Applied Ocean Research 47 (2014) 2 0 4 - 2 1 8
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Applied Ocean Research
joutnal honnepage: www.elsevier.com/locate/aporTwo-dimensional moonpool resonances for interface and
surface-piercing twin bodies in a two-layer fluid
Xinshu Zhang^'*, Piotr Bandyk'^
^ Department of Naval Architecture and Marine Engineering, University ofMicliigan, Ann Arbor, MI 4SW9, United States " DRS Technologies, Inc., Advanced Marine Technology Center, Stevensville, MD 21666, United States
CrossMark
A R T I C L E I N F O A B S T R A C T
Article history:
Received 22 November 2013 Received i n revised f o r m 18 May 2014 Accepted 18 M a y 2014 Available o n l i n e 2 July 2014 Keywords: Two-layer f l u i d Moonpool resonance Internal wave Hydrodynamic coefficients interface and surface-piercing
We study the moonpool resonances of two interface and surface-piercing rectangular bodies in a two-layer fluid due to forced harmonic heave morion. The problem is solved by employing a domain decomposition scheme with an eigenfuncrion matching approach. Heave added mass and damping coefficients, as well as inner and outer region (far-field) radiated wave elevations, are computed to exam-ine the hydrodynamic behavior ofthe twin floaring bodies. The numerical solutions have been compared with those for the case investigated by Zhang and Bandyk [1 ], where the floaring bodies remain in the upper layer fluid. The present analyses reveal that there exist both Helmholtz and higher-order, also called sloshing mode, resonances in the two-layer fluid. It is found that, for an interface and surface piercing twin bodies, the higher-order resonances are closely related with both the free surface and internal waves inside the moonpool gap. Moreover, it is also found that low frequency forced motion can excite higher-order resonances through forming standing internal waves inside the moonpool. Parametric studies have been performed to idenrify the dependence of hydrodynamic behavior and resonant chatacterisrics on geometry and density strarificarion.
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1. I n t r o d u c t i o n
This p a p e r a i m s t o s t u d y t h e m o o n p o o l w a v e r e s o n a n c e p h e -n o m e -n o -n i -n a t w o - l a y e r f l u i d s y s t e m d u e t o t h e o s c i l l a t i -n g heave m o t i o n o f t w o i d e n t i c a l r e c t a n g u l a r bodies (also c a l l e d t w i n b o d -ies). I n c o n t r a s t t o a p r e v i o u s p a p e r [ 1 ] , w h i c h addresses t h e case w h e r e the f l o a t i n g t w i n b o d i e s o n l y r e m a i n i n t h e u p p e r l a y e r f l u i d , t h e p r e s e n t s t u d y w i l l f o c u s o n t h e m o o n p o o l r e s o n a n c e d u e t o i n t e r f a c e a n d s u r f a c e p i e r c i n g bodies. The m o o n p o o l s t u d i e d i n t h e p r e s e n t p a p e r m o d e l s t h e o p e n i n g / g a p b e t w e e n t w o f l o a t i n g b o d -ies, s u c h as a l i q u i f i e d n a t u r a l gas ( L N G ) c a r r i e r a n d t e r m i n a l i n t h e case o f s i d e - b y - s i d e a r r a n g e m e n t s , o r b e t w e e n t h e i n d i v i d u a l h u l l s o f a m u l t i - h u l l vessel. I n o f f s h o r e o p e r a t i o n , t h e f l u i d m o t i o n i n s i d e t h e m o o n p o o l a n d t h e d y n a m i c b e h a v i o r o f t h e h u l l s are c r i t -ical t o t h e d e s i g n a n d analysis o f ships, a n d also v e r y i m p o r t a n t f o r t h e d e v e l o p m e n t o f a n e f f i c i e n t a n d r e l i a b l e p r o c e d u r e f o r o f f s h o r e o p e r a t i o n s s u c h as d r i l l i n g , p i p e l i n e l a y i n g , floatover i n s t a l l a t i o n , a n d cargo o r c r e w t r a n s f e r r i n g . I n a d d i t i o n , t h e m o t i o n d y n a m i c s o f t h e LNG s h i p d u r i n g o f f l o a d i n g is also c r u c i a l f o r t a n k s l o s h i n g analysis o f t h e l i q u i f i e d gas. * Corresponding a u t h o n Tel.: +1 281 721 2369.
E-mail addresses: x i n s h u z ® u m i c h . e d u (X. Zhang), pbandyk@drs.com (P. Bandyk). 0141-1187/$ - see f r o n t m a t t e r ® 2014 Elsevier Ltd. A l l rights reserved.
http://dx.doi.org/10.1016/J.apor.2014.05,005
M o l i n [ 2 ] a n d M c l v e r [ 3 ] d e m o n s t r a t e d a resonance i n s i d e t h e m o o n p o o l u s i n g l i n e a r i z e d w a t e r w a v e t h e o r y . Y e u n g a n d Seah [ 4 ] s t u d i e d m o o n p o o l resonance f o r t w o s y m m e t r i c r e c t a n g u -lar bodies i n finite w a t e r d e p t h u s i n g a n e i g e n f u n c t i o n m a t c h i n g m e t h o d . Faltinsen [ 5 ] s t u d i e d t h e t w o - d i m e n s i o n a l p i s t o n - l i k e w a v e s l o s h i n g i n s i d e a m o o n p o o l based o n a d o m a i n d e c o m p o s i -t i o n s c h e m e a n d -the G a l e r k i n m e -t h o d , c o m p a r i n g n u m e r i c a l resul-ts w i t h e x p e r i m e n t a l m e a s u r e m e n t s . Later, K r i s t i a n s e n a n d F a l t i n -sen [ 6 , 7 ] i n v e s t i g a t e d o n t h e s a m e p r o b l e m u s i n g a v o r t e x t r a c k i n g m e t h o d a n d CFD, a n d f o u n d t h e d a m p i n g o n t h e near r e s o n a n t w a v e is m a i n l y due t o t h e v o r t e x s h e d d i n g f r o m t h e s h a r p c o r n e r p o i n t s . M o l i n [ 8 , 9 ] also e x t e n s i v e l y r e s e a r c h e d o n t h e d a m p i n g e f f e c t s o n gap resonances b y e x p e r i m e n t s a n d n u m e r i c a l s t u d i e s . M o s t o f t h e p r e v i o u s m o o n p o o l h y d r o d y n a m i c s t u d i e s h a v e b e e n f o c u s e d o n a s i n g l e - l a y e r fluid; v e r y f e w h a v e b e e n c a r r i e d o u t f o r s t r a t i f i e d flow, w h i c h occurs i n t h e m a r i n e e n v i r o n m e n t [ 1 0 - 1 2 ] . I n s t r a t i f i e d flow, t h e d e n s i t y changes w i t h t h e v a r i a t i o n s i n s a l i n i t y o r t e m p e r a t u r e i n t h e v e r t i c a l d i r e c t i o n . For e x a m p l e , a p y c n o c l i n e is d e f i n e d as the l a y e r w h e r e d e n s i t y change s i g n i f i c a n t l y . The fluid d e n s i t y above a n d b e l o w t h e p y c n o c l i n e is a l m o s t c o n s t a n t . T h e r e f o r e , w e can m o d e l t h e s t r a t i f i e d fluid s y s t e m as a t w o l a y e r fluid a s s u m i n g the p y c n o c l i n e is i n f i n i t e l y s m a l l .
W h e n t h e w a v e s t r u c t u r e i n t e r a c t i o n is t r e a t e d u s i n g p o t e n -tial t h e o r y , t h e b o u n d a r y v a l u e p r o b l e m m a y h a v e a n e i g e n v a l u e t o s a t i s f y t h e c o n d i t i o n o f n o r a d i a t e d w a v e s at i n f i n i t y . The cor-r e s p o n d i n g e i g e n f u n c t i o n is c a l l e d a ' t cor-r a p p i n g s t cor-r u c t u cor-r e ' . A t the
X Zhang, P. Bandyk/Applied Ocean Research 47 (2014) 204-218
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Fig. 1. Definition o f the p r o b l e m and coordinate systems.
