Experimental and numerical simulations of sloshing behaviour in liquid cargo tanks and its effect on ship motions

EXPERIMENTAL AND NUMERICAL SIMULATIONS OF SLOSHING BEHAVIOUR I N L I Q U I D CARGO TANKS AND OF I T S E F F E E C T ON SHIP MOTIONS

by

N.E. M i k e l i s and J.M.J. Journée R e p o r t n r . 640-P J u l y 1984 I n t e r n a t i o n a l C o n f e r e n c e on N u m e r i c a l Methods f o r T r a n s i e n t a n d C o u p l e d P r o b l e m s . To be h e l d i n V e n i c e , I t a l y , 9 t h - 1 3 t h J u l y , 1984

Delft University of Technology

Ship Hydromechanics Laboratory Mekelweg 2

2628 CD Delft The Netherlands Phone 015 - 78 68 82

EXPERIMENTAL AND NUMERICAL SIMULATIONS OF SLOSHING BEHAVIOUR IN LIQUID CARGO TANKS AND OF ITS EFFECT ON SHIP M O T I O N S

N.E. MIKELIS *, J.M.O. JOURNEE #

* S h i p S u r v e y o r , L l o y d ' s R e g i s t e r o f S h i p p i n g , U.K.

# S c i e n t i f i c O f f i c e r , D e l f t U n i v e r s i t y o f T e c h n o l o g y , H o l l a n d SUMMARY

A t w o - d i m e n s i o n a l f i n i t e - d i f f e r e n c e t r a n s i e n t s o l u t i o n has been a d a p t e d f o r t h e p r e d i c t i o n o f l i q u i d m o t i o n s and i n d u c e d p r e s s u r e s i n p a r t i a l l y f i l l e d s h i p t a n k s . I n t e g r a -t i o n s o f -t h e p r e s s u r e s a r o u n d -t h e -t a n k w a l l s y i e l d -t h e o v e r a l l f o r c e s and moment t r a n s m i t t e d by t h e l i q u i d o n t o t h e t a n k s t r u c t u r e and c o n s e q u e n t l y t o t h e s h i p . The l i q u i d i n d u c e d moment i s c o u p l e d t o an e q u a t i o n d e s c r i b i n g t h e s h i p r o l l i n g m o t i o n and t h i s c o u p l e d e q u a t i o n now p r o v i d e s t h e e x c i t a t i o n f o r t h e s l o s h i n g c o m p u t e r p r o g r a m . E x p e r i m e n t s have been c o n d u c t e d on s c a l e d t a n k s and t h e m e a s u r e d p r e s s u r e s and moment a r e compared w i t h t h e n u m e r i c a l p r e d i c t i o n s . F u r t h e r e x p e r i m e n t s on a s h i p model w h i c h i n c o r p o r a t e d p a r t i a l l y f i l l e d l i q u i d t a n k s p r o v i d e d a t a f o r c o m p a r i s o n s w i t h t h e c o u p l e d s l o s h i n g and s h i p m o t i o n s c o m p u t e r p r o g r a m . 1. INTRODUCTION L i q u i d c a r g o c a r r i e d i n p a r t i a l l y f i l l e d t a n k s r e s p o n d s t o s h i p - m o t i o n s and when t h e e x c i t a t i o n i s n e a r t h e n a t u r a l p e r i o d o f t h e l i q u i d c a r g o , v i o l e n t waves c a n f o r m w h i c h i m p a r t d y n a m i c and i m p a c t p r e s s u r e s t o t h e t a n k w a l l s and c e i l i n g . S l o s h i n g l o a d s c a n be s e v e r e enough t o c a u s e s t r u c t u r a l damage ( e . g . see r e f . / I / ) . I t i s t h e r e f o r e n e c e s s a r y t o p r e d i c t t h e s l o s h i n g r e s p o n s e a t t h e d e s i g n s t a g e . I t must be p o i n t e d o u t however t h a t r e a l i s t i c p r e d i c t i o n s a r e made p a r t i c u l a r l y d i f f i c u l t b y t h e n o n - l i n e a r n a t u r e o f t h e phenomenon and b y t h e l a r g e number o f p a r a m e t e r s a f f e c t i n g i t , such as t a n k g e o m e t r y , l i q u i d - f i l l h e i g h t , p e r i o d and a m p l i t u d e o f e x c i t a t i o n and p o s i t i o n o f t h e c e n t r e o f r o t a t i o n . L l o y d ' s R e g i s t e r o f S h i p p i n g a d o p t e d i n t h e p a s t an a n a l y t i c a l model based on a n o n l i n e a r t h e o r y f o r s h a l l o w -f i l l c a s e s HI w h i l e f o r h i g h - f i l l s ( i . e . r a t i o o f l i q u i d h e i g h t t o c h a r a c t e r i s t i c l e n g t h g r e a t e r t h a n 0.2) t h e s o l u t i o n i s o b t a i n e d f r o m l i n e a r p o t e n t i a l f l o w t h e o r y /3/. The t a n k

