L i s s a m a n , the b e s t of m y t e a c h e r s

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TRANSVERSE HYDRODYNAMIC F O R C E S ON SLENDER BODIES I N F R E E

-

SURFACE FLOWS A T LOW S P E E D

T h e s i s by

' J o h n S e y m o u r L e t c h e r

In P a r t i a l F u l f i l l m e n t of the R e q u i r e m e n t s F o r the D e g r e e of

D o c t o r of P h i l o s o p h y

' C a l i f o r n i a Institute of Technology P a s a d e n a , Calif o r nia

1966

( S u b m i t t e d M a y 5, 1966)

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ACKNOWLEDGMENTS

I w i s h t o e x p r e s s m y g r e a t e s t a p p r e c i a t i o n t o Prof. S t e w a r t f o r his continued i n t e r e s t i n my w o r k and f o r the g r e a t amount of time he h a s contributed i n the l a s t y e a r to enlightening d i s c u s s i o n s of it.

The b r o a d n e s s and depth of h i s i n t e r e s t s i n m a t h e m a t i c a l fluid mechanics have been valuable t o me, and t h i s contact with h i m h a s been pleasant and inspiring.

S p e c i a l thanks a r e a l s o due t o P r o f . L i s s a m a n , the b e s t of m y t e a c h e r s . He h a s been both a d v i s o r and p e r s o n a l f r i e n d to me during

m y y e a r s of g r a d u a t e study, and i t w a s he who encouraged me t o t u r n m y attention t o the p r o b l e m s that l e d t o t h i s thesis. 9: hope when I teach, i t w i l l be with a p a r t of the e n t h u s i a s m and insight that make h i m s o outstanding.

I thank P r o f . Cole f o r t i m e spent i n d i s c u s s i o n s of p e r t u r - bation methods; and I e x p r e s s appreciation t o P r o f . Wu f o r his

c r i t i c i s m s and suggestions, which led to much b e t t e r understanding of the m a t e r i a l t r e a t e d in Chapter 111.

Mrs. Vivian Davies, with the g r e a t e s t patience and good nature, did the handsome job of typing the pages that follow; except f o r the f i r s t t h r e e appendices, which M r s . Virginia Connor w a s kind enough t o do. Mr. Kelly Booth w a s h i r e d f o r the tedious but n e c e s s a r y job of checking the manipulations of C h a p t e r s X and XI and the l a s t f o u r appendices; this he did efficiently.

Thanks t o m y fellow students h e r e f o r t h e i r patience and

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sympathy during the trying w e e k s when t h e r e w e r e waves u p s t r e a m , waves i n a l l directions, o r no w a v e s a t all, and the deadline approach- ing; and thanks t o m y fianc'e, P a t r i c i a Ryan, f o r the i n s p i r a t i o n t o make a n end t o the job.

I have been a m p l y supported by a National Science Foundation Cooperative Fellowship during m y t h r e e y e a r s of g r a d u a t e study.

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ABSTRACT

The f o r c e s and m o m e n t s on a moving body p a r t i a l l y i m m e r s e d i n the s u r f a c e of a d e e p ocean of heavy fluid a r e c o n s i d e r e d i n the l i m i t of s m a l l F r o u d e number, F. Asymptotic e x p r e s s i o n s f o r velocity potential a n d f r e e s u r f a c e elevation a r e developed. The choice of the f i r s t t e r m s of the a s y m p t o t i c sequence i s indicated by the behavior, a t

s m a l l F, o f t h e c l a s s i c a l r e s u l t s of l l s m a l l d i s t u r b a n c e theory"

-

a n a l y s i s s t a r t i n g f r o m the l i n e a r i z e d f r e e - s u r f a c e boundary conditions.

It is found t h a t the leading t e r m s depend on the l o c a l disturbance, which c a n be expanded a s a power s e r i e s i n F. The wave p a t t e r n contrib- u t e s h i g h e r - o r d e r t e r m s which a r e not analytic about F = 0; only e s t i m a t e s of the o r d e r of t h e s e t e r m s a r e obtained. Consequently the p r e s e n t w o r k does not e s t i m a t e d r a g but i s confined t o c o n s i d e r a t i o n of t r a n s v e r s e f o r c e s and moments.

Once the a s y m p t o t i c sequence i s a s s u m e d , p e r t u r b a t i o n of the e x a c t equations and boundary conditions about F = 0 is s t r a i g h t - forward. The z e r o - o r d e r potential i s that of the "reflection-plane"

model of Davidson. F o r a r e s t r i c t e d c l a s s of shapes, the s l e n d e r body t h e o r y is applied t o the z e r o - o r d e r and f i r s t - o r d e r p r o b l e m s . A

g e n e r a l method i s developed using c o n f o r m a l mapping t o solve the first- o r d e r p r o b l e m f o r sufficiently s l e n d e r s h a p e s of a r b i t r a r y c r o s s -

section. T h i s method i s applied to two p a r t i c u l a r shapes, viz. a wing of z e r o thickness and a h a l f - s u b m e r g e d body of revolution, both i n sideslip. The c o r r e c t i o n t o the reflection plane model i s found t o be g e n e r a l l y quite s m a l l i n the r a n g e of F f o r which t h i s t h e o r y i s expected t o apply.

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v

T A B L E O F CONTENTS

I INTRODUCTION

I1 EQUATIONS AND BOUNDARY CONDITIONS

I11 T H E F O R M O F SOLUTIONS IV P E R T U R B A T I O N EQUATIONS

4. 1 Z e r o O r d e r

v

VI VII

VIII IX

4. 2 F i r s t O r d e r

4. 3 S e c o n d a n d H i g h e r O r d e r s SUBMERGED CYLINDER

SLENDER BODY APPROXIMATION

STEADY SIDESLIP-BODY - F I X E D COORDINATES 7. 1 Z e r o O r d e r

SYMMETRY CONSIDERATIONS

SOLUTIONS BY CONFORMAL MAPPING 9 . 1 Z e r o O r d e r

SLENDER WING O F Z E R O THICKNESS

HALF-SUBMERGED BODY O F REVOLUTION CONCLUSIONS

A P P E N D I C E S

I T h e P o t e n t i a l of a n E l e m e n t a r y S u b m e r g e d H o r s e s h o e V o r t e x 57

I1 Modified Method of S t a t i o n a r y P h a s e 62

I11 Mapping of a S o u r c e D i s t r i b u t i o n 6 4

IV P l a n a r Wing: I n t e g r a t i o n o v e r S o u r c e s 66 V P l a n a r Wing: S p a n w i s e I n t e g r a t i o n 70

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v i

T A B L E O F CONTENTS ( C o n t ' d ) VI Body of Revolution: I n t e g r a t i o n o v e r S o u r c e s VII Body of Revolution: C i r c u m f e r e n t i a l I n t e g r a l s R E F E R E N C E S

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I. IN TR OD UC TI ON

The d e t e r m i n a t i o n of the f o r c e s and m o m e n t s on a p a r t i a l l y i m m e r s e d body is i m p o r t a n t f o r the prediction of p e r f o r m a n c e , c o n t r o l and s t a b i l i t y i n the design of s u r f a c e s h i p s and other i n t e r f a c e vehicles.

Historically, by f a r the g r e a t e s t concentration of attention has been on the d r a g problem. The i n t e r e s t i n d r a g is c e r t a i n l y justified by the f a c t that c o m m e r c i a l and m i l i t a r y v e s s e l s spend most of t h e i r time in

steady, r e c t i l i n e a r motion a t z e r o sideslip, and b y the d i r e c t r e l a t i o n of d r a g t o the economically and m i l i t a r i l y i m p o r t a n t f a c t o r s of speed and fuel consumption. However, f o r some types of ope r a t i o n m o r e g e n e r a l motions m u s t be considered.

