Applied Ocean Research 54 (2016) 3 9 - 5 2
ELSEVIER
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Applied Ocean Research
journal homepage; www.elsevier.com/locate/aporBody nonlinear time domain calculation of vertical ship responses in ( f h
extreme seas accounting for 2nd order Froude-Krylov pressure
S. Rajendran, N. Fonseca, C. Guedes Soares*
Centre of Marine Teclinology and Engineering (CEmEC), Instituto Superior Técnico. Universidade de Lisboa. Portugal
CrossMark
A R T I C L E I N F O A B S T R A C T
Article history: Received 11 July 2015 Received in revised form 19 September 2015 Accepted 26 October 2015 Available online 7 December 2015
Keywords:
Body nonlinear time domain method Strip theory
Cummins formulation
Second-order Froude-Krylov pressure Cruise ship
Extreme seas
A b o d y n o n l i n e a r t i m e d o m a i n m e t h o d b a s e d on strip t h e o r y w i t h a w e a k l y n o n l i n e a r f o r m u l a t i o n for the free s u r f a c e c o n d i t i o n is p r o p o s e d a n d u s e d to a n a l y z e the v e r t i c a l r e s p o n s e s of a c r u i s e s h i p i n r a n d o m seas, T h e t e r m 'body n o n l i n e a r i t y ' i m p l i e s the d e p e n d e n c e of the h y d r o d y n a m i c f o r c e s o n t h e i n s t a n -t a n e o u s w e -t -t e d s u r f a c e a r e a of -the s h i p h u l l . T h e r a d i a -t i o n forces a r e c a l c u l a -t e d by m e a n s of C u m m i n s f o r m u l a t i o n a n d are c a l c u l a t e d for t h e w e t t e d h u l l u n d e r the i n c i d e n t w a v e profile. A p r a c t i c a l e n g i n e e r -i n g a p p r o a c h -is f o l l o w e d for the e s t -i m a t -i o n of the b o d y n o n l -i n e a r r a d -i a t -i o n a n d d -i f f r a c t -i o n forces. T h e h y d r o s t a t i c forces are c a l c u l a t e d for the w e t t e d h u l l u n d e r the i n c i d e n t w a v e profile. T h e free surface c o n d i t i o n is r e p r e s e n t e d u s i n g a w e a k l y n o n l i n e a r f o r m u l a t i o n a n d the i n c i d e n t i r r e g u l a r w a v e s h a v e n o n l i n e a r i t i e s up to s e c o n d order. T h e w a v e p r e s s u r e is c a l c u l a t e d f r o m the first- a n d s e c o n d - o r d e r w a v e potentials u s i n g the B e r n o u l l i e q u a t i o n a n d is integrated over the e x a c t w e t t e d s u r f a c e to c a l c u l a t e the F r o u d e - K r y l o v forces. T h e n u m e r i c a l r e s u l t s are c o m p a r e d w i t h the e x p e r i m e n t a l r e s u l t s m e a s u r e d in a w a v e tank. T h e c o m p a r a t i v e s t u d y is c a r r i e d out b y a n a l y z i n g the statistics of the w a v e s a n d t h r o u g h p r o b a b i l i t y of e x c e e d a n c e of t h e w a v e peaks. © 2 0 1 5 E l s e v i e r Ltd. A l l rights r e s e r v e d .
1. Introduction
T h e e x t r e m e loads a c t i n g o n a s h i p d u r i n g i t s l i f e cycle is a n e s s e n t i a l p a r a m e t e r f o r t h e s t r u c t u r a l design o f t h e ships. N o w a d a y s , h i g h f i d e l i t y h y d r o d y n a m i c n u m e r i c a l m e t h o d s a n d t e c h n i q u e s , l i k e 3 D p a n e l m e t h o d s [ 1 - 3 ] a n d c o m p u t a t i o n a l fluid d y n a m i c s (CFD) [ 4 ] , are e x t e n s i v e l y u s e d , p a r t i c u l a r l y b y t h e researchers a n d t h e c l a s s i f i c a t i o n societies, f o r t h e c a l c u l a t i o n o f t h e s h i p responses. Since these m e t h o d s c o n s i d e r t h e 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 a n d t h e u n s t e a d y c o m p o n e n t o f t h e m o t i o n s a l m o s t c o m p l e t e l y , t h e y give p r o m i s i n g r e s u l t s f o r t h e responses o f t h e ships w i t h h i g h Froude n u m b e r riding o v e r s m a l l t o m o d e r a t e w a v e s . H o w e v e r , f o r t h e c a l c u l a t i o n o f t h e e x t r e m e loads a c t i n g o n s h i p w h e r e t h e s h i p m o v e s w i t h r e d u c e d service speed, t h e y d o n o t c l a i m a n y s u p r e m a c y o v e r t h e s i m p l e r m e t h o d s l i k e s t r i p t h e o r y [1 ] . The e s t i m a t i o n o f e x t r e m e loads is a c h a l l e n g i n g task d u e t o t h e u n c e r t a i n t y associated w i t h t h e ocean e n v i r o n m e n t , p a r t i c u l a r l y w i t h t h e o c e a n w a v e s i n w h i c h t h e s h i p o p e r a t e s . The e x t r e m e loads are g e n e r a l l y c a l c u l a t e d e i t h e r f r o m t h e e x t e n s i v e M o n t e C a r i o -t y p e s i m u l a -t i o n s o r f r o m -t h e l o n g - -t e r m d i s -t r i b u -t i o n o f -t h e loads w h i c h is o b t a i n e d b y f i t t i n g a p r o b a b i l i s t i c m o d e l t o t h e s h o r t - t e r m d i s t r i b u t i o n o f the ship responses i n selected e x t r e m e sea states.* Corresponding author. Tel.: +351 218 417607; fax; +351 218 474015. E-mail address: c.guedes.soares@centec.tecnico.ulisboa.pt (C. Guedes Soares).
0141-1187/$ - see front matter ® 2015 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.apor.2015,10.008
T h e s h i p responses i n these e x t r e m e c o n d i t i o n s are h i g h l y n o n -l i n e a r , p a r t i c u -l a r -l y f o r ships w i t h -large b o w f -l a r e . T h e r e f o r e , i t s a c c u r a t e p r e d i c t i o n r e q u i r e s t h e n u m e r i c a l m e t h o d s w i t h t h e b o d y n o n l i n e a r i t y o f l e v e l 4 or h i g h e r . The b o d y n o n l i n e a r i t y o f l e v e l 4, as d e f i n e d b y I n t e r n a t i o n a l s h i p a n d o f f s h o r e S t r u c t u r e C o m m i t -tee [ 5 ] , i n v o l v e s t h e c a l c u l a t i o n o f t h e i n c i d e n t a n d d i s t u r b a n c e p o t e n t i a l at t h e exact w e t t e d h u l l u n d e r t h e i n c i d e n t w a v e . This d e m a n d s r e - g r i d d i n g o f t h e w h o l e d o m a i n at e a c h i n s t a n t o f t i m e a n d r e c a l c u l a t i o n o f t h e p o t e n t i a l s i n t h e 3 D time d o m a i n m e t h -ods, w h i c h is h i g h l y t i m e c o n s u m i n g . P r a c t i c a l a p p l i c a t i o n o f such a m e t h o d f o r t h e c a l c u l a t i o n o f t h e e x t r e m e loads, p a r t i c u l a r l y d u r -i n g t h e c o n c e p t d e s -i g n 'stage, -is q u -i t e u n p o p u l a r . H o w e v e r , -i t -is easier to a c c o m p l i s h t h i s task u s i n g s i m p l i f i e d t h e o r i e s l i k e s t r i p t h e o r y w i t h i n t h e accuracy acceptable f o r t h e p r a c t i c a l e n g i n e e r i n g a p p l i c a t i o n s . D i f f e r e n t k i n d s o f s t r i p t h e o r y have b e e n p r o p o s e d so far, a m o n g w h i c h t h e t h e o r y p r o p o s e d b y Salvesen e al. [ 6 ] is q u i t e p o p u l a r , because i t is c o n s i s t e n t a n d has b e e n w i d e l y t e s t e d a n d v a l i d a t e d . C u m m i n s ( 7 ] p r o p o s e d t h e h y d r o d y n a m i c f o r m u l a t i o n i n t h e t i m e d o m a i n , a n d t h e r a d i a t i o n forces w e r e c a l c u l a t e d u s i n g i n f i n i t e f r e -q u e n c y a d d e d mass, c o n v o l u t i o n o f t h e m e m o r y f u n c t i o n w i t h t h e s h i p v e l o c i t y , a n d t h e h y d r o d y n a m i c r e s t o r i n g c o e f f i c i e n t f o r ships u n d e r w a y . Fonseca a n d Guedes Soares [ 8 , 9 ] p r o p o s e d t h e p a r t i a l l y b o d y n o n l i n e a r time d o m a i n m e t h o d i n w h i c h t h e F r o u d e - K r y l o v a n d h y d r o s t a t i c f o r c e s w e r e c a l c u l a t e d f o r t h e e x a c t w e t t e d h u l l u n d e r t h e i n c i d e n t w a v e . The r a d i a t i o n forces w e r e c a l c u l a t e d b y
40 S. Rajendran et al./Applied Ocean Research 54 (2016)39-52 m e a n s o f t h e C u m m i n s f o r m u l a t i o n a n d w e r e c a l c u l a t e d a t t h e m e a n w a t e r l e v e l . T h e d i f f r a c t i o n forces w e r e c a l c u l a t e d u s i n g 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 based o n Salvesen e t a l . [ 6 ] . M i k a m l a n d K a s h i w a g i [ 1 0 ] p r o p o s e d a b o d y n o n l i n e a r t i m e d o m a i n code b a s e d o n s t r i p t h e o r y . T h e y a p p l i e d t h e firstorder B e r n o u l l i p r e s -s u r e e q u a t i o n t o t h e p o t e n t i a l cau-sed b y a n i m p u l -s e . Ba-sed o n r e l a t i v e m o t i o n concept, 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 f o r c e s w e r e c a l c u l a t e d . M o r t o l a e t a l . [ 1 1 ] p r o p o s e d a t i m e d o m a i n code based o n s t r i p t h e o r y i n w h i c h t h e r a d i a t i o n forces are e x p r e s s e d u s i n g t h e C u m m i n s f o r m u l a t i o n . The r a d i a t i o n p r o b l e m w a s t r e a t e d n o n -l i n e a r -l y a n d is s o -l v e d i n t h e f r e q u e n c y d o m a i n i n w a y o f t h e rea-l w e t t e d h u l l g e o m e t r y u s i n g b o u n d a r y e l e m e n t s . T h e h y d r o d y n a m i c c o e f f i c i e n t s w e r e c o n v e r t e d to m e m o r y f u n c t i o n t h r o u g h F o u r i e r t r a n s f o r m a n a l o g y . R a j e n d r a n e t a l . [ 1 2 , 1 3 ] c o n d u c t e d s t u d i e s o n t h e responses o f t h e ships w i t h large b o w flare i n e x t r e m e seas a n d f r e a k w a v e s u s i n g t h e p a r t i a l l y n o n l i n e a r t i m e d o m a i n m e t h o d [ 8 , 9 ] , 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 r e s u l t s w e r e c o m p a r e d a n d a f a -i r l y g o o d a g r e e m e n t w a s f o u n d b e t w e e n t h e v e r t i c a l m o t i o n s a n d t h e h o g -g i n -g peaks, h o w e v e r t h e sa-g-gin-g peaks w e r e o v e r e s t i m a t e d . The p a r t i a l l y n o n l i n e a r m e t h o d w a s f u r t h e r e x t e n d e d b y R a j e n d r a n e t a l . [ 1 4 , 1 5 , 1 6 ] t o take i n t o a c c o u n t t h e b o d y n o n l i n e a r r a d i a -tion a n d d i f f r a c t i o n f o r c e s . C e r t a i n p r a c t i c a l e n g i n e e r i n g t e c h n i q u e s w e r e f o l l o w e d f o r t h e c a l c u l a t i o n o f t h e b o d y n o n l i n e a r r a d i a t i o n f o r c e s , a n d t h e F r o u d e K r y l o v a n d h y d r o s t a t i c f o r c e s w e r e c a l c u -l a t e d f o r t h e exact w e t t e d h u -l -l u n d e r t h e -l i n e a r i n c i d e n t w a v e e l e v a t i o n . R a j e n d r a n e t a l . [ 1 7 ] f u r t h e r i m p r o v e d t h e b o d y n o n -l i n e a r time m e t h o d b y r e p -l a c i n g t h e -l i n e a r F r o u d e - K r y -l o v p r e s s u r e w i t h a s e c o n d - o r d e r pressure, a n d t h e c a l c u l a t i o n s w e r e c a r r i e d o u t i n r e g u l a r w a v e s , a n d t h e l o a d c a l c u l a t i o n s i n i r r e g u l a r w a v e s are b r i e f l y p r e s e n t e d i n R a j e n d r a n e t a l . [ 1 8 ] . M a n y o f t h e state o f t h e a r t n u m e r i c a l m e t h o d s use i n t h e i r f o r -m u l a t i o n s l i n e a r i n c i d e n t i r r e g u l a r w a v e s , w h i l e t h e r e a r e s o -m e e x p e r i m e n t a l evidences t h a t t h e a s y m m e t r i e s a n d n o n l i n e a r i t i e s i n t h e i n c i d e n t w a v e s i n d u c e n o n l i n e a r e f f e c t s o n t h e v e r t i c a l s h i p responses [ 1 9 ] . I n this paper, t h e s h i p responses i n i r r e g u l a r e x t r e m e seas are s t u d i e d i n d e t a i l u s i n g t h e a f o r e m e n t i o n e d b o d y n o n l i n e a r m e t h o d , b u t w i t h a m o d i f i e d f r e e s u r f a c e t h a t is r e p -r e s e n t e d b y a w e a k l y n o n l i n e a -r f o -r m u l a t i o n based o n t h e Stokes e x p a n s i o n . T h e w e a k l y n o n l i n e a r f r e e s u r f a c e f o r m u l a t i o n is based o n t h e a s s u m p t i o n t h a t t h e w a v e s l o p e is s m a l l a n d t h e fluid is i n c o m p r e s s i b l e a n d i r r o t a t i o n a l . T h e u n k n o w n f r e e s u r f a c e c a n be a p p r o x i m a t e d a b o u t t h e m e a n w a t e r l e v e l u s i n g a series e x p a n s i o n f o r w h i c h T a y l o r series e x p a n s i o n is g e n e r a l l y u s e d . I n t h e s e c o n d -o r d e r f -o r m u l a t i -o n , t h e first- a n d s e c -o n d - -o r d e r t e r m s -o f t h e v e l -o c i t y p o t e n t i a l s a n d t h e w a v e e l e v a t i o n s t h a t c o n s t i t u t e t h e b o u n d a r y v a l u e p r o b l e m s are t h e n e x t r a c t e d u s i n g a p e r t u r b a t i o n s c h e m e b a s e d o n a p e r t u r b a t i o n p a r a m e t e r s u c h as w a v e slope. Stokes, [ 2 0 ] , w a s p r o b a b l y t h e first o n e t o d e v e l o p s u c h a p e r t u r b a t i o n s c h e m e . S i m i l a r l y , t h e b o d y forces o f a n y o r d e r c a n b e c a l c u l a t e d f r o m t h e p e r t u r b a t i o n o f t h e b o d y f o r c e s a b o u t t h e m e a n p o s i t i o n o f t h e b o d y . H o w e v e r , t h i s is e x c l u d e d f r o m t h e scope o f t h e s t u d y since s u c h a m e t h o d w i l l b e h i g h l y time c o n s u m i n g . Based o n t h e l i n e a r a s s u m p t i o n , t h e w a v e - b o d y p r o b l e m c a n be d e c o m p o s e d i n t o r a d i a t i o n , d i f f r a c t i o n a n d i n c i d e n t w a v e p r o b l e m s . T h e r e f o r e i n t h e p r e s e n t p r o b l e m , t h e s e c o n d - o r d e r F r o u d e - K r y l o v p r e s s u r e is d e r i v e d f r o m t h e flrst a n d s e c o n d o r d e r i n c i d e n t w a v e p o t e n -t i a l a n d -t h e d i s -t u r b a n c e p o -t e n -t i a l sa-tisfies -t h e l i n e a r f r e e s u r f a c e c o n d i t i o n , h o w e v e r , takes a c c o u n t o f t h e g e o m e t r i c a l n o n l i n e a r i t y . T h e r e s u l t s f r o m t h e m o d i f i e d m e t h o d w i l l be r e p r e s e n t e d b y t h e a c r o n y m 'TDNL-STOKES' i n t h e f o l l o w i n g figures a n d t h e ones f r o m t h e l i n e a r a i r y w a v e s w i l l be r e p r e s e n t e d b y 'TDNL-AIRY' f o r t h e sake o f c o n v e n i e n c e . I n t h i s paper, t h e n u m e r i c a l s i m u l a t i o n s car-r i e d o u t o n a ccar-ruise vessel i n e x t car-r e m e seas u s i n g 'TDNL-STOKES' acar-re c o m p a r e d w i t h t h e r e s u l t s f r o m 'TDNL-A1RY', a n d t h e ones o b t a i n e d f r o m t h e w a v e t a n k .
2. Theory
2.1. Wave-induced ship motions
A c o o r d i n a t e s y s t e m fixed w i t h respect t o m e a n p o s i t i o n o f t h e s h i p is d e f i n e d f o r t h e h y d r o d y n a m i c p r o b l e m . The o r i g i n is i n t h e plane o f t h e u n d i s t u r b e d f r e e s u r f a c e . C o n s i d e r i n g a s h i p a d v a n c i n g i n w a v e s a n d o s c i l l a t i n g as a n u n r e s t r a i n e d r i g i d b o d y , t h e o s c i l l a -t o r y m o -t i o n s w i l l consis-t o f -t h r e e -t r a n s l a -t i o n s a n d -t h r e e r o -t a -t i o n s . The p r e s e n t w o r k is r e s t r i c t e d t o h e a d w a v e s , t h u s t h e o s c i l l a t o r y m o t i o n s t o be s t u d i e d are t h e surge a n d heave d i s p l a c e m e n t s a n d t h e p i t c h r o t a t i o n . Surge m o d e is i n c l u d e d i n t h e e q u a t i o n o f m o t i o n t h r o u g h a s e m i - e m p i r i c a l m e t h o d [ 2 1 ]. Based o n t h e p o t e n t i a l t h e o r y a n d l i n e a r i z a t i o n o f t h e v e l o c -i t y p o t e n t -i a l , t h e h y d r o d y n a m -i c f o r c e s can be s e p a r a t e d -i n several i n d e p e n d e n t c o m p o n e n t s , n a m e l y : r a d i a t i o n forces, w a v e e x c i t i n g forces ( c o m p o s e d b y t h e F r o u d e - K r y l o v p a r t a n d t h e d i f f r a c t i o n p a r t ) a n d h y d r o s t a t i c f o r c e s . These forces are c o m b i n e d t o o b t a i n t h e e q u a t i o n s o f w a v e - i n d u c e d s h i p m o t i o n s a n d g l o b a l s t r u c t u r a l loads. For surge, h e a v e a n d p i t c h , t h e e q u a t i o n o f m o t i o n c o u l d b e w r i t t e n as: {M + A f ^ M t ) + j K i i ( t - T ) ^ , ( r ) d T + C i i ? , ( t ) + ( M . Z c g - h / l i 5 ) ? 5 J - 0 0 + ƒ K i 5 ( t - r ) ? 5 ( r ) d r + C i 5 ? 5 ( t ) = F i E ( f ) ( 1 ) {M +
Af^Mt)+
I
K 3 3 ( f - T ) ? 3 ( T ) d T + C 3 3 ? 3 ( 0 + ^ 3 l ? 5 J-OO + j K 3 5 ( t - r ) ? 5 ( T ) d T + C 3 5 § 5 ( t ) + f 3 " ( t ) J - C O -Mg = F^\t) + F f { t ) ( 2 )il55+Af^)Ut)+
I
K 5 5 ( f - T ) ? 5 ( T ) d r + C55?5(f) + /ir3?3 J-OQ + I K 5 3 ( f - r ) ^ 3 ( T ) d r - f C 5 3 § 3 ( f ) + ( M . Z c g + y \ f = i ) § l + f Kslit - r ) ? , ( r ) d r + Cs,^,{t) + F»{t) = F | ( t ) + F f ( t ) w h e r e ^ , , ^3 a n d §5 r e p r e s e n t , r e s p e c t i v e l y , surge, heave a n d p i t c h m o t i o n s a n d dots o v e r t h e s y m b o l s r e p r e s e n t d i f f e r e n t i a t i o n w i t h r e s p e c t t o t i m e . M is t h e s h i p mass, g is a c c e l e r a t i o n o f g r a v i t y , Zcg is t h e v e r t i c a l d i s t a n c e f r o m t h e s h i p c e n t e r o f g r a v i t y a n d t h e h e i g h t w h e r e t h e v e r t i c a l b e n d i n g m o m e n t is t o b e c a l c u l a t e d a n d ƒ55 r e p r e s e n t t h e s h i p i n e r t i a a b o u t t h e y - a x i s . T h e h y d r o s t a t i c f o r c e a n d m o m e n t , a n d F^, are c a l c u l a t e d a t each time step b y i n t e g r a t i o n o f t h e h y d r o s t a t i c p r e s s u r e o v e r t h e w e t t e d h u l l u n d e r t h e u n d i s t u r b e d w a v e p r o f i l e , i.e. u n d e r t h e i n c i d e n t w a v e p r o f i l e e x c l u d i n g t h e w a v e c o m p o n e n t s i n d u c e d b y t h e s h i p m o t i o n s o r s c a t t e r e d b y t h e h u l l . The e x c i t i n g f o r c e s d u e t o t h e i n c i d e n t w a v e s , F f , F | a n d F | , are d e c o m p o s e d i n t o a d i f f r a c t i o n p a r t , F f , F^^andF^, a n d t h e w e l l - k n o w n F r o u d e - K r y l o v part, F f , F|andF^^. A J 0'. ' < : = ! . 3, 5 ) are t h e i n f i n i t e f r e q u e n c y a d d e d masses, % r e p r e s e n t t h e m e m o r y f u n c t i o n s a n d C,™ is t h e r a d i a t i o n r e s t o r a t i o n c o e f f i c i e n t s . Ff"mdFf" are g r e e n w a t e r f o r c e a n d m o m e n t . T h e d e s c r i p t i o n o f t h e c o e f f i c i e n t s a n d t h e f o r c e c o m p o n e n t s are g i v e n i n t h e f o l l o w i n g sections.5, Rajendran et al. / Applied Ocean Research 54 (2016)39-52 41
2.1.1. First-oriler Froude-Krylov Pressure
The F r o u d e - K r y l o v f o r c e w h i c h is associated w i t h t h e i n c i d e n t w a v e p o t e n t i a l is c a l c u l a t e d f r o m t h e i n t e g r a t i o n o f t h e associated pressure o v e r the i n s t a n t a n e o u s w e t t e d h u l l u n d e r t h e u n d i s t u r b e d w a v e p r o f i l e . T h e F r o u d e - k r y l o v forces a n d m o m e n t i n i r r e g u l a r seas f o r surge, heave a n d p i t c h m o t i o n s are g i v e n b e l o w .
