Verification of a new radiation condition for two ships advancing in waves

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Applied Ocean Research 48 (2014) 186-201

ELSEVIER

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Applied Ocean Research

journal homepage: www.elsevier.com/locate/apor

O C E A N R E S E A R C H

Verification of a new radiation condition for two sliips advancing

in waves

Z h i - M i n g Yuan*, Atilla Incecik, Day Alexander

Department of Naval Architecture, Ocean and Marine Engineering, University of Strathclyde, Glasgow, UK

d)

CrossMark

A R T I C L E I N F O

Article history: Received 4June 2014

Received in revised form 15 August 2014 Accepted 19 August 2014

Available online 16 September 2014

Keywords:

Hydrodynamic interaction Ranlcine source method Radiation condition Wave pattern Forward speed

A B S T R A C T

3-D Ranlcine source method is used to investigate the hydrodynamic interactions between two ships arranged side by side w i t h forward speed. The radiation condition is satisfied by using a modified Som-merfeld radiation condition w h i c h takes into account the Doppler shift of the scattered waves. This new radiation condition is applicable to a w i d e range of forward speeds, including very l o w forward speed problem where the parameter T (T = U(u/g) is smaller than 0.25. The numerical solution is evaluated by applying the present method to t w o pau's of models and compared w i t h experimental data and Green function method. Through the comparison study, we verify the new radiation condition and examine the wave patterns for a f u l l range of f o r w a r d speeds. Discussions are highlighted on the effect of the radiation conditions.

1. I n t r o d u c t i o n

H y d r o d y n a m i c i n t e r a c t i o n b e t w e e n t w o or m o r e ships occurs i n h a r b o r area a n d w a t e r w a y s w i t h dense s h i p p i n g t r a f f i c as t h e vessels h a v e t o pass each o t h e r i n close p r o x i m i t y ; b e t w e e n tugs a n d vessels d u r i n g e s c o r t i n g or m a n e u v e r i n g a n d b e r t h i n g o p e r a -t i o n s as w e l l as d u r i n g s h i p - -t o - s h i p o p e r a -t i o n s f o r cargo -t r a n s f e r s d u r i n g o i l a n d gas o f f l o a d i n g o p e r a t i o n s . The b e h a v i o r o f t w o s h i p s i n w a v e s w i t h speed e f f e c t is o f special c o n c e r n t o t h e N a v y , t h a t is, f o r u n d e r w a y r e p l e n i s h m e n t , a n d f o r o t h e r c o m m e r c i a l p u r p o s e s .

Because o f 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 , e v e n r e l a t i v e l y s m a l l w a v e can i n d u c e large m o t i o n s o f the s m a l l e r s h i p due t o t h e n e a r -ness o f t h e l a r g e r s h i p . W h e n t h e ships are t r a v e l i n g w i t h f o r w a r d speed, 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 c o m e m o r e c o m p l i c a t e d . Fang a n d K i m [ 1 ] firstly t o o k f o r w a r d speed i n t o c o n s i d e r a t i o n i n s h i p - t o - s h i p p r o b l e m . T h e y u t i l i z e d a 2 - D p r o c e d u r e , i n c l u d 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 a n d a n i n t e g r a l e q u a t i o n m e t h o d , to p r e -d i c t t h e c o u p l e -d m o t i o n s b e t w e e n t w o ships a -d v a n c i n g i n o b l i q u e seas. They f o u n d t h a t t h e r o l l m o t i o n w a s r e d u c e d w h i l e t h e ships w e r e a d v a n c i n g . H o w e v e r , d u e to t h e 2 D a s s u m p t i o n s , s o m e d e f i -ciencies i n c l u d i n g t h e special t r e a t m e n t o f t h e c o n v e c t i v e t e r m s t i l l exist. K a s h i w a g i [ 2 ] used a u n i f i e d t h e o r y t o i n v e s t i g a t e t h e h e a v e a n d p i t c h m o t i o n s o f a c a t a m a r a n a d v a n c i n g i n w a v e s . I w a s h i t a a n d

* Corresponding author at: Department of Naval Architecture, Ocean & IWarine Engineering, University of Strathclyde, Henry Dyer Building, G4 OLZ Glasgow, UIC Tel.: +44 0141 548 2288; fax: +44 0141 552 2879.

E-mail address: zhiming.yuan@strath.ac.uk (Z.-IW. Yuan).

K a t a o k a [ 3 ] used t h e 3 D t r a n s l a t i n g a n d p u l s a t i n g G r e e n - f u n c t i o n m e t h o d to analyze 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 b e t w e e n steady a n d u n s t e a d y fiows f o r a c a t a m a r a n . Chen a n d Fang [ 4 ] e x t e n d e d Fang's m e t h o d [1 ] t o 3 - D . T h e y used a 3 - D G r e e n f u n c t i o n iTiethod t o i n v e s t i g a t e t h e h y d r o d y n a m i c p r o b l e m s b e t w e e n t w o m o v i n g ships i n waves. I t w a s f o u n d 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 c a l c u l a t e d b y 3 - D m e t h o d w e r e m o r e r e a s o n a b l e i n t h e resonance r e g i o n , w h e r e t h e responses w e r e n o t so s i g n i f i c a n t p r e d i c t e d by 2 - D m e t h o d . H o w e v e r , t h e i r m e t h o d w a s o n l y v a l i d a t e d b y m o d e l tests w i t h zero speed. M o r e r i g o r o u s v a l i d a t i o n s h o u l d be m a d e by f u r t h e r e x p e r i m e n t s . The first m o d e l t e s t o f t w o ships a d v a n c i n g i n w a v e s w a s c o n d u c t e d b y Li [ 5 ] . B o t h ships w e r e r e s t r a i n e d i n surge, s w a y a n d y a w , as w e l l as t h e f r e e m o t i o n s i n heave, r o l l a n d p i t c h . M c T a g g a r t et al. [ 6 ] a n d Li [ 7 ] used t h a t m o d e l test data t o v e r i f y t h e i r n u m e r i c a l p r o g r a m s , w h i c h w a s based o n 3 D G r e e n f u n c -t i o n m e -t h o d . The n u m e r i c a l p r e d i c -t i o n s a n d e x p e r i m e n -t s s h o w e d t h a t t h e presence o f a l a r g e r s h i p c o u l d s i g n i f i c a n t l y i n f l u e n c e the m o t i o n s o f a s m a l l e r s h i p i n close p r o x i m i t y . B u t the n u m e r i c a l p r e -d i c t i o n o f r o l l m o t i o n w a s n o t accurate. A n o t h e r m o -d e l test o f t w o ships a d v a n c i n g i n w a v e s w a s c o n d u c t e d b y R o n s s s [ 8 ] at M A R I N -TEK. The e x p e r i m e n t s w e r e p e r f o r m e d at d i f f e r e n t speeds a n d w i t h d i f f e r e n t l o n g i t u d i n a l distance b e t w e e n t h e ships. The n u m e r i c a l p r o g r a m based o n u n i f i e d t h e o r y w a s v e r i f i e d . It w a s f o u n d t h a t heave a n d p i t c h m o t i o n s c o u l d be p r e d i c t e d w e l l w h i l e t h e r o l l m o t i o n w a s h a r d t o p r e d i c t d u e t o t h e viscous e f f e c t s . Ronsss's m o d e l t e s t data w a s u s e d b y X u a n d F a l t i n s e n [ 9 ] t o v e r i f y t h e i r n u m e r i c a l p r o g r a m based o n 3 - D Ranlcine source m e t h o d . They a p p l i e d a n a r t i f i c i a l n u m e r i c a l b e a c h t o s a t i s f y t h e r a d i a t i o n c o n -d i t i o n . T h e y f o u n -d t h a t t h e h y -d r o -d y n a m i c peaks a n -d spikes w e r e

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Z.-M. Yuan et al./Applied Ocean Researcix 48 (2014) 186-201 187 r e l a t e d to t h e resonance m o d e s i n t h e w a t e r gap b e t w e e n t h e h u l l s . H o w e v e r , t h e y also f a i l e d t o p r e d i c t t h e r o l l m o t i o n precisely. Recently, w i t h i n t h e f r a m e w o r k o f G r e e n f u n c t i o n , X u a n d D o n g [10] d e v e l o p e d a 3-D t r a n s l a t i n g - p u l s a t i n g (3DTP) source m e t h o d to c a l c u l a t e w a v e loads a n d f r e e m o t i o n s o f t w o ships a d v a n c i n g in w a v e s . M o d e l tests w e r e c a r r i e d o u t t o m e a s u r e t h e w a v e loads a n d t h e f r e e m o t i o n s f o r a p a i r o f s i d e - b y - s i d e a r r a n g e d s h i p m o d e l s a d v a n c i n g w i t h a n i d e n t i c a l s p e e d i n h e a d r e g u l a r w a v e s . B o t h the e x p e r i m e n t a l a n d t h e n u m e r i c a l p r e d i c t i o n s s h o w e d t h a t h y d r o -d y n a m i c i n t e r a c t i o n e f f e c t s o n w a v e loa-ds a n -d f r e e m o t i o n s w e r e s i g n i f i c a n t . T h e y also p o i n t e d o u t t h a t t h e p r e d i c t i o n accuracy o f the 3DTP m e t h o d w a s m u c h b e t t e r t h a n t h a t o f 3DP, especially f o r peak values o f the f r e e m o t i o n responses.