I n t e r e s t i n sideslipping a n d yawing motion of s h i p s has been motivated by c o n s i d e r a t i o n s of maneuvering and turning c h a r a c t e r

-

i s t i c s . Davidson and Schiff (Ref. 1, 1946) i s a d i s c u s s i o n of m a n e u v e r - ing p r o b l e m s and a s u m m a r y of e a r l i e r work. It w a s a p p a r e n t l y

Davidson who f i r s t suggested the "reflection plane" model, which a p p e a r s i n the c u r r e n t w o r k a s the z e r o - o r d e r theory: the f r e e s u r - f a c e i s r e g a r d e d a s a r i g i d f l a t wall, in which the s u b m e r g e d p a r t of hull m a y be reflected. Advances i n the a e r o d y n a m i c s of low-aspect- r a t i o wings and s l e n d e r bodies have been applied by Tsakonas (Ref. 2, 1959) t o calculate hydrodynamic coefficients of ships, s t i l l b a s e d on the reflection-plane model. To date no t h e o r y h a s a p p e a r e d on yawing and sideslipping which t a k e s into account the changes i n elevation of the f r e e s u r f a c e i n the vicinity of the ship, and no t h e o r e t i c a l j u s t - ification h a s been p r e s e n t e d f o r the reflection-plane model, o r i t s range of validity. It is the purpose of the c u r r e n t w o r k t o make s o m e

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2

contribution t o a c l a r i f i c a t i o n of t h e s e m a t t e r s . The p r e s e n t a u t h o r ' s initial motivation c a m e not f r o m any love of m o d e r n ships

-

i n fact he h a s good p e r s o n a l r e a s o n s f o r a definite a v e r s i o n to t h e m

-

but r a t h e r f r o m p r o b l e m s a r i s i n g i n the d e s i g n of sailing yachts, c o n s i d e r e d i n a previous p a p e r (Ref. 3,1965). The yacht hull n o r m a l l y o p e r a t e s i n side- s l i p to develop a side f o r c e equal t o that of the s a i l s ; c o n t r o l problems demand a knowledge of the yawing moment involved.

The significant d i m e n s i o n l e s s p a r a m e t e r f o r motion of geo- m e t r i c a l l y s i m i l a r s h a p e s n e a r the f r e e s u r f a c e of a semi-infinite inviscid fluid i s the F r o u d e number F Z C ~ / ~ B w h e r e c i s a c h a r - a c t e r i s t i c s p e e d of the motion, g i s a c c e l e r a t i o n due to gravity, and 1 i s a c h a r a c t e r i s t i c length of the body, taken h e r e a s the length m e a s u r e d a t the w a t e r - l i n e . Typical upper l i m i t s of F a r e 0.15 f o r a f a s t s t e a m e r , 0.10 for a f a s t yacht close-hauled. P l a u s i b l e

qualitative a r g u m e n t s have been p r e s e n t e d by Davids on f o r the validity of the reflection model i n the l i m i t of vanishing F r o u d e number. ( I t i s most useful t o think of t h i s l i m i t a s g becoming v e r y l a r g e while c, 1, and g e o m e t r y a r e held fixed. ) It w a s felt, then, t h a t a n expansion of a l l flow quantities a s power s e r i e s i n the s m a l l p a r a m e t e r F might lead t o significant r e s u l t s . When t h i s p e r t u r b a t i o n a n a l y s i s is applied t o the e x a c t equations and the boundary conditions on the f r e e s u r f a c e ,

on the s o l i d s u r f a c e , and i n the distant field, the z e r o - o r d e r s e t of equations and boundary conditions a r e found t o be e x a c t l y those of t h e reflection model. O r d e r F yields a mixed boundary value p r o b l e m f o r L a p l a c e D s equation i n the portion of the lower half-space outside the body. N o r m a l f i r s t- o r d e r velocity on the plane of the undisturbed

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s u r f a c e i s p r e s c r i b e d by the z e r o - o r d e r p r e s s u r e on that s u r f a c e . In the c a s e of sufficiently s l e n d e r s h a p e s and s m a l l angles of sideslip, two simplifications e m e r g e : the slender-body t h e o r y p r o - vides a n m a l y s i s f o r t h e z e r o - o r d e r p r e s s u r e s ; and the f i r s t o r d e r p r o b l e m c a n be solved i n the c r o s s - f l o w plane. T h i s w o r k d e a l s p r i m a r i l y with s o m e g e n e r a l r e s u l t s and some specific solutions of the s e c o n d - o r d e r p r o b l e m f o r s l e n d e r geometry.

11. EQUATIONS AND BOUNDARY CONDITIONS

The governing r e l a t i o n s w i l l be f o r m u l a t e d f i r s t i n t e r m s of a r e c t a n g u l a r C a r t e s i a n c o - o r d i n a t e s y s t e m fixed i n the distant fluid.

The X, Z

-

plane coincides with t h e undisturbed s u r f a c e f a r away, and the Y-axis i s v e r t i c a l l y upward; T i s the t i m e m e a s u r e d f r o m s o m e a r b i t r a r y r e f e r e n c e . The following a s s u m p t i o n s a r e made:

1) The flow i s i n c o m p r e s s i b l e , i r r o t a t i o n a l , and inviscid.

The existence of a velocity potential cf, is thus a s s u r e d , s u c h that t h e components of velocity i n the X, Y, Z-directions a r e

a @ a@ a @

T X T Y ' T Z

respectively.

2 ) The f r e e s u r f a c e i s given by Y = H(X, Z, T), a s i n g l e - valued function over the r e g i o n of the X, Z-plane e x t e r i o r t o the body. This a s s u m p t i o n i s uniformly valid a t sufficiently

s m a l l values of F; it m a y b r e a k down i n the vicinity of the bow f o r l a r g e r

I?,

depending on the g e o m e t r y ( a breaking bow wave). Such c a s e s a r e beyond the scope of t h i s t r e a t - me nt

.

3, The s u r f a c e of the r i g i d body i s given b y E(X,Y,Z,T)=O,

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with the e x t e r i o r E

>

^0.

4) No wave t r a i n s a r e incident, coming f r o m f a r away. In other w o r d s , the only disturbance p r e s e n t is that c a u s e d by the body's motion.

5 ) Velocity i s finite i n the vicinity of a s h a r p t r a i l i n g edge.

This is the well-known Kutta condition of a i r f o i l theory, the only manifestation of v i s c o s i t y c o n s i d e r e d h e r e .

6 ) The Weber number pc2.1/(surface tension) i s v e r y l a r g e , s o t h a t s u r f a c e t e n s i o n effects a r e negligible.

Then t h e following r e l a t i o n s g o v e r n the flow: the f i e l d e q - uation, which e x p r e s s e s continuity:

the boundary conditions e x p r e s s i n g tangency of the flow on t h e solid surface:

a n d on the f r e e s u r f a c e :

the dynamic condition r e q u i r i n g constant p r e s s u r e ( t a k e n z e r o ) on the f r e e s u r f a c e , using B e r n o u l l i ' s equation f o r unsteady flow:

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a d i s t a n t boundary condition f a r below:

S e v e r a l difficulties a r e apparent:

1) N o n l i n e a r i t y of boundary conditions ( 2 - 3 ) a n d ( 2 - 4 ) 2) B o u n d a r y conditions ( 2 - 3 ) a n d ( 2 - 4 ) a r e applied on t h e s u r f a c e Y = H(X, Z, T ) which i s not known a t t h e outset and m u s t a p p e a r a s p a r t of the solution. Indeed, e x i s t e n c e a n d uniqueness p r o o f s a r e a p p a r e n t l y lacking f o r s u c h unknown boundary p r o b l e m s i n m o r e t h a n one dimension.

3 ) A f u r t h e r d i s t a n t boundary condition i s n e c e s s a r y t o s a t i s f y condition 4) above ( n o incident wave t r a i n s ) while s t i l l pe rrnitting p e r s i s t e n t w a v e s t o be g e n e r a t e d by the body's motion. F o r s t e a d y motion i n the - X d i r e c t i o n ,

i s a p p r o p r i a t e .

T h i s c a s e of s t e a d y motion i n the -X d i r e c t i o n i s of s p e c i a l i n t e r e s t , a n d i t is u s e f u l t o i n t r o d u c e a moving C a r t e s i a n f r a m e

( X I , Y, Z, T ) w h e r e X I = X ^fcT. The moving body i s a t t h e o r i g i n of t h e moving f r a m e .

