SNAME The Wageningen B-Screw Series.pdf

The Wageningen B-Screw Series By W . P. A. van L a m m e r e n ] and M. W . C. Oosterveld, 2

Member, J. D. van M a n e n / Member, Member

The Wageningen B-screw series have been extended gradually to 21 screw series having blade numbers ranging from 2 to 7 and blade-area ratios between 0.30 and 1.05. Recently, the existing screw series were correlated with the new screw series with an upto-date fairing technique. This correlation was made by means of a regression analysis. Further, the influence of the Reynolds number (scale effect) on the test results was taken into account. For ship maneuvering studies, it is necessary to know the propeller thrust and torque characteristics over a wider range of operating conditions. Therefore, openwater tests with B-series screws over the entire region of operation were: carried out. In order to obtain systematic knowledge on the cavitation characteristics of the B-screw series in a uniform flow, tests have been conducted. The curves for cavitation inception were established and the influence of cavitation on propeller thrust and torque was

determined.

Introduction AN IMPORTANT m e t h o d of screw design is t h a t b a s e d on t h e results of o p e n - w a t e r t e s t s w i t h s y s t e m a t i c a l l y v a r i e d series of screw models. T h e s e screw series c o m p r i s e m o d e l s whose c h a r a c teristic dimensions, such as p i t c h r a t i o P/D, n u m b e r of b l a d e s Z, b l a d e - a r e a r a t i o A u/Ao, b l a d e o u t line, s h a p e of b l a d e sections, a n d b l a d e t h i c k n e s s a r e s y s t e m a t i c a l l y varied. A m o n g t h e w e l l - k n o w n screw series as d e v e l o p e d b y Schaffran, T a y l o r , G a w n a n d others, t h e W a g e n i n g e n B-screw series of t h e N e t h e r l a n d s S h i p M o d e l B a s i n t a k e an i m p o r t a n t place. T h e B-series screw t y p e is f r e q u e n t l y used in p r a c t i c e a n d possesses s a t i s f a c t o r y efficiency as well as reasonable cavitation properties. T h e first t e s t s w i t h s y s t e m a t i c series of screw propellers were p e r f o r m e d a t t h e N e t h e r l a n d s S h i p M o d e l Basin in 1936. F r o m m o d e l experiDirectors, Netherlands Ship Model Basin, Wageningen, The Netherlands. =Head of Research Department, Netherlands Ship Model Basin, Wageningen, The Netherlands. Presented at the Annual Meeting, New York, N.Y., November 12-14, 1969, of THE SOCIETYOF NAVALARCHITECTS AND M A R I N E ]~NGINEERS.

merits c a r r i e d o u t b y B a k e r a n d R i d d l e [1]a a n d B a k e r [2] it h a d b e c o m e e v i d e n t t h a t screws used so far w h i c h h a d c i r c u l a r - b a c k b l a d e sections a n d elliptical b l a d e o u t l i n e ( T a y l o r a n d Sehaffran) were in m a n y cases inferior to screws w i t h airfoil sections. T e s t s p e r f o r m e d at t h e N e t h e r l a n d s S h i p M o d e l Basin confirmed t h e s e conclusions in m a n y cases. B a s e d on t h e s e results a series of f o u r - b l a d e d m o d e l screws of a s h a p e similar to B a k e r ' s was m a n u f a c t u r e d a n d t e s t e d in t h e open condition. T h i s series was called t h e A 4:-40 series (thus 4 - b l a d e d series w i t h 0.40 b l a d e a r e a r a t i o ) . T h e r e s u l t s of t h e o p e n - w a t e r tests w i t h this series were given b y T r o o s t [3]. L a t e r on it was f o u n d t h a t t h e A 4-40 series screws, owing to t h e n a r r o w b l a d e t i p s a n d t h e a i r f o i l - s h a p e d sections o v e r t h e whole blades, were o n l y s u i t a b l e for use in cases w h e r e no c a v i t a t i o n d a n g e r was present. I n a d d i t i o n , t h e A 4-40 series screw t y p e did h a v e t m f a v o r a b l e b a c k i n g c h a r a c t e r i s t i c s . T h e s e con,dderations led to t h e design of screws w i t h wider b l a d e tips, circularb a c k seetions n e a r t h e b l a d e tips, a n d airfoils h a p e d seetions n e a r t h e hub. Screws of t h i s t y p e a Numbers in brackets designate References at end of paper.

269

Table 1 Btade

number Z

Btade

Summary of the Wageningen B-Screw Series ratio

area

A~/Ao

o.3o

2 3

0.55

#

0.140

1 . oo 0.45

_

0.55

o.6o

0.70

0.75

5

1.05

0.80

6

7

were called B-series screws. In this way the B 4 - 4 0 s c r e w s e r i e s w a s d e s i g n e d a n d g r a d u a l l y extended to further series with larger blade-area ratios and various blade numbers. These extensions were partly (mainly concerning the higher n u m b e r of b l a d e s a n d t h e l a r g e r b l a d e - a r e a r a t i o s ) s p o n s o r e d b y t h e ( f o r m e r ) B u r e a u of S h i p s of t h e United States Navy. T h e r e s u l t s of t h e o p e n w a t e r t e s t s w e r e g i v e n i n a n u m b e r of p u b l i c a t i o n s b y T r o o s t [3], a n d o t h e r s [4], [5], a n d [6]. A t p r e s e n t , a b o u t 120 s c r e w m o d e l s of t h e B series screw type have been tested at the Netherlands Ship Model Basin. Table 1 gives a summ a r y of t h e series. I n g e n e r a l , t h e r e s u l t s of t h e t e s t s w e r e g i v e n i n t h e f o r m of K T a n d K o coeffic i e n t s e x p r e s s e d a s a f u n c t i o n of t h e a d v a n c e c o e f f i c i e n t dr f o r a n a l y t i c a l w o r k , a n d in t h e f o r m of B p - 6 a n d B u - 6 d i a g r a m s f o r d e s i g n p u r p o s e s . From a correlation between the available diag r a m s of t h e B - s c r e w s e r i e s i t a p p e a r s t h a t s m a l l

differences exist. However, during the last years the B-series have been extended considerably and a c r o s s - f a i r i n g of t h e B - s c r e w s e r i e s d i a g r a m s f o r d i f f e r e n t b l a d e - a r e a r a t i o s a n d p r o b a b l y f o r diff e r e n t n u m b e r s of b l a d e s m u s t b e p o s s i b l e n o w . R e c e n t l y w e h a v e s t a r t e d t h e f a i r i n g of t h e B s c r e w s e r i e s t e s t r e s u l t s b y m e a n s of a r e g r e s s i o n a n a l y s i s . A s a r e s u l t of t h i s a n a l y s i s , t h e t h r u s t a n d t o r q u e c o e f f i c i e n t s K T a n d K o of t h e B - s e r i e s will b e e x p r e s s e d a s p o l y n o m i a l s of t h e a d v a n c e ratio J, the pitch ratio P/D, the blade area ratio A R / A o , a n d t h e n u m b e r of b l a d e s Z . I n a d d i t i o n , t h e e f f e c t of R e y n o l d s n u m b e r o n the test results was taken into account by using t h e m e t h o d d e r i v e d b y L e r b s [7] f r o m s i m i l a r m e t h o d s u s e d f o r t h e c a l c u l a t i n g of t h e p e r f o r m a n t e c h a r a c t e r i s t i c s of a i r s c r e w s f r o m t h e c h a r a c t e r i s t i c s of e q u i v a l e n t b l a d e s e c t i o n s . In the f u t u r e , t h e e f f e c t of R e y n o l d s n u m b e r m a y b e t a k e n i n t o a c c o u n t i n t h e p o l y n o m i a l s as well.

Nomendature By = loading coefficient, B,c CD Cf CL CT

= = = = =

N P ~/~ VA %

chord length of blade section drag coefficient skin-friction drag coefficient lift coefficient t h r u s t coefficient, T CT -711 / 2 p V A 2 --4 O

Cr* = t h r u s t coefficient, CT* = T 7r

~/2p[Va 2 -t- (0.77rnD) 2] ~ D 2 Co* = torque coefficient,

Q C o * = V..p[VA 2 + (0.77rnD)q -~ D~D 4

270

d = hub diameter D = propeller diameter J = advance coefficient, J -

VA nD

K~, = t h r u s t coefficient, KT--

T on2D ~

K o = torque coefficient,

O Ko = -pn2D - 5

n,N

= number

p = pv = p~ = _P O r R Re

= = = = =

of revolutions per second and per minute pressure vapor pressure static pressure of undisturbed streanl power torque radius propeller radius Reynolds number

The Wageningen B-Screw Series

t = maximum thickness of blade section T = thrust V,I = undisturbed stream velocity Z = number of screw blades AE/Ao = blade area ratio of screw .P/D = pitch ratio of screw d / D = hub diameter ratio = angle of attack of blade section /3 = hydrodynamic pitch angle a t 0.7 R, /3 = aretan VA/O.7rrnD = speed ratio, ~ = 101.27/J p = specific mass of water ao = cavitation number, p -- pv 170 - -

~/2. VA2

no = open-water efficiency, J Kr 2~ KQ = kinematic viscosity of water

,

~R

1__--~-4x ! o~

\

!

_

\

!

~

Pitch dlstibution °Ji

L __

"',

~ 4 =

I

__

\

02R

,,~

--t--n----

~.¢_--4---~------./

, - -B 4',- 7 0"

,~'--B zr- 5 "= - 1

s~-~o

Fig. 1

-7

'~ _-- ----J-- - - T - - - - ' ~ . /

T - - ~~ =4 --6-5~ - - ¢ ~ : ; , - . , , o

\ ....

7 - - - - q -- - - - - ~ / u ~.-nxu

'

2'

~so'~.:

~

822 °/

G e n e r a l plan o f B 4 s c r e w series Pitch distribution

o ~ l

02R

:____&~

~_k [ I

c31!x

I

I

'~

' I

"m 85-45

B5-60

85-105 L

~ 2 2 ~-2N~u-

General plan of B 5 s c r e w series

T h e p o l y n o m i a l s a n d the corrected design diag r a m s of t h e four- a n d five-bladed B-series screws are g i v e n in this paper. I n order to o b t a i n d a t a for a n a l y z i n g the m a n e n v e r s of ships, submersibles, drilling vessels, a n d so on, o p e n - w a t e r tests with a p a r t of the Bscrew series were c o n d u c t e d over a wider r a n g e of a d v a n c e coefficients. T h i s e x p e r i m e n t a l p r o g r a m covered t h e following i t e m s : ahead, ahead, astern, astern,

rpm rpm rpm rpm

ahead astern astern ahead

=:~

>/

85-75

Fig. 2

speed speed speed speed

(

",

!

(lst quadrant) (2nd q u a d r a n t ) (3rd q u a d r a n t ) (4th q u a d r a n t )

T h e effect of p i t c h ratio P / D , blade area ratio A E / A o, a n d n u m b e r of blades Z on the characteristics of the B-screw series in the four q u a d r a n t s h a v e been d e t e r m i n e d . I n addition, t h e results of the f o u r - q u a d r a n t m e a s u r e m e n t s are a n a l y z e d in the form of a F o u r i e r series a n d the F o u r i e r coefficients are given. T o o b t a i n s y s t e m a t i c knowledge of the c a v i t a tion characteristics of the B-screw series in a uniform flow, tests were carried o u t in t h e large N S M B c a v i t a t i o n t u n n e l with a p a r t of the Bscrew series. T h e influence of c a v i t a t i o n on propeller t h r u s t a n d t o r q u e was measured. I n addition, i n c e p t i o n curves were established for tip vortex cavity, sheet c a v i t a t i o n a t suction a n d pressure sides, a n d b u b b l e c a v i t a t i o n a t back a t midchord. T h e results of all these i n v e s t i g a t i o n s concern-

Table

2

Dimensions of Four B-Screw Series

c~ Z D Az/Ao 1.662 1.882 2.050 2. 152 2.187 2. 144 1.970 1,582 ...

Y~

R 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

ar c~ 0.617 0.613 0.601 O. 586 0.56t O. 524 0.463 O. 351 0

and

Five-Bladed

tr D 0.0366 0,0324 0.0282 O, 0240 0,0198 O. 0156 0.0114 O, 0072 0.0030

b~ c~ 0.350 0.350 0.350 O. 350 0.389 O. 443 0.479 O. 500 ...

distance between leading edge and centerline at radius r b r = distance between leading edge and maximum thickness of blade profile at radius r chord length of blade profile at radius r Cr D = screw diameter r radius R = tip radius AL./Ao = expanded blade area ratio Z = number of blades t r = maxinmm thickness o1' blade profile at radius r a r

ing the W a g e n i n g e n B-screw series are given a n d discussed in this paper. G e o m e t r y of B-Series Screws

A s y s t e m a t i c screw series is formed b y a n u m b e r of screw models of which o n l y the pitch ratio J P / D is varied. All other characteristic screw d i m e n sions, such as d i a m e t e r D, n u m b e r of blades Z , blade area ratio A e / A o , blade outline, shape of b l a d e sections, b l a d e thicknesses, a n d h u b - d i a m eter ratio d / D are the same. T h e results of tests


271

[

/

[dynamometer --7

IRPM t---~(propetter thrust[

/a"d torque) J

Fig. 3

Measuring equipment for performing open-water test

with the four-bladed B-screw series with bladearea ratios of 0.40, 0.55, 0.70, 0.85, and 1.00 and the five-bladed series with blade area ratios of 0.45, 0.60, 0.75, and 1.05 are given in this paper. Figs. 1 and 2 show the general plans of the fourand five-bladed screw propellers respectively. The dimensions of these screws are given in Table 2. The diagrams show clearly that the B-series screws have relatively wide blade tips, circularback blade sections near the tip, and airfoil sections near the hub. The four-bladed screws have a decrease in pitch at the hub of 20 percent in order to adapt the screw better to the velocity distribution behind a ship. A large number of experiments has shown, however, that the difference in efficiency due to this decrease in pitch in comparison with a screw with constant pitch is insignificant. The five-bladed screws have a constant pitch. The hub-diameter ratio of the fourand five-bladed B-series screws was d / D = 0.167.