W e find t h a t m o s t o f the p u b l i c a t i o n s o n t w o ships w i t h f o r -w a r d speed p r o b l e m are based o n G r e e n f u n c t i o n t h a t satisfies the K e l v i n f r e e surface c o n d i t i o n , as w e l l as t h e r a d i a t i o n c o n d i t i o n . I t is a n e f f e c t i v e m e t h o d f o r t h e zero speed p r o b l e m s , b u t i f t h e vessel is t r a v e l i n g w i t h f o r w a r d speed, t h i s m e t h o d s t i l l has s o m e l i m i t a -tions. F i r s t i y , i t c o u l d n o t a c c o u n t f o r t h e n e a r - f i e l d flow c o n d i t i o n . A l t h o u g h s o m e researchers [ 1 1 , 1 2 ] e x t e n d e d i t t o i n c l u d e t h e near-field f r e e surface c o n d i t i o n , t h e s o - c a l l e d i r r e g u l a r f r e q u e n c y s t i l l c a n n o t be a v o i d e d . A n d i t w i l l b r i n g s i n g u l a r i t y to t h e c o e f f i c i e n t m a t r i x e q u a t i o n . Secondly, i t is i m p o s s i b l e f o r t h e Green f u n c t i o n to account f o r t h e e f f e c t s o f t h e s t e a d y flow o n t h e u n s t e a d y p o t e n t i a l . In t h e p r e s e n t study, t h e R a n k i n e source a p p r o a c h w i l l be a p p l i e d , w h i c h uses a v e r y s i m p l e G r e e n f u n c t i o n i n t h e b o u n d a r y i n t e g r a l f o r m u l a t i o n . This m e t h o d r e q u i r e s the sources d i s t r i b u t e d n o t o n l y on t h e b o d y surface, b u t also o n t h e f r e e s u r f a c e a n d c o n t r o l sur-face. T h e r e f o r e , a flexible choice o f f r e e - s u r f a c e c o n d i t i o n s can be r e a l i z e d i n these m e t h o d s . The c o u p l e d b e h a v i o r b e t w e e n steady a n d u n s t e a d y w a v e p o t e n t i a l c o u l d be expressed i n a d i r e c t f o r -m u l a . M e a n w h i l e , t h e n o n l i n e a r i t y o n t h e f r e e s u r f a c e c o u l d also be a d d e d i n t h e b o u n d a r y c o n d i t i o n .

The R a n k i n e source a p p r o a c h has b e e n used b y m a n y i n v e s -t i g a -t o r s since i -t has b e e n firs-t p r o p o s e d b y Hess a n d S m i -t h [ 1 3 ] . I n v e s t i g a t o r s f r o m M I T [ 1 4 - 1 6 ] a p p l i e d t h e R a n k i n e source a p p r o a c h to m o d e l s t e a d y a n d u n s t e a d y w a v e s as a s h i p m o v e s i n w a v e s . A n analysis t e c h n i q u e d e v e l o p e d b y Scalvounos a n d Nal<os [ 14 ] f o r t h e p r o p a g a t i o n o f g r a v i t y w a v e s o n a p a n e l i z e d f r e e surface s h o w e d t h a t a R a n k i n e m e t h o d c o u l d a d e q u a t e l y p r e d i c t t h e s h i p w a v e p a t t e r n s a n d forces. T h e i r w o r k l e d t o t h e d e v e l o p m e n t o f a f r e q u e n c y - d o m a i n f o r m u l a t i o n f o r s h i p m o t i o n s w i t h a c o n s i s t e n t l i n e a r i z a t i o n based u p o n t h e d o u b l e b o d y steady flow m o d e l w h i c h assumes s m a l l a n d m o d e r a t e F r o u d e n u m b e r s . A p p l i c a t i o n s w e r e r e p o r t e d b y Nakos a n d Sclavounos [ 1 5 ] . This m o d e l w a s e x t e n d e d to t h e t i m e d o m a i n b y K r i n g [ 1 6 ] w h o also p r o p o s e d a p h y s i c a l l y r a t i o n a l set o f K u t t a c o n d i t i o n s at a ship's t r a n s o m s t e r n . Recently, Gao a n d Z o u [ 1 7 ] d e v e l o p e d a h i g h - o r d e r Rankine p a n e l m e t h o d based o n N o n - U n i f o r m R a t i o n a l B-Spline (NURBS) t o solve t h e 3¬ D 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 w i t h f o r w a r d speed. T h e i r results h a d v e r y g o o d a g r e e m e n t w i t h t h e e x p e r i m e n t a l data. H o w -ever, t h e r e are s t i l l s o m e l i m i t a t i o n s f o r t h e e x t e n s i v e use o f t h e Rankine source a p p r o a c h . First o f all, t h e R a n k i n e source m e t h o d r e q u i r e s m u c h m o r e panels w h i c h w i l l c o n s i d e r a b l y increase t h e c o m p u t a t i o n t i m e , especially w h e n t h e m a t r i x e q u a t i o n is f u l l range m a t r i x . H o w e v e r , t h e c o m p u t a t i o n time w i l l s t r o n g l y d e p e n d o n t h e n u m e r i c a l m e t h o d a n d c o m p u t e r language. As t h e p e r f o r m a n c e o f c o m p u t e r s increase r a p i d l y , i t o n l y takes less t h a n 1 m i n t o solve a l O ' ' X 1 0 ^ f u l l r a n g e m a t r i x u s i n g M a t l a b . T y p i c a l l y , t h e n u m b e r o f p a n e l s w i l l be n o m o r e t h a n 10,000. The c o m p u t a t i o n time is acceptable i n e n g i n e e r i n g a p p l i c a t i o n s . Besides, t h e R a n k i n e source m e t h o d r e q u i r e s a s u i t a b l 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 t o a c c o u n t f o r t h e s c a t t e r e d w a v e s i n c u r r e n t . A v e r y p o p u l a r r a d i a t i o n c o n d i -tion f o r t h e f o r w a r d speed p r o b l e m , w h i c h is s o - c a l l e d u p s t r e a m r a d i a t i o n c o n d i t i o n , w a s p r o p o s e d b y Nakos [ 1 8 ] , The f r e e surface w a s t r u n c a t e d at s o m e u p s t r e a m p o i n t s , a n d a q u i e s c e n t b o u n d a r y

Fig. 1. An example vessels and coordinate system.

c o n d i t i o n w a s i m p o s e d at these p o i n t s t o e n s u r e t h e c o n s i s t e n c y o f t h e u p s t r e a m t r u n c a t i o n o f t h e f r e e s u r f a c e . A n o t h e r m e t h o d to d e a l w i t h t h e r a d i a t i o n c o n d i t i o n is t o m o v e t h e source p o i n t s o n t h e f r e e s u r f a c e at s o m e distance d o w n s t r e a m [ 1 9 ] . The results f r o m these t w o m e t h o d s s h o w v e r y g o o d a g r e e m e n t w i t h p u b l i s h e d e x p e r i m e n t a l data w h e n t h e p a r a m e t e r r ( r = u w / g ) is g r e a t e r t h a n 0.25, since t h e y are b o t h based o n t h e a s s u m p t i o n t h a t t h e r e is n o s c a t t e r e d w a v e t r a v e l i n g ahead o f t h e vessel. H o w e v e r , w h e n t h e f o r w a r d s p e e d o f t h e vessel is v e r y l o w , r w i l l be s m a l l e r t h a n 0.25. W h e n t h i s case occurs, t h e s c a t t e r e d w a v e s c o u l d t r a v e l ahead o f t h e vessel, a n d these t r a d i t i o n a l r a d i a t i o n c o n d i t i o n s c o u l d n o l o n g e r be v a l i d . For s h i p t o s h i p p r o b l e m , t h e f o r w a r d speed is u s u -a l l y l i m i t e d t o -a l o w l e v e l f o r t h e s-afe o p e r -a t i o n . T h e r e f o r e , -a n e w e x t e n s i v e r a d i a t i o n c o n d i t i o n s h o u l d be p r o p o s e d t o deal w i t h t h e v e r y l o w f o r w a r d speed p r o b l e m . Das a n d Cheung [ 2 0 , 2 1 ] p r o v i d e d an a l t e r n a t e s o l u t i o n t o 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 f o r w a r d speeds above a n d b e l o w t h e g r o u p v e l o c i t y o f t h e s c a t t e r e d w a v e s . T h e y c o r r e c t e d t h e S o m m e r f e l d r a d i a t i o n c o n d i t i o n b y t a k i n g i n t o a c c o u n t t h e D o p p l e r s h i f t o f t h e s c a t t e r e d w a v e s at t h e c o n t r o l s u r f a c e t h a t t r u n c a t e s the i n f i n i t e fluid d o m a i n . T h e y c o m p a r e d t h e i r r e s u l t s w i t h the e x p e r i m e n t a l data, a n d g o o d a g r e e m e n t w a s a c h i e v e d . T h e y also c o m p u t e d t h e w a v e e l e v a t i o n o n t h e f r e e s u r -face, a n d a reasonable w a v e p a t t e r n w a s o b t a i n e d at r < 0 . 2 5 b y u s i n g t h e i r n e w r a d i a t i o n c o n d i t i o n . Y u a n e t al. [ 2 2 ] a p p l i e d Das a n d Cheung's r a d i a t i o n c o n d i t i o n t o a W i g l e y III h u l l a d v a n c i n g i n w a v e s , a n d v e r y g o o d a g r e e m e n t h a d b e e n a c h i e v e d b e t w e e n t h e i r p r e d i c t i o n s a n d m e a s u r e m e n t s . In t h e p r e s e n t s t u d y , w e w i l l e x t e n d Das a n d Cheung's r a d i a t i o n c o n d i t i o n t o t h e s h i p - t o - s h i p p r o b l e m . A 3 - D p a n e l code based o n R a n k i n e source m e t h o d w i l l be d e v e l o p e d t o i n v e s t i g a t e 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 b e t w e e n t w o vessels a r r a n g e -d si-de b y si-de w i t h f o r w a r d speed. The m o t i o n responses o f b o t h ships w i l l be c a l c u l a t e d a n d c o m p a r e d to Li's a n d Ronaess' e x p e r i m e n t a l r e s u l t s . Discussions w i l l be h i g h l i g h t e d o n t h e w a v e p a t t e r n s at f u l l range o f f o r w a r d speeds. 2. M a t h e m a t i c a l f o r m u l a t i o n s of the p o t e n t i a l s 2 . 3 . Coordinate systems The c o r r e s p o n d i n g r i g h t - h a n d e d c o o r d i n a t e s y s t e m s are s h o w n i n Fig. 1. The b o d y c o o r d i n a t e s y s t e m s Oa-XayaZa a n d 0i,-XbybZb are fixed o n Ship_a a n d Ship_b, r e s p e c t i v e l y w i t h t h e i r o r i g i n s o n t h e m e a n f r e e surface, c o i n c i d i n g w i t h t h e c o r r e s p o n d i n g c e n t r e o f g r a v i t y (CoG) i n r e s p e c t t o x a n d y c o o r d i n a t e s w h e n b o t h o f t h e

