Hydrodynamics of Ship Propellers (Cambridge Ocean Technology Series)

This book deals with flows over propellers operating behind ships, and the hydrodynamic forces and moments which the propeller generates on the shaft and on the ship hull. The first part of the text is devoted to fundamentals of the flow about hydrofoil sections (with and without cavitation) and about wings. It then treats propellers in uniform flow, first via advanced actuator disc modelling, and then using lifting-line theory. Pragmatic guidance is given for design and evaluation of performance, including the use of computer modelling. The second part covers the development of unsteady forces arising from operation in non-uniform hull wakes. First, by a number of simplifications, various aspects of the problem are dealt with separately until the full problem of a noncavitating, wide-bladed propeller in a wake is treated by a new and completely developed theory. Next, the complicated problem of an intermittently cavitating propeller in a wake and the pressures and forces it exerts on the shaft and on the ship hull is examined. A final chapter discusses the optimization of efficiency of compound propulsors. The authors have taken care to clearly describe physical concepts and mathematical steps. Appendices provide concise expositions of the mathematical techniques used. The book will be of interest to students, research workers and professional engineers (naval architects) in propeller dynamics.

CAMBRIDGE OCEAN TECHNOLOGY SERIES 3 General Editors: I. Dyer, R. Eatock Taylor, J. N. Newman, W. G. Price

HYDRODYNAMICS OF SHIP PROPELLERS

Cambridge Ocean Technology Series 1. Faltinsen: Sea Loads on Ships and Offshore Structures 2. Burcher & Rydill: Concepts in Submarine Design 3. Breslin & Andersen: Hydrodynamics of Ship Propellers

HYDRODYNAMICS OF SHIP PROPELLERS

John P. Breslin Professor Emeritus, Department of Ocean Engineering, Stevens Institute of Technology

and Poul Andersen Department of Ocean Engineering, The Technical University of Denmark

CAMBRIDGE UNIVERSITY PRESS

PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge CB2 1RP, United Kingdom CAMBRIDGE UNIVERSITY PRESS The Edinburgh Building, Cambridge CB2 2RU, United Kingdom 40 West 20th Street, New York, NY 10011-4211, USA 10 Stamford Road, Oakleigh, Melbourne 3166, Australia © Cambridge University Press 1994 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press First published 1994 Reprinted 1996 First paperback edition 1996 A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication data Breslin, John P. Hydrodynamics of ship propellers / John P. Breslin, Poul Andersen, p. cm. - (Cambridge ocean technology series; 3) Includes bibliographical references and index. 1. Propellers. 2. Ships-Hydrodynamics. I. Andersen, Poul, 1951- . II. Title. III. Series. VM753.B68 1993 623.8'73-dc20 93-26511 CIP ISBN 0 521 41360 5 hardback ISBN 0 521 57470 6 paperback

Transferred to digital printing 2003

Contents

Preface

xi

Notation

xiv

Abbreviations

xxiv

1

Brief review of basic hydrodynamic theory Continuity Equations of motion Velocity fields induced by basic singularities Vorticity

1 1 2 7 17

2

Properties of distributions of singularities Planar distributions in two dimensions Non-planar and planar distributions in three dimensions

26 26 33

3

Kinematic boundary conditions

42

4

Steady flows about thin, symmetrical sections in two dimensions The ogival section The elliptical section Generalization to approximate formulae for families of two-dimensional hydrofoils A brief look at three-dimensional effects

46 51 54

5

Pressure distributions and lift on flat and cambered sections at small angles of attack The flat plate Cambered sections

6

Design of hydrofoil sections Application of linearized theory Application of non-linear theory

7

Real fluid effects and comparisons of theoretically and experimentally determined characteristics Phenomenological aspects of viscous flows Experimental characteristics of wing sections and comparisons with theory

57 62 66 66 74 86 87 103 111 111 117 vii

viii

Contents

8

Cavitation Historical overview Prediction of cavitation inception Cavitating sections Partially cavitating hydrofoils Modification of linear theory Supercavitating sections Unsteady cavitation

128 128 130 140 142 151 156 159

9

Actuator disc theory Heavily loaded disc Lightly loaded disc

162 166 187

10 Wing theory

196

11 Lifting-line representation of propellers Induced velocities from vortex elements Generalization to a continuous radial variation of circulation Induction factors Forces acting on the blades and the equation for the circulation density

