The wave field scattered by a vertical cylinder in a
narrow wave tank
p. Mclver
Department of Mathematical Sciences, Loughborough Umversity of Technology, Loughborough, Leicestershire, UK, LEll 3TU
(Received 5 March 1992; accepted 9 October 1992)
When a plane wave is incident on a fixed vertical cylinder standing in a narrow wave tank, the scattered wave field will be influenced by reflections f r o m the side walls. This situation is investigated using an approximate solution for scattering by a cyhnder of arbitrary cross-section derived under the fundamental assumptions that the cylinder diameter is much less than the tank width and the wavelength. The solution is used to examine i n detail the wave field around the cylinder and to improve understanding of many of the tank-confinement eff'ects observed i n other work. Particular attention is paid to the hydrodynamic forces and pressures on the cylinder surface and situations where the tank walls may have a particularly significant effect are highUghted.
1 I N T R O D U C T I O N
W h e n c a r r y i n g o u t physical m o d e l tests i n a n a r r o w wave t a n k i t is i m p o r t a n t t o be able t o assess the effects o f wave reflections f r o m the t a n k walls. .To this e n d , a n u m b e r o f authors'"^ have given theoretical descriptions o f the scattering a n d r a d i a t i o n o f water waves b y a v e r t i c a l , c i r c u l a r c y l i n d e r i n a n a r r o w wave t a n k . One o f the m o s t notable features o f a l l these results is t h a t m o s t h y d r o d y n a m i c quantities, w h e n considered as a f u n c t i o n o f f r e q u e n c y , display resonant 'spikes' c o r r e s p o n d i n g t o cross-tank s t a n d i n g waves i n the e m p t y t a n k . F o r a t a n k o f w i d t h 2b a n d d e p t h /?, standing waves m a y occur at r a d i a n frequencies w = (g/ctanh/cA)'/^ where, f o r waves a n t i s y m m e t r i c a b o u t the t a n k centre-plane, k = (« — l / 2 ) 7 r / è , a n d f o r symmetric waves, k = nTr/b, f o r a n y positive integer n. I n general, h y d r o d y n a m i c q u a n t i t i e s have spikes close to a l l o f these standing-wave frequencies t h o u g h i f there is a n t i s y m m e t r y ( s y m m e t r y ) o f the fluid m o t i o n a b o u t the t a n k centre-plane, o n l y the a n t i s y m m e t r i c ( s y m m e t r i c ) resonances can be excited.
A w a y f r o m the standing-wave frequencies, i t has been f o u n d t h a t some h y d r o d y n a m i c quantities are r e l a t i v e l y insensitive t o the presence o f the t a n k walls, w h i l e others s h o w m a r k e d a n d a p p a r e n t l y u n p r e d i c t a b l e v a r i a t i o n f r o m t h e i r open sea values. F o r example. S p r i n g a n d M o n k m e y e r ' a n d Thomas'* made detailed c a l c u l a t i o n s
Applied Ocean Research 0141-1187/93/S06.00 © 1993 Elsevier Science Publishers L t d .
f o r the scattering o f an i n c i d e n t plane wave b y a c y l i n d e r s y m m e t r i c a l l y placed a b o u t the t a n k centre plane a n d extending t h r o u g h o u t the d e p t h . T h e i r results were given i n terms o f ratios c o m p a r i n g values i n the w a v e t a n k w i t h the c o r r e s p o n d i n g open sea values; i f there were n o effects due t o r e f l e c t i o n f r o m the t a n k walls t h e n such a r a t i o w o u l d be u n i t y . I t was f o u n d t h a t the i n - l i n e d i f f r a c t i o n f o r c e r a t i o usually v a r i e d l i t t l e f r o m u n i t y , w h i l e , o n the other h a n d , the r a t i o f o r the h y d r o d y n a m i c pressure measured o n the c y l i n d e r surface o f t e n showed considerable v a r i a t i o n f r o m u n i t y ' b u t n o t i n a w a y t h a t is easily predictable.'* I t is the purpose o f t h e present w o r k t o g a i n f u r t h e r u n d e r s t a n d i n g o f these results b y e x a m i n i n g i n d e t a i l an a p p r o x i m a t e s o l u t i o n .
T h e geometry t o be considered here is t h a t o f a v e r t i c a l c y l i n d e r o f a r b i t r a r y cross-section e x t e n d i n g t h r o u g h o u t the d e p t h o f the wave t a n k . T h i s g e o m e t r y includes the case o f m u l t i p l e cylinders. T h e a p p r o x i m a t e s o l u t i o n is derived, b y the m e t h o d o f m a t c h e d a s y m p t o t i c expansions, under the a s s u m p t i o n t h a t a t y p i c a l h o r i z o n t a l d i m e n s i o n o f the c y l i n d e r is m u c h less t h a n the distance f r o m the nearest t a n k w a l l ( a n d hence also m u c h less t h a n the t a n k w i d t h ) . T h e w a v e l e n g t h is t a k e n t o be o f the order o f the t a n k w i d t h so t h a t i t t o o is m u c h greater t h a n the b o d y d i m e n s i o n . These assumptions t u r n o u t t o be n o t t o o restrictive a n d comparisons w i t h k n o w n solutions f o r the c i r c u l a r cylinder show that the approximate s o l u t i o n gives reason-able accuracy even outside its strict range o f v a l i d i t y .
T h e p l a n o f the paper is as f o l l o w s . T h e scattering p r o b l e m is f o r m u l a t e d i n Section 2 a n d the s o l u t i o n is
f u l l y described i n Section 3. T h e paper has been w r i t t e n so t h a t Section 3 m a y be o m i t t e d by a reader n o t interested i n the details o f the s o l u t i o n procedure. A n a l t e r n a t i v e , i n f o r m a l , account o f the s o l u t i o n is g i v e n i n Section 4 a n d this leads to a simple i n t e r p r e t a t i o n o f flume c o n f i n e m e n t effects i n terms o f the flow generated by the images i n the t a n k walls. T h e i m p l i c a t i o n s o f the s o l u t i o n f o r pressures a n d forces are discussed i n detail i n Sections 5 a n d 6, respectively.
2 F O R M U L A T I O N
A n i n f i n i t e l y l o n g wave t a n k o f u n i f o r m d e p t h h has p a r a l l e l walls a distance 2b apart. Cartesian coordinates are chosen w i t h the o r i g i n i n the m e a n free surface a n d m i d w a y between the channel walls so t h a t the x'-axis is directed a l o n g the channel a n d the z-axis vertically u p w a r d s . A v e r t i c a l cylinder o f a r b i t r a r y cross section T extends t h r o u g h o u t the d e p t h a n d {x,y) = {0,d) is chosen as a reference p o i n t w i t h i n T . Plane polar c o o r d i n a t e s r a n d 6 w i t h o r i g i n at this reference p o i n t are d e f i n e d by
rcosd a n d y-d=rsm9. (2.1)
A p l a n v i e w o f the geometry is g i v e n i n F i g . 1.
A plane wave o f a m p l i t u d e A a n d f r e q u e n c y u is i n c i d e n t f r o m large negative x. U n d e r the usual assumptions o f linear water-wave t h e o r y the time-h a r m o n i c flow m a y be described by a v e l o c i t y p o t e n t i a l
igA cosh /r(z + li) > cosh Ich
(2.2) where Ic is the wavenumber satisfying the dispersion relation
to g / c t a n h / d ï , (2.3)
g is the acceleration due to g r a v i t y a n d Re indicates that the real p a r t is to be t a k e n . T h e f o r m o f the p o t e n t i a l has been chosen to satisfy the linearised free-surface c o n d i t i o n , o n z = 0, a n d the bed c o n d i t i o n , o n z = -/?. T h e p o t e n t i a l $ must satisfy the three-dimensional L a p l a c e e q u a t i o n so t h a t , o n s u b s t i t u t i n g the f o r m
/ / / / / / / / / / / / / / /
2b
/ / / / / / / / / / / / / / /
Fig. L Definition sketch.
