Ocean Engineering 84 (2014) 45-53
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Ocean Engineering
j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / o c e a n e n g
Hydrodynamics of a 2D vessel including internal sloshing flows
Wenhua Zhao Jianmin Yang ^•*, Zhiqiang Hu ^, Longfei Xiao ^, Longbin Tao ^•'^
'State Key Laboratoiy of Ocean Engineering, Slianghai Jiao Tong University, 800 Dongchuan Road, 200240 Shanghai, China^Faculty of Engineering, Computing and Mathematics, The University of Western Australia, 35 Stirling Highway Crawley WA 6009, Australia '^School of Marine Science and Technology, Newcastle University, Newcastle upon Tyne NEt 7RU, United Kingdom
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A R T I C L E I N F O
Article instory: Received 5 August 2013 Accepted 1 March 2014 Available online 16 April 2014 Keywords: FLNG Hydrodynamics Sloshing Filling levels Motion responses A B S T R A C T
A series of t w o - d i m e n s i o n a l m o d e l tests has b e e n carried out to s t u d y the h y d r o d y n a m i c p e r f o r m a n c e of a floating liquefied natural gas ( F L N G ) section i n c l u d i n g i n t e r n a l s l o s h i n g oscillations. T h e reference F L N G section is ballasted w i t h f r e s h w a t e r a n d equivalent solid w e i g h t s respectively, to clarify the c o u p l i n g effects. I n a d d i t i o n , five different ballasting c o n d i t i o n s of the F L N G s e c t i o n w e r e c o n s i d e r e d , to investigate the i n f l u e n c e of filling levels a n d natural frequencies. R e s p o n s e a m p l i t u d e operators ( R A O s ) of both m o t i o n r e s p o n s e s a n d i n t e r n a l s l o s h i n g f l o w s are c a l c u l a t e d b a s e d on m e a s u r e d data. T h e i n n e r -tank s l o s h i n g exhibits obvious effects o n s w a y a n d roll motions, w h i l e little effects o n h e a v e motion. It is o b s e r v e d that the first m o d e of s l o s h i n g c a n significantly affect the global m o t i o n s of t h e vessel, w h i l e that i n h i g h e r m o d e s s h o w s little effects. T h e c o u p l i n g effects are found to be sensitive to the filling levels of the t a n k a n d the roll frequencies of the v e s s e l . W h e t h e r the i n t e r n a l s l o s h i n g a m p l i f i e s or r e d u c e s the global m o t i o n s is related to the difference b e t w e e n t h e first m o d e of s l o s h i n g f r e q u e n c y a n d the roll f r e q u e n c y o f t h e v e s s e l . T h e outcome of this s t u d y w o u l d offer better u n d e r s t a n d i n g on the c o u p l e d h y d r o d y n a m i c s of s h i p m o t i o n s a n d s l o s h i n g flows.
© 2 0 1 4 P u b l i s h e d by E l s e v i e r Ltd.
1. I n t r o d u c t i o n
Due t o t h e g r o w i n g d e m a n d s f o r clean energy such as L i q u e f i e d N a t u r a l Gas (LNG), s t r a n d e d o f f s h o r e gas fields, w h i c h w e r e c o n s i d e r e d t o be q u i t e c h a l l e n g i n g i n t h e past, are b e c o m i n g m o r e a n d m o r e a t t r a c t i v e . Those s t r a n d e d gas flelds are c o m m o n l y located i n deep w a t e r area a n d are isolated f r o m o n s h o r e i n f r a -s t r u c t u r e -s or -s o m e o t h e r o f f -s h o r e pipeline-s, w h i c h i m p e d e t h e i r e x p l o i t a t i o n . To o v e r c o m e those d i f f i c u l t i e s , f l o a t i n g l i q u e f i e d n a t u r a l gas (FLNG), a c o n c e p t o f o f f s h o r e u n i t t h a t consists o f a s h i p - t y p e f l o a t i n g p r o d u c t i o n storage a n d o f f l o a d i n g (FPSO) h u l l e q u i p p e d w i t h LNG storage t a n k s a n d l i q u e f a c t i o n plants, has b e e n p r o p o s e d r e c e n t i y . The w o r l d ' s first n e w l y - b u i l t FLNG s y s t e m is e x p e c t e d t o c o m e o u t i n t h e v e r y near f u t u r e . O n its heels, m o r e t h a n t w e n t i e s are w i l l i n g t o be b u i l t , w h i c h w i l l be w i d e l y used i n t h e e x p l o i t a t i o n o f s t r a n d e d o f f s h o r e gas flelds. This n e w t y p e o f p l a t f o r m is u s u a l l y d e s i g n e d t o be t u r r e t - m o o r e d at t h e t a r g e t gas field d u r i n g its p r o d u c t i o n cycle, a n d t h u s w o u l d be s u b j e c t e d t o v e r y c o m p l e x sea states w h i c h m a y i n d u c e severe m o t i o n responses. The w a v e i n d u c e d m o t i o n s o f the FLNG vessel w o u l d r e s u l t i n v i o l e n t s l o s h i n g o f t h e l i q u i d inside t h e storage t a n k s . I n
* Corresponding author. T e l : -t-86 21 34207053 2101. E-mail addresses: w e n h u a . z h a o ® u w a . e d u . a u (W. Zhao), jmyang@sjtu.edu.cn 0. Yang). http://dx.doi.Org/10.1016/j.oceaneng.2014.03.001 0029-8018 © 2014 Published by Elsevier Ltd. r e t u r n , t h e v i o l e n t s l o s h i n g o s c i l l a t i o n s r e s u l t i n h i g h l y l o c a l i z e d i m p a c t pressures o n t h e w a l l s o f t h e t a n k s , w h i c h m a y cause s t r u c t u r a l damages a n d m a y e v e n s i g n i f i c a n t l y a f f e c t t h e g l o b a l m o t i o n s o f the vessel. This coupled p h e n o m e n o n is one o f the design concerns and is essential f o r FLNG systems i n p r o d u c t i o n o r o f f l o a d i n g operations i n real sea states. M e a n w h i l e , t h e investigation is also o f great importance f o r LNG carriers ( M o l i n et al., 2002). Thus, establish-i n g a systematestablish-ic procedure to s m d y the coupled effects b e t w e e n shestablish-ip m o t i o n s and sloshing flows is necessary.
I n t h e c o u p l e d p r o b l e m b e t w e e n s h i p m o t i o n s a n d s l o s h i n g flows, t h e ship m o t i o n p r o b l e m is one o f t h e classics i n m a r i n e h y d r o d y n a m i c s , w h i l e t h e s l o s h i n g p r o b l e m is s t i l l a c h a l l e n g i n g task due to its s t r o n g n o n i i n e a r i t y . T h e r e have been several studies r e g a r d i n g t h e d y n a m i c s o f n o n l i n e a r s l o s h i n g flows. The m a j o r c o n c e r n o f those earlier studies is t h e s l o s h i n g - i n d u c e d loads such as local i m p u l s i v e pressure d u e t o f o r c e d t r a n s l a t i o n a l o r r o t a t i o n a l e x c i t a t i o n s ( W u et al., 1998; K i m , 2 0 0 1 ; Alcyildiz a n d U n a l , 2 0 0 5 ; Graczyk et al., 2 0 0 7 ) . Readers are d i r e c t e d t o the r e p o r t b y Zhao et al, (2011) f o r m o r e i n f o r m a t i o n a b o u t t h e n o n i i n e a r i t y o f sloshing. A l t h o u g h several studies f o c u s i n g o n t h e n o n l i n e a r s l o s h i n g flows have b e e n available, t h e c o n c l u s i o n s c a n n o t be d i r e c t l y e x t e n d e d t o t h e c o u p l e d p r o b l e m . I n fact, studies w h i c h d i r e c t l y focus o n t h e c o u p l i n g m e c h a n i s m are r e q u i r e d .
