floating cylinder

7/30/2019 Floating Cylinder

1/35

Ocean Engng, Vol. 15, No 6, pp. 549-583, 1988. 0029-8018/88 $3.00 + .00Printed in Gre at Britain. Pergamo n Press plc

H Y D R O D Y N A M I C I N T E R A C T I O N S I N F L O A T I N GC Y L I N D E R A R R A Y S - - I . W A V E S C A T T E R I N G

A . N . W I L L I A M SDepartmen t of Civil Engineering, University of Houston, Houston, TX 77004 , U.S.A.

a n dZ. DEMIRBILEKProduction Research Division, Conoco Inc., Ponca City, OK 74603, U.S.A.

Ab stract--T he hydrodynamic interactions due to wa ve scattering between the numbers of anarray o f stationary, truncated circular cylinders simulating the colum ns of an idealized tension-leg platform (TLP) are investigated. Th e m ethod of solution fo r the fluid velocity potentialinvolves replacing scattered waves by equivalent plane w aves together with non-planar correctionterms. This technique is, therefore, essentially a large spacing approximation. Use of thisapproach makes it poss ible to determine the hydrod ynamic interactions between the arraymembers utilizing only the diffraction characteristics of an isolated cylinder.Numerical results are presented for six array configurations consisting of 2-6 cylindersrepresenting the le gs of idealized TLPs. Calculations of the wave loads on these cy linders havebeen perform ed for a range of wave and structural parameters. It is found that, for certainparameter combinations, the influence of neighbouring bodies on the total wave field leads tohydrodyn amic loading on individual colum ns which is significantly greater than the loadingthey would experience in isolation. Th e presented results demon strate the significance ofhydrodynamic interactions between TLP columns and clearly indicate that these effects shouldbe considered b y the designers and researchers associated with TLP s.

1. I N T R O D U C T I O NT H E HY DR OD YN AM IC i n t e r a c t i o n s b e t w e e n t h e v a r i o u s m e m b e r s o f c o m p l e xm u l t i c o m p o n e n t o c e a n s t r u c t u r e s s u b j e c t e d t o w a v e a c t i o n is o f c o n s i d e r a b l e p r a c ti c a li m p o r t a n c e . I n t h is c a s e o f a te n s i o n - l e g p l a t f o r m ( T L P ) w h i c h c o n t a i n s s e v e ra l v e r t ic a lc o l u m n s o f r e l a t i v e l y l a r g e d i a m e t e r , i t is e x p e c t e d t h a t t h e i n f lu e n c e o f t h e n e i g h b o r i n gm e m b e r s o n t h e t o t a l w a v e f ie ld m a y l e a d t o h y d r o d y n a m i c l o a d i n g o n in d i v id u a lc o l u m n s w h i c h d i ff e r s si g n if i ca n tl y f r o m t h e l o a d i n g t h e y e a c h w o u l d e x p e r i e n c e i ni s o l a t i o n .

I n p ri n c ip l e , t h e h y d r o d y n a m i c l o a d i n g o n a n y t h r e e - d i m e n s i o n a l s t r u c t u r e o f a r b i t r a r yg e o m e t r y m a y b e c a lc u l a t e d n u m e r i c a l ly t h r o u g h t h e u s e o f s o u r c e d i s tr i b u t io n o rG r e e n ' s f u n c t i o n t e c h n i q u e s ( s e e , f o r e x a m p l e , S a r p k a y a a n d I s a a c s o n , 1 98 1 ). H o w e v e r ,s u c h a n a p p r o a c h m a y b e c o m p u t a t i o n a l l y e x p e n s i v e s in c e th e w e t t e d s u r f a c e o f t h ee n t i r e s t r u c t u r e m u s t b e d i s c r e t i z e d i n t o a n u m b e r o f sm a l l f a c e t s a n d a l a r g e s y s t e mo f a l g e b r a i c e q u a t i o n s s o l v e d t o o b t a i n t h e f lu i d v e l o c i t y p o t e n t i a l o n t h e s t r u c t u r a ls u r f a ce . A l s o , o n e o f se v e r a l a l te r n a t i v e s e m i - a n a l y t i ca l a p p r o a c h e s m a y b e u s e d t os o l v e t h e h y d r o d y n a m i c i n t e r a c t io n p r o b l e m ; f o r e x a m p l e , a d i r e ct m e t h o d i n v o l v i n gt h e s o l u ti o n o f a m a t r i x e q u a t i o n t o d e t e r m i n e t h e u n k n o w n c o e ff i c i e n ts o f a n a s s u m e df o r m f o r t h e t o ta l s c a t t e r e d w a v e f ie ld ( S p r i n g a n d M o n k m e y e r , 1 9 74 ) o r a m u l ti p le

549


2/35

550 A.N. WU.I.IAMSand Z. DI~MIRBIL~Kscattering technique in which each scattering event within the array is treatedsuccessively (Ohkusu, 1974). Recently, Kagemoto and Yue (1986) have presented anapproach which combines the features of the direct matrix method and the multiplescattering technique and which is algebraically exact within the context of the linearizedwave theory.However, when both the member spacing to incident wavelength ratio and themember spacing to cylinder diameter ratio are sufficiently large, then the full diffractionproblem which includes the consideration of the evanescent wavefield in the interactionsolution need not be solved and a theory based on a large-spacing assumption mayreasonably be used (Martin, 1984). One such method is the modified plane-wavetechnique in which a cylindrical wave scattered by one body is replaced in the vicinityof another body by an equivalent plane wave together with a non-planar correctionterm, and evanescent waves are neglected when computing interference effects (Mclverand Evans, 1984; McIver, 1984; Abul-Azm and Williams, 1987). The method is anextension of the plane wave approach of Simon (1982) and has been shown to providea major saving in computational effort when compared to the direct method or themultiple scattering technique (Mclver and Evans, 1984; Mclver, 1984). In commonwith the method of Kagemoto and Yue (1986) and Ohkusu (1974), the modified plane-wave method enables the hydrodynamic loading on a structural array to be determinedfrom a knowledge of the wave scattering characteristics of the individual members only.

In the present paper the modified plane wave method is utilized to calculate thehydrodynamic interactions between the members of an array of stationary truncatedcircular cylinders of equal radius. The fluid is assumed to be inviscid and incompressibleand to undergo small-amplitude, irrotational motion. Numerical results are presentedfor arrays of 2-6 cylinders for a range of wave and structural parameters. Comparisonsare made with published results obtained from the potentially more accurate sourcedistribution technique and good agreement with the present method is shown. It isfound that for certain parameter combinations individual cylinders may experiencehydrodynamic loading significantly greater than that which they would experience inisolation, even at a relatively large spacing.

2, THEORETICAL DEVELOPMENTAn array of N stationary truncated circular cylinders of equal draft b and radius aare situated in water of uniform depth d, the clearance beneath each cylinder will be

denoted by h ( = d - b ) . Cartesian coordinates will be employed with the x- and y-axesin the horizontal plane of the sea-bed and the z-axis pointing vertically upwards froman origin on the sea-bed directly below the center of cylinder 1. The center of eachcylinder, at point ( x j , y fl j = 1 ,2 . . . . N is taken as the origin for a local polar coordinatesystem (r j , Oj ) where 0 is measured counterclockwise from the positive x-axis. Thecenter of the kth cylinder has polar coordinates (R~k, e~j~) relative to that of the jthcylinder for j , k = 1 ,2 ,. . .N . A horizontal section showing the coordinate relationshipsbetween the jth and kth cylinders is shown in Fig. 1. The cylinder array is subjectedto an incident train of regular surface waves of amplitude H (crest-to-trough height2H), angular frequency to and wavenumber ko propagating at an angle 13 to the positivex-direction.


3/35

Hydrodynamic in te rac t ions in f loa t ing cyl inder a r rays - - - Ip k

rjR~k

J

Fro. 1 . H or izonta l c ross -sec t ion showing coordina te re l a t ionships be tw een cyl inders j a nd k .

55 1

A s s u m i n g t h a t t h e f l u i d i s i n v i s c i d a n d i n c o m p r e s s i b l e , t h e s m a l l a m p l i t u d e ,i r r o t a t i o n a l f l u i d m o t i o n m a y b e d e s c r i b e d i n t e r m s o f a v e l o c i t y p o t e n t i a l ~ b( x, y , z , t ) ,w h e r e t h e f l u id v e l o c i t y v e c t o r q -- V ~b . I t f o l l o w s t h a t t h i s fl u i d p o t e n t i a l m u s t s a t i s f yL a p l a c e ' s e q u a t i o n

V2~b = 0 (1 )i n t h e r e g i o n o f f l o w . T h e s e a - b e d b o u n d a r y c o n d i t i o n i s

O~_~= 0 o n z = O , ( 2 )O zw h i l s t t h e l i n e a r i z e d f r e e - s u r f a c e b o u n d a r y c o n d i t i o n i s

a2,1, + a ~0 t- T g ~ z = 0 o n z = d , ( 3 )w h e r e g i s t h e a c c e l e r a t i o n d u e t o g r a v i t y . T h e b o u n d a r y c o n d i t i o n s o n t h e i m m e r s e dc y l i n d e r s u r fa c e s a r e b e s t e x p r e s s e d i n t e r m s o f t h e i r l o c a l p o l a r c o o r d i n a t e s , t h a t i s

O r j = 0 o n r j = a , h < - z < d , ( 4 a )0~_~ = 0 o n z = h , O < - r j < a , ( 4 b )O z

f o r j = 1 , 2 . . . . N . In a d d i t i o n t h e r e i s a r a d i a t i o n c o n d i t i o n o n t h e s c a t t e r e d c o m p o n e n to f 6 w h i c h e n s u r e s t h a t t h e s c a t te r e d w a v e s p r o p a g a t e a w a y f r o m t h e d i f f r a c t i n gc y l i n d e r .


4/35

5 5 2 A . N . W . J a A M S a n d Z . [ )E M ,R m L t! K

Y

~ X

Z =~7(r, o, t) + d ~

2 I dII - 2 a - II I h

I // / / / / / / / / / / / / / / / ~ / / / / / / / / / / /FIG. 2. De finition sketch for isolated cylinder.

3 . S O L U T I O N F O R A N I S O L A T E D C Y L I N D E RB e f o r e i n v e s t i g a t i n g t h e h y d r o d y n a m i c i n t e r a c t i o n s b e t w e e n t h e v a r i o u s m e m b e r s

o f a c y l i n d e r a r r a y i t i s f i r s t n e c e s s a r y t o o b t a i n a d e s c r i p t i o n o f t h e i n t e r a c t i o n o f as in g l e s tr u c t u r e a n d t h e i n c i d e n t w a v e f i e ld . T h i s s o l u t i o n w a s f ir st o b t a i n e d b y G a r r e t t( 1 9 7 1 ) .