(2.2), the c o m p l e x - v a l u e d f u n c t i o n ^j{x,y) m a y be seen to satisfy the H e l m h o l t z e q u a t i o n . T h e i n c i d e n t wave is described b y
Jkr cos 9
(2.4) a n d the t o t a l p o t e n t i a l (px is decomposed as
cPT='f>i + 'P- (2.5) T h e p o t e n t i a l (/> describing the scattered wave field w i l l
t h e n satisfy the H e l m h o l t z e q u a t i o n .
• +
-dx^ ' dy^ + k^<f) = Q (2.6)
i n the fluid d o m a i n , the c o n d i t i o n s o f n o flow t h r o u g h the channel wafls,
d(f> dy 0 o n y — ±b, (2.7) (2.8) a n d the c y l i n d e r surface, ^ = - ? f ^ o n r , dn dn
where n is a n o r m a l c o o r d i n a t e directed i n t o the fluid, a n d also the r a d i a t i o n c o n d i t i o n s p e c i f y i n g that the scattered waves m u s t be o u t g o i n g as oo.
Let fl be a t y p i c a l d i m e n s i o n o f the b o d y cross-section r . A s o l u t i o n o f the above p r o b l e m w i l l be f o u n d under the assumptions t h a t a is m u c h smaller t h a n b o t h the w a v e l e n g t h a n d the distance f r o m the nearest t a n k w a f l , t h a t is ka < 1 and a b — \d\ respectively. T h e second i n e q u a h t y i m p l i e s t h a t the b o d y d i m e n s i o n a is also m u c h less t h a n the t a n k w i d t h b. T h e t a n k w i d t h a n d w a v e l e n g t h are t a k e n t o be o f the same order o f m a g n i t u d e so t h a t kb = 0 ( 1 ) .
3 S O L U T I O N P R O C E D U R E
T h e s o l u t i o n is by the m e t h o d o f m a t c h e d a s y m p t o t i c expansions a n d requires t h a t the flow d o m a i n be d i v i d e d i n t o t w o regions. T h e inner r e g i o n s u r r o u n d s the cylinder to r a d i a l distances - c /c~' a n d , because o f the assumptions a <^ b — \d\ and kb = 0 ( 1 ) , does n o t c o n t a i n the t a n k walls. T h e outer r e g i o n is external t o this at distances » a, w h e r e a is a t y p i c a l h o r i z o n t a l d i m e n s i o n o f the b o d y , a n d includes the t a n k walls. F r o m the p o i n t o f view o f an observer i n the outer r e g i o n the c y l i n d e r appears as a p o i n t disturbance. S o l u t i o n s c o n t a i n i n g u n k n o w n constants are c o n -structed separately i n the t w o regions a n d , w i t h the a s s u m p t i o n e = /co < 1, they can be f u l l determined b y m a t c h i n g i n a n overlap r e g i o n .
F o r the inner r e g i o n define scaled coordinates w i t h o r i g i n w i t h i n the b o d y c o n t o u r F b y
T h e i n n e r scattered p o t e n t i a l ^i{^,rj) = (j){x,y) satisfies the field equation (2.6) a n d the b o d y b o u n d a r y c o n d i t i o n (2.8) w h i c h , i n terms o f t h e i n n e r coordinates, are
0 (3.2)
a n d
^ = - ^ ( e ' - ^ ^ )
| ^ ( / < - i . V - ^ ' . ¥ ) + 0(e^) o n r,
respectively.Scaled outer coordinates are defined b y
X=kx, k { y - d ) , R = kr.
(3.3)
(3.4) T h e o u t e r r e g i o n s o l u t i o n ^g{x,Y) = 4'{x,y) m u s t satisfy a l l the c o n d i t i o n s o f the p r o b l e m except f o r the b o d y b o u n d a r y c o n d i t i o n (2.8). T h e outer s o l u t i o n w i l l be c o n s t r u c t e d f r o m 'channel m u l t i p o l e s ' d e r i v e d b y M c l v e r a n d Bennett."" These m u l t i p o l e s are solutions o f the H e l m h o l t z e q u a t i o n (2.6) s a t i s f y i n g the t a n k - w a l l c o n d i t i o n (2.7) a n d the r a d i a t i o n c o n d i t i o n ; t h a t is, a l l the c o n d i t i o n s o f the p r o b l e m except the b o d y b o u n d a r y c o n d i t i o n .
T h e i n n e r a n d outer solutions w i l l each c o n t a i n a n u m b e r o f u n k n o w n constant coefficients a n d these w i l l be d e t e r m i n e d b y m a t c h i n g . T h e structure o f the i n n e r a n d outer solutions f o r a n a r b i t r a r y c o n t o u r F is s i m i l a r to the case o f a c i r c u l a r c o n t o u r treated i n M c l v e r a n d Bennett^ a n d so, to s i m p l i f y the p r e s e n t a t i o n , the c o r r e c t f o r m o f the expansions w i l l be assumed f r o m the outset.
Leading-order terms in the inner solution
T h e i n n e r s o l u t i o n has a n e x p a n s i o n
= + I n e * , - 2 i + £"^i,2 + e M n e ^-,-31 + e^*;^3
(3.5) where \ l / [ ' ' denotes the inner s o l u t i o n e x p a n d e d u p t o o r d e r e'. T h e field e q u a t i o n a n d b o d y b o u n d a r y c o n d i t i o n g o v e r n i n g each o f the terms o n t h e r i g h t -h a n d side o f eqn (3.5) m a y be f o u n d b y s u b s t i t u t i n g i n t o eqn (3.2, 3.3) a n d e q u a t i n g l i k e terms i n e. I t is f o u n d t h a t the l e a d i n g - o r d e r t e r m i n the inner s o l u t i o n , is h a r m o n i c a n d satisfies
dn - i l ^ i p c o s f f ] on
r.
(3.6)T h e s o l u t i o n 1 is the disturbance to a u n i f o r m stream o f speed / i n the ^ d i r e c t i o n flowing past the c o n t o u r F w h e n i n an u n b o u n d e d fluid. T h e l e a d i n g - o r d e r inner s o l u t i o n is t h e r e f o r e w r i t t e n
(3.7) where x is the response to a u n i f o r m flow o f u n i t speed so t h a t dx/dn = ~d^/dn o n T. F r o m B a t c h e l o r , ' Section 2.10, X = Mr cos( + A s i n ö c o s 2 ö , s i n 2 ö + At2— 7 - + h — T -+ 0 as p (3.8)
T h e t e r m \I>,-2i i n eqn (3.5) is also h a r m o n i c a n d , f r o m eqn (3.3), satisfies a h o m o g e n e o u s b o u n d a r y c o n d i t i o n o n r. A s i n M c l v e r a n d Bennett,^ the o n l y h o m o g e n e o u s s o l u t i o n suitable f o r m a t c h i n g is a constant, so w r i t e
* / , 2 i = ^ 0 - (3.9) ( T h r o u g h o u t this section P „ , , where m is an integer, is
used t o denote a constant i n the inner s o l u t i o n ) . T h e t h i r d t e r m i n eqn (3.5) is again h a r m o n i c b u t this satisfies the b o u n d a r y c o n d i t i o n
dn 2dn^^ ' o n F . (3.10)
T h i s n o n - h o m o g e n e o u s b o u n d a r y c o n d i t i o n generates a net flow across F o f
' dn dl = (3.11)
by the divergence t h e o r e m , where / is arc l e n g t h a n d S is the ( d i m e n s i o n a l ) cross-sectional area o f T. T h i s net flow implies t h a t ^Pj^ m u s t be source-like at large distances f r o m F . W r i t e *,-,2 = ^ l + f ^ l ( p , Ö ) , where is a constant a n d (3.12) lira as ^ , ^ c o s ( \np + D s i n ö O (3.13) T h e source s t r e n g t h i n eqn (3.13) has been chosen t o compensate f o r the net flow g i v e n b y eqn (3.11) a n d the r e m a i n i n g terms again f o U o w f r o m B a t c h e l o r , ' Section 2.10.