W i t h t h e a s s u m p t i o n o f l i n e a r s l o s h i n g flow i n t h e tanks, M o l i n et al., ( 2 0 0 2 ) i n t r o d u c e d a s e m i - a n a l y t i c a l a p p r o a c h a n d f u l l 3 - D
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W. Zhao et al / Ocean Engineering 84 (2014) 45-53p o t e n t i a l f l o w c a l c u l a t i o n s o t i a barge w i t h a p a r t i a l l y - f i l l e d w a t e r t a n k o n deck. Based o n t h e same a s s u m p t i o n , N e w m a n ( 2 0 0 5 ) a d o p t e d a u n i f i e d a p p r o a c h f o r t h e p a n e l code W A M I T t o analyze t h e c o u p l e d p r o b l e m b e t w e e n s h i p m o t i o n a n d s l o s h i n g i n f r e q u e n c y d o m a i n . To c o n s i d e r t h e n o n l i n e a r s l o s h i n g flows, m e t h o d s based o n c o m p u t a t i o n a l fluid d y n a m i c s (CFD) w e r e i n t r o d u c e d . For e x a m p l e , Lee et a l . ( 2 0 0 7 ) d e v e l o p e d a t i m e d o m a i n s i m u l a t i o n scheme, i n w h i c h t h e l i n e a r flow is a s s u m e d for t h e e x t e r n a l waves a n d t h e i n t e r n a l s l o s h i n g is s i m u l a t e d based o n a CFD s c h e m e w i t h a flnite d i f f e r e n c e m e t h o d . I n t h e i r study, o n l y s w a y a n d r o l l m o t i o n s w e r e i n v e s t i g a t e d . Based o n a finite e l e m e n t m e t h o d f o r t h e s i m u l a t i o n o f n o n l i n e a r s l o s h i n g , M i t r a et a l . ( 2 0 1 2 ) c a r r i e d o u t n u m e r i c a l s i m u l a U o n s i n rime d o m a i n w h e r e a s t h e n o n l i n e a r s h i p m o t i o n is a s s u m e d . I n t h e i r study, several factors a f f e c t i n g t h e h y d r o d y n a m i c s o f s h i p m o t i o n s are i n v e s t i g a t e d , a n d s l o s h i n g i n t h e t a n k s is f o u n d t o change s i g n i f i c a n t l y as t h e s i g n i f l c a n t w a v e h e i g h t increases. A n o b v i o u s s h o r t a g e o f t h e CFD m e t h o d is q u i t e time c o n s u m i n g . To o v e r c o m e t h e i n a c c u r a c y o f l i n e a r s l o s h i n g w h i l e m a i n t a i n i n g t e m p o r a l e f f l c i e n c y Rognebakke a n d Faltinsen ( 2 0 0 3 ) e x t e n d e d t h e n o n l i n e a r m e t h o d o f Faltinsen et a l . ( 2 0 0 0 ) a n d d e v e l o p e d a m u l t i -m o d a l -m e t h o d t o p r e d i c t t h e c o u p l e d responses o f an LNG t a n k a n d n o n l i n e a r s l o s h i n g flows. The LNG t a n k is e x c i t e d b y r e g u l a r w a v e s i n sway, a n d t h e n u m e r i c a l s i m u l a t i o n s s h o w reasonable 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 data. H o w e v e r , t h i s m e t h o d can n o w be easily used o n l y f o r t h e t a n k s w i t h s i m p l e a n d r e g u l a r shapes. The available l i t e r a t u r e s h o w s t h a t accurate m o d e l i n g o f t h e c o u p l e d p r o b l e m t h r o u g h n u m e r i c a l s i m u l a t i o n is s t i l l a c h a l l e n g i n g task, because o f t h e s t r o n g n o n l i n e a r s l o s h i n g flows. To b e t t e r u n d e r s t a n d t h e c o u p l e d m e c h a n i s m b e t w e e n s h i p m o t i o n s a n d t h e n o n l i n e a r s l o s h i n g i n s i d e t h e t a n k s , w e s h o u l d also r e s o r t t o t h e t e c h n i q u e o f scaled m o d e test.
T h e eariy e x p e r i m e n t a l studies can be t r a c e d back t o t h e r e p o r t by M i k e l i s e t a l . ( 1 9 8 4 ) , i n w h i c h t h e c o u p l e d p r o b l e m was i n v e s t i g a t e d o n a p r o d u c t carrier s h i p m o d e l c o n s i d e r i n g t h e s l o s h i n g e f f e c t i n p a r t i a l l y filled t a n k s . Francescutto a n d C o n t e n t o ( 1 9 9 4 ) c o n d u c t e d a n e x p e r i m e n t to i n v e s t i g a t e t h e c o u p l i n g e f f e c t b e t w e e n t h e r o l l m o t i o n response o f a s h i p a n d t h e s l o s h i n g flow in a floodable c o m p a r t m e n t , w i t h t h e t a n k s u b j e c t e d t o a b e a m sea c o n d i t i o n . T h e r e s o n a n t peak o f t h e r o l l m o t i o n is f o u n d t o s h i f t t o w a r d s l o w e r f r e q u e n c i e s w i t h a n increase i n fllling levels, a n d a s e c o n d a r y peak is o b s e r v e d at t h e n a t u r a l f r e q u e n c y o f i n t e r n a l s l o s h i n g . M o l i n e t al. ( 2 0 0 2 ) c o n d u c t e d an e x p e r i m e n t o n a barge w i t h a p a r t i a l l y - f i l l e d w a t e r t a n k o n deck. T h e barge w a s m o o r e d by a e r i a l lines a n d a l l o w e d t o oscillate i n t h r e e degrees o f f r e e d o m m o t i o n s such as sway, heave a n d r o l l . A series o f m o d e l tests have b e e n c a r r i e d o u t b y Rognebald<e a n d Faltinsen ( 2 0 0 3 ) , i n w h i c h t h e t a n k m o d e l w i t h d i f f e r e n t fllling c o n d i t i o n s w a s e x c i t e d i n sway b y r e g u l a r waves. T h r o u g h t h e c o m p a r i s o n b e t w e e n n u m e r i c a l s i m u -l a t i o n s a n d e x p e r i m e n t a -l data, t h e y c o n c -l u d e d t h a t t h e use o f a s i m p l i f l e d e q u a t i o n o f m o t i o n w i t h c o n s t a n t c o e f f l c i e n t s is s u f f l -c i e n t t o m o d e l t h e steady-state m o t i o n . N a m a n d K i m ( 2 0 0 7 )
c a r r i e d o u t a series o f m o d e l tests t o i n v e s t i g a t e t h e effects o f s l o s h i n g o n t h e m o t i o n responses o f FLNG. I n t h e i r s m d y , t h e y c o n c l u d e d t h a t t h e c o u p l i n g effects do n o t a l w a y s r e s u l t i n t h e increase o f s l o s h i n g - i n d u c e d pressure a n d t h a t t h e increase o r decrease o f pressure is d e p e n d e n t o n r e s o n a n t c o n d i t i o n . Nasar e t al. ( 2 0 0 8 , 2 0 1 0 ) c o n d u c t e d a n e x p e r i m e n t t o s t u d y t h e p h e n o m -e n o n o f l i q u i d s l o s h i n g i n p a r t i a l l y fill-ed t a n k s m o u n t -e d o n a LNG c a r r i e r exposed t o r e g u l a r b e a m waves. I n t h e i r study, t h r e e b a l l a s t i n g c o n d i t i o n s o f t h e vessel w i t h d i f f e r e n t filling levels, d r a f t s a n d c e n t e r o f gravities w e r e s t u d i e d .
I n t h i s study, m o d e l tests o f a n FLNG s e c t i o n e x c i t e d i n s w a y heave a n d r o l l c o n s i d e r i n g i n n e r - t a n k s l o s h i n g are c a r r i e d o u t a t a scale o f 1:50. Responses o f b o t h t h e vessel m o t i o n s a n d t h e i n t e r n a l s l o s h i n g surface are m e a s u r e d . Spectral analyses h a v e b e e n c o n d u c t e d f o r t h e m e a s u r e d data, t o o b t a i n t h e c o r r e s p o n d -ing response a m p l i t u d e o p e r a t o r s (RAOs). I n t h e e x p e r i m e n t s , t h e FLNG s e c t i o n is b a l l a s t e d w i t h f r e s h w a t e r a n d e q u i v a l e n t s o h d w e i g h t s respectively, f o r t h e p u r p o s e o f c o m p a r i s o n . Based o n t h e c o m p a r i s o n results, t h e c o u p h n g effects b e t w e e n s h i p m o t i o n s a n d sloshing flows are c l a r i f i e d . I n a d d i t i o n , some possible explanations for the c o u p l i n g m e c h a n i s m have also been illusft-ated i n this s t u d y Furthermore, m o d e l tests o f t h e R N G section i n flve d i f f e r e n t ballasting c o n d i t i o n s are conducted, t o s m d y t h e i n f l u e n c e o f filhng levels o f t h e t a n k a n d n a m r a l frequencies o f the vessel.