C o n s i d e r a s t a t i o n a r y , i s o l a te d , t r u n c a t e d c y l i n d e r ] o f d r a f t b a n d r a d i u s a s i t u a t e di n w a t e r o f u n i f o r m d e p t h d a s d e p i c t e d i n Fi g. 2 . T h e f lu id d o m a i n is d i v i d e d i n t o 2r e g i o n s , o n e b e n e a t h t h e c y l i n d e r , t h e i n t e r i o r re g i o n ( 1 ) , th e o t h e r e x t e r i o r t o t h ec y l i n d e r a n d e x t e n d i n g t o i n f in i ty in t h e h o r i z o n t a l p l a n e ( 2 ). T h e p o t e n t i a l i n th ee x t e r i o r r e g i o n is d e c o m p o s e d i n to i n c i d e n t a n d s c a t te r e d c o m p o n e n t s , t h a t is

d~ = ~bl fo r O < - r < - a, O < - z < - h , ( 5 a )d~ = d ~ 2 1 + d ~ f o r a < - - r < - ~ , O < ~ z< ~ d , ( 5 b )

w h e r e , f o r c o n v e n i e n c e t h e s u p e r s cr i p t j h a s b e e n d r o p p e d . T h e m a t c h i n g c o n d i ti o n sa t t h e i n t e r f a c e b e t w e e n t h e t w o r e g i o n s r e p r e s e n t i n g c o n t i n u i t y o f p r e s s u r e a n d m a s sf l u x a c r o s s t h e i n t e r f a c e m a y b e w r i t t e n

( ~ 1 = ( I ) 2 1 + l ~ )2 s f o r O < - z < - h , o n r = a , ( 6 a )~- for O < - z < - h , o n r = a . ( 6 b )a r O r O r


5/35

Hy drodynam ic interactions in floating cylind er arrays---I 553I n e q u a t i o n s ( 5 ) a n d ( 6 ) t h e i n c i d e n t p o t e n t i a l +2 i i s t h a t c o r r e s p o n d i n g t o a r e g u l a rw a v e o f a m p l i t u d e H p r o p a g a t i n g i n t h e p o s i t i v e x - d i r e c t i o n a n d i s g i v e n i n p o l a rc o o r d i n a t e s b y

g H c o s h k o z ~ 2 1R ~ ~ c o ~ s h - k ( ~ m / ~ = O m i m + l J m ( k r ) co s m O e - i ~ t , (7 )i n w h i c h Jm d e n o t e s a B e s s e l f u n c t i o n o f t h e f i r s t k i n d o f o r d e r m a n d R e d e n o t e s t h er e a l p a r t o f a c o m p l e x e x p r e s s i o n . I n e q u a t i o n ( 7) e, , is N e u m a n n ' s n u m b e r , % = 1 ,e,~ = 2 , m --- 1 a n d t h e w a v e f r e q u e n c y a n d w a v e n u m b e r a r e l i n k e d t h r o u g h t h ed i s p e r s i o n r e l a t i o n

o 2 = g k o t a n h k o d . (8 )T h e v e l o c it y p o t e n t i a ls i n e a c h d o m a i n a r e n o w e x p r e s s e d in t h e f o r m

~cd ?l - g n ~ a t t tm ) ( r ,z ) C O S m O e - R t, (9a )R e o m =O~ c~b2 = ~ ~ ~ [ m ) ( r , z ) c o s m O e - i ~ ' . ( 9 b )R e w m : O

I n r e g i o n ( 1 ) t h e d i m e n s i o n l e s s s p a t i a l s o l u t i o n i s e x p r e s s e d a sA ' g { r_ ] : I m ( n ~ rr /h )~ m ) ( r , z ) = 2 - ~ a] + z .,~ A(m)--, ( n lr z/ h ) (10), = ~ i m ( n T r a/ h ) c o s

w h e r e Im d e n o t e s a m o d i f ie d B e s s e l fu n c t i o n o f t h e f i rs t k i n d o f o r d e r m . T h e f o r m o fe q u a t i o n ( 1 0 ) e n s u r e s t h a t +1 s a ti s fi e s t h e g o v e r n i n g L a p l a c e e q u a t i o n a n d t h e s e a - b e db o u n d a r y c o n d i t i o n i n r e g i o n ( 1 ) . S e t t i n g r = a , e q u a t i o n ( 1 0 ) b e c o m e s a c o s i n e F o u r i e rs e r i e s w i t h c o m p l e x c o e f f i c i e n ts A ~~) w h i c h a r e g i v e n b yA ( ,m ) = ~ ~ ( f O ( a , z) c o s ( n ~ r z / h ) d z (11)

f o r n = 0 , 1 , 2 . . .I n t h e e x t e r i o r r e g i o n ( 2 ) t h e d i m e n s i o n l e s s s p a t i a l s o l u t i o n m a y b e w r i t t e n a s, ~ m ) ( r , z ) = e m i m + ] C o s h k (r z { j m ( k o r ) J '~ ( k o a) }c o s h k o~ d H ' ~ ( k o a ) H m ( k ( )

H m ( k o r ) q ~ l B (q m ) g r n (k q F )+ n 6 r n ) n ' m ( k o a ) Z o ( z ) + = g ' ~ ( k q a ) q ( z ) , (12)w h e r e H , , i s th e H a n k e l f u n c t io n o f th e f i rs t k i n d o f o r d e r m , c h o s e n i n o r d e r t h a t t h es c a t t e r e d c o m p o n e n t o f + 2 s a ti sf ie s t h e a p p r o p r i a t e r a d i a t i o n b o u n d a r y c o n d i t i o n . I ne q u a t i o n ( 1 2) t h e f u n c t i o n s K , , d e n o t e t h e m o d i f i e d B e s s e l f u n c t io n o f t h e s e c o n d k i n do f o r d e r m , p r i m e s d e n o t e d i f f e r e n t ia t i o n w i t h re s p e c t t o a r g u m e n t a n d t h e e i g e n v a l u e sk q f o r q = 1 ,2 . . . . a r e t h e p o s it iv e r ea l r o o t s o f

(o2 + gk q t a n k q d = 0. (13)


6/35

554 A .N . WILLIAMSand Z. DEMIRBILEKT h e f u n c t i o n s Z q ( Z ) a r e o r t h o n o r m a l o v e r t h e i n t e r v a l [ O ,d ] a n d a r e d e f i n e d b y

7 -~ 2 c o s h k o zZ o ( z ) = [ f ~ s i n h 2 k o d / 2 k ~ d ] ~ for q -> 0 , (14)

~2"COS k q zZ q ( z ) - [1 + sin 2 k q d / 2 k q d ] or q - I . (15)

T h e c o m p le x c o e f f i c i e n t s B~ qm ) a p p e a r in g i n e q u a t i o n ( 1 2 ) a r e , a t p r e s e n t , u n k n o w n .T h e f o r m o f e q u a t i o n ( 1 2 ) e n s u r e s t h a t qb2 s a t is f ie s t h e g o v e r n in g L a p l a c e e q u a t i o nt o g e t h e r w i t h t h e r a d i a t i o n , s e a - b e d a n d l i n e a r i z e d f r e e - s u r f a c e b o u n d a r y c o n d i t i o n s i nr e g io n ( 2 ) . D i f f e r e n t i a t i n g e q u a t i o n ( 1 2 ) w i th r e s p e c t t o r a n d s e t t i n g r = a , t h e nm u l t i p ly in g b o th s i d e s b y Z q ( z ) a n d i n t e g r a t i n g o v e r [ 0 ,d ] , l e a d s t o t h e f o l l o win ge x p r e s s io n f o r t h e c o m p le x c o e f f i c i e n t s B~ qm),

1 I d O ~ t ,~ 'mB


7/35

Hydrodynamic nteractions in floating cylinder arrays---I 555

F~ = - p Jo Jo 0~- r dr dO (20b)

M = - 2 a o i S f [ ( z - h ) ,= cos(xr-O) d0 dz(27 (. 0~bl ~=hCOS 0r_0 ) rE dr dO (20c )-Jo JoT h e f o r c e s a n d m o m e n t o n a n i s o l a t e d c y li n d e r t a k e t h e f o l lo w i n g f o r m s

F,, ~,o, 2i (s in h k o d - s inh koh)p g~ - w a 2 - e - ~ ~ - - ~ o d

i B ~ l ) H l ( k o a ) V r 2 (s inh kod - s inh k ,h )k o a H ~ ( k o a ) [1 + sinh 2 k o d / 2 k o d ] ~

= B~ q l )K l ( k q a ) ~ - ( s i n k q d - s in k q h ) ]- ~ k q a I ( i ( k q a ) [1 + sin 2 k q d / Z k q d ] ~ j , (21a)q=l

F ~ . ( i A ~ ) 2 ih = ( - 1 ) " l l ( m r a / h ) lp g ~ - a 2 - e ' 0 " 1 ~ + - - Z A # ') (21b)a n , = x n lo(n~ra/h ) J '

[cos2k,g/(Jl(ka) H'l(koa)Jl(ka)Hl(/q,a))i B [ ~ ) H ~ ( k o a) ~ - 2

H~(koa)[1 + sinh 2 k o d / 2 k o d Pg ( 1 ) g l ( k q a ) ~ ] i A h 1 )

- i Z g ~ ( k q a) [ 1 + sin 2 k q d / 2 k q d ] J 8q=l A . ( - X ) h- t ~ . . . . . ( - - l o (n ~ r a/h ) 2 l l (n ' r r a / h ) . (21c), ,= 1 t l ( n 'c r a m ) \ m r a n

4 . E X T E N S I O N T O C Y L I N D E R A R R A YTh e d i f f icu l ty in general iz ing the resu l ts fo r an i so la ted cy l inde r to the case o f a cy l indera r ray a r i ses f ro m th e mu tu a l h y d ro d y n am ic in t e r fe ren ce e ffec t s b e tw een th e cy l in d er s an dt h e p r o b l e m o f h o w t o s u m t h e v a r i o u s s c a t t e r e d w a v e c o m p o n e n t s o n e a c h i n d i v i d u a l

cy l inder . In o rd er to s im pl i fy the so lu t ion , the spacings R ~ k ] , k = 1 ,2 . . . . . N b e t w e e n t h e


8/35

556 A . N . WII.LIAMS nd Z. DEMIRBILEKc y l i n d e r s w i l l b e a s s u m e d t o b e m u c h l a r g e r t h a n t h e i n c id e n t wa v e l e n g th , i . e . k o R j k1 . U n d e r t h i s a s s u m p t i o n t h e d i v e r g i n g w a v e s e m a n a t i n g f r o m o n e c y l i n d e r m a y b er e p l a c e d i n t h e v i ci n it y o f a n o t h e r c y l i n d e r b y a p l a n e w a v e t o g e t h e r w i t h a n o n - p l a n a rc o r r e c t io n t e r m ( M c l v e r a n d E v a n s , 1 9 8 4 ; M c l v e r , 1 9 8 4 ; A b u l - A z m a n d W i l li a m s , 1 98 7) .