t r a p p e d m o d e f r e q u e n c y , t h e a d d e d mass c o e f f i c i e n t s can be i n f i -n i t e . E x t e -n s i v e r e s e a r c h has b e e -n c o -n d u c t e d t o i -n v e s t i g a t e o -n t h e p o s s i b i l i t y o f ' w a v e t r a p p i n g ' , a p h e n o m e n a f o r b o t h fixed a n d floaring bodies. The .so-called ' t r a p p e d m o d e ' has been r e s e a r c h e d by M c l v e r [ 1 3 - 1 5 ] a n d K u z n e t s o v [ 1 6 ] f o r t w o - d i m e n s i o n a l a n d t h r e e - d i m e n s i o n a l cases, r e s p e c t i v e l y . N e w m a n [ 1 7 ] also s t u d i e d t h e t r a p p e d w a v e r e s o n a n c e i n a floating t o r u s u s i n g W A M I T . K u z n e t s o v et a l . [ 1 8 ] i n v e s t i g a t e d the w a v e t r a p p i n g m o d e s f o r t w o -d i m e n s i o n a l bo-dies i n a t w o - l a y e r fliii-d. H o w e v e r , t h e e x i s t e n c e o f w a v e t r a p p i n g m a y be a r a r e e x c e p d o n f o r a c t u a l m o o n p o o l s , i n c l u d i n g t h e p r e s e n t l y s t u d i e d d u a l h e a v i n g r e c t a n g u l a r bodies. Nevertheless, i d e n t i f y i n g t h e t r a p p e d r e s o n a n t c h a r a c t e r i s t i c s f o r a g i v e n m o o n p o o l s t r u c t u r e is c r i t i c a l ; a n d a m o t i v a t i o n f o r t h e p r e s e n t s t u d y . L i n t o n a n d M c l v e r [ 1 9 ] s t u d i e d t h e w a v e r a d i a t i o n a n d scat-t e r i n g o f a h o r i z o n scat-t a l c y l i n d e r i n a scat-t w o - l a y e r fluid. Y e u n g a n d N g u y e n [ 2 0 , 2 1 ] d e r i v e d a G r e e n f u n c t i o n f o r a s t e a d i l y t r a n s l a t -i n g source a n d a t w o - d -i m e n s -i o n a l t r a n s -i e n t G r e e n f u n c t -i o n f o r a n o s c i l l a t i n g source. T h e l a t t e r can be used f o r t h e c o m p u t a t i o n o f w a v e - s t r u c t u r e i n t e r a c t i o n i n t h e time d o m a i n . A l a m et a l . [ 2 2 ] i n v e s t i g a t e d t h e t h r e e d i m e n s i o n a l G r e e n f u n c t i o n f o r a n o s c i l l a t -i n g source t r a n s l a t -i n g w -i t h s t e a d y s p e e d a n d c o m p a r e d t h e f a r - f -i e l d r a d i a t e d w a v e s w i t h t h e r e s u l t s u s i n g d i r e c t n u m e r i c a l s i m u l a t i o n . The m e t h o d d e v e l o p e d i n t h i s p a p e r is a p p l i c a b l e t o t w i n r e c t a n g u -lar bodies. I n o r d e r t o s t u d y t h e case f o r a g e n e r a l i z e d g e o m e t r i e s , a f r e e s u r f a c e Green's f u n c r i o n o r Rankine p a n e l m e t h o d m a y be e m p l o y e d . T e n a n d K a s h i w a g i [ 2 3 ] a n d K a s h i w a g i e t al. [ 2 4 ] s t u d i e d t h e w a v e r a d i a t i o n a n d d i f f r a c t i o n p r o b l e m s f o r a t w o - d i m e n s i o n a l b o d y o f a r b i t r a r y shape u s i n g G r e e n f u n c t i o n , a n d c o m p a r e d t h e i r n u m e r i c a l results w i t h e x p e r i m e n t s . A n e i g e n f u n c t i o n m a t c h i n g m e t h o d is d e v e l o p e d i n t h e p r e s e n t s t u d y t o solve t h e b o u n d a r y v a l u e p r o b l e m f o r a t w o - l a y e r fluid sys-t e m . T h e e i g e n f u n c sys-t i o n m a sys-t c h i n g m e sys-t h o d has b e e n w i d e l y u s e d sys-t o s t u d y w a v e - b o d y i n t e r a c t i o n p r o b l e m s ( Y e u n g [ 2 5 ] , S h i p w a y a n d Evans [ 2 6 ] ) f o r e i t h e r t w o - d i m e n s i o n a l o r t h r e e - d i m e n s i o n a l cases. M a v r a k o s [ 2 7 ] also a p p l i e d t h e m e t h o d to c o m p u t e t h e h y d r o -d y n a m i c c o e f f i c i e n t s f o r t w o c o n c e n t r i c s u r f a c e - p i e r c i n g t r u n c a t e -d c i r c u l a r c y l i n d e r s . M o r e r e c e n t i y , M a v r a k o s a n d C h a t j i g e o r g i o u [ 2 8 ] used t h e m e t h o d t o s t u d y t h e s e c o n d o r d e r w a v e d i f f r a c t i o n p r o b -l e m f o r t w o c o n c e n t r i c c i r c u -l a r c y -l i n d e r s . T h e p r i n c i p a l f o c u s o f t h e p r e s e n t p a p e r is t h e w a v e r a d i a t i o n due t o heave e x c i t a t i o n o f t h e t w i n bodies i n b o t h u p p e r a n d l o w e r fluid layers. W e first g i v e t h e m a t h e m a t i c a l f o r m u l a t i o n , f o l l o w e d b y a n u m e r i c a l s c h e m e t o s o l v e t h e d i s c r e t i z e d l i n e a r s y s t e m . T h e h y d r o d y n a m i c b e h a v i o r o f t h e floating t w i n bodies near H e l m h o l t z (also c a l l e d p i s t o n m o d e ) a n d h i g h e r - o r d e r (also called s l o s h i n g
m o d e s ) r e s o n a n t m o d e s is e x a m i n e d . The o u t e r r e g i o n ( f a r field) r a d i a t e d f r e e surface a n d i n t e r n a l w a v e s are c o m p u t e d a n d d i s -cussed. P a r a m e t r i c s t u d i e s are p e r f o r m e d t o e x a m i n e t h e e f f e c t s o f m o o n p o o l g e o m e t r y a n d d e n s i t y s t r a t i f i c a t i o n o n t h e r e s o n a n t fluid m o t i o n a n d h y d r o d y n a m i c c o e f f i c i e n t s . I t s h o u l d be n o t e d t h a t t h e a s s u m p t i o n o f p o t e n t i a l flow d u r i n g r e s o n a n t b e h a v i o r g e n e r -a l l y o v e r - p r e d i c t s w -a v e -a m p l i t u d e s , -a n d v i s c o u s e f f e c t s ( s u c h -as v o r t e x s h e d d i n g f r o m t h e c o r n e r s ) m u s t be c o n s i d e r e d . H o w e v e r , t h e m e t h o d p r o p o s e d h e r e is e f f e c t i v e i n q u i c k l y p r e d i c t i n g reso-n a reso-n t c h a r a c t e r i s t i c s g i v e reso-n a m o o reso-n p o o l c o reso-n f i g u r a t i o reso-n , w h i c h m a y be used i n d e s i g n a n d o p t i m i z a t i o n . 2. M a t h e m a t i c a l f o r m u l a t i o n The s u r f a c e a n d i n t e r n a l w a v e s caused b y f o r c e d s m a l l a m p l i -t u d e v e r -t i c a l (heave) m o -t i o n o f -t w o r e c -t a n g u l a r h u l l s w i -t h i d e n -t i c a l g e o m e t r y i n a t w o - l a y e r fluid are s t u d i e d . T h e p r o b l e m s k e t c h is s h o w n i n Fig. 1. The w i d t h o f e a c h r e c t a n g u l a r b o d y is 2B. The d i s -tance b e t w e e n t h e t w o centers o f t h e b o d i e s is 2 W . T h e d r a f t o f e a c h floating c y l i n d e r is d. The d e p t h o f u p p e r a n d l o w e r fluid is / i i a n d Ii2, r e s p e c t i v e l y , w i t h a t o t a l w a t e r d e p t h ft = h i + h2. The fluid is a s s u m e d t o be ideal a n d t h e flow i r r o t a t i o n a l . T h e fluid d e n s i t y f o r t h e u p p e r a n d l o w e r fluid are p i a n d P2. r e s p e c t i v e l y . Since o n l y heave m o t i o n is s t u d i e d here, t h e h y d r o d y n a m i c p r o b l e m is s y m -m e t r i c a b o u t x = 0 ; t h e r e f o r e o n l y t h e d o -m a i n x > 0 is c o n s i d e r e d . T h e p r e s e n t s t u d y c o n s i d e r s t h e case o f d > h i , w h i c h assumes t h e t w i n bodies p e n e t r a t e t h e i n t e r f a c i a l s u r f a c e at z = - h i .
Let t h e heave m o t i o n o f t h e t w o i d e n t i c a l b o d i e s be f cos(a)f), w h e r e a) is t h e a n g u l a r f r e q u e n c y a n d f is t h e m o t i o n a m p l i t u d e . W e assume f « ; 0 ( 1 ) . T h e v e l o c i t y p o t e n t i a l w i t h i n t h e t w o - l a y e r fluid can be w r i t t e n as $ ( ' " ' ( x , z, f ) = W [ - i w ? < ^ ( ' " ' ( x , z ) e - ' ' " f ] ( 1 ) w h e r e (p'-"'\x, z) is t h e s p a t i a l v e l o c i t y p o t e n t i a l , m = 1 r e p r e s e n t s t h e s o l u t i o n i n t h e u p p e r fluid a n d m = 2 r e p r e s e n t s t h e s o l u t i o n i n the l o w e r fluid. The g o v e r n i n g e q u a t i o n f o r ^ f " " ' is t h e Laplace e q u a t i o n
«
The 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 is w r i t t e n as -Krp^'^'i = 0 at z = 0 " ( 3 ) w h e r e K=(a^jg, a n d g is t h e g r a v i t a t i o n a l a c c e l e r a t i o n a n d<P^ = d(pldz.