00~

/

CD 0.03

0.02

" ~ - ~ O rnln t x p ~ "X~O min t h e 0 r

0.01

0

025

0

0.025

0 050 0<. (rad)

0.075

0.100

Fig. 4 Relation between drag coefficient CD and angle of attack ~xof equivalent blade profile of B 4-70 screw with P / D = 1

Open-Water Tests Reo.wR = c°vs~X/lZa2 + (0"75~rnD)2

Test Procedure The open-water tests with the B-series screws were carried out with the usual apparatus shown in Fig. 3. The immersion of the propeller shaft was equal to the screw diameter. Before the tests were carried out, the system friction and d u m m y hub torque and thrust were determined so that the measured propeller thrust and torque could be corrected accordingly. The usual routine of open-water tests was followed; the rpm of the screw was kept constant, and by varying the speed of advance the desired value of the advance coefficient J was obtained. Usually the rpm was chosen as high as possible to obtain a high Reynolds number. The rpm was chosen in accordance with the maximum speed of the towing carriage and the capacity of the d y n a m o m e t e r used for the thrust and torque measurements. Most of the open-water tests were made at 450 rpm. The Reynolds number for the B-series screws, based on the chord length of the screw blades at 0.75 R, m a y be written as

272

v

where c0.7~ = 2.073. AF,__._,ID A0 Z v = kinematic viscosity of water The diameter of the B-series screw models was chosen to be 0.240 m. At a screw rpm of 450 and with the advance coefficient J varying from 0 to 1.5, the Reynolds number varies between Reo.7~R = (2.1 to 2.5) 1 . A ~ vAoZ

1

This variation in Reynolds number is small and therefore it is permissible to base the Reynolds number of the different screw series on a mean value of J. The mean value of J is chosen to be 1.0. With a kinematic viscosity of the water equal to v -- 1.141.10 -6 m2/sec, the Reynolds number for the different screw series becomes:


1.0

0.g

10 KG. 08 KT ~0

0.7 Q6

0.5

- _

_i

0,8

09

0.4

Q3

Q2

0.1

0

L Ol

02

03

Fig. 5

0/.

0.5

05

07

O

1.0

1.1

1.2

1.3

1~

15

16

Open-water test results of B 4-40 screw series

10KQ KT

~o

'

v0

01

02

03

Fig. 6

Reo.7~

=

2.10

~.

"-°.

Ot

05

05

07

J

08

Q9

10

13

12

13

14

15

16


1

A0 Z

A c c o r d i n g to this definition, t h e B 4-70 series screws, for instance, were t e s t e d a t a R e y n o l d s n u m b e r e q u a l to 3.5 • 105.

F i n a l l y , it m u s t be n o t e d t h a t t h e t e s t s w i t h t h e B-series screws were c o n d u c t e d o v e r a p e r i o d of m o r e t h a n t h i r t y years. T h e s e t e s t s were c a r r i e d o u t in different basins of N S M B a n d d y n a m o m eters w i t h different c a p a c i t i e s were used. T h e r e fore, t h e R e y n o l d s n m n b e r a t w h i c h t h e different


273

O~

10K: E KT

?o OE

05

O~

01

0 0

01

02

03

Fig. 7

04

g5

06

07 3

Analysis of Test Results Usually the open-water test results of a series of screws were faired (parameters: advance coefficient J, and pitch ratio P / D ) and plotted in the conventional way with the coefficients:

KQ--

T pn2D 4

Q

pn2D 5 J Kr 27r Ko

as functions of the advance coefficient J = VA/nD. By interpolating in the K T - K Q - J diagram of a screw series, most problems which arise when de274

09

10

11

12

13

1/,

15

1G


screw series were tested varied considerably. This fact must be taken into account when the results of different screw series are compared with each other.

KT --

08

signing or analyzing screw propellers can be solved. From a correlation between the available design diagrams of the B-screw series, it appears that small differences exist. This is partly caused by Reynolds number effects, the degree of turbulence in the towing tank, and so on. Before making a cross fairing between the different B-screw series to blade area ratio and eventually to number of blades, we want to correct the test results for these effects. The effect of the Reynolds number on the test results can be taken into account by using the method developed by Lerbs [7] from the characteristics of equivalent blade sections. This method has been followed also by Lindgren [8], Lindgren and Bj~trne [9], and Newton and Rader [10]. Assuming, according to Lerbs, that the profile of the blade sections at 0.75 R is equivalent for the blade, simple relations between the coefficients K~, KQ, and J from the propeller tests, the corresponding lift and drag coefficients CL and CD and


!

/ 15

. . . . . .

i

-

B/--

-

- -

1.7

- -

- -

0.9

- -

"13

.

.

.

.

.

-

-

85

.

10KQO~

.

.

.

KT

. ~

\

7Fi2----q\y_/~,

°o

al

d2

~

~3 Fig. 8

04

05

06

0.7 O

09

_

I0

_

1.1

12

13

1/.

15

16

O p e n - w a t e r test results of B 4 - 8 5 s c r e w series

the profile angle of attack a can be deduced. As an example, the drag coefficients CD of the B 4-70 series screw with pitch ratio P / D = 1.0 is given in Fig. 4 on a base of angle a. From these results the values of minimum drag coefficients C~.ml, of the equivalent blade sections can be determined. In addition, the drag coefficients C~,m~. of the equivalent blade sections can be determined theoretically according to Hoerner [11 ] with : C,,,,,, = 2C~-[1 + 2

08

_

/o.w~

co.7~a_] where Cf = drag coefficient of a fiat plate in a turbulent flow to.v~R/Co.v~,u = thickness ratio of the blade section at 0.75 R The drag coefficient Cv depends on the Reynolds number and can be determined according to the 1TTC line. The thickness ratio of the blade section of B-series screws at 0.75 R is equal to

to.v~,~/Co.7~

0.0185 -- 0.00125Z =

As 1

2.073 . . . . A0 Z

Hence the open-water test results of a screw propeller can be corrected for Reynolds number effects by shifting the experimentally obtained drag curve in such a way that the CD,mlnaccording to the test results coincides with the theoretically calculated drag coefficient CD.,,,I.. This theoretical drag coefficient corresponds with the assumed value of the Reynolds number. The concept is shown in Fig. 4. Once the lift and corrected drag coefficients C~. and CD and profile angle of attack a are known, the corrected thrust and torque coefficients KT and KQ and the advance coefficient J can be deduced. All the open-water test results with the B-series screws are corrected in this way. The assumed Reynolds numbers and drag coefficients C~,mi= are given in Table 3. To show the scale effect, the open-water test results with the B 4-70 screw


275

B 4-100

m

I I I

i

IOKQ

F

KT

%

~0

O?

02

03

Fig. 9

OL

05

06

O?

J

0.8

09

1,0

1.1

12

13

14

15

O p e n - w a t e r test results of B 4 - 1 0 0 s c r e w series

1.0

Q9

IOKoOg KT '~0 1?7

I ,2 J

~

Ji

JI

~i

E

i

I

Og

05

-

O4

O3 O2

01 0

01

02

03

Fig. 10

276

04

05

06

07

,3

08

0£

10

11

12

O p e n - w a t e r test results of B 5-45 s c r e w series


13

1~

15

16

12

---

11

10

og

08 IOKQ KT 07

'~O

06

05

0~

03

0~

01

0

01

02

03

Fig. I I

O~

05

06

07

O

08

0g

10

1.1

12

13

1~.

15

16

O p e n - w a t e r test results o f B 5 - 6 0 s c r e w series

12

.~.~_~___

11

-

-

1.0

09

10KQO8 KT

I

NO 0.'7

O6

05

0/.

03

O2.

01

0

b

0.1

02

83

Fig. 12

04

0.S

0g

07

J

0.8

0g

10

II

12

~3

I.L

15

16



277

\Z

__l . . . . . . .

1 0 - - - -

88. l0

,~--

.

.

.

- -

~.

.

.

.

.

KQ

KT

/0

°.6

OS

84

.... X ~" " ~ 7 ~ - -

8.2--

x'v --

~--~

" - - 1 "-q

r~ 1 / O,

0.1

8.2

03

Fig. 13

O&

85

86

87

An

4

5

10

11

12

13

14

15

Table 4 Reynolds Numbers and Corresponding Drag Coefficients CD, m~. to Which Results of B 4-70 Screw Series Are Extrapolated

10.75 R

As I0.40 10.55 t0"70 /0.85 ~1.00 I0.45 Jo.60 ~0.75 [1.05

Reo.v,n 0.572 X 106 0.787 X 106 1.000 X 106 1.214 X 106 1.429 X 106 0.515 X 106 0.687 X 108 0.859 X 106 1.202 X 106

2 CI 0.01060 0.00985 0.00931 0.00892 0.00862 0.01090 0.01017 0.00964 0.00894

c6.v, R 0.0652 0.0473 0.0372 0.0305 0.0261 0.0658 0.0493 0.0395 0.0282

CD, ,ni~ 0.0120 0.0108 0.0100 0.0095 0.0091 0.0123 0.0112 0.0104 0.0094

Reo.v, R = 5.72AE/A6" 1 / Z . 106 CD, rain = 2 Cp[1 -F- 2 [0.TaR/tO.TaR] to.v6n CO.7SR

(0.0185-- 0.00125 Z ) Z 2.073 AE/Ao

series are given for different values of the R e y nolds n u m b e r (see T a b l e 4). T h e fairing of the corrected o p e n - w a t e r test results with the B-screw series was performed with t h e aid of a C D C 3300 c o m p u t e r b y m e a n s of a 278

09

O p e n - w a t e r test results of B 5 - 1 0 5 s c r e w series

Table 3 Assumed Reynolds Numbers and Drag Coefficients CD, nl~. of Different Screw Series Z

,2

08

A E

Z

10.Ta R

Ao

Reo.Tr, R

( l

4

0.3 1.0 0.70 3.0 10 ,lO0

X X X X X

106 106 106 10G 106

2 Cf

c0.75

R

0,01240 0.0372 0,00931 0.0372 0.00738 0.0372 0.00582 0.0372 0.00401 0,0372

CD.

mln

0.0133 0.0100 0.0078 0.0062 0.0043

regression analysis, U p till now o n l y t h e foura n d five-bladed B-screw series were a n a l y z e d in this way. T o begin with, the results of t h e diff e r e n t screw series ( p a r a m e t e r s : a d v a n c e ratio J a n d pitch ratio P / D ) were analyzed, T h e t h r u s t a n d t o r q u e coefficients were expressed as polyn o m i a l s of a d v a n c e coefficient J a n d p i t c h ratio P/D: KT = Ao,o

+ Aoa


J - t - Ao,2

d2 + . . + Ao,,

js +

........ i7".......1__12

f Z[~-Z~

,

~'li 0.[ 10K 0 KT

I

~ 0 0.5

0.t

I

I

Ot

03

Or 0

Fig. 14

O1

0.2

03

0.4

05

06

07

.

10

11

12

l,r*

Reo.75 R :=

1.5

16

0.3 X lOe;

2 -t- A2,6~? j 6 .

6 - d~6

KQ=

J+i3o,,.,

13

-]--Zll,6~- j6_[_

......................

J~+..+Bo,6

j6+

oP o J) ff6_[_ Bl,O~-- -1- BI,1]~ ff -[- B 1,-/~.J"-]-..-1B1.6~p2

.B~,o~ +

09

P

.

pe

A2,0~2 -~- . . . . . . . . . . . . . . . . . . . . . .

+Bo,1

,3

Open-water test results of B 4-70 screw series extrapolated to ( C . . min -- 0.0133)

P P Ax,o~-- + AI,I~) J-]- A1,2~) J2 +

Bo,o

08

p 6 if6 + B6,6D~

/36,0P66 + . . . . . . . . . . . . . . . . . . . . . .

With the aid of a regression analysis the significant terms of the polynomials and the values of the corresponding coefficients were determined. From this analysis it was :found that the results of both the four- and the five-bladed B-series screws could be approximaLed very well by the following series : P K r = Ao,o + AoaJ + Ao,2:2 + A o , J 3 + A , , o ~ -1- A I , I ~ )

P

J -t- - ¢ 1 1 , 3 b J

....................................

3

p2 p~ -lr-A2,1D~J ...[-Ao,o~

p6 + A~a~J The Wageningen B-Screw Series

279

10K{ KT

~o

~0

01

02

83

04

f18

08

87

08

09

0

10

1.1

12

13

14

15

18

Fig. 15.~ O p e n - w a t e r test r e s u l t s o f B 4 - 7 0 s c r e w s e r i e s e x t r a p o l a t e d to Reo.7a n = 3.0 X 106; (CD,mln : 0 . 0 0 7 8 )

KQ = BO,O+ BoaJ + Bo,~J 2 + Bo,aJ a + B ,,l P ~ J p2

p2

Kr

=

p6

+ B~,a ~P J" + B2,0 ~-2 + B~,a ~ ja + B6,o D6-p6 + Bs,I~?J A EP 6 M o s t of the terms with high powers of J and P / D were insignificant. Secondly, the results of the screw series with the same n u m b e r of blades b u t with different blade area ratios were correlated with each other. T h e t h r u s t and torque coefficients of the four- as well as the five-blade propellers were expressed as polynomials of J, P / D , and A E/A 8. As far as J and P / D are concerned, the choice of the terms was based on the foregoing analysis. T h e following polynomials were used for the calculation: 280

+ Co,o,oDP~66+ G,6,0 -~-0~

A E2.P6 + G,~,O--A o2 D 6-

A EaP 6

+ Ca,~,o-Ao 3 D 6

+ Co,o,,

p6

+ c,,o,,

+ Ce,
ditto


A E p6

:

CaG1A~ aP6

16 15 14 13

~.

v

a

. . . .

i .

.

.

.