C o n s i d e r t h e j th c y l i n d e r i n t h e a r r a y , c e n t e r e d a t (x j,y y) . T h e v e l o c i t y p o t e n t i a l d u et o a r e g u l a r w a v e p r o p a g a t i n g a t a n a n g l e 13 t o t h e p o s i t i v e x - a x is r e f e r r e d t o t h e ( x , y )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 i sg H c o s h k o zbzt - ~ cosslak o d sin (kox cos13 + k o y sin13 - ~ot). (22)

T h i s p o t e n t i a l m a y b e r e f e r r e d t o t h e l o c a l p o l a r c o o r d i n a t e s y s t e m ( r j , 0 j ) c e n t e r e d a t( x j , y j ) a sg H c o s h k o z

+~ ' R e ~ ~ P i i e x p {ik ,)r jcos(0 i -13)}e - i ' ' (23 )w h e r e P j i s a p h a s e f a c t o r d e f i n e d b y P j = e x p {i(k~rxjcos13 + k o y j s in13)} . Th is express ionm a y b e r e w r i t t e n a s

g H c o s h k o zd)~, R e ~ co~shkod v] ~ ~ .mim+|Jm(kor]) COS m(O j--13)e i 't . (24)m : 1}

R e f e r r e d t o a n o r ig i n a t t h e c e n t e r o f c y l i n d e r j t h e i n t e r io r a n d e x t e r i o r p o t e n t i a l sh a v e t h e f o r m s

_ g Hf~){R e ~ P ] ~ ~ I ttm ) ( r] 'z) cos m (O j-13 )e i ,, ,, (25a)m=O

oc+~ =~ g H p ~ ~ , ~ m ) ( r j , z ) c o s m ( O i - 1 3 ) e - i '% ( 2 5 b )/~e 00 m=O

I t c a n b e s e e n t h a t t h e e q u a t i o n s w h i c h d e t e r m i n e t h e p o t e n t i a l c o e f f i c i e n t s A (,m) a n dB(~m) w i l l b e i d e n t i c a l t o t h o s e o b t a i n e d p r e v i o u s l y in t h e i s o l a t e d c y l i n d e r c a s e , i .e . t h ep o t e n t i a l c o e f f ic i e n ts a re i n d e p e n d e n t o f t h e p a r t i c u l a r c y l i n d e r , ] . T h u s t h e s c a t t e r e dp o t e n t i a l i n t h e e x t e r i o r d o m a i n c a u s e d b y a w a v e i n c i d e n t o n c y l i n d e r j h a s t h e f o r m

w h e r e( D ~ ( r J ' O ] ' z ' 1 3 ) ; e ~ - PJ m= o ~Zt~sm)(rJ'z) s m ( O j - 1 3 ) e - i ' " (26)

~ ) ( r i , z ) = _ +1 co sh k o z J ' , . ( k o a ) Hm( k o r j )e , . t " c o s h k o d H ~ ( k ( ~ )H , ,,( ko rj) ~ K m ( k q r j )+ B~ ") H ' ., ( koa) Zo (z ) + ~ B(qm) Z o ( z ) .q = l g'~(kqa) (27)


9/35

Hydrodynam ic nteractions in floating cylind erarrays---I 557C o n s i d e r a n o t h e r c y l i n d e r , k , a d i s t a n c e R j k a w a y f r o m c y l i n d e r j a s s h o w n i n F ig . 1 .T h e w a v e s c a t t e r e d f r o m c y l i n d e r j m a y h e r e f e r r e d t o a n o r i g i n a t c y l in d e r k b y t h eB e s s e l f u n c t i o n a d d i t i o n t h e o r e m ( W a t s o n , 1 9 4 4 ) . B y t h e a d d i t i o n t h e o r e m ,

oo

H , , , ( k o r i ) c o s m ( O j - a i k ) = E

H , , , ( k o r i ) s in m ( 0 j - % , ) = EH , , ,+ e ( k o R j k ) J e ( k o r k ) cos e (0tk/--0k) (28a)

H m + e ( k o R j k ) J e ( k o r k ) sin e (akj--Ok). (28 b)S o r e f e r r e d t o a n o r i g in a t c y l i n d e r , k , t h e s c a t t e r e d w a v e e m a n a t i n g f r o m c y l i n d e r j i sg i v e n a p p r o x i m a t e l y b y

+ ~ . ,( r , O t , ,z , f t ) - g Hm = O

. ( _ ~mim+lco sh k o z J ' , , ( k o a ) B ( ~ ' ) o ( z ) ]co sh k o d H m ( k o a ) + H " ( k o a ) / "(, , ,+ e ( k o R i k ) J e ( k o r k ) cos m(%k--13) COS e(a kj--0 k) (29)

- s i n m ( % k - f i ) s i n , ( o t k j - O k ) ) } e - i ' t ,s i n c e t h e e v a n e s c e n t m o d e s m a y b e n e g l e c t e d f o r l a r g e k o R i k . A lso , fo r l a rg e k o R j , ,H , , , + e ( k , r R , ) _ ( _ i ) ~ H m ( k o R j ~ , ) { 1 + i (, 2 + 2 m , ) }2 k , ~ j k + O [ (k o R j k ) 2 ] (30)a n d s o t h e s c a t t e r e d p o t e n t i a l a t c y l i n d e r k c a u s e d b y a w a v e i n c i d e n t o n c y l i n d e r j m a yb e w r i t t e n a s

l ~ ( r k , O k , z , f 3 ) R e O ) P j S ( R j , , a j k , f3 ) e-ik,"ks(~*F%

in w h ich+ D(r,,0k,13)} cosh k o z e - i " ' ,

m = 0 H~,(koa)[1 + sinh 2 k o d / 2 k o d ] ~ m m + l J 'm ( k ( l ) I- co sh k o d H ~ , (k o a)J n , , , ( k , r R j , ) co s m ( a j , - ~ )

i s t h e a m p l i t u d e o f t h e e q u i v a l e n t p l a n e w a v e a n d

(31)

(32)


10/35

558 A . N . WILLIAMS and Z. DEMIRBILEKDk j ( f3 ) = D ( rk , Ok , f 3 )

= lq;i~,jk Ski(13) ~ (- -1 ) e g2je (kork ) COS W OLkj--Ok) (33)#-4 )

+ rk / (~) ~ e t ( - - i ) ~'t J e ( k o r k ) sin e (~kj--Ok)[ =11

i s t h e n o n - p l a n a r c o r r e c t i o n t e r m , w h e r e

T k j( [3 = T ( R j k , % k , [ 3 ) = m=o 1 + sin h 2 k o d / 2 k o d ] ~e m m + l : /~ ,, (koa) / . . . . . s inm(% k--f3 ) . (34)co sh k o d H m ( k o a ) ] n ,. tX ( ~ X j k ) m

S in ce th e ex p ress io n fo r ~ i s v a l id t o O [ ( k o R / k ) - 3 / 2 ] , i t i s co n s i s t en t t o ap p ro x imateH m ( k o R j k ) b y { im a }H , . ( k o R j k ) ~ H o ( k ( ~ i k ) 1 + 2 ~ + O [ ( k R j k )- 2 ] (35)a n d t o s u b s t i t u t e t h e a s y m p t o t i c f o r m o f t h e H a n k e l f u n c t i o n o f z e r o o r d e r , ( W a t s o n ,1944) to y ie ld a s impl i f ied fo rm for Sk i ( f 3 ) , n a m e l y

{ B ,~ _ _ , , H ' . , ( k o a ) [1 + sinh 2 k o d / 2 k o d ] ~

(36)i ( 4 m 2 - 1 ) te , ,, i " + ' J ' ~ ( k o a ) / 1 + c o s m ( O t / k - - ~ ) e i ( k o R / k - ' r / 4 ) .

- co sh k o d H ~ ( k ( a ) ] ~ }T h e to t a l i n c id en t p o ten t i a l o n cy l in d er j i n an a r ray o f N cy l in d er s co n s i s ts o f t h reec o m p o n e n t s :( 1) t h e i n c i d e n t p l a n e w a v e f r o m o u t s i d e t h e c y l i nd e r g r o u p w h i c h h a s a p o t e n t i a l ,m e a s u r e d r e l a ti v e t o c y l i n d e r j , o f

g H co sh k(,z e,,,i,,,+ 1 Jm (k0rj) c os m ( O j - ~ ) e - i j t , (37)(2 ) th e p l an e w av e ap p ro x imat io n to th e w av e f i e ld a t t h e j t h cy l in d er d u e tos c a t t e r i n g f r o m t h e r e m a i n i n g ( N - l ) c y l i n d e r s , t h i s h a s t h e p o t e n t i a l

o / 4O~ Z CJ e--iko~fs(~Jk--OPcos h koz e-i~ '0O k=lk g * /

(38)


11/35

Hydrod ynam ic interactions in floating cylinder arr ay s-- I 559w h e r e c jk is th e ( u n k n o w n ) c o m p l e x a m p l i t u d e o f t h e r e s u l t a n t a p p r o x i m a t i n g p l a n ew a v e a t t h e j t h c y l i n d e r , r e p r e s e n t i n g s c a t t e r in g f r o m t h e k t h c y l i n d e r ,( 3 ) t h e n o n - p l a n a r f i r s t - c o r r e c t i o n t o t h e s c a t t e r e d p l a n e w a v e s . T h i s d e p e n d s o n t h ep l a n e w a v e s s c a t t e r e d b y a l l o f t h e c y l i n d e r s ( t y p i f i e d b y t h e k t h c y l i n d e r ) a n d c o n s i s t so f tw o p a r t s , ( a ) t h e c o r r e c t i o n d u e t o t h e s c a t te r in g o f th e i n c i d e n t w a v e f r o m o u t s i d et h e c y l in d e r g r o u p b y th e o t h e r ( N - 1 ) c y l in d e r s an d ( b ) t h e c o r re c t i o n t o t h e s c at te r in go f t h e p l a n e w a v e s i n (2 ) fr o m t h e o t h e r ( N - l ) c y l in d e r s, i .e . t h e c o r r e c t io n s t o th e" b a c k - s c a t t e r e d ' w a v e s . T h i s p o t e n t i a l , c o n s i s t i n g o f p a r t ( a ) a n d p a r t ( b ) , i s g i v e n b y

P g D ( r j ,O j ,~ ) + C k e D ( r j ,O j ,e t ,~ ) c o s h k o z e - i t . (39)k g : j e ~ k

T h e e q u i v a l e n t p l a n e w a v e a m p l i t u d e Cjk i s t h e t o t a l s c a t t e r e d w a v e f r o m t h e k t hc y l in d e r e v a l u a t e d a t t h e j t h c y l in d e r a s d is c u s se d a b o v e a n d m a y b e d e t e r m i n e d a sf o l lo w s . T h e p l a n e w a v e a m p l i t u d e cjk c o n s i s t s o f t w o p a r t s , o n e d u e t o t h e s c a t t e r i n go f t h e i n c i d e n t w a v e b y c y l i n d e r k a n d t h e o t h e r d u e t o t h e s c a t t e r i n g b y c y l i n d e r k o ft h e p l a n e w a v e s s c a tt e r e d b y all t h e o t h e r c y l i n d e rs i .e . t h e p l a n e w a v e c o m p o n e n t s o ft h e b a c k - s c a t t e r e d w a v e s . S o ,

Nc jk = P k S ( R k j , a k j , ~ ) + ~ C ke S ( R k j ,e t k i, ae g ) (40)

~ =1 ~ k

a n d a p p l y in g t hi s co n d i t i o n f o r j , k = 1 ,2 . . . . N , j : /: k , g iv e s N ( N - 1 ) e q u a t i o n s f o r t h ee q u i v a l e n t p l a n e w a v e a m p l i t u d e s c j k .T h e t o t a l w a v e p o t e n t i a l f o r e a c h c y l i n d e r m a y n o w b e o b t a i n e d b y r e - s o l v i n g t h ei s o l a t e d c y l i n d e r p r o b l e m d i s c u s s e d i n t h e p r e v i o u s s e c t i o n w h e r e i n e a c h c a s e t h ei n c i d e n t w a v e f i e l d h a s b e e n m o d i f i e d a c c o r d i n g t o t h e a b o v e a n a l y s i s t o t a k e i n t oc o n s i d e r a t i o n h y d r o d y n a m i c i n t e r f e r e n c e e f f e c t s .