i s assumed t w o - d i m e n s i o n a l , w i t h no• i n t e r n a l o b s t r u e t i o r v s - a n d no c e i l i n g . Because o f t h e l i m i t a t i o n s o f t h i s s o l u t i o n , i n the b e g i n n i n g o f 1982 a r e - e x a m i n a t i o n o f t h e p r o b l e m was i n i t i a t e d . I t was t h e n d e c i d e d t o e m p l o y a n u m e r i c a l r a t h e r t h a n a n a l y t i c a l method t o e n s u r e t h e g e n e r a l i t y o f m o d e l l i n g t a n k s h a p e , w h i l e t h e r e q u i r e m e n t t o model t h e t a n k c e i l i n g i n d i c a t e d t h a t a t r a n s i e n t s o l u t i o n was needed i n p r e f e r e n c e t o a s t e a d y s t a t e one. A l i t e r a t u r e s u r v e y /4,5/ soon, i n d i c a t e d t h e most p r o m i s i n g a r e a o f r e c e n t d e v e l o p m e n t . N a v i c k a s e t . a l - / 6 / a p p l i e d t h e SOLA-SURF v a r i a t i o n o f t h e MAC method /7,8/ t o t h e s l o s h i n g p r o b l e m . N a v i c k a s m o d e l l e d a t w o -d i m e n s i o n a l p r i s m a t i c t a n k w i t h a c e i l i n g i n t r a n s l a t i o n a l o s c i l l a t i o n and e x t e n d e d t h e MAC code t o model l i q u i d com-p r e s s i b i l i t y d u r i n g i m com-p a c t s on t h e c e i l i n g a s s u m i n g s m a l l changes o f d e n s i t y . I t was r e p o r t e d / 6 / t h a t c o m p a r i s o n s o f c o m p r e s s i b l e and i n c o m p r e s s i b l e t y p e s o f p r e s s u r e w i t h e x p e r i m e n t s showed q u a l i t a t i v e a g r e e m e n t w h i l e v e r y good a g r e e m e n t was o b s e r v e d on c o m p a r i s o n s o f f r e e s u r f a c e m o t i o n s . F u r t h e r m o r e t h e s o l u t i o n was c l a i m e d t o be n u m e r i c a l l y s t a b l e . A c o p y o f t h e c o m p u t e r code was a c q u i r e d and s u b s e q u e n t l y m o d i f i e d and e x t e n d e d as d e s c r i b e d i n t h e n e x t s e c t i o n . In t h e m i d d l e o f 1983 a s e r i e s o f s l o s h i n g e x p e r i m e n t s was c o n d u c t e d a t t h e S h i p H y d r o m e c h a n i c s L a b o r a t o r y o f t h e D e l f t U n i v e r s i t y o f T e c h n o l o g y . The e x p e r i m e n t s w e r e j o i n t l y c o m m i s s i o n e d by B o e l w e r f n.v. S h i p y a r d o f B e l g i u m and L l o y d ' s R e g i s t e r o f S h i p p i n g . These e x p e r i m e n t s e x a m i n e d t h e e f f e c t s o f l i q u i d - f i l l h e i g h t , a m p l i t u d e and p e r i o d o f h a r m o n i c m o t i o n . The t a n k was e x c i t e d i n r o l l , i n p i t c h and i n r o t a t i o n a b o u t an a x i s a l o n g t h e t a n k d i a g o n a l . T y p i c a l c o m p a r i s o n s between e x p e r i m e n t a l and c o m p u t e d p r e s s u r e s and f r e e s u r f a c e m o t i o n s a r e p r e s e n t e d i n t h i s p a p e r f o r t h e p u r e r o l l and f o r t h e d i a g o n a l e x c i t a t i o n s .

The c o m p u t e d p r e s s u r e s a t each t i m e s t e p o f t h e t r a n s i e n t s o l u t i o n a r e i n t e g r a t e d a r o u n d t h e w a l l s o f t h e t a n k t o p r o -duce t h e o v e r a l l f o r c e s and moment t r a n s m i t t e d f r o m t h e l i q u i d t o t h e t a n k and t h u s t o t h e s h i p . Such c o m p u t a t i o n s f o r t h e r e s u l t i n g moment, a r e a l s o compared i n t h i s p a p e r w i t h a v a i l -a b l e e x p e r i m e n t -a l m e -a s u r e m e n t s . The -a g r e e m e n t t h -a t i s f o u n d i s v e r y s a t i s f a c t o r y and n a t u r a l l y s u g g e s t s i n t e r e s t i n g e x -t e n s i o n s o f -t h e c o m p u -t a -t i o n w i -t h a p p l i c a -t i o n s i n -t h e d e s i g n and t u n i n g o f p a s s i v e a n t i - r o l l t a n k s , i n t h e p r e d i c t i o n o f d y n a m i c f o r c e s on i n t e r n a l s t r u c t u r a l members, and i n a s t u d y of c o u p l i n g between s h i p - m o t i o n s and s l o s h i n g . I n o r d e r t o p r o v i d e a t o o l f o r a s s e s s i n g t h e s e v e r i t y o f t h i s c o u p l i n g when s y n c h r o n i s m between s h i p - m o t i o n s and t h e n a t u r a l p e r i o d o f t h e l i q u i d c a r g o o c c u r s , a s i m p l i f i e d model o f t h e c o u p l e d p r o b l e m has been s e t up, w h e r e b y t h e one d e g r e e o f f r e e d o m r o l l e q u a t i o n o f t h e s h i p ' d r i v e s ' t h e s l o s h i n g p r o g r a m w h i c h i n t u r n c o n t r i b u t e s t h e a d d i t i o n a l l i q u i d c a r g o moment t o t h e e q u a t i o n o f m o t i o n . An e x t e n s i o n o f t f i i s s o l u t i o n t o a c c o u n t f o r o t h e r s h i p m o t i o n s ( e . g . sway and h e a v e ) i s p r e s e n t l y