2. E x p e r i m e n t a l setups
To p h y s i c a l l y u n d e r s t a n d t h e c o u p l i n g m e c h a n i s m b e t w e e n s h i p m o r i o n s a n d s l o s h i n g flows, t w o - d i m e n s i o n a l e x p e r i m e n t s o f an FLNG s e c t i o n c o n t a i n i n g t a n k s w e r e c a r r i e d o u t i n t h e Deep W a t e r Basin at S h a n g h a i Jiao T o n g U n i v e r s i t y .
The t w o - d i m e n s i o n a l m o d e l tests w e r e c a r r i e d o u t at a scale o f 1:50 w i t h d u r a t i o n o f 2 6 m i n ( c o r r e s p o n d i n g t o t h r e e h o u r s i n p r o t o t y p e ( I T I C , 2 0 0 8 ) ) . T h e d i m e n s i o n s o f t h e b a s i n are 50 m X 4 0 m X 10 m a n d t h e w a t e r d e p t h w a s set as 1.5 m c o r r e -s p o n d i n g t o t h e p r o t o t y p e d w a t e r d e p t h o f 7 5 m . T h i -s b a -s i n i-s e q u i p p e d w i t h 2 2 2 m u l t i f l a p w a v e m a k e r panels o n t w o n e i g h -b o r i n g sides. T h e o t h e r t w o n e i g h -b o r i n g sides o f t h e -b a s i n are e q u i p p e d w i t h w a v e a b s o r b i n g beach f o r passive w a v e d i s s i p a t i o n , in o r d e r t o a b s o r b t h e r e f l e c t e d w a v e e n e r g y f r o m t h e b o u n d a r i e s of t h e b a s i n . I n t h i s case, n o p r o b l e m w i t h w a v e r e f l e c t i o n s w i l l be e n c o u n t e r e d d u r i n g t h e m o d e l tests.
A n FLNG s e c t i o n c o n s i s t i n g o f a h u l l s e c t i o n a n d a n i n n e r t a n k has b e e n selected as t h e reference, as s h o w n i n Fig. 1. Five d i f f e r e n t b a l l a s t i n g c o n d i t i o n s f o r t h e FLNG s e c t i o n have b e e n i n v e s t i g a t e d , to i n v e s t i g a t e t h e i n f l u e n c e o f t h e fllling levels o f t h e t a n k a n d n a t u r a l f r e q u e n c i e s o f t h e vessel. As a consequence, a l l t h e p a r a m e t e r s o f t h e FLNG s e c t i o n i n t h e flve b a l l a s t i n g c o n d i t i o n s are d e s i g n e d to k e e p t h e s a m e value e x c e p t t h e fllling levels a n d t h e r a d i u s o f r o l l g y r a t i o n . T h e m a i n p a r t i c u l a r s o f t h e r e f e r e n c e FLNG s e c t i o n i n t h e flve b a l l a s t i n g c o n d i t i o n s h a v e b e e n l i s t e d m Wove probe
-SOOmni-Inner lonk
-1000mm Vessel Ë E oin
W. Zhao et al. / Ocean Engineermg 84 (2014) 45-53 47
Table 1. The i n t e r n a l s l o s h i n g f r e q u e n c i e s &;„ i n d i f f e r e n t surface modes are calculated t h r o u g h t h e linear a p p r o x i m a t i o n e q u a t i o n ( I b r a h i m , 2 0 0 5 )
I = y^n;rgtanh(nw/i//)/(, n •-
1, 2, 3. (1) w h e r e , / denotes t h e l e n g t h o f t h e tank, n indicates t h e surface m o d e n u m b e r , a n d h r e p r e s e n t s t h e f i l l i n g d e p t h o f the t a n k .The m o d e l o f t h e FLNG s e c t i o n is m a d e o f t r a n s p a r e n t p l e x i -glass a n d r e d color is a d d e d t o t h e w a t e r i n t h e tank, f o r t h e sake o f o b s e r v i n g t h e i n t e r n a l s l o s h i n g . A p h o t o o f t h e FLNG s e c t i o n is s h o w n i n Fig. 2. The space b e t w e e n t h e h u l l and t h e i n n e r t a n k can be used t o a c c o m m o d a t e s o l i d w e i g h t s , f a c i l i t a t i n g t h e a d j u s t m e n t of t h e i n e r t i a p a r a m e t e r s o f t h e m o d e l . T h r o u g h t h e a d j u s t m e n t o f t h e l o c a t i o n s o f t h e s o l i d w e i g h t s , w e can keep t h e c e n t e r o f g r a v i t y c o n s t a n t i n d i f f e r e n t filling c o n d i t i o n s . D u r i n g t h e m o d e l tests, the FLNG s e c t i o n m o d e l w a s c o n n e c t e d w i t h a n e q u i p m e n t at t h e c e n t e r o f g r a v i t y (CoG), i n w h i c h case t h e FLNG s e c t i o n is a l l o w e d t o m o v e i n sway, heave a n d r o l l f r e e l y u n d e r w a v e excitations, w h i l e t h e surge, p i t c h a n d y a w m o t i o n s are r e s t r a i n e d . T w o w a v e p r o b e s are fixed a t t h e p r e d e f i n e d locations at each side o f t h e t a n k , f o r t h e m e a s u r e m e n t o f the i n t e r n a l s l o s h i n g flows. Tlie s c h e m a t i c o f t h e e x p e r i m e n t a l setup has b e e n s h o w n i n Fig. 3. As s h o w n i n Fig. 3, t w o s t r i p s o f Plexi-glass w i t h t h e d i m e n s i o n s w e r e a d d e d o n t o each side o f t h e FLNG m o d e l to f o r m a flume. A n e q u i p m e n t consists o f m a n y trusses w e r e also a d o p t e d to p r o v i d e e n o u g h s t i f f n e s s f o r t h e t w o s t r i p s . The w i d t h o f t h e gap b e t w e e n t h e m o d e l a n d t h e w a l l is o n l y 1.0 c m , w h i c h is v e r y s m a l l c o m p a r e d t o the b r e a d t h ( 3 5 c m ) o f t h e FLNG s e c t i o n m o d e l , e n s u r i n g t h e flow c o n d i t i o n s close t o t w o - d i m e n s i o n . Given t h a t the m o d e l is a s e c t i o n o f a n FLNG vessel, t h e sway, heave a n d r o l l m o t i o n s are d e f i n e d i n t h e w a y s h o w n i n Fig. 3(a). To p r e v e n t t h e
m o d e l f r o m d r i f t i n g o f f , a s y s t e m o f s p r i n g s w i t h t h e s t i f f n e s s o f 7.2 N / c m is a d o p t e d . The i n d u c e d e i g e n f r e q u e n c y o f t h e s w a y m o t i o n is 0.28 rad/s, w h i c h is o u t s i d e t h e f r e q u e n c y range o f i n t e r e s t ( t h e i n t e r e s t e d f r e q u e n c i e s are larger t h a n 0.4 rad/s). As s h o w n i n Fig. 3, t h e f r i c t i o n forces i n sway, heave a n d r o l l m o t i o n s are m a i n l y i n d u c e d by r o l l i n g o f b a l l bearings, w h i c h are s m a l l e n o u g h t o be i g n o r e d . D u r i n g t h e m o d e l tests, a p r e t e n s i o n f o r c e o f 2 N is also a p p l i e d t o the springs.