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The p e r t u r b a t i o n potential and f r e e - s u r f a c e elevation a r e

The function r e p r e s e n t i n g the solid s u r f a c e i s independent of t i m e f o r s t e a d y motions:

Under t h i s t r a n s f o r m a t i o n (2-1) t o ( 2 - 6 ) become

V I Q 1

2

= O

in Y Q H l

,

E l ²0 (2-10)

a @ aH

3% ₍ _ax,) ^- _{a x ,} _a* ⁺ a z , a z

⁰ ^on

^{Y = H ~}

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IIL THE FORM OF SOLUTIONS

The p r o b l e m d e s c r i b e d i n the previous section, (2-1) t o (2-6).

involving a f r e e boundary, i s notoriously i n t r a c t a b l e ; not a single e x a c t solution i s available.

The u s u a l approximation, leading t o what m.ay be called s m a l l - d i s t u r b a n c e theory, a s s u m e s vanishingly s m a l l displaceme nt of the f r e e s u r f a c e and vanishingly s m a l l velocity p e r t u r b a t i o n s , s o that the f r e e - s u r f a c e boundary conditions ( 2 - 3 ) and ( 2 - 4 ) a r e l i n e a r i z e d t o

applied on Y = 0. S m a l l - d i s t u r b a n c e t h e o r y c a n be d e r i v e d a s a r a t i o n a l power s e r i e s expansion (Wehausen, Ref.4) i n a s m a l l geo- m e t r i c p a r a m e t e r , e.g. r a t i o of amplitude t o length f o r s u r f a c e waves;

o r t h i c k n e s s r a t i o i n Michell's thin s h i p theory, which t r e a t s a s y m - m e t r i c p l a n a r wing penetrating the f r e e s u r f a c e . The s m a l l - d i s t u r - bance solutions a r e c h a r a c t e r i z e d by two different kinds of t e r m s : (1) a l o c a l disturbance, which d i e s out r a p i d l y with distance f r o m the body, and ( 2 ) a superposition of s u r f a c e waves of the f o r m

a

= A sin(^^ xtwl

T I

s i n ^{( K ~}

z +

^w2^{T )}^exp^{( K :}

+ ¹ 'h

^-y ( 3 - 2 )

which follow a t r a v e l i n g d i s t u r b a n c e ; p r e s u m a b l y s i m i l a r waves would be p r e s e n t i n solutions of the e x a c t equations. The e n e r g y r a d i a t e d by

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the waves i s provided by w o r k done against "wave r e s i s t a n c e 1 ' , the calculation of which is the object of most s m a l l - d i s t u r b a n c e p r o b l e m s ,

In the p r e s e n t a n a l y s i s quite a different approximation i s made. The p a r a m e t e r which i s a s s u m e d to be s m a l l i s the F r o u d e number F = c 2 / g l . It i s m o s t useful and convenient to think of the l i m i t F-0 a s g becoming v e r y l a r g e while c , l

,

and g e o m e t r y a r e held fixed. Then t h e f r e e s u r f a c e i s clamped v e r y tightly to Y=O;

a finite H over a finite r e g i o n of t h e X, Z-plane would r e q u i r e infinite energy. Another w a y t o r e g a r d the l i m i t F-cO i s t o r e c o g - nize c ' / ~ a s the wavelength of the s u r f a c e w a v e s which k e e p up with the d i s t u r b a n c e a t s p e e d c, which becomes much s m a l l e r than I

.

But since the wavelength a p p e a r s in the exponent i n ( 3 - 2 ) , ^wave effects become v e r y s m a l l a t depths below a wavelength. F o r v e r y s m a l l F, the waves a f f e c t only a v e r y thin l a y e r of fluid n e a r the f r e e surface.

The only a p p e a r a n c e of the p a r a m e t e r F i n the governing relations is i n the p r e s s u r e boundary condition (2-41, which c a n be w r i t t e n

F

a p p e a r s a s the r a t i o of typical i n e r t i a l f o r c e s t o the gravitational f o r c e , o r the r a t i o of typical fluid a c c e l e r a t i o n s t o g r a v i t a t i o n a l a c c - eleration. In the l i m i t F-+8 gravity dominates. If t h i s i s t r u e uniformly t h r oughsut the flow, the a p p r o p r i a t e f i r s t approximation i s to neglect the right-hand s i d e of ( 3 - 3 ) , which now b e c o m e s H=O.

Consequently ( 2 - 3 ) b e c o m e s @ = 0 on Y=O a n d t h e fieldequation Y

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(2-1) is valid i n Y =S 0

,

^E^E^0. This i s the reflection-plane model, on which t h e r e w i l l be s o m e f u r t h e r d i s c u s s i o n i n l a t e r sections. A

f u r t h e r approximation could be calculated by using the reflection-plane potential t o evaluate the right-hand side of (3-3), a r r i v i n g a t a f i r s t approximation f o r H; using t h i s H the full s e t of equations be- c o m e s l i n e a r a n d c a n b e s o l v e d f o r a s e c o n d a p p r o x i m a t e

.

^This

i t e r a t i o n p r o c e s s r e p e a t e d w i l l g e n e r a t e a n a s y m p t o t i c expansion of Qi.

In t h i s a n a l y s i s the equivalent and m o r e s y s t e m a t i c p e r t u r - bation p r o c e d u r e i s used. The choice of a n a s y m p t o t i c sequence f o r the expansion i s guided by r e s u l t s of s m a l l - d i s t u r b a n c e theory, expanded for s m a l l F. The s i m p l e s t choice

-

a power s e r i e s i n F

-

is a t f i r s t d i s c o u r a g e d by the frequent a p p e a r a n c e of nonanalytic t e r m s , e.g. e - 2 P F / ~ 3 i n the wave r e s i s t a n c e of a s u b m e r g e d cylinder a s given by Lamb, Art.249 (Ref. 5 ) ; o r F 5/2 s i n

4/F

i n Havelock's r e s u l t s for r e s i s t a n c e i n thin s h i p t h e o r y (Ref. 6,1923). In fact, it quickly b e c o m e s a p p a r e n t in c a r r y i n g out the expansion in powers of F that no wavelike behavior a p p e a r s a t a n y o r d e r and, consequently, no wave r e s i s t a n c e shows up a t all.

The distinction between local d i s t u r b a n c e a n d wave p a t t e r n i s i m p o r t a n t here, and another a s p e c t of i t w i l l be pointed out by consid- e r a t i o n of the fundamental s o u r c e - l i k e solution, f o r a s o u r c e of

s t r e n g t h m a t d e p t h f , a s g i v e n b y H a v e l o c k ( R e f . 7 , 1 9 5 1 ) .

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~ / 2 00

- +

K o m S s e c ' e S K ( ~ - ~ ) C ~ ~ ( K X , cos ~ ) c o s ( K ~ s i n 0) dKde

0 K

-

^K~ s e c Z 8

-

^n/2

t 2 ~ ~ mK 0 ( Y - f s e c 2 'sin(Ko

xl

s e c e ) c o s ( ~ ~ Z s i n 8 s e c 2 9)sec"dB

( 3 - 5 ) w h e r e K~ = g / c 2 ,

R I

Z ₌_X: _t(Ytf)'

+

^Zz

,

^and

The f i r s t i n t e g r a l , r e p r e s e n t i n g the l o c a l disturbance, i s r e a d i l y e x - panded i n p o w e r s of F = l / ~ ~ l :

while the second integral, r e p r e s e n t i n g a s u p e r p o s i t i o n of plane s u r f a c e waves, cannot be placed in this form. The s a m e conclusion holds f o r m o r e c o m p l e x flows r e s u l t i n g f r o m s u p e r p o s i t i o n s of sources and other s i n g u l a r i t i e s d e r i v e d f r o m the s o u r c e b y differentiation and integration: while waves do not a p p e a r in a n expansion i n powers of F, s u c h a n expansion i s a valid r e p r e s e n t a t i o n of the l o c a l disturbance.

It i s i n t e r e s t i n g to s e e w h e r e the waves do a p p e a r . If ( 3 - 4 )

-

w e r e a uniformly valid asymptotic r e p r e s e n t a t i o n of 9

,

^{and wave-}

like t e r m s did not a p p e a r i n (3-41, then wavelike t e r m s , and hence wave r e s i s t a n c e , would have t o be s m a l l e r than a n y power of F;

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11

t h i s conclusion, however, is a t v a r i a n c e with the r e s u l t s of thin ship t h e o r y ( f o r example, Ref. 6) w h e r e t e r m s i n the powers of F do appear. The nonuniformity o c c u r s i n the vicinity of stagnation points on the f r e e s u r f a c e

-

f o r instance, a t the bow and s t e r n of a s h i p

-

w h e r e a c c e l e r a t i o n i s no longer s m a l l c o m p a r e d with t h a t of gravity, and the right-hand side of ( 3 - 3 ) i s no longer negligible. In the e x - pansion ( 3 - 4 ) s u c c e s s i v e t e r m s become l a r g e r and m o r e singular.