,

12 11-10--

Qg~ 10K: 8 KT "r~O 07

\4) /

0O ~01

Q2

03

0/,

0IS"~

06

07

08 O 09

10

1.1

12

13

"l~

1.S

16

Fig. 16 Open-water test results of B 4-70 screw series extrapolated to Re0.~5 n = 10¢; (CD.mi~ = 0.0062)

Again with a regression analysis the significant terms of the polynomials and the values of the corresponding coefficients were determined. For the four- and five-bladed B-series screws these results are given in Tables 5 and 6 respectively. Presentation of Test Results

The results of the open-water tests with the four- and five-bladed screw series were extrapolated and expressed in polynomials. In these polynomials, either KT or K o were the dependent variable with the advance coefficient J, the pitch ratio P/D, and the blade area ratio A E/Ao as the independent variables. The form of these series together with their coefficients were given in Tables 5 and 6. These results can be used directly for solving problems which arise when designing and analyzing screw propellers if a computer is available. Also, we will give the results of the analysis in graphical form. With the aid of a tape-controlled drawing machine the coefficients Kr, Ko, and no

were drawn in the conventional way as a function of J. The diagrams of the B 4-40, B 4455, B 4-70, B 4,-85, and B 4-100 ,;crew series are given in Figs. 5 through 9. Figs. 1.0 through 13 show the diagrams of the B 5-45, B 5-60, B 5-75, and B 5-105 screw series. The results of the 13 4-70 screw series were extrapolated to different Reynolds numbers. The CD.,,in values of the equivalent blade section corresponding with the assumed Reynolds numbers were given in Tat)le 4. The KT-KQ-J diagrams are given in Figs. 14, 7, 15, 16, and 17. By interpolating in the'. K•-KQ-J diagram of a screw series most problems which arise when designing or analyzing screw propellers can be solved. For design purposes various types of more practical diagrams can be derived from the Kr-Ko-J diagram. The most widely encountered design problem is that where the speed of ;advance of the screw VA, the power to be absorbed b y the screw P, and the number of revolutions n are given. The


281

"

i

-

]. . . . .

B

10 =0004:

i

I i i

....... [.......................

13

---

t2

~ . . . .

1 t __.

11 .

.

.

.

.

.

.

10 . . . . . .

i .......... i. . . . . . . . . .

[. . . . . . . . . . . . . . . . . .

. . . .

I

. . . . . . . .

09~ KT ~0

07 06

--i! I

0 L ~

"

_

"t~.

01

.

I

_\

_

!

.

Y

o~ Fig.

17

oi

02

83

Open-water

0.4

0,5

06

07

KQ1/2/J °/2 : : nPl/~/Va 5/~ The Taylor variable Bp is related to this dimensionless variable by the equation B~ = 33.07 KQ1/2/J 6/2 = NP'/"-/Va V~ where N = number of revolutions per minute ff = power in hp VA = speed of advance in knots. In the usual diagram, the design coefficient By is the base and a new speed ratio 6 is used. This speed ratio is defined as

282

09

10

1.1

12

test results of B 4-70 screw series extrapolated (CD, m~ = 0 . 0 0 4 3 )

diameter D is to be chosen such that the greatest efficiency can be obtained. The problem of the optimum diameter can be solved in an easy way by plotting 7o and J as a function of

6 --

08

ND

101.27

v~

y

13

14

to Reo.vsn

15

=

16

10s;

in which D = screw diameter in feet The Bp-6 diagrams of all the four- and five-bladed B-screw series are given in Figs. 18 through 26. As a check of the correlation between the different diagrams, the efficiency 7o, speed ratio 6, and pitch ratio .P/D corresponding to optimum diameter are given in Figs. 27, 2S, and 29 on the base of B~. The curves of the four-bladed Bscrew series are given in Fig. 27. Figs. 28 and 29 show the results of the five-bladed B-screw series and the B 4-70 screw series extrapolated to different Reynolds numbers. In cases where, Va, T, and n or VA, ]', and D are given, the problem of determining the optimum diameter or the optimum number of revolutions can be solved by plotting 7o and J as functions of

KT/J 4 The Wageningen B-Screw Series

and

KT/J 2

Form of Polynomial and Coefficients of Four-Bladed B-Screw Series

Table 5 Cx,y,z

Dx,y,z

x :y g

--.719975 ~ --.790916 ~ --.179541 ~ -.625748 # --.311639 # 4.. 143160 ~ 4..531326 ~ --.114389 ~ 4..625376 # 4.. 125537 ~ --.523821 ~ --.207108 ¢ 4..270781 ~ 4 . . 134182 ~ --.121086 ¢ --.189764 ¢ --.439535 ~ --.624937 ~ --.496939 ~ ++.115986 ~

--2 --1 0 --1 0 0 0 0 --i 0 --i 0 0 0 0 1 --1 --I --2 --i

0 1 0 1 0 2 0 i 0 0 I 0 1 0 1 3 3 0 2 2

0 0 0 0 0 0 1 1 1 1 1 2 2

2 2 2 2 2 6 6

0 0 1 1 2 3 0 I 2 3 3 0 0 1 1 1 2 3 0 1

xyz

4 . . 964375 - - . 104103 4 . . 512431 ++. 109936

--2 0 0 0 --1 1 0 0 --2 200

--.453419

--2 0 0 1 --1 1 0 1 --1 O 0 2 --1 1 0 2 --1 3 0 3 --1 2 1 0 --1011 0111 --1 3 1 2 -1 213 --1 0 2 0 0 120 --I 121 --1 2 2 1 --1 022 --1 1 2 2 --1 222 --1 322 --1 123 --2 060 --2 3 60 --2 161

4..216078 - - . 507337 4" .377970 - - . 549486 --.507319 4 . . 368649 - - . 106520 +.465315 4..883010 4..112619 4 . . 10482;5 -- .449154 4..378780 4..177304 --.164687 --.344328

--.249132 --.233007 --.120209 - - .118997 ++ .458094

--i

¢ ~ ¢ ~ ~ ~ ~ ~ ~ ~ ¢ ¢ ~ ~

300

K T = ZC~,>:[A~'/A,,]~[P/D]'[J] ~ KQ = ZD ..... [Au/Ao]~[P/DIY[J] -"

Form of Polynomial and Coefficients of Five-Bladed B-Screw Series

Table 6 Cx,y,z

T)x,y,z

X 3' Z

--.191226 ~ --1 --.860251 ~ --1 --.598290 ~ --2 --.181301 ~ 0 --.457217 ~ 0 4..562903 ~ --1 4..990584 ~ --1 4..431314 ~ 0 4 . . 189210 ~ 0 --.254103 ~ 0 4...413600 ~ 0 --.126380 ~ 0 - - . 8 6 5 9 1 4 ¢ --1 4 . . 856485 ~ - - 1 4 . . 113060 ~ 0 --.146526 ~ 0 --.792780 ~ --2 4..669368 ~ --2

0 2 3 0 0 3 2 0 1 1 0 2 0 1 0 0 0 1

0 0 0 0 0 0 0 1 1 1 1 1 2 2 2 2 6 6

0 0 0 1 2 2 3 0 0 1 2 2 0 0 1 2 0 1

4 . . 520666 4 . . 627259 --.337780 - - . 518101 4 . . 396494 - - . 848886 4..317571 4 . . 130016 --.340161

4 . . 624821 --. 239657

4..801838 ++. 119589 - - . 477460 4 . . 133699 - - . 382898 4-. 270494 --.381466

4..241740

xyz --2 0 0 0 -I 201 --1301

--1 002 --1 102 --2 003 --1 3 0 3 -1 010 --1110 --1011 0 111 --1211 --1 0 1 2 --1312 0120 / --1 220 --1 121 --2 160 ¢ --2 360

KT, = EC ..... [AE/Aolx[P/D]~[J] • KQ

For

the

optimum

~D:,u,~[AE/Ao].~[P/D]y[J] •

different diameter

propeller (on

groups

base

of

the curves

for

~ ¢ / K T / J 4) a n d

optimum r p m ( o n b a s e o f v / K r / . l 2) a r e g i v e n i n Figs. 30, 31, and 32 and Figs. 33, 34, and 35 respectively.

°

~_ Fig.

18

Open-water


test results

of B 4-40

screw

series

283

\A

\ix

_ . . .

Fig. 19 984

O p e n - w a t e r test r e s u l t s o f B 4 - 5 5 s c r e w s e r i e s

Fig. 20

x x

,.

O p e n - w a t e r test r e s u l t s o f B 4 - 7 0 s c r e w s e r i e s


Fig. 21

O p e n - w a t e r test r e s u l t s o f B 4 - 8 5 s c r e w series

Fig. 22

O p e n - w a t , e r test r e s u l t s o f B 4 - 1 0 0 s c r e w s e r i e s


285

o

i o

D to

o

o°

o

h

x'~

\4 g

o

#

f

Fig. 23 286

O p e n - w a t e r test results o f B 5-45 s c r e w s e r i e s

Fig. 24



O

Fig. 2 5

O p e n - w a t e r test r e s u l t s o f B 5-75 s c r e w series

Fig. 26



287

0

0

0

?o O

500

500

O,

;00

40q

0

3O0

300

O:

20O

200

JO0

100

3

0

&,-

I.(

P/c

D(

3

4

Fig. 2 7

5

10

20 9p

30 40 50

100

200

300

Curves for optimum diameter of four-bladed Bscrew series

3

4

5

10

20

Bp

30 40 50

100

200

300

Fig. 29 Curves for optimum diameter of B 4-70 screw series extrapolated to different Reynolds numbers

5O0

3OO 8 -~00

I00

J

3

4

Fig. 28

5

10

8p

20

30 40 50

100

200

\/ , wd4

300

Curves for optimum diameter of five-bladed B-screw series

Fig. 3O Curves for optimum diameter of four-bladed B.screw series

Test Procedure

propeller operations. D a t a covering the entire operating region are scarce and m o s t of the available data are related to isolated applications only.

T o obtain data for ship m a n e u v e r i n g studies, it is necessary to k n o w the propeller thrust and torque characteristics over t h e entire region of

Therefore, part of the B-series was tested over the entire range of operating conditions. Conse-

Four-Quadrant Measurements

288

(See, for instance,


[11], [12], [13], and [14].)

o~[

., T

', . . . . .

07

I

I

B 4-40

)

4 -55 4 -70

i - -

- -

!

L -SO

4 -100

05

~o 05 04

03 02 12

10

%

08 i

05 C

02

03 04 0S

1 ~

2

3

4

5

~0

20

-

02

030405

2

~_K~T1

"

?-

I i

i

3

4

5

10

20

,3

Fig. 31

Curves for optimum diameter o f five-bladed B-screw series

Fig. 33

0~

08

0

07

'% 0

Curves for optimum rpm o f four-bladed Bscrew series

$

06

i i I/C

70 O5

7

04

6

03

5

02

4

% 0

O~

0!

Oe

0,

02

03 0~ 05

,

2

3

4

5

,0

20

Fig. 32 Curves for optimum diameter of B 4 - 7 0 screw series extrapolated to different Reynolds numbers

02

Fig. 34

I 03

~2

2

3

4

5

10

20

Curves for optimum rpm of five-bladed B-screw series

quently, tile screw propellers were tested at the following combinations of speed and r p m : First quadrant: speed ahead, r p m ahead. The h y d r o d y n a m i c pitch angle 3 (tan 3 = Vx/O.TrrnD) varies between 0 and 90 deg.

i,

04 05

o°

Second quadrant:


~ 3 4

9o °

speed ahead, r p m astern

o° ~ ~ ~

lSO °

289

1 B 4-70 series screw with P / D = 1.0 2 B 4-70 series screws with P / D = 0.5, 0.6, 0.8, 1.0, 1.2, 1.4 (influence of pitch ratio) 3 B 4-40, B 4-55, B 4-70, B 4-85, and B 4-100 series screws, all with P / D = 1.0 (influence of

0~ 0:

To

blade-area ratio)

0'

4 B 3-65, B 4-70, B 5-75, B 6-80, B 7-85 series screws, all with P / D = 1.0 (influence of number

0,

of blades) 0.

T h e test results are given in Figs. 36, 37, and 38. In these diagrams the nondimensional thrust and torque coefficients: T

CT*

1

~r D2 ½o[V,fl + (0.7~nD) 2] ~-

0

Co* 0~ 02

0X 0 a 0 S

\ K/-~T t

2

]

a

5

10

0

---

1 2 + (0.7~nD) 2] ~ .D2"D ~p[VA 4

20

+7 Fig. 35 Curves for optimum rpm of B 4-70 screw s e r i e s e x t r a p o l a t e d to different Reynolds numbers

are given as function of the h y d r o d y n a m i c pitch angle

v~ 0.7rrnD

fl = a r c t a n Third quadrant: speed astern, r p m astern 180 ° ~ ~ ~ 270 ° F o u r t h quadrant:

speed astern, r p m ahead

270 ° 4 ~ ~< 360 ° The measurements were performed with the usual apparatus for open-water tests. T h e tests with the negative propeller advance velocities were conducted with reversely mounted model propellers and without change in the carriage motion direction. Within the normal test range of a screw propeller it is generally possible to perform the tests at a constant rpm. Outside this range, the tests m u s t be performed at different r p m and speed of advance to operate within the range of capabilities of the d y n a m o m e t e r and on account of the m a x i m u m speed of the towing carriage. Usually the rpm and speed were chosen as high as possible to obtain a high Reynolds number. Presentation of Test Results

T h e influence of pitch ratio P/D, blade area ratio A J A o , and number of blades Z on the characteristics of the B-series screws in the four quadrants have been determined. Therefore, open-water tests were performed with the following propeller models: 290

In Fig. 35 the influence of the pitch ratio on the characteristics in the four quadrants is given. Figs. 36 and 37 show the influence of blade area ratio and number of blades respectively. In ship maneuvering studies, it is necessary to have a m a t h e m a t i c a l representation of the data suitable for use on a computer. Therefore, calculations were carried out to represent the results of the four-quadrant measurements b y a Fourier series. T h e thrust and torque coefficients Ce* and CO* were approximated b y the following series: Cr* =

~

[A(K) cosflK+B(K)sinSK]

K=0

CO* = ditto In Figs. 39, 40, and 41 the measured results and the approximation of these results by a Fourier series of 2, 5, 10, and 20 terms are given for the B 4-70 series screws with pitch ratios of 0.6, 1.0, and 1.4 respectively. F r o m the results given in these diagrams it can be seen t h a t the characteristics of the model propellers in the four quadrants can be adequately represented by a Fourier series of 20 terms. Tile values of the Fourier coefficients of the different screw propellers are given in Tables 7, 8, and 9.


c; -ll)CQ 08

//:,%,,

,.',

,

/t

04 "-..