B a n d - l i m i t e d w h i t e noise waves, w h i c h e x h i b i t g o o d l i n e a r i t y , w i t h t h e s i g n i f l c a n t w a v e h e i g h t o f 3.0 m a n d peak f r e q u e n c i e s r a n g i n g f r o m 0.22 rad/s to 1.60 rad/s are selected as t h e e x c i t a t i o n w a v e s . T i m e series o f the i r r e g u l a r w a v e s s h o u l d be p r e - g e n e r a t e d t h r o u g h t h e a l g o r i t h m o f fast Fourier t r a n s f o r m , b e f o r e t h e m o d e l test. The c a l i b r a t i o n results o f t h e w h i t e noise w a v e s are g i v e n i n Fig. 4 . As s h o w n i n Fig. 4, t h e r e is a d i f f e r e n c e b e t w e e n t h e m e a s u r e d s p e c t r u m a n d t h e t a r g e t o n e . A t s o m e f r e q u e n c i e s , w a v e steepness is a l i t t l e larger t h a n 1/20, w h i c h i n d u c e s a c e r t a i n degree o f n o n i i n e a r i t y H o w e v e r , i t w i l l n o t a f f e c t t h e conclusions d r a w n i n t h i s study, because t h i s s t u d y focuses o n t h e effects o f i n t e r n a l s l o s h i n g o n t h e g l o b a l m o t i o n s , i n w h i c h t h e m o d e l is b a l l a s t e d w i t h d i f f e r e n t m a t e r i a l s ( w a t e r a n d steel w e i g h t s , r e s p e c t i v e l y ) b u t excited b y t h e same i n p u t w a v e s . M o r e o v e r , i t w i l l n o t reduce t h e r e l i a b i l i t y o f t h e e x p e r i m e n t a l results. This is because t h e RAOs are calculated f r o m t h e r e s p o n s e s p e c t r u m s d i v i d e d b y t h e m e a s u r e d s p e c t r u m o f t h e w h i t e noise waves, w h i c h can be expressed t l i r o u g h t h e f o l l o w i n g e q u a t i o n :
H{co) = ^SA(m)/Sf(w)
(2)
w h e r e , H(o)) indicates t h e RAOs, SA(W) p r e s e n t s t h e response s p e c t r u m o f t h e m o t i o n s o r i n t e r n a l s l o s h i n g , S^(o)) m e a n s t h e m e a s u r e d s p e c t r u m o f t h e w h i t e n o i s e waves.
Table 1
Principal scantlings of the reference FtNG section in prototype.
Designation Case A Case B Case C Case D Case E
Length (m) 50
Breadth (m) 17.5
Depth (m) 27.5
Draft (m) 13.72
CoG (m) 10.47
Radius of roll gyration (R^) (m) 13.6 13.6 13.6 15.8 20.0
Filling height (h) (m) 18 10 6 6 6
Mass ratio of sloshing water to displaced mass of the hull (r) 0.6 0.33 0.2 0.2 0.2
Roll frequency (<h^) (rad/s) 0.64 0.64 0.64 0.58 0.48
Sloshing frequency in first mode (aji) (rad/s) 0.83 0.71 0.58 0.58 0.58
Fig. 2. Snapshot of the FLNG section model: (a) the liquid ballasting condition, red color has been added to the water in the tanlfli ballasting condition. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of th
48
W. Zhao et al. / Ocean Engineering 84 (2014) 45-53Basin Carriage
Basin Carriage
Water Basih
Solid iqUi
Fig. 3. Scliematic of experimental setups: (a) front view and (b) side view (pliysical are not to scale). The wave probe located at upstream of the tank is defined as "Upstream", and the wave probe at downstream of the tank is defined as "Downstream".
T i m e ( s ) F r e q u e n c y ( r a d / s ) Fig. 4. The white noise wave; (a) time history and (b) power spectrum density function.
D u r i n g t h e m o d e l tests, b o t h t h e m o t i o n responses o f t h e FLNG s e c t i o n a n d t h e t i m e traces o f t h e s l o s h i n g surface i n s i d e t h e t a n k w e r e c a p t u r e d a t a s a m p l i n g f r e q u e n c y o f 25 H z . A n o n - c o n t a c t o p t i c a l s y s t e m c o n s i s t i n g o f t w o i d e n t i c a l l a m p s are fixed o n t h e sides o f t h e w a t e r basin t o m e a s u r e t h e m o t i o n responses o f t h e FLNG section. T h e e r r o r o f t h i s n o n - c o n t a c t o p t i c a l s y s t e m i n t h e m o t i o n m e a s u r e m e n t is w i t h i n 0.1 m m . Resistance t y p e d w a v e p r o b e s w i t h m e a s u r i n g e r r o r less t h a n 1 m m w e r e used i n t h i s study. One o f t h e w a v e probes s h o u l d be m o u n t e d at t h e l o c a t i o n o f t h e vessel m o d e l d u n n g t h e c a l i b r a t i o n o f w a v e s p e c t r u m s , w h i l e be e l i m i n a t e d d u r i n g t h e m o d e l tests. T w o o f t h e m are fixed o n each side o f t h e t a n k t o m e a s u r e t h e surface o f t h e s l o s h i n g f l o w s d u r i n g t h e m o d e l tests. Tension t r a n s d u c e r s are flxed at t h e e n d o f t h e s p r i n g s to m e a s u r e t h e forces a c t i n g o n t h e m . The m e a s u r e m e n t range o f t h e transducers is f r o m 0 t o 100 N , a n d t h e i r m e a s u r i n g e r r o r is w i t h i n 0.4 N . 3. R e s u l t s a n d d i s c u s s i o n E x p e r i m e n t a l w o r k has b e e n c a r r i e d o u t f o r an FLNG s e c t i o n b a l l a s t e d w i t h f r e s h w a t e r a n d e q u i v a l e n t s o l i d w e i g h t s , respec-t i v e l y . The c o m p a r i s o n s o f respec-t h e RAOs o f respec-t h e vessel m o respec-t i o n s are i l l u s t r a t e d i n t h i s section, t o g e t h e r w i t h t h e s l o s h i n g surface at t h e p r e d e f l n e d l o c a t i o n s i n s i d e t h e t a n k . A t t e m p t s are m a d e t o c l a r i f y t h e c o u p l i n g e f f e c t s b e t w e e n ship m o t i o n s a n d s l o s h i n g flows. F u r t h e r m o r e , i n f l u e n c e s o f t h e fllling levels o f t h e i n s i d e t a n k a n d t h e distances b e t w e e n t h e s l o s h i n g f r e q u e n c y a n d r o l l f r e q u e n c y o f t h e FLNG section, w h i c h are o f g r e a t i m p o r t a n c e f o r t h e o p e r a t i o n o f an FLNG s y s t e m , are also expressed i n t h i s s e c t i o n .
3.1. Coupling effects
The responses o f t h e FLNG s e c t i o n ballasted w i t h f r e s h w a t e r a n d e q u i v a l e n t solid w e i g h t s are c o m p a r e d . The FLNG s e c t i o n i n b a l l a s t i n g c o n d i t i o n "Case A " is selected as t h e reference, t h e d e t a i l e d p a r a m e t e r s of w h i c h have b e e n s h o w n i n Table 1. Statistic analyses of t h e responses have b e e n c o n d u c t e d . T h e a b s o l u t e m a x i m u m s w a y a m p l i t u d e is 2.56 m i n t h e solid case, 2.41 m i n t h e l i q u i d case; t h e absolute m a x i m u m r o l l a m p l i t u d e is 1.79° i n t h e solid case a n d 3 . 6 3 ° i n t h e l i q u i d case. T h e m o d e l is a l l o w e d t o m o v e f r e e l y i n sway, heave a n d r o l l u n d e r t h e e x c i t a t i o n o f w h i t e noise i r r e g u l a r w a v e s d u r i n g t h e iTiodel tests. T h e c o m p a r i s o n results o f t h e RAOs f o r t h e vessel m o t i o n s a n d i n n e r - t a n k s l o s h i n g o s c i l l a t i o n s are s h o w n i n Fig. 5.
Fig. 5(a) s h o w s t h e effects o f t h e i n n e r - t a n k s l o s h i n g o n t h e g l o b a l m o t i o n responses i n s w a y As s h o w n i n t h i s flgure, t h e s w a y RAO i n t h e l i q u i d case is s m a l l e r t h a n t h a t i n t h e s o l i d case at t h e e x c i t a t i o n f r e q u e n c i e s s l i g h t l y l a r g e r or s m a l l e r t h a n t h e first m o d e o f s l o s h i n g f r e q u e n c y (0.83 rad/s). W h e n t h e e x c i t a t i o n f r e q u e n c y equals t h e first m o d e o f s l o s h i n g f r e q u e n c y , t h e s w a y response i n t h e l i q u i d case is a l m o s t zero. H o w e v e r , at t h e r e g i o n w h e r e t h e e x c i t a t i o n f r e q u e n c y is larger b u t n o t close t o t h e first m o d e o f s l o s h i n g f r e q u e n c y , t h e p r e s e n t e x p e r i m e n t s c o n f i r m e d t h a t responses i n t h e l i q u i d case are c l e a r l y l a r g e r t h a n t h o s e i n t h e solid case. S i m i l a r p h e n o m e n o n has also b e e n o b s e r v e d b y Rognebakke a n d Faltinsen ( 2 0 0 3 ) a n d Nasar et a l . ( 2 0 1 0 ) . I t is w o r t h n o t h i n g that, based o n t h e s i m i l a r r e s p o n s e o f vessel r o l l m o t i o n t o t h e i n t e r n a l l i q u i d t a n k o b s e i v e d p r e v i o u s l y , a n t i - r o l l t a n k s have b e e n d e v e l o p e d as a p r a c t i c a l s o l u t i o n t o r o l l m o t i o n s u p p r e s s i o n ( G a w a d et al., 2 0 0 1 ) . This i n t e r e s t i n g p h e n o m e n o n
W. Zhao et al. / Ocean Engineering 84 (2014) 45-53 49 Liquid case Solid c a s e 0.4 0.8 1.2 1.6 F r e q u e n c y (rad/s) 2.0 1.0 0.8 0.6 <u 0.4 > CD 03 X 0.2 0.0 Liquid c a s e Solid c a s e 0.4 0.8 1.2 1.6 F r e q u e n c y (rad/s) 2.0 5.0
I "
S. 3-0 Og
2.0 Ö 1.0 0.0 0.0 -Liquid c a s e •Solid c a s e 0.4 0.8 1.2 1.6 F r e q u e n c y (rad/s) 2.0 Upstream Downstream 0.4 0.8 1.2 1.6 F r e q u e n c y (rad/s) 2.0Fig. 5. lïAOs of the vessel motions and inner-tank sloshing oscilladons for the FLNG secdon ballasted in Case A (/i=18 m, r=0.6, (u^=0.64 rad/s, and a, =0.83 rad/s); the model can move freely in sway, heave and roll.