5 . N U M E R I C A L R E S U L T S A N D D I S C U S S I O NI n t h i s s e c t i o n , n u m e r i c a l r e s u l t s f o r a n u m b e r o f t e s t c a s e s a r e p r e s e n t e d . F o r e a c ho f t h e t e s t c a s e s a n a l y z e d , t h e r e s u l t s w i l l b e p r e s e n t e d a s d i m e n s i o n l e s s f o r c e o rm o m e n t c o e f f i ci e n ts w h e r e t h e l o a di n g o n t h e a r ra y m e m b e r h a s b e e n n o r m a l i z e d b yt h e c o r r e s p o n d i n g l o a d i n g c o m p o n e n t o n a n i s o l a t e d c y l i n d e r .F i g u r e 3 p r e s e n t s s i m p l e s k e t c h e s o f a l l s i x a r r a y c o n f i g u r a t i o n s n u m e r i c a l l yi n v e s t i g a t e d i n th i s s t u d y . D u e t o t h e d i r e c t a p p l i c a t i o n o f th e p r e s e n t w o r k t o t h et e n s i o n l e g p l a t f o r m d e s i g n c o n c e p t t h e a c r o n y m T L P p r e c e d e s t h e s e c a s e s .F o r c o n v e n i e n c e , a l l c o l u m n s a r e a s s u m e d t o b e o f e q u a l r a d iu s a a n d d r a f t b = 3 a ,t h e w a t e r d e p t h i n a ll c a s e s w a s t a k e n t o b e e q u a l t o t e n r a d ii , i . e . d = 1 0a . A l s o , t h ec e n t e r - t o - c e n t e r s p a c i n g b e t w e e n a d j a c e n t c y l i n d e r s w a s t a k e n t o b e u n i f o r m , R = 5 a .T h e o n l y e x c e p t i o n t o t h e a b o v e o c c u r s i n F i g . 4 w h e r e t h e n u m e r i c a l r e s u l t s o b t a i n e du s in g t h e p r e s e n t a p p r o a c h a r e c o m p a r e d t o t h o s e o f M a t s u i a n d T a m a k i ( 1 98 1 ). F o rt h is c a s e d = 1 0 a , b = 0 . 5 a a n d a n a d d i t i o n a l s p a c i n g o f R = 3 a is a l s o c o n s i d e r e d .O n e c o n s e q u e n c e o f t h e d r a f t b = 3 a c h o s e n f o r t h e n u m e r i c a l e x a m p l e s is t h a t t h ev e r t i c a l ( h e a v e ) f o r c e c o m p o n e n t i s v e r y m u c h l e s s t h a n t h e h o r i z o n t a l ( s u r g e , s w a y )c o m p o n e n t s w h e n e v e r t h e s e a r e n o n - z e r o . T h u s t h e n o r m a l i z e d p i t c h a n d r o l l


12/35

560 A.N. WILLIAMS and Z, DEMIRBII.EK

R

TLP 2

R//, I

J R"-,,# ITLP 5 2 L ~ R

# 'I RI

" " I~ - . . .TL P 3 t ) z

X/ \" / , , "

/ \

\ /\ /R\ / R\ /

TLP 6

I II I R

T L P 4 I Ii I

I I II I IR I I I

Flo. 3. Geo me tric configurations investigated in this study.TLP6R

o v e r t u r n i n g m o m e n t s a r e i n d is t in g u i sh a b l e f r o m t h e n o r m a l i z e d s u rg e a n d s w a y f or c e si n t h e f o l l o w i n g f i g u r e s . I t i s e m p h a s i z e d t h a t t h i s i s s i m p l y a c o n s e q u e n c e o f t h ee x a m p l e g e o m e t r y c h o s e n a n d , i n g e n e r a l , t h e p it c h a n d r o ll m o m e n t s w i ll c o n t a i ns ig n if ic a nt c o n tr i b u t io n s f r o m t h e h y d r o d y n a m i c p r e s s u r e c o m p o n e n t a c ti n g b o t h o nt h e s i d e w a l l s a n d o n t h e u n d e r s i d e o f t h e s t r u c t u r e . T h e c o n t r i b u t i o n f r o m t h e v e r t i c a l


13/35

Hyd rodyna m ic in te rac t ions in f loa t ing cy l inder a r ray s - -I 561

2 5

2

tC

P R E S E N TS T U D Yox

0.'.

M & TS E P A R A TION D IS TA N C E R = 50S E P A R A TION D IS TA N C E R = 50

. . . . . S I NG L E B O D Y

/ 3 , ' , . 4 " . , ' - X t .. . . .I I I I I

0.5 1 .0 1 .5 2 .0 2 .5k o O

I5 ,0

2 .5

2 . 0

1.5o l

i. o

0.,5

0 0

P R E S E N TS T U D Y M S T ~ SEPARATION DISTANCE R=~k: l

I , X ~ ,~ S E P A R A TION ;D ISTA N C E R = 50" .~k , . . . . . S IN GLE B OD Y'.~~ , _ . - - - @

"-~..

I I I I I I0.5 1 .0 1 .5 2 . 0 2 .5 3 .0%0

C y l i n d e r 1

2~

2,C

N o 1.5:=

N I . Gu-

Q ~

P R E S E N TS T U D Y M a T SEPARATION DISTANCE R = 50x ~ S E P A R A TION D IS TA N C E R = 5a

. . . . . . . . S I N G L E B O D Y

- - - , - @ . .. .. . . . . .. z ; ; t . . . . . . . . .

0 O.D 1 .0 1 .5 2 .0 2 .5k o O

I3 ,0

2 ~

2

t , . " I.C

0. ~

P R E S E N TS T U D Y M S T

. . . . . . . . S I N G L E BO D Y

I I I I I0 5 I .O 1 . 5 2 D 2 . 5 5 0

k o O

C y l i n d e r 2FiG. 4 . Su rge and hea ve exc i t ing fo r c e s on two cy l inder a r ray fo r 13 = 0 , b /a = 0 .5 , d /a = 10 . Compar i s ono f p r e s e n t m e th o d w i th r es u l t s o f M a t s u i a n d T a m a k i ( 1 9 81 ) .

p r e s s u r e c o m p o n e n t t o t h e o v e r t u r n i n g m o m e n t m a y b e e x p e c t e d t o b e m o s t s i g n i f i c a n tf o r b o d i e s o f s h a l l o w d r a ft .5. 1 . T w o c y lin d e r a r r a y ( T L P 2 )

O n e o b j e c t i v e i n s t u d y i n g t h i s g e o m e t r y w a s t o c o m p a r e t h e r e s u l t s f r o m t h e p r e s e n tt h e o r e t i c a l s t u d y w i t h t h o s e o b t a i n e d b y M a t s u i a n d T a m a k i ( 1 9 8 1 ) , w h o s o l v e d t h e


14/35

562 A. N, WILLIAMS and Z. DFMIRBII.[ K

t w o c y li n d e r p r o b l e m u s in g a n a x i s y m m e t r i c G r e e n ' s f u n c ti o n a p p r o a c h . T h e o v e r a l la g r e e m e n t b e t w e e n t h e t w o a p p r o a c h e s h a s b e e n f o u n d t o b e v er y g o o d , as c a n b cs e e n f r o m F ig . 4 . F o r t h e s m a l l e r s e p a r a t i o n d i s t a n c e ( R = 3 a ) , th e p r e s e n t m e t h o do v e r e s t i m a t e s t h e n e t s u r g e f o r c e o n c y l i n d e r 2 e s p e c i a ll y i n t h e m i d f r e q u e n c y r a n g e .H o w e v e r , a t a s e p a r a t i o n d i s t a n c e o f R = 5 a t h e p r e d i c t io n s e x h i b it m u c h b e t t e ra g r e e m e n t , v a l i d a t i n g t h e p r e s e n t t h e o r y .

F u r t h e r a n a l y s is o f t h e T L P 2 c o n f i g u r a t i o n h a s b e e n m a d e f o r c o n s t a n t R / a , b / a a n dd /a w i t h 13 = 0 , 4 5 a n d 9 0 , t h e d e t a i l e d r e s u l t s o f w h i c h a r e p r e s e n t e d i n F i g s 5 - 7 .I t is n o t e d t h a t t h e h y d r o d y n a m i c in t e r a c t i o n e f f e c t s d e p e n d s t r o n g ly o n t h e w a v ef r e q u e n c y . F r o m t h e f i g u r e s it c a n b e s e e n t h a t i n t e r a c t i o n e f f e c t s a r e n o t s ig n i f ic a n t a tv e r y lo w f r e q u e n c i e s a n d t h e r e f o r e m i g h t r e a s o n a b l y b e n e g l e c t e d w h e n e s t i m a t i n g d r if tf o r c e s .

I t is a l s o n o t e d f r o m t h e f i g u r e s t h a t t h e a n g l e o f w a v e i n c i d e n c e p l a y s a m a j o r r o l ein d e t e r m i n i n g t h e d e g r e e o f h y d r o d y n a m i c in t e r fe r e n c e . T h e m a x i m u m s u rg e f o rc ea n d p i t c h m o m e n t o c c u r a t 13 = 0 , b u t t h e i r m a x i m a f o r t h e u p s t r e a m ( 1 ) c y l i n d e r a r er e d u c e d b y a b o u t 3 0 - 4 0 % w h e n 13 = 4 5 a n d b y a p p r o x i m a t e l y 7 0 % w h e n 13 = 9 0 .A l t h o u g h t h e r e i s a c o r r e s p o n d i n g i n c r e a s e in th e s a m e f o r c e a n d m o m e n t f o r t h ed o w n s t r e a m ( 2 ) c y l i n d e r w i t h i n c r e a s i n g 13, t h i s i n c r e a s e is n o t a s s u b s t a n t i a l . A se x p e c t e d , t h e s w a y f o r c e a n d r o l l m o m e n t e x h i b i t a n o p p o s i t e t r e n d t o t h e s u r g e f o r c ea n d p i t c h m o m e n t b e h a v i o r . F o r o b l i q u e w a v e s w i t h a n i n c i d e n t a n g l e o f 45 , t h ed o w n s t r e a m ( 2 ) c y l i n d e r e x p e r i e n c e s a l a r g e r s w a y f o r c e t h a n t h e u p s t r e a m ( 1 ) c y l i n d e ro v e r n e a r l y a ll t h e f r e q u e n c y r a n g e o f i n te r e s t . O b v i o u s l y , b o t h c o l u m n s w i ll h a v ee q u a l s w a y f o r c e s a n d r o l l m o m e n t s f o r 13 = 9 0 a s d e p i c t e d i n F ig . 7 .

A l t h o u g h t h e h e a v e f o r c e s o n th e u p s t r e a m ( 1 ) a n d d o w n s t r e a m ( 2 ) c y l i n d e rs a r e n o te q u a l f o r a n y g i v e n k o a a n d 13 t h e y a t t a i n a p p r o x i m a t e l y t h e s a m e m a x i m u m v a l u e so v e r t h e r a n g e o f k o a i n v e s t i g a t e d . F o r 13 = 4 5 , th e h e a v e f o r c e o n t h e d o w n s t r e a m( 2) c o l u m n d e c r e a s e s b y a b o u t 2 0 % a t m o s t f o r k o a < 1 .2 5 . H o w e v e r , f o r k o a > 1 .25 ,t h is f o r c e i n c r e a s e s o v e r t h a t f o r [3 = 0 a n d f o r k ~ a - - 1 . 9 , th i s i n c r e a s e is o f t h e o r d e ro f 3 5 % . F o r t h e c a s e 13 = 9 0 , t h e h e a v e f o r c e o n b o t h c o l u m n s b e c o m e s i d e n t ic a l a ta ll f r e q u e n c i e s , a s e x p e c t e d , a n d m a y d i f f e r f r o m t h a t o n a n i s o l a t e d c o l u m n b y a sm u c h a s 3 0 % .5 . 2 . T h r e e c y lin d e r a r r a y ( T L P 3 )