?

I -0~

.

/

~08

\--..,//

\ \/ / ", \

/

/ / /

,,' ,,,,'

-12

, -2C

J 1 , i

.,j

-2~0°

40°

80°

120°

160° /3

200°

Fig. 36 Open-water test results with B 4-70 s c r e w Cavitation

Test Results

Test Procedure To obtain systematic knowledge on the cavitation characteristics of the B-series screws in a uniform flow, tests have been carried out in the large N S M B cavitation tunnel with the B 4-$5, B 4-100, B 5-75, and B 5-105 screw series. The N S M B cavitation tunnel has a 90 cm X 90 cm closed working section and a uniform flow. The material used for the propeller models was normal propeller bronze. The surfaces were polished to a high-quality smooth finish. For reliable cavitation testing of propellers, the leading edges of the propeller blades must be very accurately finisbed to the correct shape. Therefore, all screw models were carefully corrected with a blade-edge microscope. A detailed description of this apparatus is given in [15].

240° series

I J

280~---~~

;0 °

in four quadrants

The tests with the screw models were carried out at a constant water velocity in the test section of 5.5 m/see. The required range of cavitation numbers was obtained by wtrying the pressure in the tunnel. By varying the rpm of the model propellers a range of adwmee coefficients was covered between 0, and about 50 percent slip. Due to the normal restrictions of a cavitation tunnel it was not possiblle to cover the slip range up to 100 percent (zero ]). The air contenl: of the tunnel was kept within certain limits throughout the complete testing and ranged between about a / a s = 0.20 and a/a~ = 0.33, according to Van Slijke. The cavitation inception curves were established by determining, as a function of the propeller load and cavitation number, the point at which cavitation just disappeared (desinent cavitation). This procedure was followed because the


291

Table 7 K 0 1 2

3 4

CT*

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

[20 0 1

CQ*

2 3 4 5 6 7 8 9 10 11 12 13 14 15 ~16 17 18 19 20

, P / D .= 0.6 A (K) B(K) +.23680 ~ --1 +.00000 ~ + 0

-P/D = 0.8 A (K) B(K) +.32155 ~ --1 +.00000 /

+.15260 +.66146 +.97879 +.60002

. +0

.P/D = A (K) +.25350 / --1

~ ~ ~ ~ --.14447 ~ --.19974 ~ +.16256 ~ --.20861 ~ +.10054 ¢ --.59153 ~ +.50710 ~ +.11960 ~ +.13316 ~ +.43638 ~ +.49284 ¢ +.46434 ~ +.42182 ~ +.30748 ;~ +.55746 ¢ --.96705 ~

0 --'2 --3 --2 --i --2 --2 --2 --i --3 --2 --2 --2 --3 --2 --3 --2 --3 --3 --4

--.45175 --.24044 --.21257 --.68887 +.67897 +.38679 +.15350 +.25304 --.39916 --.18073 +.68506 +.13320 +.64404 +.17048 --.10662 --.18045 --.13301 --.17601 --.16006 +.66625

~ ~ ¢ ¢ ¢ ¢ ¢ ¢ ~ ~ ~ ~ ~ ~ ¢ ¢ ¢ ¢ ¢ ¢

0 --2 --i --2 --I --3 --i --2 --2 --2 --2 --2 --2 --2 --2 --2 --2 --2 --3 --3

+.1609.5 +.14351 +.28507 --.42671 --.53118 +.10643 +.37410 --.60685 --.13500 +.46384 +.13219 --,29566 +.64063 +.42304 +.16274 --.22250 +.60626 +.26669 --.11136 +.21718

~ ¢ ¢ ~ ~ ~ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ~ ¢ ~ ~ ¢

0 --2 --I --2 --i --2 --i --2 --1 --2 --i --2 --2 _o --2 --2 --'2 --2 --2 --4

--.63525 --.11844 +.52180 --.97199 +.63724 +.93564 --.70870 --.42418 +.17681 +.32632 --.14014 +.14379 +.99927 --.95800 --.66572 +.19305 +.45298 --,20864 --,51539 +.12468

¢ ~ ¢ ~ ;~ ~ ¢ ¢ ~ ¢ ¢ ;~ # ~ ¢ ;~ ¢ ¢ ¢ ¢

0 --i --i --2 --1 --2 --2 --2 --i --2 --2 --2 --2 --3 --2 --2 --2 --2 --2 --2

+,17820 +.14674 +.28054 --.16328 --.53041 +.60605 +.36823 --.25429 --.17689 +.27331 +.21436 --.24782 +.12317 +.50980 +.78076 --.37816 +.35353 +.53014 +.21940 --.28306

¢ ;~ ¢ ¢ ¢ ~ ~ ¢ ¢ ¢ # ¢ ¢ ¢ ¢ ;~ ¢ ¢ ¢ ¢

0 --i --i --i --I --3 --I --2 --1 --2 --I --2 --2 --2 --2 --2 --2 --2 --2 --2

+.32412 +. 22769 --.27622 +.41402 +.12264 --.13100 +.24133 --.72338 --.47773

--t 0 --1 --2 --1 --2 --2 --2 --3

+,00000 --. 62757 +.35503 --.41952 --.87073 +.11366 --.63308 +.31429 --.51207

¢ ~ ¢ ¢ ~ ¢ ¢ ¢ ¢

+0

+.47734 +, 24469 --,34342 +.48075 +.14028 --.47355 --.13162 +.20234 --.26110

¢ ¢ ¢ ¢ ~ ~ ¢ ¢ ¢

--I 0 --I --1 --1 --I --2 --1 --I

+.00000 --. 88736 +.23609 +.54166 --.318~36 +,12457 +.14249 --.27189 --.12249

¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢

+0

+.24645 +. 26718 +.16056 +.65822 --.22497 --.78062 +.24126 +.61475 --.16065

~ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢

--I 0 --i --i -i --i --2 --i --1

~" ~ ~ ~ ~ ~ ~ ¢ ~

0 --I --i --2 0 --2 --i --3

0 --t --I --1 0 --i --I --i

+.16189 ~ --1 --.53801 # _ o

--.12663 ~ --1 --.20723 ~ --2

+.10196 ~ --1 +.20437 ¢ --1

+,26454 ~ --1 +.46334 ~ --2

--.33291 ~ --1 +.12311 ¢ --1

+.14161

+.10887 --.24566 +.10271 +.15357 --.68530 --.11526 --.20622 --.13874 +.14868 +.25205

--.13622 --.12381 +.24253 --.59326 --,68631 +.55097 +.66757 --.51424 +,29728 +.68255

+.13838 +.68735 +,58492 --,13377 --,11603 +.13264 --.42987 --.76084 +,77164 --,37948

+.31123 --.12559 +.13948 +.88397 +.50358 --.79990 +.13345 +.11928 --.13556 --.70825

~ # # ~ ~ ¢ ~ ~ ~ ¢

--.46008 +.25050 --.19800 +.57280 --.40970 +.75132 +.50548 --.57,544 +.12067

--i --3 --3 --2 --2 --2 --2 --3 --4 --2

~ ~ ~ ¢ ¢ ~ ~ ~ ¢ ~

--I --2 --1 --2 --2 --2 --2 --2 --2 --2

d e t e r m i n a t i o n of t h e p o i n t of i n c i p i e n t c a v i t a t i o n does n o t give c o u s i s t e n t t e s t results w h er eas desin e n t c a v i t a t i o n is fairly r e p e a t a b l e .

Analysis of Test Results T h e n e t p r o p e ll e r t h r u s t s a n d t o r q u e s m e a s u r e d in t h e t u n n e l were c o n v e r t e d to t h e c o n v e n t i o n a l n o n d i m e n s i o n a l t h r u s t a n d t o r q u e coefficients KT a n d KQ. T h e s e coefficients were c a l c u l a t e d as a f u n c t i o n of t h e a d v a n c e coefficient J = V f f n D an d t h e c a v i t a t i o n n u m b e r p~ 0- 0

pv

--

T h e effieiencies were calculated from the relation ~0

~

J

Kr

27r

KQ

using t h e KT and K o values f r o m t h e c u r v e s faired t h r o u g h t h e m e a s u r e d values.

292

Coefficients of Fourier Series by Which Results of Four-

~ ¢ ¢ ~ ~ # ~ ~ ¢ ¢

--2 --i --I --3 --3 --2 --2 --2 --2 --2

~ ¢ ~ ¢ ~ ;~ ~ ~ ¢ ¢

--i --2 --3 --i --2 --i --2 --2 --4 --2

~ ¢ ¢ # ~ ¢ ¢ ~ ¢ ¢

--i --i --1 --2 --3 --2 --i --i --2 --2

T h e c a v i t a t i o n i n c e p t i o n curves for t h e tip v o r t e x c a v i t y , t h e sheet c a v i t a t i o n on pressure and suction sides, an d t h e m i d c h o r d b u b b l e c a v i t a tion were g i v en on a base of t h e p h y s i c a l l y realistic t h r u s t coefficient 1"

~PV~2 ~r

D2

8

Kr

7r

.]2

an d t h e c a v i t a t i o n n u m b e r ao. C o r r e c t i o n s were ap p l i ed to t h e w a t e r v e l o c i t y an d t h e pressure in t h e w o r k i n g section for t u n n e l wall interference. T h e s e co r r ect i o n s were based on t h e i d e n t i t y of t h e t h r u s t coefficient KT for t h e screw m o d e l s in t h e t u n n e l and t h a t f r o m t h e o p e n - w a t e r tests. T h e a p p l i e d c o r r e c t i o n s ar e in p r i n ci p l e t h e same as i n d i c a t e d b y Bu r r i l l [t6]. T h e c o r r e c t i o n leads i n i t i a l l y to t h e r e l a t i o n J t = k J . The f a c t o r k is d e t e r m i n e d for e v e r y screw as a function of J for eq u al KT, f r o m t h e o p e n - w a t e r K T - J


Quadrant Measurements with B 4-70 Screw Series Are Approximated 1.0

, B(K)

.P/D = 1.2--. A(K)

P,/D = 1.4 B(K)

B(K)

t0 0 --1 0 --1 --1 --1 --3 --1 --1 --2 --2 --2 --2 --3

--.59772

~

--2

+.12499

~

--I

+.63688 +.59988 +.97174 --.82817 --.28370

~ ~ ~ ¢ ~

--3 --2 --3 --2 --2

--.5794:1 --.19753 +.72177 +.72116 --.40411

~ ~ ~ ~ ~

--'2 _o -2 --2 --2

# ~ ~ ~ ~ ~ ~ ~ ~ ~ # ¢ ~ ~ ~ ~ ~ ~ # ~ ~

tO +1 --2 0 --1 --1 --1 -- 1 --1 --1 --1 --:) --2

+.55885 +.33553 -- .55351 +.295(;5 --.39907 --.59824 +.87083 +.38511 --.21977 --.16701 +.74012 +.411__ +.14304 +.24615 +.54771 +.990q7 -- .11674 +.28215 +.13405 +.61852 --.10103

~ ~ ~ ¢ ~ ~ ~ ~ ¢ ~ ~ ~ ¢ ~ ~ ~ # ~ ~ ~ ~

--1 0 --'2 -I --1 --1 -2 -- 1 --1 -1 --2 --1 _o -2 --o -2 --1 --2 --1 -- ") --1

+.00000 --.13087 -- .13050 +.22932 --.39610 +.35474 +.30131 + .22306 --.19345 +.31211 +.24556 +.82519 --.10001 +.29153 --.14795 --.93816 +.31241 +.82514 +.68793 --.14969 --.15397

~ t0 ~ 0 ~'~ --1 ~ 0 ~ --1 ~'~ --1 ~ --1 ~'~ --2 ~ --2 ~'~ --1 ~ --2 ~'~ --3 ~# --'2 ~'~ --2 ~'~ --'2

+.24182 +.19526 +.24861 +.20096 --.27491 --.49417 +.10719 +.31616 --.60703 --.160(,)7 +.36134 +.23123 --.38763 +.15034 +.73161

--.65683 --.61655

~

--2

+.10954

~

--i

+.51033 --.60263 --.82244 --.63789

~# ~ ~'~ ~'~ ~'~

--3 --2 --3 --2 --3

--.62652 --.47583 +.64641 +.36168 --.41148

~ ~ ~ ~ #

--2 --5 --2 --2 _o

+.00000 --.11081 + . 15909 +.16455 --.20601 +.85343 +.87856 -- .31327 --.96650 +.43190 +.12453 +.95986 --.79986 +.15073 +.24595 --.16918 +.51603 +.11504 --.47976 --.14566 +.23280

~'~ t O ~ +1 ~'~ --2 ~ 0 ~'~ --1 ~'~ --1 ~'~ --2 ~ --1 ~ --2 ~'~ --1 ~a --1 ~'~ --3 ~'~ --'2 ~'~ --1 ~ --2 ~ --I ~ --2 ~'~ --1 ~'~ --'2 ~ --1 ~-* --2

+.38330 +.30834 + . 13009 +.37977 --.42771 --.62278 +.11654 +.46308 --.20672 --.25171 +.93390 +.40552 --.36183 +.36580 +.79060 +.8.3125 - - . 121(,)9 +.53102 +.13882 +.35419 --.10801