underv/ay. F i n a l l y , t h e p r e d i c t i o n s f o r t h e c o u p l e d p r o b l e m a r e compared w i t h u n p u b l i s h e d measurements f r o m e x p e r i m e n t s c o m m i s s i o n e d by B o e l w e r f n . v . S h i p y a r d on a s h i p model w i t h t h r e e c a r g o t a n k s p a r t i a l l y f i l l e d w i t h l i q u i d , i n f r e e r o l l and i n beam waves a t z e r o f o r w a r d s p e e d . A g a i n t h e c o m p a r i s o n i s f a v o u r a b l e . 2. THEORETICAL METHOD The t w o - d i m e n s i o n a l f i n i t e - d i f f e r e n c e a p p r o a c h u s e d , i s o f t e n r e f e r r e d t o as t h e 'Marker and C e l l ' t e c h n i q u e d e v e l o p e d o r i g i n a l l y a t Los Alamos / 7 , 8 / . E s s e n t i a l l y t h e N a v i e r - S t o k e s e q u a t i o n s a r e s o l v e d f o r each c e l l o f t h e c o m p u t a t i o n a l mesh i n . c o n j u n c t i o n w i t h t h e a p p r o p r i a t e b o u n d a r y c o n d i t i o n s a n d a n c i l l a r y e q u a t i o n s . The s o l u t i o n i s a d v a n c e d t h r o u g h t i m e u s i n g a ' s n a p s h o t ' p r i n c i p l e and e n a b l e s v i s c o u s t r a n s i e n t f l u i d f l o w p r o b l e m s t o be t r e a t e d . A d d i t i o n a l b o u n d a r y c o n -d i t i o n s are. use-d t o i n t r o -d u c e f r e e s u r f a c e s i n t o a s i m u l a t i o n . TI ME- 21 .01 S E C T I M E - 2 3 . 8 S E C T I M E - 2 3 . 2 3 S E C T I M E - 2 0 . 9 2 S E C F i g . 1 N u m e r i c a l s i m u l a t i o n s a t q u a r t e r p e r i o d i n t e r v a l s S i n c e t h e i n t e n t i o n a t L l o y d ' s R e g i s t e r was t o p r o d u c e a c o m p u t e r code t h a t c o u l d c o p e w i t h a w i d e v a r i e t y o f t a n k

shapes a n d a n a l y s e t h e r e s u l t i n g d a t a f r o m many a s p e c t s , many m o d i f i c a t i o n s and e x t e n s i o n s t o t h e s o u r c e code by N a v i c k a s e t a l / 6 / , were i n t r o d u c e d . B r i e f l y : t h e f o r m u l a t i o n o f b o u n d a r y c o n d i t i o n s was a l t e r e d and as a r e s u l t t a n k s w i t h c h a m f e r e d c o r n e r s , t a n k s w i t h a p p r e c i a b l e i n t e r n a l s t r u c t u r e and U-tanks d e s i g n e d f o r p a s s i v e r o l l s t a b i l i s a t i o n c a n now be i n v e s t i g a t e d . A l s o , r o t a t i o n a l e x c i t a t i o n was i n t r o d u c e d s i n c e r o l l i s a most r e l e v a n t m o t i o n t o l i q u i d c a r g o s l o s h i n g . The f o r c e s and moment i n d u c e d by t h e l i q u i d on a t a n k b o u n d a r y ; o r on t h e c o m p l e t e t a n k , a r e now c o m p u t e d b y i n t e -g r a t i n -g p r e s s u r e s a l o n -g t h e r e l e v a n t b o u n d a r i e s . F o r s t a b l e and e f f i c i e n t s i m u l a t i o n , a f a c i l i t y has been added t o c h o o s e a u t o m a t i c a l l y t h e t i m e s t e p by i n s p e c t i n g t h e v e l o c i t y d i s t r i b u t i o n and t h u s c h o o s i n g a t i m e i n c r e m e n t a c c o r d i n g l y . A p o s s i b l e b u i l d i n g up o f n u m e r i c a l e r r o r s has been a v o i d e d by s a t i s f y i n g t h e c o n t i n u i t y c o n d i t i o n b e f o r e and a f t e r u p -d a t i n g t h e f r e e s u r f a c e . Computer g r a p h i c s co-de has been i n t r o d u c e d ) ^e'-.g.'"- see F i g . 1 ) , and f i n a l l y i t s h o u l d be m e n t i o n e d t h a t t h e t h e o r y d e v e l o p e d by N a v i c k a s e t a l / 6 / t o h a n d l e f r e e s u r f a c e i m p a c t s on t h e t a n k c e i l i n g has been r a t i o n a l i s e d and i m p r o v e d . T h i s enhancement has y e t t o be f u l l y t e s t e d and c o u l d be t h e s u b j e c t o f a f u t u r e p a p e r .

As a f i r s t s t a g e o f d e v e l o p m e n t f o r a n u m e r i c a l model o f c o u p l e d shii3 m o t i o n s and l i q u i d s l o s h i n g , t h e s h i p i s assumed t o r e s p o n d i n r o l l o n l y . The e q u a t i o n o f m o t i o n i n t h e absence o f i n t e r n a l l i q u i d ( f r o z e n c o n d i t i o n s a y ) i s :

l'(^ + N ^ ( i ) + N 2 * ^ + R^<()+R3(()^ = K ^ s i n a i ^ t ( 1 )

w h e r e , <i> i s t h e r o l l a n g l e and a d o t d e n o t e s a t i m e

d e r i v a t i v e . I , N-j, N 3 , R ] , R 3 a r e t h e i n e r t i a o f t h e

