Analysis of the forward speed effects on the radiation forces on a fast ferry

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Ocean Engineering

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Analysis of the forward speed effects on the radiation forces on a Fast Ferry

R. Datta, N. Fonseca, C. Guedes Soares *

Centre for Marine Tedmology and Engineering (CENTEC), Instituto Superior Técnico, Tecimical University of Lisbon, Av. Rovisco Pais, 1049-001 Lisbon, Portugal

A R T I C L E I N F O

Article ttistory:

Received 29 June 2012 Accepted 25 November 2012 Available online 26 January 2013

Keywords:

Boundary integral method Panel code Strip theory Radiation forces Effect o f speed Experiments A B S T R A C T T h e p a p e r p r e s e n t s a n a n a l y s i s o f t h e r a d i a t i o n f o r c e s o n a s h i p a d v a n c i n g w i t h f o r c e d h e a v e a n d p i t c h m o t i o n s . T h i s a n a l y s i s is b a s e d o n e x p e r i m e n t a l d a t a o b t a i n e d w i t h m o d e l t e s t s a n d o n n u m e r i c a l s i m u l a t i o n s w i t h t w o c o d e s , o n e b a s e d o n 3 D t i m e d o m a i n G r e e n s f u n c t i o n m e t h o d a n d t h e o t h e r is a 2 D l i n e a r s t r i p t h e o r y m e t h o d . T h e case s t u d y c o n s i s t s o f a h u l l s h a p e f r o m a Fast F e r r y a n d s e v e r a l s p e e d s a r e i n v e s t i g a t e d b e t w e e n 0 a n d 4 0 k n o t . T h e o b j e c t i v e is t o q u a n t i f y t h e f o r w a r d s p e e d e f f e c t s o n t h e r a d i a t i o n f o r c e s a n d t o v e r i f y t h e sl<ill o f t h e c o n v e n t i o n a l s t r i p t h e o r y a n d o f a t i m e d o m a i n p a n e l m e t h o d s i n d e a l i n g w i t h l a r g e f o r w a r d s p e e d s . T h e c o n s e q u e n c e s o f a p p l y i n g s i m p l i f i c a t i o n s o n t h e l i n e a r b o u n d a r y c o n d i t i o n s o f t h e c o m m o n s e a k e e p i n g codes are i n v e s t i g a t e d . I t is c o n c l u d e d t h a t f o r w a r d s p e e d has a l a r g e e f f e c t o n t h e r a d i a t i o n f o r c e s , e s p e c i a l l y o n t h e c o u p l i n g t e r m s , a n d t h u s i t is i m p o r t a n t t o c o n s i d e r t h e f u l l l i n e a r i n t e r a c t i o n s b e t w e e n t h e s t e a d y a n d u n s t e a d y f l o w s 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 s as t h e s p e e d o f t h e s h i p i n c r e a s e s a n d t h e t i m e d o m a i n p a n e l c o d e is a b l e t o r e p r e s e n t t h e s e e f f e c t s . © 2 0 1 2 E l s e v i e r L t d . A l l rights r e s e r v e d . 1. I n t r o d u c t i o n P r e d i c t i n g h y d r o d y n a m i c loads a n d m o t i o n s o f ships w h e n t h e y progress t h r o u g h a w a v e field is o f f u n d a m e n t a l i n t e r e s t . T h e s t u d y o f w a v e i n d u c e d loads a n d t h e r e l a t e d m o t i o n s c a l c u l a t i o n is i m p o r t a n t because i t is n e e d e d f o r t h e safe a n d e f f i c i e n t d e s i g n o f s h i p h u l l s . T h e r e f o r e , f o r m a n y decades, p r e d i c t i o n o f w a v e loads a n d m o t i o n s is one o f t h e m a i n research areas i n t h e field o f m a r i n e h y d r o d y n a m i c s . A l t h o u g h t h i s p r o b l e m has b e e n s t u d i e d f o r m a n y decades, a robust, e f f i c i e n t a n d c o m p l e t e s o l u t i o n is y e t t o be d e v i s e d because o f t h e c o m p l e x i t y o f t h e t h e o r y and o f t h e n u m e r i c a l s o l u t i o n . I n t h e c u r r e n t l i t e r a t u r e , m a n y t w o - d i m e n s i o n a l ( 2 D ) a n d t h r e e - d i m e n s i o n a l ( 3 D ) t h e o r i e s are a v a i l a b l e f o r t h e s o l u t i o n o f t h e s e a k e e p i n g p r o b l e m . A m o n g t h e 2 D t h e o r i e s , t h e f r e q u e n c y d o m a i n s t r i p t h e o r y d e v e l o p e d b y Salvesen et a l . ( 1 9 7 0 ) is v e r y p o p u l a r i n t h e i n d u s t r y . Fonseca a n d Guedes Soares ( 1 9 9 8 ) d e v e l o p e d a t i m e d o m a i n s o l u t i o n based o n t h e s t r i p t h e o r y o f Salvesen e t al. ( 1 9 7 0 ) w i t h t h e o b j e c t i v e o f i n c l u d i n g s o m e n o n l i n e a r e f f e c t s . T h e y s t u d i e d t h e w e l l - k n o w n S I 7 5 h u l l a n d o b t a i n e d g o o d r e s u l t s i n large a m p l i t u d e w a v e s (Fonseca a n d Guedes Soares, 2 0 0 2 , 2 0 0 4 a , b ) . I t is i n t e r e s t i n g t h a t t h i s t h e o r y w a s s t r e t c h e d t o deal w i t h v e r y large a b n o r m a l w a v e s a n d p e r f o r m e d w e l l as s h o w n b y v a r i o u s 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 results (Guedes Soares et al., 2 0 0 8 ; Fonseca e t al., 2 0 1 0 ) ,

' C o r r e s p o n d i n g author. Tel.: + 3 5 1 218417957.

E-mail address: guedess@mar.ist.ud.pt (C. Guedes Soares),

b u t h a d p r o b l e m s t o deal w i t h h i g h speeds (Fonseca a n d Guedes Soares, 2 0 0 4 c ; Fonseca et al., 2 0 0 5 a ) .

The p r o b l e m o f w a v e i n d u c e d loads i n h i g h speed ships w a s s t u d i e d i n a EU p r o j e c t t h a t c o n s i d e r e d t h r e e d i f f e r e n t t y p e s o f h u l l s a n d used v a r i o u s codes t o c o m p a r e t h e r e s u l t s as s u m -m a r i s e d b y S c h e l l i n a n d Guedes Soares ( 2 0 0 4 ) . I n t h i s p r o j e c t e x p e r i m e n t s w e r e m a d e o n a h i g h - s p e e d f e r r y , a f a s t m o n o h u l l , a n d a c o n t a i n e r s h i p . The h i g h - s p e e d f e r r y a n d t h e m o n o h u l l r e p r e s e n t e d large s i n g l e h u l l f a s t ships a n d t h e c o n t a i n e r s h i p served as a r e f e r e n c e case f o r t h e h y d r o d y n a m i c t h e o r i e s . The e x p e r i m e n t s a n d t h e c a l c u l a t i o n s f o r t h e f a s t m o n o h u l l ( M a r ó n e t a l , 2 0 0 4 ; Fonseca a n d Guedes Soares, 2 0 0 4 c ) a n d f o r t h e Fast Ferry (Fonseca et al., 2G05a) s h o w e d p r o b l e m s i n t h e p e r f o r m a n c e o f t h e code f o r h i g h speeds.

To u n d e r s t a n d t h e o r i g i n o f t h e p r o b l e m s o f t h e s t r i p t h e o r y code several s t u d i e s have b e e n m a d e l o o k i n g at t h e a s s e s s m e n t 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 a n d t h e m o t i o n o f s i m p l i f i e d m o d e l s (Fonseca et al., 1995, 1997, 2 0 0 5 b ) a n d t h e p r e s e n t s t u d y is a f o l l o w - u p , a n a l y s i n g i n d e t a i l t h e r a d i a t i o n f o r c e s i n t h e Fast F e r r y a n d c o m p a r i n g t h e m also w i t h t h e p r e d i c t i o n s o f a t i m e d o m a i n p a n e l code t h a t is able t o deal w i t h t h i s e f f e c t .

T h e o r e t i c a l l y , t h e s t r i p t h e o r y is s i m p l e a n d r e l a t i v e l y easy t o i m p l e m e n t . A l t h o u g h i t gives v e r y c o n s i s t e n t a n d n u m e r i c a l l y e f f i c i e n t r e s u l t s f o r m a n y c o n d i t i o n s , t h e m e t h o d s d l l h a v e s o m e p r o b l e m s d u e t o t h e s i m p l i f i c a t i o n s i n b o u n d a r y c o n d i t i o n s . S t r i p t h e o r y is m a i n l y a l o w speed and h i g h f r e q u e n c y t h e o r y . I t is m o r e a d e q u a t e f o r s l e n d e r ships. Hence f o r t h e c o m p u t a t i o n o f t h e n o n - s l e n d e r s h i p w i t h h i g h speed a n d l o w f r e q u e n c y s i t u a t i o n , a m o r e s o p h i s t i c a t e d t o o l is r e q u i r e d . The p a n e l m e t h o d is m o r e

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s o p h i s t i c a t e d since i t accounts f o r t h e 3 D e f f e c t s o n t h e f r e e surface a n d b o d y b o u n d a r y c o n d i t i o n s . T h e 3 D p a n e l m e t h o d b e c a m e p o p u l a r since e a r l y 80's a f t e r t h e i m p r o v e m e n t o f c o m p u t e r speed a n d m e m o r y p o w e r because i t d e m a n d s h i g h m e m o r y r e q u i r e m e n t s . D e p e n d i n g o n t h e choice o f t h e Green's f u n c t i o n , 3 D p a n e l m e t h o d s can be c l a s s i f i e d i n t h r e e basic types, one is t h e s o l u t i o n i n t h e f r e q u e n c y d o m a i n u s i n g a z e r o o r a f o r w a r d speed Green's f u n c t i o n ( G u e v e l a n d Bougis, 1 9 8 2 ) , a n o t h e r is t h e s o l u t i o n u s i n g a Rankine Green's f u n c t i o n (Nakos a n d Sclavounos, 1 9 9 0 ) , a n d t h i r d one is t h e s o l u t i o n i n t i m e d o m a i n u s i n g a t r a n s i e n t f r e e surface Green's f u n c t i o n ( K i n g , 1 9 8 7 ) .