T h e inner s o l u t i o n t o o r d e r e^, \ l / f ' , w i l l n o w be m a t c h e d w i t h t h e outer s o l u t i o n . T h e b e h a v i o u r o f ^I/p^ at large distances f r o m F f o l l o w s f r o m eqns (3.5), ( 3 . 7 ¬ 3.9), (3.12-3.13^. T h e l e a d i n g - o r d e r terms i n the o u t e r e x p a n s i o n o f ^ are f o u n d b y expressing \I/p^ i n terms o f the outer variables a n d f u r t h e r e x p a n d i n g i n e. F o l l o w i n g this p r o c e d u r e a n d w r i t i n g the result b a c k i n terms o f the i n n e r variables gives
,T,(2,2) . / cos( A
sin 0
- F e M n e P , 0
V2™2 (3.14)
H e r e denotes the i n n e r s o l u t i o n t o order / r e w r i t t e n i n terms o f the outer variables a n d e x p a n d e d t o order tn. A similar n o t a t i o n is used f o r the o u t e r
p o t e n t i a l T h u s ^f^'"^ denotes the o r d e r « j s o l u t i o n w h i c h , w h e n r e w r i t t e n i n terms o f the inner variables a n d expanded to o r d e r /, is d e n o t e d b y ^ ^ " ' ' ' \ T h e m a t c h i n g p r i n c i p l e requires t h a t ^f"'^ = \l/("''') (see, f o r example, C r i g h t o n a n d L e p p i n g t o n ^ ) .
Leading-order outer solution and matching
T h e outer e x p a n s i o n o f the i n n e r s o l u t i o n eqn (3.14) contains b o t h source- a n d d i p o l e - l i k e terms a n d this m u s t be reflected i n t h e outer s o l u t i o n . T h e leading-order outer s o l u t i o n is w r i t t e n i n terms o f the source-a n d d i p o l e - h k e c h source-a n n e l m u l t i p o l e s source-as
=e\A,<j), + A,<t>,+B,,lj,), (3.15)
where f r o m M c l v e r a n d Bennett,^ eqns (3.7) a n d ( 3 . 1 4 ) ,
(j)„ = H„[R) COSWÖ • n - l I -27r iXl '2jkd 7 sinh X e •' c o s h « T dt a n d -ip,, = H„{R) sinnO 1 2 ^ (3.16) -27Arf\ 7 sinh 27A;6 X e ' ^ ' s i n h 7 ! r dt, (3.17) where H„ denotes a H a n k e l f u n c t i o n o f the first k i n d , 7 = (^2 _ 1)1/2 ^ _ ^2^/2 c o s h r = t. T h e p a t h o f i n t e g r a t i o n here, a n d b e l o w , is i n d e n t e d beneath the poles o f the i n t e g r a n d i n o r d e r t o satisfy the correct r a d i a t i o n c o n d i t i o n . T h e H a n k e l f u n c t i o n terms are the singular solutions o f the H e l m h o l t z e q u a t i o n a p p r o -p r i a t e to scattering i n o -p e n w a t e r w h i l e the i n t e g r a l terms are the c o r r e c t i o n s due t o the presence o f the channel walls. T o o b t a i n the i n n e r e x p a n s i o n o f the outer s o l u t i o n the f o l l o w i n g results f r o m M c l v e r a n d Bennet,^ eqns ( 3 . 1 9 ) - ( 3 . 2 4 ) are used:
h^H„{R) c o s « Ö + £ (
«;=0
, cos mO + P„i„ sin m9 ) / , „ ( i ? ) ,
(3.18) where
em( _.yn-n+l
y •oo e-^-r'^b + coshljkd A — OO 7 sinh 2'ykb 2 ( ^
-0"'-"
'°° smh2jkd - o o 7 sinh 2'jkb cosh m r cosh 7!T dt, (3.19) sinh H 7 r c o s h « r dt, (3.20) a n d Ip,, = H„{R) smnO + £ ( f l ™ , c o s « 7 0 + b,„„ smme)J,„{R), (3.21) where-0"
2 ( - 0
TT m—n—\ sinh 27/cö?, 7 sinh 2'ykb cosh mr s i n h nr dt, (3.22)
TT -27/tft
cosh 2')kd
7 sinh 2 7 / c è sinh mr sinh « r dt. (3.23) I n the above EQ = 1 a n d e,„ = 2 f o r m> 1. N o t e t h a t AiO = t>iiO = 0 f o r a l l n. T h e expansions (3.18) a n d (3.21) are v a l i d f o r Q < R< 2k{b - d). T h e coefficients a,„„ a n d b,„„ are zero i f m + n is o d d w h i l e /3,„„ a n d a,„„ are zero i f m + n is even; these zero coefficients are r e t a i n e d i n expansions (3.18) a n d (3.21) i n o r d e r t o have a c o m p a c t n o t a t i o n . T h e n u m e r i c a l e v a l u a t i o n o f these coefficients is described i n appendix A o f M c l v e r a n d Bennett."^
U s i n g the k n o w n series expansions o f Bessel f u n c t i o n s ( A b r a m o w i t z a n d S t e g u n , ' p . 360) i n expansions (3.18) a n d (3.21), the i n n e r expansion o f the o u t e r s o l u t i o n (3.15) is f o u n d to be (2,2) TT V 2 / c o s Ö " TT ep 2 / s i n 6» TT ep - + a 10 (3.24) where ^ — 0.57721 . . . is Euler's constant. T h i s m a y n o w be m a t c h e d w i t h the outer e x p a n s i o n o f the i n n e r s o l u t i o n , eqn (3.14) to get - ( ' g ' - l n 2 ) + a o o + 5 , f l , o = P i , 1 TT An- = Pn 2'na 2 ' 2i - = ipi. -B, 2i --iXy. (3.25) T a k i n g account o f eqn (3.25), the i n n e r a n d outer solutions to o r d e r m a y n o w be w r i t t e n as S (2) eix- I n e 2 ™ 2 -he-iS_
•47
2i 1 + - ( ' ^ - l n 2 ) + aoo TT 1 a n d 2 ^ - ^ 2 ' 27ri3' •</'0 + A l l / * ! (3.26) (3.27) respectively.Extension of the inner solution
R e f e r r i n g b a c k t o eqn (3.5), the f o r m o f the f u r t h e r terms i n the i n n e r s o l u t i o n over those i n eqn (3.26) are g u i d e d b y the c o n t i n u e d inner expansion o f the outer s o l u t i o n eqn (3.27). I t is f o u n d t h a t ^(2,3) ^ 2i
+
-TT 27ra2 2/ — I TT In 2) + aoo lne/9 + siné*- \ ep 2i 1 i , 1- - ep I n ep TT ep TT lepi 1 - - ( 1 - 2 ' ^ + 2 1 n 2 ) + a „ cos 9 + A, 2i 1 , _ _ _ + _ e p l n e p + i e p 2<ë' + 2 1 n 2 ) + è i , s i n ö + ö '10 (3.28) T h e r e m a i n i n g terms t o be considered i n the i n n e r s o l u t i o n are those at orders e M n e a n d e^ i n eqn ( 3 . 5 ) . S u b s t i t u t i n g eqn (3.5) i n t o the g o v e r n i n g eqns ( 3 . 2 ) -(3.3) f o r the i n n e r p r o b l e m a n d e q u a t i n g l i k e terms i n e it m a y be seen that * , - 3 i is a h a r m o n i c f u n c t i o n s a t i s f y i n g a homogeneous b o u n d a r y c o n d i t i o n o n T. T h e large p b e h a v i o u r o f this s o l u t i o n m u s t m a t c h w i t h the terms at o r d e r e ^ n e i n eqn (3.28) w h i c h m a y be i d e n t i f i e d as ' u n i f o r m flow' terms p r o p o r t i o n a l t o p c o s 9 a n d p sin 6. T h e a p p r o p r i a t e f o r m is t h e r e f o r e*,-31 = P 2 + P 3 [ P C 0 S Ö + X ( P , Ö ) ]
+ P4 [p sin ö + r ( p , 6»)] (3.29) where T is the response t o a u n i f o r m flow o f u n i t speed i n the 77 d i r e c t i o n so t h a t dr dn a n d ^1 ^ t • cos> T = OTi ' , s i n ö ^ + I] + O P P \P' T h e field e q u a t i o n f o r \!/,-3 is V2vl/,-,3 = = -ix w i t h the b o u n d a r y c o n d i t i o n _ d dn dn as
00.
o nr.