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188 Z.-M. Yuan et al. /Applied Ocean Research 48 (2014) 186-201

ships are at t h e i r static e q u i l i b r i u m p o s i t i o n s . Oa-Za a n d Ob-Zn are b o t h p o s i t i v e u p w f a r d . The i n e r t i a c o o r d i n a t e s y s t e m oxyz w i t h o r i -g i n l o c a t e d o n t h e c a l m f r e e s u r f a c e coincides w i t h Oa-XaVaZa w h e n t h e s l i i p has no u n s t e a d y m o t i o n s . OXYZ is t h e e a r t h f i x e d c o o r -d i n a t e s y s t e m w i t h its o r i g i n l o c a t e -d o n t h e c a l m f r e e s u r f a c e a n -d OZ axis p o s i t i v e u p w a r d . T h r e e c o m p o n e n t s o f t r a n s l a t i o n m o t i o n s i n c l u d e s u r g e ()?^ a n d } ] \ , w h i c h are p a r a l l e l t o x - a x i s ) , s w a y (/;^ a n d w h i c h are p a r a l l e l t o y - a x i s ) a n d heave (T/^ a n d r]^, w h i c h are p a r a l l e l to zaxis). A n o t h e r t h r e e r o t a t i o n a l m o t i o n c o m p o -n e -n t s are r o l l (7;^ a -n d w h i c h r o t a t e a r o u -n d x-axis), p i t c h (/;^ a -n d ;?5, w h i c h r o t a t e a r o u n d y - a x i s ) a n d y a w and w h i c h r o t a t e a r o u n d z - a x i s ) . The i n c i d e n t w a v e d i r e c t i o n is d e f i n e d as t h e angle b e t w e e n t h e w a v e p r o p a g a t i o n d i r e c t i o n a n d X a x i s . ,0 = 1 8 0 ° c o r r e -sponds t o head sea; j8 = 9 0 ° c o r r e s p o n d s to b e a m sea. d t d e n o t e s the t r a n s v e r s e d i s t a n c e b e t w e e n t w o s h i p s w h i l e dl is t h e l o n g i t u d i n a l distance. UQ is t h e f o r w a r d speed.

I n t h e c o m p u t a t i o n , t h e m o t i o n s a n d forces o f Ship-a a n d S h i p - b are c o n c e r t e d t o t h e local c o o r d i n a t e s y s t e m i n w h i c h the o r i g i n is at t h e c e n t e r o f g r a v i t y o f each s h i p .

2.2. Diffraction wave potential

I t is a s s u m e d t h a t the s u r r o u n d i n g fluid is i n v i s c i d a n d i n c o m -pressible, a n d t h a t t h e m o t i o n is i r r o t a d o n a l , the t o t a l v e l o c i t y p o t e n t i a l exists w h i c h satisfies t h e Laplace e q u a t i o n i n t h e w h o l e f l u i d d o m a i n . Let t d e n o t e t i m e a n d x = (x, y, z ) t h e p o s i t i o n v e c t o r . A c o m p l e x v e l o c i t y p o t e n t i a l p r o v i d e s a d e s c r i p t i o n o f t h e flow as

6

ir{x, t ) = 'iio[^ös(x) -x] + ReY^[,]f(p9{x)e~"'^' + i]^cp^[x)e-"'^'^] j = i

+ Re[»?o^o(x)e-''''^f] + Re[),7?'7(x)e-''"^'], j = 1, 2, . . . , 6 ( 1 ) w h e r e (ps is t h e s t e a d y p o t e n t i a l a n d i t is n e g l e c t e d i n t h e p r e s e n t s t u d y ; (p9 a n d cpP ( / = 1,2 6) are t h e s p a t i a l r a d i a t i o n p o t e n t i a l i n six degrees o f f r e e d o m c o r r e s p o n d i n g t o t h e o s c i l l a t i o n s o f Ship.a a n d Ship-b r e s p e c t i v e l y a n d ly (j'= 1 , 2 , . . . , 6 ) is t h e c o r r e s p o n d i n g m o t i o n a m p l i t u d e surge; >?2.sway; 773, h e a v e ; )/4, r o l l ; ??5, p i t c h ; );6, y a w ) ; rn = rjQ is t h e i n c i d e n t w a v e a m p l i t u d e ; cpj is t h e s p a t i a l d i f f r a c t i o n p o t e n t i a l ; cpo is t h e s p a t i a l i n c i d e n t w a v e p o t e n t i a l a n d oje is t h e e n c o u n t e r f r e q u e n c y . G e n e r a l l y , t h e b o d y b o u n d a r y c o n -d i t i o n s c a n be t r e a t e -d s e p a r a t e l y b y t h e -d i f f r a c t i o n a n -d r a -d i a t i o n p r o b l e m as f o l l o w s : ( 1 ) B o d y b o u n d a r y c o n d i t i o n s f o r t h e d i f f r a c t i o n p r o b l e m : dtpo dn dn 0<P7 d(po dn dn So St ( 2 ) ( 3 ) ( 2 ) B o d y b o u n d a r y c o n d i t i o n s f o r t h e r a d i a t i o n p r o b l e m ( S h i p . a is o s c i l l a t i n g w h i l e S h i p . b is fixed): = -icoeiij + LiQinfis, 9 < dn ( 4 ) ( 5 ) ( 3 ) Body b o u n d a r y c o n d i t i o n s f o r t h e r a d i a t i o n p r o b l e m ( S h i p . b is o s c i l l a t i n g w h i l e Ship.a is fixed): 9^? - g i = - ! a ) c n j ' + i / o m f l s , ( 6 ) dn = Ols„ ( 7 ) w h e r e n = ( n i , 112, n 3 ) i s t h e u n i t n o r m a l v e c t o r d i r e c t e d i n w a r d o n b o d y s u r f a c e . The mj d e n o t e s t h e j - t h c o m p o n e n t o f the so-c a l l e d m - t e r m a n d f o r t h e s l e n d e r vessels, i t so-can be expressed b y ( n i l , 1712, 1113) = ( 0 , 0, 0 ) ( m 4 , Ills, me) = {0, 113, - / 1 2 ) ( 8 ) The f r e e s u r f a c e b o u n d a r y f o r b o t h d i f f r a c t i o n a n d r a d i a t i o n p r o b l e m can be w r i t t e n as: ^ dz 9 . 3(pi nd'^CPi .(oi<pj + 2wJeUo^ + 4 - ^ : 0 , J = l , 2 , 7 ( 9 ) 2.3. Radiation condition

Fig. 2 s h o w s the D o p p l e r S h i f t o f the s c a t t e r e d w a v e field b y a vessel t r a v e l i n g w i t h c o n s t a n t f o r w a r d speed UQ i n t h e p o s i t i v e X d i r e c t i o n . W h e n a vessel is m o v i n g f r o m p o i n t B t o p o i n t 0, the t r a v e l i n g time s h o u l d be t=BOjuo. D u r i n g t h i s p e r i o d o f t i m e , the vessel p r o d u c e s s c a t t e r e d w a v e s a l l a l o n g BO ( t h e first s c a t t e r e d w a v e s h o u l d arise at p o i n t B). The c o n t r o l s u r f a c e h e r e is d e f i n e d as a c i r c l e w i t h its c e n t r o i d o n p o i n t 0 a n d its radius as BO. The v e l o c i t y o f t h e s c a t t e r e d w a v e is d e f i n e d as c, BOIuo=BDIc. A c c o r d i n g to the sine t h e o r e m , i t can be easily t r a n s f e r r e d t o

( 1 0 ) UQ s i n 9 c ~ s i n a The s c a t t e r e d w a v e v e l o c i t y a t D can be e x p r e s s e d as c^ = f - tan/i/<sd '<s w h e r e « s is t h e a n g u l a r f r e q u e n c y o f t h e s c a t t e r e d w a v e s f r o m a fixed r e f e r e n c e p o i n t g i v e n as ( 1 1 ) 0 ) 5 = coe + U Q / C S cos ( a - 9 ) <Ws =g/<stan/i/<sd (12) ( 1 3 ) i n w h i c h Ics is t h e local w a v e n u m b e r at a n y p o i n t o n t h e f r e e or c o n t r o l surface, a n d d is t h e w a t e r d e p t h .

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C o m b i n i n g Eqs. ( 1 0 ) ( 1 3 ) , w e can o b t a i n 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 cos^ o' - s i n -I- | 2 T C O S v ^ s i n f f y t a n l i ( / c / f ^ ) ^ / V K s i n a a - s i n ^ t a n / i ( / f / F 2 ) ( 1 4 )

A t i n f i n i t e w a t e r d e p t h , dôo, Eq. ( 1 4 ) can be r e d u c e d t o cosÎa; - sin'^'îVK sin a)]K^

+ { 2 T cos [a - sin"^ ( , / } ? s i n a ) ] - 1 }/c - f = 0 ( 1 5 ) w h e r e Xs = 2n/ks is t h e local w a v e l e n g t h , y = Xsg/ul is t h e

d i m e n s i o n l e s s local w a v e l e n g t h , f h = uo/^/gd is t h e d e p t h F r o u d e n u m b e r , ic = I n / y is t h e d i m e n s i o n l e s s local w a v e n u m b e r , a n d p a r a m e t e r r =

uoOe/g-Let's discuss t h e d i m e n s i o n l e s s local w a v e l e n g t h o n x - a x i s . A t a = 0oT7t, sin~^(/f s i n a ) = 0 Eq. ( 1 5 ) b e c o m e s

cos^ (a)/c^ -I- [ 2 T cos (o') - 1 ]/c -i- = 0 The s o l u t i o n s f o r Eq. ( 1 6 ) can be w r i t t e n as

( 1 6 ) ( 1 7 ) y i = Y2 = _ 1 - 2 T C O S Q ; ± V I

-4Tcosffi

2 cos2 a

A t a = 0 a n d T < 0.25, t w o s o l u t i o n s can be o b t a i n e d f r o m Eq. ( 1 7 )

47r

1 - 2 T - V T ^ ^

47r

( 1 8 ) ( 1 9 ) A t a; = jr, a n o t h e r g r o u p o f t w o s o l u t i o n s can be o b t a i n e d f r o m E q . ( 1 7 )

47r

Y3

Y4 =

1 - h 2 T - V T T 4 T

4k

l+2r + VTT4r

( 2 0 ) ( 2 1 )