~ ~ ~ ~ ~ ¢ ~ ~ ~ ~ # ~ # # ~ ~ # ~ # ~ ~

--1 0 -- 1 --1 --1 --1 --1 -- 1 --1 --1 --2 --1 _o --9 --'2 --2 -- 1 --9 --1 --2 --1

+.00000 --.12379 -- .76857 •" ) 0 +._1_30 --.34173 +.51927 +.25757 - - . 13343 --.16388 +.36642 +.17869 +.54887 --.72657 +.52610 --.32526 --.10724 +.58753 +.90872 --.33639 --.15324 +.15551

non-cavitation

--1 0 --1 --1 --1 --1 --1 --1 --2 --1 --3 --1 --4 --2 --2

~ ~ ~ ~ # ~ # # ~ # ~ # # ~ ~

+.00000 --.74777 --.13822 +.10077 --.11318 +.47186 +.10666 --.90239 --.78452 +.23941 +.80787 --.14942 --.31925 +.92620 +.15527

relationship and

~ ~ ~ ~ # # # # ~ ~ ~ # ~ ~ ~

A(K) +.34168 ~ +.21336 ~ +.159cff # +.18491 ~ --.28473 # --.57102 # +.13258 # +.31091 # --.61834 ~ --.1251.8 # +.14889 ~ +.24639 # --.43034 # --.21774 ~ +.62884 ~

+.00000 --.83513 --.17626 +.12925 --.16164 +.35073 +.17977 +.94574 --.10011 +.18813 +.96321 +.26472 --.34897 +.74367 +.70138

influenced Kr,-

oft r e l a t i o n s h i p m e a s u r e d in t h e c a v i t a t i o n t u n n e l . ( T h e s u b s c r i p t t refers to t h e c a v i t a t i o n tunrtel experiments.) F r o m t h e f o r e g o i n g c o m p a r i s o n of o p e n - w a t e r and cavitation tunnel test results the relationship b e t w e e n KT,,/Jt 2 a n d K T / J 2 w a s d e t e r m i n e d f o r c o n s t a n t v a l u e s of t h e c o r r e c t i o n f a c t o r k. T h i s r e l a t i o n s h i p was used as a b a s e for t h e c o r r e c t i o n s of J a n d ao in t h e r e g i o n s w h e r e c a v i t a t i o n influenced the thrust delivered by the propeller. A n extra, c o r r e c t i o n h a d t o b e a p p l i e d t o t h e K o t - J , r e l a t i o n s h i p for identification w i t h t h e open-water KQ-J relationship. For the correct i o n of t h e c a v i t a t i o n - i n f l u e n c e d Kot c u r v e s it w a s assumed that the ratio between the influenced and not-influenced torque coefficients for a given cavitation number remained the same, after app l i c a t i o n of t h e t o t a l c o r r e c t i o n . B e c a u s e of t h e r e l a t i v e l y low r a t i o of p r o p e l l e r

--'2 --'2 --1 --'2 --2 --2 --1 --2

--1 0 --1 --1 --1 --1 --1 --1 --2 --1 --2 --1 -3 -2 --2

+.00000 --.89037 --.19243 +.14229 --.18926 +.26496 +.19985 +.95880 --.15497 +.16351 +.12108 +.60098 --.40057 +.66508 +.26481

~ ~ ~ ~ ~ # # # ~ # ~ # # ~ ~

t0

--.65500

~

--2

--.25089 +.51549 +.34309 --.80479 --.28547

~ ~ ~ ~ ~

--2 --'2 --2 --'2 --2

~ ~ ~ ¢ ~ ~ ~ ~ ¢ ~ ~ ~ ~ ~ ~ ~ # ~ ~ ~ ~

+0 +1 -- 1 0 --1 --1 --1 --2 --1 --1 --1 --2 --1 --2 --2 --2 --2 --2 --3 --1 --2

0 --1 0 --1 --1 --1 --2 --1 --1 --i

--2 --2 --2 --2

d i s k a r e a / a r e a of t h e w o r k i n g :section of t h e t u n n e l ( a b o u t 0.071), t h e c o r r e c t i o n s o n v e l o c i t y a n d p r e s s u r e f o r t u n n e l w a l l interferec,Lee w e r e s m a l l .

Presentation of Test Results T h e i n f l u e n c e of c a v i t a t i o n o n p r o p e l l e r p e r f o r m a n c e of t h e B 4-85, B 4-1(}0, B 5-75, a n d B 5-105 s c r e w s e r i e s i~ g i v e n in ]Figs. 42, 43, 44, a n d 45 r e s p e c t i v e l y . [ n t h e s e d i a g r a m s KT, Ko a n d 7o a r e s h o w n as f u n c t i o n s of J w i t h t h e c a v i t a t i o n n u l n b e r ao as p a r a m e t e r . T h e f a i r e d e u r w : s f o r t h e o n s e t of t i p v o r t e x cavitation, suction- and pressure-side sheet cavit a t i o n , a n d m i d c h o r d b u b b l e c a v i t a t i o n of t h e s e s e r i e s a r e g i v e n as a f u n c t i o u of t h e c a v i t a t i o n n u m b e r ¢o a n d t h e t h r u s t c o e f f i c i e n t C r in F i g s . 46 t h r o u g h 49. 111 p r e s e n t i n g t h e s e r e s u l t s t h e f o l l o w i n g a b b r e v i a t i o n s f o r t h e d i f f e r e n t t y p e s of cavitation were used: tvc = tip vortex cavitation


293

Table 8 K 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 [20 0

Cr*

1

2 3 4 5 6 7 8 9

tlo

12

18

I20

Coefficients of Fourier Series by Which Results Are Approximated of Four-Quadrant

ssc = s h e e t c a v i t a t i o n o n s u c t i o n s i d e psc = sheet cavitation on pressure side b m c = b u b b l e c a v i t a t i o n a t m i d c h o r d of s u c t i o n side. F u r t h e r t o t h e o v e r a l l p r e s e n t a t i o n in F i g s . 46 t h r o u g h 49, F i g s . 50 t h r o u g h 53 s h o w t h e c o m p l e t e r e s u l t s of t h e c a v i t a t i o n i n c e p t i o n m e a s u r e m e n t s f o r all p r o p e l l e r s w i t h ] ) / D = 1.0. T h e suction-side cavitation developed from the tip vortex cavity and spread down over the blade w h e n t h e l o a d i n g of t h e s c r e w w a s i n c r e a s e d . T h e f o l l o w i n g i n d i c a t i o n s w e r e u s e d t o d e s c r i b e t h e ext e n t of t h e c a v i t a t i o n : ssc

1.0 R :

ssc

0.9 R :

294

AE/Ao = 0.55 ~.4E/Ao = A (K) B(K) A (K) +.31922 ~ --1 -}-.00000 # -}-0 +.16114 ~ 0 --.63528 ~ 0 -}-.99198 ¢ --3 --.11945 ~ --1 -}-.28523 ~ --1 -}-}-.52443~ --1 --.46824 ~ --2 --.96855 ¢ --2 --.53162 ¢ --1 +.63872 ~ --1 -}-.74323 ~ --3 -}-.94490 ~ --2 -}-.37786 ~ --1 --.70557 ~ --2 --.59009 ~ --2 --.41397 ~ --2 --.13650 ~ --1 -}-.17913 ~ --1 -}-.48123 ~ --2 -}-.32751 ~ --2 SEE -}-.13356 ~ --1 --.15649 ¢ --2 --.28356 ~ --2 -}-.12139 ~ --2 -}-.66011 ~ --2 +.98431 ~ _ o -}-.41218 ~ --2 --.11306 ~ _ o +.12833 ~ --2 --.68633 ~ --'2 --.24525 ~ --2 +.16502 ¢ --2 -}-.61399 ~ --2 -}-.44140 # --2 -}-.26417 ~ --2 --.20713 ~ --2 --.96108 ~ --3 --.51416 ~ --2 -}-.22652 ~ --5 +.11161 ~ --2 +.47745 ~ --1 -}-.00000 ~ + 0 -}-.24474 ¢ 0 --.88736 ~ 0 --.33679 ~ --1 +.23048 ~ --1 -}-.48213 ¢ --1 -}-.54388 ~ --1 -}-.14481 ~ --1 --.30945 ~ --1 --.47373 ~ --1 -}-.12533 ~ 0 --.23258 :~ --2 -}-.15077 ~ --1 + . 1 9 6 0 8 ~ --1 --.27471 ~ --1 --.26832 ~ --1 --.12804 ~ --1 -}-.10348 ~ --1 -}-.25570 ~ --1 -}-.20883 ~ --1 -}-.43878 ~ --2 --.78899 ¢ --3 -}-.13814 ¢ --1 --.12592 ¢ --1 -}-.69907 ~ --'2 -}-.24307 7~ --1 +.69677 ¢ --3 --.43386 ¢ --3 --.14066 ¢ --1 --.40164 ~ --3 --.14160 ~ --2 -}-.60723 ~ --2 -}-.13130 ~ --1 -}-.69114 ~ --2 --.35743 ~ --2 --.47407 ¢ --2 --.68982 ~ --2 -}-.22412 ¢ --2 + . 6 0 4 0 8 ~ --3 +.62279 ~ --2 --.35655 ~ --2

A~'/Ao = 0.40 A (K) B(K) -}-.23451 # --1 -}-.00000 # + 0 -}-.15234 ~ 0 --.45189 ~ 0 -}-}-.65422 ~ --2 --.24681 ~ --2 -}-.10679 ¢ --'2 --.21427 # --1 -}-}-.59102 ¢ --2 --.70269 ¢ --2 --.14467 ¢ --1 +.67759 ~ --1 --.19208 ¢ --'2 -}-.22291 ¢ --3 +.16123 ~ --2 -}-.15526 ~ --1 --.20361 ~ --2 -}-.25736 ~ --2 -}-.98661 ~ --2 --.39835 # --2 --.38284 # --3 --.18292 ~ --2 + . 4 9 3 5 8 ~ --2 +.67522 ~ --2 +.97493 ~ --3 +.12664 ~ --2 -}-.13744 ~ --2 +.65164 ~ --2 -}-.36877 ~ --3 -}-.15957 ¢ --2 +.47643 ~ --2 --.11204 ~ --2 + . 5 3 9 0 5 ~ --3 --.18836 ~ --2 + . 4 1 9 8 0 ~ --2 --.14484 ~ --2 -}-.57987 ~ --3 --.16466 ~ --2 + . 5 1 9 5 7 ~ --3 --.41572 ~ --3 --.15836 ~ --4 -}-.75927 ~ --3 -}-.32922 ~ --1 -}-.00000 ~ + 0 -}-.22818 ¢ 0 --.62722 # 0 --.27076 ~ --1 +.35740 ~ --1 -}-.40533 ~ --2 --.41363 ~ --1 +.11865 ~ --1 --.81833 ~ --2 --.16010 ~ --2 +.11410 # 0 -}-.16364 ~ --2 --.58649 7~ --2 --.75320 ~ --2 +.31366 ~ --1 --.10827 ~ --2 --.64464 ~ --3 +.16051 ~ --1 --.13427 ~ --1 --.54749 ~ --2 --.25659 ~ --2 +.14265 ¢ --1 -}-.10360 ~ --1 --.85995 ¢ --4 --,27531 ¢ --2 + . 4 0 8 6 8 ¢ --3 -}-.10084 ~ --1 --.12955 ~ --2 -}-.18382 ¢ --2 +.59046 ~ --2 --.67291 ~ --2 --.38788 ~ --2 --.70726 ¢ --3 .}..73,531 ~ --2 --.17222 ~ --2 -}-.41273 ~ --3 --.10638 ~ --2 --.54923 ~ --3 +.17638 ¢ --2 -}-.10712 ~ --2 -}-.27112 ~ --2

b e g i n n i n g of s u c t i o n - s i d e c a v i t a tion at the blade tip suction-side sheet cavitation spreading down from blade tip to 0.9R

ssc

0.7 R :

suction-side sheet cavitation spreading down from blade tip to 0.7R. The pressure-side cavitation started at the leading e d g e b e t w e e n 0.6 _R a n d 0.7 R a n d e x t e n d e d along the leading edge to both sides when the l o a d i n g of t h e s c r e w w a s d e c r e a s e d .

Conclusion 1 T h e d e r i v e d p o l y n o m i a l s of t h e t h r u s t a n d t o r q u e c o e f f i c i e n t s of b o t h t h e f o u r - a n d fiveb l a d e d B - s e r i e s s c r e w s a n d t h e F o u r i e r a n a l y s i s of the "four-quadrant" measurements enable design calculations and analyses with a computer. 2 The cavitation inception diagrams and the d i a g r a m s s h o w i n g t h e i n f l u e n c e of c a v i t a t i o n o n t h r u s t a n d t o r q u e can be used for m a k i n g quasis t e a d y p r e d i c t i o n s of t h e c a v i t a t i o n c h a r a c t e r i s t i c s of a s c r e w in a n o n u n i f o r m w a k e flow.