(3.30) (3.31) (3.32) (3.33) L e t ifl2{p,9) be a p a r t i c u l a r s o l u t i o n o f eqn (3.32) (the i m a g i n a r y u n i t has been i n t r o d u c e d to ensure t h a t Ü2 is real). T h e f a r - f i e l d f o r m o f ^ 2 m a y be deduced f r o m eqn (3.32) u s i n g the k n o w n b e h a v i o u r o f x g i v e n b y e q n (3.8) a n d , e x c l u d i n g solutions o f the h o m o g e n e o u s e q u a t i o n , i t m a y be v e r i f i e d b y s u b s t i t u t i o n i n eqn (3.32) t h a t O2 - ( - 5 ( M I COS Ö + Al sin Ö )P In P + |(/i2Cos26i + A 2 s i n 2 6 i ) ) ^ 0 as p ^ 00. (3.34) T h e s o l u t i o n O2 generates a flow across the b o d y c o n t o u r F w h i c h m a y be calculated as f o l l o w s . L e t L denote a n enclosing circle at large p a n d F the fluid d o m a i n between F a n d L; a p p l i c a t i o n o f the divergence t h e o r e m givesV^n2dA
ruL
Ö0_2
dn dl. (3.35)
N o w because o f the angular dependence i n e q n (3.34), the c o n t r i b u t i o n to the r i g h t - h a n d side o f e q n (3.35) f r o m L is zero so the flux across F is
d^7 „ ( { 0
.V on } ip XdA (3.36)
( T h e last step f o l l o w s because iÜ2 satisfles e q n (3.32).) N o w w r i t e *,-3 = / ( 0 2 + 0 3 ) (3.37) so t h a t f r o m eqns ( 3 . 3 2 ) - ( 3 . 3 3 ) ^ 3 is h a r m o n i c a n d satisfies 0 ^ 3 dn ^ ^ - ^ 2 ) o n F . (3.38)
T h e flow across F r e s u l t i n g f r o m the b o u n d a r y c o n d i t i o n (3.38) w i f l be reflected i n a source-like b e h a v i o u r at large distances so t h a t , a g a i n e x c l u d i n g s o l u t i o n s o f the homogeneous p r o b l e m ,
O3
- Q In p 0 as p00
(3.39)where, f r o m eqns (3.38), (3.36) a n d the divergence t h e o r e m .
e
27r r dn 2n ^dA xdA). (3.40)
H e r e S is the cross-sectional area o f F . F i n a l l y , i n c l u d i n g solutions o f the homogeneous p r o b l e m needed t o m a t c h w i t h ( 3 . 2 8 ) ,
*;,3 = (O2 + O3) + P5 + Pg [p cos 9 + x ( p , 9)]
+ Pj[psm9 + T{p,9)]. (3.41)
T h e outer expansion o f m a y n o w be calculated t o give, i n terms o f the i n n e r variables,
(3,3) .ƒ cose^ , s i n ö cos 20 , sin 261
+ e^ I n e 2'Ka' J 5 '4a^ P TT -iTrAiflio 27ra -S" , ^ cos 9 \np + D \-E P I n 2) sin 9 « 0 0
+ I n e{P2 +P3P cos 6 + P^p sin 6}
+ e "I - - ( ^ 1 COS0 + Al sm9)p\np
+ - ( ^ 2 C0S2Ö + A 2 s i n 2 ö )
+ iQ In p + P 5 + P(,p cos 9 + PTP sin ö j (3.42)
T h e singular terms i n p"^ at order e w h e n expressed i n terms o f the outer variable R = p/e indicate the need f o r s i m i l a r singular terms i n the outer s o l u t i o n so t h a t the outer expansion continues as
= ^-P^ + e\A2<Po + ^ 3 ^ 1 + ^ 4 0 2 + 52V'i + ^ 3 ^ 2 ) (3.43) where \1/P^ is given b y eqn (3.27). T h i s has a n inner e x p a n s i o n 1 + — ('^ - I n 2 ) + «00 + - I n ep TT TT 2;cos6'' A , + B2 TT
ep
2 / s i n 6» TT e p + AA «10 4/COS2Ö i TT e2p' + 5 3 TT 4 / s i n 26» TT e^p^ IT COS 29 + Q'20 sin 29 (3.44) where ^ ^ ' ^ ^ is given b y eqn (3.28). M a t c h i n g eqn (3.44) w i t h e q n (3.42) determines a l l o f the u n k n o w n constants to give a n inner s o l u t i o n = e/x + e^lne-—=• 1 — iS2Tva^
2i
^ { - Y ^ 1 + - ( ^ - l n 2 ) + Q'oo - i T r A i f l i o + O 4fl^+ e' I n e{iQ - \ip,
(^ + x ) -
i ' A i (?] + r ) }
+ e^ <j ( ^ 2 + O3) + 5TTÖ ( 1 + 5 - I n 2 ) + aoo 2i - ( TT -\'ïïp2O!-20 + \'^iEa 10 - i 7 r M i ( l - - ( l - 2 ' g ' + 2 1 n 2 ) + a i i ) (^ + x ) k A i 1 - - ( 1 - 2 ^ + 2 1 n 2 ) + è i i TT (5' 8fl 2/^01 (^7 + ^ ) (3.45) a n d a n outer s o l u t i o n (3) e ^ W l 1-.2-ïïa
•(po + Pxcpi + A i ' 0 ie'i7r(2e</)o + 2//),^i - /x2</>2 + 2iE^y - X2^2)-(3.46)
4 I N T E R P R E T A T I O N O F T H E S O L U T I O N B e f o r e g o i n g o n to l o o k i n d e t a i l at the consequences o f the results o b t a i n e d i n Section 3, a n alternative, p a r t i a l , d e r i v a t i o n w i l l be g i v e n using i n f o r m a l arguments i n order t o a i d u n d e r s t a n d i n g o f the s o l u t i o n .
Consider, f i r s t the w e l l - k n o w n s o l u t i o n f o r scattering b y a vertical cyhnder o f radius a s t a n d i n g i n open water. D e f i n i n g the scattered wave field 0 as i n eqns (2.2) a n d (2.5), i t is f o u n d t h a t i n o p e n water
- T e J " S ^ H „ i k r ) cos,19 (4.1)
(see, f o r example. M e i , ' " p . 312) where {r,9) are plane p o l a r coordinates w i t h o r i g i n o n the c y l i n d e r axis, eo = 1 a n d e„ = 2, n>\. T h e s o l u t i o n o f the previous section was d e r i v e d using m a t c h e d a s y m p t o t i c expansions under the a s s u m p t i o n t h a t the w a v e l e n g t h is m u c h longer t h a n the cyhnder radius, t h a t is /ca - C 1. Expansions o f the p o t e n t i a l i n terms o f ka were f o u n d b u t , because o f the singular n a t u r e o f the p e r t u r b a t i o n p r o b l e m , n o single expansion c a n be v a l i d t h r o u g h o u t the fluid d o m a i n a n d i t was necessary t o consider separate expansions i n inner a n d o u t e r regions close t o a n d f a r f r o m the b o d y , respectively.
T h e inner r e g i o n is t h a t fluid at distances m u c h less t h a n a w a v e l e n g t h f r o m t h e c y l i n d e r so t h a t kr <^ 1. E x p a n d i n g t h e terms i n e q n (4.1) (using resuhs i n A b r a m o w i t z a n d Stegun,^ p . 360) u n d e r the a s s u m p t i o n t h a t b o t h kr a n d ka are s m a l l , w i t h ;• = 0{a), gives a n inner r e g i o n a p p r o x i m a t i o n t o the scattered field
kai-cos 9 -\- \{Ica) In ka
+ Wcaf I n - -
la'2a 2 ; cos26l + ' ^ - T (4.2) where = 0 . 5 7 7 2 1 . . . is Euler's constant. T h e inter-p r e t a t i o n o f this scattered field is aided by c o n s i d e r a t i o n o f a s i m i l a r inner expansion o f the i n c i d e n t wave field (2.4) 1
\+kai---{lea
= 1 - I - A : f l ! - c o s ö - ^ ( / c a ) ^ - ^ ( l + C 0 S 2 Ö 'a
4 ^ «2 V (4.3) T h e t e r m at 0{ka) i n (4.3) represents a n o s c i f l a t o r y u n i f o r m flow i n the x d i r e c t i o n a n d the c o r r e s p o n d i n g t e r m i n (4.2) is the d i p o l e response t o t h a t flow (recall t h a t a n o s c i l l a t o r y t i m e dependence has been r e m o v e d i n e q n (2.2)). T h e t e r m at 0{{lca)^) i n t h e i n c i d e n t wave e x p a n s i o n (4.3) generates a non-zero flux across the c y l i n d e r surface a n d this is compensated f o r b y the l o g a r i t h m i c source t e r m i n eqn (4.2). I n a s i m i l a r w a y t o the u n i f o r m flow a n d d i p o l e terms at 0{ka), w h e n t a k e n together the terms i n cos 20 at 0[{kaY) i nthe t w o equations have zero n o r m a l derivative o n the cylinder surface. T h e constant terms i n eqn (4.2) cannot be explained simply b u t they clearly satisfy the n o - f l o w c o n d i t i o n o n the cylinder surface.