2 0 6 X. Zhang. P. Bandyk/ Applied Ocean Research 47 (2014) 204-218 The b o u n d a r y c o n d i t i o n s at t h e i n t e r f a c i a l s u r f a c e can be w r i t t e n as (see [ 2 1 ] a n d [ 2 9 ] ) at z = - / i i y ( 4 i ^ - / C 0 n ) ) = 4 2 ) _ / f 0 ( 2 ) at 2 = - h i (4) (5) w h e r e y = p i / p 2 is t h e d e n s i t y r a t i o . Because o n l y s y n c h r o n i z e d p i s t o n t y p e m o d o n o f the t w i n b o d -ies is c o n s i d e r e d , t h e h y d r o d y n a m i c p r o b l e m is s y m m e t r i c a b o u t t h e x = 0, i m p l y i n g /)i'") = 0 o n x = 0 The b o u n d a r y c o n d i t i o n a t t h e seabed is g i v e n b y = 0 a t z = - / 7 , ~-h2 = -h (6) (7) I n a d d i t i o n , m u s t s a t i s f y t h e n o - f l u x c o n d i t i o n o n t h e s o l i d b o d y b o u n d a r i e s . L e v e r a g i n g t h e s y m m e t r y p r o p e r t y , t h e side face b o u n d a r y c o n d i t i o n s are 4 " ' ^ = 0 a t X = I V T B , - d < z < 0 A n d t h e b o t t o m face b o u n d a r y c o n d i t i o n (/>p^ = 1 at z = -d,W-B<x<W + B (8) (9)
2. J. Division of computational domain
T h e r i g h t side o f t h e f l u i d d o m a i n {x > 0 ) can b e d e c o m p o s e d i n t o t h r e e r e c t a n g u l a r s u b - d o m a i n s w h i c h i n c l u d e I , I I , a n d i l l , as s h o w n i n Fig. 1. T h e v e l o c i t y p o t e n t i a l s i n t h o s e r e g i o n s are d e n o t e d b y 0f{m)^ 0//(2) g j ^ j j 0m(m)^ r e s p e c t i v e l y . 0 " ( 2 ) is t h e v e l o c i t y p o t e n t i a l o f t h e s u b d o m a i n u n d e r n e a t h t h e b o d y . Hence, t h e f l u i d w i t h i n r e g i o n I I s h o u l d s a t i s f y t h e f o l l o w i n g g o v e r n i n g e q u a t i o n (10) ( 1 1 ) (12) 0 S . ^ ' + 0 ^ ( 2 ) = O w i t h t h e f o l l o w i n g b o u n d a r y c o n d i t i o n s fQj. W-B<x<W-i-B,z = -d 0"'^^ = O f o r W-B <x <W-\-B,z==-li T h e v e l o c i t y p o t e n t i a l (t)'M f o r r e g i o n s 1 s h o u l d s a t i s f y
<Axi'"' + 0 z ^ ' = O ( 1 3 )
w i t h t h e f o l l o w i n g b o u n d a r y c o n d i t i o n s (f,P-K4>'W^0 a t z = 0 ( 1 4 ) A'(2) . : 0 a t z = - / i (15)T h e t h r e e c o n d i t i o n s above, Eqs. ( 1 3 ) - ( 1 5 ) , are t r u e i n r e g i o n III ( s u b s t i t u t e ( / ) " ' f o r </>'). O n t h e a d j o i n i n g b o u n d a r y x = W - B 0 ' ( 2 ) = </,"(2) at j ; ^ V V - B , -]i<z<-d ( 1 6 ) 0 i < 2 ) ^ 0 / / ( 2 ) a t x = W - B , -\i<z<-d ( 1 7 ) (/>i^"" = 0 a t x = M / - B , ~ d < z < 0 ( 1 8 ) O n t h e a d j o i n i n g b o u n d a r y x = W + B, 0 " ' ( 2 ) = 0 " ( 2 ) a t x = W + B, - / ! < z < - d ( 1 9 ) 0 f 2 ) = 0j;'(2) at x = W + B, - / t < z < - d ( 2 0 ) = O a t x = W + B, - d < z < 0 (21) Eqs. ( 1 6 ) , ( 1 7 ) , ( 1 9 ) a n d ( 2 0 ) , are n e e d e d t o e n s u r e t h e c o n t i n u i t y o f t h e f l u i d v e l o c i t y a n d p o t e n t i a l a t t h e c o m m o n b o u n d a r y o f t h e n e i g h b o r i n g r e g i o n s . T h e s y m m e t r y c o n d i t i o n a b o u t t h e z-axis a t x = 0 f o r r e g i o n I gives 0 f o r X = 0, - / i < z < 0 (22) I n r e g i o n 111, t h e r a d i a t i o n c o n d i t i o n f o r t h e w a v e e l e v a t i o n i m p l i e s t h a t
y{x,t) = r){x)e-'""t as x^co ( 2 3 ) w h e r e Y{x, t) is t h e o u t g o i n g f r e e surface w a v e e l e v a t i o n . T h e s o l u -tions i n a l l t h e t h r e e regions h a v e t o be m a t c h e d a t t h e a d j o i n i n g b o u n d a r i e s . 2.2. Solution in subdomain I The v e l o c i t y p o t e n t i a l i n r e g i o n I c a n b e w r i t t e n as, oo 0'(™)(x, z ) = ^ B ] A ' ( / < j " , x ) z ( " " ( / < j ^ ' , z ) J=0 • ^ B f A ' { l f \ x ) Z ^ " ' \ l j ,(2) (24) j=o w h e r e A ' a n d Z^'"' are t h e e i g e n f u n c t i o n s i n t h e h o r i z o n t a l a n d v e r t i c a l d i r e c t i o n s , r e s p e c t i v e l y . Bl a n d B? are c o e f f i c i e n t s w h i c h w i l l b e d e t e r m i n e d b y m a t c h i n g f l u i d v e l o c i t y a n d p o t e n t i a l a t t h e j u n c t u r e b o u n d a r y b e t w e e n r e g i o n s I a n d I L / < | " \ j = 0 , 1 , . . . , ooare t h e e i g e n v a l u e s w h i c h c a n be c a l c u l a t e d b y s o l v i n g t h e d i s p e r s i o n r e l a t i o n s ( 2 5 ) a n d ( 2 9 ) . For j = 0, t h e d i s p e r s i o n r e l a t i o n can b e s h o w n t o be
fc
2 ( l + ) t , t 2 ) f l + t 2 + ( - ! ) " - , + t 2 r - 4 « l t 2 ( l + )/t,t2) (25) w h e r e n = 1 r e p r e s e n t s t h e d i s p e r s i o n r e l a t i o n f o r t h e s u r f a c e w a v e a n d n = 2 r e p r e s e n t s t h e i n t e r n a l w a v e . Y e u n g [ 2 1 ] d e r i v e d t h e d i s -p e r s i o n r e l a t i o n s h o w n i n ( 2 5 ) f o r o u t g o i n g w a v e s u s i n g t h e m e t h o d o f s e p a r a t i o n o f variables, w i t h t h e f o l l o w i n g d e f i n i t i o n s e = l - y t l = tanh(;<:/ii) t2 = t a n h ( W i 2 ) For j > 1, t h e d i s p e r s i o n r e l a t i o n is 0)2 -k g 2 ( l - ) / t i t 2 ) w i t h f l = t a n ( W i i ) f2 = tan(/i/72) t i + t 2 + ( - i r + t 2 r - 4 e t , t 2 C l - y t i t 2 ) (26) (27) (28) (29) (30) (31) T h e d i s p e r s i o n r e l a t i o n ( 2 5 ) is i l l u s t r a t e d i n Fig. 2. As s h o w n i n t h e flgure, i n c o n t r a s t t o f r e e surface w a v e m o d e , t h e i n t e r n a l w a v e n u m b e r s h o w s m o r e sensitive t o t h e d e n s i t y r a t i o y. By s e l e c t i n g y = 1, t h e t w o l a y e r fluid s y s t e m b e c o m e s a s i n g l e -l a y e r f-luid, a n d t h e d i s p e r s i o n r e -l a t i o n ( 2 5 ) r e d u c e s t o ^ = /<tanh(/</i), OJI=0; W h i l e t h e o t h e r d i s p e r s i o n r e l a t i o n ( 2 9 ) r e d u c e s t o - / c t a n ( / d j ) , ft)? = 0; (32) (33) w h i c h are c o n s i s t e n t w i t h t h e d i s p e r s i o n r e l a t i o n s f o r s i n g l e - l a y e r fluid s y s t e m (see Y e u n g [ 4 ] a n d W e h a u s e n [ 3 0 ] ) .X. Zhang, P. Bandyk /AppUed Ocean Research 47 (2014) 204-218 207
UJ
Fig. 2. Dispersion relation of a two-layer f l u i d system, K is wave number, a) is wave frequency, and y is the density ratio.
The e i g e n f u n c t i o n i n t h e v e r t i c a l d i r e c t i o n Z^""' is w r i t t e n as (Ü;2
'^Vg/<[,"))sinh;<f,'"z-hcosh/4"'z
2 0 ) ( ; f , z ) = {co'^/gkf'>)smlf^z + c o s l f h « 2 c o s h / 4 " ' ( z - ^ l i )for j = 0
f o r j > l ( 3 4 ) g;c[,"'sinh/4"'h2 c t ) 2 c o s ; < j " ' ( z - f h ) t g f c f s i n f c f h 2 f o r i = 0 f o r j > l ( 3 5 ) w h e r e a'(fcj"') is t h e a m p l i t u d e r a d o d e f i n e d b y ; ) / , , ) [ 1 _ ^ t a n h 1 f o r j = 0 cosh(/<; in) ( 3 6 ) cos(/<"Jh,) - 1 i ^ t a n ; < " J / i i f o r j > l The e i g e n f u n c t i o n i n t h e h o r i z o n t a l d i r e c t i o n A ' f o r r e g i o n I is w r i t t e n as c o s ( ; 4 " ' x ) A ' ( k ( ' " , x ) = c o s ( / < [ | " ' ( W - B ) ) cosh(;<]"'x) c o s h ( / < j " ' ( W - B ) )for J = 0
f o r j > l ( 3 7 ) 2.3. Solution in subdomain UI The s o l u t i o n i n r e g i o n 111 can be w r i t t e n as w h e r e A j a n d A? are c o e f f i c i e n t s w h i c h 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 b o u n d a r y c o n d i t i o n at t h e j u n c t u r e b o u n d a r y b e t w e e n regions 11 a n d I I I . A ' " is t h e e i g e n f u n c t i o n i n t h e h o r i z o n t a l d i r e c t i o n f o r r e g i o n 111 A ' " ( / < f ' , x ) = ( 3 9 ) 2.4. Solution in subdomain II The s o l u t i o n i n r e g i o n 11 can be d e c o m p o s e d i n t o a h o m o g e n e o u s s o l u t i o n 4>"^ a n d a p a r t i c u l a r s o l u t i o n </)'*, w h i c h is c o n s t r u c t e d t o s a t i s f y t h e i 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 d e s c r i b e d i n Eq. ( 9 ) . Thus t h e t o t a l s o l u t i o n ó" is w r i t t e n as 6 " = ^()'"'-h0"P The h o m o g e n e o u s s o l u t i o n c a n be e x p r e s s e d as è " ' ' ( x , z ) = £ ^ H ( A , . , x ) y ( X j , z ) ( 4 0 ) ( 4 1 ) 1=0 w i t h H ( A , - , x ) : Co + Do x-W B f o r ! = 0 c o s h A K x W ) s i n h A , < x - W ) . ^ ^ ' c o s h X j B ' s m h A f Bw h e r e A.,- are t h e eigenvalues w h i c h can be c o m p u t e d u s i n g
in f o r 1 = 0 , 1 , . . . , , ( 4 2 ) ( 4 3 ) 0"'('")(x,z) = ^ / \ ; A ' " ( / < j ' \ ; ( ) Z ( ' " ' ( k ] ' ^ z ) + ^ / l / A ' ' ' ( ; ^ ^ ^ ^ ^ ( 3 8 ) y ( X j , z ) = ƒ ^ i-o 1-0 " ' • - / t - d The e i g e n f u n c t i o n i n t h e v e r t i c a l d i r e c t i o n Y(Xi, z ) is w r i t t e n as f o r 1 = 0 V 2 c o s A ( ( z - f / i ) f o r ! > 1 ( 4 4 )
208 X. Zhang, P. Bandyk/Applied Ocean Research 47(2014) 204-218 -O—Added mass, KB =0.09001 —0— Added mass, KB =• 0.13108 —Dampitig, KB = 0.0Ü0Ü1 - a - Dwnpino. KB = 0.13108
10'
log(iVO
Fig. 3. Convergence tests for h y d r o d y n a m i c coefficients (heave added mass and damping), W = 5.0, B = 1.0, hi =2.0, hi =2.0, d = 2 . 5 , ^ " O . ? , K=cü^lg, N, is number o f terms for A|, N j is n u m b e r o f terms f o r A:j"', Nj = 20.