These f o u r s o l u t i o n s are s h o w n i n Fig. 3, w h i c h are i d e n t i c a l t o Becker's [ 2 3 ] results. I t has b e e n f o u n d b y u s i n g t h e Green f u n c -t i o n m e -t h o d -t h a -t a-t r &l-t; 0 . 2 5 , -t h e r e are -t h r e e w a v e s y s -t e m s : o n e r i n g w a v e s y s t e m a n d t w o K e l v i n f a n w a v e systems w i t h d i f f e r e n t w e d g e angle [ 2 3 , 2 4 ] , A t T > 0.25, t h e r e are o n l y t w o w a v e systems, one o f w h i c h is t h e w a v e s y s t e m f o r m e d b y t h e o u t e r f a n w a v e s . F r o m Fig. 3, w e find t h a t at T < 0.25, t h e r e are f o u r w a v e l e n g t h s i n X - a x i s : yi a n d ys c o r r e s p o n d s t o t h e r i n g w a v e s y s t e m , y2 a n d y4 c o r r e s p o n d s to t h e i n n e r a n d o u t e r K e l v i n f a n w a v e s y s t e m s respec-t i v e l y . Irespec-t can also be f o u n d respec-t h a respec-t arespec-t T &grespec-t; 0.25, respec-t h e r e are o n l y respec-t w o w a v e l e n g t h s i n x - a x i s : y^ c o r r e s p o n d s t o t h e r i n g w a v e s y s t e m a n d 5/4 c o r r e s p o n d s t o o u t e r K e l v i n f a n w a v e s y s t e m . W e n o t i c e t h a t t h e w a v e l e n g t h o f t h e r i n g w a v e s y s t e m is m u c h larger t h a n t h a t o f K e l v i n f a n w a v e s y s t e m s . I n t h e n u m e r i c a l s t u d y , the f r e e s u r f a c e is u s u a l l y t r u n c a t e d at 2 L 3 L u p s t r e a m a n d d o w n s t r e a m . This t r u n -c a t i o n l e n g t h is i n t h e same o r d e r as t h e l e n g t h o f t h e r i n g w a v e s y s t e m . B u t f o r t h e K e l v i n f a n w a v e s y s t e m s , t h i s t r u n c a t i o n l e n g t h is m u c h larger, and i t c a n be r e g a r d e d as i n f i n i t y . I n R a n k i n e s o u r c e m e t h o d , i f t h e t r u n c a t i o n l e n g t h is v e r y large (R -> 00), t h e r a d i a t i o n c o n d i t i o n is n o t necessary since t h e n u m e r i c a l d a m p i n g m a y d i s -sipate t h e scattered w a v e s b e f o r e t h e y r e a c h t h e t r u n c a t e d c o n t r o l surface. A n d also, as d e m o n s t r a t e d b y N a k o s [ 1 8 ] , t h e s h o r t w a v e s y s t e m c a r r i e d i n s i g n i f i c a n t e n e r g y . T h e r e f o r e , i n the p r e s e n t s t u d y , t h e r e is n o r a d i a t i o n c o n d i t i o n i m p o s e d t o K e l v i n f a n w a v e s y s t e m s . The r a d i a t i o n c o n d i t i o n p r o p o s e d i n t h i s p a p e r is o n l y a p p l i c a b l e t o solve t h 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 o f t h e r i n g w a v e s y s t e m . T h e r e f o r e , t h e p a r a m e t e r ks o n l y r e f e r s t o t h e local w a v e n u m b e r o f t h e r i n g w a v e s y s t e m .

Let's d e f i n e a p o i n t D, w h i c h is used t o d i v i d e the c o n t r o l s u r f a c e i n t o t w o parts, a n d Sc2. I f w e c a n n o t find t h e s o l u t i o n s f r o m

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190 Z.-M. Yuan et al./Applied Ocean Research 48 (2014) 186-201

e q u a t i o n s y s t e m ( 1 0 ) - ( 1 3 ) , tiiese p o i n t s m u s t be o n t h e c o n t r o l s u r f a c e Sc2. O t h e r w i s e , t h e y are o n 5 ^ . T h e c r i t i c a l 0 at p o i n t D c a n be d e r i v e d a n a l y t i c a l l y . The s c a t t e r e d w a v e r e a c h i n g p o i n t D is p r o d u c e d b y the vessel at p o i n t B. N o t i c e t h a t a = 2 Ö , Eq. ( 1 0 ) can be w r i t t e n as ijo c - ( 2 2 ) 2COS0 S u b s t i t u t i n g Eqs. ( 1 1 ) a n d ( 2 2 ) i n t o Eq. ( 1 2 ) , w e c a n o b t a i n t h e f o l l o w i n g e q u a t i o n at i n f i n i t e w a t e r d e p t h g " 4 cos 6 N o t i c e t h a t r •-(23) i i o « e / g , Eq. ( 2 3 ) b e c o m e s c o s Ö = } -4 T (24) F r o m Eq. ( 2 4 ) , w e f i n d t h a t i . r < 0 . 2 5 , n o s o l u t i o n c a n be f o u n d f o r t h e c r i t i c a l 9 since t h e s c a t t e r e d w a v e s can r e a c h a n y p o i n t s o n t h e w h o l e c o n t r o l sur-face. A t t h i s case, t h e s c a t t e r e d w a v e p r o d u c e d at p o i n t B s h o u l d reach s o m e w h e r e ahead o f p o i n t A. C o r r e s p o n d i n g l y , t h e w a v e g r o u p w i l l t r a v e l ahead o f t h e vessel.

i i . r = 0.25, the c r i t i c a l r o t a t e d angle 0 = 0. A t t h i s case, t h e scat-t e r e d w a v e p r o d u c e d ascat-t p o i n scat-t B is p r o p a g a scat-t i n g scat-t o p o i n scat-t A. C o r r e s p o n d i n g l y , t h e w a v e g r o u p is r e a c h i n g p o i n t 0 .

i i i . T > 0 . 2 5 , t h e c r i t i c a l 9 c a n be f o u n d at p o i n t D . A t t h i s case, t h e c o n t r o l surface c o u l d d i v i d e i n t o arc DB ( S ^ ) a n d arc DA [Sal I n t h e n u m e r i c a l c a l c u l a t i o n , t h e c o o r d i n a t e s o f a n y a r b i t r a r y p o i n t o n t h e c o n t r o l s u r f a c e are g i v e n , a n d t h e n t h e u n k n o w n s 9 a n d fcs c o u l d be o b t a i n e d b y s o l v i n g t h e n o n l i n e a r e q u a t i o n sys-t e m ( 1 0 ) - ( 1 3 ) . The r a d i a sys-t i o n c o n d i sys-t i o n is d e f i n e d as sys-t w o d i f f e r e n sys-t e q u a t i o n s o n a n d Scj i n d e p e n d e n t i y . : 0 0' = 1 , 2 , . , 7 ) o n

Sc2

7) o n Sc (25) (26) 9<»i - ikscpj cos 9 •-V<pj = 0 0 ' = 1 , 2 , Eq. ( 2 5 ) is a n u p d a t e d S o m m e r f e l d r a d i a t i o n c o n d i t i o n w i t h f o r -w a r d speed c o r r e c t i o n . If t h e f o r -w a r d speed is zero, ks = k, 0 = 0 a n d Eq, ( 2 5 ) c o u l d reduce t o t h e S o m m e r f e l d r a d i a t i o n c o n d i t i o n as

an

^--ikcpj^O 0 ' = 1 , 2 , 6) o n Sc (27) The r a d i a t i o n c o n d i t i o n ( 2 5 ) a n d ( 2 6 ) can also be a p p l i e d t o s h i p -t o - s h i p p r o b l e m , as s h o w n i n Fig. 4 . I-t is a s s u m e d -t h a -t -t w o ships a r e a d v a n c i n g i n w a v e s w i t h t h e same f o r w a r d speed. The t r a n s -verse a n d l o n g i t u d i n a l distances b e t w e e n t w o ships are dt a n d dl, r e s p e c t i v e l y . The i n e r t i a c o o r d i n a t e s y s t e m is s h o w n i n Fig. 4 w i t h its o r i g i n l o c a t e d o n t h e c e n t r a l l i n e b e t w e e n t w o s h i p s . Fig. 5 is a n u m e r i c a l case o f t w o o s c i l l a t i n g sources a d v a n c 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 . The f r e e s u r f a c e is t r u n c a t e d b y a c i r c l e . To s i m p l i f y t h e p r o b l e m , o n l y 4 0 nodes are d i s t r i b u t e d o n t h e c o n t r o l s u r f a c e ( 2 0 n o d e s o n t h e u p p e r h a l f c i r c l e a n d 2 0 nodes o n t h e l o w e r h a l f c i r -cle). Fig. 6 is t h e c a l c u l a t e d local w a v e n u m b e r a n d r o t a t e d angle r e s p e c t i v e l y at T = 0.2. The s o l u t i o n s o f ks a n d 0 can be f o u n d a t a n y n o d e s o n t h e c o n t r o l surface, w h i c h i l l u s t r a t e t h a t t h e s c a t t e r e d w a v e s c o u l d reach a n y p o i n t s o n t h e t r u n c a t e d surface. Due t o t h e D o p p l e r e f f e c t , t h e s c a t t e r e d waves u p s t r e a m have s h o r t e r w a v e -l e n g t h s . As a r e s u -l t , t h e -l o c a -l w a v e n u m b e r u p s t r e a m is greater t h a n t h a t d o w n s t r e a m , w h i c h is s h o w n i n Fig, 6a. B u t t h e m a x i m u m v a l -ues o f r o t a t e d angle appear a r o u n d y = 0, a n d i t decreases u p s t r e a m a n d d o w n s t r e a m g r a d u a l l y . I t is v e r y i n t e r e s t i n g to f i n d t h a t o n t h e u p p e r h a l f circle, t h e r o t a t e d angle 0.a is close t o zero at N o d e 1 a n d

Fig. 4. Sketch of Doppler shift and radiation condition of two ships advancing in waves.