can be e x p l a i n e d b y t h e r e a s o n t h a t t h e r e is a phase s h i f t i n g b e t w e e n i n t e r n a l s l o s h i n g forces a n d t h e e x t e r n a l e x c i t a t i o n forces. As r e p o r t e d i n t h e n u m e r i c a l s i m u l a t i o n s b y K i m et a l . ( 2 0 0 7 ) , t h e phases near t h e first m o d e o f s l o s h i n g f r e q u e n c y s h o w a l m o s t 1 8 0 ° d i f f e r e n c e b e t w e e n i n t e r n a l s l o s h i n g forces a n d e x t e r n a l e x c i t a t i o n forces, i n w h i c h case t h e s l o s h i n g can e f f e c -t i v e l y reduce -t h e g l o b a l m o -t i o n s . As -the e x c i -t a -t i o n f r e q u e n c i e s m o v e a w a y f r o m t h e first m o d e s l o s h i n g frequencies, t h e phase d i f f e r e n c e s t e n d t o a p p r o a c h 0 ° , i n w h i c h case t h e s l o s h i n g w o u l d a m p l i f y t h e g l o b a l m o t i o n s . H o w e v e r , i t is d i f f i c u l t t o m e a s u r e t h e t o t a l forces a c t i n g i n s i d e o r o u t s i d e t h e t a n k d u r i n g t h e m o d e l tests. H o w e v e r , t h e w a v e e l e v a t i o n s i n s i d e o r o u t s i d e t h e t a n k w h i c h are related t o t h e forces have been r e c o r d e d d u r i n g t h e m o d e l tests. Fig, 6 p l o t s t h e phase d i f f e r e n c e s b e t w e e n t h e i n t e r n a l s l o s h i n g oscillations a n d t h e i n c i d e n t w a v e elevations. As s h o w n i n Fig. 6, t h e phase o f t h e i n t e r n a l s l o s h i n g surface s h i f t s 1 8 0 ° w h e n t h e e x c i t a t i o n p e r i o d moves f r o m above t o b e l o w t h e first m o d e o f s l o s h i n g f r e q u e n c y . S i m i l a r p h e n o m e n o n has also been o b s e r v e d b y Francescutto a n d C o n t e n t o ( 1 9 9 9 ) a n d G r a h a m a n d R o d r i g u e z ( 1 9 5 2 ) , A n o t h e r p o i n t t o n o t i c e i n Fig. 5(a) is t h a t t h e s w a y RAOs h o l d s a l m o s t t h e same u n d e r t h e e x c i t a t i o n s w i t h f r e q u e n c i e s m u c h s m a l l e r t h a n t h e first m o d e o f s l o s h i n g f r e q u e n c y f o r b o t h t h e t w o cases.
A n i n s i g n i f i c a n t d i f f e r e n c e i n t h e heave RAO is n o t e d f o r b o t h t h e cases w i t h t h e FLNG s e c t i o n b a l l a s t e d w i t h f r e s h w a t e r a n d e q u i v a l e n t s o l i d w e i g h t s . This m a y p o s s i b l y due t o t h e reason t h a t t h e l i q u i d inside t h e tanks translates w i t h t h e heave m o t i o n o f t h e vessel. Fig. 5 ( b ) also s h o w s t h a t t h e heave RAO t e n d s t o be e q u a l t o 1 i f t h e e x c i t a t i o n f r e q u e n c i e s b e c o m e smaller. This indicates t h a t t h e FLNG s e c t i o n h y d r o s t a t i c a l l y f o l l o w s t h e i n c i d e n t w a v e f r e e s u r f a c e f o r e x c i t e d waves w i t h s m a l l e r f r e q u e n c i e s .
Fig. 5(c) presents large e f f e c t s o f t h e i n n e r - t a n k s l o s h i n g o n t h e g l o b a l m o t i o n responses i n r o l l . The RAO peak o f t h e r o l l m o t i o n s
180 r 120 U p s t r e a m O ) CD 2 , a> (/) cc .c Q. D) c O
OT
•120 -180 L F r e q u e n c y (rad/s)Fig. 6. Phase of the wave elevation at upstream for the FLNG section ballasted in Case A (/!=18 m, r=0.6, a)#=0.64 rad/s, and coi =0.83 rad/s); The x-axis represents the wave excitation frequencies, and the y-axis represents the phases of the internal sloshing relative to external wave elevations.
i n t h e l i q u i d case is observed t o be s i g n i f i c a n t l y r e d u c e d , c o m p a r e d t o t h a t i n t h e s o l i d case. As discussed above, t h e r e a s o n f o r t h i s p h e n o m e n o n is r e l a t e d to t h e p h a s i n g b e t w e e n t h e i n t e r n a l s l o s h i n g a n d t h e i n c i d e n t w a v e elevations ( G a w a d e t al., 2 0 0 1 ; Francescutto a n d C o n t e n t o , 1999; G r a h a m a n d R o d r i g u e z , 1952). Fig. 5 ( d ) i l l u s t r a t e s t h e c o m p a r i s o n r e s u l t s o f t h e i n t e r n a l s l o s h i n g m e a s u r e d at u p s t r e a m a n d d o w n s t r e a m , respectively. T h e RAO o f t h e i n t e r n a l s l o s h i n g at d o w n s t r e a m o f t h e t a n k is l a r g e r t h a n t h a t at u p s t r e a m . This m e a n s t h e i n n e r w a l l at d o w n -s t r e a m o f t h e LNG t a n k w o u l d experience l a r g e r i m p a c t pre-s-sure-s t h a n t h a t a t u p s t r e a m . It can be seen f r o m Fig. 5 ( d ) t h a t t h e RAO o f t h e i n t e r n a l sloshing exhibits obvious peaks at t h e n t h m o d e sloshing frequencies irrespective o f the first m o d e o f sloshing frequency.