By using expanded c o - o r d i n a t e s i t is possible t o make a c l o s e r investigation of the r e g i o n of nonuniformity. If, i n a s t e a d y motion, t h e choice of i n n e r v a r i a b l e s is X = X ~ / F B , y = ~ / ~ l , Z=Z/FI, h =H/FI, q = @ I / ~ ~ c , the f i r s t - o r d e r s e t of equations and boundary conditions f r o m (2-10) t o (2-15) a r e

which a r e the s m a l l - d i s t u r b a n c e equations. The solutions depend s t r o n g l y on the d e t a i l s of the shape of t h e body d e s c r i b e d by the function e n e a r the s i n g u l a r point, but always involve the typical f r e e wave p a t t e r n d a w n s t r e a m , f a r out side the r e g i o n of nonuniformity.

This p i c t u r e of w a v e s being g e n e r a t e d a t t h e s i n g u l a r point and ^.

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12

propagating into the r e s t of t h e solution is i n good a g r e e m e n t with a g e n e r a l r e s u l t of t h i n - s h i p theory, due t o Inui (Ref.8,1962). By con- s i d e r i n g t h e wave p a t t e r n due t o a continuous d i s t r i b u t i o n of s o u r c e s , Inui h a s d e m o n s t r a t e d t h a t f o r s m a l l F the w a v e s a l l originate a t t h e ends of the distribution, and that the s t r e n g t h of the waves i s s t r o n g l y dependent on the d e t a i l s of the distribution n e a r the ends. It should a l s o be pointed out that i n those c a s e s of wave motion t h a t have been c a r r i e d out f o r completely s u b m e r g e d bodies, w h e r e no singular stagnation points occur, ( p a r t i c u l a r l y , the e n t i r e l y g e n e r a l c a s e of Lamb, Art. 25a (Ref.5) ) the wave r e s i s t a n c e is exponentially s m a l l .

The question that m u s t now be c o n s i d e r e d i s the extent of effects of t h e wavelike p a r t of ^je on the t r a n s v e r s e f o r c e distribution on a body. Within s m a l l - d i s t u r b a n c e theory, the only component of f o r c e on a body due t o i t s waves is a drag, since i n f i n i t e s i m a l waves c a n c a r r y away e n e r g y but not momentum. T h i s c o n s i d e r a t i o n does not, however, r u l e out p u r e couples due t o w a v e s ; a pitching moment is p r e s e n t in g e n e r a l and has been t r e a t e d by Lunde (Ref.9). The s i m p l e s t c a s e that might involve yawing moments i s that of a p l a n a r wing of z e r o thickness penetrating the f r e e s u r f a c e . ( T h e thickness c a s e , which c a n be superimposed, i s c a l l e d "thin s h i p theory.") The lifting s u r f a c e i s r e p l a c e d by a distribution of e l e m e n t a r y h o r s e s h o e v o r t i c e s , whose potential i s calculated i n Appendix I, having density proportional t o the wing loading.

The co-ordinate s y s t e m and v a r i a b l e s s e t up f o r s t e a d y motion a r e u s e d

-

^{s e e}( 2 - 7 ) t o (2-15). The lifting wing i s r e p r e s e n t e d by a v o r t e x distribution over the plane Z = 0, of s t r e n g t h y ( X 1 , Y), S O

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the p r e s s u r e difference between the two s i d e s of the wing is pcy.

The potential a s s o c i a t e d with s u c h a v o r t e x s y s t e m i s

w h e r e V is the e l e m e n t a r y v o r t e x potential d e r i v e d i n Appendix I. If w(X,, Y ) is the downwash velocity on the plane Z = 0, defined by

t h e n

0 00

Defining the k e r n e l function

we w r i t e (3-14) a s

which i s in the f o r m of the fundamental i n t e g r a l equation of lifting- s u r f a c e theory, except that the k e r n e l i s m o r e complex in this c a s e . The k e r n e l c a n be w r i t t e n a s the s u m of two p a r t s :

K 1

a r i s i n g f r o m the f i r s t t h r e e t e r m s of V, r e p r e s e n t i n g the l o c a l disturbance, and Kw a r i s i n g f r o m the l a s t t e r m , the wave disturbance. F o r the

p r e s e n t purpose of investigating the magnitude of the wave effects, the

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i n t e r e s t i s i n Kw:

and i n the a s s o c i a t e d component of the downwash

We investigate the behavior of Kw f o r l a r g e values of

K~ = g/c2 by t h e method of s t a t i o n a r y phase. Since c o s [ ~ ~ ( X ~ - i ) s e c 81 is a rapidly oscillating function, the p r i n c i p a l contribution t o the i n - t e g r a l (3-17) c o m e s f r o m the vicinity of points f o r which the phase K ~ ( X ~ - 5 ) s e c ,0 i s stationary. The only s t a t i o n a r y point i n the r a n g e of integration i s 0 = 0; however, the point 8 = 0 i s a z e r o f o r the r e s t of the integrand, s o the o r d i n a r y method r e q u i r e s modification.

The r e q u i r e d t r e a t m e n t i s c a r r i e d out i n Appendix 11. In t h i s c a s e we have f(O) = Ko(Xl-6)r f l ' ( 0 ) = Ko(X1-$), fU'(O) = O , @ ' ( O ) = 0,

@ " ( 0 ) = 2eKo(Ytrl) and (3-17) becomes approximately

w h e r e the

-

s i g n i s the s i g n of ( X I -

6 ) .

This e x p r e s s i o n h a s a nonintegrable s i n g u l a r i t y a t

X I

=

5;

however, the application of s t a t - i o n a r y phase i s valid only i f i c o ( X 1 - t ) i s l a r g e . At X I = & , (3-17) gives

K (y*?)sec2

e

^{d e}

Kw(XlpYi Xlrq) = ~ K O sin2 8 s e c 5 8 e O

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F o r l a r g e K~ this i n t e g r a l c a n b e evaluated a p p r o x i m a t e l y by the method of s t e e p e s t descent, done i n a fashion quite analogous t o the method of Appendix 11. Again the p r i n c i p a l contribution i s f r o m value s of 8 n e a r the s t a t i o n a r y point 8 = 0; the r e s u l t is

s o the k e r n e l i s finite and integrable.

A s s u m e [ y ( X 1 , Y ) I < M f o r O S X p < , ( and y ( X 1 , Y ) = O outside that region. Then f r o m (3-18)

w h e r e f(X1-e) is finite and integrable. S o

and using F

=

KO,(

,

The contribution t o the downwash f r o m the wave p a r t of the k e r n e l i n (3-16) is exponentially s m a l l a t any finite depth Y below the f r e e surface. Any i n t e g r a t e d effect of t h i s downwash f r o m the f r e e s u r f a c e downward i s of o r d e r

F3Iz.

The o s c i l l a t o r y behavior of f(Xl-

5 ) ,

displayed i n ( 3 -19), indicates that i n t e g r a t e d f o r c e s and moments connected with the wave p a r t sf the k e r n e l a r e e v e n s m a l l e r than F 3/~

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This r e s u l t of s m a l l - d i s t r i b u t i o n t h e o r y i s u s e d t o justify neglecting the wave t e r m i n the k e r n e l while calculating f o r c e s and moments of o r d e r unity and o r d e r F

.

IV. PERTURBATION EQUATIONS

T o obtain the p e r t u r b a t i o n expansion, the potential and the f r e e - s u r f a c e elevation a r e expanded i n p o w e r - s e r i e s form:

Throughout t h i s p a p e r the notation of s u p e r s c r i p t s i n p a r e n t h e s e s i s used t o denote the coefficients of powers of

F

i n s i m i l a r s e r i e s expansions f o r v a r i o u s quantities.