207 209 219 222

12 Propeller design via computer and practical considerations Criteria for optimum distributions of circulation Optimum diameter and blade-area-ratio determinations Calculation procedures Pragmatic considerations

227 227 235 239 252

13 Hull-wake characteristics Analysis of the spatial variation of hull wakes Temporal wake variations

262 264 270

14 Pressure fields generated by blade loading and thickness in uniform flows; comparisons with measurements Pressure relative to fixed axes Comparisons with measurements

272 272 281

15 Pressure fields generated by blade loadings in hull wakes

290

16 Vibratory forces on simple surfaces

301

17 Unsteady forces on two-dimensional sections and hydrofoils of finite span in gusts Two-dimensional sections Unsteady lift on hydrofoils of finite span Implications for propellers

315 315 327 332

224

Contents

ix

18 Lifting-surface theory Overview of extant unsteady theory Blade geometry and normals Linear theory A potential-based boundary-element procedure

334 334 337 340 368

19 Correlations of theories with measurements

374

20 Outline of theory of intermittently cavitating propellers A basic aspect of the pressure field generated by unsteady cavitation Pressure field due to cavitating propeller Numerical solution of the intermittently- by

1 dv T

80 ~Ut~

r 2WT

Integrating

Figure 1.4 Flow due to a vortex.

(j)v — — 6 + a constant

(1.37)

27T

Again, the constant can be ignored as we are only interested in derivatives of v.

10

Brief Review of Basic Hydrodynamic Theory

In rectangular coordinates, r y y = — tan"1 2TT x

y ; —7T < tan"1 — < K x

To make 0V single valued we define it in the region excluding the cut, cf. Figure 1.5. Then on the upper bank of the cut

r 2 and on the lower bank

r 2 Figure 1.5 Cut along negative x-axis.

.'. the jump in (j)v across the CUt is

A(j)y = (j)y+ — (j)y- = T

(l.38)

To show that 0V satisfies the Laplace Equation, consider the form in polar coordinates, namely

r#

ia

[dr*

r

(1.39)

= 0

Q\

As <j)y is independent of r, we have

LfLfL r 2 Off* [2r which is clearly zero for all r ^ 0. The Dipole (In two dimensions) The dipole2 is defined to be the limit of the sum of a source and a sink as they are brought together in such a way that the product of their common strength and the distance between them is a constant defined to be the dipole moment strength. Thus, dipoles have magnitude and directivity. Consider a source located at x = 0, y = e and a sink of equal strength at the origin, Figure 1.6. Then the sum of these potentials is M

1

<j> = — l n J x 2 + (y - e) 2 2 7T 2 7T

2 dipoles are often referred to as doublets.

M 2l 2 l

1

Injx2 + y2

(1.40)

Velocity Fields Induced by Basic Singularities

11

Write (y _

- 2ey + e2 2ey -M as e —* 0;

Figure 1.6 Source and sink of equal strength.

then

M jx2+(y-e) 2 M = — i n —^-— = — = — 2 7T

I

V

9i,r9

2ey y2

2 7

+..

(L41)

The expansion of the logarithm gives, in general, ln(l - z ) = -

fz + z2 1 — + ...

|z| < 1

(1.42)

Then to order e Me 2TTX 2 +

(1.43)

y2

Defining Me = Ey, the y-directed dipole strength, we have

2TT

X2

+

(1.44)

y2

This result may be obtained more adroitly by differentiating the unit source displaced along the y-axis. Thus, for a source of unit strength, 1

d y'= 0

y - y1

1 2TTX2 +

(y

-y1)2

y'=0

27TX2

(1.45)

For a dipole of strength Sy, then s

y

y

(1.46)

12

Brief Review of Basic Hydrodynamic Theory This gives a vertically upwarddirected dipole at the origin. In polar coordinates 0yd = -

E y sin0

(1.47)

27TT

then d(f>yd/dr = SySin0/27rr2, showing that the radial velocity is an odd function of 0 directed outward (away from the "source" in the region 0 < 0 < IT and inward — toward the "sink" in the region -w < 0 < 0), cf. Figure 1.7. This agrees with our exFigure 1.7 Flow due to a vertical upward—directed dipole. pectations. For a vertically downward directed dipole, change the sign. Consider a line distribution of downward dipoles along the negative xaxis having uniform strength d = M

ex

• +

— lim

0

47r

|jx

2

+ r

2

2

^x + r

2

2

+ O(£2) 2 3 2

[x + r ] /

Defining the dipole moment strength by (1.66)

we obtain Sxx

(1.67)