s h i p and added i n e r t i a , l i n e a r and n o n - l i n e a r d a m p i n g and r e s t o r i n g f o r c e c o e f f i c i e n t s r e s p e c t i v e l y . K^^ r e p r e s e n t s the wave e x c i t a t i o n a t f r e q u e n c y w and t h e s u b s c r i p t w d e n o t e s t h e f o r c i n g wave. A l l c o e f f i c i e n t s , e x c e p t t h e r e s t o r i n g f o r c e o n e s , a r e d e p e n d e n t on t h e f r e q u e n c y o f e x c i t a t i o n and c a n be o b t a i n e d by h y d r o d y n a m i c a n a l y s i s . I n t h e s i m p l i f i e d s t u d y p r e s e n t e d h e r e t h e c o e f f i c i e n t s were assumed t o be c o n s t a n t , e x c e p t w h i c h was o b t a i n e d by t h e f o l l o w i n g s i m p l e e x p r e s s i o n f r o m r e f . /9/ ( 2 ) where d i s t h e s h i p ' s d r a u g h t w h i l e a , c , A a r e t h e s l o p e , a m p l i t u d e and l e n g t h o f t h e wave r e s p e c t i v e l y . When t h e s h i p c a r r i e s l i q u i d c a r g o , t h e e q u a t i o n o f m o t i o n t a k e s t h e f o r m :

C l - I f c ) ' ^ ' + N l ^ + N 3 $ 3 + ( R l ^ - ? ^ ^ ) ( ^ + R 3 f ^ = K ^ s i n 6 J t + C ( 3 )

w h e r e I f ^ , W and QG" a r e t h e c a r g o ' s moment o f i n e r t i a , w e i g h t and v e r t i c a l d i s t a n c e f r o m i t s c e n t r o i d t o t h e s h i p ' s c e n t r e o f g r a v i t y r e s p e c t i v e l y , a l l c a l c u l a t e d s t a t i c a l l y . The t e r m C ( ^ v l , ( { ) ) i s t h e s l o s h i n g i n d u c e d moment, computed a t e v e r y t i m e s t e p by t h e p r o g r a m d e s c r i b e d e a r l i e r , ^ and m u H i p l i e d b y t h e t o t a l l e n g t h o f p a r t i a l l y f i l l e d t a n k s .

Thes o l u t i o n t o t h i Thes e q u a t i o n y i e l d Thes t h e v a l u e Thes o f a n g u l a r d i Thes -p l a c e m e n t , v e l o c i t y and a c c e l e r a t i o n w h i c h a r e used t o e x c i t e t h e s l o s h i n g a n a l y s i s i n t h e s u b s e q u e n t t i m e - s t e p . The t i m e phase between t h e r e s u l t i n g m o t i o n and t h e computed l i q u i d i n d u c e d moment d e t e r m i n e s t h e p o r t i o n s o f t h e moment t h a t s h o u l d be d i s t r i b u t e d between t h e t w o d y n a m i c and one s t a t i c t e r m s i n t h e e q u a t i o n o f m o t i o n . The p r e d i c t e d r e s p o n s e i n t h e l i q u i d c a r g o c o n d i t i o n i s d e p e n d e n t on t h e p o s i t i o n o f t h e assumed c e n t r e o f r o t a t i o n when e m p l o y i n g t h e ' a r t i f i c i a l ' model o f t h e o n e d e g r e e o f f r e e d o m e q u a t i o n . H e r e , i n an a t t e m p t t o s i m u l a t e t h e e q u i v a l e n t c e n t r e o f r o t a t i o n o f a s h i p , s a i d t o be somewhere b e t w e e n t h e c e n t r e o f g r a v i t y and t h e f r e e s u r f a c e , t h e wave i n d u c e d moment i n t h e l i q u i d c o n -d i t i o n i s s e t p r o p o r t i o n a l t o t h e r e s t o r i n g f o r c e r e -d u c e -d by t h e f r e e s u r f a c e c o r r e c t i o n t o t h e m e t a c e n t r i c h e i g h t ( K ^ ) . 3. EXPERIMENTS AND COMPARISONS WITH COMPUTATIONS

3.1 P r e s s u r e s and Free S u r f a c e M o t i o n s

F i g . 2 d e p i c t s t h e t r a n s p a r e n t model ( s c a l e 1:40) o f a P r o d u c t s c a r r i e r p r i s m a t i c t a n k used i n t h e e x p e r i m e n t s . DRUCK PDCR-42 m i n i a t u r e s e m i c o n d u c t o r p r e s s u r e t r a n s d u c e r s w i t h a range o f 10 p s i d were e m p l o y e d . The n a t u r a l f r e q u e n c y o f t h e p r e s s u r e t r a n s d u c e r s was l o w e r e d by m o u n t i n g t h e i r h o u s i n g on a s t e e l mass, e l a s t i c a l l y g l u e d t o t h e o u t e r s i d e o f t h e t a n k .

F i g . 2 Model t a n k , p r e s s u r e t r a n s d u c e r s , wave h e i g h t r e c o r d e r ana p a r t o f c o m p u t a t i o n a l g r i d ( i n mm)

-3000 T - T i m e o f f s e t 4 6 t ( s e c ) HR ( m ) 4 6 t ( s e c ) F i g . 3 E x p e r i m e n t a l and computed p r e s s u r e s i n P a s c a l s a t t r a n s d u c e r s R2, R3, R4 and f r e e s u r f a c e h e i g h t : r o l l , h/D = 0 . 6 1 , T = 1.112 s e c , $ = 0.1 r a d s R i . D , (Pa) 30001 T i m e o f f s e t I f Diagonal 1 5 0 0 i 1000 500 0 HR (m)

mm

4 6 t ( s e c ) F i g . 4 E x p e r i m e n t a l and computed p r e s s u r e s i n P a s c a l s a t t r a n s d u c e r s R2,D2,R4,D4,R5,D5 and f r e e s u r f a c e h e i g h t : r o l l and d i a g o n a l , h/D = 0.75, T = 1.057 s e c , $ = 0.1 r a d s 6