When (4-1) and ( 4 - 2 ) a r e substituted into (2-1) t o ( 2 - 5 ) and the coefficients of the v a r i o u s powers of

F

a r e collected, t h e r e r e s u l t s e t s of equations and boundary conditions f o r the and

The conditions on t h e s u r f a c e

Y

= H m u s t be t r a n s f e r r e d t o the Y = 0 plane, by expanding Cf, i n t h e f o r m of (4-l), and i t s derivatives, i n

Taylor s e r i e s about

Y =

0:

t O ( F 3 ) e t c ( 4 - 3 )

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4.1 Z e r o O r d e r

Together with a f u r t h e r distant boundary condition on

vdO)

n e a r Y = 0, and the Kutta condition on t r a i l i n g edges, t h e s e relations d e t e r m i n p @(O'. They a r e p r e c i s e l y the r e l a t i o n s governing the

motion i n a n unbounded fluid, of a body ( c a l l e d the Isreflected body") whose s u r f a c e i s given by E(X,Y, Z,T) = 0,

Y<

0, and i t s r e f l e c t i o n i n the Y = 0 plane, c o n s t r a i n e d t o move on that plane. S o the reflection-plane model of Davidson e m e r g e s , a s expected, a s the l i m i t f o r vanishingly s m a l l Fr oude number.

Heaving, rolling and pitching motions produce changes i n the g e o m e t r y of the r e f l e c t e d body, while f o r t r a n s l a t i o n s i n the

Y

= 0 plane ( f o r w a r d motion and s i d e s l i p ) and yawing motions the body's shape i s t i m e - i n v a r i a n t . The z e r o - o r d e r p r o b l e m with fixed g e o m e t r y is a f a m i l i a r and fundamental p r o b l e m i n a e r o d y n a m i c s , and much attention has been given t o i t s solution. The z e r o - o r d e r p r o b l e m with v a r i a b l e g e o m e t r y i s not s o w e l l developed; h a v e v e r , m o s t of t h e techniques developed f o r unsteady motion of bodies having fixed geo- m e t r y a r e suitable for extension t o t i m e - v a r i a b l e geometry.

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18

Equations ( 4 - 4 ) t o ( 4 - 8 ) with a n additional d i s t a n t boundary condition have a unique solution i f the potential i s a s s u m e d single- valued. That solution, however, i n g e n e r a l involves infinite velocities around the t r a i l i n g edges, i f a n y a r e p r e s e n t . Satisfaction of the Kutta condition r e q u i r e s the p r e s e n c e i n the fluid of f r e e v o r t e x s h e e t s , s u r - f a c e s a c r o s s which

m6)

i s discontinuous

-

the f a m i l i a r t r a i l i n g

v o r t e x s h e e t s of wing theory, e s s e n t i a l t o the development of lift.

The z e r o - o r d e r f o r c e s and moments a r e calculated by i n - t e g r a t i n g p r e s s u r e s d e r i v e d f r o m the z e r o - o r d e r approximation t o Bernoulli's equation ( 2-4):

T h e t e r m -gY i s omitted, since it contributes only the uninteresting hydrostatic f o r c e a n d mome nts.

4.2 F i r s t O r d e r

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H(') c a n be e l i m i n a t e d between (4-12) and (4-13) t o yield a single condition on i n t e r m s of 53'0). F r o m ( 4 - 9 ) i t i s a p p a r e n t t h a t the quantity H(') i s d i r e c t l y r e l a t e d to P'o): f r e e - s u r f a c e height, t o o r d e r F, i s p r e c i s e l y the head of fluid supported by the z e r o - o r d e r p r e s s u r e o c c u r r i n g on the r e f l e c t i o n plane.

S o a(') i s a harmonic function of the s p a c e v a r i a b l e s , with n o r m a l d e r i v a t i v e s given on the solid boundary and on the plane

Y

₌0 and d e r i v a t i v e s v a n i s h i ng f a r ahead. Again the continuous solution f a i l s i n g e n e r a l t o s a t i s f y the Kutta condition, and v o r t e x s h e e t s must be p r e s e n t i n a lifting problem. Now i n the e x a c t p r o b l e m t h e r e i s

only one v o r t e x s h e e t f r o m e a c h trailing edge; its position should r e s p o n d t o a l l o r d e r s of p e r t u r b a t i o n velocities. Strictly, it should be t r e a t e d a s another f r e e s u r f a c e which i s slightly p e r t u r b e d f r o m i t s s t r e n g t h and position i n the z e r o - o r d e r solution. However, the avail- a b l e wing t h e o r i e s f o r solution of t h e z e r o - o r d e r p r o b l e m neglect even the inductions of the bound v o r t i c e s i n locating t h e t r a i l i n g v o r t e x sheet;

c o m p a r e d t o these the velocity contributions f r o m 53") a r e O(F).

It i s consistent with the a c c u r a c y of wing-theory solutions f o r

t o allow t h e v o r t e x s h e e t for a l l o r d e r s t o coincide with t h e z e r o - o r d e r sheet.

In t h e m o s t g e n e r a l c a s e this f i r s t - o r d e r p r o b l e m does not a d m i t of e a s i l y computed solutions, on account of the complex shape of the boundary on which the n o r m a l d e r i v a t i v e s a r e specified; however, f o r a c e r t a i n c l a s s of sufficiently s l e n d e r s h a p e s the p r o b l e m becomes a two-dimensional one and c a n be approached by complex analysis.

When

a(')

h a s b e e n found, f i r s t - o r d e r f o r c e s a r e computed f r o m the

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p r e s s u r e s given by the f i r s t - o r d e r Bernoulli's equation:

4 . 3 Second and Higher O r d e r s

The f i r s t o r d e r p r o b l e m i s typical of a l l the h i g h e r - o r d e r ones.

All a r e governed by Laplace1s equation

V2 di)

⁼⁰ i n the lower half- s p a c e outside the body, with the s a m e condition

V E * V @ ( ~ )

= 0 on E = O and

vdi)

vanishing f a r ahead. The n o r m a l velocity over the Y = 0 plane i s p r e s c r i b e d by a functional of the $ j ) and ~ ( j ) , j < i ; f o r example

1 (H(1)) 2

-

^$I) ^H(l).+

@('k(I(')

⁺^@(o)^{H(l) H(l)}

- z

_{Y Y Y} _{Y Y}

z z

Z Y Z w h e r e

on the plane Y = 0

.

The methods developed for the f i r s t - o r d e r p r o b l e m a r e equally applicable t o higher o r d e r s , since the problems f a l l into the s a m e f o r m ; the computations a r e a p p a r e n t l y m o r e c o m - plicated.

V, SUBMERGED CYLINDER

The p e r t u r b a t i o n equations i n Section IV have b e e n e s p e c i a l l y developed t o d e a l with t r a n s v e r s e f o r c e s . It would be r e a s s u r i n g a t this point t o find t h e m i n a g r e e m e n t with the r e s u l t s of s m a l l - d i s t u r - bance t h e o r y under c i r c u m s t a n c e s w h e r e both a r e applicable.

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One c a s e i n which t r a n s v e r s e f o r c e s have b e e n calculated by s m a l l - d i s t u r b a n c e t h e o r y i s t h a t of the lift on a s u b m e r g e d cylinder n o r m a l t o the s t r e a m ( o r m o r e a c c u r a t e l y , a s u b m e r g e d line doublet pointed up-

s t r e a m ) t r e a t e d by Havelock (Ref. 10, 1928 and Ref. 11, 1936). The r e s u l t i s valid f o r a n y s p e e d provided the depth of submergence is sufficiently g r e a t , c o m p a r e d t o the radius.

Using the p r e s e n t method i t i s possible t o compute the lift on a cylinder a t any submergence, t o t a l o r p a r t i a l , provided the s p e e d is low enough. T o do s o would only be a n uninteresting e x e r c i s e ; f o r p u r p o s e s of c o m p a r i s o n we want t o calculate the combination of deep

s u b m e r g e n c e and low speed.

X

.4-.