F it): Fa' ( t ) : / / y^^gfƒe'•('*=jf+^•'e'V+¥ Nsd/dx ( 5 ) Ju JcX \ : , f 5 ' \ t ) = Re ^ysgfJe'^'^^Z+S-'e'V+V Nsd/dx ( 6 ) L* JCx w h e r e p represents t h e d e n s i t y o f t h e f l u i d , g is t h e g r a v i t y accel-e r a t i o n , is t h accel-e a m p l i t u d accel-e o f t h accel-e i n c i d accel-e n t w a v accel-e h a r m o n i c a n d t h accel-e i n t e g r a t i o n is o v e r t h e w e t t e d cross s e c t i o n c o n t o u r , Q u n d e r t h e i n c i d e n t w a v e e l e v a t i o n , k is t h e w a v e n u m b e r dl is t h e i n c r e m e n -t a l l e n g -t h a l o n g -t h e g i r -t h o f s h i p s e c -t i o n , 'z' is -t h e v e r -t i c a l dis-tance b e t w e e n t h e s t r i p a n d t h e p o i n t w h e r e t h e v e r t i c a l b e n d i n g m o m e n t is t o be c a l c u l a t e d a n d is t h e phase angle o f t h e i n c i d e n t w a v e , coe is t h e e n c o u n t e r f r e q u e n c y b e t w e e n t h e s h i p a n d t h e w a v e s w h i c h is r e l a t e d t o t h e w a v e f r e q u e n c y , COQ, i n head seas b y t h e r e l a t i o n -s h i p cüe = COQ + (colU/g), w h e r e U i-s t h e v e l o c i t y o f t h e -s h i p . T h e s u m m a t i o n is c a r r i e d o u t o v e r t h e N n u m b e r o f h a r m o n i c s . JVi a n d N3 are t h e u n i t v e c t o r n o r m a l c o m p o n e n t s i n t o t h e s e c t i o n e l e m e n t i n X a n d z - d i r e c t i o n i n t h e i n e r t i a l f r a m e o f reference m o v i n g w i t h t h e s h i p . These c o m p o n e n t s are r e l a t e d t o t h e u n i t n o r m a l v e c t o r c o m p o n e n t s , n\ a n d ns, c a l c u l a t e d w i t h r e f e r e n c e t o t h e b o d y f i x e d c o o r d i n a t e s y s t e m t h r o u g h t h e t r a n s f o r m a t i o n m a t r i x D . c o s f s sin?5 - s i n ? 5 cos §5 ( 7 ) a n d , N , = D n i i = l , 3 The n o r m a l c o m p o n e n t f o r t h e p i t c h m o m e n t i n t h e i n e r t i a l f r a m e o f r e f e r e n c e m o v i n g w i t h t h e s h i p is d e f i n e d as N5 =XxNi, 1 = 1,3, w h e r e X is t h e c o o r d i n a t e o f t h e e l e m e n t w i t h respect t o t h e i n e r t i a l f r a m e o f r e f e r e n c e m o v i n g w i t h t h e s h i p .
The pressure above t h e m e a n w a t e r l i n e ( z = 0) a n d b e l o w t h e w a v e crest is c a l c u l a t e d b y ripg, w h e r e 77 is t h e l o c a l w a v e e l e v a t i o n .
2.1.2. Second-order Froude Krylov Pressure
I n t h i s paper, t h e n u m e r i c a l m e t h o d is m o d i f i e d t o i n c l u d e t h e s e c o n d - o r d e r F r o u d e - K r y l o v pressure. I n a f r a m e o f reference t h a t m o v e s w i t h t h e s h i p a n d fixed a t t h e m e a n p o s i t i o n o f t h e s h i p , d y n a m i c p r e s s u r e d u e t o w a v e s at a n y p o i n t i n t h e fluid c a n be w r i t t e n as: dt dx 2 ( 8 ) p 2 - = p N N L i n e a r i z i n g t h e t e m p o r a l a n d spatial d e r i v a t e o f t h e i n c i d e n t w a v e p o t e n t i a l a b o u t t h e m e a n w a t e r l e v e l u s i n g T a y l o r series e x p a n s i o n r e s u l t s i n Eq. ( 9 ) P = - p 9 ^ 9 2 0 at ' dzdt dx ^ 'dzdx
+
( 9 ) The stokes e x p a n s i o n o f t h e v e l o c i t y p o t e n t i a l a n d w a v e eleva-t i o n can be w r i eleva-t eleva-t e n as:7; = 7?' -t-e??^ + e2,j3 ^ ^
( 1 0 ) w h e r e Ê is t h e w a v e steepness m u c h s m a l l e r t h a n one. S u b s t i t u t i n g Eqs. ( 1 0 ) i n ( 9 ) a n d r e t a i n i n g t h e t e r m s u p t o s e c o n d - o r d e r , t h e first- a n d s e c o n d - o r d e r p r e s s u r e c a n be w r i t t e n p d ) . , P(2) = - p + 2 9 ( 0 1 at ajf] dx ( 1 1 ) 9 ( 0 2 ) ^ ^ , 9 2 ( 0 ^ ) at dzdt 9 ( 0 2 ) ^ ^ , 9 2 ( 0 1 ] dx dzdx ( 1 2 ) The s e c o n d - o r d e r w a v e e l e v a t i o n c a n be c a l c u l a t e d f r o m t h e s e c o n d - o r d e r d y n a m i c - f r e e surface c o n d i t i o n o n z = 0. 7 ? 2 = -+ 2 1 9 ( 0 2 ) , 9 2 ( 0 1 ) 9t + ?7 9z9t 9 ( 0 2 ) , 9 2 ( 0 1 ) dx dzdx ( 1 3 ) w h e r e 0 i a n d cjP are first a n d s e c o n d o r d e r i n c i d e n t w a v e p o t e n -t i a l a n d Tji is -t h e firs-t-order i n c i d e n -t w a v e e l e v a -t i o n . N b] = - ^ J - ^ s i n (/<(X + ü)e,t - S() exp(/<|Z)1=1 '
N !?] = ^ f j ? cos (;<jX + COejt - Sj) J=l ( 1 4 ) ( 1 5 )w h e r e e is t h e phase angle a n d t h e s u m m a t i o n is o v e r each h a r -m o n i c s i n t h e i r r e g u l a r seas. S e c o n d - o r d e r p o t e n t i a l is n e g l e c t e d because o f t h e deep w a t e r a s s u m p t i o n . T h e s u m f r e q u e n c y c o m p o n e n t arises f r o m t h e i n t e r -a c t i o n b e t w e e n t h e first-order w -a v e e l e v -a t i o n -a n d t h e d e r i v -a t i v e o f first-order i n c i d e n t w a v e p o t e n t i a l . T h e d i f f e r e n c e f r e q u e n c y c o m p o n e n t results f r o m t h e i n t e r a c t i o n o f t h e first-order w a v e v e l o c i t i e s . The final e q u a t i o n f o r t h e s u m a n d f r e q u e n c y c o m p o n e n t o f t h e i n c i d e n t w a v e pressure can be w r i t t e n as:
l ^ ^ A - A j t ó ? [cos(Xi + Xj)] exp(fc,z)
i=i j = i N N
l^Yl
(M-"^'^
U< " ^}) ^^P(.'<iZ)-AiAjCüj^ COS ( x i - Xj) exp((/<i + kj)z))1=1
j=i
42 S, Rajendran et al./Applied Ocean Research 54 (2016) 39-52
w h e r e A, =A{oji), Aj =A(^Wj) are t h e a m p l i t u d e s o f a m o n o c h r o -m a t i c w a v e a n d s,- a n d sj are the r a n d o -m phases, a n d Xj = +
T h e s u m a n d d i f f e r e n c e f r e q u e n c y c o m p o n e n t o f t h e s e c o n d -o r d e r w a v e e l e v a t i -o n can be w r i t t e n as: >1 ^ N N ^ N N ^Y^AjAjCOi^ cosiXi + X j ) ( 1 7 ) l=\ j = i A,v4j ( o j i ^ - o ) / ) cos ( x i - X j ) T h e s e c o n d - o r d e r F r o u d e - K r y l o v pressure is i n t e g r a t e d o v e r t h e i n s t a n t a n e o u s w e t t e d h u l l surface. p ( 2 ) M . d ; d x ! = 1,3 p ( 2 ) ( X x N i ) d / d x Ls JCx ( 1 8 ) h u l l w e t t e d s u r f a c e is d e f i n e d , at e a c h t i m e step, t h e h y d r o d y n a m i c h u l l p r o p e r t i e s ( t h e m e m o r y f u n c t i o n s , i n f i n i t e f r e q u e n c y - a d d e d masses a n d r a d i a t i o n r e s t o r i n g c o e f f i c i e n t s ) are c a l c u l a t e d f r o m 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 o f t h e sections u n d e r t h e i n c i d e n t w a v e e l e v a t i o n . These s e c t i o n a l c o e f f i c i e n t s are c a l c u l a t e d t h r o u g h i n t e r -p o l a t i o n o f t h e -p r e - c a l c u l a t e d s e c t i o n a l c o e f f i c i e n t s c a l c u l a t e d f o r a n u m b e r o f d r a f t s . The s e c t i o n a l c o e f f i c i e n t s are i n t e g r a t e d t o c a l -c u l a t e t h e g l o b a l -c o e f f i -c i e n t s a n d Eqs. ( 2 0 ) a n d ( 2 1 ) are e m p l o y e d o n t h e g l o b a l c o e f f i c i e n t s f o r t h e c a l c u l a t i o n o f t h e m e m o r y f u n c -t i o n a n d -t h e h y d r o d y n a m i c r e s -t o r i n g c o e f f i c i e n -t . Pseudo n o n l i n e a r r a d i a t i o n f o r c e s are c a l c u l a t e d w i t h Eq. ( 1 9 ) . 2.J.4. Diffraction force