T l i e p a r t i c u l a r s o l u t i o n 0'*, s u b j e c t t o b o u n d a r y c o n d i t i o n s ( 7 ) a n d (9), is g i v e n by
( 4 5 )
By m a t c h i n g the v e l o c i t y p o t e n t i a l a t x = W - B a n d x = W+B, a n d u s i n g t h e o r t h o g o n a l i t y c o n d i t i o n f o r 7,, w e o b t a i n
Q-Di = ^l<(p',Yi> ~ <4>"P,Yi>] it x = W - B ( 4 6 )
Q + = ] ^ [ < 0'". ^ i > - < 0'*, y,- > ] a t X = M/ + B ( 4 7 ) w h e r e W, = / w ( z ) y 2 ( A i , z ) d z = ; 7 - d w ( z ) w i t h t h e w e i g h t f u n c t i o n w ( z ) d e f i n e d b y K - ' l l £ z < 0 ^^ 1 - / i < z < - / i ] M a t c h i n g the f l u i d v e l o c i t y at x = W - B y i e l d s Y^BjA''il<f\ x)Z('"\kf\ z)-i-Y,BfA''{kp, x)Z«^\lf\ z ) k=0 j=0 oo = Y^H'iki,x)Y{ki,z) + <l)'iPix,z) i=0 By u s i n g the o r t h o g o n a l i t y o f Z ( ' " ' ( / c j ^ ' , z), w e o b t a i n ( 4 8 ) ( 4 9 ) (50) BjA''(l<f\x)hé.^'' = / w ( z ) V H ' ( A i , x ) n A , - , z ) Z ( " ' ) ( / < ] ^ ' , z ) d z + / w(z)0j;'P(x,z)Z(""(k('',z)d, ' h 1=0 B^?A''(/<j^',x)JW|i^ = ^ H ' ( A i , x ) L | ^ ^ ^ + / w(z)0i"'(x,z)Z('")(/<(i',z)d 1=0 (51) 1=0 - d B 2 A''(/<t2), = ^ H ' ( A , - , X)L|,^' + ƒ w ( z ) 0 f ( X , z)Z("')(/<]2), z)dz w h e r e i , [ j ' ' a n d i w j " ' are g i v e n i n A p p e n d i c e s B a n d C, respectively. w h e r e ' d e n o t e s t h e d e r i v a t i v e o f the f u n c t i o n s S i m i l a r l y , b y u s i n g the o r t h o g o n a l i t y c o n d i t i o n f o r Z'-'^\lj^\ z), Eq. ( 5 0 ) y i e l d s (52) BfA>\kf\x)M^'^^= / t v ( z ) £ ] H ' ( A i , x ) n A , , z ) Z ( " " ( / < ] 2 ) , z ) d z + / w ( z ) 0 f ( x , z)Z('"'(/<:]2), z ) d J^h J-h
r
M a t c h i n g the v e l o c i t y a t x = W + B
0 0 0 0
X. Zhang, P. Bandyk/AppUed Ocean Research 47 (2014) 204-218
3. N u m e r i c a l details a n d c o n v e r g e n c e 209 ;<=o ^ H ' ( A ( , x ) y ( X , - , z ) + 0 i " ' ( x , z ) (=0 T a i d n g advantage o f the o r t h o g o n a l i t y o f Z ( ' " ) ( / < j ' \ z ) The i n h o m o g e n e o u s s y s t e m o f t h e six i n t e g r a l e q u a t i o n s i n c l u d -i n g ( 4 6 ) , ( 4 7 ) , ( 5 1 ) , ( 5 2 ) , (54), a n d ( 5 6 ) c o u p l e t h e s -i x g r o u p s o f c o e f f i c i e n t s Q, D,, A j B j , a n d B?. The associated e i g e n v a l u e series ^g^-) is t r u n c a t e d u s i n g N j t e r m s , w h i l e / < ! " ' ( f o r n = 1 , 2 ) is t r u n c a t e d u s i n g Nj t e r m s . Hence, t h e s i x i n t e g r a l e q u a t i o n s are d i s c r e t i z e d i n t o a l i n e a r s y s t e m o f r a n k 2Ni + 4Nj. The l i n e a r s y s t e m is s o l v e d AjA"''(kf\x)My' = j I - h z ) ^ H ' ( A , - , x)Y{Xi, z)Z^"%if\ z)dz + / i ; A ' " ' ( / f U ) M ] " = Y ^ H ' i X t , x ) f + f wiz)<p''''{x,z)ZOnKlj'\z)dz w i t h . f i / < W f o r j = 0 ( 5 5 ) w(z)4>?(x,z)Z^'"\kf\z)dz A " ' \ k f , W + B) = ( 5 4 ) - / < ! " ' f o r j > l S i m i l a r l y , b y u s i n g the o r t h o g o n a l i t y o f Z('"'(;<ï^', z ) u s i n g LU d e c o m p o s i t i o n f o r each f o r c e d m o t i o n f r e q u e n c y . Once t h e six g r o u p s o f c o e f f i c i e n t s are d e t e r m i n e d , t h e h y d r o d y n a m i c c o e f f i c i e n t s a n d b o t h t h e f r e e surface a n d i n t e r f a c i a l surface w a v e e l e v a t i o n s c a n be c o m p u t e d u s i n g Eqs. ( 5 7 ) , ( 5 8 ) a n d ( 6 0 ) . Convergence tests h a v e b e e n c a r r i e d o u t f o r a t y p i c a l m o o n -p o o l c o n f i g u r a t i o n . T w o f o r c i n g f r e q u e n c i e s w i t h n o n - d i m e n s i o n a l
AfA<"\kf\x)M]^^= I "w{z)J2H'{Xi,x)Y{Xi,z)Z^"'\kf\z)dz+ f " wiz)(p'^{x, z)Z('"\kfKz)dz
J-h J-h
AjA"''{k'f\x)M]^^ = Y,^^'(Xi,x)L[ (2) i=0
-d ( 5 6 )
w ( z ) 0 f ( x , z ) Z ( ' " ) ( / < ] 2 ' , z ) d z
2.5. Evaluation of wave elevations and liydrodynamic coefficients The f o r m o f t h e f r e e surface w a v e e l e v a t i o n ( p e r u n i t f o r c e d m o t i o n a m p l i t u d e f ) a t z = 0 can be w r i t t e n as if{x} = K4>('\x,0) ( 5 7 ) w h e r e K=co'^lg T h e i n t e r n a l w a v e e l e v a t i o n (per u n i t f o r c e d m o t i o n a m p l i t u d e f ) a t z = - / i i is d e f i n e d b y „ / r „ ^ _ - / i i ) - K 0 ( " ( x , - f t , ) ]
V ( x )
= 1 - K ( 5 8 ) T h e r e f o r e , all t h e p r e s e n t e d ) f a n d n' are n o n - d i m e n s i o n a l t h r o u g h o u t this paper. T h e n o n - d i m e n s i o n a l heave a d d e d mass a n d d a m p i n g c o e f f i c i e n t s o f t h e t w i n bodies are c o m p u t e d u s i n g t h e v e l o c i t y p o t e n t i a l o n t h e w e t t e d b o d y surface t h r o u g h 033 + iÖ33 B2 W+B <p"{x, - d ) d x w W+B /CB = 0 . 0 9 0 0 1 a n d / f B = 0.13108 are selected. /fB = 0 . 0 9 0 0 1 c o r r e -s p o n d -s t o a n o r m a l w a v e f r e q u e n c y a n d 1CB = 0.13108 i-s clo-se t o t h e first h i g h e r - o r d e r r e s o n a n t f r e q u e n c y . The 'Error' s h o w n i n Figs. 3 a n d 4 is d e f i n e d as (Vjv, - VNmax)/{VNmaxl w h e r e N j ( o r Nj) are t h e n u m b e r o f t e r m s f o r i ( o r j ) a n d ' V r e p r e s e n t s a d d e d mass o r d a m p i n g . I n these studies, Nmax = 5 0 has b e e n selected as t h e u p p e r l i m i t . As i l l u s t r a t e d i n t h e l o g l o g p l o t s , t h e p r e d i c t i o n o f t h e h y d r o -d y n a m i c c o e f f i c i e n t s has a c h i e v e -d g o o -d convergence, e v e n a r o u n -d t h e r e s o n a n t f r e q u e n c y . T h e r e f o r e , N( = N j = 4 0 has b e e n used f o r c o m p u t a t i o n s t h r o u g h o u t t h i s paper, unless o t h e r w i s e s p e c i f i e d . It s h o u l d be n o t e d t h e l o c a l convergence o f the v e l o c i t y a t t h e s h a r p c o r n e r p o i n t s is s l o w d u e t o t h e s i n g u l a r i t y i n f r a m e o f p o t e n t i a l flow t h e o r y , b u t i t does n o t i m p a c t t h e c o n v e r g e n c e o f t h e g l o b a l h y d r o d y n a m i c b e h a v i o r . The c o n t i n u i t y o f fluid v e l o c i t y at t h e i n t e r f a c i a l b o u n d a r y is i l l u s t r a t e d a n d v e r i f i e d i n Figs. 5(a) a n d ( b ) f o r t w o d i f f e r e n t f o r c -i n g f r e q u e n c -i e s . B o t h t h e real a n d -i m a g -i n a r y parts o f v e l o c -i t y d u e t o u p p e r l a y e r fluid 0 ^ ' ' m a t c h t h e v e l o c i t y o f t h e l o w e r l a y e r fluid ^f' at t h e i n t e r f a c i a l s u r f a c e . The s y m m e t r y c o n d i t i o n a b o u t x = 0 is a u t o m a t i c a l l y e n s u r e d b y a p p l y i n g t h e s y m m e t r i c h o r i z o n t a l f u n c -tion g i v e n i n Eq. ( 3 7 ) . 2 rVV+B = • 5 2 / l<P"''{x,-d) + <p"P{x,-d)]dx ( 5 9 ) 4. P r e d i c t i o n o f l i y d r o d y n a m i c c o e f f i c i e n t s ° Jw-B w h e r e 033 a n d Ö33 r e p r e s e n t t h e h e a v e a d d e d mass a n d d a m p i n g c o e f f i c i e n t s o f t h e bodies, r e s p e c t i v e l y . By s u b s t i t u t i n g ( 4 1 ) and ( 4 5 ) i n t o Eq. ( 5 9 ) a n d i n t e g r a t i n g , i t y i e l d s«33 + itl33 = •
E
H{k,, x)Y{X,, -d)dx + Bih - d) - B33 ( / ] - d ) ( 6 0 )
It s h o u l d be n o t e d t h a t t h e d i m e n s i o n a l a d d e d mass a n d d a m p -i n g c o e f f -i c -i e n t s are n o n - d -i m e n s -i o n a l -i z e d b y p2B'^ a n d p2o:>B^, r e s p e c t i v e l y .