50 W. Zhao et al. / Ocean Engineering 84 (2014) 45-53
I n fact, a response p e a k cari,,be o b s e r v e d at t h e f r e q u e n c y o f 0.96 rad/s, w h i c h is 15.7% l a r g è r t h a n t h e first m o d e o f s l o s h i n g f r e q u e n c y (0.83 rad/s). C o r r e s p o n d i n g l y , t h e r e is also a response peak at t h e same e x c i t a t i o n f r e q u e n c y i n t h e s w a y RAOs ( s h o w n i n Fig. 5(a)), w h i c h w e call as " t h e s l o s h i n g - i n d u c e d peak" i n t h i s paper. T h r o u g h t h e c o m p a r i s o n o f Fig. 5(a), (c) a n d ( d ) , one can conclude t h a t t h e i n f l u e n c e o f t h e i n t e r n a l sloshing o n t h e g l o b a l m o t i o n s is m a i n l y d e r i v e d f r o m t h e first m o d e . To p r o v i d e a q u a n t i t a t i v e u n d e r s t a n d i n g , t h e n t h m o d e o f sloshing f r e q u e n c i e s o b t a i n e d f r o m b o t h t h e c a l c u l a t i o n s a n d e x p e r i m e n t s have b e e n l i s t e d i n Table 2. The e x p e r i m e n t a l s l o s h i n g f r e q u e n c i e s are d e r i v e d f r o m t h e f r e q u e n c y analysis results s h o w n i n Fig. 5 ( d ) . As s h o w n i n Table 2, t h e largest d i s c r e p a n c y i n the h i g h e r m o d e s is w i t h i n 0.7%, i n d i c a t i n g t h e l i t t i e c o u p l i n g b e t w e e n g l o b a l m o t i o n s a n d t h e i n t e r n a l s l o s h i n g i n h i g h e r m o d e s . I t can be seen i n Fig. 5 ( d ) t h a t f o r t h e response u n d e r w h i t e noise waves, a l t h o u g h t h e peak values o f s l o s h i n g RAO i n h i g h e r m o d e s are t h e same l e v e l w i t h t h a t o f t h e first m o d e , t h e e n e r g y i n h i g h e r m o d e s is m u c h smaller t h a n t h a t a r o u n d t h e first m o d e sloshing. A s a conse-quence, t h e i n t e r n a l s l o s h i n g i n h i g h e r m o d e s shows l i t t i e e f f e c t o n t h e g l o b a l m o t i o n responses o f t h e vessel. It s h o u l d be n o t e d
Table 2
Frequencies of the internal sloshing In different modes for Case A (in prototype). 1st mode 2nd mode 3rd mode 4th mode 5th mode
m, (s) 0,2 (s) 0)3 (s) fl)4 (s) Ols (s) Calculations 0,83 1.24 1.52 1.75 1.96 Experiments 0.96 1.24 1.53 1.76 1.97 Discrepancy 15.7 0.0 0.7 0.6 0.5 (%) t h a t t h e s l o s h i n g i n h i g h e r m o d e s m a y i n d u c e s i g n i f i c a n t i m p a c t forces o n t h e w a l l s o f the tanks, w h i c h is o f g r e a t c o n c e r n f o r t h e d e s i g n o f t h e t a n k s .
3.2. Influences of filling levels and roll frequencies
As i l l u s t r a t e d i n t h e above section, l i t t l e e f f e c t o f s l o s h i n g can be obsei-ved i n heave m o t i o n o f t h e FLNG s e c t i o n , t h u s t h e section focuses o n l y o n the s w a y a n d r o l l m o t i o n s . Consequently, t h e FLNG s e c t i o n m o d e l is a l l o w e d t o m o v e f r e e l y o n l y i n s w a y a n d r o l l d u r i n g t h e m o d e l tests, w h i l e t h e heave m o t i o n o f t h e m o d e l is r e s t r a i n e d t h r o u g h a l o c k i n g a p p a r a t u s . To c l a r i f y t h e i n f l u e n c e o f filling levels a n d r o l l frequencies, responses o f t h e FLNG section i n five d i f f e r e n t b a l l a s t i n g c o n d i t i o n s ( s h o w n i n Table 1) are inves-tigated. As s h o w n i n Table 1, t h e o n l y d i f f e r e n c e b e t w e e n Cases A , B a n d C lies i n t h e filling levels, w h i c h i n d u c e d i f f e r e n t i n t e r n a l s l o s h i n g f r e q u e n c i e s . W h i l e t h e o n l y d i f f e r e n c e b e t w e e n Cases C, D a n d E lies i n t h e radiuses o f r o l l g y r a t i o n , w h i c h results i n d i f f e r e n t r o l l f r e q u e n c i e s . A l l the o t h e r p a r a m e t e r s o f t h e FLNG section, s u c h as t h e t o t a l mass, d r a f t and c e n t e r o f g r a v i t y , keep t h e same v a l u e i n t h e five b a l l a s t i n g c o n d i t i o n s .
3.3. Influence of filling levels
Fig. 7 shows t h e i n f l u e n c e o f d i f f e r e n t fllling levels o f t h e t a n k . I t can be seen f r o m Fig. 7(c) a n d ( d ) t h a t t h e i n t e r n a l s l o s h i n g o s c i l l a t i o n s b e c o m e m o r e i n t e n s e as t h e filling levels decrease. I t s h o u l d be n o t e d t h a t t h e r e are also i n t e r n a l s l o s h i n g responses as t h e f r e q u e n c i e s are larger t h a n 0.16 rad/s. T h i s is because t h e r e is also e n e r g y d i s t r i b u t i o n w i t h i n t h i s f r e q u e n c y area f o r t h e i n c i d e n t w h i t e noise w a v e s . To f u r t h e r i l l u s t r a t e t h e i n f l u e n c e o f t h e fllling • AJiquid BJiquid C J i q u i d -Solid iiiJr-.^y.-:'-1.2 1.6 F r e q u e n c y (rad/s) 2.0
c
6.0 [5.0-e
5.0 4.0-o
3.0 -O ) '.E 2.0 : «5 OOT
1.0 0.0 -U p s t r e a m A J i q u i d B J i q u i d — C J i q u i d 0.4 0.8 1.2 1.6 F r e q u e n c y (rad/s) 2.0 _ A J i q u i d • B J i q u i d C J i q u i d - S o l i d 0.8 1.2 1.6 Frequency (rad/s) 2.0 d 6.0 r 5.0 4.0 -O<
CC 3.0 D) • £ 2.0 -v> OOT
1.0 0.0 -D o w n s t r e a m _- A J i q u i d • - B J i q u i d C J i q u i d 0.4 0.8 1.2 1.6 F r e q u e n c y (rad/s) 2.0Fig. 7. Response amplitude operators for the FLNG s e c ü o n (heave motion is restrained): (a) sway motions, (b) roll motions, (c) sloshing oscillations at upstream, and (d) sloshing oscillations at downstream. Case A: h = \8 m, r=0.6, w^=0.64 rad/s, and a-,=0.83 rad/s; Case B: h = lO m, r=0.33, « # = 0 . 6 4 rad/s, and a,, =0.71 rad/s; Case C: f i = 6 m , r = 0 . 2 , w^=0.64 rad/s, and [B]=0.58 rad/s.
W. Zhao et al. / Ocean Engineering 84 (2014) 45-53 51
levels a n d p r o v i d e a q u a n t i t a t i v e d e s c r i p d o n , t h e response e n e r g y o f t h e i n t e r n a l s l o s h i n g o s c i l l a d o n s i n d i f f e r e n t f i l l i n g levels is i n v e s t i g a t e d . As a g o o d m e a s u r e o f t h e response energy, z e r o -o r d e r m -o m e n t s -o f t h e s l -o s h i n g -oscillati-ons are c a l c u l a t e d as t h e i n t e g r a t e d area u n d e r t h e spectral d e n s i t y curve, s h o w n i n Table 3. As i l l u s t r a t e d i n Table 3, t h e t o t a l e n e r g y o f t h e i n t e r n a l s l o s h i n g increases s i g n i f i c a n t l y by 103.7% f r o m 0.8267 m ^ i n Case A t o 1.6841 m ^ i n Case B. W h i l e i t increases b y 2.7% f r o m 1.6841 m ^ i n Case B t o 1.7291 m ^ i n Case C. The v a r i a t i o n s o f t h e z e r o - o r d e r m o m e n t s versus t h e d i f f e r e n t f i l l i n g levels have also b e e n p l o t t e d i n Fig. 8. Fig. 8 s h o w s o b v i o u s g r o w t h o f response e n e r g y o f t h e i n t e r n a l s l o s h i n g o s c i l l a d o n s as t h e fllling levels decrease. I t can be f o u n d f r o m Table 3 a n d Fig. 8 t h a t t h e f i l l i n g levels have s i g n i f l c a n t e f f e c t o n t h e i n t e r n a l s l o s h i n g o s c i l l a d o n s .
Fig. 7(a) expresses t h e i n f l u e n c e o f fllling levels o n t h e s w a y RAOs. This figure s h o w s t h a t t h e s l o s h i n g - i n d u c e d peak i n s w a y m o t i o n s s h i f t s t o w a r d s l o w e r f r e q u e n c i e s as t h e fllling levels decrease. This is r e l a t e d t o t h e reason t h a t t h e first m o d e o f s l o s h i n g f r e q u e n c y becomes s m a l l e r as t h e filling levels decrease. One can see t h a t a t t h e first m o d e o f s l o s h i n g f r e q u e n c y , t h e s w a y RAOs i n Case A ( R A O = 0 . 0 4 ) a n d Case B ( R A O = 0 . 1 0 ) t e n d t o be zero d u e to t h e e f f e c t o f t h e i n t e r n a l s l o s h i n g oscillations, w h i l e l i t t l e s i m i l a r t r e n d s can be f o u n d i n Case C ( R A O = 0 . 2 8 ) . This m a y be r e l a t e d t o t h e r e a s o n t h a t t h e r a t i o o f l i q u i d mass t o solid mass is s u c h s m a l l i n Case C ( r = 0.2), c o m p a r e d t o t h a t i n Case A ( r = 0.6) a n d Case B ( r = 0.33), t h a t i t c a n n o t s i g n i f i c a n t l y a f f e c t t h e s w a y RAO at t h e first m o d e o f s l o s h i n g f r e q u e n c y .