T h e outer region is the fluid m a n y cylinder r a d i i f r o m the o r i g i n so that ;• 3> a. E x p a n d i n g the terms i n eqn (4.1) f o r smafl ka, b u t w i t h kr = 0 ( 1 ) , gives an outer r e g i o n a p p r o x i m a t i o n to the scattered field
order outer s o l u t i o n is
(4.4)
T h e leading-order terms i n the o u t e r field consist o f the source a n d dipole terms f r o m the complete expansion a n d the strengths o f the singularities i n these terms are the same as the c o r r e s p o n d i n g terms i n the i n n e r expansion o f the scattered field, eqn (4.2).
T h e scattering b y a v e r t i c a l c y l i n d e r o f a r b i t r a r y cross section w i t h i n a c o n t o u r T has been considered b y L a m b , " A r t . 305 a n d the results m a y also be o b t a i n e d using m a t c h e d a s y m p t o t i c expansions. T h e inner region scattered field
kaix{-,9) + {Icay In/ca-^
+ ( k a f \ n A - , 9 ] + Ina^
1
ni — In 2
(4.5) where S is the cross-sectional area o f the c y l i n d e r and a is n o w a t y p i c a l h o r i z o n t a l d i m e n s i o n . T h e t e r m at 0{ka) is the response t o the u n i f o r m flow t e r m i n the incident wave expansion (4.3) a n d satisfles
dx dn X - Pi-Idx adn a cos 9 o n A, asinO as -r a (4.6)
A t large distances the leadingorder b e h a v i o u r is d i p o l e -like; a f u l l discussion o f the f a r - f i e l d b e h a v i o u r is given by B a t c h e l o r , ' Section 2.10. ( F o r a c y l i n d e r s y m m e t r i c a b o u t y = d, A j = 0 w h i l e f o r a c i r c u l a r c y l i n d e r o f radius a, x = a c o s 0 / r . ) T h e response t o the 0{{kaf) terms i n the i n c i d e n t wave is described b y fii w h i c h satisfies a f i l ~d^ fi, 1 dx^ ''2^~d^ r l n -2 ™ ^ o n 0 as (4.7)
T h e strength o f the l o g a r i t h m i c source t e r m i n fij compensates f o r the flow F i n d u c e d by the b o u n d a r y c o n d i t i o n i n eqn (4.7). T h e f u n c t i o n s x and fij have been expressed i n terms o f r/a, rather t h a n ;•, f o r consistency w i t h the usage i n Section 3. T h e
leading--{-Kikaf
i-—2-^o(/c'') + Hi{kr)[px cos 0 + A j sin 0
(4.8) A g a i n the strength o f the source i n the outer s o l u t i o n corresponds w i t h t h a t i n the i n n e r s o l u t i o n , as i n e q n (4.7), w h i l e the d i p o l e strengths c o r r e s p o n d t o those i n the u n i f o r m flow response, eqn (4.6).
T h e presence o f the t a n k walls is equivalent t o a set o f image cylinders at y = Amb + d a n d y = 2{2m - \)b - d where m is any integer {m = 0 i n the first f o r m u l a corresponds t o the o r i g i n a l c y l i n d e r ) . L e t {rj, 9j) be plane p o l a r coordinates relative t o image j where the cylinders have been n u m b e r e d i n ascending order o f y c o o r d i n a t e w i t h J = 0 as the o r i g i n a l cylinder. I n the a p p r o x i m a t i o n considered here, the p r i m a r y effect o f the t a n k walls is f o u n d b y assuming t h a t each image c y l i n d e r scatters the incident wave as i f i t were i n the open sea w i t h subsequent scatterings between the images considered t o be negligible. S u m m i n g over the images d e r i v e d f r o m eqn (4.8) b y a p p r o p r i a t e changes o f o r i g i n gives a f a r -field p o t e n d a l S ^ o ( ^ ' > ) + M i 2 Hx{krj) COS9j + \x ^ ( - l ) - ' 7 7 i ( / c ; > ) s i n 0 ; = - o o j=-oo J ^ n { k a f ( i - ^ M k > ; 9 ) + fi,Mkr,9) + \iMkr,9) (4.9)
where the H a n k e l f u n c t i o n series have been s u m m e d a n d w r i t t e n i n terms o f the 'channel m u l t i p o l e ' representa-t i o n s defined i n equarepresenta-tions ( 3 . 1 6 ) - ( 3 . 1 7 ) a n d d e r i v e d b y M c l v e r and Bennett.^ T h u s (pQ is the s u m o f the source terms, 4>i the sum o f the i n - l i n e dipoles a n d V i the s u m o f the cross-tank dipoles. N o t e t h a t the o r i e n t a t i o n o f successive image cross-tank dipoles is reversed, hence the appearance o f (—1)-' i n the s u m m a t i o n i n eqn (4.9).
T h e near-field effects o f the image cylinders m a y be deduced f r o m the expansions o f the m u l t i p o l e potentials given b y eqns (3.18) a n d (3.21). I n the i n n e r r e g i o n
0 o ( / c / - , 0 ) - / / o ( / « - ) - E ^ o ( f c o )
= Y^[ao,2,>,cos2m9J2,„{kr)
m=0
= aoo + A;a^/3oi ^ s i n ö + 0{{kaf) - aoo
+ A:a:r^oi^^ , 2 fl
6i (fo-, 9 ) - Hi {kr) cos 0 = ^ i ^ i (fcj)) cos 9j
OQ
. = X l [ " i . 2 m + i COS ( 2 w + \)9J2,n+x{kr)
r a t i o o f i n c i d e n t plus scattered wave fields a n d is
(4.10)
+ A,2m Sin277t0/2m(^'-)]
= = / c f l ^ a i i - c o s 0 + O ( ( / c f l ) ^ ) - / c f l i a i i - (4.11) 2 " f l 2 " f l
a n d
ipi{kr,9) - Hi{kr) sm9 = ^ ( - l ) - ' / / i ( / c ; y ) smOj
OO [fll,2m cos2m9J2,n{kr) + ^l,2,«+l sin (2/77 + l ) 0 / 2 m + l ( ^ ' - ) ] 1 r flio + ka-bu - s i n 0 + O { { k a f ) 2
a
~ fl]o + ka-bxx 2 fl (4.12)where t h e series expansions o f the Bessel f u n c t i o n s J,„ have been used t o o b t a i n t h e i n n e r r e g i o n a p p r o x i m a -tions. F o r a c y l i n d e r o n t h e t a n k centre plane, t h a t is w h e n ( i = 0, t h e e x p a n s i o n coefficients /?oi a n d AJO are b o t h i d e n t i c a l l y zero. I n (4.10) a n d (4.12), t h e l o c a l effect near t o the c y l i n d e r o f the image systems f o r t h e sources a n d t h e cross-tank dipoles is t o give a net constant increase t o the p o t e n t i a l a n d t o generate a cross-tank o s c i l l a t o r y f l o w . F r o m (4.11), the l o c a l effect o f the image i n - l i n e dipoles o r i e n t e d a l o n g the t a n k is t o generate a n o s c i l l a t o r y flow a l o n g the t a n k .