F r e q u e n c y d o m a i n p a n e l m e t h o d w i t h zero speed Green's f u n c t i o n has g a i n e d p o p u l a r i t y i n t h e o f f s h o r e i n d u s t r y . The f i r s t c o m m e r c i a l code u s i n g t h i s a p p r o a c h is p r o b a b l y t h a t b y G a r r i s o n ( 1 9 7 8 ) , w h i l e one o f t h e m o s t w i d e l y used is p r o b a b l y W A M I T ( K o r s m e y e r et al., 1 9 8 8 ) . These codes have b e e n s u b s e q u e n t l y e n h a n c e d t o i n c l u d e second o r d e r m e a n d r i f t a n d s l o w l y v a r y i n g forces.

The t i m e d o m a i n panel m e t h o d is adequate w h e n t h e seakeeping of ships is concerned. This is m a i n l y because c o m p u t a t i o n s o f f r e q u e n c y d o m a i n Green's f u n c t i o n become e x t r e m e l y d i f f i c u l t w h e n f o r w a r d speed is considered. Also i t is n o t easy to i m p l e m e n t n o n - l i n e a r effects i n 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 . Therefore t h e t i m e d o m a i n f o r m u l a t i o n is useful i n such classes o f problems. There are f e w s i g n i f i c a n t d e v e l o p m e n t i n this d o m a i n as f o l l o w s : Liapis and Beck ( 1 9 8 5 ) , w h o i n t r o d u c e d the t i m e d o m a i n Green f u n c t i o n based s o l u t i o n m e t h o d f o r the 3 D linear f o r w a r d speed p r o b l e m , w h i l e K i n g (1987), L i n a n d Yue (1990), B i n g h a m et al. ( 1 9 9 4 ) a n d Korsmeyer a n d B i n g h a m (1998), a m o n g others, pursued variants o f the same m e t h o d f o r d i f f e r e n t classes o f 3 D f o r w a r d speed p r o b l e m s .

A l t h o u g h the t i m e d o m a i n panel theory and t h e calculation o f w a v e i n d u c e d ship m o t i o n s and loads have been w e l l developed f o r the last couple o f decades, a robust solution and i m p l e m e n t a t i o n o f the panel m e t h o d approach is still a great challenge. M o s t o f the theories w o r k e d very w e l l f o r simple structures like an hemisphere or the W i g l e y hull, b u t t h e y do not w o r k w e l l f o r t h e complicated ship hulls o f f i s h i n g vessels. T i l l date, to the best o f the authors' knowledge, no such c o n v i n c i n g evidence is reported except i n Datta et al. ( 2 0 1 1 , 2012a, b), w h i c h was able to deal w i t h short a n d n o t very slender f i s h i n g vessel hulls. This is one o f the reasons w h y s t r i p theory is still w i d e l y used by the industry, since the solution procedure is easy compared to other more sophisticated theories. A d d i n g to this, i t is k n o w n to p r o v i d e good results even f o r some h u l l shapes t h a t do n o t completely cope w i t h its restrictive assumptions.

D a t t a a n d Sen ( 2 0 0 7 ) d e v e l o p e d a h i g h e r o r d e r t i m e d o m a i n s o l u t i o n w h i c h w o r k e d w e l l f o r s i m p l i s t i c h u l l shapes, b u t f a i l e d t o p e r f o r m w h e n m o r e c o m p l i c a t e d h u l l s are c o n s i d e r e d as f o r e x a m p l e f r o m f i s h i n g vessels. D a t t a et al. ( 2 0 1 1 a ) s t u d i e d t h i s i n c o n s i s t e n c y a n d p r o p o s e d a possible s o l u t i o n t o t h i s p r o b l e m . I n t h i s a p p r o a c h , t h e y f o u n d t h a t the l o w e r o r d e r m e t h o d is m o r e a d e q u a t e f o r realistic c o m p l i c a t e d h u l l shapes. T h e y also p r o p o s e d a t e c h n i q u e t o r e f i n e t h e m e s h i n s p e c i f i c areas o f t h e h u l l s . A f t e r i n c o r p o r a t i n g such changes, t h e y s h o w e d t h a t t h e time d o m a i n a l g o r i t h m , w i t h s u c h changes, gives e f f e c t i v e results f o r the fishing vessels i n head seas. Later t h e y s t u d i e d t h e ship m o t i o n p r o b l e m m o r e r i g o r o u s l y w i t h a v a r i e t y o f ships w i t h d i f f e r e n t h e a d i n g angles a n d speeds a n d o b t a i n e d s i m i l a r r e s u l t s ( D a t t a et al., 2 0 1 2 a , b ) . One o f t h e s t r e n g t h s o f t h i s t i m e d o m a i n p a n e l m e t h o d is t h e c o n s i s t e n t t r e a t m e n t o f t h e f r e e surface b o u n d a r y c o n d i t i o n a n d b o d y k i n e m a t i c b o u n d a r y c o n d i t i o n . I n fact, besides t h e l i n e a r i z a -tion, t h e m e t h o d assumes n o f u r t h e r s i m p l i f i c a t i o n s o n these c o n d i t i o n s . For t h i s reason t h e m e t h o d solves t h e c o m p l e t e l i n e a r s o l u t i o n o f t h e s e a k e e p i n g p r o b l e m . I n c o n t r a s t , t h e e x i s t i n g

f r e q u e n c y d o m a i n pa f i e l m e t h o d s need t o i n t r o d u c e s o m e s i m -p l i f i c a t i o n o n t h e l i n e a r b o u n d a r y c o n d i t i o n s i f r e l i a b l e n u m e r i c a l results are t o be o b t a i n e d f o r r e a l i s t i c h u l l shapes. This need is r e l a t e d t o t h e i n t e r a c t i o n s b e t w e e n t h e s t e a d y flow associated w i t h t h e ships' f o r w a r d speed a n d t h e o s c i l l a t o r y flow d u e t o w a v e s a n d ship m o t i o n s . Such i n t e r a c t i o n s are r e l a t i v e l y s i m p l e t o c o n s i d e r i n t h e t i m e d o m a i n , b u t v e r y c o m p l e x i n t h e f r e q u e n c y d o m a i n . The s i m p l e r f r e q u e n c y d o m a i n p a n e l m e t h o d s c o n s i d e r t h e f o r w a r d speed e f f e c t s i n a w a y s i m i l a r t o t h a t used b y c o n s i s t e n t s t r i p t h e o r i e s ( P a p a n i k o l a o u a n d S c h e l l i n , 1 9 9 2 ) .

A r e l a t i v e l y h i g h f o r w a r d speed case is i d e a l t o test t h e a d v a n t a g e s o f t h e p r e s e n t t i m e d o m a i n p a n e l m e t h o d . T h e r e f o r e , i n t h e p r e s e n t paper, one Fast F e r r y is c o n s i d e r e d a n d e x p e r i -m e n t a l r e s u l t s f r o -m -m o d e l tests are c o -m p a r e d w i t h t i -m e d o -m a i n code d e v e l o p e d by D a t t a et al. ( 2 0 1 1 a ) . The F r o u d e n u m b e r s t e s t e d go f r o m 0 u p t o 0.6, t h e r e f o r e c l e a r i y o u t s i d e t h e range o f " s m a l l F r o u d e n u m b e r " a s s u m p t i o n . F u r t h e r m o r e , t h e Fast F e r r y h u l l shape is a c h a l l e n g e f o r t h e robustness o f t h e n u m e r i c a l m e t h o d . This p o i n t is discussed f u r t h e r ahead i n t h e t e x t .

I n t h i s w o r k t h e m a j o r p a r t o f t h e analysis is f o c u s e d o n t h e h y d r o d y n a m i c s o f t h e ship's f o r c e d m o t i o n p r o b l e m , m e a n i n g o n t h e h y d r o d y n a m i c r a d i a t i o n forces w i t h f o r w a r d speed. T h e a d v a n t a g e s a n d l i m i t a t i o n s o f the s e a k e e p i n g f o r m u l a t i o n s are m u c h b e t t e r assessed b y a n a l y s i n g p a r t i a l r e s u l t s , l i k e t h e r a d i a -tion forces, t h a n a n a l y s i n g g l o b a l r e s u l t s l i k e t h e o s c i l l a t o r y s h i p m o t i o n s . G l o b a l m o t i o n s m a y c o m p a r e w e l l w i t h e x p e r i m e n t s even w h e n t h e r a d i a t i o n ( o r d i f f r a c t i o n ) forces are i n a c c u r a t e l y c a l c u l a t e d . T h e r e f o r e , instead o f c o m p u t i n g o n l y o s c i l l a t o r y m o t i o n s i n w a v e s , d u e t o t h e a v a i l a b i l i t y o f t h e e x p e r i m e n t a l data, r a d i a t i o n f o r c e r e s u l t s are also c o m p a r e d f o r v a r i o u s speeds. A d d i t i o n a l l y , t h e i n f l u e n c e o f the s t e a d y h y d r o d y n a m i c forces d u e t o f o r w a r d speed is discussed. This is a g a i n one b e n e f i t o v e r t h e f r e q u e n c y d o m a i n s o l u t i o n as i n f r e q u e n c y d o m a i n s o l u t i o n t h e i n f l u e n c e o f steady forces is u s u a l l y i g n o r e d i n t h e c a l c u l a t i o n o f t h e f o r c e d m o t i o n p r o b l e m . The c o m p a r i s o n w i t h e x p e r i m e n t a l results s h o w t h e e f f i c i e n c y o f t h e p r e s e n t t i m e d o m a i n f o r m u l a -tion e v e n f o r t h e c o m p l i c a t e d s h i p h u l l s , w h i c h is e n c o u r a g i n g . The e x p e r i m e n t a l data a n d t i m e d o m a i n p a n e l m e t h o d r e s u l t s are c o m p a r e d w i t h s t r i p t h e o r y p r e d i c t i o n s based o n t h e m e t h o d o f Salvesen et al. ( 1 9 7 0 ) . This is o f i n t e r e s t t o d e m o n s t r a t e t h e consequences o f s i m p l i f y i n g t h e f o r w a r d speed e f f e c t s o n t h e b o u n d a r y c o n d i t i o n s , as w e l l as o f a s s u m i n g h i g h f r e q u e n c y o n t h e f r e e s u r f a c e c o n d i t i o n . M a r ó n e t al. ( 2 0 0 4 ) describe t h e e x p e r i m e n t a l p r o g r a m m e i n w h i c h t h e r a d i a t i o n forces p r e s e n t e d i n t h e p r e s e n t p a p e r w e r e m e a s u r e d . The e x p e r i m e n t a l setup a n d t h e p r o c e d u r e s f o r a n a l y -sis o f t h e signals a n d e s t i m a t i o n o f t h 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 are p r e s e n t e d i n d e t a i l b y M a r ó n e t a l . ( 2 0 0 4 ) . T h e n t h e i r analysis o f e x p e r i m e n t a l data is f o c u s e d o n :

(a) The n o n l i n e a r e f f e c t s o n the r a d i a t i o n forces, n a m e l y t h e m e a n v a l u e s a n d t h e h i g h e r h a r m o n i c c o n t e n t o f t h e m e a s u r e d signals, ( b ) The s t e a d y f o r w a r d speed effects, n a m e l y t h e m e a n r a d i a t i o n forces a n d t h e q u a s i - s t e a d y r a d i a t i o n f o r c e s .