2.1.4.1. Nonlinear diffraction force. Since t h e c u r r e n t s t u d y is r e s t r i c t e d to h e a d seas, t h e d i f f r a c t i o n f o r c e s are c a l c u l a t e d u s i n g 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 . For h e a d seas, ^ = 1 8 0 ° , t h e l i n e a r d i f f r a c t i o n f o r c e a n d m o m e n t c a n be w r i t t e n as f o l l o w i n g . Surge d i f f r a c t i o n f o r c e i n t i m e d o m a i n f o r h e a d sea c o n d i t i o n is c a l c u l a t e d as g i v e n b e l o w . T h e first- a n d s e c o n d - o r d e r i n c i d e n t w a v e p o t e n t i a l s d o n o t a l l o w t h e c a l c u l a t i o n s above the m e a n w a t e r l e v e l . T h e r e f o r e , a b o v e t h e m e a n w a t e r l e v e l , t h e p r e s s u r e is a s s u m e d to be h y d r o s t a t i c a n d is c a l c u l a t e d f r o m t h e w a v e e l e v a t i o n w h i c h i n c l u d e s b o t h first- a n d s e c o n d o r d e r w a v e e l e v a t i o n s . The results f r o m t h e b o d y n o n l i n -ear t i m e c o d e w i t h t h e s e c o n d - o r d e r F r o u d e - K r y l o v p r e s s u r e are d e n o t e d as 'TDNL-STOKES' i n t h e f o l l o w i n g figures 2.1.3. Radiation forces
2.1.3.1. Time domain nonlinear radiation force. The r a d i a t i o n f o r c e s are r e p r e s e n t e d b y i n f i n i t e f r e q u e n c y a d d e d mass, r a d i a t i o n r e s t o r a t i o n c o e f f i c i e n t s a n d c o n v o l u t i o n i n t e g r a l s o f m e m o r y f u n c -tion. T h e time d o m a i n r a d i a t i o n f o r c e s are c a l c u l a t e d b y : F f [ t ) = Re N j = i e'¥{icoejau + bu)dx (wej.af, - b f i ) ( 2 2 ) Heave d i f f r a c t i o n f o r c e i n t i m e d o m a i n is c a l c u l a t e d u s i n g e q u a -tion: F3°(t) = Re
N
f-e¥
: •. I e"'j\a)e.a33 - ib33)dxu
ikjX'^+kjZ^, f f c ( f ) = A ^ ^ j ( t ) + / K ™ ( t - r ) ? j ( T ) d T + C , ^ ^ j ( t ) /<,i = l , 3 , 5 ( 1 9 ) ( 2 3 ) P i t c h d i f f r a c t i o n m o m e n t i n t i m e d o m a i n is c a l c u l a t e d u s i n g e q u a t i o n : Fm) = Re N j = i ( « e j Q a s -11^33)xdx + ( - ! z ( o j e j a i i - ! Ö n ) ) d x "I \ i i e ' V + ¥ ( ü ) e , a 3 3 - i Ö 3 3 ) d x ( ( 2 4 ) The- m e m o r y f u n c t i o n s , K,"?, a n d r a d i a t i o n r e s t o r a t i o n c o e f f i c i e n t s , C ^ , are c a l c u l a t e d u s 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 ( f ) = | / ( B k j ( o j ) c o s 6 J t ) d a ) J 0 A^-A,jico) ,0 I {K^[T)sm{o)t))Ax lo /c,j = 1 , 3 , 5 ( 2 0 ) ( 2 1 ) w h e r e /4(-j((») a n d B ; y ( w ) r e p r e s e n t f r e q u e n c y - d e p e n d e n t g l o b a l - a d d e d masses a n d d a m p i n g c o e f f i c i e n t s .For t h e b o d y n o n l i n e a r m e t h o d , the r a d i a t i o n f o r c e s are c a l -c u l a t e d at t h e e x a -c t w e t t e d s u r f a -c e o f t h e s h i p h u l l . A p r a -c t i -c a l e n g i n e e r i n g a p p r o a c h is f o l l o w e d i n s t e a d o f s o l v i n g t h e exact b o d y b o u n d a r y c o n d i t i o n . This f a c i l i t a t e s faster c o m p u t a t i o n a n d easy i m p l e m e n t a t i o n o f t h e m e t h o d . It consists o n u p d a t i n g t h e h y d r o -d y n a m i c p r o p e r t i e s at each time i n s t a n t base-d o n t h e h u l l w e t t e -d s u r f a c e u n d e r the i n c i d e n t w a v e e l e v a t i o n . Once t h e i n s t a n t a n e o u s
w h e r e 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 , z is t h e v e r t i c a l d i s -t a n c e b e -t w e e n c e n -t r o i d o f -t h e u n d e r w a -t e r s e c -t i o n a n d -t h e p o i n -t w h e r e t h e v e r t i c a l b e n d i n g m o m e n t is t o be c a l c u l a t e d , We is t h e e n c o u n t e r f r e q u e n c y b e t w e e n t h e s h i p a n d t h e w a v e s , OJ is t h e w a v e f r e q u e n c y a n d 0i, is t h e phase angle o f t h e i n p u t w a v e . Surge a n d heave s e c t i o n a l - a d d e d mass c o e f f i c i e n t s are d e n o t e d b y a n a n d 033, r e s p e c t i v e l y , a n d t h e surge a n d heave p o t e n t i a l d a m p i n g are d e n o t e d b y b n a n d 033. The s u p e r s c r i p t 'A' associated w i t h t h e sec-t i o n a l h y d r o d y n a m i c c o e f f i c i e n sec-t s s h o w s sec-t h e c o e f f i c i e n sec-t s o f sec-t h e a f sec-t m o s t s e c t i o n .
The d i f f r a c t i o n forces are c a l c u l a t e d f o r t h e i n s t a n t a n e o u s w e t -t e d s u r f a c e area o f -t h e s h i p u n d e r -t h e i n c i d e n -t w a v e p r o f i l e , as i n -t h e case o f t h e r a d i a t i o n forces, a n d t h u s b e c o m e s b o d y n o n l i n e a r f o r t h e f u l l y b o d y n o n l i n e a r c a l c u l a t i o n . The a d d e d mass a n d d a m p -i n g c o e f f -i c -i e n t s are u p d a t e d f o r t h e s u b m e r g e d s e c t -i o n f o r each time s t e p . Use o f these h y d r o d y n a m i c c o e f f i c i e n t s u p d a t e f o r each time s t e p are u s e d i n Eqs. ( 2 2 ) - ( 2 4 ) i n o r d e r t o c a l c u l a t e t h e b o d y n o n l i n e a r surge, heave a n d p i t c h d i f f r a c t i o n f o r c e s a n d m o m e n t s , r e s p e c t i v e l y .
S. Rajendran etal/Applied Ocean Researcii 54 (2016)39-52 43
(m)
Fig. 1. Body plan of the cruise vessel.
2.1.5. Hydrostatic and green water force
B o t h t h e p a r t i a l a n d f u l l y n o n l i n e a r m e t h o d s c a l c u l a t e t h e h y d r o s t a t i c forces f o r t h e v a r y i n g i n s t a n t a n e o u s w e t t e d surface area u n d e r t h e i n c i d e n t w a v e p r o f i l e u s i n g Eq. ( 2 5 ) F " ( M ) = - p g ^ / z . ( x , 0 . n ; d , d . , ^ 3 ^ ^ ^^^^ Zr = fe- ( x f s + Z f l ) w h e r e is the h y d r o s t a t i c f o r c e c a l c u l a t e d f o r s u b m e r g e d sec-t i o n s a n d Zr is sec-t h e s u b m e r g e d d e p sec-t h o f sec-t h e s sec-t r i p c a l c u l a sec-t e d f o r each t i m e step, ^ i , ^3, § 5 , r e s p e c t i v e l y , are t h e surge, heave a n d p i t c h m o t i o n c a l c u l a t e d f o r each s t r i p . V e r t i c a l forces a c t i n g o n t h e deck d u r i n g t h e presence o f green w a t e r is c a l c u l a t e d u s i n g m o m e n t u m m e t h o d [ 2 2 ] .