5 T H E P R E S S U R E F I E L D
T o ihustrate h o w the t a n k w a f l s i n f l u e n c e the pressures a n d forces o n the c y l i n d e r , a p p r o p r i a t e r a t i o s c o m p a r i n g t a n k a n d open-sea values are deflned. A pressure r a t i o P is d e f l n e d as the m o d u l u s o f the d y n a m i c pressure o n the c y l i n d e r w h e n i n the t a n k d i v i d e d b y the c o r r e s p o n d i n g value w h e n t h e c y l i n d e r is i n open water. I n t h e scattering p r o b l e m the d e p t h dependence m a y be r e m o v e d as i n e q n (2.2), so t h a t this r a t i o is independent o f d e p t h , b u t i t w i l l depend o n p o s i t i o n a r o u n d t h e cross-section. A n alternative i n t e r p r e t a t i o n o f t h e pressure r a t i o is t h a t i t is the r a t i o o f t h e a m p l i t u d e s o f the free surface oscillations a t t h e c y l i n d e r . T h i s pressure r a t i o is calculated f r o m t h e
|e''''^ +<?!'tankl |e*^ + .
(5.1) 'open sea I
where a l l terms are evaluated o n the c y l i n d e r surface. T h e scattered field 0tank f o r tbe inner r e g i o n , w h e n t h e c y l i n d e r is i n a t a n k , is given t o O { { k a f ) b y t h e i n n e r p o t e n t i a l i n e q n (3.45) where e = ka a n d t h e n o n d i m e n s i o n a l r a d i a l c o o r d i n a t e p = r/a. T h e c o r r e s p o n d -i n g result ''open sea is recovered b y setting a l l o f t h e expansion coefficients a „ „ „ P,,,,,, a,,,,,, b„„, (see eqns ( 4 . 1 0 ) - ( 4 . 1 2 ) ) t o be zero. E x p a n d i n g the i n c i d e n t w a v e p o t e n t i a l t o o r d e r 0{{kx)^), s u b s t i t u t i n g i n (5.1) a n d e x p a n d i n g the d e n o m i n a t o r leads t o
-- r iS* 1
— J I m aoo - n TTAI R e «lo
1 +
{ka)'
4fl2+ (/cfl)^-| l 7 r [ ö R e aoo - \ p 2 ^ e azo - £ I m fljo]
1 / S - npi Re « 1 1 + ^ R c "00 4 \ fl'' \ i^ \ ){-a + ^ ) 2 ? • + T (5.2) T h e terms i n square braces are a l l c o n s t a n t terms, t h a t is they d o n o t depend o n p o s i t i o n a r o u n d t h e c y l i n d e r surface. U s i n g the results o f eqns (4.10) a n d ( 4 . 1 2 ) , t h e terms at 0{{kaf) i n v o l v i n g «oo a n d A J O c a n b e seen t o arise f r o m t h e sets o f image sources a n d image cross-c h a n n e l dipoles, respecross-ctively. These terms are t h e leading c o n t r i b u t i o n t o the a d d i t i o n a l m e a n pressure field due t o the image system.
T u r n i n g t o the terms a t O { { k a f ) , the constants p2, E a n d Ö are d e f i n e d b y eqns (3.8), (3.13) a n d (3.40) respectively. I n p r i n c i p l e these m a y be c a l c u l a t e d f o r a n y geometry b y s o l v i n g t h e b o u n d a r y - v a l u e p r o b l e m s i n d i c a t e d i n Section 3 ( f o r example, t h i s c a n be done f o r a c y l i n d e r o f e l l i p t i c a l cross section, at a n a r b i t r a r y o r i e n t a t i o n t o t h e i n c i d e n t wave, u s i n g e l l i p t i c c o o r d i -nates a n d i n this case p.2 = E = Q = 0). T h e terms i n r o u n d e d braces describe t h e v a r i a t i o n o f P a r o u n d t h e c y l i n d e r . T h e f u n c t i o n x , d e f i n e d b y e q n ( 4 . 6 ) , is t h e response t o a u n i f o r m flow i n the x d i r e c t i o n a n d r is the response t o a flow i n the y d i r e c t i o n a n d satisfies
Ö T _ _ l ö > ;
dn a dn
o na cos 9 a sin 9
T ~ nil 1" '1 as
-fl (5.3)
T h e terms i n v o l v i n g a ^ , b^ a n d /?oi arise f r o m t h e u n i f o r m flows generated b y the image sets g i v e n i n eqns ( 4 . 1 0 ) - ( 4 . 1 2 ) ; t h e t e r m i n v o l v i n g /?oi is zero w h e n t h e offset d =0. T h e r e m a i n i n g t e r m at O { { k a f ) i n v o l v i n g « 0 0 bas n o simple i n t e r p r e t a t i o n , i t arises f r o m cross
O.asI I I I I I I [ I I I I I I I I I I I I I I I I I I r I I I I I I I I I I I I I I I I
Vi'TT-vr-i-0.0 0.1 0.2 C.3 C.4 0.5 0.6 0.7 0,8 0 9 1.0
Fig. 2. Pressure ratio Pv. d/n f o r ka = 0.3 and various kb where 9 is angular position on surface o f circular cylinder and Ö = TT is up-wave direction; comparison o f accurate ( ) and
approximate ( ) theories.
terms w h e n s u b s t i t u t i n g f o r the i n n e r p o t e n t i a l i n eqn (5.1) a n d e x p a n d i n g i n ka.
F o r a b o d y t h a t is s y m m e t r i c w i t h respect t o t w o v e r t i c a l planes aligned w i t h a n d p e r p e n d i c u l a r to the wave d i r e c t i o n , the results given b y B a t c h e l o r , ' Section 2.10 m a y be used t o s h o w t h a t Ai a n d a l l o f the O { { k a f ) constant terms are zero. T h e pressure r a t i o then reduces to P = \ + { k a f ^ I m a o o {ka S Tr^i Re ^ Re QQO
)
( - + X / \a la 2 I m ^ o i l ^ ^ — - + (5.4)again /3oi = 0 i f the b o d y is c e n t r a l l y placed. F o r a c i r c u l a r c y l i n d e r o f radius a, S = -na^ a n d p.\ — \.
B e l o w the first standing-wave f r e q u e n c y {kb = TF i f the c y l i n d e r is s y m m e t r i c w i t h respect t o the t a n k centre plane, kb = ix/l o t h e r w i s e ) ,
Re aoo = l / / c è - 1 a n d Re Q „ = 2 / / c è - 1. (5.5) T h u s , b e l o w the first s t a n d i n g wave f r e q u e n c y a n d f o r a c e n t r a l l y placed, c i r c u l a r c y l i n d e r
1 1
1 + {ka) I m aoo + {ka)'
kb cosO.
(5.6) Results f o r this s i t u a t i o n are displayed i n F i g . 2. T h e pressure r a t i o is given as a f u n c t i o n o f angle 9 a r o u n d the c y l i n d e r w i t h 9 = TT the u p - w a v e d i r e c t i o n a n d , because o f the s y m m e t r y , o n l y values i n the range 0 < 0 < TT are displayed. T h e sohd curves are calcula-t i o n s m a d e using calcula-the accuracalcula-te m e calcula-t h o d described b y M c l v e r a n d Bennett,'' a n d reproduce some o f the results given i n F i g . 4 o f Thomas,'* the dashed curves are
I
"0.0 0.5 10 1.5
kb/n
2.5 3.0
Fig. 3. Real ( ) and imaginary ( ) parts o f aoo v. kb/ir for offset d=0.
calculated u s i n g eqn (5.6) above. T h e w a v e n u m b e r p a r a m e t e r ka = 0.3, so the present t h e o r y w o u l d be expected to describe this s i t u a t i o n reasonably w e l l . T h e largest errors are f o r kb = 3 w h i c h is close t o the resonance at kb = TT where the present series s o l u t i o n breaks d o w n . H o w e v e r , here a n d i n m a n y other c a l c u l a t i o n s n o t presented, the q u a h t a t i v e features o f the t w o solutions are the same.