The analysis o f t h e e x p e r i m e n t a l data d e s c r i b e d i n t h e p r e s e n t p a p e r is n e w a n d consists o f d e t e r m i n i n g t h e l i n e a r c o m p o n e n t s o f t h e r a d i a t i o n f o r c e i n : heave d u e t o heave, heave d u e t o p i t c h , p i t c h d u e t o heave a n d p i t c h d u e t o p i t c h , a l l f o r f o u r F r o u d e n u m b e r s . The paper f r o m M a r ó n et al. ( 2 0 0 4 ) p r e s e n t s o n l y o n e i l l u s t r a t i v e e x a m p l e w h i c h is t h e heave r a d i a t i o n f o r c e d u e t o heave f o r one Froude n u m b e r a n d t h e o b j e c t i v e w a s t o s h o w t h a t t h e n o n l i n e a r e f f e c t s o n t h e first h a r m o n i c a m p l i t u d e s w e r e s m a l l .

T h e n e w c o n t r i b u t i o n s f r o m t h i s paper are:

(a) N e w set o f t h e first h a r m o n i c r a d i a t i o n forces i n t h e e x p e r i -m e n t a l data f o r a Fast Ferry w i t h d i f f e r e n t f o r w a r d speeds o f u p t o 4 0 k n .

(3)

(b) A demonstration, by comparison w i t h experimental data, o f the consequences o f s i m p l i f y i n g the f o r w a r d speed effects o n the free surface a n d body boundary conditions o n the seakeeping f o r -mulations. A l t h o u g h this topic has been discussed before, even since several decades ago, there are only a f e w 3D codes w h e r e the linear seakeeping p r o b l e m is f u l l y solved ( w i t h o u t s i m p l i f i c a -tions i n t h e linear boundary condi-tions) and comparisons w i t h the results f r o m this type o f codes w i t h measured radiation forces is scarce. Furthermore there are no comparisons i n the literature for t h e t y p e o f realistic fast hull presented here.

2. M a t h e m a t i c a l f o r m u l a t i o n a n d n u m e r i c a l i m p l e m e n t a t i o n o f the p a n e l c o d e

In the m a t h e m a t i c a l f o r m u l a t i o n , o n l y a brief description o f the general f o r m u l a t i o n is given, as a detailed discussion is available i n other sources (Lin and Yue, 1990). The w h o l e f o r m u l a t i o n is based o n earth f i x e d ordinate system. Let ( 0 , x, y, z) be the earth fixed co-ordinate system w h e r e o r i g i n is placed i n m e a n w a t e r surface w i t h z positive u p w a r d and ( 0 , x', y , z") be the b o d y fixed system w h e r e the o r i g i n o f t h e b o d y fixed system is situated at the w a t e r l i n e (at z = 0 ) above CG o f the ship. If the vessel is m o v i n g w i t h the v e l o c i t y 11 i n the direction o f positive x axis, t h e n b o t h the co-ordinate system is related by t h e f o l l o w i n g relation:

x = x! + Ut; y = y'; z = z' (1)

For t h e f o r m u l a t i o n o f t h e h y d r o d y n a m i c p r o b l e m , a s s u m e t h a t a l i n e a r ( s i n u s o i d a l ) w a v e is a p p r o a c h i n g an a n g l e a w i t h t h e p o s i t i v e X axis. T h e n t h e fluid m o t i o n can be d e f i n e d b y t h e v e l o c i t y p o t e n t i a l < / , r ( X ; f ) = < / . , ( X ; f ) + 0 ( X ; t ) (2) w h e r e 0 , ( ^ X ; t j is d e n o t e d as i n c i d e n t w a v e p o t e n t i a l a n d et>(X;t^ = •,i^-ipi{x -,1^ is c a l l e d t o t a l d i s t u r b e d p o t e n t i a l . The t o t a l d i s t u r b e d p o t e n t i a l 4>(X;tj satisfies 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 s , b o u n d a r y c o n d i t i o n , r a d i a t i o n c o n d i t i o n s a n d i n i t i a l c o n d i t i o n s : V^()!>(>f;t) = 0 ; X eQ dz : 0 o n z = 0 ^ = I / , , - o n So dn dn Vtp-^O as R H - » O O . o n z = 0 ^ , V ( ^ - * 0 a s t ^ O (3) (4) (5) (6) (7) I n t h e a b o v e expressions, Ü represents t h e fluid d o m a i n . So is t h e m e a n w e t t e d b o d y surface, V„ is t h e n o r m a l v e l o c i t y o f t h e b o d y surface, g is t h e a c c e l e r a t i o n due to g r a v i t y . For t h e r a d i a t i o n p r o b l e m , t h e b o u n d a r y c o n d i t i o n ( 5 ) is m o d i f i e d as f o l l o w s : If = ü n , - H / l ( t ) ï ï f c f o r ;<:= 1,2,3 : U n i - l - / l ( t ) ( f x % _ 3 for;<: = 4,5,6 o n So (8) I n Eq. ( 8 ) , n is t h e u n i t n o r m a l v e c t o r a n d A ( t ) r e p r e s e n t s t h e time h a r m o n i c i m p u l s i v e f u n c t i o n .

The s o l u t i o n o f the above initial boundary value p r o b l e m is based on the transient Green's f u n c t i o n approach o f Lin a n d Yue (1990).

The disturbed potential can be obtained by f o l l o w i n g t w o integral equations: éiP.t) = - ^ • / a i q . t ) G ' ' ( p , q ) d S + dr JJff(q,t)C{(p,q; t - T ) d S U o ( t ) - 1 / aiq,t)G{(p.q;t-i)VNV„dL SJrm D i f f e r e n t i a t i n g (9) w i t h respect to n gives 84>{p\t) 1 J ff„,„,,SG°ip.q)^ (9) t)-dn„ '-dS 1 So(0 / / , ( q , 0 ^ f e ^ d S JJ onp a G { ( p , q ; t - T ) , I f a(q,t)-"^::'-''V,V„dL g Jrm (10)

Eq. (9) represents the t o t a l disturbed potential (p at any point p at any instant o f t i m e t i n terms o f the d i s t r i b u t i o n o f sources over the b o d y surface a n d w a t e r i i n e contour. Here, q=q{^,i]Z) is t h e location o f a p o i n t source o n the b o d y surface So, cr(q,T) is the associated source strength a t . r e t a r d e d time t , and G°(p,g) and (f(p,q;t~x) are t h e Rankine a n d regular parts o f the transient free surface Green's f u n c t i o n G(p,t,q,T):

G(p,t,q,T) = G°-i-G-''with p T ^ q , t > T (11) r(T)represents the curve d e f i n e d b y t h e instantaneous intersection o f

the hull surface w i t h t h e z = 0 plane, a n d corresponds to the t w o dimensional n o r m a l velocity i n the z = 0 plane o f a p o i n t o n F. and V,, are related b y = ( V n / N ) ri".

For t h e n u m e r i c a l s o l u t i o n o f t h e above p r o b l e m , a l o w e r o r d e r p a n e l m e t h o d a p p r o a c h is t a k e n . T h e details s o l u t i o n process are g i v e n b y Sen ( 2 0 0 2 ) , D a t t a a n d Sen ( 2 0 0 7 ) a n d D a t t a e t al. ( 2 0 1 1 ) , a n d hence n o t r e p e a t e d here.

The m e t h o d p r o p o s e d b y D a t t a a n d Sen ( 2 0 0 7 ) faced s o m e n u m e r i c a l i n s t a b i l i t y f o r t h e h y d r o d y n a m i c s o l u t i o n w h e n t h e m o t i o n is s t u d i e d f o r c o m p l i c a t e d h u l l s l i k e fishing vessels. D a t t a e t a l . ( 2 0 1 1 ) p r o p o s e d a s o l u t i o n s c h e m e t o r e c o v e r f r o m t h e n u m e r i c a l i n s t a b i l i t y f o r t h e h y d r o d y n a m i c s o l u t i o n . The s o l u t i o n is g i v e n i n t e r m s o f m o d i f i c a t i o n o f t h e m e s h n e a r free surface. I n t h e p r e s e n t paper, t h e s a m e s c h e m e is used a n d hence t h e d e s c r i p t i o n o f the s c h e m e is n o t r e p e a t e d here.

3. L i n e a r f r e q u e n c y d o m a i n s o l u t i o n s

W h i l e the t i m e d o m a i n m e t h o d presented i n the previous section is f o r m u l a t e d i n an earth fixed coordinate system, the linear f r e q -uency d o m a i n seakeeping p r o b l e m is usually f o r m u l a t e d i n an i n e r t i a l reference system w h i c h advances w i t h the ship constant f o r w a r d speed. I n t h e scope o f the present section only, this m o v i n g reference system is defined as (0,x,y,z). The reference system is parallel to t h e earth fixed reference system and also parallel t o the b o d y fixed reference system w h e n t h e ship has no oscillatory motions.