2.2. Structural dynamic wave loads
The w a v e - i n d u c e d s t r u c t u r a l d y n a m i c loads at a s h i p s e c t i o n are o b t a i n e d f r o m t h e d i f f e r e n c e b e t w e e n the i n e r t i a f o r c e a n d the s u m o f t h e h y d r o d y n a m i c f o r c e a c t i n g o n the p a r t o f the h u l l f o r -w a r d o f t h e section. The m e t h o d t o c a l c u l a t e t h e c o m p o h e n t s o f t h e forces are c o n s i s t e n t w i t h t h e m e t h o d s discussed i n t h e Section 2 . 1 , w h e r e t h e discussion w a s h e l d t o calculate t h e s h i p m o t i o n s . Here, t h e forces are c a l c u l a t e d o n l y u p t o t h e p a r t i c u l a r s e c t i o n o f i n t e r e s t i n s t e a d o f t h e e n t i r e s h i p h u l l . The l o n g i t u d i n a l shear f o r c e , v e r t i c a l shear f o r c e a n d v e r t i c a l b e n d i n g m o m e n t s are c a l c u -l a t e d f o r t h e sections o f i n t e r e s t . For t h e b e n d i n g m o m e n t s , p o s i t i v e a n d n e g a t i v e m o m e n t s d e n o t e t h e h o g g i n g a n d sagging m o m e n t s , r e s p e c t i v e l y .
3. Experimental setup
T h e s h i p m o d e l w a s tested 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 o f t h e El Pardo M o d e l Basin (CEHIPAR) i n M a d r i d . The l a b o r a t o r y is a w e l l - e q u i p p e d f a c i l i t y p r e p a r e d t o c o n d u c t sea k e e p i n g a n d m a n e u v e r i n g test o f s h i p m o d e l s a n d o f f s h o r e s t r u c t u r e s . The t a n k is 1 5 0 m l o n g , 3 0 m w i d e a n d 5 m deep. The w a v e g e n e r a t o r is able t o p r o d u c e r e g u l a r w a v e s a n d i r r e g u l a r waves, b o t h l o n g crested a n d s h o r t crested. W a v e gauges w e r e f i x e d o n t h e carriage 5.42 m i n f r o n t o f t h e m o d e l a n d m o v e d a l o n g w i t h the m o d e l .A s e l f - p r o p e l l e d m o d e l w a s used f o r t h e test; t h e r e f o r e i t w a s p o s s i b l e t o m e a s u r e surge m o t i o n . Fig. 1 s h o w s t h e b o d y p l a n o f t h e m o d e l . M o d e l scale w a s 1:50 a n d t h e m o d e l c o n s i s t e d o f f o u r seg-m e n t s . The absolute seg-m o t i o n s w e r e seg-m e a s u r e d b y a n o p t i c a l systeseg-m, r e s i s t i v e w a v e sensors fixed o n t h e carriage m e a s u r e d t h e w a v e e l e v a t i o n a n d resistive w a v e sensors fixed o n t h e m o d e l m e a s u r e d r e l a t i v e m o t i o n . S t r a i n gauges m o u n t e d o n t h e b a c k b o n e at m i d -s h i p m e a -s u r e d v e r t i c a l b e n d i n g m o m e n t . The-se -s t r a i n gauge-s w e r e
Table 1
Main particulars of the cruise vessel.
Length between perpendiculars [m], Lpp 216.8
Breadth (m],B„i 32.2
Draught [ m l . T 7.2
Displacement [t], A 34081
Block coefficient [-], Cb 0.66
Longitudinal center of gravity from AP |m]
Xcg (w.r.t aft perpendicular) 99.6
Vertical center of gravity from base line[m]
Zcg (w.r.t keel line) 15.3
Transversal metacentric height [m], GMt 2.5
Scale ofthe model 1:50
Moment of inertia for roll (w.r.t center of gravity) (kg.m^) 8.2647e + 09 Moment of inertia for pitch (w.r.t center of gravity) (kg.m^) 1.0483e + 011
m o u n t e d at t h e same h e i g h t as t h e ship's v e r t i c a l center o f g r a v -i t y p o s -i t -i o n (15 m f r o m basel-ine), t h e r e f o r e t h e v e r t -i c a l b e n d -i n g m o m e n t is m e a s u r e d at VCG o f t h e s h i p . Each o f t h e s e g m e n t s w a s c a l i b r a t e d i n o r d e r to o b t a i n t h e d e s i r e d g l o b a l w e i g h t , LCG a n d i n e r t i a . Table 1 gives details a b o u t t h e m a i n p a r t i c u l a r s o f t h e cruise vessel. The w e i g h t a n d mass m o m e n t o f i n e r t i a f o r each s e g m e n t o f t h e m o d e l is g i v e n i n Table 2. The m o m e n t s o f i n e r t i a are w i t h respect t o t h e center o f the g r a v i t y o f e a c h s e g m e n t . LCG a n d VCG o f each s e g m e n t are w i t h respect t o t h e a f t p e r p e n d i c u l a r o f t h e s h i p a n d t h e keel l i n e , r e s p e c t i v e l y .
4. Results
4.1. Incident wave analysis
I n t h i s section, the statistics o f t h e i n c o m i n g w a v e s m e a s u r e d i n t h e t a n k a n d t h e ones f r o m t h e n u m e r i c a l m e t h o d are a n a l y z e d , a n d t h e p r o b a b i l i t y o f exceedance o f t h e i r peaks is c o m p a r e d . The e x p e r i m e n t a l w a v e s r e p r e s e n t t h e e x t r e m e seastates t h a t t h e s h i p w o u l d e n c o u n t e r d u r i n g its 20 y e a r l i f e span. The final sea states are c h o s e n f r o m t h e lACS scatter d i a g r a m c o r r e s p o n d i n g t o t h e N o r t h sea w a v e c l i m a t o l o g y . The w a v e s w e r e m e a s u r e d 5.42 m ahead o f t h e c e n t e r o f g r a v i t y o f t h e m o d e l u s i n g t h e r e s i s t i v e w a v e sensors fixed o n t h e carriage t h a t m o v e d at t h e speed o f t h e m o d e l . The tests w e r e c o n d u c t e d i n l o n g crested head seas w i t h a F r o u d e n u m b e r o f 0.067.
The n u m e r i c a l w a v e s are g e n e r a t e d f r o m t h e JONSWAP spec-t r u m w i spec-t h spec-t h e characspec-terisspec-tics v a l u e s s a m e as spec-t h e m e a s u r e d w a v e s . Table 3 gives t h e s i g n i f i c a n t w a v e h e i g h t (Hs), peak p e r i o d (Tp) a n d t h e peakedness p a r a m e t e r ( 7 ) o f t h e m e a s u r e d a n d t h e n u m e r i -cal i r r e g u l a r sea states. The n u m e r i c a l w a v e s c o n s i s t o f l i n e a r a n d s e c o n d - o r d e r Stokes w a v e s a n d are r e p r e s e n t e d b y t h e a c r o n y m ' N u m l ' a n d ' N u m 2 ' , r e s p e c t i v e l y , i n t h e tables. Table 4 c o m p a r e s t h e m e a n , variance, ske,wness a n d k u r t o s i s o f t h e m e a s u r e d a n d t h e n u m e r i c a l w a v e s . Hcmax a n d Hmax d e n o t e t h e l a r g e s t w a v e crest h e i g h t a n d t h e largest w a v e h e i g h t , r e s p e c t i v e l y .
One can observe t h a t t h e m e a n values o f t h e m e a s u r e d w a v e s are s l i g h t l y n e g a t i v e a n d are a l w a y s less t h a n 1% o f t h e s i g n i f i c a n t w a v e h e i g h t . This d e v i a t i o n is v e r y s m a l l a n d is p r o b a b l y d u e t o t h e s m a l l w a v e s created b y t h e resistive w a v e sensors w h i c h m o v e at t h e speed o f t h e vessel. The n u m e r i c a l l i n e a r w a v e s have a zero m e a n since t h e y are c r e a t e d f r o m t h e z e r o m e a n G a u s s i a n process. Due t o the deep w a t e r a s s u m p t i o n , n o set d o w n is e x p e r i e n c e d is b y t h e s e c o n d - o r d e r Stokes w a v e s a n d t h u s t h e y h a v e a z e r o m e a n v a l u e .
The percentage o f e r r o r b e t w e e n t h e v a r i a n c e c a l c u l a t e d f r o m t h e n u m e r i c a l a n d t h e m e a s u r e d i n p u t w a v e time series is a l w a y s less t h a n 10%. V a r i a n c e c a l c u l a t e d f r o m t h e time series, w h i c h is also m e a s u r e o f t h e area u n d e r t h e e n e r g y s p e c t r u m , s h o w s t h e q u a n -t i -t a -t i v e a g r e e m e n -t b e -t w e e n -t h e e n e r g y s p e c -t r u m c a l c u l a -t e d f r o m t h e m e a s u r e d w a v e s a n d t h e t a r g e t s p e c t r u m (JONSWAP) used f o r
44 S, Rajendran et al. /Applied Ocean Researcii 54 (2016) 39-52
Table 2
Weight distribution of the cruise vessel.
Section 1 Section 2 Section 3 Section 4 Total
M(l(g) Zcg /xx /vv 8.S812e + 006 32.03 0.00 13.53 1.5993e + 009 3.0675e + 009 1.0042e + 007 81.85 0,00 17.63 1.9401 e +009 2.9688e + 009 1.0426e + 007 133.76 0.00 15.48 1.5071e + 009 2.6256e + 009 5.0321e + 006 196.24 0.00 15.89 9.0809e + 008 2.105e + 009 3.4081e + 007 102.09 0.00 15.68 6.0286e + 009 1.0483e + 011 Table 3
Characteristic values of the irregular sea state.