I f the channel w a l l s h a d n o effect o n the scattering p r o p e r t i e s o f the c y l i n d e r t h e n P w o u l d be u n i t y . H o w e v e r , b o t h the m e a n value, w i t h respect t o 9, o f P a n d the v a r i a t i o n s a b o u t this m e a n s h o w considerable dependence o n kb. F r o m eqn (5.4), the b e h a v i o u r o f P i n the s i t u a t i o n o f F i g . 2 is g o v e r n e d b y the expansion coefficients aoo a n d a n a n d these are p l o t t e d as a f u n c t i o n o f kb i n F i g s 3 a n d 4 f o r the offset d=0;in this s y m m e t r i c s i t u a t i o n , cross-tank resonances o c c u r o n l y at
kb = nTT. T h e m e a n value o f P is p r o p o r t i o n a l to I m aoo
a n d w i t h i n each i n t e r v a l between resonances i t is a m o n o t o n i c a h y decreasing f u n c t i o n o f kb a n d i n a c c o r d w i t h F i g . 2; this s i m i l a r i t y o f b e h a v i o u r between successive standing-wave frequencies was observed b y
Fig. 4. Real ( ) and imaginary ( ) parts o f a n v. kb/Tr for offset = 0.
P 1.05
Fig. 5. Pressure ratio P v. Ö/TT f o r ka = 0.3 and various kb where d is angular position on cylinder surface and 9 = TT is up-wave direction; comparison o f circular cylinder ( ) w i t h eUiptic cylinders having major axis parallel (• • •) and perpendicular ( ) to channel walls (see text for further
details o f geometry).
Thomas."* A s each such f r e q u e n c y is a p p r o a c h e d f r o m b e l o w I m aoo is singular; h o w e v e r the expansions i n ka w i l l t h e n break d o w n so this does n o t m e a n the s o l u t i o n to the exact linear p r o b l e m behaves i n this w a y . T h e v a r i a t i o n o f P a r o u n d the b o d y depends w i t h r o u g h l y equal w e i g h t o n the r e a l p a r t s o f aoo a n d a n . F r o m F i g . 4, a n is n o t singular at the standing-wave frequencies a n d , f o r kb > n, is generally o f smaller m a g n i t u d e t h a n aoo. I n other w o r d s , the image dipoles have less influence o n the l o c a l flow t h a n the image sources.
F o r the case o f a c i r c u l a r c y l i n d e r w i t h its axis o n the t a n k centre plane, the deviations f r o m the m e a n value o f
P g i v e n b y the final t e r m i n eqn (5.6) are weakest f o r kb = 1.5 (see also F i g . 4 o f Thomas'*). F r o m F i g . 3,
I m aoo is zero at kb ~ O.VTF ~ 2.2 so the pressures o n a c i r c u l a r c y l i n d e r w i f l be close t o the o p e n sea values f o r 1.5 < kb < 2.2. U n f o r t u n a t e l y , this is close t o the first a n t i s y m m e t r i c channel resonance at kb = TT/I w h i c h m a y cause p r o b l e m s i n an e x p e r i m e n t a l s i t u a t i o n .
F o r a slender c y l i n d e r aligned w i t h the i n c i d e n t wave d i r e c t i o n b o t h S a n d /xj are s m a l l so, b y eqn (5.4), P w i f l deviate h t t l e f r o m u n i t y . F o r a slender c y l i n d e r aligned across the t a n k , S is s t i l l s m a l l b u t p i w i l l n o t be; t h u s the m e a n d e v i a t i o n w i l l be close to zero b u t , depending on w h e t h e r o r n o t Re a ^ is close t o zero, the fluctuations a b o u t the mean m a y be s i g n i f i c a n t . T h i s last p o i n t is i l l u s t r a t e d i n F i g . 5 where P is p l o t t e d as a f u n c t i o n o f the p o l a r angle 0 f o r a c i r c u l a r cross-section a n d a n elliptic cross-section, w i t h m a j o r axis b o t h p a r a l l e l and p e r p e n d i c u l a r to the walls. B o t h cross-sections have been chosen t o have the same area S a n d the l e n g t h scale a is defined b y a^ = S/TT. T h e c i r c u l a r c y l i n d e r has radius a, w h i l e the elliptic c y l i n d e r has semi-m a j o r axis A = Via a n d s e semi-m i - semi-m i n o r axis B = a/VÏ so that the ellipse axes have lengths i n the r a t i o 2 : 1 . W h e n
a 0.5
Fig. 6. Imaginary part ( ) o f «QO V . kb/w for offset d = 0; also pfotted is (/cè)-i/2 ( ) .
the m a j o r axis is ahgned w i t h the t a n k w a l l s the d i p o l e s t r e n g t h
Ml
BjA
+
B)la^
(5.7)w h i l e w h e n the m a j o r axis is p e r p e n d i c u l a r t o the w a l l s
A and B m u s t be interchanged. T h e s o l u t i o n f o r t w o
-d i m e n s i o n a l flow past a n ellipse, r e q u i r e -d t o evaluate eqn (5.4), is described b y M i l n e - T h o n i s o n , ' ^ p p . 1 6 7 - 7 1 . I n F i g . 5, results are p l o t t e d f o r t w o channel w i d t h s . F o r the smaller w i d t h , kb = 1 a n d , as m a y be seen f r o m F i g . 4 a n d eqn (5.5), Re an is non-zero so t h a t w h e n the elliptic c y ü n d e r has its m a j o r axis across the channel there are substantial fluctuations a b o u t the m e a n value o f P. T h e second set o f curves is f o r kb = 2 w h e n , f r o m eqn (5.5), Re a n = 0 a n d the pressure r a t i o is s i m i l a r f o r a l l three geometries.
O n e o f the features n o t e d by Thomas'* is the s l o w decay o f t a n k effects w i t h increasing w i d t h (see, f o r example, his F i g . 5 ) . T o f u r t h e r i l l u s t r a t e t h i s , I m «O O J w h i c h c o n t r o l s the m e a n value o f P, is p l o t t e d f o r a larger range o f kb i n F i g . 6. A l s o p l o t t e d f o r c o m p a r i s o n is a curve o f {kb)~^^'^; u s i n g the m e t h o d described i n M c l v e r a n d Bennett,^ Section 6, i t m a y be s h o w n t h a t this is the c o n t r o l l i n g b e h a v i o u r as kb ^ oo o f a l l o f the expansion coefficients. T h i s m i m i c s the large a r g u m e n t b e h a v i o u r o f the H a n k e l f u n c t i o n s i n the image sets.
I f the b o d y is m o v e d o f f the centre hne o f the channel then the s i t u a t i o n is c o n s i d e r a b l y m o r e c o m p l e x . T h e s y m m e t r y a b o u t the t a n k centre plane is destroyed a n d , i n general, resonances w i l l occur at b o t h a n t i s y m m e t r i c a n d s y m m e t r i c frequencies. T h i s is i l l u s t r a t e d i n F i g . 7 f o r I m aoo w i t h d/b = 0.3. A s a f u n c t i o n o f the offset
d/b, I m aoo w i l l behave i n a reasonably consistent
f a s h i o n f o r values o f kb between any t w o successive standing-wave frequencies. F o r example, i n F i g . 8, i t is a m o n o t o n i c a l l y decreasing f u n c t i o n o f d/b w h e n
0 < kb < TT/I b u t has a single m a x i m u m w h e n
Fig. 7. Real (- -) and imaginary ( for offset rf= 0.3.
' parts of ctoo v. kb/n
B u t l e r and T h o m a s ' ^ have made c o m p u t a t i o n s f o r t w o c i r c u l a r cylinders s y m m e t r i c a l l y placed i n a n a r r o w wave t a n k and m a k e c o m p a r i s o n s w i t h results f o r a single c y l i n d e r i n their F i g . 2. F o r t w o cylinders the cross-sectional area S is d o u b l e d a n d the d i p o l e coefficient p^ a p p r o x i m a t e l y d o u b l e d w h e n c o m p a r e d w i t h a single cyhnder. Consequently, eqn (5.4) predicts t h a t the mean d e v i a t i o n o f P f r o m u n i t y and the a m p l i t u d e o f the v a r i a t i o n s a b o u t t h a t m e a n w i l l also be d o u b l e d a n d this is consistent w i t h the results o f B u t l e r a n d T h o m a s . ' ^ I t s h o u l d be n o t e d t h a t i n those results the w a v e l e n g t h is c o m p a r a b l e w i t h the spacing between cylinders a n d the t h e o r y given here is n o t strictly v a l i d . T h e present ' s m a l l - b o d y ' t h e o r y m a y be extended t o cover the case o f m u l t i p l e cylinders where the w a v e -l e n g t h is o f the o r d e r o f the spacing a n d this w i -l -l be the subject o f f u t u r e w o r k .