Node 19 w h i l e o n t h e l o w e r h a l f circle, 9.b t u r n s t o be zero at Node 1 a n d N o d e 19. This is because these t w o nodes are a l m o s t o n the t r a j e c t o r y o f source a a n d source b, w h i c h can be s h o w n i n Fig. 5. A t these p o i n t s , t h e s c a t t e r e d w a v e d i r e c t i o n is p a r a l l e l t o x axis and i t w i l l n o t be r o t a t e d at a l l . Since t h e o r i g i n o f t h e c o n t r o l surface is l o c a t e d o n t h e c e n t r a l l i n e b e t w e e n t w o sources, t h e s y m m e t r y c a n n o t be a c h i e v e d a b o u t t h e t r a j e c t o r y o f source a a n d source b. T h e r e f o r e , t h e r e s u l t s o n u p p e r a n d l o w e r h a l f c i r c l e are d i f f e r e n t . The r e s u l t s o f source a a n d source b are also n o t i d e n t i c a l t o each o t h e r . Fig. 7 is t h e c a l c u l a t e d l o c a l w a v e n u m b e r a n d r o t a t e d angle r e s p e c t i v e l y at T = 0.6. I n n u m e r i c a l c a l c u l a t i o n , i f t h e r e is n o s o l u -t i o n f o r e q u a -t i o n s y s -t e m ( 1 0 ) - ( 1 3 ) , -t h e ks a n d 0 are l a b e l e d as O. W i t h r e g a r d t o s o u r c e a, t h e s c a t t e r e d w a v e s can o n l y p r o p a g a t e to N o d e 7 o n the u p p e r h a l f circle, w h i l e Node 6 is t h e f u r t h e s t p o i n t on t h e l o w e r h a l f c i r c l e . A h e a d o f these t w o nodes, t h e r e is n o scatter

1 0 11

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Z.-M. Yuan etal./Applied Ocean Research 48 (2014) 186-201 191

Node serier Node serier

Fig. 6. Results at r=0.2. (a) Local wave number and (b) rotated angle.

6 8 10 12

Node serier

" 0.7 ? 0-6 •S 0-5 0.3 « 0.2

I

0.1 o ^ 0 1 ( -— 1 1 1 1 1

i ^JUpper half circle) •j • ^ (Lower half circle)

* (Upper half circle) ^(Lower half civ cle) _

- - - y-- rjA- (

-— 1 1 1 1 1

i ^JUpper half circle) •j • ^ (Lower half circle)

* (Upper half circle) ^(Lower half civ cle) _

L-1\-

j j

L-1\-1 L-1\-1 1 1 1 1 1 1 1 1 \ i 1 1

j 1

\

1 1 1 1 0 6 8 10 12

Node serier

16 18 20

Fig. 7. Results at T = 0.6. (a) Local wave number and (b) rotated angle.

w a v e a n d ;<s a n d 9 are l a b e l e d as 0. Since source b is l o c a t e d at s o m e distance a f t e r w a r d , its s c a t t e r e d w a v e s can o n l y r e a c h N o d e 5 o n t h e u p p e r h a l f circle w h i l e N o d e 6 is t h e f u r t h e s t p o i n t o n t h e l o w e r h a l f c i r c l e . W e also c a l c u l a t e ks a n d 9 at r = 0.4. The c r i t i c a l nodes are s h o w n i n Fig. 5. I t can be c o n c l u d e d t h a t t h e q u i e s c e n t r e g i o n e x p a n d s as t h e increase o f T , since t h e s c a t t e r e d w a v e s are c o n -v e c t e d b e h i n d the sources. It can also b e easily d e m o n s t r a t e d t h a t t h e t r u n c a t i o n o f f r e e surface c o u l d be a r b i t r a r y ( c i r c u l a r , r e c t a n g u -lar o r e l l i p s e ) and ks a n d 9 are o n l y d e t e r m i n e d b y t h e c o o r d i n a t e s o f t h e p o i n t s o n t h e c o n t r o l s u r f a c e .

3. E q u a t i o n of m o t i o n

Once t h e u n k n o w n d i f f r a c t i o n p o t e n t i a l <p^ a n d r a d i a t i o n p o t e n -t i a l cpj are s o l v e d , t h e t i m e - h a r m o n i c pressure can be o b t a i n e d f r o m B e r n o u l l i ' s e q u a t i o n :

P j = prij UOe(pj + 1(0

ax j = 0 , l 7

( 2 8 )

w h e r e p is t h e fluid d e n s i t y . The h y d r o d y n a m i c f o r c e p r o d u c e d b y t h e o s c i l l a t o r y m o t i o n s o f t h e vessel i n t h e six degrees o f f r e e d o m

can be d e r i v e d f r o m t h e r a d i a t i o n p o t e n t i a l s as [ 2 5 ] 6 Sa

=

J2[colAf

+ icoBf]r^^ 6

+ Y^[o:ilAf + icoBfjijP, i = 1, 2 6

7=1 ( 2 9 ) 6

= J2[wlA>^l- + io,eB>^l']lj^

6 + Y^[colAf + ioJeBf]ril / = 1,2, ( 3 0 ) j=i

(7)

Z.-M. Yuan et all Applied Ocean Research 48(2014)186-201 192

T h e a d d e d mass a n d d a m p i n g can be expressed r e s p e c t i v e l y as:

, . . ^ J L f f

(.l-^±'-?)n,s

w h e r e A?." is t h e a d d e d m a s s o f Ship_a i n i - t h m o d e w h i c h is i n d u c e d b y t h e m o t i o n o f S h i p . a i n j - t h m o d e ; is t h e a d d e d mass o f Ship.a i n i - t h m o d e w h i c h is i n d u c e d b y t h e m o t i o n o f S h i p . b i n j - t h m o d e ; A^." is t h e a d d e d mass o f S h i p . b i n i - t h m o d e w h i c h is i n d u c e d b y t h e m o t i o n o f S h i p . a i n j - t h m o d e ; A'?!' is t h e a d d e d mass o f S h i p . b i n ! - t h m o d e w h i c h is i n d u c e d b y t h e m o t i o n o f S h i p . b i n j - t h m o d e ; B is t h e a d d e d d a m p i n g a n d the d e f i n i t i o n t h e s u b s c r i p t is t h e s a m e as t h a t o f a d d e d m a s s ; (p^j is t h e real p a r t o f j - t h p o t e n t i a l , a n d (pij is t h e i m a g i n a r y p a r t . Tlie w a v e e x c i t a t i o n f o r c e can be o b t a i n e d b y t h e i n t e g r a t i o n o f i n c i d e n t a n d d i f f r a c t i o n pressure as = j j ^{Po+PiWS ( 3 3 ) F^'^ j j ^{Po + P7)nidS ( 3 4 ) A p p l y i n g N e w t o n ' s second l a w , t h e 12 c o m p o n e n t s o f s h i p m o t i o n s i n t h e f r e q u e n c y d o m a i n can be o b t a i n e d b y s o l v i n g t h e f o l l o w i n g e q u a t i o n s y s t e m : 6

^ { [ - a ) 2 ( i W g +Ap + ico.B^f +

cm

+ [-o^Af +

meBf]jp

j = i

= F,)^°, 1 = 1 , 2 , 6 ( 3 5 )

6

Y^{[-wjA^ + kOeB^'l'^ + [-«2(M,5 + A f ) + itóeBg-" + Cfj]r]^]

= F j ^ ^ i = l , 2 , 6 ( 3 6 ) w h e r e M f j a n d M y r e p r e s e n t t h e g e n e r a l i z e d mass m a t r i x f o r S h i p . a and S h i p . b ; C9. a n d CPj r e p r e s e n t t h e r e s t o r i n g m a t r i x o f S h i p . a a n d Ship-b. The s t a n d a r d m a t r i x s o l u t i o n r o u t i n e p r o v i d e s t h e c o m p l e x a m p l i t u d e o f t h e o s c i l l a t o r y m o t i o n s f r o m Eqs. ( 3 5 ) to ( 3 6 ) . T h e

w a v e e l e v a t i o n o n t h e f r e e surface t h e n can be o b t a i n e d f r o m the d y n a m i c f r e e surface b o u n d a r y c o n d i t i o n i n t h e f o r m & = + + l ^ ^ f s - uox) • V()??^? -f

n1<pf)

= ?/!j + i f ö , j = 0 , l , . . . , 7 ( 3 7 ) w h e r e is t h e r e a l p a r t o f j - t h m o d e l , a n d is t h e i m a g i n a r y p a r t . 4. N u m e r i c a l i m p l e m e n t a t i o n I n t h e n u m e r i c a l s t u d y , t h e b o u n d a r y is d i v i d e d i n t o a n u m b e r o f q u a d r i l a t e r a l panels w i t h c o n s t a n t source d e n s i t y a{i) ( i = l , 2, . . . , N ) , w h e r e N i s t h e p a n e l n u m b e r . The p o t e n t i a l at t h e i t h p a n e l ( t h e c e n t r o i d c o o r d i n a t e can be d e n o t e d as (x,-, y,-, Z j ) ) i n d u c e d b y t h e j t h p a n e l ( t h e c e n t r o i d c o o r d i n a t e can be d e n o t e d as {xj,yj, Zj)) c a n be e x p r e s s e d b y

<Pi.i = Gi,jaj, i,j = 1,2, ...,N ( 3 8 ) w h e r e (p d e n o t e s t h e s t e a d y p o t e n t i a l cps o r t h e u n s t e a d y p o t e n t i a l

(Pj, G j j is t h e R a n k i n e - t y p e Green f u n c t i o n t h a t satisfies t h e sea bed b o u n d a r y c o n d i t i o n t h r o u g h t h e m e t h o d o f m i r r o r i m a g e

^ J i X i - X j f + { y i - y j f + { Z i - Z j f

+ ^ ( 3 9 ) \ / ( X i - xj f + (y,- - yj f + (Zf + 2 d + Zj f

W h e n t h e i t h p a n e l a n d t h e j t h p a n e l are close t o each other, G j j can be c a l c u l a t e d w i t h a n a l y t i c a l f o r m u l a s l i s t e d b y Prins [ 2 6 ] , W h e n t h e distance b e t w e e n t h e i t h p a n e l a n d t h e j t h p a n e l is large, these c o e f f i c i e n t s are c a l c u l a t e d n u m e r i c a l l y .

The f i r s t d e r i v a t i v e s o f t h e p o t e n t i a l are e v a l u a t e d w i t h a n a l y t -ical f o r m u l a s f o r t h e first d e r i v a t i v e s o f t h e R a n k i n e source s h o w n i n Hess a n d S m i t h [ 1 3 ] .