P a r t i a l submergence, f

<

a _c

IC---

-

Deep submergence, f > > a Fig. 5.1

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In t h i s c a s e the cylinder of r a d i u s a i s r e p r e s e n t e d by a doublet of s t r e n g t h M

=

c a 2 . The co-ordinates used i n t h i s s e c t i o n only a r e shown i n Fig. 5.1. F i s r e f e r r e d t o the t y p i c a l dimension of the body, a:

F

F i r s t , the z e r o - o r d e r f o r c e i s calculated by the reflection- plane model. In this approximation a n i m a g e doublet of s t r e n g t h c a 2 i s located a t ₍0,

+

f ) and i t ^isn e c e s s a r y t o calculate the f o r c e on one doublet due t o the other. The f o r c e on a doublet i s given by Milne- Thompson (Ref. 12) as:

w h e r e OL is the inclination of the doublet's a x i s ( h e r e z e r o ) and f ( z ) i s the velocity with the doublet a t z removed. In the p r e s e n t

application t h i s b e c o m e s

The velocity g r a d i e n t au/ay i s calculated f r o m the potential of a doublet, w i t h the r e s u l t

The fir s t o r d e r is calculated f r o m the p e r t u r b a t i o n equations (4-10) t o (4-14), using f o r the z e r o - o r d e r potential

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F r o m (4-13) i t is found

t h e n f r o m (4-12)

,

This condition, p r e s c r i b i n g the n o r m a l velocity over y = 0, is s a t i s

-

fied by a s o u r c e distribution along the x - a x i s with s t r e n g t h m ( x )

=

2$) (x,

1 ,

^{o r}

Fig. 5.2 F i r s t - o r d e r P r o b l e m

Now i t is n e c e s s a r y to find t h e velocity g r a d i e n t au(l)/ay a t ( 0 , - f ) due t o the s o u r c e distribution rn(x). The contribution t o

&/ay

f r o m the p a r t of m between x and x t dx is s i n 8 c o s 8

nRz

^mdx

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Consequently a u (1)

/ay

i s found by integrating ( 5 - 8 ) f r o m -a t o

$ a ; or, using x = f t a n 8 ,

~ / 2

The integration i s done with the help of Dwight (Ref.l3), No.858.514, w i t h the r e s u l t

Putting t h i s i n (5-2), we find

Combining t h e r e s u l t s ( 5 - 3 ) and (5-111,

Havelock's r e s u l t is given i n the f o r m (Ref.10)

w h e r e K g/c2 and l i i s the logarithmic integral. Using the identity between the l o g a r i t h m i c i n t e g r a l and the exponential integral, and using the asymptotic expansion of the l a t t e r given b y Jahnke and E m d e (Ref. 14):

(5-13) c a n be r e p r e s e n t e d f o r l a r g e ~f by

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2 5

which i s gratifyingly identical with the p r e s e n t r e s u l t (5-12).

No sufficiently simple solution involving a t r a i l i n g v o r t e x s h e e t is available f o r comparison.

VI. SLENDER BODY APPROXIMATION

In the s l e n d e r - b o d y t h e o r y ( s e e , f o r example, A d a m s and S e a r s (Ref.l5,1953) and S a c k s (Ref.16, 1954) we have a n analytic a p p r o x i m a t e solution f o r the z e r o - o r d e r p r o b l e m f o r @(o). f o r a g e n e r a l c l a s s of bodies, denoted a s "slender1', and motions restricks3 t o slow m a n e u v e r s and s m a l l angles of attack. T o fit i n t o the o r d i n a r y s l e n d e r -body theory, a shape m u s t have i t s l a t e r a l dimensions s m a l l c o m p a r e d with d i s t a n c e f r o m the nose, s l o p e s of i t s s u r f a c e s m a l l c o m p a r e d with unity, and c u r v a t u r e s i n the s t r e a m w i s e d i r e c t i o n s m a l l c o m p a r e d with the r e c i p r o c a l of distance f r o m the nose. Under t h e s e c i r c u m s t a n c e s , conditions change s o slowly along the length that i n e a c h lscroseflow plane1# n o r m a l t o the body the flow i s e s s e n t i a l l y the two-dimensional flow p a s t a cylinder having the s a m e c r o s s - s e c t i o n a s the body.

F o r a f u r t h e r r e s t r i c t e d s u b c l a s s of s l e n d e r bodies the s a m e r e a s o n i n g l e a d s t o a t r e a t m e n t of the s e c o n d - o r d e r p r o b l e m i n the c r o s s flow plane. The r e s t r i c t i o n i s t h a t the z e r o - o r d e r p r e s s u r e distribution P(O)(X,O, Z , T ) be a l s o effectively " s l e n d e r " ; that is.

1) the l a t e r a l extent of significant p r e s s u r e d i s t u r b a n c e s m u s t be s m a l l c o m p a r e d with d i s t a n c e f r o m the nose, and sf the s a m e o r d e r a s the l a t e r a l dimensions of the body; and

2) p r e s s u r e d i s t u r b a n c e s m u s t change slowly along the length.

If t h e s e conditions obtain, then the boundary condition (4-13) changes

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slowly along the length; appealing to the continuous dependence of on i t s boundary conditions, we conclude that the s e c o n d - o r d e r flow changes slowly along the length and s o i s e s s e n t i a l l y the two- dimensional flow i n the c r o s s - f l o w plane.

The g e o m e t r i c a l r e s t r i c t i o n s that w i l l g u a r a n t e e s a t i s f a c t i o n of r e s t r i c t i o n s 1) and 2) on p r e s s u r e a r e not i m m e d i a t e l y c l e a r , and t h e i r investigation i s postponed until l a t e r in the paper, when s o m e a n a l y s i s relating t o s l e n d e r configurations has been c a r r i e d out.

VII. STEADY SIDESLIP

-

BODY-FIXED COORDINATES

F o r the s p e c i a l c a s e of s t e a d y f o r w a r d motion and sideslip, two f a c t o r s r e s u l t i n a simplified treatment:

1) G e o m e t r y of the r e f l e c t e d body is fixed.

2 ) The f r e e - s u r f a c e boundary conditions a s s u m e s i m p l e forms i n body-fixed co-ordinates.

A r e c t a n g u l a r C a r t e s i a n c o - o r d i n a t e s y s t e m (x8y8z,t) fixed i n the body and t r a n s l a t i n g w i t h r e s p e c t t o the (X,Y,Z,T) s y s t e m i s introduced (Fig. 7J). At T = t = 0, the two s y s t e m s a r e co-incident.

The origin of the (x,y,z,t) s y s t e m i s chosen to be t h e f o r e m o s t point of the i n t e r s e c t i o n of the body s u r f a c e with Y = 0 ; i.e., the nose of the r e f l e c t e d body.

The x - a x i s i s c h o s e n m o r e o r l e s s along the length of the body.

In c a s e s w h e r e the body h a s a v e r t i c a l plane of s y m m e t r y , the x - a x i s w i l l always be taken i n t h a t plane. The origin has the s t e a d y velocities

- c c o s & T C sins i n the X

-

^and ^Z

-

d i r e c t i o n s respectively. Then the t r a n s f o r m a t i o n between moving and s t a t i o n a r y f r a m e s is

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Z

Fig. 7.1 Moving Co-ordinate S y s t e m

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The following functions a r e introduced f o r the p e r t u r b a t i o n potential, f r e e - s u r f a c e elevation, and p r e s s u r e :

under t h e t r a n s f o r m a t i o n (7-1). By t h i s definition

cp

is the " p e r t u r - bation potential1'; i t s d e r i v a t i v e s a r e the " p e r t u r b a t i o n velocities"

-

the fluid velocity components at a point (x,y,z,t), minus the f r e e - s t r e a m components. ~p and h a r e a s s u m e d f o r the p r e s e n t t o bei functions of t; w h e r e a s t h e function r e p r e s e n t i n g the s o l i d s u r f a c e

is independent of t for the motions considered.

The z e r o - o r d e r r e l a t i o n s ( 4 - 4 ) t o ( 4 - 8 ) now become

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$ ) - o

^{a s} ^{x + - m} ^{o r} ^{y - - m} ( 7 - 9 )

(0 )

'C7q

^finite, t r a i l i n g edge (7-10) It is t o be o b s e r v e d by differentiating ( 7 - 6 )

-

( 7 - 9 ) w i t h r e s p e c t to

(0) *

time, t h a t t h e p r o b l e m f o r

cgt .

i s homogeneous, and h a s only the solution

qt

( 0 ' = 0 i n y < O , e 3 0 . Thus t h e z e r o - o r d e r solution

8)

i s independent of t i m e , and s o the v a r i a b l e t i s omitted i n the following.