Measurements with B 4-40, B 4-55, B 4-70, B 4-85, and B 4-100 Series Screws with P/D = 1.0 0.70~

B(K)

A (K) +.24744 +.19698 +.24688 +.19970 --.29298

-1

+.12086 +.23772 +.14059 +.31704 +.66002 +.97627 --.59991 +.22851 +.73574 +.53360

# ~~~ ¢ ~ ~ ~ ~ ~ ~ ~ # ~ ~ ~ ~ ¢ # ~

--.36541 +.38821

-.50489

+.92885 +.31605 --.51017 --.14934

TABLE 7

"AE/A(,

=

0.85-

B(K)

"Az/Ao = 1.00

A (g)

B(K)

0 --1 --1 --1 --1 --2 --1 --2 -1 _o --1 --3 --3 --2 --2 --2 --3 --2 --')

+.00000 --.83446 --,15875 +,13064 --,15776 +.35044 +.16119 +.1.5147 --.11703 +.18128 +.10143 +.37940 --.25960 +.86001 +.75758 --.63507 --.36952 +.42495 +.12208 --.82948

# # ¢ ~ ~ ¢ ¢ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ¢

+0 0 --1 0 --1 -1 --1 -3 -1 -1 --1 --2 --2 --2 --3 --2 --3 --2 --2 --2

+.34061 +.21365 +.15858 +.18327 --.28700 --.57304 +.12995 +.30856 --.64453 --.12383 +.13141 +.24622 --.621C1 --.19550 +.68171 +.12593 --.57680 --.18257 +.74443 +.72116

# ¢ ¢ # ¢ # ¢ ¢ ~ 7s ~ ~ ¢ ~ ~ ~ ~ ~ ~ #

--1 0 --1 --1 -1 --1 --1 --1 --2 -1 --2 --1 --3 --2 --2 --1 --2 --2 --2 --2

+.00000 --.89003 --.19294 +.14235 --.18767 +.26340 +.20316 +.94515 --.15484 +.16308 +.11972 +.59032 --.43512 +.67547 +.28778 --.67282 --.20573 +.47559 +.33164 --.79023

# ¢ ~ ¢ ¢ ~ ~ ¢ ~ ~ ~ ~ ~ ~ ~ ~ ¢ ~ ~ ~

+0 0 --1 0 --1 --1 --1 -2 -1 -1 --1 --2 --2 --2 --2 --2 --2 --2 --2 --2

~

--2

--.19871

~

--2

--.41164

~

--2

--.24768

~

--2

~

--I

+.00000

~

+0

+.558c,5

~

--1

+.33.57'5 ~

0

~ ~ ~

+1

~ ~ ~ ¢ ~

0 --1 --1 --1 --2

+.12968

~

--i

--.69162

~

--2

--.54483

~

--2

+.00000 --.13089 --.12986

+.37596 --.43202 --.62517 +.11711 +.46338

~ ~ ~ 7~ ~

--1 --1 --1 --1 --1

+.21242 --.34391 +.51711 +.25874 --.12915

~ ~ ~ ~ ~

0 --1 --1 --1 --1

+.30159 --.39788 --.59767 +.85676 +.38556

~ ~ ~ ~ ~

--1 --1 --1 --2 --1

+.22943 --.39545 +.35275 +.29736 +.19505

--.20997

¢

--i

--.16229

~

--i

--.2'2022

¢

--i

--.19625

¢

--i

--.25065 +.94712 +.40479 --.34106 +.34406 +.72600 +.84629 --.12383 +.55770 +.14548 +.31655 --.10461

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ¢ ~

--1 --2 --1 --2 --2 --2 --2 --1 --2 -1 -2 --1

+.36854 +.17505 +.53011 --.66787 +.51614 --.27707 --.11040 +.50134 +.87478 -.33660 -.14750 +.1976l

~ --1 ~ --1 ~ --2 ~ --2 ¢ --2 ~ --2 ~ --I 7+- --2 ~ --2 ~ -2 ¢ -1 ~ --2

--. 16580 +.768;~,2 +.40968 +.143(17 +.19131 +.565~3 +.99527 --.11082 +.27185 +.132(15 +.637(;8 --.10405

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ¢ ~

--1 --2 --1

+.31577 +.24466 +.80744 --.10030 +.27097 --.14513 --,94578 +.27592 +.82455 +.74715 -.15220 --.14567

~ ~ ¢ ¢ ~ ¢ ~ ~ ~ ~ ¢ ~

--1 --1 --2 --1 --2 --2 --2 --2 --2 -3 -1 --2

+ . 3 0 8 5 7 7~

0

--.12374 ~ + 1

3 The cross-correlation between the design d i a g r a m s , a s h a s b e e n d o n e i n t h i s p a p e r f o r different blade-area ratios at constant blade number, can be extended to different blade numbers. In a d d i t i o n , t h e e f f e c t of t h e R e y n o l d s n u m b e r c a n be taken into account in the polynomials for Kr a n d KQ. 4 H e n c e , t h e e f f e c t of t h e R e y n o l d s n u m b e r on the screw characteristics being known, the c o n s e q u e n c e s of t h i s e f f e c t c a n b e i n t r o d u c e d i n t o trial predictions. 5 Further progress on these subjects to be m a d e a t t h e N S M B will b e d i s c u s s e d i n a s e c o n d paper.

Acknowledgments T h e a u t h o r s a r e i n d e b t e d t o m a n y m e m b e r s of t h e t e c h n i c a l s t a f f of N S M B f o r t h e i r c o n t r i b u t i o n t o t h e c o n t e n t s of t h i s p a p e r . In particular, they

--'2 --2 --2 --') -1 --2 -1 -2

-1

+0 --I

e x p r e s s t h e i r a p p r e c i a t i o n t.o D r . H . le G r a n d f o r performing the numerical calculations. S p e c i a l m e n t i o n s h o u l d b e m a d e of t h e c o n t r i b u t i o n of M r . H . N i j d i n g , w h o d i d o v e r a c o u p l e of y e a r s a l a r g e p a r t of t h e w o r k of c a l c u l a t i n g and preparing the diagrams.

References 1 G . S. B a k e r a n d A. W . R i d d l e , " S c r e w P r o p e l l e r s of V a r y i n g B l a d e S e c t i o n s in O p e n W a t e r , " Trans. I N A , 1932 a n d 1934. 2 G . S. B a k e r , " T h e D e s i g n of S c r e w P r o pellers with Special Reference to the Single-Screw S h i p s , " Trans. IJVA, 1 !-)34. 3 L. T r o o s t , " O p e n - W a t e r T e s t s w i t h M o d e r n P r o p e l l e r F o r m s , " Frans. N E C I , 1938, 1940, a n d 1951. 4 W . P. A. v a n L a m m e r e n , L. T r o o s t , a n d


295

Table 9 K 0 I 2 3 4 5 6 7 8 9

Cr*

11o 12 13 14 15 16

L20 I0

G,

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Coefficients of Fourier Series by Which Results Are Approximated of Four-Quadrant

A(K) 4.17533 ¢ + . 185O2 ¢ +.15280 ¢ +.29186 ¢ --.14518 ~ --.72663 ¢ +.18334 ~ 4.58557 ~ --.12843 ~ --.22873 ~ 4.62426 ~ 4.17177 ~ --.39038 ~ +.14212 ~ 4.55205 ~ --.77101 ~ --.74621 ~ +.12434 ~ 4.42080 ~ --.49419 ~ +.15046 ~ 4.36082 ~ 4 . 27478 ¢ --.60351 ~ +.70069 ~ --.79315 ~ - - . 10096 ~ --.11682 ~ 4.80251 ~ --.17612 ~ --.25992 ~ 4.14525 ~ +.13287 ~ --.13192 ~ 4.31817 ~ 4.11369 ~ --.91563 ~ --.12451 ~ 4.20853 ~ +.13686 ~ --.66659 ~ --. 74024 ~

-B 3-65 --1 0 --1 --1 --1 --1 --2 --I

--1 --1 --2 -1 --2 -i --2 --3 --3

--1 --2

--2 --2 --1 0 --2 --1 --2 0 --i

--1 --1 --1 --1 --1 --1 --1 --1 --2 --1 -I

--1 --2 -- 2

A (K)

+.00000 --.73186 --.10683 +.83232 --.67230 4.74907 --.57188 --.29067 +.69646 4.31902 4.11766 --.67097 +.27271 4.14362

#

-

¢ ~ # ¢ ~ ~ ~ ~ ~ ~

--2 --1 --2 --1 --2 -- 1 --2 --2 --2 --1

--.16196 --.14496 +.47854 +.42777 --.54757 --.88629 +.24095

# ~ ~ # # # ~

+.00000

--.10801 4.40219 4.12286 --.28156 +.12603 +.16670 --.56992

~ ~ ~ ~ ~ # ~

4-70--~

B(K)

¢ +0 ~ 0 ~ --1

A (K)

-B 5-75

+.23478 4.18045 +.20771 +.2848o --.17734 --.44127 +.26768 -+.29516 +.84623 --.13558 4.20876 4.21133 --.17863 +.17519

~

-

# ~ # ¢ ~ ¢ ~ # ~ ~

--1 --1 --2 --1 --3 --1 --2 --1 --2 --2

-2 --I -2 --2 --2 --2 --2

+.60328 +.97352 --.36217 4.38472 4.44636 +.32672 --.40864

# ~ ~ # # ~ #

--2 --2 --2 --2 --2 --2 --2

~

40

+.33515

#

--I

~ ~ ~ ¢ ~ ~

+1 --2 0 --1 0

4.28220 ~

1

SEE TABLE 7

# --1 ~ 0 ~ --1 1

0 --1 4.48943 ~ --1 --.24185 # --1 -.64821 # -1 +.70829 # --2 +.10059

~

--2 --1 --2 --i --I --2 -1 --I

+.62483 --.88086 --.32211 +.74303 4.29956 -.111o1 4.98516

~ ~ ~ ¢ ~ # ~

--i --2 --I --2 --I -I --2

~ ~ ~ ~ ~ --.93035 #

_o --1 --2 --2 --2 --2

4.99828 4.99234 --.13624 4.37064 +.81717 +.37055

~ ~ ~ ~ # ~

--2 --2 --1 --2 --2 --2

+.84916

--2

--.58415

~

--2

--.39189 +.48374 +.14034

--.66918

-.11141 4.20220

+.53074 --.20058 4.66949 4.57476 --.75793

J. G. K o n i n g , Resistance, Propulsion and Steering of Ships, H. S t a r e H a a r l e m , N e t h e r l a n d s , 1948. 5 PrinciplesofNavalArchitecture, S N A M E , N e w Yo r k , 1967. 6 J. D. v a n M a n e n , " F u n d a m e n t a l s of Ship R e s i s t a n c e a n d Propulsion, P a r t B , " P u b l i c a t i o n No. 132a, N S M B . 7 H . W . Lerbs, " O n t h e E f f e c t s of Scale an d R o u g h n e s s on F r e e R u n n i n g P r o p e l l e r s , " Journ. ASNE, 1951. 8 H. L i n d g r e n , " M o d e l T e s t s w i t h a F a m i l y of T h r e e an d F i v e - B l a d e d P r o p e l l e r s , " P u b l i c a tion No. 47, S S P A , 1961. 9 H. L i n d g r e n a n d E. Bj~trne, " T h e S S P A Standard Propeller Family Open-Water Charact e r i s t i c s , " P u b l i c a t i o n N o . 60, S S P A , 1967. 10 R . N . N e w t o n a n d H. P. R a d e r , " P e r f o r m a n c e D a t a of Propellers for H i g h - s p e e d C r a f t , " Trans. R[NA, 1961.

296

~ B

B(K)

~

11 H . F . N o r d s t r o m , " S c r e w P r o p el l er C h a r a c t er i st i cs," P u b l i c a t i o n No. 9, SSPA , 1948. 12 I. Ya. M i n i o v i c h , " I n v e s t i g a t i o n of H y d r o d y n a m i c C h a r a c t e r i s t i c s of Screw Propellers U n d e r C o n d i t i o n s of R e v e r s i n g an d C a l c u l a t i o n M e t h o d s for B a c k i n g of S h i p s , " B U S H I P S T r a n s lation 697, 1960. 13 K . I . M e y n e , " U m s t e u e r e i g e n s c h a f t e n y o n S eh i f f sp r o p el l er n ," Schiff und Hafen, H e f t 5, 1964. 14 Sv. Aa. H a r v a l d , " W a k e an d T h r u s t Ded u ct i o n a t E x t r e m e P r o p e l l e r Loadings,"Publication No. 61, S S P A , 1967. 15 J. H. W i t t e an d J. E s v e l d t , " T h e B l a d e E d g e M i c r o s c o p e of t h e N e t h e r l a n d s Ship M o d e l B a s i n , " ISP, Vol. 13, 1966. 16 L . C . Burrill, " T u n n e l W a l l I n t e r f e r e n c e , " A p p e n d i x 7, S e v e n t h I n t e r n a t i o n a l C o n f e r e n c e on Ship H y d r o d y n a m i c s , P u b l i c a t i o n No. 34, S S P A , 1955.