Special a t t e n t i o n s s h o u l d be p a i d o n t h e second d e r i v a t i v e s of t h e p o t e n t i a l o n t h e f r e e surface. G e n e r a l l y , t h e d i f f e r e n c e schemes can be d i v i d e d i n t w o classes: u p w i n d d i f f e r e n c e schemes a n d cen-t r a l d i f f e r e n c e schemes. A l cen-t h o u g h c e n cen-t r a l d i f f e r e n c e s c h e m e s are s u p p o s e d to be m o r e accurate, t h e s t a b i l i z i n g p r o p e r t i e s o f the u p w i n d d i f f e r e n c e schemes are m o r e d e s i r e d i n t h e f o r w a r d speed p r o b l e m [ 2 7 ] . P h y s i c a l l y this can be e x p l a i n e d b y t h e face t h a t n e w i n f o r m a t i o n o n t h e w a v e p a t t e r n m a i n l y c o m e s f r o m t h e u p s t r e a m side, e s p e c i a l l y at h i g h speeds, w h e r e a s t h e d o w n s t r e a m side o n l y c o n t a i n s o l d i n f o r m a t i o n . T h e first-order u p w i n d d i f f e r e n c e scheme f o r t h e second d e r i v a t i v e o f t h e p o t e n t i a l t o x c a n be w r i t t e n as f o l l o w s < = ^ [ ^ ( . j + 2 - 2 ^ , j + i + ^ i . j ] (40) By s u b s t i t u t i n g t h e first a n d second d e r i v a t i v e s o f t h e p o t e n t i a l i n t o t h e b o d y - , f r e e - a n d c o n t r o l - 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 , the f o l l o w i n g set o f l i n e a r e q u a t i o n s f o r t h e values o f t h e source density c a n be o b t a i n e d

N

X

^fijo;/= Q l . 1 = 1 , 2 jv (41)

For t l i e c o n s t a n t p a n e l m e t h o d , t h e s e c o n d d e r i v a t i v e s o f the p o t e n t i a l can be expressed a n a l y t i c a l l y [ 2 8 ] . H o w e v e r , w e find that w h e n t h e a n a l y t i c a l e x p r e s s i o n is used, t h e c o n d i t i o n n u m b e r o f the c o e f f i c i e n t m a t r i x Py is e x t r e m e l y l a r g e a n d t h e c o e f f i c i e n t m a t r i x t e n d s to be a n i l l - c o n d i t i o n e d m a t r i x . The d i a g o n a l e l e m e n t s (/ = j ) is v e r y l a r g e d u e t o t h e f a c t o r t h a t t h e field p o i n t is w i t h i n the source

(8)

Z.-M. Yuan et al./Applied Ocean Research 48 (2014) 186-201 193 p a n e l . A s a result, e x c e p t t h e c i i f f r a c t e d a n d r a d i a t e d w a v e m o d e s , s o m e s p u r i o u s w a v e m o d e s can also be o b s e r v e d o n t h e f r e e s u r -face, w h i c h c o u l d i n t e r f e r e w i t h t h e p h y s i c a l w a v e s a n d e v e n t u a l l y d e s t r o y t h e c r e d i b i l i t y o f t h e s o l u t i o n c o m p l e t e l y . T h i s is a p u r e l y n u m e r i c a l p h e n o m e n o n t h a t f i r s t l y discussed b y L o n g u e t - H i g g i n s a n d Col<elet [ 2 9 ] . T h e y f o u n d t h a t t h e s a w - t o o t h l i k e w a v e s w e r e s u p e r i m p o s e d o n t h e p h y s i c a l w a v e s s u c h t h a t t h e w a v e s w e r e z i g z a g a l i k e i f n o p r e v e n t i v e m e a s u r e is t a k e n . The g e n e r a l c o n s e n -sus o n t h e cause o f t h e p r o b l e m is t h a t t h e r e is h i g h c o n c e n t r a t i o n o f f l u i d p a r t i c l e s w i t h h i g h speed i n c e r t a i n r e g i o n s e s p e c i a l l y near t h e w a v e crests. T h e same p h e n o m e n o n w a s also e n c o u n t e r e d b y X u a n d Y u e [ 3 0 ] i n t h e i r 3 - D s o l u t i o n . A t y p i c a l t r e a t m e n t t o r e m o v e t h e s a w - t o o t h i n s t a b i l i t y is b y i n t r o d u c i n g a l o w - p a s s n u m e r i c a l f i l t e r [ 2 9 - 3 1 ] . A l t h o u g h i t c a n p r o v i d e a s m o o t h w a v e p a t t e r n , i t is b e l i e v e d t h a t t h i s n u m e r i c a l f i l t e r c a n b r i n g s o m e i n f l u e n c e t o t h e r e a l w a v e e l e v a t i o n . W e f i n d t h e d i f f e r e n c e s c h e m e o n t h e f r e e s u r -face b o u n d a r y c o n d i t i o n is t h e m a i n r e a s o n f o r t h e s p u r i o u s w a v e m o d e s . Even w i t h o u t n u m e r i c a l f i l t e r i n g , t h e u p s t r e a m d i f f e r e n c e s c h e m e c a n o b t a i n a s m o o t h w a v e p a t t e r n . W h e n t h e u p w i n d d i f f e r -ence s c h e m e is used, t h e d i a g o n a l e l e m e n t s o f t h e c o e f f i c i e n t m a t r i x c a n be r e s t r a i n e d a n d as a result, t h e c o n d i t i o n n u m b e r decreases a n d t h e s o l u t i o n t e n d s to be stable. I t s h o u l d also be n o t i c e d t h a t t h e s i n g u l a r i t y d i s t r i b u t i o n does n o t h a v e t o be l o c a t e d o n t h e f r e e s u r f a c e itself, i t c a n also be l o c a t e d at a s h o r t distance above the f r e e surface, as l o n g as t h e c o l l o c a t i o n p o i n t s , w h e r e the b o u n d a r y c o n d i t i o n has t o be s a t i s f i e d , stay o n t h e f r e e s u r f a c e . In practice, a distance o f m a x i m a l t h r e e t i m e s t h e l o n g i t u d i n a l size o f a p a n e l is possible [ 2 7 ] . I n t h e p r e s e n t s t u d y , t h e r a i s e d distance A z j = -y/sj^, w h e r e Sj is t h e area o f the i t h p a n e l .

5. V a l i d a t i o n s a n d d i s c u s s i o n I n r e a l i s t i c c o n d i t i o n s , a s h i p c a n n o t be c o n s i d e r e d as a p o i n t s o u r c e a n d d i f f e r e n t p a r t s o f its h u l l u s u a l l y p r o d u c e several w a v e s y s t e m s . G e n e r a l l y , o n l y s t e r n w a v e s have m a g n i t u d e c o m p a r a b l e w i t h t h a t o f b o w w a v e s . The d i v e r g i n g b o w a n d s t e r n w a v e s m a y t r a v e l i n d e p e n d e n t l y i f t h e s h i p is l o n g e n o u g h [ 3 2 ] . T h e l o n g e r w a v e s a n d t h e s h i f t o f t h e o r i g i n d o w n s t r e a m o f t h e vessel r e q u i r e t h e r e p l a c e m e n t o f t h e c o n t r o l s u r f a c e f u r t h e r a w a y t o s a t i s f y t h e p o i n t s o u r c e a s s u m p t i o n . H o w e v e r , Das a n d C h e u n g [21 ] c a r r i e d o u t t h e c o n v e r g e n c e s t u d y a n d f o u n d t h a t t h e p r e s e n t m o d e l a p p a r -e n t l y h a n d l -e d thos-e p h -e n o m -e n a w -e l l w i t h c o n s t a n t p a n -e l s a n d p r o v i d e d a c c u r a t e results w i t h a d o m a i n o f r e a s o n a b l e d i m e n s i o n s a n d s o l u t i o n c o m m o n l y used i n s h i p a n d o f f s h o r e p l a t f o r m design. I n t h i s p a p e r , w e w i l l v e r i f y t h e p r e s e n t n u m e r i c a l p r o g r a m t h r o u g h t w o p a i r s o f m o d e l s . M o d e l 1 is a b o u t a f u l l scale s u p p l y s h i p a n d f r i g a t e m o d e l , a n d Li's m o d e l test r e s u l t s [ 7 ] w i l l be u s e d t o v e r -i f y t h e p r e s e n t n u m e r -i c a l c a l c u l a t -i o n . M o d e l 2 -is a t a n k e r - L N G s h -i p m o d e l i n m o d e l scale, a n d R o n s s s ' e x p e r i m e n t s [ 8 ] w i l l p r o v i d e t h e m o t i o n responses f o r v a l i d a t i o n . 5.1. Results of ModeU The m a i n p a r t i c u l a r s o f s u p p l y s h i p ( S h i p . a ) a n d f r i g a t e ( S h i p . b ) are s h o w n i n Table 1 . The t r a n s v e r s e a n d l o n g i t u d i n a l d i s t a n c e s b e t w e e n t w o ships are 5 2 . 7 0 2 m a n d O m , r e s p e c t i v e l y . A t y p i c a l case is s i m u l a t e d h e r e : h e a d sea w i t h f o r w a r d s p e e d o f 6.18 m / s ( f n = 0 . 1 5 ) . T o be c o n s i s t e n t w i t h t h e m o d e l tests c o n d i t i o n , b o t h s h i p s are r e s t r a i n e d i n surge, s w a y a n d y a w w h i l e t h e m o t i o n s i n heave, r o l l a n d p i t c h are f r e e . I n o r d e r t o m a k e c o m p a r i s o n , w e also p r e s e n t t h e n u m e r i c a l r e s u l t s o f t w o ships at z e r o f o r w a r d speed. T h e c o m p u t a t i o n a l d o m a i n is s h o w n i n Fig. 8. T h e f r e e s u r f a c e is t r u n c a t e d at La u p s t r e a m , 2La d o w n s t r e a m . La i n t h e s u p p l y s h i p s i d e w a r d a n d L/, i n t h e f r i g a t e s i d e w a r d . T h e r e are 3 7 8 p a n e l s o n

Table 1

Main particulars of supply ship and frigate [51.

Supply ship Frigate Length between perpendicular L . - = 180m ii, = 122m

Breadth By = 30.633 m Bb = 14.78 m

Draught T,-= 8.5m rb = 4.5m

Displacement = 28,223.31 l/b= 4023.71

Blocl< coefficient = 0.588 C ' = 0.484 Longitudinal CoG (ret. midship) = - 1 . 6 8 8 m X ° = 3.284 m Vertical CoG (rel. calm waterline) ya

G = 3.925 m Z° = 2 . 0 4 9 m Radius of inertia for roll r°

.14

= 8.047 m r544.921 m Radius of inertia for pitch ' 5 5 = 4 5 m r | 5 = 30.5 m

Radius of inertia for yaw ..a _{' 6 6}= 4 5 m r*. = 3 0 . 5 m

66

Fig. 8. Computational domain of Model 1.

t h e b o d y s u r f a c e o f s u p p l y s h i p , 5 4 0 0 o n f r e e surface, 2 4 3 2 o n t h e c o n t r o l s u r f a c e a n d 4 1 4 o n t h e b o d y s u r f a c e o f f r i g a t e .