I n o r d e r t o v a l i d a t e t h e p r e s e n t c o m p u t a t i o n a l m o d e l , a l i m i t i n g case is selected t o c o m p a r e w i t h t h e results f o r a s i n g l e - l a y e r fluid b y Y e u n g a n d Seah [ 4 ] . The d e p t h s o f t h e u p p e r a n d l o w e r l a y -ers are ft, = 0 . 2 a n d ft2=20.0, r e s p e c t i v e l y . The d r a f t o f t h e t w i n bodies is d = 1 . 0 a n d d e n s i t y r a t i o is y = 0.9. Since ft2/fti = 1 0 0 , t h e p r e s e n t results are e x p e c t e d t o r e d u c e t o t h e s o l u t i o n f o r a s i n g l e -l a y e r f-luid w i t h w a t e r d e p t h ft = 2 0 . As i -l -l u s t r a t e d i n Fig. 6, t h e p r e s e n t c o m p u t a t i o n a l results agree w e l l w i t h t h e s o l u t i o n s f o r t h e s i n g l e - l a y e r fluid case. There are t h r e e b o u n d e d spikes o n t h e c u r v e o f b o t h a d d e d mass a n d d a m p i n g i n t h e l o w f r e q u e n c y r e g i o n b e t w e e n KB=0.02 a n d 0.15. These spikes are r e l a t e d t o h i g h e r - o r d e r
210 X Zhang, P. Bandyk/Applied Ocean Research 47(2014) 204-218 Added mass, / f S = 0.09001 Added mass, /-STS = 0.1.3108 —•Xr— Damping, KB = 0.09001 -a-. Damping, JfJ? = 0.13108 10 10 10 l o g ( A r , )
Fig. 4 . Convergence tests f o r h y d r o d y n a m i c coefficients (heave added mass and d a m p i n g ) , W = 5 . 0 , B = 1.0, h i =2.0, /12 =2.0, d = 2.5, y = 0.7, K=(o^lg, Nt is n u m b e r of terms f o r Xl, N] is n u m b e r o f terms f o r k j " ' , N, = 20.
resonances o r c a l l e d s l o s h i n g m o d e resonances due t o i n t e r n a l weaves, w h i c h w i l l be f u r t h e r e x p l o r e d i n t h e n e x t s e c t i o n .
Fig. 7 s h o w s t h e h e a v e a d d e d mass a n d d a m p i n g c o e f f i c i e n t s o f t h e t w i n b o d i e s as a f u n c r i o n o f KB f o r a t y p i c a l m o o n p o o l c o n f i g u -r a U o n . The h y d -r o d y n a m i c c o e f f i c i e n t s a-re also l i s t e d i n Table 1. W e seek t h e c r i r i c a l f r e q u e n c i e s w h e r e t h e r e is m i n i m a l w a v e r a d i a -t i o n i n -t h e o u -t e r r e g i o n . As can be seen i n Fig. 7, c r i -t i c a l f r e q u e n c i e s are f o u n d at /CB= 0.102, 0 . 1 3 1 , 0.278, a n d 0.785. The H e l m h o l t z m o d e can be i d e n t i f i e d a r o u n d /CB = 0.102, w h e r e t h e a d d e d mass changes s i g n a n d d a m p i n g a p p r o a c h e s zero. A t t h e o t h e r r e s o -n a -n t f r e q u e -n c i e s , t h u s c a l l e d h i g h e r - o r d e r reso-na-nces, t h e a d d e d mass a n d d a m p i n g b o t h s h o w b o u n d e d spikes. T h e a d d e d mass changes s i g n n e a r t h o s e r e g i o n s a n d t h e d a m p i n g c o e f f i c i e n t s s u d -d e n l y increases a n -d t h e n -d r o p , a p p r o a c h i n g zero. The spikes s h o w n i n Fig. 7 are a l l b o u n d e d since t h e c o m p l e x d e t e r m i n a n t associated
w i t h t h e u n k n o w n c o e f f i c i e n t s Q, D,-, A j , A?, Bj, a n d B?, c o u p l e d b y ( 4 6 ) , ( 4 7 ) , ( 5 1 ) , ( 5 2 ) , ( 5 4 ) , a n d ( 5 6 ) , r e m a i n s n o n - z e r o . 5. F r e e s u r f a c e a n d i n t e r f a c i a l s u r f a c e e l e v a t i o n s B o t h f r e e s u r f a c e a n d i n t e r n a l w a v e e l e v a t i o n s h a v e b e e n c o m p u t e d u s i n g Eqs. ( 5 7 ) a n d ( 5 8 ) , r e s p e c t i v e l y . T h e m o o n p o o l c o n -figuration a n d s t r a t i f i c a t i o n p a r a m e t e r s are t h e s a m e as t h o s e f o r Fig. 7. Fig. 8 s h o w s t h e r e a l a n d i m a g i n a r y c o m p o n e n t s o f t h e c o m -p l e x m o o n -p o o l f r e e s u r f a c e w a v e e l e v a t i o n r]^ a n d i n t e r n a l w a v e e l e v a t i o n rj' i n s i d e the m o o n p o o l n e a r t h e H e l m h o l t z r e s o n a n t f r e q u e n c y l f B = 0.102. As o b s e r v e d i n these figures, t h e i m a g i n a r y c o m p o n e n t changes s i g n n e a r t h e r e s o n a n t f r e q u e n c y , c o n s i s t e n t w i t h t h e a d d e d mass a n d d a m p i n g c h a r a c t e r i s t i c s i l l u s t r a t e d i n ( a ) - - -ffe(0l") - 0 fle(,/,(-)) • • ƒ ? ) ) . ( < / ; ' ! " ) ( b ) ' Re.{(l>{^^) - 0 fle(0p)) Im{<!>{^>) X / m ( 0 ( 2 ' ) l l l l 0.0 o.e - - -ffe(0l") - 0 fle(,/,(-)) • • ƒ ? ) ) . ( < / ; ' ! " ) 0 8 0.6 Re.{(l>{^^) - 0 fle(0p)) Im{<!>{^>) X / m ( 0 ( 2 ' ) 0.4 • 0.4 - -0.2 02' -1 0 To c > -0.2 ' 1 ° s ^ 0 -1 0 To c > -0.2
-
-0.2 3., -0.4 - -0.4 ~ ~ 0 - — 0 0 Q — 0 S — ^ -0.6 - -0.6-
-- 0 . 8 - i-
-0 8 -1 1 1 1 1 L _ -0.1 0.2 0.3 0.4 0.5 0.6 0.7 .x/(W-B) 0 4 0.5 x/(W-B)C O
C O
CO C3
10
X. Zhang. P. Bandyk/ AppUed Ocean Research 47 (2014) 204-218 111
2 0 f 15 • A d d e d m a s s - p r e s e n t results D a m p i n g - p r e s e n t results A d d e d m a s s ( Y e u n g a n d S e a h 2 0 0 6 ) D a m p i n g ( Y e u n g a n d S e a h 2 0 0 6 ) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
K B
Fig. 6. H y d r o d y n a m i c coefficients o f ttie t w i n floating bodies i n heave m o t i o n , 033 and 633, w i t h W = 5 . 0 , B = 1.0, hi =0.2, hi =20.0, d = 1.0, y = 0 . 9 , K=co^lg
Fig. 7. As can b e i d e n d f i e d f r o m t h e p l o t o f Re{7f}, t h e r e is a n o n z e r o m e a n f r e e s u r f a c e w a v e e l e v a t i o n (?j^) near /fB = 0.102.