There is a n o t h e r i m p o r t a n t d i f f e r e n c e b e t w e e n t h e time d o m a i n a n d t h e 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 s : w h i l e t h e first solves t h e b o u n d a r y v a l u e p r o b l e m d i r e c t l y f o r t h e t o t a l l i n e a r p e r t u r b a t i o n p o t e n t i a l , w i t h i n t h e f r e q u e n c y d o m a i n a p p r o a c h the p e r t u r b a t i o n p o t e n t i a l is d e c o m p o s e d i n t o steady, cj), p l u s o s c i l l a t o r y p o t e n t i a l s a n d t h e s e c o n d is f u r t h e r d e c o m p o s e d i n t o

(4)

r a d i a t i o n , 0'* and d i f f r a c t i o n , 0°, p o t e n t i a l s :

(12)

The i n c i d e n t w a v e s are a s s u m e d h a r m o n i e and, w i t h i n t h e l i n e a r a p p r o a c h , t h e o s c i l l a t o r y p o t e n t i a l a n d a l l s h i p r e l a t e d responses are h a r m o n i c as w e l l . T h e l i n e a r b o u n d a r y v a l u e p r o b l e m , e q u i v a l e n t t o t h e t i m e d o m a i n one p r e s e n t e d b y Eqs. ( 3 ) - ( 7 ) w h i c h is e s t a b l i s h e d b y : X ; t = 0 ; X eQ .^ \ 2 iR.D dn dn •• iomj+Unij, j = 1 , . , „ 6 , o n So

50'

'dn' o n So R a d i a t i o n c o n d i t i o n as RH - » o o , o n z = 0 (13)

(14)

(15) (16) (17)

w h e r e i is the i m a g i n a r y u n i t n u m b e r , co is the frequency o f oscillation, h is the generalised o u t w a r d u n i t vector perpendicular t o the h u l l surface So and the radiation c o n d i t i o n is imposed so t h a t the waves generated b y the body radiate a w a y to i n f i n i t y {Rn is the radial distance w i t h respect to the body). The ifi vector represents, i n a condensed way, the linear interactions b e t w e e n the steady a n d oscillatory f l o w s i n t h e body 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 for/< = n = m = rik ^1,2,3 (r X n))(_3 for/(: = 4 , 5 , 6 ( m „ m 2 , m 3 ) = - t ™ (18) (19)

I n Eqs. ( 1 2 ) and (19) <ji represents the steady p o t e n t i a l related to the ship constant f o r w a r d speed, a n d this is f u r t h e r decomposed i n t o </> = - U n , + ( / ) s w h e r e Un, is the p o t e n t i a l due t o the u n i f o r m i n c i d e n t f l o w and (f)^ represents the m o d i f i c a t i o n o f the u n i f o r m f l o w due to the ship's hull.

The e q u a t i o n s above s h o w t h a t t w o b o u n d a r y v a l u e p r o b l e m s n e e d t o be s o l v e d t o o b t a i n t h e u n k n o w n r a d i a t i o n a n d d i f f r a c t i o n p o t e n t i a l s . T h e r a d i a t i o n p r o b l e m is r e l a t e d t o t h e h u l l a d v a n c i n g w i t h c o n s t a n t speed a n d f o r c e d h a r m o n i c m o t i o n s i n o t h e r w i s e c a l m w a t e r ( n o i n c i d e n t w a v e s ) . T h e d i f f r a c t i o n p r o b l e m is r e l a t e d t o t h e h u l l a d v a n c i n g t h r o u g h t h e i n c i d e n t h a r m o n i c w a v e f i e l d a n d r e s t r a i n e d i n t h e m e a n p o s i t i o n ( n o o s c i l l a t o r y m o t i o n s ) .

There are t w o i m p o r t a n t differences b e t w e e n the b o u n d a r y value p r o b l e m s f o r m u l a t e d under Sections 2 and 3: (a) the time d o m a i n f o r m u l a t i o n considers the steady effects due t o f o r w a r d speed directly i n the disturbed velocity potential and therefore i n the body b o u n d a r y and free surface boundary conditions, w h i c h means t h a t the steady a n d oscillatory problems are solved simultaneously, w h i l e the f r e -quency d o m a i n approach decomposes the p o t e n t i a l i n t o steady and oscillatory potentials and the b o u n d a i y value p r o b l e m is f o r m u l a t e d f o r t h e oscillatory potential only; (b) the d i f f r a c t i o n effects i n the time d o m a i n f o r m u l a t i o n are coupled to the oscillatory m o t i o n s o f the h u l l , w h i l e the frequency d o m a i n approach assumes t h a t the radiation and d i f f r a c t i o n problems are decoupled m e a n i n g t h a t the d i f f r a c t i o n forces are independent o f the ship oscillatory m o t i o n s .

Some a u t h o r s s o l v e d t h e l i n e a r p r o b l e m i n t h e f r e q u e n c y d o m a i n as presented i n Eqs. ( 1 2 ) - ( 1 7 ) , b o t h a p p l y i n g t h e Green's f u n c t i o n p a n e l m e t h o d s (as f o r e x a m p l e I n g l i s a n d Price, 1 9 8 1 a , b ) a n d t h e R a n k i n e p a n e l m e t h o d s ( N a k o s et al., 1 9 9 3 ) .

A l t h o u g h the p r o b l e m stated above is f u l l y linear, the n u m e r i c a l solution is still d i f f i c u l t to obtain and f o r this reason m a n y o f the existing seakeeping codes w h i c h are used f o r practical applications s i m p l i f y f u r t h e r the b o u n d a r y conditions. The objective is t o s i m p l i f y the interactions b e t w e e n the steady and oscillatory fiows and the threedimensional effects o f the flow. The s i m p l i f i c a t i o n s are i n t r o -duced b y i m p o s i n g restrictions o n some parameters g o v e r n i n g the solution, n a m e l y : the speed o f the ship, the slendemess o f the h u l l and the f r e q u e n c y o f oscillation o f the boundaries. T h e selection and c o m b i n a t i o n o f simplifications t o introduce d i s t i n g u i s h the d i f f e r e n t seakeeping theories. T h e b o d y b o u n d a r y c o n d i t i o n c a n be f u r t h e r s i m p l i f i e d b y a s s u m i n g t h a t t h e s h i p is ( v e r y ) s l e n d e r a n d t h e s t e a d y p o t e n t i a l is s i m p l i f i e d t o = - U n , , m e a n i n g t h a t t h e m o d i f i c a t i o n o f t h e u n i f o r m flow d u e t o t h e h u l l can be n e g l e c t e d . W i t h t h i s a s s u m p t i o n t h e g r a d i e n t s o f t h e s t e a d y p o t e n t i a l are v e r y s i m p l e t o c a l c u l a t e a n d t h e Eq. ( 1 9 ) is s i m p l i f i e d t o : m = ( r n i , m 2 , m 3 ) = (0,0,0) ( m 4 , m 5 , m 6 ) = ( 0 , - n 3 , n 2 ) (20)

This a p p r o a c h w a s used b y C h a n g ( 1 9 7 7 ) , w h o first s o l v e d t h e seakeeping p r o b l e m a p p l y i n g the p a n e l m e t h o d a n d t h r e e d i m e n -s i o n a l Green'-s f u n c t i o n -s . Several o t h e r a u t h o r -s f o l l o w e d a -s i m i l a r a p p r o a c h . I n o r d e r t o s i m p l i f y t h e l o n g i t u d i n a l g r a d i e n t s o f t h e o s c i l l a -t o r y p o -t e n -t i a l s i n -t h e f r e e surface b o u n d a r y c o n d i -t i o n ( 1 4 ) a s -t r i p t h e o r y a s s u m p t i o n is o f t e n used, n a m e l y t h e h i g h f r e q u e n c y a s s u m p t i o n . T h e r e s u l t i n g c o n d i t i o n is. ,R.D dz : 0 , o n z = 0 ( 2 1 ) T h e p o t e n t i a l is s t i l l 3 D , h o w e v e r t h e f o r w a r d s p e e d e f f e c t s have b e e n r e m o v e d f r o m t h e f r e e s u r f a c e c o n d i t i o n b y a s s u m i n g t h e f r e q u e n c y is h i g h ( a n d t h e s h i p speed is l o w ) . T h i s s o l u t i o n has b e e n a p p l i e d , f o r e x a m p l e , b y P a p a n i k o l a o u a n d S c h e l l i n ( 1 9 9 2 ) , S c h e l l i n a n d R a t h j e ( 1 9 9 5 ) a n d Chan ( 1 9 9 3 ) .

There is a s m a l l step between the b o u n d a r y c o n d i t i o n s ( 1 8 ) and (19) and the ones used b y consistent strip t h e o r y m e t h o d s (as f o r example t h e one f r o m Salvesen et al., 1970), w h i c h is t h e a s s u m p t i o n t h a t the u n i t n o r m a l vector to the h u l l surface, w h i c h is at this p o i n t already assumed as (very) slender, is t w o - d i m e n s i o n a l , t h e r e f o r e the l o n g i t u d i n a l c o m p o n e n t can be neglected. The l i m i t a t i o n s o f the panel methods w h i c h use Eqs. ( 1 8 ) and ( 1 9 ) and strip theories are s i m i l a r and this is t h e reason w h y the predicted m o t i o n s a n d global structural loads i n waves are similar as w e l l .

T h e m o s t s i m p l i f i e d level o f accuracy consists o n u s i n g t h e f r e q u e n c y o f e n c o u n t e r a p p r o a c h , m e a n i n g t h a t t h e m v e c t o r o f Eq. ( 1 8 ) is zero i n 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 a n d t h e f r e e surface c o n d i t i o n is g i v e n b y ( 1 9 ) . T u i t m a n a n d M a l e n i c a ( 2 0 0 9 ) a p p l i e d t h i s a p p r o a c h , t o g e t h e r w i t h a 3 D p a n e l m e t h o d a n d zero speed Green's f u n c t i o n , t o calculate t h e c o u p l e d s e a k e e p i n g a n d w h i p p i n g responses o f ships.