Fig. 9 s h o w s t h e response a m p l i t u d e s o f t w o ships i n heave, r o l l a n d p i t c h m o t i o n s . The c o m p a r i s o n s w i t h e x p e r i m e n t a l d a t a a n d G r e e n f u n c t i o n m e t h o d [ 7 ] are also i n c l u d e d . The n u m e r i c a l results c a l c u l a t e d b y the p r e s e n t 3 D R a n k i n e source m e t h o d g e n -e r a l l y agr-e-e w i t h th-e -e x p -e r i m -e n t a l data. I n o r d -e r t o i n v -e s t i g a t -e the s p e e d e f f e c t , w e also p r e s e n t t h e r e s u l t s o f t w o ships w i t h -o u t f -o r w a r d speed, I t can be -o b s e r v e d t h a t t h e increase -o f t h e r e s p o n s e a m p l i t u d e o p e r a t o r s w i t h f o r w a r d s p e e d is c o n s i d e r a b l e , e s p e c i a l l y f o r t h e s m a l l e r s h i p ( S h i p . b ) . Roll m o t i o n o f S h i p . a is o b v i o u s l y r e d u c e d , d u e to t h e f o r w a r d speed. B u t f o r S h i p . b , t h e r o l l m o t i o n increases d r a m a t i c a l l y at A / L > 1 d u e t o t h e f o r w a r d s p e e d . In h e a v e a n d p i t c h m o t i o n s , t h e r e are also s o m e d i s c r e p -ancies b e t w e e n t h e p r e d i c t i o n s a n d m e a s u r e m e n t s , especially i n t h e l o n g w a v e case. T h e r e are t w o aspects t o e x p l a i n these d i s -crepancies. T h e f i r s t r e a s o n s h o u l d be the m o d e l t e s t s e t - u p . F r o m the p u b l i s h e d w o r k , o n l y t h r e e m o d e l tests can be f o u n d o n s h i p -t o - s h i p w i -t h f o r w a r d speed p r o b l e m [ 5 , 8 , 3 3 ] . I-t w a s f o u n d -t h a -t -t h e m o d e l test s e t u p w a s v e r y c h a l l e n g i n g , e s p e c i a l l y f o r the m e a s u r e -m e n t o f r o l l -m o t i o n . T h e s e c o n d r e a s o n is t h e n u -m e r i c a l p r o g r a -m . The p r e s e n t p o t e n t i a l f l o w p r o g r a m is b a s e d o n t h e l i n e a r a s s u m p -t i o n . I-t c a n be f o u n d i n Fig. 9 -t h a -t -t h e g r e a -t e s -t discrepancies b e -t w e e n m e a s u r e d a n d p r e d i c t e d m o t i o n s g e n e r a l l y o c c u r at l o n g w a v e l e n g t h . I n t h e s e c o n d i t i o n s , t h e m o t i o n s o f S h i p . b are v e r y large, e s p e c i a l l y i n r o l l m o t i o n . I t v i o l a t e s t h e l i n e a r a s s u m p t i o n . Even f o r t h e m o d e l test, as d e m o n s t r a t e d b y Li [ 6 ] , t l i e e x p e r i m e n t s c o u l d n o t be c o m p l e t e d f o r t h e h i g h e s t t w o w a v e l e n g t h s d u e t o excessive m o t i o n s o f t h e S h i p . b ( r o l l a m p l i t u d e exceeds 3 0 ° ) . F u r t h e r m o r e , 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 s e t w o ships are also v e r y i m p o r t a n t . The m o t i o n s o f t h e J a r g e r s h i p ( S h i p . a ) c o u l d i n f l u -ence t h e m o t i o n s o f s m a l l e r s h i p ( S h i p . b ) s i g n i f i c a n t l y . The large a m p l i t u d e r o l l m o t i o n is c o u p l e d w i t h t h e h e a v e a n d p i t c h m o t i o n s , w h i c h is d i f f e r e n t f r o m t h e s i n g l e s h i p p r o b l e m . T h e u n p r e d i c t a b l e

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194 Z.-M. Yuan et al./Applied Ocean Research 48 (2014) 186-201 a 1 0.8 ^ 0.6 # 0.4 0.2 0 0

c

0.5 0.4 O 0.3 CO T £ • 0.2 0.1 0

c

e

1 0.8 o 0.6 0.4 0.2 « Experiments (Li, 2007) — Present (Rankine source)

Li (2007) (Green function) -a— Zero speed (Tm sliips)

1 1 1 1 1 1 ..M 1

'-y-r-1

'-y-r-0.4 0.8 X/la 1.2 1.6 1.5 1.2 0.9 0.6 0.3 0 1 I • • [ ' " 1 1 1 1 1 1 1 1 I 0 0.4 0.8 1.2 X/Lb 1.6 1 1

: /

'b-°>a ' /---. i ^ 1 : 2 0.4 0.8 X/La 1.2 1.6 0.4 0.8 1.2 X/Lb 1.6 1 1 1 1 1

/

1

r

; i 7

-a—o—

Fig. 9. Response amplitude operators of two ships at Fn = 0.15. (a) Heave of Ship-a; (b) heave of Ship.b; (c) roll of Ship.a; (d) roll of Ship.b; (e) pitch of Ship-a; (f) pitch of Ship.b.

r o l l m o t i o n i n l o n g w a v e l e n g t h c o u l d also i n f l u e n c e t h e p r e d i c t i o n s o f heave a n d p i t c h m o t i o n s .

W e also find t h e r o l l m o t i o n s o f b o t h ships are s i g n i f i c a n t l y i n f l u e n c e d b y the r o l l d a m p i n g c o e f f i c i e n t . It is f o u n d t h a t the d a m p i n g i n r o l l c a n n o t be p r e d i c t e d w e l l b y t h e r a d i a t i o n c o m p o -n e -n t o -n l y [ 3 4 ] . The d i f f i c u l t y i -n p r e d i c t i -n g t h e r o l l m o t i o -n arises f r o m t h e n o n l i n e a r c h a r a c t e r i s t i c s o f r o l l -due t o t h e e f f e c t o f fluid v i s c o s i t y . I n s h i p - t o - s h i p problemt, t h e r o l l m o t i o n is a l w a y s r e m a r k a b l e d u e t o 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 b e t w e e n t w o ships. The p r e s e n t p o t e n t i a l flow t h e o r y is based o n t h e a s s u m p t i o n t h a t t h e s u r r o u n d i n g fluid is i n v i s c i d a n d i t c a n n o t p r e d i c t t h e r o l l d a m p i n g p r e c i s e l y . To c o m p l e m e n t t h e v i s c i d c o m p o n e n t , an

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Z.-M. Yuan etal./Applied Ocean Researcli 48 (2014) 186-201 195

Table 2

Main particulars of tanl<er and LNG ship [8].

Tanker LNG ship

Length between perpendicular L„= 3.76 m i t = 2.28 m

Breadth B„ =0.625 m B|, =0.387 m

Draught r„ = 0.232 m r6=0.124m

Displacement 1^0=0.4355 1 l/t =0.074 t

Block coefficient CJ0.83 Ö = 0.68

Water plane area coefficient C° = 0.90 C° =0,7D Longitudinal CoG (rel. midship) X ° = 0.086 m X'i = -0.01 m Vertical CoG (ret. calm waterline) 2° = - 0 . 0 5 2 m = 0.012 m Radius of inertia for roll r j , = 0 . 1 7 5 m rj^ = 0 . 1 0 3 m Radius of inertia for pitch r ^ 5 = 1.008 m r ° 5 = 0 , 6 0 4 m Radius of inertia for yaw rje = 1.008 m

4 = 0-604 m

Table 3

Corrections for model set-up of Ship-b, non-dimensionalized using p, Vb, h, i tand

An*

Component, l j 33 55 35, 53

Additional inertia,/|j 1.6E--01 5.5E--02 - 7 . 2 E - 04 Additional damping, Bij 5.2E--03 1.6E-•03 - 2 . 6 E - 03 Additional restoring, Qj 4.8E--04 1.5E--04 - 2 , 1 E - •06

e q u i v a l e n t l i n e a r d a m p i n g c o e f f i c i e n t is a p p l i e d i n t h e p r e s e n t s t u d y . The n o n - d i m e n s i o n a l r o l l d a m p i n g c o e f f i c i e n t , A:, is g i v e n b y _ ^44 + ^ 4 4 » ( 4 2 ) 2 v ' ( / 4 4 + A 4 4 ) C 4 4 w h e r e 044^ is t h e viscous d a m p i n g . This d a m p i n g c o e f f i c i e n t is w r i t t e n as a f r a c t i o n b e t w e e n the a c t u a l d a m p i n g c o e f f i c i e n t , 844 + 844^, a n d t h e c r i t i c a l d a m p i n g c o e f f i c i e n t , 2y'{l44+A44)C44-Fig. 10 is t h e n u m e r i c a l results o f r o l l m o t i o n a m p l i t u d e s o f t w o ships at d i f f e r e n t d a m p i n g c o e f f i c i e n t s . W e find t h a t Ka =0.2 a n d /Ci, = 0.6 agree w i t h t h e e x p e r i m e n t a l results b e t t e r t h a n o t h e r values. This is because t h e r o l l m o t i o n o f Ship.a is r e l a t i v e l y s m a l l , w h i l e the r o l l m o t i o n o f S h i p . b is e x t r e m e l y large. C o r r e s p o n d i n g l y , t h e n o n l i n e a r viscous c h a r a c t e r i s t i c s o f r o l l m o t i o n o f S h i p . b are m o r e o b v i o u s . A larger e q u i v a l e n t l i n e a r d a m p i n g c o e f f i c i e n t s h o u l d be used i n t h e n u m e r i c a l s i m u l a t i o n s . 5.2. Results of Model 2 The m a i n p a r t i c u l a r s o f t a n k e r ( S h i p . a ) a n d LNG ( S h i p . b ) are s h o w n i n Table 2. The details o f m o d e l test s e t - u p is e l a b o r a t e d b y R o n s s s [8], She a n a l y z e d the bias sources a n d c a r r i e d o u t c o m -p a r a t i v e s t u d y . She f o u n d t h a t t h e e x -p e r i m e n t a l set u -p c o r r e c t i o n s w e r e necessary a n d i n t h e p r e s e n t c a l c u l a t i o n , such c o r r e c t i o n s w i l l be used, as s h o w n i n Table 3. The f o r c e d r o l l c e n t e r is talcen t o be 0.032 m b e l o w t h e m e a n w a t e r l e v e l f o r Ship.a a n d 0.104 m above t h e m e a n w a t e r l e v e l f o r S h i p . b . To be c o n s i s t e n t w i t h t h e m o d e l tests c o n d i t i o n , S h i p . a is r e s t r a i n e d i n surge a n d s w a y w h i l e t h e m o t i o n s i n heave, r o l l , p i t c h a n d y a w are f r e e ; Ship.b is r e s t r a i n e d i n surge, s w a y a n d y a w w h i l e t h e o t h e r degrees o f f r e e d o m are set f r e e . A t y p i c a l case is s i m u l a t e d h e r e : h e a d sea w i t h f o r w a r d speed o f 0 . 9 1 2 m / s (Froude n u m b e r Fn = i i o / ^ ^ ï ^ = 0.15). The t r a n s -verse a n d l o n g i t u d i n a l distances b e t w e e n t w o ships are 1.25 m a n d 0.09 m , r e s p e c t i v e l y , w h i c h i n d i c a t e s t h a t t h e l o n g i t u d i n a l c e n t e r o f these t w o ships are a p p r o x i m a t e l y t h e same. I n o r d e r t o m a k e c o m p a r i s o n , w e also p r e s e n t t h e results o f single s h i p w i t h t h e s a m e f o r w a r d speed a n d t w o ships at zero f o r w a r d speed. The c o m p u -t a -t i o n a l d o m a i n is s h o w n i n Fig. 1 1 . The f r e e s u r f a c e is -t r u n c a -t e d at l.OSLa u p s t r e a m , 1.84La d o w n s t r e a m , 1.051a i n t h e t a n k e r s i d e -w a r d a n d 1.3Lfa i n t h e LNG s h i p s i d e -w a r d . There are 4 2 0 panels o n t h e b o d y surface o f t a n k e r , 9 0 2 0 o n f r e e surface, 2 4 6 4 o n t h e c o n t r o l s u r f a c e a n d 4 2 0 o n t h e b o d y s u r f a c e o f LNG s h i p .