R e s u l t s f o r t h i s p a r t i c u l a r T L P c o n f i g u r a t i o n a r e s h o w n i n F i g s 8 - 1 0 . C o m p a r i s o n sw i t h t h e c o r r e s p o n d i n g f i g u r e s f o r t h e T L P 2 c o n f i g u r a t i o n s h o w t h a t t h e l o a d i n g s o nc y l i n d e r N o . 1 e x h i b i t s i m i la r b e h a v i o u r i n b o t h c a s es . H o w e v e r , f o r t h e T L P 3 c a s et h e r e is a m o r e p r o n o u n c e d i n f lu e n c e o f th e i n t e r a c t i o n e f f e c t s . I t is n o t e d t h a t t h es u r g e f o r c e o n c y l i n d e r N o . 1 is i n c r e a s e d b y a p p r o x i m a t e l y 1 0 - 2 0 % o v e r th a t o f t h eT L P 2 c a s e w h e n 1 3 - - - 9 0 o v e r m o s t o f th e f r e q u e n c y c o n s i d e re d . A l s o , a s e x p e c t e d ,t h e s u r g e l o a d s d e c r e a s e s u b s t a n t i a l l y w i t h i n c r e a s i n g 1 3 . H o w e v e r , t h e s u r g e l o a d o nc y l i n d e r 1 m a y b e a s m u c h a s 9 5 % o f t h e i s o l a t e d c y l i n d e r v a l u e f o r 13 = 4 5 a n d 4 0 %fo r 13 = 90 . It is n o t e d t h a t i n c r e a s e s o f 6 0 % o r m o r e m a y o c c u r i n t h e h e a v e f o r c ea t c e r t a i n f r e q u e n c i e s d u e t o i n t e r f e r e n c e e f f e c t s .5 . 3 . F o u r c y li n d e r a r r a y ( T L P 4 )

T h i s c o n fi g u r at io n m o d e l s a c o n v e n t i o n a l , s q u a re , f o u r c o l u m n T L P . T h e c o m p u t e dw a v e l o a d s o n a l l c o l u m n s a r e d e p i c t e d i n F i g s 1 1 - 1 3 f o r t h r e e d i f f e r e n t i n c i d e n t w a v e


15/35

- 1.0

0.1~

2 .00. (0. 0

1. 4

L2

I.C

u . 0

O~

0. ~

OC0. 0

, i , i i ,

' , + ' ' , 4 , ' ~e. o.o- /l e - . - - - e - - - e , I R , 5 0 /~:,~

I i I I I i0 .5 1 .0 1 .5

k o a

I2 C YLB . O "~'~d 1~

., \0. 5 I I I I J1.0 1.5

k o 02. 0

I IA I2 C YL/ 9 o 0 . 0 "R m S Ob , 3 ad , 1 0 o

12

0.~

0. ~ I I0 . 0 0 . 5

H y d r o d y n a m i c i n t e r a c t i o n s i n f l o a ti n g c y l i n d e r a r r a y s - - I 5 6 3

I I I I I1.0 1 ,5

k o o2.0

F IG . 5 . N o r m a l i z e d e x c i t in g f o r c e s a n d m o m e n t s o n t w o c y l i n d e r a r r a y f o r ~ = 0 , R/a = 5 , b/a = 3 ,d/a = 10.


16/35

564 ~ l N. WI[.I.IAMN an d Z. DEMIRBI IIK

c

12 1 I I ' I 1 1 ]

R 5 o| ~'~o ~_.~"I . d IOaO B

0 6

i 2Oko O

080 ' ICYL.0 45 R : 5 ab 3 0d I 0 o

0 6 5 -

i ! I i I i I LO . O 0.5 ID 1.5 20k o OI I 2 / Y LB - 4 5 *R , 5 1b - 3 o

O t

OJ I I I I L I iO O 05 1.0 15 2 Dko o

F IG . 6 . N o r m a l i z e d e x c i t i n g f o r c e s a n d m o m e n t s o n t w o c y l i n d e r a r r a y f o r 13 = 4 5 , R / a = 5 , b / a = 3 ,d/a = 10.


17/35

H y d r o d y n a m i c i n t e r a c t i o n s i n f l o a ti n g c y l i n d e r a r r a y s - - I 5 6 5O.t~,

0 2

0

0 OO

, , '?CYL ( ~~ ' 9 0 "R " ~b " ~ Rd IOo .

05 1 . 0 1.5k o o

2.0

I ' I I , ' I ' I ,CTLj e . 9 0 oR . S ab . 3 od . l O o

9 0 i I i i i I ,O. 0 0.5 1.0 L5 2.0% 0

l-41 u I I I5) CYL,8 90"R ' 5 Ob 3 0

0 8

06 t I i I , I iO0 05 1.0 1.5 2.0k ~F IG . 7. N o r m a l i z e d e x c i t i n g f o r c e s a n d m o m e n t s o n t w o c y l i n d e r a r r a y f o r [3 = 9 0 , R/a = 5 , b/a = 3 ,d/a = 10.


18/35

566 A . N , WILLIAMS and Z . DEMIRBILEKI I [ t -

3 CYL/~,0 ~R,Sab,3od lOo13

I.I

0 9

o7

+, u ' ' + -

O ~ I ] I , i0.0 0.5 1.0 15 20

koo

I I I3 C Y LB ' O *R 5 0b ,3a

0 l

02

O~

OJ

O O ~ t I I ~ I J I iO D Q5 I. O 15 2 0

k a ~

1 8 I I I i3 CYLB 'O"

IE R 50b ,30d I0014

0.(

O!

i I i I ~ I iOWo.o 0.5 LO I.S 2DkoQ

F IG . 8. N o r m a l i z e d e x c i t i n g f o r c e s a n d m o m e n t s o n t h r e e c y l i n d e r a r r a y f o r 13 = 0 , R l a = 5 , b / a = 3 ,d / a = 10.


19/35

H y d r o d y n a m i c i n t e r ac t i o n s i n f l oa t in g c y l i n d e r a r r a y s - - I 5 6 71,0 I I I i I i

3 C YLD , e o "R . S ab - 3 o0 8 d IO Q

OJ6

O2

0 .0 I I I l I I0 . 0 0 . 5 1 .0 1 5 2 D

k o O

1.2

- ioz

06

I I i3 CYL,8 ee,R - 5ob -3o

C~ J J I ~ IO D 0 5 1 . 0 1 . 5 2 , 0

ko o

I.JE I I J i3 C Y L

I.E ~ " 60=f l , 5 ob , 3 od - I C o

0. (

0.~

02~ i I ~ I I I JO 0 0 .5 1 .0 l , 5 2 .0

ko .F IG . 9 . N o r m a l i z e d e x c i t i n g f o r c e s a n d m o m e n t s o n t h r e e c y l i n d e r a r r a y f o r 13 = 6 0 , R / a = 5 , b / a = 3 ,

d / a = 10.


20/35

05

oo

o4

- o ~ .

OI

O.C

I i i , |3 CYL~= 90 'R=Sab - S Qd , t O o

I I I I

CYLR fx~b ,3o

05 1,0 I 5 2 0w@

12

IC

OE

O(O 0

5 6 8 A , N . W I L LI A M S a n d Z . D E M I RB I I. EK

i , , ,0 5 I 0 1 5.ko O

2 0

I0

O 0

60O

i i i3 C Y LB = 9 0 R 5~b " 5 o

05 1.0 1.5 20k@

F IG . 1 0 . N o r m a l i z e d e x c i t i n g f o r c e s a n d m o m e n t s o n t h r e e c y l i n d e r a r r a y f o r 13 = 9 0 , R / a = 5 , b / a = 3 ,d / a = 10 .


21/35

H y d r o d y n a m i c i n t e r ac t i o n s i n f l oa t in g c y l i n d e r a r r a y s - - I 5 6 9

t~

i I.C

0.~

02

O~

0.1

0~ 'O O

' I ' I I4 C Y LB o O 'R , S Ob J ~ l ad tOo

, , , ,05 1.0 1.5 2D% *

o,~ , , , i4 C Y LB , O *R , ~ ob , S o0 .4 d , IO o

O 00 . 0 0 . 5

% 0

I ' I

I i I i1.0 1.5 2Dko a

i I ' I i I4 C Y LB , O *d I00

( i ) .005 10 1 .5 2 .0to *

F I ~ . 1 1. N o r m a l i z e d e x c i t i n g f o r c e s a n d m o m e n t s o n f o u r c y l i n d e r a r r a y f o r 13 = 0 % R / a = 5 , b / a = 3 ,d / a = 10.


22/35

57(I A . N . WILIAAMS an d Z . I)EMIRBILI~K

I1 2

1. 0

~E0 8

~ o

I I I 4 C Y L~ 3 0 ~R 5ob ~a

I t i0 . 5 I i II 0 1 5ko O

2 0

- O (

0 4

Coo

I4 CYLB , 3 0 ~b . 3 ad , l O o

0.5 1.0 1.5 20koO

4 CY L I, 3 0 ~R . 5 ob - 3 od - ~ O o1 2

0 8

OE

0 4

I 1O ~ ~ I 0 5 I 0 I 5 2 0ko a

F ] o . 1 2 . N o r m a l i z e d e x c i t i n g f o r c e s a n d m o m e n t s o n f o u r c y l i n d e r a r r a y f o r 1 3 = 3 0 , R / a = 5 , b / a = 3 ,d / a = 1 0 .


23/35

H y d r o d y n a m i c i n t e r a c t i o n s i n f l o at i n g c y l i n d e r a r r a y s- - - I 5 7 1

I

= , ." O (

O ~

O;

O 0 O 0

4 5 "R ' 5 ~

0 . 5 I 0 1 ,5 2 0k o O

4 CYL I ! I~ , 4 5 b . 3 o

i '0604

02O 0 0 5 I 0 1 .5 2 0kO

2.C4 C Y L

LiE d IOg

1 .4

1.2~ I.C

O !

0, (

0,~

O |O , C n I I I n I iO ,O 0.5 1.0 1.5 20

k ~

F xG. 1 3 . No r m a l i z e d e x c i t in g f o r c e s a n d m o m e n t s o n f o u r c y l in d e r a r r a y f o r 13 = 4 5 , R / a = 5 , b / a = 3 ,d / a = 1 0 .


24/35

572 A . N . Will.tAMS and Z. DI-;MIRRtH~Ka n g l e s . I t is o b s e r v e d t h a t f o r [3 - 0 . t h e n o r m a l i z e d s u r g e f o r c e a n d p i t c h i n g m o m e n tf o r t h e u p s t r e a m (1 a n d 3 ) c o l u m n s f l u c t u a t e b e t w e e n 1 .3 a n d 0 . 6 w h i le t h o s e f o r t h ed o w n s t r e a m ( 2 a n d 4 ) c o l u m n s h a v e a n o r m a l i z e d v a l u e o f a b o u t l .I ) t h r o u g h o u t t h ef r e q u e n c y r a n g e c o n s i d e r e d .

F o r [3 = 3 0 , th e s u r g e f o r c e a n d p i t c h m o m e n t c o e f f i c i e n t s o f c o l u m n N o s 2 a n d 4a r e a p p r o x i m a t e l y e q u a l f o r k o a < 1 . 25 , w i th s i g n if i c a n t d i f f e r e n c e s b e y o n d t h i s v a l u e .A s e x p e c t e d , t h e s a m e c o e f f ic i e n t s f o r c o l u m n N o . 1 s li g h tl y d e c r e a s e w h i l e t h o s ea s s o c i a t e d w i t h c o l u m n N o . 3 d o n o t d i s p l a y a n y s i g n i f i c a n t c h a n g e w i t h i n c r e a s e i n [ 3 .W i t h a f u r t h e r i n c r e a s e i n 13 t o 4 5 , th e e x p e c t e d s y m m e t r y i n t h e f o r c e c o e f f i c i e n t s f o re a c h o f th e c o l u m n s m a y b e o b s e r v e d .

W he n 13 = 0 , f o r k o a < 1 .2 , t h e s w a y f o r c e a n d r o l l m o m e n t c o e f f ic e n t s o f t h e f r o n tc o l u m n s a r e l a r g e r t h a n t h o s e o f t h e r e a r c o l u m n s w h i l e f o r k ~ a > 1 .2 t h e r e v e r s e ist r u e . W h i l e t h e s e c o e f f ic i e n t s s h o w o s c i l la t o r y b e h a v i o r f o r c o l u m n N o s 1 a n d 3 , it isr a t h e r i n t e r e s t i n g t o n o t e t h a t f o r t h i s c a s e a s t e a d i l y i n c r e a s i n g t r e n d w i t h i n c r e a s i n gf r e q u e n c y is o b s e r v e d f o r c o l u m n N o s 2 a n d 4 . A v e r y n o t i c e a b l e i n c re a s e i n t h e s w a yf o r c e a n d r o l l m o m e n t c o e f f i c i e n t s is v is i b le w h e n 13 is c h a n g e d f r o m 3 0 t o 4 5 . T h el a r g e s t i n c r e a s e o c c u r s f o r k o a < 1 .75 .