I n t h e n e x t sections a s t r i p t h e o r y code based o n t h e f o r m u l a -t i o n f r o m Salvesen e-t a l . ( 1 9 7 0 ) is used -t o c o m p u -t e -t h e h e a v e a n d p i t c h r a d i a t i o n f o r c e s a n d c o m p a r e w i t h t h e e x p e r i m e n t a l d a t a a n d w i t h t h e time d o m a i n panel code results. T h e o b j e c t i v e is t o d e m o n s t r a t e t h e l i m i t a t i o n s o f i n t r o d u c i n g t h e s i m p l i f i c a t i o n s o n t h e l i n e a r b o u n d a r y c o n d i t i o n s , n a m e l y t h o s e r e l a t e d t o t h e i n t e r a c t i o n s b e t w e e n t h e s t e a d y a n d t h e u n s t e a d y p o t e n t i a l s a n d t o v e r i f y h o w t h e 3 D p a n e l code deals w i t h t h e p r o b l e m . 4. E x p e r i m e n t a l p r o g r a m m e w i t h a F a s t F e r r y m o d e l T h e e x p e r i m e n t a l p r o g r a m m e w a s c a r r i e d o u t at t h e El Pardo M o d e l Basin (CEHIPAR) a n d i t i n c l u d e d f o r c e d m o t i o n tests t o

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L e n g t h o v e r a l l ( m ) 125.0 L e n g t h b e t w e e n perpendiculars ( m ) 110.0 L e n g t h w a t e r l i n e ( m ) 110.0 B e a m m a x ( m ) 18.7 B e a m w a t e r l i n e ( m ) 14.7 D r a u g h t ( m ) 2.80 T r i m ( m ) 0 . 0 0

Fig. 1. Fast Ferry bodylines.

o b t a i n t h e r a d i a t i o n forces a n d f r e e m o t i o n tests i n i n c i d e n t r e g u l a r w a v e s . A d e s c r i p t i o n o f t h e s e t u p is g i v e n i n M a r ó n et a l . ( 2 0 0 4 ) , a l t h o u g h i n t h e c o n t e x t o f t e s t i n g a n o t h e r s h i p h u l l . Four d i f f e r e n t speeds w e r e c o n s i d e r e d f o r b o t h g r o u p s o f tests. Some h y d r o d y n a m i c c o e f f i c i e n t results a n d v e r t i c a l m o t i o n s have been p r e s e n t e d b y Fonseca et a l . ( 2 0 0 5 a ) . M o s t o f t h e f o r c e d m o t i o n tests w e r e p e r f o r m e d at t h e S t i l l W a t e r Tank, n a m e l y a l l those w i t h f o r w a r d speed. Zero speed tests w e r e c a r r i e d o u t a t t h e L a b o r a t o r y o f Ship D y n a m i c s , w h o s e t a n k is w i d e r a n d i n t h i s w a y t h e w a l l e f f e c t s can be m i n i m i s e d . E x p e r i m e n t s i n w a v e s w e r e c o n d u c t e d at t h e L a b o r a t o r y o f Ship D y n a m i c s as w e l l . T h e S t i l l W a t e r T a n k is 3 2 0 m l o n g , 12.5 m w i d e a n d 6.5 m deep, w h i l e t h e L a b o r a t o r y o f Ship D y n a m i c s has a t a n k 145 m l o n g f r o m t h e face o f t h e w a v e g e n e r a t o r t o t h e i n t e r s e c t i o n o f t h e beach w i t h t h e w a t e r f r e e surface, 3 0 m w i d e a n d 5 m deep.

The e x p e r i m e n t s w e r e c a r r i e d o u t w i t h a m o d e l o f a Fast Ferry. The s h i p is 125.0 m o v e r a l l a n d has service speed o f 4 4 k n o t . Fig. 1 presents t h e h u l l b o d y l i n e s a n d T a b l e 1 t h e s h i p m a i n p a r t i c u l a r s . The m o d e l w a s c o n s t r u c t e d i n fibre r e i n f o r c e d plastic (FRP), t h e l e n g t h is 4 . 4 m b e t w e e n p e r p e n d i c u l a r s (scale o f 1:25) a n d i t is m a d e u p o f f o u r s e g m e n t s c o n n e c t e d by a r i g i d a l u m i n i u m b a c k b o n e . The cuts b e t w e e n s e g m e n t s are l o c a t e d i n s e c t i o n 5, 10 a n d 1 5 . V e r t i c a l cross s e c t i o n a l loads w e r e m e a s u r e d at t h e s e p o s i t i o n s . The n e u t r a l axis o f t h e " a l m o s t r i g i d " b a c k b o n e is a p p r o x i m a t e l y at t h e w a t e r l i n e l e v e l .

One observes t h a t t h e s h i p h u l l has a t r a n s o m stern, w h i c h f o r ships w i t h f o r w a r d s p e e d i n t r o d u c e s local viscous e f f e c t s , n a m e l y t h e flow s e p a r a t i o n a t t h e h a r d c h i n e . I n t h e p r e s e n t p a p e r t h e m a i n f o c u s i n o n p o t e n t i a l flow r a d i a t i o n forces, t h e r e f o r e t h e t r a n s o m s t e r n v i s c o u s e f f e c t s are n o t t a k e n i n t o a c c o u n t n o r discussed. The f o r c e d o s c i l l a t i o n s w e r e p r o d u c e d b y m e a n s o f t w o v e r t i c a l l i n e a r a c t u a t o r s . T h e a f t a c t u a t o r w a s i n s t a l l e d at s t a t i o n 7'/2, u s i n g t h e usual r e f e r e n c e s y s t e m w i t h s t a t i o n 0 at t h e a f t p e r p e n d i c u l a r a n d 2 0 a t t h e f o r w a r d p e r p e n d i c u l a r . This a c t u a t o r w a s c l a m p e d t o t h e t o w i n g carriage so t h a t i t r e m a i n e d a l w a y s Table 1

Fast Ferry m a i n characteristics.

Fast Ferry

Length overall Loa(m) 125.0

Length betwfeen perp. L(m) 110.0

Breadth overall B(m) 18.7

Depth D(m) 16.7

Draught T(m) 2.80

Displacement d(ton) 2267

Service speed V(lin) 40

Longitudinal position of CG LCC (m) - 1 1 . 4 7

Vertical posidon of CG VCC (m) 4.475

Pitch radiation o f gyr. Kyy/ipp 0.248

v e r t i c a l p r o v i d i n g t h e t o w i n g f o r c e necessaiy t o m a i n t a i n t h e m o d e l at speed. The f o r w a r d a c t u a t o r w a s l o c a t e d a t s t a t i o n 121/2. T h e c o n n e c t i o n t o t h e carriage w a s t h r o u g h a h i n g e a l l o w i n g s o m e r o t a t i o n i n t h e l o n g i t u d i n a l v e r t i c a l p l a n e s u c h t h a t t h e m o d e l c o u l d p i t c h . Fig. 2 s h o w s a s k e t c h w i t h t h e m o d e l a n d t h e p o s i t i o n o f t h e actuators, as w e l l as a l l t h e i n s t r u m e n t a t i o n used d u r i n g t h e tests, n a m e l y l o a d cells t o m e a s u r e t h e forces o n t h e a c t u a t o r s , p o t e n t i o m e t e r s t o m e a s u r e t h e f o r c e d m o t i o n s a n d l o a d cells i n s t a l l e d a t t h e b a c k b o n e f o r t h e s t r u c t u r a l loads at t h e s e g m e n t e d cross sections.

For t h e tests i n waves t h e m o d e l w a s t o w e d b y a s y s t e m c o n n e c t e d t o t h e carriage w h i c h a l l o w s f r e e heave, p i t c h a n d r o l l m o t i o n s . O n l y head waves w e r e t e s t e d a n d t h e m o t i o n s w e r e m e a s u r e d w i t h an o p t i c a l s y s t e m .

5. R e s u l t s a n d d i s c u s s i o n

5. Ï. Radiation force for a lieinispliere

I n i t i a l l y , t o check t h e correctness o f t h e p r o p o s e d s c h e m e , heave r a d i a t i o n f o r c e o f a h e m i s p h e r e is c o m p u t e d f o r a heave

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f o r c i n g f u n c t i o n (F33) f o r u n i t a m p l i t u d e m o t i o n . T h e f o r c e r e s u l t s are c o m p u t e d f o r a r a n g e o f f r e q u e n c i e s a n d t h e n c o m p a r e d w i t h WAIVIIT. The f o r c e results a n d f r e q u e n c i e s are n o n -d i m e n s i o n a l i s e -d b y |F33|/pgzl a n -d kR, r e s p e c t i v e l y , w h e r e I F 3 3 I r e p r e s e n t s t h e r a d i a t i o n f o r c e a m p l i t u d e , A r e p r e s e n t s t h e u n d e r -w a t e r v o l u m e , k a n d R represents t h e -w a v e n u m b e r a n d radius o f t h e h e m i s p h e r e , r e s p e c t i v e l y . Fig. 4 c o n f i r m s t h e e x c e l l e n t agree-m e n t b e t w e e n t h e i n d u s t r i a l l y accepted W A M I T code a n d t h e p r e s e n t t i m e d o m a i n p a n e l code. F r o m t h e figure, i t m a y be n o t e d t h a t p r e s e n t s c h e m e is w o r k i n g w e l l a n d p r o d u c e v e r y g o o d results f o r t h e h e m i s p h e r e . 0.2 0.15 0.1 0.05 0 -0.05 -0.1 -0.15 -0.2 Heave F o r c e History T T X 4

\ ,/ T7TT ;

_{l V 120 / 3'o\ / 4 ' o 50}

VT V

- T P time history |

5.2. Radiation force for the Fast Ferry

Fig. 3 r e p r e s e n t s t h e i n i t i a l m e s h o f t h e Fast F e r r y ( w i t h o u t i n t r o d u c i n g t h e v e r t i c a l p a n e l ) a n d i t can be seen t h a t t h e h u l l is n o t s i m p l i s t i c . The h u l l is h i g h l y u n s y m m e t r i c a l o n b o w a n d s t e r n parts, w i t h "v" shaped cross s e c t i o n o n t h e b o w w h i c h g r a d u a l l y b e c o m e flatter a n d s h a l l o w d r a f t n e a r t h e s t e r n ( f o r s e m i -p l a n n i n g c h a r a c t e r i s t i c s ) . The d e t a i l s o f t h e h u l l are g i v e n i n Table 1. F r o m t h e table, one m a y n o t i c e t h a t t h e l o n g i t u d i n a l p o s i t i o n o f t h e c e n t r e o f g r a v i t y (LCG) o f t h e s h i p is 11.5 m a f t o f m i d s h i p , w h i c h s i m p l y c o n f i r m t h e h i g h a s y m m e t r y l e v e l o f t h e h u l l .