Fig. 11. Computational domain of Model 2.

5.2, J. Motion responses

Fig. 12 is t h e response a m p l i t u d e s o f t w o ships. The c o m p a r -isons w i t h e x p e r i m e n t a l data a n d u n i f i e d t h e o r y are also i n c l u d e d . The p r e s e n t results i n heave a n d p i t c h m o t i o n o f b o t h ships g e n e r -a l l y h-ave -a s -a t i s f i e d -a g r e e m e n t w i t h t h o s e o f e x p e r i m e n t -a l d-at-a. A n o t i c e a b l e d i s c r e p a n c y c a n be o b s e r v e d i n Fig. 12a a n d e at A/La = 1.2 a n d X/Lo = 1.3, w h i c h c o r r e s p o n d s t o t h e r e s o n a n t f r e q u e n c y o f heave a n d p i t c h o f Ship.a, r e s p e c t i v e l y . But t h e r e s o n a n t f r e q u e n c y i n t h e n u m e r i c a l c a l c u l a t i o n is a r o u n d XjLa = 1 f o r b o t h heave a n d p i t c h o f Ship.a. This d i f f e r e n c e is a t t r i b u t e d t o t h e t r i m s u s p e n s i o n s i n the m o d e l test setup [8]. W h e n i t c o m e s t o r o l l , t h e p r e s e n t p r e -d i c t i o n , as w e l l as Ronasss' [8] c a l c u l a t i o n , is n o t s a t i s f a c t o r y . The m a i n r e a s o n f o r the discrepancies is a b o u t t h e d a m p i n g c o e f f i c i e n t , w h i c h has b e e n discussed p r e v i o u s l y i n t h e v a l i d a t i o n o f M o d e l 1. A c c o r d i n g to R o n s s s [8], t h e r o l l viscous d a m p i n g o f Ship.a is tal<en as 844^ = 2844 f o r the f o r w a r d speed case a n d 844^ = 844 f o r t h e zero speed case. For Ship.b, i t is t a k e n as 844^ = 844 f o r t h e f o r -w a r d s p e e d case a n d B44„ = 4B44 f o r t h e zero speed case. The l i f t d a m p i n g is a n o t h e r f a c t o r , w h i c h w i l l increase w h e n t h e r o l l c e n -t e r is above -t h e m e a n w a -t e r l e v e l [ 3 5 ] . Besides, -t h e m e a s u r e m e n -t o f r o l l m o t i o n i n t h e m o d e l t e s t is f u l l o f challenges. T h e devices used t o m e a s u r e t h e r o l l m o t i o n c o u l d b r i n g a d d i t i o n a l f r i c t i o n a n d u p w a r d forces, as d e m o n s t r a t e d b y R o n s s s [8]. W e also i n c l u d e t h e results o f a s i n g l e s h i p w i t h f o r w a r d speed. F r o m t h e c o m p a r -ison, w e find 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 has m u c h g r e a t e r i n f l u e n c e o n t h e m o t i o n s o f t h e s m a l l e r s h i p . For h e a v e a n d p i t c h m o t i o n s o f t h e larger s h i p (Ship.a), t h e i n f l u e n c e f r o m t h e s m a l l e r s h i p ( S h i p . b ) is n o t n o t i c e a b l e . B u 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 is the essential r e a s o n t h a t induces t h e r o l l m o t i o n f o r b o t h ships. T h e r e is n o r o l l m o t i o n i n h e a d sea c o n d i t i o n f o r a s i n g l e s h i p due t o t h e s y m m e t r i c a l c h a r a c t e r i s t i c .

5.2.2. Effect of radiation condition

Fig. 13 s h o w s t h e r e a l p a r t o f d i f f r a c t e d a n d r a d i a t e d w a v e s o f t w o ships w i t h h i g h f o r w a r d speed. Fig. 1 4 s h o w s t h e t o t a l w a v e e l e v a t i o n , w h i c h is n o n - d i m e n s i o n a l i z e d b y t h e i n c i d e n t w a v e a m p l i t u d e ÏJQ. I t is o b s e r v e d t h a t t h e s y m m e t r i c a l c h a r a c t e r i s t i c o f w a v e p a t t e r n p r o d u c e d b y single s h i p has b e e n m o d i f i e d i n t h e p r e s e n c e o f t h e o t h e r one. A V - s h a p e r e g i o n is c l e a r l y c o n v e c t e d d o w n s t r e a m . The d i f f r a c t e d w a v e s f r o m t h e t w o sides i n t e r a c t w i t h t h o s e f r o m t h e gap t h r o u g h a s y s t e m o f t r a n s v e r s e w a v e s a n d a p p r o a c h t h e d o w n s t r e a m b o u n d a r y at an o b l i q u e angle. The r a d i -a t e d w -a v e s p r o p -a g -a t e s i d e w -a r d i n d e p e n d e n t l y -a n d -a p p r o -a c h t h e d o w n s t r e a m b o u n d a r y p a r a l l e l . N o r e f l e c t i o n s c a n be f o u n d o n t h e

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Z.-M. Yuan et al./Applied Ocean Research 48(2014) 186-201

Fig. 14. Total wave elevation on the free surface of two ships in head seas: A/lb = 1-08. F,. = 0.25, T = 1.35. (a) Real part and (b) imaginary part.

Fig. 15. Real part of diffracted waves of two ships in head seas by using Sommerfeld and present radiation condition: A/lb = 1.08, F„ = 0.05, T = 0.2. (a) Wave pattern in the portside of Ship.a and (b) wave pattern in the starboard of Ship.a.

Fig. 16. Real part of radiated waves of two ships in head seas by using Sommerfeld and present radiation condition: A/ib = 1.08, F„ = 0.05, T = 0.2. (a) Wave pattern in the portside of Shlp.a and (b) wave pattern in the starboard of Ship.a.

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198 Z.-M. Yuan et al./Applied Ocean Research 48 (2014) 186-201

a

T]

V/ ////•Zr^

-0.4 -0.3 -0.1 -0.0 0.1 0.3 0.4 0.5 0.7 0.8 0.9 -0.4 -0.3 -0.1 -0.0 0.1 0.3 0.4 0.5 0.7 0.8 0.9

J~l r I Present radiation condlBonl

' • ) • ) ) . /

'•W-m{ 1 H . " \ \ [Upstream radiation conditioril

I ^-^Presentradiatibnconditron

wmym:=x

rupsfream radiation condrtionj

rv,\VTnrT77f[i,( •) \

-6 -4

Fig. 17. Real part of diffracted waves of two sliips in head seas by using upstream boundary condition of Nakos [18] and present radiation condition: A/Lb = 2.15, F„ =0.1,

T = 0,27. (a) Wave pattern in the portside of Ship.a and (b) wave pattern in the starboard of Ship.a.

a

-1.3 -1.1 -0.9 -0.7 -0.4 -0.2 -0.0 0.2 0.4 0.6 -1.7 -1.4 -1.2 -0.9 -0.6 -0.3 -0.1 0.2 0.5 O.i

-4

: U P r e s e n t radiation conditioni

( )

MS- I I I ƒl ü ' ^ - \ \ i', ' ^ ^ ' ^ l U p s t r e a m radiation conditiörï]

-4

I Present radiation conditioni

- / . • ~^^-a^-*X\\\\ Z / ? ^ ^ ? ' : ^ [Upstreamradlaflonconditioiil

Fig. 18. Real part of radiated waves of two ships in head seas by using upstream boundary condition of Nakos [ I S ] and present radiation condidon: A/Lb = 2.15, Fn =0.1i r = 0.27. (a) Wave pattern in the portside of Ship.a and (b) wave pattern in the starboard of Ship.a.

a

-0.6 -0.6 -0.4 -0.3 -0.1 0.1 0.3 0.5 0.6 0.8 1.0 P r e s e n t radiation condition

S ~ r > ^ _ ; r : = ~ ^ a^^^^sife—'"^ I U p s t r e a m radiation condition

-0.8 -0.6 -0,4 -0.3 -0.1 0.1 0.3 0.5 0.6 0.8 1.0

7 '^Present radiation conditioni

Upstream radiation condilion -4 -2 0

Fig. 19. Real part of diffracted waves of two ships in head seas by using upstream boundary condition of Nakos [18] and present radiation condition: A/JLb =0.75, F„ =0.1, r = 0,51, (a) Wave pattern in the portside of Ship.a and (b) wave pattern in the starboard of Ship.a.