F o r [3 = 0 , th e n o r m a l i z e d h e a v e f o r c e c o e f f i c i e n t s f o r t h e w e a t h e r s i d e (1 a n d 3 )c o l u m n s o s c i l l a t e s b e t w e e n 0 . 6 a n d 1 . 5 w i t h t w o l o c a l m a x i m a o c c u r r i n g a t k o a ~ 0 . 7 5a n d 1 .7 5 . C o l u m n s N o s 1 a n d 3 e x p e r i e n c e th e l a r g e st h e a v e f o r c e o v e r t h e f r e q u e n c yr a n g e o f t h i s s t u d y f o r a l l t h r e e f l o w a n g l e s c o n s i d e r e d . C o l u m n N o . 4 e x p e r i e n c e s ah e a v e f o r c e w h i c h d e c r e a s e s w i t h i n c r e a s in g f r e q u e n c y a t [3 = 3 0 . I n t h i s c a s e , t h eh e a v e f o r c e o n c o l u m n N o . 2 f l u c tu a t e s a r o u n d t h a t o f c o l u m n N o . 4 w i t h a n o r m a l i z e dm a g n i t u d e o f _ + 0 .2 . F o r 13 = 4 5 , c o l u m n N o . 1 e x p e r i e n c e s a h e a v e f o r c e a p p r o x i m a t e l y8 0 % g r e a t e r t h a n t h a t o n a n i s o l a t e d c y l in d e r at k o a ~ 1 . 0 .5 . 4 . F i v e c y l i n d e r ar r a y ( T L P 5 )

T h i s p a r t i c u l a r c o n f i g u r a t i o n c o n s is t s o f fi ve c o l u m n s p o s i t i o n e d s o t h a t t h e y f o r mt h e v e r t i c e s o f a r e g u l a r p e n t a g o n . R e s u l t s f o r t h is s p e c if ic g e o m e t r y a r e d e p i c t e d i nF igs 14 and 15 .A s e x p e c t e d , f o r [3 = 0 th e s u r g e f o r c e a n d p i t ch m o m e n t c o e f f ic i e n ts o f c o l u m nN o s 2 a n d 5 a r e t h e s a m e . S i m i l a r l y , c o l u m n N o s 3 a n d 4 h a v e i d e n t i c a l c o e f f i c i e n t ss i n c e t h e y a r e s y m m e t r i c w i t h r e s p e c t t o t h e x - a x i s . M o r e o v e r , i t i s n o t e d t h a t c o l u m nN o s 3 a n d 4 h a v e t h e s m a l l e s t s u r g e a n d p i t c h c o e f f i c i e n t s o f al l t h e c o l u m n s f o r [3 =0 . A s a n t i c i p a t e d , c o l u m n N o . 1 h a s t h e l a r g e s t s u r g e a n d p i t c h c o e f f i c i e n t s i n th i sc a s e .F o r [3 = 9 0 , a s u b s t a n t i a l r e d u c t i o n i n t h e s e c o e f f i c i e n t s i s c l e a r l y s e e n , i n c l u d i n gt h o s e c o l u m n s o n t h e w e a t h e r s i d e o f t h e a p p r o a c h i n g w a v e s . C o l u m n N o . 5 h a s t h es m a l l e s t s u r g e a n d p i t c h c o e f f i c i e n t s f o r [3 = 9 0 ( as i t i n t u i ti v e l y s h o u l d ) , f o l l o w e d b yc o l u m n N o . 4 .C o l u m n s N o s 2 a n d 5 e x h i b i t t h e l a r g e s t s w a y f o r c e a n d p i t c h m o m e n t c o e f f i c i e n t sf o r [3 = 0 f o l l o w e d b y c o l u m n N o s 3 a n d 4 . T h e f r o n t c o l u m n , c o l u m n N o . 1 , h a s z e r ol o a d i n g i n t h i s c a s e . F o r [3 - - 9 0 , v e r y l a r g e i n c r e a s e s a n d f l u c t u a t i o n s f o r t h e s ec o e f f ic i e n t s a r e p r e d i c t e d . C o l u m n N o s 1 , 2 , a n d 3 e x p e r i e n c e t h e l a rg e s t s w a y a n dp i t ch l o a d i n g a s a n t i c ip a t e d . I t s h o u l d a l so b e p o i n t e d o u t t h a t t h e c o e f f ic i e n ts f o rc o l u m n N o s 4 a n d 5 e x h i b i t t h e l e a s t f l u c t u a t i o n a n d t h e i r m a x i m u m n o r m a l i z e d v a l u e sd o n o t e x c e e d 1 . 0 f o r a l l f r e q u e n c i e s .


25/35

H y d r o d y n a m i c i n t e r a c t i o n s i n f l o a t in g c y l i n d e r a r r a y s - - I 5 7 3i I i I

/ 9 . 0 "3 0d - I O o

, ' ,' ~ 9 . 113

Q 6

0E

O ~

o r OO 0.5 I0 1.5 2 0ko O

0.,

%o

I I I5 CYLB ' O "R = 5 o

,

0.5 I 0 1.5 20kO0

2.0

18

12

- o , e lQ 6

o 4 10. ~

OCQ O

F I G . 1 4 . N o r m a l i z e d e x c i t i n g

~."~L I I I '5R , S ad , I 0 o

I I I I I0. ~ ID i .5 2 Dk ~

f o r c e s a n d m o m e n t s o n f i ve c y l i n d e r a r r a y f o r 13 = 0 , R/a = 5 , b/a = 3 ,d/a = 10.


26/35

5 7 4 A . N . W I L L IA M S a n d Z , D E M I RB I LE K

O 6

0 .2

o0 ,0

I , t '

5 C Y L. ~ , 9 0 'R " 5 0b . 3 ad . I O Q

0 5 1 .0 1 5 2 0koO

04O0

I I I5 C T L, e . 9 0 ~ 3o

0 5 I O 1 5 2 0k o ~

I i I I5 C Y L~ e . 9 0 ~

R , 5 0b . 3 014

1 2

OE

0 .(

O 4

0 - - 0 0 . 5 1 .0 1 5 2 . 0k@

F IG . 1 5 . N o r m a l i z e d e x c i t in g f o r c e s a n d m o m e n t s o n f iv e c y l i n d e r a r r a y f o r 13 = 9 0 , R / a = 5 , b / a = 3 ,d / a = 10.


27/35

Hy drodynam ic interactions in floating cylinder arrays---I 575F o r 13 = 0 , t h e l a r g e s t h e a v e c o e f f ic i e n t a n d v a r i a t i o n w i t h f r e q u e n c y o c c u r s f o r

c o l u m n N o . 1 . T h e s e c o e f f ic i e n t s f o r c o l u m n N o s 3 a n d 4 d e c r e a s e w i t h i n c re a s i n gf r e q u e n c y w h i l e t h o s e o f c o l u m n N o s 2 a n d 5 o s c il la te a r o u n d a n o r m a l i z e d a m p l i tu d eo f 1 . 0 w i t h a v a r i a t i o n o f - --0 .3 . F o r 13 = 9 0 , c o l u m n N o . 5 h a s t h e s m a l l e s t h e a v e f o r c ec o e f f ic i e n t w h i l e c o l u m n N o . 3 d is p l ay s t h e g r e a t e s t e x c u r s i o n s a b o u t 1 .0 f o r t h isc o e f f ic i e n t o v e r th e f r e q u e n c y r a n g e c o n s i d e r e d .I t i s i n s t r u c t i v e a t t h i s s t a g e t o c o m p a r e t h e n u m e r i c a l r e s u l t s o b t a i n e d f o r t h e t h r e ec y l i n d e r a n d f iv e c y l i n d e r c o n f i g u r a t i o n s . C o m p a r i n g F i g . 8 t o F ig . 1 4 , i t c a n b e s e e nt h a t f o r t h e c o r r e s p o n d i n g c o l u m n s i n T L P 5 , a t l e as t a 1 0 % i n c r e a s e i n t h e s u r g e f o r c ea n d p i t c h m o m e n t i s s e e n . T h i s i n c re a s e m u s t s o l e ly b e d u e t o t h e h y d r o d y n a m i ci n t e r f e r e n c e o f t h e a d d i t i o n a l c o l u m n s . A s i m il a r t r e n d a l so e x is t s b e t w e e n t h e s u r g ef o r c e a n d p i t c h m o m e n t c o e f f i c i e n ts o f t h e s e t w o T L P c o n f i g u r a t i o n s w h e n 13 = 9 0 , a sc a n b e s e e n b y c o m p a r i n g F i g s 1 0 a n d 1 5. A s f a r a s t h e s w a y f o r c e a n d r o l l m o m e n tc o e f f ic i e n t s a r e c o n c e r n e d , a n e x a m i n a t i o n o f th e f i g u re s re v e a l s t h a t t h e l o a d i n g o nt h e c o r r e s p o n d i n g c o l u m n s b e t w e e n t h e s e c o n f i g u r a t io n s is , i n m a n y c a s e s l o w e r f o rt h e f i v e c y l i n d e r c o n f i g u r a t i o n , T L P 5 . T h e s e c o e f f i c i e n t s a t 13 = 0 d o n o t m o n o t o n i c a l l yi n c r e a s e a n d , i n f a c t , h a v e s u b s t a n t ia l l y lo w e r v a l u e s fo r T L P 5 c o m p a r e d t o t h o s e f o rT L P 3 a t s o m e f r e q u e n c i e s . F o r t h is w a v e h e a d i n g , t h e s e c o e f f ic i e n t s a r e l a r g e r f o r T L P 5t h a n t h o s e o f T L P 3 o n l y a t v e r y h ig h f r e q u e n c i e s , i . e . b e y o n d kt~ u > 1 .7 5 . A c o m p a r i s o nf o r t h e 13 = 9 0 c a s e i n d i c a t e s m a n y s i m i l a ri t ie s i n t h e s w a y a n d p i t c h l o a d i n g s b e t w e e nt h e t h r e e a n d f iv e c y l i n d e r c o n f i g u ra t i o n s . T h e h e a v e f o r c e c o e f f ic i e n t s o f c o l u m n N o s2 a n d 5 i n T L P 5 a r e , i n g e n e r a l , h i g h e r th a n t h o s e o f t h e i r c o u n t e r p a r t s i n T L P 3 e x c e p tf o r 0 . 4 < k o a < 0 . 6. F o r t h e s e c o l u m n s , t h e l a r g e s t d i f f e r e n c e is a b o u t 5 0 % a t k ~1 .7 f o r 13 = 0 . W h e n 13 = 9 0 , t h e h e a v e f o r c e c o e f f i c i e n t s b e t w e e n t h e c o r r e s p o n d i n gc o l u m n s o f t h e s e T L P s a r e s i m il a r.5 . 5 . S i x c y l in d e r ar r a y -h e x a g o n (T L P 6 )

F o r t h is c o n f i g u r a t i o n t h e c y l i n d e r s w e r e a r r a n g e d w i t h t h e i r c e n t e r s a t t h e v e r t ic e so f a r e g u l a r h e x a g o n . F i g u r e s 1 6 a n d 1 7 p r e s e n t t h e f o r c e a n d m o m e n t c o e f f ic i e n ts f o rt h e h e x a g o n c o n f i g u r a t i o n .