5.2. J. Steady radiation force

I n i t i a l l y , t h e t i m e h i s t o r y f o r t h e f o r c e d m o t i o n is d e m o n -s t r a t e d f o r t w o d i f f e r e n t -speed-s ( i n t h i -s ca-se, 0 a n d 4 0 k n o t w h i c h

Fig. 5. Heave radiation force t i m e history f o r Fast Ferry due to Heave forcing f u n c t i o n . Zero speed case, /i:i=2.58.

S 0.015 E I 0.01 S 0.005 K n 0 c • i -0.005 -0.01 -0.015

Pitch Moment History

A A A 1 TD time History] A / \ / \ / \ 1 TD time History] \ / \ \ \ / \ - / \ / 1 TD time History] ) \ ^h \ 20 / 3^ / 40 5 3

\ / \ / \ /

3 A A 3

Fig. 6. Pitch radiation m o m e n t time history for Fast Ferry due to p i t c h f o r c i n g f u n c t i o n . Zero speed case, l<L=2.58.

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0.6 0,5 0.4 0.3 0.2 0.1 O -0.1 -0.2 -0.3 -0.4 -0.5 H e a v e f o r c e h i s t o r y ( 4 0 k n o t )

n

A . - Without removing steady contribution " removing steady contribution

Fig. 7. Heave radiation force t i m e history f o r Fast Ferry due to Heave forcing f u n c t i o n . 40 l<not speed, k f , = 2 . 5 8 . 0.25 O O u . 0.2 (1) >

ro

0.15 a> I _0.1 "rö O 0.05 'üi c OJ 0 e '•5 c -0.05 O z _-0.1

Steady heave force history

10 20 30 40 6 »

- steady heaveforcel

Fig. 9. Steady heave force, 40 l<not speed.

g

0.05

1

0 -0.05 -0.1 -0.15 -0.2

Pitch moment hlstory(40 knot)

I^Ai'-,^ 10 / • ' A ^ O 4 ö ~ I^Ai'-,^ 10 / • ' A ^ O 4 ö ~

Without remoyng steady conlribution - - - . RemoMng steady

conlribution Without remoyng steady conlribution - - - . RemoMng steady

conlribution Without remoyng steady conlribution - - - . RemoMng steady conlribution 0.01 0 -0.01 -0.02 -0.03 -0.04 -0.05 -0.06 -0.07 -0.08 -0.09

Steady pitch moment time history

20 40

- steady Pilch moment |

Fig. 8. Pitch radiation m o m e n t t i m e history f o r Fast Ferry due to pitch forcing

f u n c t i o n . 40 l<not speed, (:L=2.58. Fig. 10. Steady pitch moment, 40 k n o t speed.

is a l m o s t t h e service speed o f t h e vessel) (Fig.4). Figs. 5 a n d 6 r e p r e s e n t t h e r a d i a t i o n f o r c e h i s t o r y f o r zero speed, at / d = 2 . 5 8 , w h e r e L is t h e l e n g t h b e t w e e n p e r p e n d i c u l a r s , a m p l i t u d e o f f o r c e d heave m o t i o n a n d f o r c e d p i t c h m o t i o n is 1 m a n d 1°, r e s p e c t i v e l y . Fig. 7 r e p r e s e n t s t h e t i m e h i s t o r y ( t i m e axis is g i v e n i n second scale f o r a l l t i m e h i s t o r y g r a p h s ) f o r n o n - d i m e n s i o n a l heave f o r c e d u e t o heave f o r c e d h a r m o n i c f u n c t i o n a n d Fig. 8 p r e s e n t s t h e t i m e h i s t o r y f o r t h e n o n - d i m e n s i o n a l p i t c h m o m e n t d u e t o f o r c e d p i t c h . T h e f o r c e a n d m o m e n t s are n o n -d i m e n s i o n a l i s e -d b y F33(t)/pgzl a n -d F55(t)/pgzlL, r e s p e c t i v e l y .

F r o m Figs. 5 and 6 i t m a y be n o t e d t h a t f o r the zero speed case responses are very steady a n d p e r f e c t l y s y m m e t r i c a l about the X-axis. Figs. 7 and 8 represent the heave and p i t c h force a n d m o m e n t f o r the 4 0 k n o t speed ( a l m o s t service speed) o f the vessel. I n Figs. 7 and 8, the continuous line represents the r a d i a t i o n force history w i t h o u t r e m o v i n g the effect o f steady force component. The dod:ed line represents the r a d i a t i o n force history a f t e r r e m o v i n g the steady linear force component. The steady c o m p o n e n t is d e f i n e d here as the h y d r o d y n a m i c force generated o n the h u l l advancing w i t h constant f o r w a r d speed w i t h o u t f o r c i n g oscillatory m o t i o n s (and w i t h o u t i n c i d e n t waves as w e l l ) . A l t h o u g h the h u l l is advancing, i t is restrained at the zero speed static sinkage and t r i m , m e a n i n g t h a t t h e steady h y d r o d y n a m i c force represents the effects o f t h e f o r w a r d speed o n the steady pressure d i s t r i b u t i o n on the h u l l . M a t h e m a t i c a l l y i t is calculated by m o d i f y i n g the b o d y boundary c o n d i t i o n (8) as f o l l o w s :

^ = U n \ o n So (22) Bn

It s i i n p l y means t h a t to calculate the steady forces, the code is r u n i n the absence o f any waves or i m p u l s e and computes the heave and p i t c h m o m e n t . Figs. 9 and 10 give the steady heave force and p i t c h m o m e n t c o m p o n e n t using b o u n d a i y c o n d i t i o n (22). The heave force

and p i t c h m o m e n t are negative, m e a n i n g the ship w o u l d t e n d t o submerge under speed a n d rotate the b o w u p .

F r o m t h e Figs. 7 a n d 8, i t can be n o t e d t h a t t h e c o n t i n u o u s time series is n o t s y m m e t r i c a l a b o u t t h e x - a x i s , w h i c h m e a n s t h a t i t is i n f l u e n c e d by some steady forces. C o m p a r e d t o t h e o s c i l l a t o r y a m p l i t u d e o f t h e r a d i a t i o n forces, o n e c o n c l u d e s t h a t t h e s t e a d y e f f e c t s are v e r y p r o n o u n c e d f o r t h e ship's s e r v i c e speed. The d o t t e d lines o f Figs. 7 a n d 8 g i v e t h e f o r c e h i s t o r y a f t e r r e m o v i n g t h e s t e a d y c o m p o n e n t . I t can be o b s e r v e d t h a t , a f t e r r e m o v i n g the s t e a d y e f f e c t , t h e f o r c e h i s t o r y a s y m m e t r y reduces, b u t does n o t d i s a p p e a r s . This m e a n s t h a t , besides t h e n o n - o s c i l l a t o r y h u l l s t e a d y h y d r o d y n a m i c f o r c e c o m p o n e n t , t h e r e is also a steady r a d i a t i o n f o r c e i n d u c e d b y f o r w a r d speed e f f e c t s . W h i l e t h e time d o m a i n m e t h o d is able t o i n c l u d e t h e s t e a d y f o r c e c o n t r i b u t i o n , i t is v e r y d i f f i c u l t t o i n c l u d e t h e s t e a d y c o n t r i b u t i o n i n t h e case o f t h e f r e q u e n c y d o m a i n s o l u t i o n . This is o n e m a j o r a d v a n t a g e o f u s i n g t h e time d o m a i n s o l u t i o n o v e r f r e q u e n c y d o m a i n s o l u t i o n .

5.2.2. Amplitude of the radiation forces

T h e results f o r r a d i a t i o n forces d u e t o heave f o r c i n g h a r m o n i c f u n c t i o n a n d p i t c h f o r c i n g h a r m o n i c f u n c t i o n are p r e s e n t e d f o r t h e d i f f e r e n t speeds i n Figs. 1 1 - 2 7 . The f o r c e d m o t i o n a m p l i t u d e s are 1.0 m f o r heave a n d 1.0° f o r p i t c h . The c o m p u t e d results w i t h t h e s t r i p t h e o r y a n d 3 D t i m e d o m a i n p a n e l m e t h o d are c o m p a r e d w i t h e x p e r i m e n t a l d a t a . I n t h e f i g u r e s , t h e a m p l i t u d e o f forces a n d m o m e n t s are n o r m a l i s e d b y | F | / p g z l a n d | M | / p g A L , respec-t i v e l y . The f r e q u e n c i e s are n o n - d i m e n s i o n a l i s e d b y w-^/lfg. I n respec-t h e legends, " E x p " r e p r e s e n t s t h e results f r o m e x p e r i m e n t s , " T D " r e p r e s e n t s the results f r o m p a n e l code c a l c u l a t i o n s a n d "ST" is t h e r e s u l t s f r o m s t r i p t h e o r y code. It m a y also be n o t e d t h a t F 3 3 , F 5 3 , F35 a n d F55 r e p r e s e n t t h e f o l l o w i n g cases:

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c £ '•5 c O 1.2 1 0.8 0.6 0.4 0.2 0 Experimental R e s u l t s (F33) 2 4 6 n o n d i m e n s i o n a l F r e q u e n c y «• S p e e d - Oknot • S p e e d = 2 0 knot A, S p e e d = 30 knot X S p e e d = 4 0 knot 0.3 0.25 0.2 0.15 0.1 0,05 O Experimental R e s u l t s (F35) X X X X A 1 D • • A t * * * «

*

2 4 6 Nondimensional frequency • Speed = O knot • Speed = 20 knol k Speed = 30 knot < Speed = 40 knot 0.045 0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 Experimental R e s u l t s (F53) X A • X xx^=^ X X >*^AA A 1 • • • •

*

• • • 2 4 6 Nondimensional F r e q u e n c y • S p e e d = 0 knot • S p e e d = 2 0 knot A S p e e d = 30 knot X S p e e d = 4 0 knot 0.14 0,12 0,1 0,08 0.06 0,04 0,02 0 Experimental R e s u l t s {F55)

It

2 4 6 Nondimensional Frequency • Speed = 0 knot • Speed = 20 knot X Speed - 30 knot >c Speed = -iO knot

Fig. 11. (a) Comparison o f radiation force amplitude i n d i f f e r e n t speeds f o r f33, ( b ) comparison of radiation m o m e n t amplitude i n d i f f e r e n t speeds f o r F53, (c) comparison o f radiation force amplitude i n d i f f e r e n t speeds f o r F35, ( d ) comparison o f radiation m o m e n t amplitude in d i f f e r e n t speeds f o r F55.