6 W A V E F O R C E S
T h e h o r i z o n t a l c o m p o n e n t s o f the wave f o r c e are calculated by i n t e g r a t i o n o f the pressure over the
l l l l l l l l l l l l l l l l l l l l l l : kh = 7 r / 5 1 1 I 1 1 1 1 1 1 1 1 1 1 M > 1 1 1 1 < i M > 1 1 -Jó6 = 3 7 r / 5 ^ 1 kb = 4 7 r / 5 ;^ , , , • 1 1 iNi 1 ' 0.0 0.1 0,2 0.3 0.+ 0.5 0,6 0,7 0.8 0,9 1.0
Fig. 8. Imaginary part o f V . d / b for various values of kb.
c y l i n d e r surface. U s i n g the d e c o m p o s i t i o n o f the v e l o c i t y p o t e n t i a l i n eqns ( 2 . 2 ) - ( 2 , 5 ) , the j {= x o r y) c o m p o n e n t o f the f o r c e is Re { f j e " ' " ' } where ƒ . = - ^ t a n h / c / 7 k J r Jkx )nj dl (6.1) a n d nj is the a p p r o p r i a t e c o m p o n e n t o f the o u t w a r d n o r m a l t o the c y l i n d e r surface. T o evaluate the integrals d e f i n e d b y eqn (6.1), the expansion o f the i n c i d e n t wave p o t e n t i a l to o r d e r {kxf and the i n n e r scattered p o t e n t i a l (3.45) are used. A f o r c e r a t i o
P _ f j , tank
t-j - 7 (6.2) J j , open sea
c o m p a r i n g the t a n k values w i t h open-sea values w i h be used f o r i l l u s t r a t i v e purposes. T h e o n l y terms i n e q n (3.45) t h a t c o n t r i b u t e t o this r a t i o , a n d so the o n l y ones t h a t need be evaluated explicitly, are those i n v o l v i n g the expansion coefficients a „ „ „ a„„„ b,„„. B y a n a p p l i c a t i o n o f Green's theorem over the fluid r e g i o n between the b o d y surface F a n d an enclosing circle at large distances i t m a y be s h o w n t h a t . r rx + x - d . r a - d r V a x + x)nydl = 2TTX, + T \ n^dl = lirnh + T Jiydl : 2A, (6.3)
where the f a r - f i e l d f o r m s i n eqns (4.6) a n d (5.3) have been used. F o r the i n - l i n e force the r a t i o is
1 + (kaf]-\mpLian +7Ti^^^bi 4
I
Pi la^fii 13, '01(6.4) R e f e r r i n g back t o eqns ( 4 . 1 0 ) - ( 4 . 1 2 ) , the three O { { k a f ) terms m a y be i d e n t i f i e d as a r i s i n g f r o m , respectively, the image i n - l i n e dipoles, cross-tank dipoles a n d sources. E i t h e r , o r b o t h , o f the last t w o terms m a y be i d e n t i c a h y zero f o r certain geometries. I f the b o d y is s y m m e t r i c a b o u t a plane a h g n e d w i t h the d i r e c t i o n o f w a v e advance then Aj is zero while i f i t is s y m m e t r i c a b o u t a cross-tank plane t h e n nii is zero. F o r zero offset d t h e n
Poi is zero.
F o r the simpler case o f a b o d y w i t h t w o v e r t i c a l planes o f s y m m e t r y , one o f w h i c h is aligned w i t h the i n c i d e n t wave d i r e c t i o n , then
F ^ = l + {ka)^\TTipiaii (6.5) C o m p a r i s o n s o f eqn (6.5) w i t h accurate results f o r a c i r c u l a r cyhnder {pi = \) are made b y M c l v e r a n d B e n n e t t p curves are given f o r fixed channel w i d t h b/a w i t h excellent agreement f o r values o f ka u p t o a b o u t 0.8. I n the present F i g . 9, the effect o f increasing channel w i d t h is i l l u s t r a t e d f o r ka = 0.3 s h o w i n g the r e l a t i v e l y r a p i d a p p r o a c h t o u n i t y o f the f o r c e r a t i o .
kb/TT
Fig. 9. Force ratio \F^\ v. kb/n f o r ka = 0.3; comparison of accurate ( ) and approximate ( ) theories for a
circular cylinder.
T h e p r i m a r y t a n k effect o n the in-line f o r c e is due t o the image in-line dipoles. A s p o i n t e d o u t i n the previous section, the m a g n i t u d e o f is generally s i g n i f i c a n t l y less t h a n t h a t o f aoo, w h i c h governs the m e a n t a n k effects o n the pressure field, and consequently the influence o f the t a n k walls o n the in-line f o r c e is m u c h less t h a n t h a t o n the mean pressure a r o u n d the cyhnder. Because o f this dependence o n the dipole coefficient pi, the f o r c e r a t i o w i f l be smaller i n m a g n i t u d e f o r a slender b o d y aligned w i t h the flow t h a n i f t h a t b o d y is o r i e n t e d across the t a n k . T h i s is i l l u s t r a t e d i n F i g . 10 where results are c o m p a r e d f o r a c i r c u l a r c y l i n d e r a n d e l l i p t i c cylinders o r i e n t e d p a r a l l e l and p e r p e n d i c u l a r to the w a l l s . T h e geometries are m o r e f u l l y described i n the p r e v i o u s section i n the discussion o f F i g u r e 5.
F o r the cross-tank force r a t i o
Fy
=
\ + {kaf\^^miiian+mhbn" ^ ^ A i j
(6.6) where the O
{{kaf)
terms arise i n the w a y described a f t e r eqn (6.4). F o r a b o d y t h a t is s y m m e t r i c a b o u t a v e r t i c a l plane aligned w i t h the d i r e c t i o n o f wave advance, so t h a t A i = 0, t h e n fy^ open = 0 a n d Fy is u n d e f i n e d . I n this case, the cross-tank wave f o r c e m a y be c o m p a r e d w i t h the in-line o p e n sea force t o getA } ^ = - { k a f ^ l 3 , , (6.7)
Jx, open sea " M l
w h i c h w i f l be zero f o r a c e n t r a l l y placed b o d y . A slender b o d y aligned w i t h the walls has /[ > pi w h i l e , w h e n placed across the t a n k , /] < px so the cross-tank f o r c e is m o r e significant i n the f o r m e r case.
7 C O N C L U S I O N
A t h e o r y has been given f o r the scattering o f waves b y a
Fig. 10. Force ratio |^';^.| v. ka f o r b/a = 5; comparison o f circular cyhnder ( ) with elliptic cyhnders having major axis parallel ( . . . ) and perpendicular ( ) to channel waUs (see text f o r further details of geometry).
v e r t i c a l c y l i n d e r o f a r b i t r a r y cross-section s t a n d i n g i n a n a r r o w wave t a n k . T h e t h e o r y assumes t h a t a t y p i c a l cylinder diameter is m u c h smaller t h a n other l e n g t h scales i n the p r o b l e m . I n t e r p r e t a t i o n s i n terms o f simple properties o f the images i n the t a n k walls have led t o an i m p r o v e d u n d e r s t a n d i n g o f the w o r k o f previous authors. Because no assumptions a b o u t the cross-section o f the c y l i n d e r have been made, the results m a y be used t o predict c y l i n d e r shapes a n d o r i e n t a t i o n t h a t are l i k e l y t o experience significant t a n k - c o n f i n e m e n t effects o n pressures a n d forces.
A C K N O W L E D G E M E N T S
T h e a u t h o r is g r a t e f u l t o M r G.S. B e n n e t t a n d D r C . M . L i n t o n f o r p r o v i d i n g c o m p u t e r code a n d t o D r G . P . T h o m a s f o r h e l p f u l c o m m e n t s .
R E F E R E N C E S
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of surface waves by a vertical circular cylinder i n a channel. Phil. Trans. R. Soc. Lond. A, 338 (1992) 325-57. 6. Mclver, P. & « nnett, G.S., Scattering of water waves by axisymmetric bodies in a channel. J. Engng Math., (in press).
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Mathema-tical Functions. Dover, New Y o r k , 1965.
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