-(Pa) • T i m e o f f s e t 4 6 t ( s e c ) F i g . 5 E x p e r i m e n t a l and computed p r e s s u r e s i n P a s c a l s a t t r a n s d u c e r s R l , R3, R6 and f r e e s u r f a c e h e i g h t : r o l l , h/D = 0.90, T = 0.970 s e c , * = 0.1 r a d s R j , D j ( P a ) 4000 3000 1000 0 • T i m e o f f s e t J Ki\ nil I 0 2 4 6 8 t ( s e c ) D i a g o n a l R 7 . D , 10000 8000 + 6 0 0 0 + 4000 2000 0 HR ( m )

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I LL

t o 4 6 t ( s e c ) F i g . 6 E x p e r i m e n t a l and computed p r e s s u r e s i n P a s c a l s a t t r a n s d u c e r s R3,D3,R4,D4,R7,D7 and f r e e s u r f a c e h e i g h t : r o l l and d i a g o n a l , h/D = 0.75, T = 1.056 s e c , * = 0.25 r a d s 7

-A c o n d u c t i v e wave p r o b e was used t o m e a s u r e t h e s u r f a c e e l e v a t i o n as shown on F i g . 2. H a r m o n i c o s c i l l a t i o n s were c a r r i e d o u t w i t h an o s c i l l a t o r mechanism d e v e l o p e d a t t h e D e l f t U n i v e r s i t y o f T e c h n o l o g y and a d a p t e d t o t h e p a r t i c u l a r r e q u i r e m e n t s o f t h e t e s t s . T h r e e t y p e s o f o s c i l l a t i o n s ( r o l l , p i t c h and r o t a t i o n s a b o u t a d i a g o n a l a x i s ) w e r e p e r f o r m e d , f o r a r a n g e o f f o r c i n g p e r i o d s , f o r s i x f i l l i n g l e v e l s and i n some c a s e s f o r t h r e e d i f f e r e n t a m p l i t u d e s o f o s c i l l a t i o n . TheF c o m p a r t s o n s shown i n t h i s p a p e r a r e b e t w e e n t h e i n c o m p r e s s i b l e mode c o m p u t a t i o n ( F i g s 3 t o 6, t h i c k l i n e s ) and ther e x p e r i m e n t a l measurements ( t h i n n e r l i n e s ) . P r e d i c t e d and measured q u a n t i t i e s a r e o f f s e t i n t h e t i m e s c a l e f o r ease o f , - c o m p a r i s o n . • ATI t i m e h i s t o r y t r a c e s a r e p l o t t e d by t h e c o m p u t e r d i r e c t l y and a l l t h e c a s e s shown c o r r e s p o n d t o r e s o n a n c e . F i g s 3 t o 5 show c o m p a r i s o n s o f p r e s s u r e and f r e e s u r f a c e m o t i o n s a t d i f f e r e n t f i l l l e v e l s f o r t h e t a n k r o l l i n g W'it^ a m p l i t u d e o f 0.1 r a d s . F i g s 4 and 6 a r e c o m p a r i s o n s a t two d i f f e r e n t a m p l i t u d e s o f e x c i t a t i o n f o r t h e r o l l i n g t a n k at h/D = 0.75, where h i s t h e l i q u i d h e i g h t and D i s t h e t a n k ' s d e p t h . The same two f i g u r e s a l s o show t h e r e c o r d e d p r e s s u r e s f r o m t h e d i a g o n a l e x c i t a t i o n t e s t s a t t h e same f o r c i n g p e r i o d s as w i t h t h e p u r e r o l l t e s t s . O r i g i n a l l y i t was e x p e c t e d t h a t t h e d i a g o n a l e x c i t a t i o n w o u l d r e s u l t i n a m a g n i f i c a t i o n o f m o t i o n s and p r e s s u r e s as t h e l i q u i d r u s h e s t o t h e t a n k c o r n e r s . However on t h e s e d i a g o n a l t e s t s i t was o b s e r v e d t h a t when t h e f o r c i n g p e r i o d was away f r o m e i t h e r t h e n a t u r a l p e r i o d o f r o l l o r o f p i t c h t h e r e was v e r y l i t t l e l i q u i d m o t i o n . When t h e f o r c i n g p e r i o d was a t t h e n e i g h b o u r h o o d o f e i t h e r n a t u r a l p e r i o d s , t h e n t h e l i q u i d m o t i o n was c o n f i n e d t o t h e p l a n e o f m o t i o n whose p e r i o d was e x c i t e d . I n r e t r o s p e c t t h i s i s a t t r i b u t e d t o t h e f a c t t h a t t h e t a n k b r e a d t h t o l e n g t h r a t i o ( 1 . 8 6 ) r e s u l t s i n d i s t i n c t l y d i f f e r e n t n a t u r a l p e r i o d s o f t h e l i q u i d c a r g o i n r o l l and p i t c h , and t h u s each o f t h e s e modes i s e x c i t e d s e p a r a t e l y .