Fig. 9 s h o w s t h e r e a l a n d i m a g i n a r y parts o f b o t h f r e e s u r f a c e a n d i n t e r n a l w a v e e l e v a t i o n s near the f i r s t h i g h e r - o r d e r r e s o n a n t f r e q u e n c y , / f B = 0 . 1 3 1 . The r e a l c o m p o n e n t o f t h e i n t e r n a l w a v e changes s i g n a r o u n d /CB = 0 . 1 3 1 , w h i l e the i m a g i n a r y p a r t changes sign a l i t d e b e y o n d KB = 0.132. Fig. 10 s h o w s t h e s a m e set o f r e s u l t s f o r t h e second h i g h e r - o r d e r r e s o n a n t f r e q u e n c y , /CB = 0,278. I t is i n t e r e s t i n g t o f i n d t h a t t h e b e h a v i o r o f the f i r s t t w o h i g h e r - o r d e r resonances are c h a r a c t e r i z e d b y s t a n d i n g i n t e r n a l w a v e s e x i s t i n g i n t h e m o o n p o o l gap, as can be i d e n d f i e d i n b o t h Figs. 9 a n d 10. This i n d i c a t e s t h a t , f o r t h e p r e s e n t m o o n p o o l c o n f i g u r a d o n , t h e f i r s t t w o h i g h e r - o r d e r r e s o n a n t f r e q u e n c i e s are d i r e c t l y associ-a t e d w i t h t h e r associ-a d i associ-a t e d i n t e r n associ-a l w associ-a v e s i n s t e associ-a d o f f r e e s u r f associ-a c e w a v e s . Fig. 11 s h o w s t h e r e a l a n d i m a g i n a r y p a r t s o f b o t h f r e e s u r f a c e a n d i n t e r n a l w a v e e l e v a t i o n s n e a r t h e s i x t h h i g h e r - o r d e r r e s o n a n t f r e q u e n c y /<:B = 0.785. Based o n t h e f r e e s u r f a c e w a v e e l e v a d o n i n t h e m o o n p o o l gap, t h e r e s o n a n c e c a n be c o r r e l a t e d w i t h t h e s t a n d -i n g f r e e surface w a v e -i n t h e m o o n p o o l . T h -i s p h e n o m e n o n w a s n o t f o u n d i n t h e p r e v i o u s s t u d i e s [ 1 ] . F u r t h e r , t h e f o r m o f t h e s t a n d i n g f r e e surface w a v e suggests t h i s r e s o n a n c e m o d e is t h e f i r s t r e l a t e d t o t h e f r e e s u r f a c e . The f i r s t f i v e h i g h e r - o r d e r r e s o n a n t f r e q u e n c i e s are all associated w i t h i n t e r n a l w a v e s .
It s h o u l d be n o t e d t h a t t h e r e s o n a n t w a v e e l e v a t i o n s s h o w n i n Figs. 9 1 1 m a y be o v e r p r e d i c t e d due t o t h e a s s u m p t i o n o f p o t e n -tial flow, w i t h o u t c o n s i d e r i n g t h e v i s c o u s e f f e c t s s u c h as t h e v o r t e x s h e d d i n g f r o m t h e s h a r p c o r n e r o f t h e b o d y . These d a m p i n g e f f e c t s m a y increase w i t h t h e f o r c e d m o t i o n f r e q u e n c i e s , s i m i l a r t o t h e r o l l d a m p i n g c h a r a c t e r i s t i c s f o r a b a r g e - t y p e d m o n o - h u l l . 40 3 0 20 10 0 - 1 0 - 2 0 - 3 0 h ^33
' , . - . - 6 ,
'\
\ I'. - 4 0 0,2 0 , 4 ' 0 , 6K B
0 , 8212 X. Zhang, P. Bandyk/Applied Ocean Research 47(2014) 204-218 0.6 0.4 0.2 CO Q 0^ - 0 . 2 - 0 . 4 - 0 . 6 0 2 r 1.5 1 • 0.5 • 0 " ID -0.5 • - 1 • -1.5 • 2 -• KB = O.WO,real KB = 0.W2,real • KB = 0.105, real • 0,2 0.4 0.6
x / ( W - B )
-KB = 0.100,real KB = 0.102,real KB = 0.W5,real 0.2 0.4 0.6x / ( W - B )
0.8 0.8 0.2 0.15 0.1 0.05 0£
0 - 0 , 0 5 -0,1 -0,15 - 0 . 2 -0.25 0.15 0.1 0.05 0 -0.05 - 0 . 1 -0.15 - KB = 0.100, imag KB = 0.102, i m a ^ . KB = 0.105,imag 0.2 0.4 0.6x / ( W - B )
0,8 - KB = 0.100, imag KB = 0,102, imag • KB = 0.105, imag J 0.2 0,4 0.6x / ( W - B )
Fig. 8. Real and d = 2 . 5 , y = 0 , 7 .
imaginary components of free surface if and internal w a v e elevation r)' across the H e l m h o l t z resonant frequency KB=0,102, W= 5,0, B = 1,0, hi = 2,0, hz = 2,0,
0,5 -0.5 I 6 4 Cc; - KB = 0.128, real KB = 0.131,real KB = 0.135,real 0.2 0.4 0.6
x / ( W - B )
- M = 0.123, real KB = 0.m,real • KB = 0.i3b,real O.e 0.2 0.4 0.6 0.8x / ( W - B )
-0.5 -KB = 0.128,imag KB = 0.131,imag KB = 0.135,imag 0.4 0.6x / ( W - B )
0.8 KB =0.123,imaff . .-^ . - KB = QA3i,imag • ' 0.2 0.4 0.6 0.8x / ( W - B )
Fig. 9. Real and i maginary components of free s u r f a c e a n d i n t e r n a l w a v e e l e v a t i o n ; / across the higher-order resonant frequency KB = 0.131, I V = 5.0, B = 1.0, h i = 2,0, h2 =2.0, d = 2 . 5 , / = 0 . 7 .
X. Zhang, P. Bandyk/AppUed Ocean Research 47 (20U) 204-218 213 2,5 2 1.5 1 ^ 0,5 ^ -0.5 - 1 -1.5 -2 -2.6 60 -/<-5 = 0.2780, real A'iJ = 0.2785,rea/ • /f7J = 0.2790, rea/ 0.4 0.6
x / ( W - B )
- A " i J = 0.27aO,r A'J3 = 0.2785, r ' i i ' B = 0.2T90,r 0,2 0,4 o.ex / ( W - B )
2.5 2 1 5 0.5 V 0 -0.5 -1.5 -2 -2,5 0 s /s'B = 0.2780,irnas -. KB = 0-.2785,imag KB = 0.2790,imag 0.4 0.6x / ( W - B )
x / ( W - B )
100 80 KB = 0.2rM, i m o s . . - . / f B -0.2755, imarj KB-0.2m.im<ig • 60 • • 40 •¥
20 0 -20 / -40 ... -60 -80 • -100 ( -100 ( 0.2 0.4 0.6 0.8 1Fig. 10. Real and imaginary components o f free surface rf and i n t e r n a l w a v e elevation across the higher-order resonant frequency
/ l 2 = 2 . 0 , d = 2 , 5 , K = 0,7, KB = 0,278, W = 5 , 0 , B = 1,0, ; i , =2,0, 500 - 5 0 0 150 100 50 ^ 0 - 5 0 -100 -KB = 0.784:, real KB = 0.785,real KB = 0.787,real 0,2 0,4 0,6
x / ( W - B )
0.2 0.4 0.6x / ( W - B )
0.8 KB = 0.784,real . . . K B = 0.785,real K B = 0.787,real • 0.8 150 100 50 0 - 5 0 -100 -150 0 l - l 30 20 10 0 - 1 0 - 2 0 - 3 0 0 -KB = 0.784,imag KB = 0.785, imag KB = 0.787, imag 0.4 0.6x / ( W - B )
0.8 - K B = 0.784,imag KB = 0.785,imag • KB = 0.787, imag 0.2 0.4 0.6x / ( W - B )
Fig. 11. Real and imaginary components o f free surface rj' and i n t e r n a l w a v e elevation tf across the higher-order resonant frequency KB = 0.785, I V = 5 . 0 , B = 1.0, hi =2.0,
214 X. Zhang, P. Bandyk/Apphed Ocean Research 47 (2014) 204-218 0.5 -0.5 0,6 Real c o m p o i K i i t I m a g i n a i T component 4 5 6 x / ( W + B ) -0.4 1 2 3 4 5 6 7 8 9 x / ( W + B )
Fig. 12. Radiated wave elevation at the free surface and interfacial surface, W = 5.0, B = 1.0, / i i = 2.0, /12 = 2.0, d = 2.5, y = 0.7, KB = 0.20.
The r a d i a t e d w a v e e l e v a t i o n s at t h e f r e e s u r f a c e a n d i n t e r f a -c i a l s u r f a -c e , o u t s i d e t h e m o o n p o o l i n r e g i o n III, are i l l u s t r a t e d i n Fig. 1 2 f o r f o r c e d m o d o n f r e q u e n c y KB = 0.20, i n c l u d i n g b o t h t h e real a n d i m a g i n a r y c o m p o n e n t s . The d e p e n d e n c e o f o u t e r r a d i -a t e d w -a v e -a m p l i t u d e o n w -a v e f r e q u e n c y is i l l u s t r -a t e d i n Fig. 13. N e a r t h e H e l m h o l t z r e s o n a n t f r e q u e n c y , t h e s u r f a c e w a v e a m p l i -t u d e decreases -t o n e a r l y zero. This is c o n s i s -t e n -t w i -t h -t h e l o w w a v e d a m p i n g values near t h e H e l m h o l t z r e s o n a n t f r e q u e n c y , as i l l u s -t r a -t e d i n Fig. 7. A -t -t h e h i g h e r - o r d e r r e s o n a n -t f r e q u e n c i e s , b o -t h t h e f r e e surface a n d i n t e r n a l w a v e s s h o w r e s o n a n c e c h a r a c t e r i s -tics. I n o r d e r t o e x a m i n e t h e case w h e r e t h e b o t t o m o f t h e t w i n b o d i e s is close t o t h e i n t e r f a c i a l surface, w e p r e s e n t t h e r a d i a t e d w a v e a m p l i t u d e s f o r a ' s h a l l o w - d r a f t b o d y ' ( d r a f t o f t h e b o d y d does n o t exceed d e p t h o f t h e u p p e r f l u i d l a y e r s i g n i f i c a n t i y , i.e., 0 < ( d - / i i ) / f t i « 1 . 0 ) i n Fig. 14. As i l l u s t r a t e d , b o t h r a d i a t e d i n t e r n a l a n d f r e e s u r f a c e w a v e a m p l i t u d e s decrease a r o u n d t h e r e s o n a n t Table 1 H y d r o d y n a m i c coefficients of t w i n bodies i n a t w o - l a y e r f l u i d , W = 5 . 0 , B = 1,0, ; i i = 2 , 0 , / i 2 = 2 , 0 , d = 2 , 5 , y = 0,7. KB 033 0,2 0.004 - 7 . 2 2 3 77,268 0,4 0.016 - 8 , 1 6 5 37,249 0,6 0.037 - 9 , 7 9 6 21,896 0,8 0.065 - 1 0 , 2 2 1 9,397 1,0 0.102 - 0 . 4 1 8 0,001 1,2 0,147 - 5 . 2 7 2 1,513 1,4 0,199 2.167 0,616 1,6 0,261 3,733 1,244 1,8 0.331 3,559 0,851 2,0 0,408 4.275 0,710 2.2 0,494 4,396 0.441 2.4 0,588 4.745 0.270 2.6 0,689 5.225 0.159 2.8 0,799 3.905 0.050 3.0 0,918 5.273 0.036 3.2 1,044 5,509 0,017 m o d e s . I n t e r e s t i n g l y , i t is f o u n d t h a t t h e r a d i a t e d i n t e r n a l w a v e does n o t decay as i n t h e case i l l u s t r a t e d i n Fig. 13.