Measurements with B 3-65, B 4-70, B 5-75, B 6-80, and B 7-85 Series Screws with P/D = 1.0 B 6-80

E; 7-85

B(K)

.4 (K)

B(K)

A (K)

+.00000 ~ + 0 -.77599 # 0

+.23662 # - 1 +.18302 # 0

+.24317 ~ - 1 +.1&~99 # 0

--.93828

#

--2

+.23069

#

--I

+.11419 --.12502 +.44333 +.12637 --.15496 --.79187 +.16728 +.79242 +.43160 --.59656 +.69874 +.54287 --.46027 --.13075 +.10496 --.25852 --.74776 --.12379 +.00000 --.11542 +.11750 +.18068 --.30359 +.79442 +.21067 --.17281 --.20963 +.25310 +.14026 +.79469 --.10034 +.11763 +.19718 --.13576 --.29719 +.60037 +.16139 --.13696 --.30009

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ # ~ ~ ~ ~ ~ ~ # ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ # ~# # # ~ #

0 --I --1 --1 --2 --2 --1 --'2 -2 --2 --2 --3 --2 --2 --2 --3 --2 --2 +0 +1 --1 0 --I --1 --1 --1

+.25996 --.16170 --.36481 +.20075 +.23040 +.40578 --.11055 +.80168 +.21468 --.70636 +.7.3299 +.46487 +.90387 --.33134 +.33675 +.37209 +.39557 --.27268 +.00000 --.80123 --.83486 +.12042 --.14185 +.43515 +.13807 +.42871 --.62870 +.12691 +.77749 +.63927 --.49631 +.44461 +.49405 --.32926 --.22670 -.62284 +.62955 --.68236 --.19772

~ # ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~# ~ # # ¢ ~ ~

--1 --I --I --3 --1

+.43581 # - 1 +.28977 # 0 -.67000 # --'2 +.31618 ~ - - i --.17752 ~ - - i --.51941 ~ - - I +.42480 ~ --'2 +.58953 ~ --1 +.16834 ~ --'2 --.28462 ~ --1 --.71543 ¢ --3 +.27896 ~ - 1 --.64777 ~ --2 +.80256 ~ --2 +.70969 ~ --2 +.15564 ~ --1 --.93465 ~ --2 +.23656 # --2 %..34068 ~ --2 +.58219 ~ _ o --.18968 ~ --2 +.00000 ~ + 0 --.11890 ~ + 1 +.22405 ~ --1 + . 18961 ~ 0 --.35017 ~ --1 +.81917 ~ --1 +.28318 ~ --i --.53679 ~ --2 --.22347 ~ - I +.15508 ~ - 1 + . 10065 ~ - 1 +.14999 ~ - I --.56936 ~ --2 +.80332 ~ --'2 +.23691 ~ --2 --.82944 ~.~ --2 --.51470 # --2 +.10568 ~ --3 +.59189 # -'2 --.11771 # - 1 --.29681 # --2

-I

-1 --1 --2 --1 -I

--2 -1 --3

-2 -2 --1 --3

--'2 --1 --3 --1 --3 --3 --'2 --2 --2 --2 --2 --2 --2 +0 0 --2 0 --1 --1 --1 --2 --2 -1 --2 -2 --2 --2

--3 --2 -2 -3 -3 --2 --2

+.24!)84

+.21958 --.12528 --.29638 --.26677 +.23:_)37 +.73131 --.95965 +.91652 +.20648 --.66.t74 +.19,585 +.45277 +.80651 --.20294 +.27'931 +.2,9162 +.45340 --.24513 +.00000 --.82637 --.44265 + . 12519 --.15080 +.45207 +.12112 +.84001 --.36180 +.85657 +.50675 +.85636 --.54398 +.40116 +.15876 --.25855

#

~ # ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ # ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ --.39788 # --.77,q22 # +.14790 # -.60376 # --.26079 #

--i

--1 --i --I --';2 --1

--'2 --'2 --3 --1 --3 --2 --2

--2

--'2 --'2 --'2 _o

--2 +O 0 --2 O --I --1 --1 --2 --2 --9 -2 _o --2 --') _o _o --2 --3 --2 -2 --2

B(K) +.56690 + . 30294 --.19641 +.15382 --.14236 --.39858 +.37905 +.55971 +.11188 --.20371 --.28157 +.25070 --.61091 +.66744 +.76275 +.18124 -.76436 +.42504 +.28472 +.48341 --.16078 +.00000 -.12213 +.27934 + . 18974 --.35626 +.86348 +.32399 +.36867 --.17751 +.79699 +.25833 +.14951 --.27002 +.81931 +.36117 --.49921 --.64297 --.30697 +.62224 --.10872 --.28107

¢ ~ ~ ~ ~ ~ ~ ~ # ~ ~ ¢ ¢ ~ ¢ ¢ ¢ ~ ~ ~ ~ ~ ¢ ¢ ¢ ~ ¢ ¢ ¢ ¢ ¢ ~ ~ ¢ ~ ~ ~ ~ ~ ~ ~ ¢

--1 0 --1 --1 --1 --1 --2 -t --1 -I --2 --1 --2 --2 --2 --t --2 --2 --2 -2 --2 +0 +I --1 0 --1 --1 --1 --2 --1 --2 --2 --1 --2 --2 --2 --2 --2 --2 --2 -1 --2

Discussion Eric Bj/:irne, 4 Visitor: F i r s t of all I m u s t e x p r e s s m y s a t i s f a c t i o n o v e r t h e m a t h e m a t i c a l f a i r i n g of t h e t e s t results. T h e d i v e r g e n c e b e t w e e n differe n t i s s u e s of t h e W a g e n i n g e n p r o p e l l e r c h a r t s h a s b e e n a s o u r c e of l o n g a n d o f t e n h o t d i s c u s s i o n s a b o u t p r o p e l l e r d i m e n s i o n s a n d eflicieneies. T h e r e g r e s s i o n a n d F o u r i e r a n a l y s i s of t h e t e s t r e s u l t s a r e of g r e a t v a l u e f o r t h o s e w o r k i n g w i t h a n a l o g u e and digital computers. However I miss open-water test results pres e n t e d o n t h e b a s i s o f K T / J ~, w h i c h s o m e t i m e s can be very useful at the project work. Speaking of p a r a m e t e r s i t s e e m s i n c o n s e q u e n t t h a t t h e n o n d i m e n s i o n l e s s f a c t o r / 3 p is still u s e d w h i l e t h e fact o r B~ is n o w r e p l a c e d b y V / I ~ T r / J f l

I therefore

4 The Swedish State Shipbuilding Experimental Tank, G6teborg. Sweden.

s u g g e s t t h a t a f u r t h e r s t e p in t h e d e v e l o p m e n t will b e r e p l a c i n g B v w i t h % / - K o / . l ~. A l s o , t h e & v a l u e s a p p e a r in s o m e d i a g r a m s i n s t e a d of t h e a d v a n c e ratio J. T h e i n f l u e n c e of R e y n o l d s n u m b e r , R,,, o n t h e optimum propeller diameter, pitch, and efficiency is i n t e r e s t i n g t o n o t i c e , b u t t h e d i f f e r e n c e b e t w e e n m o d e l a n d s h i p p r o p e l l e r s w i t h r e g a r d t o d r a g coe f f i c i e n t s m u s t n o t b e o v e r e s t i m a t e d b e c a u s e of p a r t l y l a m i n a r flow in t h e p r o p e l l e r m o d e l r e g i o n of R,,, F i g . 5'4. T h e g a s c o n t e n Lof w a t e r , {n t h e c a v i t a t i o n t e s t s , ct/~t.~ = 0 . 2 0 - 0 . 3 3 , is s o m e w h a t h i g h e r t h a n w h a t is u s u a l in t h e S S P A t u n n e l . D i f f e r e n c e s in g a s c o n t e n t n m y i n f l u e n c e t h e i n c e p t i o n of c a v i t a t i o n , especially on the tip vortex cavitation.

(text continued on page 312)


297

8 4-100 P/D =10 B4-B5 84-70 B4-55 -B4-40

C[ 10C

-'4

:E

.....

Q o

B4-70

~4-55 84 -z.O

/ ,//J

i

~ ~-~-~-;'~,~:di~i

q

:

/" '~

0

\,,ii,j ,,," ,,, ,,;/

o

! 0°

Fig. 37

40 °

80 °

120 °

160 °

/3

200°

240 °

280 °

320 °

360 °

Open-water test results with B 4-40, B 4-55, B 4-70, B 4-85, and B4-100 series screws with P / D = 1.0

2400

Fig. 38

,

I

I

I

1

40 °

80 °

120 °

160 °

/3

200 ~

240 °

280 °

320 °

360 °

Open-water test results with B 3-65, B 4-70, B 5-75, B 6-80, and B7-85 series screws with P / D = 1.0

FOURIER

I

SERIES OF 20 TER~4S

]

_,0! /

/

-12

--I

I

I -10 o

Fig. 3 9

40 °

80 °

120 °

160°

/3

200 °

240 °

280 °

320 ~

866 °

R e s u l t o f four-quadrant m e a s u r e m e n t s o f B 4 - 7 0 s c r e w w i t h P / D = 0 . 6 a p p r o x i m a t e d by F o u r i e r series of 2 0 terms

20 ------CURVES OF CT AND CQ APPROXIMATED BY A FOURIER SERIES OF 2 TERMS

15

cl ~

-160o

Fig. 4 0

~---~-'~

40°

80°

120°

160°

,,3

200°

240°

2g0°

320°

360"

R e s u l t o f four-quadrant m e a s u r e m e n t s of B 4 - 7 0 s c r e w w i t h P / D = 1.O a p p r o x i m a t e d by Fourier series of 2, 5, 10, or 2 0 terms


299

FOURIER

SERIES

OF 5 T E R M S

!

~2 t

_//_ ......

o8

_~_

c~ -IOC~ 04! - - - -

I

,#

i

Jc;

-06 - -

-0 °

I I

---

40 °

80 °

120 °

150 °

Fig. 40

/3

2i

(continued)

2° i FOURIER S E R I E S

161

OF 10 TERMS

. . . . . . . . . . . . .

_

1_

c~_ !

/ ii i

_10

.

.

.

.

.

.

.

.

.

.

.

J

-18 0°

!

.

I

.

.

.

.

.

,

11[ 40 °

BO°

120 °

Fig. 40 300

_

160 °

/3

200o

2&0 e

(continued)


280 °

320 °

360 °

FOURIER SERIES OF 20 TERMS 1E .

.

.

j

,

.

!

! I

0

40Q

80°

120e

160° /3

Fig. 4 0

200°

240°

280°

320°

350°

(continued)

I L I

FOURIER SERIES OF 20 TERMS

i

c~

I/ "\ !

-10c~

,\

I

I

oz

II ! i -04 /

\

/

/

/_ / / ,YI

80 °

Fig. 41

120°

160°

:~

200 °

2~0°

f

, 280°

i 320°

Result of four-quadrant measurements of B 4 - 7 0 s c r e w with = 1.4 approximated by Fourier series of 2 0 terms


35'3:

P/D 301

0~ B z,-85

KT 1OKO

~p

~.oe

~5

I

l 0(~.3

0f,

0.5

Fig. 42

0~

0.7

I

I

'

I

i

g.8

09

I

12

1.0

Ig

13

15

Influence of cavitation on thrust and torque of screws of B 4-85 series

B z,-85 :,o

08

07

i

f~ 03

00.3

~"

0/,

0.5

06

0.7

QB

~g

1.0

1.1

J

Fig. 42

302

(continued)


1.2

13

1A

1.5

Q9

B ~-B5

%:L2

\ KT

IOKQ

%

~f

02

~

i---~

X

O 0.3

0.4

05

1g.6

(17

J

03

Fig. 42 11

0~9

1.0

1.1

12

1.1

I.~

15

(continued)

-"

B 4-85 P/D =1.4

\

t i

\

f

\

I

KT

IOKQ

%

°~o ~

I

0 02

O~

05

06

J

CLB

Fig. 42

09

1.0

1.1

12.

13

IA

(continued)


303

0E

0

5

8 4-100 =O8

~D

Io~

-

Oe7

IOKQ "gp a~

..Co o,

~

o t13

~

0,4

05

0.6

-.'b.. 0.7

Q8

0.7

10

1.1

t2

1,3

t,4

J

Fig. 43

Influence o f cavitation on thrust and torque o f screws o f B 4 - 1 0 0 series

B 4-100 P/D" 1.0 1.1

0.9

~P ~-o=~~

i~ii.~\\~ ~

0E KT 10KO

-

"Tp o.~

0 03

o~

0.&

0.5

06

0.7

Q8

09

1,0

1.1

,3

Fig 43

304

(continued)


1.2

13

14

kS

B a-IOO

%=,2

KT 10 KQ 'gp

04

05

Q6

07

08

09

~0

tl

|2

1~5

13

J

Fig. 43

(continued)

8 4-100

Qi KT 1OK0 9p 05-

%

0.5

06

O7

08

Fig. 43

Q9

~,g

12

t.3

t~

1.5

(continued)


305

B 5-75

P/Doo 8

"~p O-o >> \

06 KT IOKQ

%

05 ~

I

02

------

-~ .~.~

"-"~"-'- - ' -I ~ " 2 "

o.1

/- ' ~ . _ ~ - ~ - f ~ - - - - - - - - ~

0/.

0.5

06

""-

07

.

" " 7"--.

0.8

t0

0.9

1.1

12

1.3

1~,

0

Fig. 44


B 5-75 P/'D = tO

K!

IOKQ

%

03

04

0.5

05

07

08

0_q

1.0

1.1

J

Fig. 44

3 0 6

(continued)


12

1.3

tA

t5

OSf. . . . . . .

T" -

I

~

I

I

T

09

I0

11

B 5~75

l
Oz

O~

! ol

O

03

04

05

O7

08

12

t3

1~,

1,5

0

Fig 44

(continued)

I B 5-75 P/D 14

['(T ~o:4o ")p

'

0

03

O.4

05

Q6

07

i j

p

•

j

(18

0.9

1.0

tI

12

[3

it.

IS

J

Fig. 44

(continued)


307

I

B 5-105

P/o:o~

I % ~::

I

i

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authors' reference [9]), with different pitch ratios are plotted versus the lift coefficient, CL, for the profile in question. Faired mean curves through the points will indicate the cavitation limit and m a y be transformed to ~0-J curves by knowledge of the C L - J Correlation. T h e test results have been compared with calculated values of the local dimensionless pressure difference, Ap/q, at certain points on the equivalent profile (2-dimensional

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T h e cavitation Emit curves given by the authors were determined on the basis of desinent cavitation instead of incipient, due to the better repeatability for the former type, which means varying pressure at a constant rate of revolutions. Another method to determine these b o u n d a r y curves is to keep the pressure (i.e., cavitation number, ~0) constant and v a r y the rate of revolutions (advance ratio). T h e results of incipient cavitation tests to some extent also m a y be faired b y the aid of Lerbs' method of equivalent profile. In Fig. 55 local critical cavitation numbers, ~0.7~ = ~o/(1 + 5.552/J2), representing results from tests with the SSPA standard propeller models 5.60 (see the

S. Curtis Powell, Member: I will congratulate the authors by observing t h a t they have contributed a q u a n t u m step in our knowledge of propeller characteristics, particularly in four quadrants and under cavitating conditions. I object to the authors' statement about the most widely encountered design problem on three counts : (a) in this country the revolutions usually are not given, in contrast to Europe where internal-combustion engines are more common; (b) the diameter should be optimized, which is not the same thing as maximizing the propeller efficiency; and (c) the speed is not given, but is t h a t speed at which the ship will operate with the given power; although the diameter is not likely to be seriously mischosen if a reasonable estimate of speed in service is made. (This is not the specification speed and power.) Although the diameter m a y not be significantly in error, other

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quantities involved in the ship-propeller-engine m a t c h m a y be more seriously affected. In any case, iteration is clearly in order (see Section 17, Chapter VII, Principles of N~tw~.l Architectlt.re). I much prefer to work with the K t - J system of charts as the m a t c h achieved there is philosophically and practically more comfortable. The Bp charts in the present paper are not consistent with those of the aforementioned reference to PN.4, especially as regards the best efficiency line. Since b o t h sets of charts I understand to be derived from the same source, I would appreciate it if the authors would give us an explanation for this discrepancy. The numerical constant relating Bp to K~ and or seems to imply operation in fresh water. This is an additional complication in the use of the B~,-a charts. I would like confirmation t h a t freshwater operation is indeed implied. To those familiar with the Russian results [12] for operation in four quadrants, the use of inverted Co scales in Figs. :39 to 41 lnay be confusing at first. Would the authors confirm t h a t their conventions take thrust, 7", as positive when it is in the direction to propel the ship ahead and torque, Q, positive when it is in the direction corresponding to normal ahead propulsion ? I further assume t h a t the 20-term Fourier series coefficients are obtained from a least-meansquare fit to a number of input points much in excess of 20. Can the authors tell us what is this nmnber of inputs? T h e use of " h y d r o d y n a m i c piteh angle, fl," for presentation of four-quadrant data is a clever proeedure to eliminate some of the troubles with plots made to a base of J (and 1/or) or R P M .