F r o m t h e f i g u re s it c a n b e s e e n t h a t c o l u m n N o . 1 e x p e r i e n c e s t h e l a rg e s t s u rg e a n dp i t c h l o a d i n g f o r 13 = 0 . W h e n 13 i s i n c r e a s e d t o 9 0 , t h e l a r g e s t s u r g e l o a d i n g i se x p e r i e n c e d b y c o l u m n N o s 2 a n d 3 f o r 0 . 6 < k oa < 1 .1 , f o r c o l u m n N o s 1 a n d 4 f o r1 .1 < k o a < 1 . 6 a n d f i n a ll y c o l u m n N o s 5 a n d 6 f o r 1 .6 < ko a < 2 . 0 .

A s e x p e c t e d , t h e s w a y f o r c e a n d r o l l m o m e n t c o e f f ic i e n ts e x h i b i t a s o m e w h a t o p p o s i t et r e n d t o t h e a b o v e o b s e r v a t i o n s . T h e s w a y f o r c e a n d r o l l m o m e n t c o e f f ic i e n ts f o r al lc o l u m n s i n t h i s c o n f i g u r a t i o n s h o w s i g n i fi c a n t in c r e a s e s w i t h i n c r e a s i n g 13. W h e n 13 =0 c o l u m n N o s 3 a n d 5 e x p e r i e n c e s w a y a n d r o ll l o a d s w h i c h i n c r e a s e w i t h f r e q u e n c yu p t o ko a - 1 .6 w h i l e c o l u m n s N o s 2 a n d 6 e x p e r i e n c e l a r g e l o a d s n e a r k~ a - 1 .0 a n d2 . 0 . F o r 13 = 0 c o l u m n N o s 2 a n d 6 h a v e t h e s m a l l e s t l o a d s f o r 1 . 1 < k o a < 1 .8w h e r e a s i t is o v e r th i s ra n g e t h a t t h e m a x i m u m l o a d s o f c o l u m n N o s 3 a n d 5 o c c u r .T h e s w a y f o r c e a n d r o l l m o m e n t c o e f f ic i e n t s f o r c o l u m n N o s 1 a n d 4 a r e z e r o f o r 13 =0 . A s e x p e c t e d , w h e n 13 i s i n c r e a s e d t o 9 0 , c o l u m n N o s 2 a n d 3 e x p e r i e n c e t h e l a r g e s ts w a y a n d r o ll l o a d s , f o l l o w e d b y c o l u m n N o s 1 a n d 4 . A l s o , a t t h is i n c i d e n t w a v e a n g l e ,c o l u m n N o s 5 a n d 6 e x p e r i e n c e t h e s m a l l es t lo a d s .

F o r 13 = 0 , c o l u m n N o s 1 , 2 a n d 6 e x p e r i e n c e t h e l a r g e st h e a v e f o r c e w h i le c o l u m nN o . 4 sh o w s a n o r m a l i z e d h e a v e l o a d o f b e t w e e n 0 .7 a n d 1 .0 th r o u g h o u t t h e f r e q u e n c y


28/35

576 A , N. WILUAM S an d Z . DErvlIRBII. i~K

6 C Y Lj 3 . 0 R " 5 o~:,~I I6

.

~0 10 80 60 4 I I I

O 0 0 5

2 C

i16

Ii141

I I II 0 I 5 2 0k o 0

i i r I I I6 C Y L/ ~ = 0 ~R " 5 ob = 3 0d I O o

,, -

o~

QOO 0 0 5 I 0 I 5 ~ 0k oo

- o ~

2 { 6 C Y L I I I Ile , ~ = 0 o1 6 ; d = I 0 o

I1 4 '1 2 , ' ~

2 ~ q o0 8

0 6

0. 4

0 2

O C ~ I i I 1 ib 0 5 t O 1 ,5 2 0k ~

F IG . 1 6. N o r m a l i z e d e x c i t i n g f o r c e s a n d m o m e n t s o n s i x c y l i n d e r a r r a y ( h e x a g o n ) f o r 13 = 0 , R / a = 5 ,b / a = 3 , d / a = 10.


29/35

H y d r o d y n a m i c i n t e r a c t i o n s i n f l o a t in g c y l i n d e r a r r a y s - - -I 5 7 7

0 # 1

O 0

' ( 9 ( 9 ' '6 C Y L X ~

.@O),

0.5 1.0 1.5tO a

f

2 . 0

~ 0.6

0 2 'O 0

2.0

t 6

12

0 8

O 6

O2

I I I6 CYL

. ~: ~.

I I i I i0 . 5 I 0 15 2. 0koO

II6 CYL~ :~"b w~d , 1 0 o

O.(9.

0 5 1 . 0% 0

oo I ,QO 1.5 2.0

F IG . 1 7 . N o r m a l i z e d e x c i t i n g f o r c e s a n d m o m e n t s o n s i x c y l i n d e r a r r a y ( h e x a g o n ) f o r 13- = 9 0 , R / a = 5 ,b / a = 3 , d / a = 10.


30/35

578 A .N . WILLIAMS an d Z. DLMIRBU~t:~:

r a n g e c o n s i d e r e d . W h e n 13 = 0 t h e m a x i m u m h e a v e f o r c e o c c u r s in t h e m i d f r e q u e n c yr a n g e 0 . 8 < k o a < 1 . 6. F o r 13 = 9 0 , t h e m a x i m u m h e a v e f o r c e i s e x p e r i e n c e d b yc o l u m n N o s 2 a n d 3 w i t h t h e l a r g e s t f o r c e o c c u r r i n g a t h i g h e r f r e q u e n c i e s . F o r [3, -9 0 , t h e m i n i m u m h e a v e f o r c e f o r a ll c o l u m n s o c c u r s a t k o a -~ 1.35.5 .6 . S i x c y l i n d e r a r r a y -r e c ta n g le ( T L P 6 R )

T h e r e c t a n g u l a r a r r a n g e m e n t is a m o r e c o n v e n t i o n a l c o n f i g u r a t io n f o r a s i x- l eg g e dT L P . R e s u l t s f o r t h is g e o m e t r y a r e p r e s e n t e d i n F i g s 1 8 - 2 0 b a s e d o n t h r e e i n c i d e n tw a v e a n g l e s , 13 = 0 , 4 5 a n d 9 0 .

A c o m p a r i s o n o f t h e s e f i g u re s s h o w s t h a t t h e su r g e f o r c e a n d p it c h m o m e n tc o e f f i c i e n t s d e c r e a s e w i t h i n c r e a s i n g [3. F o r [3 - - 0 , c o l u m n N o s 1 a n d 4 e x p e r i e n c e t h el a r g e s t s u r g e l o a d s f o l l o w e d b y c o l u m n N o s 2 a n d 5 . F o r 13 = 0 c o l u m n N o s 3 a n d 6e x p e r i e n c e t h e s m a l l e s t s u r g e l o a d s . A t [3 = 4 5 , h o w e v e r , t h e r e is s o m e d e c r e a s e int h e c o e f f i c i e n t s o f t h e h e a v i l y l o a d e d c o l u m n s ( 1 a n d 4 ) f r o m t h e [3 = 0 c a s e . H o w e v e r ,c o l u m n N o . 6 r e m a i n s t h e l e a s t l o a d e d i n t e r m s o f s u r g e a n d p i t c h a t [3 = 4 5 . A t [3 =9 0 , t h e c e n t e r c o l u m n s ( i .e . c o l u m n N o s 2 a n d 5 ) e x p e r i e n c e z e r o l o a d s , w h i l e c o l u m nN o s 4 a n d 6 e x p e r i e n c e t h e l a r g e s t l o a d s f o r k o a > 1 . 2. F o r t h i s v a l u e o f [3 , c o l u m nN o s 1 a n d 3 e x p e r i e n c e t h e m a x i m u m w a v e l o a d s f o r k o a < 1 . 2 .

T h e m a x i m u m s w a y f o r c e a n d r o l l m o m e n t c o e ff i ci en t s fo r th is c o n f ig u r a t io n o c c u rf o r [3 = 9 0 a n d t h e m i n i m u m f o r [3 = 0 . F o r [3 = 0 , c o l u m n N o s 3 a n d 6 e x p e r i e n c et h e s m a l l e s t s w a y l o a d o v e r 0. 2 5 < k o a < 1 .0 w h i l e c o l u m n N o s 1 a n d 4 a r e l e a s th e a v i l y l o a d e d i n t e r m s o f sw a y o v e r 1 . 0 < k o a < 1 . 6 . F o r 13 = 4 5 , t h e l a r g e s t v a r i a t i o no f t h e s w a y a n d r o ll c o e f f ic i e n ts is s e e n f o r c o l u m n N o s 1 , 2 , 3 a n d 6 . M a x i m u m v a l u e so f t h e s w a y l o a d o c c u r b e t w e e n 0 .3 5 < k o a < 0 . 8 5 a n d 1 . 2 < k o a < 1 .7 w h i l e m i n i m ao c c u r a t k o a - 1 . 0 . F o r 13 = 9 0 , c o l u m n N o s 1 , 2 a n d 3 e x p e r i e n c e t h e l a r g e s t s w a yl o a d s w h i l e c o l u m n N o s 4 , 5 a n d 6 a r e t h e l e a s t l o a d e d .F o r [3 = 0 , c o l u m n N o s 1 a n d 4 e x h i b i t l a r g e v a r i a t i o n s i n t h e h e a v e c o e f f i c i e n t o v e rt h e e n t ir e f r e q u e n c y r a n g e c o n s i d e r e d . F o r th is w a v e h ea d i n g , t h e m i n i m u m h e a v el o a d s o c c u r a t k o a ~ 1 . 1 5 . F o r [3 = 4 5 , c o l u m n N o s 1 - 4 a r e a g a i n h e a v i l y l o a d e d i nt e r m s o f h e a v e , t h e m a x i m u m l o a ds o c c u r a r o u n d k ~ j a - 1 .1 f o r a l l c o l u m n s a n d s t e a d i l yi n c r e a s e t h e r e a f t e r w i t h f r e q u e n c y u n t i l k o a ~ 1 .5 . C o l u m n s N o s 1 , 2 a n d 3 e x h i b i tl a r g e v a r i a t i o n s i n t h e h e a v e f o r c e c o e f f i c i e n t w i t h w a v e f r e q u e n c y . F o r [3 = 9 0 , t h eh e a v e l o a d s o n a ll c o l u m n s a r e a m i n i m u m n e a r k o a = 1 . 2 .