F 3 3 , 0 s p e e d 1,2 1 - 0) 0,1 n P. F53, 0 s p e e d E < '•5 c o z 0,6

4

0,4 0,2 0 XI

<

0,0 - 1 0 . 0 -20.0 - 3 0 . 0 -40.0 -50.0 - 6 0 . 0 0 2 4 6 N o n d i m e n s i o n a l F r e q u e n c y P h a s e , F 3 3 , 0 s p e e d 2 4 6 N o n d i m e n s i o n a l F r e q u e n c y o E x p . TD •n.-- S T 0 E x p — w_ T D - S T 0 E x p — w_ T D - S T

Fig. 12. Heave force response f u n c t i o n and phase angle for Fast Ferry due to heave forcing f u n c t i o n , 0 speed. a) f 3 3 : H e a v e r a d i a t i o n f o r c e d u e t o heave f o r c i n g f u n c t i o n . b ) F53: P i t c h r a d i a t i o n m o m e n t d u e t o h e a v e f o r c i n g f u n c t i o n . c) F35: H e a v e r a d i a t i o n f o r c e d u e t o p i t c h f o r c i n g f u n c t i o n .

I

0.04 0.035 0,03 0,025 0.02 0.015 \ 0.01 0.005 0 1 5 0 . 0 1 0 0 . 0 5 0 . 0 0.0 - 5 0 . 0 - 1 0 0 . 0 - 1 5 0 . 0

0^

0 2 4 6 N o n d i m e n s i o n a l F r e q u e n c y P h a s e , F 5 3 , 0 s p e e d 2 4 6 N o n d i m e n s i o n a l F r e q u e n c y o Exp < T D - * - - S l r l p Theory ° 0 E x p —»— TD . - S T

Fig. 13. Pitch m o m e n t response f u n c t i o n and phase angle f o r Fast Ferry due to heave forcing f u n c t i o n , 0 speed.

d ) F55: P i t c h r a d i a t i o n m o m e n t d u e t o p i t c h f o r c i n g f u n c t i o n . Fig. l l a - d r e p r e s e n t s t h e c o m p l e t e e x p e r i m e n t a l r e s u l t s f o r t h e Fast Ferry. Each g r a p h p r e s e n t s t h e r a d i a t i o n f o r c e r e s u l t s f o r

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F 3 5 , O s p e e d 2 4 N o n d i m e n s i o n a l F r e q u e n c y P h a s e , F 3 5 , O s p e e d 1 5 0 . 0 - , 1 0 0 . 0 -a 01 5 0 . 0 -le (deg i 0,0¬ - 5 0 . 0 • An g _{- 1 0 0 . 0 ¬} - 1 5 0 . 0 ¬ 2 0 0 . 0 -, Exp — TD , - S T 0 2 4 6 8 N o n d i m e n s i o n a l F r e q u e n c y

Fig. 14. Heave force response f u n c t i o n and phase angle for Fast Ferry due to pitch f o r c i n g f u n c d o n , 0 speed.

II

'if

B < '•B c o z UI a O)

I

0.12 0.1 0.08 0.06 0.04 0.02 0 0.0 - 1 0 . 0 - 2 0 . 0 -30.0 -40.0 -50.0 -60.0 F 5 5 , 0 s p e e d Exp - T D - S T 2 4 6 N o n d i m e n s i o n a l F r e q u e n c y P h a s e , F 5 5 , 0 s p e e d Exp - T D - S T 0 2 4 6 8 N o n d i m e n s i o n a l F r e q u e n c y

Fig. 15. Pitch m o m e n t response f u n c t i o n and phase angle f o r Fast Ferry due to pitch forcing f u n c t i o n , 0 speed.

=

ra

= 1

Q> E E < c o z 1.2 1 0.8 0.6 0.4 0.2 0 F 3 3 , 20 knot s p e e d of Exp - T D - S T 2 4 6 N o n d i m e n s i o n a l F r e q u e n c y P h a s e , F 3 3 , 20 knot s p e e d O) CD •a cn

<

0 2 4 6 N o n d i m e n s i o n a l F r e q u e n c y

Fig. 16. Heave force response f u n c t i o n and phase angle for Fast Ferry due to heave forcing f u n c d o n , 20 k n o t speed. o 4=

1 |

O) E E < c O z 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 F 5 3 , 20 knot s p e e d o Exp — T D — . - S T 0 2 4 6 N o n d i m e n s i o n a l F r e q u e n c y P h a s e , F 5 3 , 20 knot s p e e d 0 2 4 6 N o n d i m e n s i o n a l F r e q u e n c y

Fig. 17. Pitch m o m e n t response f u n c t i o n and phase for Fast Ferry due to heave forcing f u n c d o n , 20 knot speed.

each m o d e w i t h d i f f e r e n t speed. The results are c l u s t e r e d i n t h i s w a y j u s t t o s h o w t h e e f f e c t o f speed o n r a d i a t i o n forces. F r o m Fig. 11a, i t m a y be n o t e d t h a t f o r F33 t h e v a r i a t i o n o f r a d i a t i o n f o r c e a m p l i t u d e is m i n i m u m w h e n speed is increased. The v a r i a t i o n is s i g n i f i c a n t l y increased w i t h speed f o r F55, a n d f o r F53 a n d F35, r e s u l t s l a r g e l y v a r y w h e n speed is increased.

Figs. 1 2 - 1 5 r e p r e s e n t t h e r e s u l t s f o r t h e zero speed case. F r o m t h e figures, i t c a n be seen t h a t t h e panel code c o m p u t a t i o n agrees v e r y w e l l w i t h t h e e x p e r i m e n t a l data a n d t h e s t r i p t h e o r y results.

I t is i n t e r e s t i n g t o observe t h a t t h e s t r i p t h e o r y a n d t h e p a n e l code results m a t c h v e r y w e l l f o r a l l f o u r cases i n z e r o speed w h i l e h a v e s o m e d i f f e r e n c e w i t h e x p e r i m e n t s especially f o r t h e c o m p u t a -t i o n s o f F 5 3 a n d F 3 5 .

Figs. 1 6 - 2 7 r e p r e s e n t t h e s a m e results f o r d i f f e r e n t speeds: Figs. 1 6 - 1 9 s h o w t h e r a d i a r i o n f o r c e a m p l i t u d e results w i t h phases f o r 2 0 k n o t . Figs. 2 0 - 2 3 c o r r e s p o n d t o t h e results f o r 3 0 k n o t a n d 2 4 - 2 7 are f o r 4 0 k n o t .

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F 3 5 , 20 k n o t s p e e d 0 ü O u. s i a E E < e O 0 . 1 6 0 . 1 4 0 . 1 2 0.1 0 . 0 8 0 . 0 6 0 . 0 4 0 . 0 2 O F 3 3 , 30 k n o t s p e e d O /

/

V

^ i —0 Eep T D —f - S T 0 Eep ^ i —T D —f - S T E < 0.0 - 2 0 . 0 - 4 0 . 0 - 6 0 . 0 - 8 0 . 0 - 1 0 0 . 0 - 1 2 0 . 0 - 1 4 0 . 0 0 2 4 6 N o n d i m e n s i o n a l F r e q u e n c y P h a s e , F 3 5 , 20 k n o t s p e e d Exp . T D - S T 0 2 4 6 8 N o n d i m e n s i o n a l F r e q u e n c y

Fig. 18. Heave force response f u n c d o n for Fast Ferry due to p i t c h forcing f u n c d o n , 20 knot speed. F 5 5 , 20 k n o t s p e e d 2 4 6 N o n d i m e n s i o n a l F r e q u e n c y P h a s e , F 5 5 , 20 k n o t s p e e d cn c < 0.0 - 1 0 . 0 - 2 0 . 0 - 3 0 . 0 - 4 0 . 0 - 5 0 . 0 - 6 0 . 0 - 7 0 . 0 Exp - T D - S T 2 4 6 N o n d i m e n s i o n a l F r e q u e n c y

Fig. 19. Pitch m o m e n t response f u n c d o n f o r Fast Ferry due to p i t c h forcing f u n c d o n , 20 k n o t speed.

S t a r t i n g w i t h t h e heave r a d i a t i o n forces d u e t o heave f o r c e d m o t i o n ( F 3 3 ) , t h e e x p e r i m e n t a l data s h o w s a n a l m o s t n e g l i g i b l e i n f l u e n c e o f t h e f o r w a r d speed. T h e r e is o n l y a v e r y s m a l l increase o f the force a m p l i t u d e f o r the higher speed and higher frequency o f oscillation. The agreement between the n u m e r i c a l and experimental results is very good, even f o r the strip theory at h i g h speed.

21

= 1 0) E E < c o 0 2 4 6 N o n d i m e n s i o n a l F r e q u e n c y P h a s e , F 3 3 , 30 k n o t s p e e d 2 4 . 6 N o n d i m e n s i o n a l F r e q u e n c y

Fig. 20. Heave force response f u n c t i o n and phase angle for Fast Ferry due to heave forcing f u n c d o n , 30 knot speed.