6. P a r a m e t r i c s t u d i e s
P a r a m e t r i c s t u d i e s have b e e n p e r f o r m e d t o e x a m i n e t h e d e p e n d e n c e o f r e s o n a n c e o n m o o n p o o l g e o m e t r y a n d fluid s t r a t i -fication, i n c l u d i n g t h e d e p t h s o f t h e fluid layers, d e n s i t y r a t i o , a n d b o d y spacing. The r e s o n a n t f r e q u e n c i e s are e x a m i n e d i n p a r t i c u -lar. T w o d i f f e r e n t cases haVe b e e n s t u d i e d i n c l u d i n g i n t e r m e d i a t e d r a f t (Case I ) a n d s h a l l o w d r a f t (Case II) c o n d i t i o n s . T h e i n t e r m e -d i a t e -d r a f t case is c a t e g o r i z e -d b y a s s u m i n g t h e -d r a f t o f t h e t w i n b o d i e s s i g n i f i c a n t l y exceeds t h e d e p t h o f t h e u p p e r - l a y e r fluid, i.e..
B
2
o
1 : 1
Surface wave amplitude
. Internal wave amplitude
--
j
1_ , ' 0.5 1,5 KBFig. 13. Outer r e g i o n ( l l l ) radiated wave amplitudes on f r e e surface and interfacial surface, intermediate d r a f t body, W = 5 . 0 , B = 1.0, / i , = 2.0, /12 = 2.0, d = 2.5, y = 0.7.
X. Zhang, P. Bandyk/Apphed Ocean Research 47(2014) 204-218 215
I
oI
'tk O— Surface wave amplitude
- Internal wave aiiii)litude
K B
Fig. T4. Outer region(III) radiated wave amplitudes on free surface and interfacial surface, s h a l l o w - d r a f t body, W = 5 . 0 , B = 1.0,/ii =0.2,/l2 = 1.0, d = 0 . 2 2 , y = 0.7.
d > f l ] . The s h a l l o w d r a f t case is d e f i n e d b y a s s u m i n g t h e d r a f t is n o t s i g n i f i c a n t l y larger t h a n h], i.e., ( d - / i i « 1 . 0 .
6.1. Case I
First, w e select t w i n bodies w i t h a d r a f t d s i g n i f i c a n d y l a r g e r t h a n t h e d e p t h o f u p p e r f l u i d h-i. The d e p t h o f t h e u p p e r f l u i d is f i x e d at hi = 2 . 0 a n d t h e d e p t h o f t h e l o w e r l a y e r fluid hj is v a r i e d f r o m 2.0 t o 20.0. Fig. 15 s h o w s t h e h y d r o d y n a m i c c o e f f i c i e n t s o f t h e t w i n b o d i e s f o r d i f f e r e n t d e p t h s o f t h e l o w e r fluid. The d e n -s i t y r a d o y i-s c h o -s e n a-s 0.7 a n d t h e d r a f t d = 2.5. T o e x a m i n e t h e e f f e c t o f ;i2, the s o l u t i o n s w i t h d i f f e r e n t values are c o m p a r e d . As i l l u s t r a t e d i n Fig. 15, t h e h y d r o d y n a m i c c o e f f i c i e n t s s h o w s i g n i f i -c a n t d i f f e r e n -c e s i n t h e l o w f r e q u e n -c y range f o r d i f f e r e n t d e p t h s o f l o w e r fluid. The H e l m h o l t z r e s o n a n t f r e q u e n c y has b e e n s h i f t e d d u e t o d i f f e r e n t d e p t h s o f t h e l o w e r fluid layer. I n c o n t r a s t , t h e h i g h e r -o r d e r r e s -o n a n t f r e q u e n c i e s have c h a n g e d n e g l i g i b l y . I n a d d i t i -o n , as t h e d e p t h o f the l o w e r l a y e r fluid increases, t h e H e l m h o l t z r e s o n a n t f r e q u e n c i e s also increase. Fig. 16 s h o w s t h e v a r i a t i o n o f t h e a d d e d m a s s a n d d a m p i n g c o e f f i c i e n t s w i t h t h e o s c i l l a t i o n f r e q u e n c i e s f o r d i f f e r e n t d e n s i t y r a t i o s y = 0 . 1 , 0.4, 0.7, a n d 0.9. As i l l u s t r a t e d i n these figures, t h o u g h t h e d e n s i t y r a t i o a f f e c t s t h e H e l m h o l t z m o d e s l i g h t l y , t h e "Su 6 ^2=2.0 /i.2=4.0 : /i.>=10.0 - - - ft2=20.0
-\ 1
i
i
1 i -j-i
-0 -0.1 -0.2 -0.3 0.5 0.6 0.7 0.8 0,9 1 K BFig. 15. Added mass and d a m p i n g coefficients of the t w i n bodies i n heave m o t i o n w i t h d i f f e r e n t depths of the l o w e r layer fluid hi, W = 5,0,6 = 1.0, hi =2.0, d = 2 , 5 , y = 0 , 7 .
0,1 02 0,3 0
216 X. Zhang, P. Bandyk/Applied Ocean Research 47 (2014) 204-218
W/B =
1.0- WjB
= 1.1" WjB
= 2.0---WjB = -0.0 .5
1 1 1 !• i • I ; - t ! J ^ ;!
1 1 > ! ' 1 'ƒ' 1 ' • \ •• •
. / ./ . •
I i \ 1 1 ; 1 i ' ; i ! \ _ l l l l t 1 1 1 CO Q 0 5 OS K B 1W/B =
1.0 ' t ' ' ' 1 - V ! V B = l. iW/B
= 2.0 i ' | - - - 1' I 7 0 = 5.O il ji1
tl 1\
li ,1 ij s il ] i; ii - \\- ilj
. H
fl 1 t i 1j
i l l >
l l l l_ L
) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 O.B 0.9 1 K BFig. 17. Added mass and d a m p i n g coefficients of the t w i n bodies i n heave m o t i o n w i t h d i f f e r e n t body spacings W, B = 1.0, hi = 2.0, h2 = 2.0, d = 2.5, y = 0.7.
40h'
&
7 7 7 • - 7 : 0 . 1 . 0 . 4 : 0 . 7 0.911'
' ll \! 0 1 0,2 0 3 0,5 0 6 0 7 0 8 0 9 K BX Zhang. P. Bandyk/AppUed Ocean Research 47 (2014) 204-218 217 h i g h e r - o r d e r r e s o n a n t f r e q u e n c i e s are s i g n i f i c a n t l y a f f e c t e d . T h e h i g h e r o r d e r r e s o n a n c e d u e t o i n t e r n a l w a v e m o d e s s h o w s d r a -m a t i c d i f f e r e n c e s f o r v a r y i n g d e n s i t y ratios, t h o u g h t h e s u r f a c e w a v e m o d e is n o t a f f e c t e d n o t i c e a b l y . T h i s o b s e r v e d p h e n o m e n o n can b e e x p l a i n e d b y t h e d i s p e r s i o n r e l a t i o n i l l u s t r a t e d i n Fig. 2. I n o r d e r to i d e n t i f y t h e d e p e n d e n c e o f t h e r e s o n a n c e o n t h e w i d t h o f the m o o n p o o l . Fig. 17 p r e s e n t s the a d d e d mass a n d d a m p -i n g c o e f f -i c -i e n t s versus f r e q u e n c y f o r d -i f f e r e n t m o o n p o o l w -i d t h ratios WjB. As s h o w n , b o t h t h e H e l m h o l t z a n d h i g h e r - o r d e r res-o n a n t m res-o d e s have b e e n l a r g e l y a f f e c t e d as the w i d t h res-o f m res-o res-o n p res-o res-o l varies. I n p a r t i c u l a r , f o r the case o f W / B = 1.0, t h e r e is n o resonance, w h i c h is e x p e c t e d f o r a s i n g l e b o d y case. F u r t h e r , a l l t h e r e s o n a n t f r e q u e n c i e s decrease w i t h i n c r e a s i n g W / B .
6.2. Casell
A d d i t i o n a l analysis has b e e n p e r f o r m e d f o r s h a l l o w - d r a f t case, w h e r e t h e d r a f t o f t h e t w i n b o d i e s d does n o t e x c e e d the d e p t h o f the u p p e r f l u i d layer h i s i g n i f i c a n t l y , i.e. ( d - / i i ) / f ! i «1.0. The d e p t h o f t h e u p p e r layer f l u i d is h-[ = 0 . 2 . T h e b o d y d r a f t is d = 0.22. The d e n s i t y r a t i o is y = 0.7. The b o d y s p a c i n g is W= 5.0 w h i l e t h e w i d t h of t h e b o d y B = 1.0. Fig. 18 s h o w s t h e d e p e n d e n c e o f h y d r o d y n a m i c c o e f f i c i e n t s o n t h e o s c i l l a t i o n f r e q u e n c y f o r d i f f e r e n t d e p t h s o f t h e l o w e r layer f l u i d ^2- As can b e seen i n these f i g u r e s , t h e d e p t h o f the l o w e r layer f l u i d n o t o n l y a f f e c t s t h e H e l m h o l t z r e s o n a n t f r e -q u e n c y , b u t also alters t h e h i g h e r - o r d e r r e s o n a n t f r e -q u e n c i e s d u e to s t a n d i n g surface w a v e s i n t h e m o o n p o o l . H o w e v e r , t h e h i g h e r -o r d e r res-onances ass-ociated w i t h i n t e r n a l w a v e m -o d e s are a f f e c t e d m a r g i n a l l y .