Now

f o r s l e n d e r g e o m e t r y 'p") is approximated by

(0) (0

9

(x,y,z)

= cp

^{( 2}

+

i y ; x )

= <P

(0) ( s i x ) (7-11)

w h e r e the complex v a r i a b l e s

-

^zⁱ

⁺

i y h a s been introduced. @(o) is a r e a l function of s, with x appearing a s a p a r a m e t e r labeling the c r o s s - f l o w planes. In the approximation of slender-body theory, ( 7 - 6 ) becomes

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which i s s a t i s f i e d by requiring t o be the r e a l p a r t of a n analytic

(0

1

⁽⁰

function of s, called the complex potential f ( s ; x ) =

$I

( s ; x )

+

i+(')(s;x), w h e r e

+('I

i s a l s o real.

The p r o b l e m f o r cp(0) then r e d u c e s t o finding the harmonic r e a l function @") i n e a c h c r o s s - f l o w plane, subject t o the conditions

*

(0 ⁼ - n c s i n 0 on the body c r o s s - s e c t i o n @(x)(7-14)

8n z

n = n 9 i n being the outward n o r m a l t o C(x). This i s a well-posed

z Y

mixed boundary value p r o b l e m for Laplace's equation i n a s i m p l y connected region of the plane.

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3 1

It w i l l be u s e f u l a t t i m e s t o c o n s i d e r the p r o b l e m i n the whole s-plane e x t e r i o r t o the r e f l e c t e d body c r o s s - s e c t i o n . Noting the

Cauchy-Riernann conditions e x p r e s s i n g the analyticity of f(') :

we o b s e r v e t h a t (7-17) with (7-13) r e q u i r e s t h a t vanish along the r e a l a x i s ; hence +(" = constant on z = 0 and we t a k e t h a t con-

stant to be zero. Then f(') is p u r e l y r e a l on the r e a l a x i s and m a y

( 0 )

-

-(o)

be continued into the upper half-plane by f ( s ) = f ( s )

.

The region i s now doubly connected; however the c i r c u l a t i o n about the i n t e r i o r boundary v a n i s h e s by v i r t u e of (7-13) and the s y m m e t r y of (7-14) about the r e a l axis.

The f o r m of Bernoulli's equation ( 4 - 9 ) a p p r o p r i a t e t o s l e r d e r body theory, i n the moving c o - o r d i n a t e s , is

w h e r e p(0)= p'O)(s; x ) is now a function of s , and c o s

a ,

s i n a have b e e n approximated by 1, CY respectively. On the plane y = 0, ('7-13) r e d u c e s 47-18] t o

The following r e l a t i o n s a r e found by e x p r e s s i n g (4-10)-(4-14) i n t e r m s of the new variable:

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The h") of ( 7 - 2 2 ) c a n be substituted i n t o (7-21) t o give d i r

-

e c t l y i n t e r m s of 0(0) (using (7-12) t o r e p l a c e w i t h

-%,)

⁽⁰⁾

. .

~ Y Y

Since

i(O'

w a s found t o be independent of t i m e , we m a y differentiate (7-19)-(7-23) with r e s p e c t t o t i m e and a r r i v e a t the s a m e homogemous p r o b l e m a s ( 7 - 6 ' ) - ( 7 - 9 ' ) f o r 11

rpt

; hence it i s concluded that

q

^(I) i s likewise i n d e p e ~ d e n t of time.

If the c r o s s flow a n a l y s i s c a n be applied t o the s e c o n d - o r d e r problem, a s d i s c u s s e d i n Section VI, then i t w i l l be useful t o introduce a notation s i m i l a r t o t h a t u s e d i n the z e r o - o r d e r a n a l y s i s . We define

i n analogy t o (7-11), while (7-19) r e q u i r e s that $(') be t h e r e a l p a r t of a n analytic function f (1) ( s ; x)

.

The boundary conditions on the harmonic $ ( i n the whole

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s-plane a r e

*

_{a n}(1) ⁼⁰ on the r e f l e c t e d body c r o s s - s e c t i o n C(x)

@(I)

-

⁰ f a r away (7-28)

H e r e the l i n e - s o u r c e d i s t r i b u t i o n m ( z ; x) h a s been introduced t o s a t i s f y the n o r m a l velocity r e q u i r e m e n t on the z-axis. The s t r e n g t h of rn is m i n u s twice the n o r m a l velocity @ a s given by (7-24)

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The f i r s t - o r d e r Bernoulli equation (4-15) e x p r e s s e d i n the new v a r - iables, and simplified c o n s i s t e n t with slender-body theory, is

VIII. SYMMETRY CONSIDERATIONS

The calculation of the p r e s s u r e s p(O) and p(') i s f u r t h e r simplified, i n f a c t t o t h e point w h e r e s o m e analytic r e s u l t s c a n be obtained f o r specific s h a p e s , if the body under c o n s i d e r a t i o n i s s y m -

m e t r i c about the z = 0 plane; t h i s m e a n s the r e f l e c t e d body has two planes of s y m m e t r y . Then any t e r m i n the p r e s s u r e equations which i s a n even function of z does not contribute t o the s i d e f o r c e ; a l s o the e v e n p a r t of the s o u r c e s t r e n g t h m ( z ; x ) w i l l produce the even p a r t of (P (1

,

while the' odd p a r t of m accounts f o r the odd p a r t of

cp

(1

.

^Bya n "even function of z" is meant a function f ( z ) o r f ( s )

-

with the p r o p e r t y

o r

-

^f

(-s)

=

-

^f^{( - z}t i y )

= -

^f^{( z}t i y ) =

-

^{f ( s )}

w h e r e a s f o r a n "odd function of z P 1 , g ( z ) o r g ( s ) ,

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Now consider the s y m m e t r y p r o p e r t i e s of the v a r i o u s t e r m s of the z e r o - o r d e r Bernoulli equation:

It i s c l e a r t h a t q ~ ! ) h a s the s a m e value a t c o r r e s p o n d i n g points (-z -i t i y ) on the two s i d e s , s o the second t e r m of (7-18) i s even i n m. Immediately, t h e n , (q~!))' m u s t a l s o be even. F u r t h e r m o r e

is odd, s o

(qy

(0) ) is even. S o the only t e r m that contributes t o

'PY

z e r o - o r d e r side f o r c e i s

Now the s o u r c e s t r e n g t h distribution e n t e r i n g into the s e c o n d - o r d e r p r o b l e m i s d e t e r m i n e d by (7-21). F o r this purpose h") (7-22) i s broken into odd and e v e n p a r t s :

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h(li even =

-

c 2 q!)(x. 0.z) t

i [ ( ' I qz

(x,O,z) I . > ( , - 3 ) and the odd and e v e n s o u r c e distributions a r e d e r i v e d f r o m them:

m _even( z ; x ) = - 2 ( 8 - 5 )

y=o

Now the f i r s t - o r d e r B e r n o u l l i equation ( 7 - 2 9 ) i s investigated,

(1) (1) (1)

w i t h

q

₌

qeven

%

qodd.

We s e p a r a t e out the odd p r e s s u r e t e r m s :

which a r e the only ones that contribute t o f i r s t - o r d e r s i d e f o r c e .

IX. SOLUTIONS BY CONFORMAL MAPPING.

Solutions of the two potential p r o b l e m s with boundary conditions (7--13)-(7-115) and ( 7 - 2 6 ) - ( 7 - 2 8 ) m a y be obtained by c o n f o r m a l t r a n s f o r m a t i o n of t h e s -plane into another complex plane i n which the boundaries a s s u m e s h a p e s s u i t e d t o the n e c e s s a r y computations. Since w e a r e i n t e r e s t e d i n and i t s d e r i v a t i v e s only on the i n t e r i o r

--

boundary, it s e e m s n a t u r a l t o c o n s i d e r a t r a n s f o r m a t i o n that m a p s the i n t e r i o r boundary and the r e a l a x i s , w h e r e the s o u r c e s a r e located, onto a single line ( s a y the r e a l a x i s ) i n the mapped plane. The t r a n s - f o r m a t i o n of this type that l e a v e s the plane unchanged a t infinity w i l l be chosen. The mapped plane i s c a l l e d the cr-plane (cr =

5

^$iq)* and the t r a n s f o r m a t i o n i s r e p r e s e n t e d by

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with the i n v e r s e

The complex potential i s the s a m e a t corresponding points i n

N

s - and u-planes; s o a potential f ( u ; x ) = $(o; x ) $ i +(u; x ) is introduced i n the u-plane, defined by

N

If the mapping (9-1)-(9-2) i s conformal then f i s a n analytic function of cr, and s o @

-

i s harmonic.