William B. Morgan and Geoffrey (3. Cox, Members: This paper presents very useful information for the naval architect who must make predictions of propeller performance. I t represents the most extensive and comprehensive a m o u n t of series data known to us. T h e inclusion of a rational modification to the data to allow for the effects of Reynolds number on the propulsion characteristics is most welcome. Some of us have been concerned a b o u t the problem for a long time, especially for screw series data which are based on relatively small-sized models. We think t h a t the cross-fairing technique used in the paper has greatly enhanced the Wageningen B-screw series data as there were obvious differences in some of the data presented in the past. We consider it a pity t h a t the authors did not take the opportunity when preparing this paper to present Figs. 1S through 26 in nondimensional form. The use of these diagrams m a y require a

correction for the density of the fluid medium in which the propeller will operate. For instance, are the Bp-~ diagrams presented in this paper for the propeller operating in fresh or salt water ? I t is very doubtful whether the cavitation d a t a presented in Figs. 46 through 5:3 are useful or whether Conclusion 2 regarding cavitation inception is valid. Studies of the. m a n y factors influencing correlation of cavitation inception made b y the Cavitation Committee of the I T T C would tend to invalidate any application of model propeller methodical series data to the prediction of cavitation inception on a full-scale propeller operating behind a ship. T h e authors have defined fl incorrectly in t h a t it should be referred to as advance angle and not hydrodynamic pitch angle. The h y d r o d y n a m i c pitch angle B~ includes induced effects. Incidently, why h a v e n ' t the authors used standard I T T C notation for all of their symbols? J. G. Hill, Member: I t is always a privilege to see another propeller paper front Wageningen because of their high quality and completeness. This paper is no exception and provides a wealth of information to those concerned with propeller problems. I t is noted t h a t the B4-70 series was tested at a Reynolds nnmber of 3.5 X 105 and the assumed Reynolds N u m b e r for the B4-70 chart is 1.0 M 1/)6. On the basis of Fig. 29 the efficiency for a given Bp should be higher for the higher Reynolds number of the new corrected presentation compared to the actual test results, as shown in Principles of N~twzl Architectl~re. Actually the new curves corrected to the higher Reynolds number show lower efficiency than the original curves. Figure 29 also indicates t h a t delta should be higher and the pitch ratio lower with increasing Reynolds number, but the curves show the reverse. If these discrepancies are the result of the computer fairing, it would be interesting to know how much the test data was departed from and the justification for the shift. Since the linear ratio from ship to model is frequently about 30 for large ships, the Reynolds N u m b e r changes by a factor of about 150. Figure 29 indicates about a 10-percent increase in efficiency for this change in Reynolds number. Do the authors expect t h a t this gain will be realized with the ship propeller or are there other factors t h a t will prevent this gain? In Resistance, Propulsion and Steering of Ships it is noted t h a t "this increase in efficiency is partly or wholly cancelled" ['.4.]. The cavitation data and maneuvering data are


31 3

a valuable addition to the literature and it is hoped t h a t the next paper will cover the cavitation of the lower blade areas and the 6- and 7-bladed series. T h e cavitation comparisons in Figs. g6 to 53 demonstrate t h a t each combination of pitch and camber results in o p t i m u m performance at one particular thrust coefficient. Therefore, for a n y specific design condition the camber and pitch m u s t be varied from the series values to obtain satisfactory performance. Particular attention m u s t be paid to the inner radii of propellers for high-speed, twin-screw ships, where cavitation erosion is frequently found on the pressure face of constant-pitch propellers. P. C. Pien, Member: T h e results of the Wageningen B-screw Series given in this paper are somewhat different from. those previously published. This is a consequence of the refairing and the correction of Reynolds n u m b e r effect. By comparing the model propeller test results in Fig. 7 and the extrapolated full-scale results in Figs. 16 or 17, we notice t h a t the efficiency of a full-scale propeller can be as m u c h as 20 percent, or more, higher than t h a t of a model propeller. This is a delightful surprise. Perhaps before we accept these extrapolated results with confidence, we should examine their extrapolation method rather carefully. T h e extrapolation procedure used in the paper was developed b y Lerbs almost twenty years ago. His basic idea is t h a t the efficiency of any blade section can be considered as a product of the idea efficiency and the profile efficiency. T h e idea efficiency of a blade section is a function of the loading and the advance coefficient X. I t is independent of the section profile geometry. T h e profile efficiency, on the other hand, depends upon the profile geometry as well as the loading. This is in line with the fact t h a t propeller efficiency can be considered as a product of propeller idea efficiency and the propeller blade efficiency. I t is possible to choose a blade section, the characteristics of which can be used to relate the openwater characteristics of the propeller. This can be done rather easily as shown in reference [7] of the paper. This chosen section is called the equivalent profile of the propeller. So far, there should not be any question. I t is just another way of presenting an open-water test result. Then, two assmnptions are made in order to account for the Reynolds number effect as follows : T h e first assmnption is t h a t the 0.75R section can be chosen as the equivalent profile with true physical meaning and t h a t its characteristics v a r y with Reynolds n u m b e r the same way as t h a t of 31 4

the propeller. The second assmnption is t h a t the profile drag can be divided into two parts, the tangential component due to skin friction and the normal component due to pressure distribution and t h a t only the first p a r t has Reynolds number effect. T h e first assumption m a y be reasonable since the center of propeller loading is in the neighborhood of 0.75R and the profile geometry does not change rapidly there. However, the second assumption is quite contrary to the test results of the wing section. Figure 56 shows the drag coefficient of a N A C A 66-006 wing section. T h e upper curve is for roughed section. The lower three curves are obtained at three different Reynolds numbers. T h e m i n i m u m drag coefficients of the three curves are the same while the shape of the curves are quite different. This means the minimum drag coefficient of a profile is independent of Reynolds number, and the pressure drag coefficient is highly dependent on Reynolds number. This situation seems to contradict the assumption made in the extrapolation procedure. Figure 57 shows the same tendency of a slightly cambered section. In v i e w of this I would be inclined to question the accuracy of the extrapolated results, especially for the heavily loaded conditions. J. B. Hadler, Member: I would like to add m y thanks, along with others, to the authors for their efforts in reanalyzing and expanding the test work on the 4- and 5-bladed propellers of the BScrew Series. The removal of inconsistencies in the old test results along with the inclusion of the quadrature and cavitation measurements are welcome additions. T h e inclusion of the coefficients for the equations fitted to the test results will be of material assistance to those of us who use the digital computer in design or analysis work. M y comments on this work will be restricted to questions of clarification or amplification. 1 Do the authors plan to extend this work to the 6-bladed series--an area of growing importance to the merchant ship propeller designer? 2 Do the authors plan to extend the derived polynomials for K r and K o to include the effect of cavitation ? 3 The paper indicated t h a t the equations derived for KT and t £ o approximated the experimental results " v e r y well." Could the authors give us an idea of the quality of the fit through the usual statistical terms such as standard deviation ? 4 T h e authors' study of the effect of Reynolds number upon the open-water performance, particularly the influence upon the o p t i m u m diameter and propeller efficiency, is of sufficient magnitude to make the direct use of the design charts based


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upon model open-water tests such as those in Figs. 1S through 26 questionable. I would like to ask the authors for a brief outline of how they would modify their design procedure such as contained in their reference [6] to ensure obtaining the best propeller as well as the desired full-scale perfornmnce. Maurice R, Hauschildt, Member: The authors are to be congratulated on a very thorough effort in providing a systematic series of propeller characteristics for use in future design efforts. Both the open water and cavitation tests with the higher blade numbers and larger blade area ratios were sponsored by the half of the Bureau of Ships that has now become the Naval Ship Engineering Center, and the purpose was to increase propeller characteristics information in these areas. The effect of Reynolds number does raise some questions of accuracy with these results. I understand that you have a variable-pressure towing basin under construction at NSMB. Do you plan to test these propellers in this new basin with standard ship models, to compare these cavitation results with those for propellers installed on ships? If so, when could the results of this comparison be made available? Are there any plans to develop the same type of information for ducted propellers as has been presented for conventional propellers in this paper?

Authors' Closure Before answering each discussor in detail I should like to make a general remark with respect to the effect of Reynolds number on the optimum diameter and efficiency. The results presented in Fig. 29 have all been related to a fully developed turbulent flow around the propeller blades. In m a n y cases such a turbulent flow will not occur at propeller models for a Reynolds number lower than 3-106. If we are aware that most of the open-water tests have been carried out at a Reynolds number of 3.10;" we have to accept that laminar flow effects will lead to higher et~ieieney values than can be expected for turbulent flows. So the differences between the experimental values as have been published up till now and the full-scale values will be considerably smaller than might be derived from Fig. 29. Nevertheless for a Bp-value of 50 and a fullscale diameter of 8 m there still remains a differ31 6

ence of 5 percent in efficiency and 4 to 5 percent in diameter between the up-till-now-published experimental data and the full-scale prediction. We are aware that by using these new results of the Wageningen B-Series, which have all been analyzed for turbulent flow at a Reynolds number of 1(16, we have also to look carefully to scale effect in the other propulsive components such as the wake factor, and thrust deduction in order to get the right overall correlation. Jl~rorga.n and Powell made some remarks about the B,,-value being not dimensionless and asked whether the density o is for fresh or salt water. I t is for fresh water. f2owelZ and 13jOrne asked why the conventional way of plotting the results of systematic screw series has been maintained. We have selected this conventional way of plotting because this is so familiar to the designer. On the other hand, now that the coefficients of the polynomials of the screw characteristics are known, each specialist can derive from these published coefficients his own favorite way of plotting. 13jarne asked why we kept the rpm constant and varied the pressure in determining the characteristic cavitation curves. He would prefer varying the rpm at constant pressure. In our opinion such a procedure does not make any significant difference. By varying pressure at constant rpm one moves vertically through the cavitation diagram. By varying the rpm at constant pressure one moves horizontally through the diagram. The only important point is that one determines the characteristic cavitation curves at the desinent cavitation points. A small drawback of varying the rpm might even be the varying Reynolds number. Bj/irne's suggestion to analyze the cavitation inception diagram with the equivalent profile method is in our opinion too ambitious. Threedimensional effects and tip vortices would not be treated in a proper way. In reply to Dr. P i e n we should like to mention that in our paper the profile data as published by Hoerner have been used. Especially Hoerner's functional relation between the drag coefficient C1) and the flat-plate friction coefficient and the thickness-chord ratio. We are convinced that data as shown by Dr. Pien have been included in Hoerner's summary which has been based on an enormous a m o u n t of profile data. To Morgc~n and Cox we like to make the following remarks: 1 The authors agree that in the cavitation diagram the incipient curves are not so accurate as one might desire. This is due to air content and other unknown factors. On the other hand


we are convinced that the characteristic curves for thrust and torque breakdown do not suffer from these unknown environmental conditions. 2 In the four quadrant diagrams the hydrodynamic pitch angle ~ excluding the induced velocities has been used. In our opinion this is correct. We agree that we can better denote B by advance angle. The introduction of the hydrodynamic pitch angle ~ including the induced velocities is not necessary and would lead to a rather complicated diagram. To Hill we can answer that, as already explained in the introduction of our reply, indeed the experimental results, which might be.. affected by a laminar flow, might lead to higher efficiency values than the corrected ones for turbulent flow and a Reynolds number of 106. Hill's remark concerning the restricted value of the cavitation inception diagram for screws designed with a lifting line or surface theory is correct. However, we like to underline again that even for these propellers the curw~_s for thrust and torque breakdown are valid. Powell.'s question concerning optimizing is a basic one. Indeed one can select the diameter optimum if the thrust or power, the speed, and

the rpm are fixed. If power, speed, and dimneter are fixed one can select the rpm optimum. However, a free choice of both rpm and diameter leads always to one point in the propeller chart, that is the top of the envelopment of the efficiency curves in tile K r - K Q - J diagram. Further, we can inform Powell, that 120-180 points have been used for the Fourier analysis. We would like to inform Hadler and Hill that the cross-fairing process will be continued for tile number of blades and that the Reynolds effect will be introduced ill the polynolnials. Furthermore we are considering extending the cavitation observations in order to give the cavitation diagram in polynomial form. We would like to say to 1-I~zltschiIdt that maybe the depressurized towing tank will be the best tool to carry out these cavitation tests at high loadings. And indeed we are planning the santo information as presented in this paper for dueted propellers. In future papers concerning the fairing of screw diagrams by a regression analysis we shall give exact information about the standard deviation of the series approximations. Gentlemen, we thank you for your discussions and your interest in our paper.


317

SNAME The Wageningen B-Screw Series.pdf

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