introduction ............................................................................. 318 The Concepts of Characteristics ........................................... 318 The Compatibility Relation ................................................... 319 The Numerical Computational Method ................................ 322 Theorems for Two-Dimensional Flow ................................... 330

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Contents

12.6 Numerical Computation with Weak Finite Waves .............. 332 12.7 Design of Supersonic Nozzle .................................................. 336 12.8 Summary .................................................................................. 341

13. Measurements in Compressible Flow ................................ 343–415 13.1 Introduction ............................................................................. 343 13.2 Pressure Measurements .......................................................... 343 13.3 Temperature Measurements ................................................... 350 13.4 Velocity and Direction ........................................................... 354 13.5 Density Problems .................................................................... 356 13.6 Compressible Flow Visualization ........................................... 356 13.7 High-Speed Wind Tunnels ..................................................... 374 13.8 Instrumentation and Calibration of Wind Tunnels ............. 400 13.9 Summary .................................................................................. 407 Problems .................................................................................... 415

14. Rarefied Gas Dynamics ..................................................... 416–423 14.1 14.2 14.3 14.4 14.5

Introduction ............................................................................. 416 Knudsen Number .................................................................... 417 Slip Flow ................................................................................. 420 Transition and Free Molecule Flow ...................................... 420 Summary .................................................................................. 422

15. High Temperature Gas Dynamics ..................................... 424–426 15.1 15.2 15.3 15.4

Introduction ............................................................................. 424 The Importance of High-Temperature Flows ....................... 424 The Nature of High-Temperature Flows .............................. 425 Summary .................................................................................. 426

Appendix A Table A1 Table A2 Table A3 Table A4 Table A5

........................................................... 427–495 Isentropic Flow of Perfect Gas (g = 1.4) ................... 427 Normal Shock in Perfect Gas (g = 1.4) ..................... 440 Oblique Shock in Perfect Gas (g = 1.4) .................... 450 One-Dimensional Flow with Friction ( g = 1.4) .......... 484 One-Dimensional Frictionless Flow with Change in Stagnation Temperature (g = 1.4) ............ 490

Appendix B ............................................................ 496–503 Listing of the Method of Characteristics Program ...................... 496 Appendix C ............................................................ 504–507 Output for Mach 2.0 Nozzle Contour ............................................ 504 Appendix D ........................................................... 508–509 Oblique Shock Chart 1 .................................................................... 508 Oblique Shock Chart 2 .................................................................... 509 Selected References ................................................. 511–512 Index .................................................................... 513–517

1 1.1

Some Preliminary Thoughts

GAS DYNAMICS—A BRIEF HISTORY

Until the nineteenth century very little knowledge of gas dynamics had been assimilated by man. The motion of air, its effects and power were felt by human beings only through storms or from the disturbances created for lighting fires and other similar natural phenomena. Only those who were gifted with imagination beyond their times observed the flying of birds and dreamt of flying machines. Many efforts were made in those directions, costing priceless human lives. The early manned flights like those of Icarus and Bladud were not based on any aerodynamic concept. The theory of air resistance was first proposed by Sir Isaac Newton in 1726. According to him, aerodynamic forces depend on the density and velocity of the fluid, and the shape and size of the displacing object. Newton’s theory was soon followed by other theoretical solutions to fluid motion problems. Fluid motion was assumed to occur under idealized conditions, i.e. air was assumed to possess constant density and to move in response to pressure and inertia. Interest in gaining a deep understanding of dynamics of air motion arose because of its application to hot air balloon, windmill, ballistic devices (guns and cannons), and so on. Knowledge was mostly derived by trial and error, and codes of practice did not exist. The experimental techniques introduced for measurement during the eighteenth century provided a breakthrough in the study of aerodynamics. Benjamin Robins in the UK constructed a whirling arm to determine the air resistance of bodies, and a “ballistic pendulum” to find the velocity of a bullet or shell. In the former experiment, a horizontal arm was rotated about a vertical axis by the tension of a string holding a falling weight. After a few rotations the speed of the end of the whirling arm was constant, at approximately 7.6 m/s. Test objects were mounted at the end of the arm and their air resistance altered the speed of rotation. This device was used to compare the resistance of different shapes, and to show how the resistance of 1