3.2 L i q u i d I n d u c e d Moments

The f o r c e d h a r m o n i c r o l l e x p e r i m e n t s d e s i g n e d t o measure t h e moment i n d u c e d by t h e l i q u i d m o t i o n w e r e p e r f o r m e d a t t h e D e l f t I n s t i t u t e o f T e c h n o l o g y u s i n g t h e o s c i l l a t o r mechanism d e s c r i b e d i n s e c t i o n 3.1. The moments w e r e m e a s u r e d by two e l e c t r o n i c s t r a i n gauge d y n a m o m e t e r s , w h i c h b r i n g o v e r t h e r o l l m o t i o n and moment f r o m t h e o s c i l l a t o r t o t h e p l a t f o r m where t h e t a n k was f i x e d . F r i c t i o n l o s s e s i n t h e s u p p o r t i n g a x i s a r e m i n i m i s e d by an a i r c u s h i o n b e a r i n g . To t h e o u t p u t of t h e dynamometer a h a r m o n i c a n a l y s i s was a p p l i e d , p r o d u c i n g t h e i n - and o u t - phase c o m p o n e n t s o f t h e m e a s u r e d moment s i g n a l . The i n e r t i a o f t h e t a n k and o f s u p p o r t i n g s t r u c t u r e was a l s o measured by t e s t s on t h e empty t a n k and was s u b -t r a c -t e d f r o m -t h e e x p e r i m e n -t a l v a l u e s so -t h a -t -t h e l a -t -t e r show t h e e f f e c t o f l i q u i d o n l y . For d i r e c t c o m p a r i s o n , t h e com-p u t e d t i m e h i s t o r y o f moment has. been F o u r i e r A n a l y s e d so t h a t .

100K F r o z e n c a r g o t x p e r i m e n t • . C o m p i j i n i i o n " _{• •} - 0.' .Jf. V*-) 2 4 _ 6 8 ? a ( d e g ) 100K J J q u i d c a r g o

TT'

4 F i g . 9 E x p e r i m e n t a l and computed d i m e n s i o n l e s s l o g a r i t h m i c d e c r e m e n t and computed a n g u l a r d i s p l a c e m e n t i n f r e e r o l l f r o m 10° w i t h f r o z e n and w i t h l i q u i d c a r g o . <j>(deg) 18 16 14 12 10 8 Frozen cargo * ( d e g ) . E 1 1 xperiment . o o i p u t a t i o n ' ' ^ " ^ K()eriment i , ^ . -omputation ƒ J . • i • C A Ë 1 1 xperiment . o o i p u t a t i o n ' ' ^ " ^ K()eriment i , ^ . -omputation ƒ J . • c 1 1 xperiment . o o i p u t a t i o n ' ' ^ " ^ K()eriment i , ^ . -omputation ƒ J . • •<« * o O • 0 o 0 1 10 15 20 T ( s e c ) L i q u i d cargo 25 30 . E x p e r i m e n t 4 • Computation/•J ' * Experiment \ . . ' ' C o m p u t a t i o n / J * ' - ^ ' " 10 15 20 T ( s e c ) 25 3Ü F i g . 10 E x p e r i m e n t a l and c o m p u t e d r o l l r e s p o n s e o f s h i p model i n c o r p o r a t i n g 3 t a n k s f i l l e d a t h/D = 0.45 w i t h f r o z e n and w i t h l i q u i d c a r g o i n beam s e a s a t z e r o f o r w a r d s p e e d . - 9

E x p e r i m e n t s C o m p u l a t i o n s 0 , 1 0 _—.— I 0 , 0 6 —.— o 0 , 0 4 _-—— c , ( d e g ) F i g . 7 E x p e r i m e n t a l and c o m p u t e d d i m e n s i o n l e s s a m p l i t u d e and phase a n g l e o f s l o s h i n g moment on a r e c t a n g u l a r t a n k .

E f f e c t o f f i l l l e v e l , $ = 0 . 1 r a d s . 0,020 0.016 0.012 Jo 0,008 0,004 •180 , , > , 1 1 . ~ - « - E x p e r i m e n t V A C o t n p u t a -tion j ' ' 1 ' ' I l i a * ^ 1

: 1

[

^ , , 1 , , 1,0 2 , 0 F i g . 8 E x p e r i m e n t a l and c o m p u t e d d i m e n s i o n l e s s a m p l i t u d e and phase a n g l e o f s l o s h i n g moment ( t a n k o f F i g . 2

i n r o l l , h/D = 0 . 4 5 , ^ = 0.1 r a d s ) .

t h e non d i m e n s i o n a l a m p l i t u d e - ( c = U p / P g B ^ and p h a s e angJe

of t h e f i r s t h a r m o n i c component o f t h e moment was o b t a i n e d .

F i g . 7 shows c o m p a r i s o n s between c o m p u t a t i o n and m e a s u r e -ment f o r a r e c t a n g u l a r t a n k w i t h t h r e e d i f f e r e n t s h a l l o w f i l l