Sea state 1 2 3 Hs(m) 9.7 11.0 11.0 Tp(sec) 12 12 12 y 1 1 3.3 Duration(sec) 5400 5400 .• 9000 t h e n u m e r i c a l w a v e s i m u l a t i o n . The e x p e r i m e n t a l a n d t h e s e c o n d -o r d e r St-okes w a v e s are p -o s i t i v e l y s k e w e d a n d t h e l i n e a r w a v e s have a zero s k e w n e s s . The s k e w n e s s v a l u e s h o w s t h e a s y m m e t r y i n t h e d i s t r i b u t i o n o f t h e peaks a n d is a g o o d m e a s u r e o f t h e n o n l i n e a r i t y . T h e e x p e r i m e n t a l a n d t h e s e c o n d - o r d e r Stokes w a v e s are charac-t e r i z e d w i charac-t h l a r g e r crescharac-ts a n d s h a l l o w e r charac-t r o u g h s w h i c h gives charac-t h e m a p o s i t i v e skewness. S i m i l a r l y , t h e 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 s e c o n d - o r d e r Stokes w a v e have large values f o r K u r t o s i s . A pos-i t pos-i v e v a l u e o f k u r t o s pos-i s r e p r e s e n t s l a r g e r peak a n d h e a v pos-i e r tapos-ils pos-i n t h e d i s t r i b u t i o n w h e n c o m p a r e d w i t h its Gaussian d i s t r i b u t i o n . The n u m e r i c a l l i n e a r w a v e s have l o w v a l u e o f k u r t o s i s w h i c h is a n i n d i -c a t i o n t h a t t h e peaks f o l l o w t h e Gaussian d i s t r i b u t i o n . The k u r t o s i s v a l u e o f t h e e x p e r i m e n t a l w a v e s f r o m t h e seastate 111 is 60% larger t h a n t h e n u m e r i c a l s e c o n d o r d e r Stokes w a v e w h i c h is an i n d i c a -t i o n -t h a -t u n l i k e -t h e o -t h e r -t w o seas-ta-tes, -t h e n o n l i n e a r i -t y o f -t h e sea state 111 is above t h e o r d e r o f 2 . The t i m e series o f t h e l i n e a r i n c i d e n t w a v e s , s u m a n d d i f f e r -ence f r e q u e n c y c o m p o n e n t o f t h e s e c o n d - o r d e r Stokes w a v e s a n d t h e t o t a l s e c o n d - o r d e r Stokes are p r e s e n t e d i n Fig. 2 . T w o p l o t s are g i v e n i n w h i c h t h e l e f t o n e s h o w s t h e time series c a l c u l a t e d i n a s t a t i o n a r y f r a m e o f r e f e r e n c e a n d t h e g r a p h o n t h e r i g h t s h o w s the time series e n c o u n t e r e d b y t h e s h i p w h i l e m o v i n g w i t h a Froude n u m b e r o f 0.067. The g i v e n w a v e e l e v a t i o n is c h o s e n f r o m t h e Seastate I (Hs = 9 . 7 m , Tp = 1 2 s , a n d y = 1) a n d is c a l c u l a t e d at t h e l o n g i t u d i n a l c e n t e r o f g r a v i t y o f t h e s h i p (8.8 m a f t o f t h e m i d -s h i p ) . The peak-s o f t h e l i n e a r w a v e -s c o i n c i d e w i t h t h e peak-s o f t h e s e c o n d - o r d e r w a v e c o m p o n e n t s a n d t h e s e c o n d - o r d e r Stokes w a v e is c h a r a c t e r i z e d w i t h n a r r o w e r and h i g h e r crest a n d w i d e r a n d s h a l l o w e r t r o u g h . Since t h e e n e r g y o f t h e w a v e s has t o r e m a i n t h e s a m e i n b o t h t h e f r a m e s o f r e f e r e n c e , w h i c h r e s u l t s i n change i n t h e a m p l i t u d e o f t h e h a r m o n i c s o f t h e e n c o u n t e r e d w a v e , t h e w a v e e l e v a t i o n a p p e a r d i f f e r e n t i y i n these f r a m e s o f references. Fig. 3 c o m p a r e s t h e p r o b a b i l i t y o f exceedance o f t h e m e a s u r e d w a v e peaks w i t h t h e n u m e r i c a l ones i n t h e m o v i n g f r a m e o f r e f e r -ence. '+ve' a n d ' - v e ' s y m b o l s s h o w the crest a n d t r o u g h peaks. The c o m p a r i s o n w i t h t h e n u m e r i c a l l i n e a r a n d s e c o n d - o r d e r w a v e s is g i v e n o n t h e l e f t a n d right-side o f t h e g r a p h , r e s p e c t i v e l y . One can o b s e r v e t h a t t h e peaks o f t h e l i n e a r w a v e s are s y m m e t r i c a l l y d i s -t r i b u -t e d a n d f o l l o w c l o s e l y -t h e -t r o u g h peaks o f -t h e e x p e r i m e n -t a l w a v e s , p a r t i c u l a r i y f o r seastates I a n d I I I . H o w e v e r , s l i g h t a s y m m e -t r y c o u l d be o b s e r v e d i n -t h e -t a i l o f -t h e d i s -t r i b u -t i o n w h i c h is p r o b a b l y d u e to t h e s t a t i s t i c a l u n c e r t a i n t y associated w i t h t h e l a r g e r w a v e s i n t h e w a v e series a n d c o u l d be r e m o v e d b y u s i n g m o r e r e a l i z a t i o n s o f t h e w a v e time series. H o w e v e r , t h e p r e s e n t s t u d y m a k e s o f o n l y o n e r e a l i z a t i o n o f t h e time series. Since t h e o b j e c t i v e o f t h e p a p e r is t o a n a l y z e t h e e f f e c t o f the s e c o n d o r d e r Stokes w a v e s a n d c o m -pare t h e results i n t h e l i n e a r i n c i d e n t a l waves, t h i s s h o u l d n o t be a m a j o r c o n c e r n here. Q u a l i t a t i v e a g r e e m e n t b e t w e e n t h e n u m e r i -cal s e c o n d - o r d e r Stokes w a v e peaks a n d t h e m e a s u r e d w a v e peaks are g o o d , e x c e p t a t t h e t a i l o f t h e d i s t r i b u t i o n . The n u m e r i c a l peak values are s l i g h t l y l o w e r t h a n t h e m e a s u r e d ones. T h i s is p r o b a b l y d u e to t h e s l i g h t l y l o w e r e n e r g y ( v a r i a n c e ) o f t h e n u m e r i c a l w a v e s . ( F i g . 4 )
4.2. Vertical ship motions
The v e r t i c a l s h i p m o t i o n s are c a l c u l a t e d at t h e c e n t e r o f g r a v i t y o f t h e s h i p . The statistics o f t h e heave r e s p o n s e is g i v e n i n Table 5. A p o s i t i v e a n d n e g a t i v e heave v a l u e s h o w s t h e e m e r g e n c e a n d s u b -m e r g e n c e o f t h e s h i p . The e x p e r i -m e n t a l heave -m o t i o n has n e g a t i v e m e a n values, i.e. t h e s h i p experiences a sinkage a n d i t increases as t h e sea-state increases. Since t h e s h i p m o v e s w i t h a v e r y l o w Froude n u m b e r a n d t h e sinkage v a l u e increases as t h e sea gets steeper, t h i s e f f e c t is p r o b a b l y n o t due t o t h e s t e a d y s p e e d o f t h e vessel. T h e n u m e r i c a l r e s u l t s i n l i n e a r i n c i d e n t w a v e s have p o s i t i v e m e a n v a l u e w h i l e t h e ones i n s e c o n d - o r d e r w a v e s have s l i g h t l y n e g a t i v e m e a n v a l u e s . The p e r c e n t a g e o f e r r o r b e t w e e n t h e v a r i a n c e o f t h e m e a s u r e d heave responses a n d t h e n u m e r i c a l heave responses i n s e c o n d o r d e r Stokes w a v e s are a l w a y s less t h a n 12%. T h e n u m e r i -cal a n d t h e e x p e r i m e n t a l heave responses are p o s i t i v e l y s k e w e d , i.e. t h e heave responses, w i t h t h e i r m e a n values r e m o v e d , have larger crest a n d s m a l l e r t r o u g h . The s k e w n e s s values o f t h e 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 r e s u l t s c o m p a r e s w e l l e x c e p t f o r t h e l a r g e s t sea state. 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 r e s u l t s are c h a r a c t e r i z e d
Table 4
Statistics ofthe experimental waves.
Sea state 1 Sea state 2 Sea state 3
Hs = 9.7m, Tp = 1 2 s , g = l Hs = l l m , r p = 12s,; Hs = l l m , T p = 1 2 s , g = 3.3
Exp. N u m l ' Num2'' Exp. N u m l Num2 Exp. N u m l Num2
Mean ^ 4 0.00 0.00 -0.07 0.00 0.00 -0.08 0.00 0.00 Variance 6.1 5.58 5.68 7.88 7.51 7.69 8.41 7.63 7.78 Skewness 0.21 0.00 0,27 0.29 0.04 0.35 0.299 0.00 0.29 Kurtosis 0.26 0.10 0.234 0.21 - 0 . 0 2 0.19 0.332 0.07 0.20 HCmax 11.6 9.80 10.70 14,2 10.68 12.75 13.2 9.98 11.81 Hmax 17.6 19.50 16.66 21.5 18.97 19.87 21.25 20.1 18.95 3 N u m l = linear waves.
S. Rajendran et al. /Applied Ocean Research 54 (2016) 39-52 45
Fig. 2. Time series of ttie linear incident waves, the sum and difference frequency components of the second-order Stokes waves, and the total second-order Stokes wave elevation, Graph on the left side shows the time series measured in a stationary frame of reference and the graph on the right side shows the time series encountered by the ship moving with a Froude number of 0.067. (Hs = 9.7 m, Tp = 12 s, and y = t).
w i t h p o s i t i v e l<urtosis v a l u e . The k u r t o s i s v a l u e o f the m e a s u r e d responses increases as t h e s e v e r i t y o f the sea state increases.