It i s useful to c o n s i d e r a slightly different z e r o - o r d e r p r o b k m f r o m the one posed in (7-13)-(7-15). R a t h e r than d e a l with the r i g i d boundary moving i n the negative z - d i r e c t i o n with speed

a

c, i t i s p r e f e r r e d to solve the altogether equivalent p r o b l e m of a fixed

boundary i n a s t r e a m which f a r away has velocity

a

e i n the positive

(0) (0)

z-direction. C a l l t h e potential i n the l a t t e r p r o b l e m f k ' = ^$i+l

.

Its boundary conditions a r e

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Fig. 9.1. Equivalent z e r o - o r d e r problems.

-

The potentials f o r the two p r o b l e m s a r e r e l a t e d by

Now the p r o b l e m f o r i s mapped into the w-plane, w h e r e the solution i s trivial:

Fig. 9.2. Mapping of z e r o - o r d e r problem.

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and

L e t s o = zo -t iyo r e p r e s e n t a value of s o n t h e i n t e r i o r boundary, w h e r e $(o) h a s been t a k e n equal t o zero. Then

and, by (8-l) ,

The f o r c e on a d i f f e r e n t i a l length dx of the body i s obtained b y a n integration:

dS

f i d L =

4

dx Podd ( - d y $ i d z ) = - d x 2 i p o d d ( ~ 0 ; x)d '0 (9-13)

C ( x ) C(x)

Fig. 9 . 3 F o r c e r e s o l u t i o n

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H e r e d L i s the f o r c e along the y-axis due t o

Podd' which inte- g r a t e s to z e r o by s y m m e t r y about the r e a l axis. The f r a c t i o n 2 1 i s t o make dS r e p r e s e n t the s i d e f o r c e on the a c t u a l body, that is, on the lower half of the r e f l e c t e d body. Using (9-12),

Alternatively, ~ s ( o ) c a n be obtained f r o m the v i r t u a l mome ntum c o n s i d e r a t i o n s of o r d i n a r y s l e n d e r -body theory, o r f r o m integrations i n a complex plane other than the a-plane.

Next the c o n f o r m a l mapping i s used t o investigate the d i s t - ribution of p (o) over the s u r f a c e y = 0

-

t o s e e whether i n g e n e r a l this distribution i s effectively l t s l e n d e r n . The i m a g i n a r y p a r t of f(') h a s been taken e q u a l t o z e r o here, so, f r o m (9-10) and (9-1)

(0

1

Differentiating and substituting into (7.-18) we find that p (x,O,z) h a s the f o r m of a power s e r i e s i n i n v e r s e p o w e r s of z, s t a r t i n g

The p r e s s u r e d i s t u r b a n c e d i e s off r a p i d l y w i t h - p c 2 a

--& .

with i n c r e a s i n g z; and s o we m a y expect that the l a t e r a l extent of the major p r e s s u r e d i s t u r b a n c e i s s m a l l , of the o r d e r of dAl/dx.

The mapping t o the u-plane w a s e s p e c i a l l y c h o s e n t o s i m - plify the f i r s t - o r d e r p r oblem. Under the c o n f o r m a l t r a n s f o r m a t i o n (9-2), the s o u r c e d i s t r i b u t i o n m ( z ; x ) m a p s i n t o a s o u r c e d i s t r i - bution

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r ( 5 ; x ) = m [ g ( g ; x ) ;

XI

g V ( 5 ; x ) along the 6-axis (shown i n Appendix 111).

In the rr-plane now t h e r e is the d i s t r i b u t i o n

Fig. 9.4. Mapping of second- o r d e r problem.

.

⁽ ^; ^x) of s o u r c e s along the %-axis with

1 5)

>,

5

^{l ,} a n d the potential ( need b e calculated only on the 5-axis with (

P o i s s o n ' s i n t e g r a l r e d u c e s t o a one-dimensional i n t e g r a t i o n over

gt

outside ( -

cl ^1:

w h e r e the l a t t e r f o r m h a s been obtained by substituting

- S g

^{f o r}

5'

i n the f i r s t i n t e g r a l of (9-17). When y i s e x p r e s s e d a s p = y e v e n t podd, t h e r e r e s u l t s

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H e r e Podd and I*even a r e given by (9-16) applied t o m odd and m even' equations ( 8 - 4 ) and ( 8 - 5 ) respectively.

Since the i n t e g r a l s containing l o g a r i t h m s a r e h a r d t o dealwith, i t i s useful t o obtain a l g e b r a i c f o r m s by differentiating (9-19) and (9-20) with r e s p e c t t o

c:

The r e s u l t s of (9-21) and ( 9 - 2 2 ) c a n be i n t e g r a t e d t o find

5 )

_(S;x)

e v e n

"(1

and qodd(5; X ) ; the constants of integration a r e i m m a t e r i a l constant potentials.

+(I) on the boundary is obtained by the mapping (9-1) :

s o that the f i r s t - o r d e r s i d e f o r c e i s obtained through the f i r s t - o r d e r B e r n o u l l i equation ( 8 - 6 ) i n t e g r a t e d by (9-13) :

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- - -

ⁱ ⁽¹ ⁱ

f' elX

- ² $

(@odd)xdso

- ²

^{" C}

$ ^(+('I

^{e v e n}^z^dso

(3x1 C(x)

(o\$(') )

]

^{d s o}

. '

^@y e v e n y

C ( x )

Since ^@ w a s calculated only on the boundary, it m a y a p p e a r that the z - and y- d e r i v a t i v e s i n ( 6 - 2 4 ) a r e not d e t e r m i n e d ; however knowledge of @ on the boundary plus the vanishing t h e r e of

a@(')/an ( 7 - 2 6 ) suffice t o d e t e r m i n e )@! and

X. S L E N D E R WING

OF

Z E R O THICKNESS

The f i r s t specific g e o m e t r y c o n s i d e r e d i s a f l a t plate wing of a r b i t r a r y s l e n d e r plan f o r m i s given by y = - a ( x )

.

The r e f l e c t e d body i s the f l a t plate wing with s p a n 2a(x)

.

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The i n t e r i o r boundary C ( x ) i n the c r o s s - f l o w plane i s t h e cut f r o m -ia(x) t o +ia(x). Under the

* t r a n s f o r m a t i o n

with the i n v e r s e

the boundary C ( x ) m a p s into the c u t f r o m - a ( x ) t o + a ( x ) i n the cr-plane, the z - a x i s m a p s into the 5-axis w i t h

I GI>

a , and the plane is unchanged at infinity. The s i g n of the s q u a r e root i s chosen so t h a t or is i n the s a m e quadrant a s s

.

The complex potential i s , by ( 9 -lo),

and on the boundary, w h e r e s o = iy, the r e a l potential is, by ( 9 - l l ) ,

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The z e r o - o r d e r side f o r c e distribution i s given b y (9-14) a a a

--

^x ⁱ ^idy

dx 2 dso =

- -

^pc2a aaxe 2

2

- a

(10-5) T h i s i s the f a m i l i a r r e s u l t of the J o n e s s l e n d e r wing t h e o r y (Ref. 17, 1946). The s e c t i o n of m a x i m u m s p a n is t r e a t e d a s the t r a i l i n g edge, and x = I i s i t s location.

The next s t e p i s t o calculate h(') f r o m ( 8 - 2 ) and (8-3).

The r e a l potential on the s u r f a c e y = 0 is, f r o m (10-3), (with

-

9 ⁼s i g n z )

Consequently,

a 2

h(') _even(x,z) =

-

₂ ¹z 2 e a 2 ^a

T h e s e e x p r e s s i o n s a r e u s e d in ( 8 - 4 ) and ( 8 - 5 ) t o d e t e r m i n e the s o u r c e distributions i n the p h y s i c a l plane:

-

m = 2 1 c d a ~ even

"Slendernesss' of t h i s s o u r c e distribution, a s r e q u i r e d f o r a c r o s s f l o w a n a l y s i s of the f i r s t - o r d e r problem, demands t h a t a

XX not change

v 2

t o o r a p i d l y i n x: axxx

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