Fig. 7 ( b ) i l l u s t r a t e s t h e i n f l u e n c e o f filling levels o n r o l l RAOs. It can be o b s e r v e d f r o m t h i s flgure t h a t t h e s l o s h i n g - i n d u c e d peak i n r o l l RAO curves becomes larger a n d s h i f t s t o w a r d s t h e n a t u r a l f r e q u e n c y o f t h e FLNG s e c t i o n as filling levels decrease. I n fact, d i f f e r e n t fllling levels o f t h e same t a n k i n d u c e d i f f e r e n t flrst m o d e o f s l o s h i n g f r e q u e n c i e s such as t h e f r e q u e n c y o f 0.83 rad/s i n Case A w i t h t h e fllling h e i g h t o f 18 m , t h e f r e q u e n c y o f 0.71 rad/s i n Case B w i t h t h e filling h e i g h t o f 10 m , a n d t h e f r e q u e n c y o f 0.58 rad/s i n Case C w i t h t h e fllling h e i g h t o f 6 m , respectively. Consequently, t h e first m o d e o f s l o s h i n g f r e q u e n c y is 0.19 rad/s larger t h a n t h e r o l l f r e q u e n c y i n Case A , 0.07 rad/s larger t h a n t h a t i n Case B, a n d 0.06 rad/s s m a l l e r t h a n t h a t i n Case C. A n d t h u s w e can also m a k e a d e d u c t i o n t h a t t h e degree o f t h e c o u p l i n g
Table 3
Zero order moments of the internal sloshing oscillations for the five cases. Case A Case B Case C Case D Case E Upstream 0.3549 0.6693 0.5458 1.0280 1.3240 Downstream 0.4718 1.0148 1.1833 1.4778 1.5151 Total 0.8267 1.6841 1.7291 2.5058 2.8391 2.5 r —«—Upstream — D o w n s t r e a m 2.0 - ...Total 0.0 I ' ' >
Case A Case B Case C
Fig. 8. Zero-order moments of the internal sloshing oscillations versus the filling levels.
b e t w e e n r o l l m o t i o n s a n d i n t e r n a l s l o s h i n g is i n close r e l a t i o n s h i p w i t h t h e distance b e t w e e n t h e flrst m o d e s l o s h i n g f r e q u e n c y a n d t h e r o l l f r e q u e n c y o f t h e vessel.
3.4. Influence of roll frequencies
Fig. 9 s h o w s t h e i n f l u e n c e o f r o l l m o t i o n f r e q u e n c i e s . T h e i n f l u e n c e o n t h e i n t e r n a l s l o s h i n g can be o b s e r v e d t h r o u g h t h e c o m p a r i s o n o f Fig. 9 ( c ) a n d ( d ) . The z e r o - o r d e r m o m e n t s o f t h e i n t e r n a l s l o s h i n g o s c i l l a t i o n s i n Cases C-E have been p l o t t e d i n Fig. 10, a n d t h e values are l i s t e d i n Table 3. I t can be seen t h a t t h e response e n e r g y o f t h e i n t e r n a l s l o s h i n g o s c i l l a t i o n s increases as t h e n a t u r a l f r e q u e n c i e s o f t h e r o l l m o t i o n s b e c o m e s s m a l l . This can be e x p l a i n e d t h r o u g h t h a t as the r o l l f r e q u e n c y decreases, i t gets a w a y f r o m t h e s l o s h i n g f r e q u e n c i e s o f a l l m o d e s , a n d t h u s t h e c o u p l i n g effects b e t w e e n r o l l m o t i o n s a n d s l o s h i n g get weaker. As t h e c o u p l i n g effects ( a n t i - r o l l e f f e c t s ) get w e a k e r , i n t e r n a l s l o s h i n g gets s t r o n g e r ( s h o w n i n Fig. 9(c) a n d ( d ) ) , i n d u c i n g an increase o f t o t a l energy. As i l l u s t r a t e d i n Table 3, t h e t o t a l e n e r g y o f t h e i n t e r n a l s l o s h i n g decreases s i g n i f i c a n t l y b y 40.0% f r o m 2.5058 m ^ ( i n Case D ) t o 1.7291 m ^ ( i n Case C); w h i l e i t increases b y 13.3% f r o m 2.5058 m ^ ( i n Case D ) t o 2.8391 m ^ ( i n Case E). The r o l l f r e q u e n c y i n Case D (0.58 rad/s) is 0.06 rad/s s m a l l e r t h a n t h a t i n Case C ( 0 . 6 4 rad/s); w h i l e t h e r o l l f r e q u e n c y i n Case D (0.58 rad/s) is 0.10 rad/s larger t h a n t h a t i n Case E (0.48 rad/s). I t i n d i c a t e s t h a t a l t h o u g h t h e distance f r o m t h e r o l l f r e q u e n c y i n Case C t o D ( e q u a l t o t h e first m o d e s l o s h i n g f r e q u e n c y ) is s m a l l e r t h a n t h a t f r o m Case E t o D, t h e e n e r g y v a r i a t i o n f r o m Case C t o D is m u c h l a r g e r t h a n t h a t f r o m Case D t o E. This m a y be r e l a t e d t o t h a t w h e n t h e r o l l f r e q u e n c y varies f r o m Case D to E, i t is s t i l l w i t h i n t h e f r e q u e n c y range o f t h e s l o s h i n g modes, a n d i t j u s t gets closer f r o m t h e first m o d e s l o s h i n g f r e q u e n c y t o t h e f r e q u e n c i e s o f h i g h e r m o d e s ; W h i l e w h e n t h e r o l l f r e q u e n c y varies f r o m Case D t o E, i t m o v e s f a r a w a y f r o m t h e s l o s h i n g f r e q u e n c i e s o f all m o d e s . Fig. 9 ( a ) presents t h e s w a y RAOs o f t h e FLNG section i n d i f f e r e n t b a l l a s t i n g c o n d i t i o n s ( o n l y r o l l f r e q u e n c i e s are d i f f e r e n t ) . I t can be seen t h a t t h e s l o s h i n g - i n d u c e d peak i n s w a y m o t i o n s h o l d t h e same, even t h o u g h t h e n a t u r a l f r e q u e n c i e s o f t h e r o l l m o t i o n s are d i f f e r e n t . This is d u e t o t h e f a c t t h a t t h e filling levels k e e p t h e same v a l u e i n t h e t h r e e cases, p r o d u c i n g t h e same flrst m o d e o f s l o s h i n g f r e q u e n c y . As discussed eariier, t h e s l o s h i n g -i n d u c e d p e a k -is m a -i n l y -i n d u c e d b y t h e f-irst m o d e o f slosh-ing. T h e r e f o r e , t h e s l o s h i n g - i n d u c e d peal<s i n t h e s w a y m o t i o n s are f o u n d t o a p p e a r at t h e same e x c i t a t i o n f r e q u e n c y .