.2

= 1

0) £ E < '6 c o z 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 F 5 3 , 30 knot s p e e d o Exp — T D S T 2 4 6 N o n d i m e n s i o n a l F r e q u e n c y P h a s e , F 5 3 , 30 knot s p e e d O) 0) a UI Exp - T D - S T 70.0 6 0 . 0 50.0 4 0 . 0 3 0 . 0 2 0 . 0 10.0 0.0 0 2 4 6 8 N o n d i m e n s i o n a l F r e q u e n c y

Fig. 21. Pitch m o m e n t response f u n c t i o n and phase angle f o r Fast Ferry due to heave f o r c i n g f u n c d o n , 30 k n o t speed. R e g a r d i n g t h e p i t c h m o m e n t d u e t o f o r c e d p i t c h m o r i o n (F55), t h e e x p e r i m e n t a l r e s u l t s s h o w e v i d e n c e o f s i g n i f i c a n t f o r w a r d speed e f f e c t s . I n f a c t t h e a m p l i t u d e o f t h e r a d i a t i o n m o m e n t increases w i t h t h e f o r w a r d speed b y a r o u n d 4 0 - 6 0 % b e t w e e n z e r o a n d 4 0 k n o t . T h e t i m e d o m a i n c o d e is able t o r e p r e s e n t t h e f o r w a r d speed e f f e c t s q u i t e w e l l . R e g a r d i n g t h e s t r i p t h e o r y , t h e p r e d i c t i o n s are g o o d f o r h i g h f r e q u e n c i e s , b u t n o t f o r t h e l o w t o m e d i u m f r e q u e n c y range. This r e f l e c t s t h e l i m i t a t i o n o f t h e h i g h

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O.

I

ra c O O 0.25 0.15 F 3 5 , 30 k n o t s p e e d 0.05 2 4 6 N o n d i m e n s i o n a l F r e q u e n c y P h a s e , F 3 5 , 3 0 knot s p e e d 2 4 6 N o n d i m a n s i o n a l F r e q u e n c y

Fig. 22. Heave force response f u n c t i o n and phase angle f o r Fast Ferry due to pitch forcing f u n c t i o n , 30 knot speed.

O 3

1 |

a c E < 1.2 1 0.8 0 . 6 0.4 0.2 0 F 3 3 , 4 0 k n o t s p e e d Ui 2 4 6 N o n d i m e n s i o n a l F r e q u e n c y P h a s e , F 3 3 , 4 0 k n o t s p e e d o Exp; — — T D —r - S T Fig. 24. forcing 2 4 6 N o n d i m e n s i o n a l F r e q u e n c y

Heave force response f u n c t i o n and phase angle f o r Fast Ferry due to heave f u n c t i o n , 4 0 knot speed. F 5 5 , 30 knot s p e e d 2 4 6 N o n d i m e n s i o n a l F r e q u e n c y P h a s e , F 5 5 , 30 knot s p e e d 2 4 6 8 N o n d i m e n s i o n a l F r e q u e n c y

Fig. 23. Pitch m o m e n t response f u n c d o n and phase angle for Fast Ferry due to pitch forcing function, 30 knot speed.

f r e q u e n c y h y p o t h e s i s a s s u m e d t o s i m p l i f y t h e 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 .

The e x p e r i m e n t a l c o u p l i n g r a d i a t i o n force a m p l i t u d e s (F53 and f 3 5 ) present a v e r y large dependence o f the f o r w a r d speed. These

•D 3 E < re c o 'w c E T3 0 . 1 6 0 . 1 4 0 . 1 2 0.1 0 . 0 8 0 . 0 6 0 . 0 4 0 . 0 2 F 5 3 , 4 0 k n o t s p e e d 2 4 6 N o n d i m e n s i o n a l F r e q u e n c y P h a s e , F 5 3 , 4 0 k n o t s p e e d Exp - T D -ST 2 4 6 N o n d i m e n s i o n a l F r e q u e n c y

Fig. 25. Pitch m o m e n t response f u n c d o n and phase angle f o r Fast Ferry due to heave f o r c i n g f u n c d o n , 4 0 knot speed.

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F 3 5 , 40 knot s p e e d F 5 5 , 4 0 k n o t s p e e d 2 4 N o n d i m e n s i o n a l F r e q u e n c y P h a s e , F 3 5 , 40 knot s p e e d Exp - T D •ST 0 2 4 6 8 N o n d i m e n s i o n a l F r e q u e n c y

Fig. 26. Heave force response f u n c t i o n and phase angle for Fast Ferry due to p i t c h forcing f u n c t i o n , 4 0 k n o t speed. c O 2 4 6 N o n d i m e n s i o n a l F r e q u e n c y P h a s e , F 5 5 , 4 0 k n o t s p e e d 2 4 6 N o n d i m e n s i o n a l F r e q u e n c y

Fig. 27. Pitch m o m e n t response f u n c t i o n and phase angle f o r Fast Ferry due to pitch forcing f u n c t i o n , 40 knot speed.

forces increase up t o 10 times b e t w e e n 0 and 40 knot. Interestingly, the effect o f f o r w a r d speed o n the radiation force due t o forced p i t c h m o t i o n (F35) is similar along the w h o l e frequency range, w h i l e f o r the p i t c h m o m e n t due t o forced heave m o t i o n (F53) the effects are m u c h stronger f o r l o w frequencies. Regarding the t i m e d o m a i n c o m p u t a -tions i t is possible to say that the agreement w i t h experiments is good for all speeds and frequencies. The m e t h o d is able to predict very w e l l the c o u p l i n g p i t c h m o m e n t due t o heave and reasonably w e l l the heave force d u e to p i t c h . The s t r i p theory fails to predict the c o u p l i n g forces w h e n t h e f o r w a r d speed is present.

From the figures, i t m a y be concluded that the time d o m a i n code predicts all f o u r radiation forces w e l l , there are some difference f o r coupled forces, b u t the overall behaviour is very consistent and f o l l o w the same p a t t e r n as t h a t o f the experiments, whereas f o r the coupled forces, strip theory results are f a r f r o m the experiments. Therefore i t m a y be concluded t h a t the present time d o m a i n m e t h o d produce good and robust results f o r t h e radiation forces.

6. C o n c l u s i o n s The p a p e r i n v e s t i g a t e s t h e h y d r o d y n a m i c r a d i a t i o n f o r c e s o n a Fast F e r r y m o n o - h u l l w i t h a t i m e d o m a i n t h r e e d i m e n s i o n a l p a n e l m e t h o d a n d a s t r i p t h e o r y code a n d c o m p a r e s t h e i r p r e d i c t i o n s w i t h t h e e x p e r i m e n t a l d a t a o f m o d e l t e s t s . T h e t i m e d o m a i n f o r m u l a t i o n a c c o u n t s f o r t h e f u l l l i n e a r i n t e r a c t i o n b e t w e e n t h e s t e a d y flow d u e t o t h e s h i p ' s f o r w a r d s p e e d a n d t h e o s c i l l a t o r y flow a n d t h i s is o n e o f t h e m a i n a d v a n t a g e s o f t h e m e t h o d . O n t h e o t h e r h a n d , t h e c o m m o n f r e q u e n c y d o m a i n s e a k e e p i n g t h e o r i e s i n t r o d u c e s o m e s i m p l i f i c a t i o n s i n t h e f o r m e r i n t e r a c t i o n s , i n c l u d i n g t h e p a n e l m e t h o d s . I n o r d e r t o assess t h e c o n s e q u e n c e s o f t h e s e s i m p l i f i c a t i o n s t h e s t r i p t h e o r y r e s u l t s a n d t h e t i m e d o m a i n p a n e l c o d e p r e d i c t i o n s are c o m p a r e d w i t h t h e e x p e r i m e n t a l d a t a .

The e x p e r i m e n t a l data shows that the heave r a d i a t i o n forces due to forced heave m o t i o n are a l m o s t independent o f the f o r w a r d speed. I n this case b o t h the t i m e d o m a i n m e t h o d a n d the strip t h e o r y give very good predictions o f the h y d r o d y n a m i c forces.

R e g a r d i n g t h e p i t c h r a d i a t i o n m o m e n t d u e t o f o r c e d p i t c h m o t i o n , o n e observes a s i g n i f i c a n t i n c r e a s e o f t h e m o m e n t a m p l i t u d e w i t h t h e f o r w a r d speed b e t w e e n 0 a n d 4 0 k n o t . I n t h i s case t h e t i m e d o m a i n r e s u l t s a g r e e v e r y w e l l w i t h t h e e x p e r i m e n t s f o r a l l speeds a n d f o r t h e w h o l e f r e q u e n c y r a n g e , w h i l e t h e a g r e e m e n t w i t h s t r i p t h e o r y is p o o r a t t h e h i g h speeds a n d l o w f r e q u e n c y r a n g e . T h e e x p e r i m e n t a l c o u p l i n g f o r c e s s h o w t h e l a r g e s t i n f i u e n c e f r o m t h e s h i p ' s s p e e d w i t h a m p l i f i c a t i o n o f t h e f o r c e s u p t o 10 t i m e s b e t w e e n 0 a n d 4 0 k n o t . T h e p a n e l c o d e r e p r e s e n t s t h e s e e f f e c t s w e l l , w h i l e t h e s t r i p t h e o r y f a i l s f o r t h e h i g h e r speeds. As a g e n e r a l c o n c l u s i o n , t h e p r e s e n t t i m e d o m a i n p a n e l c o d e r e s u l t 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 e x p e r i m e n t a l r e s u l t s , w h i c h s h o w s t h e e f f i c i e n c y a n d c o r r e c t n e s s o f t h e p r o p o s e d s c h e m e . I t is a l s o o b s e r v e d f r o m t h e t i m e s i g n a l s t h a t t h e p r e s e n t t i m e d o m a i n c o d e is a b l e t o g i v e v e r y s t a b l e s i g n a l s . T h e r e f o r e i t m a y be c o n c l u d e d t h a t t h e p r o p o s e d m e t h o d is v e r y r o b u s t as w e l l . A n o t h e r i n t e r e s t i n g a s p e c t o f t h e p r e s e n t t i m e d o m a i n c o d e is t o c a l c u l a t e t h e s t e a d y c o n t r i b u t i o n a u t o m a t i c a l l y f r o m t h e s i g n a l s , w h i c h is n o t p o s s i b l e w i t h f r e q u e n c y d o m a i n m e t h o d s . C o n s i d e r i n g t h e c o m p l e x i t y o f t h e h u l l , i t c a n be c o n c l u d e d t h a t p r o p o s e d t i m e d o m a i n c o d e is a b l e t o p r o d u c e v e r y g o o d r e s u l t s e v e n f o r t h e c o m p l i c a t e d h u l l shapes a n d h i g h speeds.