h e i g h t s and a range o f e x c i t a t i o n p e r i o d s . The e x p e r i m e n t a l d a t a have been p u b l i s h e d i n r e f . /9/. O t h e r c o m p a r i s o n s w i t h e x p e r i m e n t a l d a t a f r o m t h e same s o u r c e , e x a m i n i n g t h e e f f e c t o f a m p l i t u d e o f m o t i o n and p o s i t i o n o f c e n t r e o f r o t a t i o n showed, t h e same good a g r e e m e n t . F i g . 8 shows c o m p a r i s o n s w i t h u n p u b l i s h e d e x p e r i m e n t a l d a t a f o r t h e p r i s m a t i c t a n k o f F i g . 2 ( t h i s t i m e model t o f u l l s c a l e r a t i o 1 : 2 5 ) . The o b s e r v e d agreement i n a l l c a s e s i n s t i l l e d c o n f i d e n c e i n t h e computed moment, p r i o r t o d e v e l o p i n g t h e c o u p l e d m o t i o n and s l o s h i n g n u m e r i c a l m o d e l . 3.3 Coupled S h i p M o t i o n s and S l o s h i n g A 1:60 s c a l e d s h i p model o f a p r o d u c t s c a r r i e r was c o n s t r u c t e d and was e q u i p p e d w i t h t h r e e c a r g o t a n k s . A l l 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 a t z e r o f o r w a r d speed i n t o w i n g t a n k I o f t h e D e l f t S h i p H y d r o m e c h a n i c s L a b o r a t o r y and t h e model was p l a c e d t r a n s v e r s e l y i n t h e t a n k t o meet beam waves. The waves were p r o d u c e d by a p n e u m a t i c wave maker and w e r e measured by a c o n d u c t i v e wave p r o b e . D u r i n g t h e f r e e r o l l i n g t e s t s i n s t i l l w a t e r an i n c l i n o m e t e r measured t h e r o l l a n g l e t i m e h i s t o r y , w h i l e a v e r t i c a l g y r o w i t h p o t e n t i o m e t r i c o u t -p u t was e m -p l o y e d i n t h e r o l l i n g e x -p e r i m e n t s i n beam w a v e s . In t h e s e s h i p model e x p e r i m e n t s , t h r e e c o n d i t i o n s w e r e i n -v e s t i g a t e d . Each c o n d i t i o n i n c l u d e d f r e e r o l l i n g t e s t s f r o m an i n i t i a l r o l l a n g l e i n c a l m w a t e r , and f o r c e d r o l l t e s t s f o r a r a n g e o f wave e x c i t a t i o n p e r i o d s . The t h r e e t a n k s b u i l t i n t o t h e s h i p model were f i l l e d up t o 45% o f t h e i r d e p t h w i t h s o l i d ( f r o z e n ) c a r g o i n c o n d i t i o n I , w i t h l i q u i d c a r g o i n c o n d i t i o n I I , and w i t h l i q u i d c a r g o w i t h t h r e e d i f f e r e n t f i l l l e v e l s i n c o n d i t i o n I I I . I n t h e n u m e r i c a l s t u d y t h e f r e e r o l l d a t a f r o m c o n d i t i o n I ( f r o z e n c a r g o ) w e r e used t o e s t i m a t e t h e i n e r t i a c o e f f i c i e n t o f t h e s h i p ( f r o m i t s o b s e r v e d n a t u r a l p e r i o d ) and t h e l i n e a r and non l i n e a r damping c o e f f i c i e n t s ( b y a l e a s t s q u a r e s f i t on t h e o b s e r v e d l o g a r i t h m i c d e c r e m e n t ) , w h e r e a s t h e r e s t o r i n g f o r c e i s o b t a i n e d s t a t i c a l l y . I n t h e a b s e n c e o f e x p e r i m e n t a l d a t a a h y d r o d y n a m i c a n a l y s i s w o u l d p r o v i d e t h e n e c e s s a r y c o e f f i c i e n t s . A s s u m i n g t h e s e c o n s t a n t i n e r t i a and d a m p i n g t e r m s , e q u a t i o n s ( 1 ) and ( 3 ) were used t o s i m u l a t e f r e e and f o r c e d r o l l r e s p o n s e s f o r c o n d i t i o n I and I I r e s p e c t i v e l y . F i g . 9 shows t h e e x p e r i m e n t a l and m e a s u r e d f r e e r o l l r e s p o n s e f o r t h e two c o n d i t i o n s by p l o t s o f l o g a r i t h m i c d e c r e m e n t , K, a g a i n s t a v e r a g e r o l l d i s p l a c e m e n t , w h e r e K-= ( l / 2 7 r ) l n ( n t ) / < i > ( t + T n ) ) (4) i> = 0 . 5 ( < ^ ( t ) + < ( ) ( t + T n ) ) I n t h e f r o z e n c a r g o c o n d i t i o n t h e a g r e e m e n t b e t w e e n n a t u r a l p e r i o d s ( 1 3 . 7 s e e s f o r f u l l - s c a l e s h i p i n b o t h e x p e r i m e n t and

c o m p u t a t i o n ) and b e t w e e n l o g a r i t h m i c d e c r e m e n t d i s t r i b u t i o n s i s v e r y good, as e x p e c t e d . I n t h e l i q u i d c o n d i t i o n t h e o b s e r v e d n a t u r a l p e r i o d i s 21.3 sees and t h e p r e d i c t e d i s 21.4 sees. The d i s t r i b u t i o n s o f l o g a r i t h m i c d e c r e m e n t d i f f e r now by a u n i f o r m s h i f t s u g g e s t i n g t h a t t h e damping i n t h e compu-t a compu-t i o n i s u n d e r e s compu-t i m a compu-t e d . The d i s compu-t r i b u compu-t i o n o f compu-t h e compu-t w o s e compu-t s o f v a l u e s however a r e v e r y s i m i l a r and t h e c o m p u t a t i o n even p r e d i c t s t h e same s c a t t e r seen i n t h e e x p e r i m e n t . C o m p a r i s o n s between e x p e r i m e n t s and c o m p u t a t i o n s f o r t h e f o r c e d r o l l r e s p o n s e in::both. c o n d i t i o n s i s shown on F i g . 10. The a g r e e m e n t i s a g a i n v e r y r e a s o n a b l e . The n e g l i g i b l e m o t i o n s p r e d i c t e d away f r o m - . t h e , s h i p ' s m o d i f i e d n a t u r a l p e r i o d and n e a r t h e l i q u i d c a r g o n a t u r a l p e r i o d , w e r e i n s u f f i c i e n t t o e x c i t e any s e v e r e s l o s h i n g and r e l a t e d s h i p m o t i o n s . T h i s c o n c l u s i o n however a p p l i e s t o t h e c o n d i t i o n s s t u d i e d , and i t i s e a s i l y c o n c e i v a b l e t h a t f o r o t h e r l i q u i d c a r g o f i l l l e v e l s o r f o r a n o t h e r s h i p o r t a n k , s y n c h r o n i s m c o u l d o c c u r w i t h q u i t e d i f f e r e n t r e s p o n s e t o t h a t seen h e r e . F i n a l l y , i t s h o u l d be p o i n t e d o u t t h a t a l t h o u g h c o n d i t i o n I I I has n o t been s i m u l a t e d n u m e r i c a l l y , t h i s c o u l d be a c h i e v e d by s i m u l t a n e o u s l y m o d e l l i n g t h r e e t a n k s w i t h d i f f e r e n t f i l l h e i g h t s . REFERENCES