The p r o b a b i l i t y o f exceedance o f t h e 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 heave responses are g i v e n i n Fig. 5. '+ve' a n d ' v e ' s y m -b o l s s h o w s t h e e m e r g e n c e a n d s u -b m e r g e n c e o f t h e s h i p . Except at t h e t a i l o f t h e d i s t r i b u t i o n , n e g a t i v e heave peaks are l a r g e r t h a n t h e p o s i t i v e ones f o r t h e m e a s u r e d responses a n d t h e a s y m m e t r y i n t h e i r d i s t r i b u t i o n increases as t h e s e v e r i t y o f t h e sea state increases. The n u m e r i c a l heave peaks i n s e c o n d - o r d e r Stokes w a v e s f o l l o w s i m i l a r t r e n d f o r s m a l l e r sea states. The a m p l i t u d e o f t h e cross spec-t r u m c a l c u l a spec-t e d b e spec-t w e e n spec-t h e heave m o spec-t i o n s a n d spec-t h e i n c i d e n spec-t w a v e e l e v a t i o n f o r sea state 1 is s h o w n i n Fig. 5. The phase angles c a l -c u l a t e d f r o m the -cross s p e -c t r u m is also g i v e n f o r the n u m e r i -c a l w a v e s . The cross s p e c t r u m can be c a l c u l a t e d u s i n g t h e e q u a t i o n Sxyia) = (X(ct)) * Y{ui)At)/2jtN, w h e r e X* is t h e c o n j u g a t e o f t h e F o u r i e r t r a n s f o r m o f t h e i n c i d e n t w a v e , 7 is t h e F o u r i e r t r a n s f o r m o f heave response a n d N is t h e l e n g t h o f t h e data. I n this s t u d y , t h e cross s p e c t r u m is c a l c u l a t e d u s i n g B l a c k m a n - T u k e y a l g o r i t h m .
[ 2 3 ] . One c a n observe t h a t a r o u n d t h e i n p u t w a v e peak p e r i o d , n u m e r i c a l heave response is i n phase w i t h t h e i n c i d e n t w a v e e l e v a -t i o n a-t -t h e c e n -t e r o f g r a v i -t y , i.e. -t h e cruise s h i p e x p e r i e n c e s n e g a -t i v e h e a v e m o t i o n w h e n i t e n c o u n t e r s t h e i n c i d e n t w a v e crest a t t h e f o r w a r d r e g i o n a n d t h e t r o u g h close t o t h e m i d s h i p . This r e s u l t i n l a r g e r e x c i t i n g f o r c e as t h e pressure u n d e r t h e crest o f t h e s e c o n d -o r d e r St-okes is l a r g e r t h a n the l i n e a r -ones, a n d t h u s p r -o b a b l y r e s u l t s i n l a r g e r n e g a t i v e heave m o t i o n t h a n t h e r e s p o n s e i n l i n e a r w a v e s . The h e a v e peaks are b e t t e r r e p r o d u c e d i n t h e n u m e r i c a l m o d e l w i t h s e c o n d - o r d e r f r e e s u r f a c e w a v e s . T h e m e a n , v a r i a n c e , s k e w n e s s a n d t h e k u r t o s i s values o f t h e n u m e r i c a l a n d t h e e x p e r i m e n t a l p i t c h m o t i o n are c o m p a r e d i n T a b l e 6. P o s i t i v e p i t c h v a l u e d e n o t e s t h e b o w d o w n c o n d i t i o n . B o t h t h e n u m e r i c a l a n d t h e p l e a s u r e d p i t c h m o r i o n s h a v e n e g l i g i b l e m e a n v a l u e s . The largest d i f f e r e n c e i n t h e v a r i a n c e o f t h e 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 -i t c h m o t -i o n -is f o u n d f o r sea state 11 a n d t h e n u m e r i c a l v a l u e is 19% less t h a n t h e m e a s u r e d ones. B o t h t h e 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 responses are p o s i t i v e l y s k e w e d . Table 5
Statistics of the experimental and numerical heave motion..
Sea state 1 Sea state 2 . Sea state 3
Hs = 9.7m,rp = 1 2 s , g = l Hs = l l m , r p = 1 2 s , g = l Hs = l 1 m, Tp = 1 2 s £ = 3 . 3
Exp. Numl= Num2'' Exp. N u m l Num2 Exp. Numl Num2 Mean - 0 . 0 5 0.03 -0.03 - 0 . 1 5 0.02 - 0 . 0 5 - 0 . 1 7 0.06 0.00 Variance 0.51 0.59 0.57 0.78 0.81 0.79 0.65 0.73 0.72 Skewness 0.09 0.10 0.11 0.09 0.07 0.08 0.01 0.18 0.16 Kurtosis 0.08 0.13 0.28 0.34 0.12 0.20 0.94 0.24 0.40
' N u m l = linear waves.
46 S. Rajendran et al. /Applied Ocean Research 54 (2016) 39-52
Fig. 3. Probability of exceedance ofthe numerical and experimental incident wave peaks.
w^hich m e a n s t h a t t h e b o w d o w n c o n d i t i o n is severe. The n e g a -t i v e l&l-t;ur-tosis v a l u e o f -t h e e x p e r i m e n -t a l p i -t c h m o -t i o n f o r sea s -t a -t e I I denotes t h e f l a t t e r p e a k a n d l i g h t e r t a i l o f t h e d i s t r i b u t i o n c o m -p a r e d w i t h its Gaussian d i s t r i b u t i o n , i.e. t h e -p e a k v a l u e s at t h e t a i !
o f d i s t r i b u t i o n are s m a l l e r t h a n its l i n e a r d i s t r i b u t i o n . The k u r t o s i s v a l u e s are p o s i t i v e f o r t h e n u m e r i c a l responses. The p r o b a b i l i t y o f exceedance o f t h e peaks o f t h e p i t c h m o t i o n f r o m t h e m e a s u r e d a n d t h e n u m e r i c a l responses are c o m p a r e d i n Fig. 6. '+ve' a n d ' - v e '
48 5. Rajendran et al. / Applied Ocean Research 54 (2016)39-52
Fig. 5 . Cross spectrum and the phase angle of the heave motions.
Table 6
Statistics of the experimental and numerical pitch motion.
Sea state 1 Sea state 2 Sea state 3
Hs = 9.7m, Tp = 1 2 s , g = l Hs = l l m , r p = 1 2 s , g = l Hs = l l m , T p = 1 2 s , g = 3.3
Exp. N u m l ' Num2'' Exp. N u m l Num2 Exp. Numl Num2
Mean - 0 . 1 7 -0.03 -0.01 0.02 - 0 . 0 5 - 0 . 0 3 0.03 - 0 . 0 4 - 0 . 0 2 (deg) Variance 2.1 2.1 1.92 3.2 2.8 2.6 3.6 3.7 3.4 (deg2) Skewness 0.12 0.22 0.21 0.09 0.22 0.22 0.15 0.24 0.22 Kurtosis - 0 . 0 8 0.41 0.48 - 0 . 2 9 0.20 0.22 - 0 . 0 5 0.15 0.25 ' N u m l = linear waves.
Num2 =2nd order Stokes waves.
s y m b o l s s h o w s t h e b o w d o w n and u p c o n d i t i o n . The peaks are a s y m m e t r i c a l l y d i s t r i b u t e d f o r t h e m e a s u r e d a n d t h e n u m e r i c a l responses a n d t h e a g r e e m e n t is good b e t w e e n t h e m e a s u r e d a n d t h e n u m e r i c a l response i n s e c o n d - o r d e r Stokes w a v e s , p a r t i c u l a r l y f o r t h e l a r g e s t sea state.
4.3. Vertical bending moment
The s t a t i s t i c s o f t h e m e a s u r e d a n d t h e n u m e r i c a l l y c a l c u l a t e d v e r t i c a l b e n d i n g m o m e n t at t h e m i d s h i p is g i v e n i n Table 7. The n e g a t i v e a n d p o s i t i v e m o m e n t s i n d i c a t e t h e s a g g i n g a n d h o g g i n g m o m e n t s . The m e a s u r e d a n d t h e n u m e r i c a l results have n e g a t i v e m e a n v a l u e a n d t h e l a r g e s t d i f f e r e n c e i n t h e v a r i a n c e c a l c u l a t e d f o r t h e m e a s u r e d a n d t h e c a l c u l a t e d response i n s e c o n d - o r d e r w a v e s is less t h a n 6%. N e g a t i v e s k e w n e s s v a l u e o f t h e e x p e r i m e n t a l a n d n u m e r i c a l r e s u l t s s h o w s t h a t t h e sagging peaks are l a r g e r t h a n t h e h o g g i n g peaks. K u r t o s i s v a l u e is a l w a y s p o s i t i v e f o r t h e n u m e r i -cal r e s u l t s i n s e c o n d - o r d e r Stokes waves, w h i l e i t is n e g a t i v e f o r t h e e x p e r i m e n t a l results i n l a r g e r sea states. H o w e v e r , b o t h r e s u l t s s h o w large k u r t o s i s values.
Table 7
Statistics of the experimental and numerical vertical bending moment at the midship.
Fig. 7 c o m p a r e s t h e p r o b a b i l i t y o f exceedance o f t h e m e a s u r e d a n d t h e n u m e r i c a l v e r t i c a l b e n d i n g m o m e n t at t h e m i d s h i p . '+ve' a n d ' - v e ' s y m b o l s s h o w s t h e h o g g i n g a n d s a g g i n g b e n d i n g m o m e n t peaks. The v e r t i c a l b e n d i n g m o m e n t is n o n - d i m e n s i o n a l i z e d w i t h pgHsLpp^B, w h e r e p is t h e d e n s i t y o f w a t e r , g is t h e a c c e l e r a t i o n due t o g r a v i t y , Hs is t h e s i g n i f i c a n t w a v e h e i g h t , Lpp is the 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 n d B is t h e b r e a d t h o f t h e s h i p . Previous studies b y the a u t h o r s , [ 14,15,16] o n t h e v e r t i c a l b e n d i n g m o m e n t i n l i n e a r i n c i d e n t w a v e s s h o w e d t h a t t h e b o d y n o n l i n e a r h y d r o -d y n a m i c forces plays a n i m p o r t a n t r o l e i n t h e c a l c u l a t i o n o f the v e r t i c a l b e n d i n g m o m e n t a c t i n g o n t h e ships w i t h large b o w flare angle a t t h e m e a n w a t e r l e v e l . A l i n e a r r a d i a t i o n a n d d i f f r a c t i o n f o r c e results i n large o v e r e s t i m a t i o n o f t h e s a g g i n g peaks. This is m a i n l y due t o the u n d e r d a m p i n g t h a t results f r o m t h e l o w e r sub-m e r g e d b r e a d t h to d e p t h r a t i o o f t h e f o r w a r d sections. The h o g g i n g peaks f r o m t h e n u m e r i c a l r e s u l t s i n s e c o n d - o r d e r Stokes w a v e s are i n v e r y g o o d a g r e e m e n t w i t h t h e m e a s u r e d ones, w h i l e t h e n u m e r -ical r e s u l t s i n l i n e a r i n c i d e n t w a v e s s l i g h t l y o v e r e s t i m a t e t h e m . The n u m e r i c a l s a g g i n g peaks c a l c u l a t e d i n s e c o n d - o r d e r Stokes w a v e s are i n 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 results, e x c e p t a t the
Sea state 1 Sea state 2 Sea state 3
Hs = 9.7m,rp = 1 2 s , g = l Ws = l l m , Tp = 1 2 s , g = l Hs = 11 m, Tp = 12s,g=3.3
Exp. Numl= Num2i' Exp. N u m l Num2 Exp. N u m l Num2
Mean - 6 . 9 - 0 . 3 9 - 1 3 . 5 - 7 . 9 - 1 . 6 4 - 1 7 . 7 - 8 . 1 7 - 4 . 1 3 - 1 6 . 4 (KNm) X 10'' Variance 3.97 4.67 4.00 5.34 6.25 5.42 6.38 7.67 6.72 (KNm)2 X 10" Skewness -0.31 -0.15 -0.19 - 0 . 2 6 - 0 . 2 0 - 0 . 2 4 - 0 . 2 7 - 0 . 2 4 - 0 . 3 2 Kurtosis 0.27 0.00 0.25 - 0 . 2 3 - 0 . 0 8 0.29 - 0 . 1 3 - 0 . 1 9 0.33 ' N u m l = linear waves.