4TT [X2 4- r 2 ] 3 / 2

as the potential of an x-directed dipole at the origin with axis in the positive x-direction. This can be achieved directly by differentiation of the source. To secure the positively directed x-dipole at the origin place a source at x = x \ y = z = 0. Then d

M 7

dx

47r

\{x - x') 2 + r2j

x1 = 0

where the derivative is evaluated at x1 = 0 after differentiation. Then

f/2

x' = 0

Then formally replacing M by Ex we have (1.67). In general to achieve a dipole whose axis has the direction cosines n x , n y , n z at any point (x',y',z') apply the following operation =

4

d nx { O'

1- ny

d d'

1-

d1

1 (y-y')2

(1.68)

Vorticity

17

where Sx = n xS, Sy = n yS, Sz = nzS Dipoles may be distributed on lines and surfaces in the same manner as sources.

VORTICITY We have seen previously that by manipulating the equations of motion (Equations (1.8) to (1.14)), terms arise which, together, are identified as components of a so-called vorticity vector

C = fi + m + ct the components of which are given by velocity gradients, i.e., (x)-. (z)-(y) dw

dv

d\i

dw

dv

d\i

ay

dz

dz

ox

ax

ay

(x) - , (y) -

(z)

Here we see that the operations are in cyclic order and the numerators are in acyclic or reverse alphabetic order. To give a more physically based exposition of the role of vorticity we can examine the excursions in the velocity components between two neighboring points (x,y,z) and (x + &, y +

•T 1 da dw' —4 di

ftrl

I: +

5w

2

flvl I-— 0

(2.12)

x2 + y2

v -4 u -> 0

(2.13)

Vertical Dipole Distributions

The potential induced by a line distribution of vertical dipoles along the x-axis is 1 d f a 1 (2.14) (y-y 1 ) 2 dx y'=0

As (2.15) has the same structure as the transverse velocity developed by sources (2.5) then the behavior of the potential is the same, save for sign, as we seek , the jump in . For | x | > a

(2.16a)

(xfi±) = 0

(2.17)

As 0(x,y) is an odd function of y, the y-derivative is an even function of y and hence the vertical or transverse velocity is continuous through the dipole sheet. Thus l(x - »')' - y']

,

,

and v(x,0+) =

ff;y(x') 1 d [ a 0 becomes the value at the point. Inserting the limits we get

Non-Planar and Planar Distributions in Three Dimensions m(x,y)

n

n

; |n| = | r f

35

(2.42)

Now holding e fixed we descend to the surface making n -> 0+ m .-. I = -

(2.42a)

and as n -» 0-, n/1 n | -+ — 1 yielding m 1= - -

(2.42b)

Thus , m = ± 2

1 f 4TT J S

m

d 1 daR

dSf ; on the surface

(2.43)

Now d

1

cos (n,

(2.44)

where the angle (n,R) is that between the normal n and the vector R drawn from the foot of the normal to any variable point on the surface. When the surface is planar the angle (n,/2) = TT/2 and the integral term vanishes. However regardless of the curvature of the surface we have that

d]

\d)

— = m dn].

the source density

(2.45)

so when the jump in the normal velocity is specified then the source density is known. Thus for the distribution of sources • The potential itself is continuous through the surface. • The tangential velocity is also continuous. • The normal velocity is discontinuous. The presence of the integral term in (2.43) can be understood physically because on the curved surface the sources elsewhere can contribute to the normal velocity at any point in addition to the contribution of the local element as depicted in Figure 2.5.

36

Properties of Distributions of Singularities

Figure 2.5 Normal velocity induced by source element.

Distributions of Normal Dipoles We have seen that dipoles can be "constructed" by differentiating the source potential with respect to the dummy variables and in the direction desired. Hence for normal dipoles on a curved surface we have

d

If 47T J s

- I dS 1 an1 R\

(2.46)

Here an has the dimension Iength2/second. But since field point coordinates (x,y,z) and dummy point coordinates (xf,y?,zf) always enter as differences within R we can write d

d

and then the potential of the surface distribution of normal dipoles has the same structure as that of the normal velocity induced by a surface distribution of sources (except for sign) I f d = —\an— 4TTJS

fll - dS'

(2.47)

da [R\

We may then write that

or