1. BASS, R.L., BOWLES, E.B. and COX, P.A. - L i q u i d Dynamic Loads i n LNG Cargo T a n k s . T r a n s . SNAME, V o l . 8 8 , PP103-126, 1980.

2. VERHAGEN, J.H.G. and WIJNGAARDEN, L. - N o n - L i n e a r O s c i l l a t i o n s o f F l u i d s i n a C o n t a i n e r J . F l u i d Mech., V o l . 22, p a r t 4, p p 7 3 7 - 7 5 1 , 1965. "

3. BLIXELL, C A . - C a l c u l a t i o n o f W a l l P r e s s u r e s i n a Smooth R e c t a n g u l a r Tank due t o Movement o f L i q u i d s . L l o y d ' s R e g i s t e r R. & T.A.S. R e p o r t No. 5 1 0 8 , 1972. 4. I m p a c t s f r o m L i q u i d Cargo S l o s h i n g . P r o c . o f S e v e n t h

ISSC, P a r i s , p p l l . 3 9 5 - 1 0 5 , 1979. "~~

5. T r a n s i e n t Loads f r o m L i q u i d S l o s h i n g . P r o c . o f E i g h t h

ISSC, P a r i s , p p l l . 3 32-62, 1982.

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H i g h - F i l l L e v e l s i n C l o s e d Tanks. A.S.M.E. W i n t e r M e e t i n g , W a s h i n g t o n D.C, p p l 9 1 - 1 9 8 , 1 9 8 1 .

7. WELCH, J.E., HARLOW, F.H., SHANNON, J.P. and DALY, B.J. - The MAC M e t h o d , a Computing T e c h n i q u e f o r S o l v i n g V i s c o u s , I n c o m p r e s s i b l e , T r a n s i e n t F l u i d - F l o w P r o b l e m s I n v o l v i n g F r e e S u r f a c e s . Los Alamos S c i e n t i f i c L a b o r a t o r y , R e p o r t LA-3425, 1 9 6 5 .

8. HIRT, C.W., NICHOLS, B.D. and POMEROY, N.C. - SOLA A N u m e r i c a l S o l u t i o n A l g o r i t h m f o r T r a n s i e n t F l u i d F l o w s . Los Alamos S c i e n t i f i c L a b o r a t o r y , R e p o r t LA-5852, 1 9 7 5 . 9. VAN DEN BOSCH, J . J . and VUGTS, J.H. - R o l l Damping by

F r e e S u r f a c e T a n k s . N e t h e r l a n d s S h i p Research C e n t r e , S h i p b l d g D e p t . , R e p o r t No. 8 3 5 , 1966.

NOTATION B r e a d t h o f t a n k . S l ü s h i n g - i n d u c e d m o m e n t o n t a n k . = ^ g ^ £ p •'•fc' o f i n e r t i a o f s h i p a n d • f r o z e n ' c a r g o . -(2ïï)~^ln (<{) ( t ) / . J ) ( t - i - T ^ ) ) , r o l l d e e r emen t W a v e - i n d u c e d m o m e n t . L e n g t h o f t a n k . H y d r o d y n a m i c d a m p i n g c o e f f i c i e n t s . D i s t a n c e f r o m c e n t r o i d o f f r o z e n c a r g o t o s h i p ' s c e n t r e o f g r a v i t y . R e s t o r i n g m o m e n t c o e f f i c i e n t s . P e r i o d o f f o r c e d m o t i o n . W e i g h t o f l i q u i d c a r g o . S h i p ' s d r a u g h t . G r a v i t a t i o n a l c o n s t a n t . L i q u i d h e i g h t i n . t a n k . T i m e . Wave s l o p e a m p l i t u d e . P h a s e a n g l e b e t w e e n s 1 o s h i n g - i n d u c e d m o m e n t a n d r o l l d i s p l a c e m e n t , n e g a t i v e f o r l a g g i n g m o m e n t . Wave h e i g h t a m p l i t u d e L e n g t h o f i n c i d e n t w a v e . = C/(p-|^^gB L ) , n o n d i m e n s i o n a l a m p l i t u d e o f s l o s h i n g - i n d u c e d m o m e n t . 1 3

-D e n s i t y . = $ s i n ( U t , r o t a t i o n a l d i s p l a c e m e n t i n r o l l . = ( ' f ' ( t ) + ( j . ( t + T ^ ) ) / 2 , a v e r a g e r o l l d i s p l a c e m e n t i n f r e e r o l l i n g t e s t s . F r e q u e n c y o f e x c i t a t i o n . S u b s c r i p t s n w f c D e n o t e s n a t u r a l p e r i o d D e n o t e s i n c i d e n t w a v e . D e n o t e s ' f r o z e n ' c a r g o . ( 1 4