S. Rajendran et al. /Applied Ocean Research 54 (2016) 39-52 51
Rel.motion @ Bow Rel.motion @ Bow
Fig. 8. Probability of exceedance ofthe numerical and experimental relative motion peaks at the bow.
t a i l o f t h e d i s t r i b u t i o n , a n d at t h e t a i l o f the d i s t r i b u t i o n the sagging peaks are o v e r e s t i m a t e d .
As s t a t e d b e f o r e , t h e sagging peaks f r o m t h e n u m e r i c a l results i n s e c o n d - o r d e r w a v e s o v e r e s t i m a t e t h e m e a s u r e d ones at t h e t a i l o f t h e d i s t r i b u t i o n . This h a p p e n s w h e n t h e s h i p e n c o u n t e r s v e r y large a m p l i t u d e w a v e s i n t h e r a n d o m sea. Since t h e v e r t i c a l b e n d i n g m o m e n t at t h e m i d s h i p is v e r y m u c h i n f l u e n c e d b y t h e r e l a t i v e m o t i o n at t h e b o w o f t h e s h i p , t h e r e a s o n b e h i n d t h e d i s c r e p a n c y is f u r t h e r i n v e s t i g a t e d f o r t h e s h i p response i n 2 n d sea s t a t e . Fig. 8 s h o w s t h e peaks o f t h e r e l a t i v e w a v e e l e v a t i o n , ^rei. at t h e b o w o f s h i p ( 2 1 5 m f r o m t h e a f t p e r p e n d i c u l a r ) c a l c u l a t e d u s i n g t h e e q u a t i o n f rei = ? w - (§3 - ) , w h e r e is t h e w a v e e l e v a t i o n at b o w , ^sand^s are t h e heave a n d p i t c h m o t i o n s a n d 'x' is t h e distance b e t w e e n t h e b o w a n d t h e l o n g i t u d i n a l c e n t e r o f g r a v i t y ( r e f e r e n c e f r a m e ) . The '+ve' a n d ' - v e ' s y m b o l s s h o w t h e w a v e crest a n d t r o u g h peaks r e l a t i v e t o t h e s h i p . C o m p a r i s o n b e t w e e n the TDNL-STOKES a n d t h e m e a s u r e d results s h o w s a g o o d a g r e e m e n t f o r t h e n e g a t i v e peaks, i.e. t h e r e l a t i v e w a v e t r o u g h s at t h e b o w t h a t is associated w i t h t h e h o g g i n g m o m e n t . H o w e v e r , t h e p o s i t i v e peaks at t h e t a i l o f t h e d i s t r i b u t i o n are o v e r e s t i m a t e d b y t h e n u m e r i c a l m o d e l . The o v e r e s t i m a t i o n i n t h e r e l a t i v e w a v e c r e s t a t the b o w , w h i c h is associated w i t h t h e s a g g i n g m o m e n t , r e s u l t s i n a n o v e r e s t i m a t i o n o f t h e s a g g i n g peaks at the t a i l o f t h e d i s t r i b u t i o n . The n u m e r i c a l m e t h o d , TDNL-AIRY, also overes-t i m a overes-t e overes-the r e l a overes-t i v e w a v e cresoveres-t a n d overes-t r o u g h peaks aoveres-t overes-t h e overes-t a i l o f overes-t h e d i s t r i b u t i o n . H o w e v e r , t h e y do n o t r e s u l t i n o v e r e s t i m a t i o n o f t h e s a g g i n g peaks because o f the l i n e a r F r o u d e - K r y l o v p r e s s u r e u n d e r t h e crest, w h i c h is s m a l l e r t h a n t h e s e c o n d - o r d e r F r o u d e - K r y l o v p r e s s u r e . So t h e F r o u d e - K r y l o v p r e s s u r e u n d e r t h e w a v e crest, w h i c h c o n t r i b u t e s t o t h e v e r t i c a l b e n d i n g m o m e n t a t t h e m i d s h i p , is u n d e r e s t i m a t e d i n t h e TDNL-AIRY m o d e l a n d hence c o m p e n s a t e s t h e o v e r e s t i m a t i o n i n t h e m o t i o n . U n d e r t h e w a v e t r o u g h , t h e F r o u d e - K r y l o v pressure f r o m t h e TDNL-AIRY m o d e l is larger t h a n t h e s e c o n d - o r d e r m o d e l ; t h e r e f o r e t h e y r e s u l t i n a n o v e r e s t i m a t i o n o f t h e h o g g i n g m o m e n t s . I n a d d i t i o n , a n o t h e r p o s s i b l e r e a s o n is t h e c h a n g e i n the w a v e f i e l d a r o u n d t h e h u l l associated w i t h t h e r a d i a t e d a n d s c a t t e r e d w a v e s as o b s e r v e d b y t h e a u t h o r s d u r i n g
t h e i r p r e v i o u s s t u d y [ 1 4 , 1 8 ] . W a t a n a b e et a l . [ 2 4 ] c o n d u c t e d tests o n ships w i t h d i f f e r e n t b o w f l a r e angles a n d f o u n d t h a t t h e ships w i t h p r o n o u n c e d b o w f l a r e s have a t e n d e n c y to s u p p r e s s t h e w a v e field a r o u n d t h e b o w . This y i e l d s t o l o w e r r e l a t i v e m o t i o n at t h e b o w t h a n t h e n u m e r i c a l results t h a t d o n o t t a k e a c c o u n t o f t h e d i s t o r t i o n i n the w a v e field. This w i l l also have a s i g n i f i c a n t e f f e c t o n t h e c a l c u l a t i o n o f t h e m o t i o n s a n d loads a n d needs to b e f u r t h e r i n v e s t i g a t e d .
5. Conclusion
The v e r t i c a l s h i p responses i n e x t r e m e sea c o n d i t i o n are c a l c u -l a t e d u s i n g a b o d y n o n -l i n e a r time d o m a i n m e t h o d based o n s t r i p t h e o r y w i t h w e a k l y n o n l i n e a r f r e e s u r f a c e f o r m u l a t i o n . The r e s u l t s are c o m p a r e d w i t h t h e response m e a s u r e d i n t h e w a v e t a n k a n d also w i t h t h e n u m e r i c a l r e s u l t s t h a t uses t h e l i n e a r f r e e s u r f a c e c o n d i t i o n .
The m e a n , variance, s k e w n e s s a n d k u r t o s i s values o f t h e i n c i d e n t w a v e s f r o m t h e b o d y n o n l i n e a r m e t h o d w i t h t h e w e a k l y n o n l i n e a r f r e e s u r f a c e f o r m u l a t i o n c o m p a r e s w e l l w i t h t h e m e a -s u r e d re-spon-se-s, e x c e p t f o r t h e k u r t o -s i -s v a l u e o f l a r g e -s t -sea -state. The n u m e r i c a l i n c i d e n t w a v e peaks f r o m t h e l i n e a r f r e e s u r f a c e f o r m u l a t i o n is s y m m e t r i c a l l y d i s t r i b u t e d w h i l e t h e ones f r o m s e c o n d - o r d e r f r e e surface is a s y m m e t r i c a l l y d i s t r i b u t e d w i t h n a r r o w e r a n d s h a r p e r crests a n d flatter t r o u g h s . The n u m e r i c a l w a v e peaks f r o m t h e s e c o n d - o r d e r w a v e s q u a l i t a t i v e l y f o l l o w t h e m e a s u r e d peaks; h o w e v e r its values are l o w e r s t h a n t h e m e a s u r e d ones at t h e t a i l o f t h e d i s t r i b u t i o n . The n u m e r i c a l n e g a t i v e h e a v e peaks f r o m t h e s e c o n d - o r d e r f r e e s u r f a c e f o r m u l a t i o n are i n g o o d a g r e e m e n t w i t h t h e m e a s u r e d ones. U n l i k e t h e r e s u l t s i n l i n e a r f r e e s u r f a c e f o r m u l a t i o n , t h e y q u a l i t a t i v e l y f o l l o w t h e s a m e t r e n d as t h e m e a s u r e d responses w i t h t h e deeper s u b m e r g e n c e a n d s m a l l e r e m e r g e n c e , e x c e p t at t h e t a i l o f t h e d i s t r i b u t i o n . T h e p o s i t i v e peaks are s l i g h t i y o v e r e s t i m a t e d , h o w e v e r t h e r e s u l t s are i m p r o v e d c o m p a r e d w i t h t h e response i n l i n e a r f r e e surface f o r m u l a t i o n . The n u m e r i c a l results f o r the p i t c h m o t i o n are i n f a i r l y g o o d a g r e e m e n t w i t h t h e m e a s u r e d responses. The n u m e r i c a l p i t c h peaks i n t h e