Fig. 19 i l l u s t r a t e s t h e h y d r o d y n a m i c c o e f f i c i e n t s as a f u n c t i o n o f o s c i l l a t i o n f r e q u e n c y f o r d i f f e r e n t d e n s i t y ratios y. T h e o t h e r p h y s i -cal p a r a m e t e r s are t h e s a m e as t h o s e s h o w n i n Fig. 18, e x c e p t a fixed h2 = 2.0 a n d v a r y i n g y. The c o m p a r i s o n o f results w i t h d i f f e r e n t d e n -s i t y r a t i o y -sugge-st-s t h a t d e n -s i t y -s t r a t i f i c a t i o n a f f e c t -s h i g h e r - o r d e r r e s o n a n t m o d e s e i t h e r a s s o c i a t e d w i t h t h e f r e e s u r f a c e or i n t e r -nal w a v e s f o r t h e s h a l l o w - d r a f t case. A g a i n , t h e H e l m h o l t z m o d e is i m p a c t e d l i t t i e b y d e n s i t y s t r a t i f i c a t i o n as the i n t e r m e d i a t e d r a f t case. 7. C o n c l u s i o n s I n t h i s paper, t h e w a v e r a d i a t i o n p r o b l e m o f i n t e r f a c e a n d s u r -face p i e r c i n g t w i n b o d i e s i n a t w o - l a y e r fluid has b e e n i n v e s t i g a t e d . The h e a v e added mass, d a m p i n g , r a d i a t e d f r e e s u r f a c e a n d i n t e r n a l w a v e s have been c o m p u t e d f o r d i f f e r e n t m o o n p o o l c o n f i g u r a t i o n s . By e x a m i n i n g the n u m e r i c a l r e s u l t s , i t is f o u n d t h a t t h e r e e x i s t b o t h H e l m h o l t z a n d h i g h e r - o r d e r r e s o n a n c e s . The h i g h e r - o r d e r r e s o n a n t m o d e s can be e i t h e r c h a r a c t e r i z e d b y s t a n d i n g w a v e s , o n t h e f r e e s u r f a c e or i n t e r f a c i a l s u r f a c e , w i t h i n t h e m o o n p o o l gap. This b e h a v -i o r -is f u n d a m e n t a l l y d -i f f e r e n t w -i t h t h e case s t u d -i e d -i n t h e p r e v -i o u s p a p e r [ 1 ] . P a r a m e t r i c studies f o r a n i n t e r m e d i a t e d r a f t b o d y (Case I) i n d i -cate t h a t t h e r e are s i g n i f i c a n t d i f f e r e n c e s i n t h e c h a r a c t e r i s t i c s o f t h e H e l m h o l t z r e s o n a n t m o d e b e t w e e n t w o l a y e r a n d s i n g l e -l a y e r f-luid cases due t o d i f f e r e n t -l o w e r f-luid -l a y e r d e p t h s . H o w e v e r , t h e h i g h e r - o r d e r r e s o n a n t m o d e s are l i t d e a f f e c t e d b y t h e r e l a t i v e d e p t h o f u p p e r a n d l o w e r l a y e r s . T h e h i g h e r - o r d e r r e s o n a n t m o d e s are a f f e c t e d b y t h e d e n s i t y rarios, e s p e c i a l l y f o r t h e i n t e r n a l w a v e .
R e g a r d i n g s h a l l o w - d r a f t t w i n b o d i e s w i t h a d r a f t (Case II) close t o t h e d e p t h o f u p p e r l a y e r fluid, t h e d e p t h o f l o w e r l a y e r fluid can a f f e c t b o t h t h e H e l m h o l t z a n d h i g h e r - o r d e r m o d e s associated w i t h t h e f r e e surface w a v e . T h e d e n s i t y s t r a t i f i c a t i o n s h o w s i m p a c t o n b o t h h i g h e r - o r d e r resonances f r o m t h e f r e e s u r f a c e w a v e a n d the i n t e r n a l w a v e m o d e s .
T h e h y d r o d y n a m i c resonances also a p p e a r i n t h e o u t e r r e g i o n 111 ( f a r field) r a d i a t e d f r e e surface a n d i n t e r n a l s u r f a c e w a v e eleva-t i o n s , f o r b o eleva-t h i n eleva-t e r m e d i a eleva-t e - d r a f eleva-t a n d s h a l l o w - d r a f eleva-t b o d i e s . For eleva-t h e i n t e r m e d i a t e - d r a f t b o d y case, b o t h f r e e s u r f a c e a n d i n t e r n a l w a v e a m p l i t u d e s decay w i t h i n c r e a s i n g f o r c e d m o t i o n f r e q u e n c i e s . H o w -ever, f o r t h e s h a l l o w - d r a f t b o d y case, b o t h t h e o u t e r r e g i o n I I I ( f a r field) r a d i a t e d surface a n d i n t e r n a l w a v e s s h o w d i f f e r e n t t r e n d s . T h e p r e s e n t e d n u m e r i c a l m o d e l is based o n p o t e n t i a l flow t h e -ory, a n d t h e r e s o n a n t w a v e a m p l i t u d e s are l i k e l y o v e r - p r e d i c t e d . P r o p e r p r e d i c t i o n o f t h e w a v e a m p l i t u d e r e q u i r e s m o d e l i n g o f v i s c o u s e f f e c t s , s u c h as v o r t e x s h e d d i n g f r o m t h e c o r n e r s o f t h e m o o n p o o l . T h e i n t e r f a c i a l w a v e m a y b e m o r e d a m p e d t h a n t h e s u r -face w a v e , d u e t o t h e s h o r t e r r e s o n a n t w a v e l e n g t h s . T h e r e s o n a n t f r e q u e n c i e s f o r b o t h f r e e s u r f a c e a n d i n t e r f a c i a l w a v e s are accu-r a t e l y p accu-r e d i c t e d u s i n g t h e a p p accu-r o a c h p accu-r e s e n t e d , a n d m a y be q u i t e u s e f u l i n m o o n p o o l p e r f o r m a n c e assessment, d e s i g n , a n d o p t i m i z a -tion. T h e p r e s e n t s t u d y also o f f e r s clues o n d e s i g n a n d m o d e l i n g o f r e l a t e d w a v e e n e r g y c o n v e r t e r s ( W E C ) i n a t w o - l a y e r fluid. For e x a m p l e , t h e m o o n p o o l g e o m e t r y can be o p t i m i z e d based o n t h e r e s o n a n t m o t i o n o f t h e floater a n d o p e r a t i n g w a v e c o n d i t i o n s .
A p p e n d i x A . I n n e r p r o d u c t i n E q s . (46) a n d (47)
T h e first i n n e r p r o d u c t at t h e right side o f Eqs. ( 4 6 ) a n d ( 4 7 ) are c a l c u l a t e d b y
<<p',Yi>= / w(z)(p'(W-B,z)Yi{X,,z]dz
,',Yt > = '^BjA'ikfKw-B)l<:p + Y^B]A'ikfKw-B)Lf j = 0 J=0
< <p"',Yi > = J w(z)<p"'(W + B,z]Y,{k,,z)dz
•.4>"',Yi> = ' ^ A ] K ' \ k f \ W + B)ip + 'Yj']/^'''i.i<f\VJ + B)lf ( A l ) (A.2) w h e r e t h e c o e f f i c i e n t s 4 " ' (see A p p e n d i x B f o r d e t a i l s ) are w r i t -t e n as r('i) - d vj{z)Z^"'\kY\z)Y{ki,z)(iz 11 = 1 , 2 (A.3) A p p e n d i x B . E x p r e s s i o n of l ! " ' ' c a n be w r i t t e n as f o l l o w s , f o r f o u r d i f f e r e n t c o m b i n a t i o n s o f i a n d j : F o r i = 0 , j = 0 (n) _ or sinh/s:(/i - d ) 00 ~ g/<2 s i n h ; < f i 2 F o r i = 0 , j > l
(n) _ o)^s\nk{h-d)
Oj g/<2sin/<f!2 F o r ! > l , j = 0 ( B . l ) (B.2) , ( n ) g;<(A2 + k2)sinhkfi F o r i > l , j > l , ^/26)2 gk(A.2-;<2)sinkfi[X cosh k(h - d) sin A(/i - d) + /< sinh k(/] - d) cos A ( h - d ) ]
(B.3)
[A cos kih - d) sin X{h - d ) - k sin k(h - d) cos A(h - d)l
218 X. Zhang. P. Bandyk/Apphed Ocean Research 47(2014)204-218 w h e r e X a n d k r e p r e s e n t A, and,;<j"', r e s p e c t i v e l y . A p p e n d i x C . E x p r e s s i o n o f M ] " ' The M j " ' i n Eqs. ( 5 1 ) a n d ( 5 2 ) is d e f i n e d b y M j " ' = / w ( , z ) 0 " ' \ k f \ z ) f d z ( C l ) a n d can be c a l c u l a t e d b y s u b s t i t u t i n g ( 3 4 ) a n d ( 3 5 ) i n t o ( C l ) , F o r j = 0 M : ( n ) _ 1 2(g/<)^ (sinhWi2)^ /12 + sinh2Wi2 2k co'^ / / l l s i n h 2 k / i i a2 [ V 2 4k
4
sin_h2/</i 4k 2gk-,-(1 - c o s h 2 W i i ) ( C 2 ) F o r j > l 1 2(g;0'(sin/</i2)" L 6 J ' /12 + sin2Wi2 / l l 2 I T 2 k s i n 2 k / i i 4/< h-i s i n 2 / d i i T 4k •2g/c2 ( 1 - C 0 S 2 W 7 i ) ( C 3 ) R e f e r e n c e s111 Z h a n g X , Bandyk P. On t w o - d i m e n s i o n a l m o o n p o o l resonance for t w i n bodies
i n a t w o - l a y e r f l u i d . A p p l Ocean Res 2 0 1 3 ; 4 0 : 1 - 1 3 .
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