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Gas Dynamics

2 2.1

Basic Equations of Compressible Flow

THERMODYNAMICS OF FLUID FLOW

Entropy and temperature are the two fundamental concepts of thermodynamics. The energy changes associated with compressible flow, unlike low-speed or incompressible flow, are substantial enough to strongly interact with other properties of the flow. Hence, the energy concepts play an important role in the study of compressible flow. In other words, the study of thermodynamics which deals with energy (and entropy) is an essential component in the study of compressible flow. The following are the broad divisions of the fluid flow studies classified, based on thermodynamic considerations: Fluid mechanics of perfect fluids, i.e. fluids without viscosity and heat (transfer) conductivity, is an extension of equilibrium thermodynamics to moving fluids. The kinetic energy of the fluid has to be considered in addition to the internal energy which the fluid possesses when at rest. Fluid mechanics of real fluids goes beyond the scope of classical thermodynamics. The transport processes of momentum and heat are of primary interest here. But, even though thermodynamics is not fully and directly applicable to all phases of real fluid flow, it is often extremely helpful in relating the initial and final conditions. For low speed flow problems, thermodynamic considerations are not needed because the heat content of the fluid flow is so large compared to the kinetic energy of the flow that the temperature remains nearly constant even if the whole kinetic energy is transformed into heat. In modern high-speed problems, the kinetic energy content of the fluid can be so large compared to its heat content that the variations in temperature can become substantial. Hence, the emphasis on thermodynamic concepts assumes importance. 18

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GAS DYNAMICS RATHAKRISHNAN, E. EDITION: FOURTH EDITION ISBN: 978-81-2034605-5 Pages: 536 Price: R 425.00

About The Book Description: This revised and updated fourth edition continues to provide the most accessible and readable approach to the study of all the vital topics and issues associated with gas dynamic processes. At every stage, the physics governing the process, its applications and limitations are discussed in depth. With a strong emphasis on the basic concepts and problem-solving skills, this text is suitable for a course on Gas Dynamics/Compressible Flows/High-speed Aero-dynamics at both undergraduate and postgraduate levels in aerospace engineering, mechanical engineering, chemical engineering and applied physics. The elegant and concise style of the book, along with illustrations and worked examples, makes it eminently suitable for self-study by scientists and engineers working in the field of gas dynamics in industries and research laboratories. Some of the Distinguishing Features of the Book : Concise coverage of the thermodynamic concepts to serve as a revision of the background material. Logical and systematic treatment of fundamental aspects of gas dynamics, waves in the supersonic regime and gas dynamic processes. In-depth presentation of potential equations for compressible flows, similarity rule and two-dimensional compressible flows. Introduction to measurements in compressible flows and optical flow visualization techniques. Introduction to rarefied gas dynamics and high-temperature gas dynamics. Solution Manual for instructors containing the complete worked-out solutions to chapter-end problems. New to the Fourth Edition : Some vital aspects associated with the compression and expansion waves are explained, with suitable worked numerical examples. A brief section on critical Mach number is added in Chapter 8, highlighting its influence on the aerodynamic efficiency of flying mechanics. Nozzle flow process has been illustrated with worked examples focusing on the design and application aspects. A considerable number of worked examples are added, focusing attention on the design aspects. Some new problems along with answers are added at the end of many chapters. Contents: Preface • Preface to the Third Edition • Preface to the Second Edition Preface to the First Edition 1. Some Preliminary Thoughts. 2. Basic Equations of Compressible Flow. 3. Wave Propagation. 4. Steady One-Dimensional Flow. 5. Normal Shock Waves. 6. Oblique Shock and Expansion Waves. 7. Potential Equation for Compressible Flow. 8. Similarity Rule. 9. Two-Dimensional Compressible Flows. 10. Prandtl-Myer Flow. 11. Flow with Friction and Heat Transfer. 12. Method of Characteristics. 13. Measurements in Compressible Flow. 14. Rarefied Gas Dynamics. 15. High Temperature Gas Dynamics. Appendices A: Table-A1: Isentrophic Flow of Perfect Gas (γ = 1.4). A2: Normal Shock in Perfect Gas (γ = 1.4). A3: Oblique Shock in Perfect Gas (γ = 1.4). A4: One-Dimensional Flow with Friction (γ = 1.4). A5: One-Dimensional Frictionless Flow with Change in Stagnation Temperature (γ = 1.4). B: Listing of the Method of Characteristics Program. C: Output for Mach 2.0 Nozzle Contour. D: Oblique Shock Chart I, Oblique Shock Chart II Selected References • Index