C o m p a r i s o n s o f t h e r o l l RAOs o f t h e vessel i n d i f f e r e n t ballast-i n g c o n d ballast-i t ballast-i o n s have b e e n p l o t t e d ballast-i n Fballast-ig. 9 ( b ) . S ballast-i m ballast-i l a r as those o b s e r v e d i n t h e s w a y RAOs, t h e s l o s h i n g - i n d u c e d peaks i n r o l l RAOs a p p e a r at t h e same e x c i t a t i o n f r e q u e n c y i n these t h r e e cases. H o w e v e r , t h e m a g n i t u d e o f t h e peak n e a r t h e flrst m o d e o f s l o s h i n g f r e q u e n c y becomes s m a l l e r as t h e n a t u r a l f r e q u e n c y o f t h e r o l l m o t i o n s decreases. This p h e n o m e n o n i n d i c a t e s t h a t t h e c o u p l i n g b e t w e e n r o l l m o t i o n s a n d t h e i n t e r n a l s l o s h i n g can be s i g n i f l c a n t l y a f f e c t e d b y the distance b e t w e e n t h e flrst m o d e o f s l o s h i n g f r e q u e n c y a n d t h e r o l l f r e q u e n c y o f t h e vessel. A n o t h e r i n t e r e s t i n g p h e n o m e n o n can be o b s e r v e d is t h a t o n l y t h e RAO i n Case C is f o u n d t o be b i m o d a l , w h e r e a s , f o r a l l t h e o t h e r f o u r c o n d i t i o n s , i t is f o u n d t o be a u n i m o d a l . One can also see f r o m Table 1 t h a t i t is o n l y i n t h e Case C c o n d i t i o n t h a t t h e r o l l f r e q u e n c y o f t h e vessel is larger t h a n t h e flrst m o d e o f s l o s h i n g f r e q u e n c y , w h e r e a s , f o r all t h e o t h e r f o u r c o n d i t i o n s , t h e f o r m e r is n o t l a r g e r t h a n t h e l a t t e r one. T h i s m a y i n d i c a t e t h a t a b i m o d a l w i l l a p p e a r i n t h e case t h a t t h e r o l l f r e q u e n c y o f t h e vessel is l a r g e r t h a n t h e flrst m o d e o f s l o s h i n g f r e q u e n c y . S i m i l a r p h e n o m e n o n can also be f o u n d i n p u b l i s h e d r e p o r t s (Lee e t al., 2 0 0 7 ; Nasar e t al., 2 0 1 0 ) .
52 W. Zhao et al. / Ocean Engineering 84 (2014) 4 5 - 5 3 1.5 1.2 E ^ 0 . 9 O t 0.6 ro W 0.3 0.0 — C_solid • • C_Uiquid -- D_solid • D_Liquid —- E_solid • • E_Liquid 0.4 0.8 1.2 F r e q u e n c y (rad/s) 1.6 2.0 — C_Solid . - - C_Liquid — D_Solid . - . D_Liquid — - E_Solid . . . E_Liquid 0.6 0.8 F r e q u e n c y (rad/s) 6.0 5.0 ^ 4.0 3.0
1^"
o OT 1.0 0.0 Upstreann — CJiquid • • DJiquid — EJlquid 0.4 0.8 1.2 1.6 F r e q u e n c y (rad/s) 2.0 6.0 5.0 ^ 4.0 3.0 / 2.0 o OTl.c
D o w n s t r e a m — C J i q u i d • • DJiquid — EJiquid 0.4 0.8 1.2 1.6 F r e q u e n c y (rad/s) 2.0Fig. 9. Response amplitude operators for the FLNG section (heave motion is restrained): (a) sway motions, (b) roll motions, (c) sloshing oscillations at upstream, and (d) sloshing osciUations at downstream. Case C: h = 6 m, r=0.2, « ( ( = 0 . 6 4 rad/s, and =0.58 rad/s; Case D: h = 6 m, r=0.2, « ^ = 0 . 5 8 rad/s, and w, =0.58 rad/s; Case E :
h=6m, r=0.2, 0^^=0.48 rad/s, and rai =0.58 rad/s.
3.5 ^ 3.0 ¥ 2.5 h § 2.0 1.5 1,0 0.5 0.0 - Upstream - Downstream •Total
Case 0 Case D Case E
Fig. 10. Zero-order moments of the internal sloshing oscillations versus the roll frequencies of FLNG section. 4. C o n c l u s i o n s U n d e r t h e e x c i t a t i o n o f a b a n d - l i m i t e d w h i t e noise w a v e , t h e c o u p l i n g p e r f o r m a n c e o f s h i p m o t i o n s a n d i n t e r n a l s l o s h i n g has b e e n e x p e r i m e n t a l l y i n v e s t i g a t e d w i t h a n FLNG s e c t i o n b a l l a s t e d w i t h f r e s h w a t e r a n d e q u i v a l e n t solid w e i g h t s . I n f l u e n c e s o f t h e fllling levels o f t h e t a n k a n d t h e n a t u r a l f r e q u e n c i e s o f t h e vessel have b e e n c l a r i f l e d . Some conclusions o f i m p o r t a n c e d r a w n f r o m t h i s s t u d y are s u m m a r i z e d as f o l l o w s .
flrst m o d e s l o s h i n g s h o w s o b v i o u s c o u p l i n g effects w i t h t h e g l o b a l m o t i o n s o f t h e vessel. Due t o t h e c o u p l i n g effect, t h e RAO peak o f t h e i n t e r n a l s l o s h i n g , w h i c h is e x p e c t e d t o a p p e a r at t h e first m o d e f r e q u e n c y , has s h i f t e d t o w a r d s h i g h e r f r e -quencies. F u r t h e r m o r e , t h e r e is a phase s h i f t i n g o f t h e i n t e r n a l s l o s h i n g o s c i l l a t i o n s at t h e first m o d e o f s l o s h i n g f r e q u e n c y . • U n d e r t h e same w a v e e x c i t a t i o n s , t h e fluid w i t h l o w e r fllling
levels w o u l d e x h i b i t severer s l o s h i n g o s c i l l a t i o n s t h a n t h a t w i t h h i g h e r fllling levels d o .
• The c o u p l i n g b e t w e e n s h i p m o t i o n s a n d i n t e r n a l s l o s h i n g is sensitive t o t h e d i s t a n c e b e t w e e n t h e first m o d e o f s l o s h i n g f r e q u e n c y a n d t h e r o l l f r e q u e n c y o f t h e FLNG section.
• The e x p e r i m e n t a l results s h o w t h a t t h e s w a y a n d r o l l RAOs can be s i g n i f l c a n t i y r e d u c e d b y t h e i n t e r n a l s l o s h i n g w h e n t h e e x c i t a t i o n f r e q u e n c y equals t h e first m o d e o f s l o s h i n g f r e q u e n c y .
A i m e d at p r o v i d i n g a basic u n d e r s t a n d i n g o f t h e c o u p l i n g m e c h a n i s m b e t w e e n s h i p m o t i o n s a n d i n t e r n a l s l o s h i n g o s c i l l a -tions, this p r o j e c t f o c u s e d o n t h e RAOs o f a n FLNG s e c t i o n c o n t a i n i n g a t a n k u n d e r t h e e x c i t a t i o n o f b a n d - l i m i t e d w h i t e noise waves. The c o n c l u s i o n s based o n t h e e x p e r i m e n t s are e x p e c t e d t o serve as a r e f e r e n c e f o r t h e d e s i g n a n d o p e r a t i o n o f a n FLNG system. F u r t h e r s t u d y s h o u l d be c a r r i e d o u t t o i n v e s t i g a t e t h e i n f l u e n c e o f the densities o f t h e l i q u i d i n s i d e t h e t a n k . • I n s i g n i f i c a n t e f f e c t o f i n t e r n a l s l o s h i n g o n t h e heave RAO o f t h e FLNG s e c t i o n is observed. Obvious c o u p l i n g b e t w e e n s h i p m o t i o n s a n d i n t e r n a l s l o s h i n g have b e e n obsei-ved i n t h e s w a y a n d r o l l RAOs. • A l t h o u g h t h e m a g n i t u d e s o f i n t e r n a l s l o s h i n g response i n h i g h e r m o d e s are i d e n t i c a l t o t h a t i n t h e first m o d e , o n l y t h e A c l a i o w l e d g m e n t s
This w o r k w a s financially s u p p o r t e d b y t h e Science F o u n d a t i o n o f Science a n d T e c h n o l o g y C o m m i s s i o n o f Shanghai M u n i c i p a l i t y ( G r a n t no. 11ZR1417800), t h e C h i n a N a t i o n a l S c i e n t i f i c a n d
W. Zhao et al. / Ocean Engineering 84 (2014) 45-53 53
T e c h n o l o g y M a j o r P r o j e c t ( 2 0 1 1 Z X 0 5 0 2 6 - 0 0 6 - 0 5 ) , a n d also t h e Lloyd's Register E d u c a t i o n a l T r u s t (LRET) t o t h e j o i n t c e n t e r i n v o l v i n g U n i v e r s i t y College L o n d o n , Shanghai Jiao T o n g U n i v e r s i t y a n d H a r b i n E n g i n e e r i n g U n i v e r s i t y . These sources o f s u p p o r t are g r a t e f u l l y a c k n o w l e d g e d b y t h e a u t h o r s .
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