A c o m p a r i s o n b e t w e e n t h e h y d r o d y n a m i c e x ci ti n g f o r c e s a n d m o m e n t s e x p e r i e n c e db y ty p i ca l h e x a g o n a n d r e c t a n g u l a r s ix c y l in d e r T L P c o n f i g u r a t io n s m a y n o w b e m a d e .A t [3 = 0 , b o t h T L P s e x p e r i e n c e a p p r o x i m a t e l y t h e s a m e s u r g e f o r c e a n d p i t c h i n gm o m e n t . T h e s e v a l u e s a p p e a r t o d e c r e a s e m o r e r a p id l y f o r t h e h e x a g o n T L P w i t hi n c r e a s i n g w a v e f r e q u e n c y . A t 13 - - 90 , t h e r e c t a n g u l a r T L P e x p e r i e n c e s a s m a l l e rs u r g e f o rc e a n d p i tc h m o m e n t u p to k o a ~ 0 . 9. T h i s t r e n d r e v e r s e s i n f a v o r o f t h eh e x a g o n T L P f o r 1 . 0 < k o a < 1 .7 5 . I n t e r m s o f t h e s w a y f o r c e a n d r o ll m o m e n tc o e f f ic i e n t s, t h e h e x a g o n T L P i s s u p e r i o r t o t h e r e c t a n g u l a r T L P b e c a u s e it g e n e r a l l ye x p e r i e n c e s s i m i l ar o r s m a l l e r l o a d s o v e r t h e e n t i r e w a v e f r e q u e n c y r a n g e f o r [3 = 0 .T h i s a d v a n t a g e a l so h o l d s t r u e a t [3 = 9 0 f o r t h e a b o v e - m e n t i o n e d c o e f f ic i e n ts , t h eo n l y e x c e p t i o n b e i n g t h e g r e a t e r l o a d s e x p e r i e n c e d b y c o l u m n N o s 2 a n d 3 o f t h eh e x a g o n T L P a t k , ~ ~ 1 .3 8 . T h e h e a v e f o r c e s d i s p l a y e d b y b o t h t h e h e x a g o n a n dr e c t a n g u l a r T L P s g e n e r a l ly h a v e m a x i m a o f t h e s a m e o r d e r o v e r t h e f r e q u e n c y r a n g e


31/35

H y d r o d y n a m i c i n t e r a c t i o n s i n f l o a t in g c y l i n d e r a r r a y s - -- I 5 7 918 I1.6

I.C

o~ ~: 'o~

o~

0.5

I6 C Y L R E C T~ , 0 "R 5ob , 3 ed iOo

i, , @ ~ ' ~ =A R B , O ~B i ~ ~ R,5oR_.p" v db,3o

LO 15 2.0ko a1 I

o n

o z

. 013 ~ I I ,OD 0.5 LO 1.5 2.0

ko a~( . . . . I I I

6 CYL RECTR " 5 0b " 3 0I,E d I0o

,

_ J; I ~LC0, 8

iO 6; i0.4

0.2 I I I0.0 0.5 1.0 1,5 2,0%0F IG . 1 8 , N o r m a l i z e d e x c i t i n g f o r c e s a n d m o m e n t s o n s i x c y l i n d e r a r r a y ( r e c t a n g l e ) f o r 13 = 0 , R/a = 5 ,

b/ a = 3 , d/a = 10.


32/35

IC

O ~

u ~ O E

0 z

O~

o o

0 8

0 6

0 4

6 C Y L R E C I/3 45"R , 5 0b , 3 o

, e ,0 5 I 0

ko o, I I1 5 2 0

!

O C , I AO 0 0 5

' ' ' I ' I6 C T L R E C T/~ 4 5 "R = 5 ob , 5 o

I 0 15 2 0k ~

t 6

14

12

OE

OE

O i

5 8 0 A . N . W I L L IA M S a n d Z . D E MI RB IL EK

, I ' ' I ' 1 " I6 CYL RECT

b = 3 o

0 0 0 0 0 . 5 I 0 1 5 2 0

F IG . 1 9 . N o r m a l i z e d e x c i t in g f o r c e s a n d m o m e n t s o n s i x c y li n d e r a r r a y ( r e c t a n g t l e) f o r 13 = 4 5 , R / a = 5 ,b / a = 3 , d / a = 1 0 .


33/35

H y d r o d y n a m i c i n t e ra c t i o n s i n f lo a t in g c y l i n d e r a r r a y s i I 5 8 1i i i

6 C Y L R E C E

d " l O a0. !

- - o , .

O0 05 1.0 1.5ko *

i i I I~ CYI-90E C T " ,R 5ab 3o

1.6

I A

1.2

IC

i O.E

0. (

0 ~

2 0

2 0IBL61.412

u . " i UEI0 6

O~0 2OCO0

l i I I ,0.5 1.0 I 5koO

I i I6 CYL RECT.~ ' 9 0 "R ' S O

2 . 0

, I i I , I ,0.5 1.0 1.5 2Dko o

F IG . 2 0 . N o r m a l i z e d e x c i t i n g f o r c e s a n d m o m e n t s o n s i x c y l i n d e r a r r a y ( r e c t a n g l e ) f o r I~ = 9 0 , R / a = 5 ,b / a = 3 , d / a = 1 0 .


34/35

582 A . N, WILLIAMS nd Z . DEMIRBILEKc o n s i d e r e d . H o w e v e r , f o r k o a < 1 .0 , t h e h e x a g o n T L P e x p e r i e n c e s c o n s i d e r a b l y s m a l l e rh e a v e f o r c es a n d t h e p o s s i b i l i t y e x is t s o f e x p l o i t i n g th i s a p p a r e n t a d v a n t a g e o f th eh e x a g o n c o n f i g u r a t i o n in a c t u a l d e e p w a t e r T L P d e s i g n .

6 . C O N C L U S I O N SA n a p p r o x i m a t e , c o m p u t a t i o n a l l y e ff ic i e n t m e t h o d h a s b e e n p r e s e n t e d f o r e s t i m a t in gt h e h y d r o d y n a m i c i n t e r a c t i o n s d u e to w a v e s c a tt e r i n g b e t w e e n t h e m e m b e r s o f a n a r ra yo f s t a t i o n a r y t r u n c a t e d c i r c u la r c y l in d e r s s i m u l a t i n g t h e c o l u m n s o f an i d e a l i z e d T L P .T h e s o l u t i o n t e c h n i q u e i s e s s e n t i a l l y a l a r g e - s p a c i n g a p p r o x i m a t i o n a n d i n v o l v e sr e p l a c i n g d i v e r g e n t s c a t t e r e d w a v e s b y e q u i v a l e n t p l a n e w a v e s t o g e t h e r w i th n o n - p l a n a rc o r r e c t i o n t e rm s . N u m e r i c a l r e s u l ts h a v e b e e n p r e s e n t e d w h i c h s h o w t h e i n f l u e n c e o ft h e v a r i o u s w a v e a n d s t r u c t u ra l p a r a m e t e r s o n t h e h y d r o d y n a m i c l o a d i n g f o r a n u m b e ro f e x a m p l e c y l i n d e r c o n f i g u r a t io n s . I t i s f o u n d t h a t f o r c e r t a in p a r a m e t e r c o m b i n a t i o n s ,l a rg e i n c re a s e s in h y d r o d y n a m i c l o a d i n g a r e p r e d i c t e d c o m p a r e d t o th e l o a d i n g t h a t th ec y l i n d e r w o u l d e x p e r i e n c e i n i s o l a t i o n , e v e n a t r e l a t i v e l y l a rg e s p a c i n g s. T h e s e r e su l tsc l ea r ly i n d i c a te t h a t t h e h y d r o d y n a m i c i n t e r a c t i o n s b e t w e e n n e i g h b o r i n g c o l u m n s s h o u l db e c o n s i d e r e d b y d e s i g n e r s a n d r e s e a r c h e r s a s s o c i a t e d w i t h T L P s .

R E F E R E N C E SABuL-AzM , A .G . and WILLIAMS,A.N . 1987. Interference effects betwee n f lexible cylinders in waves. O c e a n

E n g n g 14 (1), 19-38.DEMIRBILEK,Z. and GASTON, J . 1985. No nlinear wave loads on a vertical cylinder. O c e a n E n g n g 12 (5),375-385.GARRETT, C.J .R. 1971. Wave force s on a cir cula r do ck. J . F l u i d M e c h . 4 6 , 129-139.KA6EMmO, H. and YU E, D.K .P. 1986. Interaction s among multiple th ree-dim ension al bod ies in water waves:an exac t a lgebra ic method. J . F l u i d M e c h . 166, 18%209.MClVER, P. 1984. Wav e forces on array s of floating bo die s. J . E n g n g M a t h . 18, 273-285.MCIVER, P. and EVANS, D.V . 1984. App roxim ation of w ave forces on cylind er arrays. A p p l . O c e a n R e s . 6 ,101-107.MARTIN, P.A . 1984. Multiple scatter ing of surface w ater w aves and the null-f ield me thod. P r o c . 1 5 th S y m p .N a v a l H y d r o d y n am i c s, H a m b u r g , G e r m a n y , pp. 119-132.MATSUI,T. and TAr~nKl,T. 1981. Hyd rodynam ic in te rac t ion b e tween groups of ve r t ica l ly axisymmetric b odiesfloating in waves. P r o c . I n t. S y r u p . H y d r o d y n a m i c s O c e a n E n g in e e r i n g , T r o n d h e i m , N o r w a y , Vol. 2, pp.817-836,OHKUSU, M. 1974. Hyd rod yn am ic forces on m ultiple cylinders in waves. P r o c . I n t . S y r u p . D y n a m i c s M a r i n eV e h i c l e s a n d S t r uc tu r e s in W a v e s , L o n d o n , U . K . , pp. 107-112.SARPKAYA,T. and ISAACSON,M. 1981. M e c h a n ic s o f W a v e F o r c e s o n O f f s h o r e S t r uc t ur e s . Van Nost randReinhold , New York.SIMON, M.J. 1982. Multiple s catter ing in array s of axisymmetric wave energy devices, p art 1 a matrix me thodusing a plane-wave approximation. J . F l u i d M e c h . 120 , 1-25.SPRING,B. H. and MONKMEYER, P.L . 1974. Inte ract ion of plan e waves with vertical cylinders. P r o c . 1 4 t h

C o n f . C o a s t al E n g in e e r i n g , C o p e n h a g e n , D e n m a r k , pp. 1828-1847.WATSON, G .N . 1944 . A T r e a ti se o n t h e T h e o r y o f B e s s e l F u n c ti o n s, 2nd edit ion. Cambridge University Press,Cambr idge .

A P P E N D I XM a t r i x c o e f f i c i e n t s

T h i s a p p e n d i x c o n t a i n s t h e e x p r e s s i o n s f o r t h e c o e f f i c ie n t s o f t h e s i m u l t a n e o u s m a t r i x e q u a t i o n s( 1 8 ) a n d ( 1 9 ) u s e d t o d e t e r m i n e t h e p o t e n t i a l c o e f f i c i e n ts i n t h e i s o l a t e d c y l i n d e r c a s e .

F ~ o ) H , ~ ( k o a ) 2 x /- 2 ko h ( - 1 ) ~ s inh koh 1 ( A . 1 )H ' , , ( k o a ) [1 + s inh 2 k o d / 2 k o d ] t ( k ~ h 2 + n 2 ~ r2 )


35/35

Hyd rodyn amic interactions in f loating cylinder array s--I 583~ , 2 ) = _ K m ( k o a ) 2 ~ r 2 k q h ( - 1 ) n s i n k q h

K', . (koa) [1 + sin 2 k q d / 2 k q d ] ( k 2q h - n2"tr2)R ~., ) = 2 ~ . , i ' + lk oh ( - 1 ) n s inh ko h { J ' , ,, (koa)

c o s h k o d (kt~h2 + n2"n 2) J m ( k ( l ) - i f m ( k o a )m sinh koh

G('2) = ~ a d k 2 [1 + sinh 2 k o d / 2 k o d ] ~m sin k q h

G~q') = , l ~ ad k2q [1 + sin 2 k q d / 2 k q d ] ~

G(o,~) -

for q >- 1,

G ~ q T =

f o r q - > 1 ,

# n r r h ( - 1 ) ~ s inh k o h l ' , , , ( m r a / h )d[1 + s inh 2 k o d / 2 k o d ] ~ I . , ( m r a / h ) ( k ~ h 2 + n 2 rr 2 )

f f n c t h ( - 1)" sin k q h [ . , , ( n w a / h )d[1 + sin 2 k q d / 2 k q d ] i l ,, , ( n 't r a/ h ) ( k 2 q h 2 - n21r2)

f o r n - > 1 ,

f o r n , q > - 1 .

( A . 2 )

( A . 3 )

( A . 4 )

( A . 5 )

( A . 6 )

( A . 7 )

floating cylinder

Documents