Applied Ocean Research 50 (2015) 119-129
E L S E V I E R
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Applied Ocean Research
journal homepage: www.elsevier.com/locate/aporWave resonances in a narrow gap betv\/een two barges using fully
nonlinear numerical simulation
X. Feng.W. Bai*
Department of Civil and Environtriental Engineering, National University of Singapore, Kent Ridge, Singapore J17576, Singapore
CrossMark
A R T I C L E I N F O Article history:
Received 27 March 2014
Received in revised form 6 January 2015 Accepted 9 January 2015
Available online 31 January 2015
Keywords: Side-by-side barges Gap resonance
Fully nonlinear simulation Stiff/soft spring
Nonlinear effect
A B S T R A C T
The traditional potential flow theory to describe fully nonlinear waves is reformulated by separating the contributions from incident and scattered waves, in order to improve the computational efficiency. The nonlinear incoming wave is specified explicitly and the modified nonlinear free surface boundary conditions for the scattered wave are expressed in the full Lagrangian description. At each time step only the scattered wave is solved using a mixed Eulerian-Lagrangian scheme by a higher-order boundary element method. The accuracy of the newly developed model is illustrated by comparisons with existing experimental and numerical data in the case of wave diffraction around an array of circular cylinders. Wave resonances in the gap between two side-by-side barges in beam seas, as in Molin et al. [1], are simulated with the barges subjected to regular waves. To clearly understand the gap resonant responses, long time simulations are performed to achieve final steady states, and the resonant mode shapes of the gap surface are presented. The gap free surface RAOs (Response Amplitude Operators) in the case of mild waves are found to agree well with linear calculations. The nonlinear effects on the resonant response due to the free surface conditions are then investigated. The first resonant frequency is found to shift but the peal< value is not changed much with increasing incoming wave steepness, which is known as stiff/soft spring behavior of a nonlinear system. Through the investigation of barges with different drafts, the stiff and soft spring behaviors are identified.
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1. I n t r o d u c t i o n
I n t h e last decade, w i t h t h e d e v e l o p m e n t o f o f f s h o r e t e c h n o l o g y and i n c r e a s i n g c o n s u m p t i o n o f o i l a n d gas, m o r e s i d e b y s i d e o p e r -ations h a v e b e e n a d o p t e d i n t h e nnarine i n d u s t r y . These o p e r a t i o n s c o u l d be, f o r i n s t a n c e , l i q u i d cargo o f f l o a d i n g f r o m a LNG-Carrier t o a FSRU i n close p r o x i m i t y , i n p r a c t i c e , s u c c e s s f u l e x e c u t i o n o f such o p e r a t i o n s r e q u i r e s n u m e r o u s c o n s i d e r a t i o n s i n c l u d i n g , f o r e x a m p l e , e n v i r o n m e n t a l c o n d i t i o n s , vessel a r r a n g e m e n t s , m o o r -i n g systems, o p e r a t -i o n p r o c e d u r e s a n d so o n . A m o n g o t h e r s , t h e r e s o n a n t p h e n o m e n o n associated w i t h t h e w a v e s i n a n a r r o w gap b e t w e e n t w o s i d e b y s i d e h u l l s m a y be a p r o b l e m . W h e n gap r e s -o n a n c e -occurs, h i g h w a v e m -o t i -o n s i n the gap c -o u l d be e x c i t e d a n d h e n c e large d r i f t f o r c e s m a y act o n t h e vessels. U r i l i z i n g t h e l i n e a r w a v e t h e o r y , M o l i n [ 2 ] d e r i v e d t h e f o r m u l a f o r e s t i m a t i n g t h e f r e q u e n c i e s o f r e s o n a n t m o d e s i n a m o o n p o o l v i a s o l v i n g a n e i g e n v a l u e p r o b l e m , w h i c h w a s e x t e n d e d t o gap resonances b y a m o d i f i c a t i o n i n t o an o p e n - e n d e d m o o n p o o l i n M o l i n et a l . [ 3 ] . Later o n t h e r e s o n a n t b e h a v i o r o f f l u i d i n t h e g a p has a t t r a c t e d m u c h
* Corresponding author, Tel,: +65 6516 2288, E-mail address: w,bai@nus,edu,sg (W, Bai),
0141 -1187/$ - see front matter © 2015 Elsevier Ltd, All rights reserved, http://dx.d0i.0rg/l 0,1016/j,apor.2015,01,003 a t t e n t i o n e s p e c i a i i y at t h e first m o d e or p i s t o n m o d e , w h e r e w a v e a m p l i f i c a t i o n s are m o r e s i g n i f i c a n t t h a n at o t h e r s . I n a b r o a d sense, t h e s i d e - b y - s i d e barges c o n f l g u r a t i o n is o n e t y p e o f s o - c a l l e d ' t r a p p i n g s t r u c t u r e ' . The e i g e n v a l u e p r o b l e m associated w i t h t r a p p i n g s t r u c t u r e s is n o t n e w , a n d p i o n e e r i n g w o r k b a s e d o n l i n e a r w a v e t h e o r y has p r o v i d e d a n a l y t i c a l s o l u -tions f o r t h e t r a p p i n g or r e s o n a n t f r e q u e n c i e s . For i n s t a n c e , L i n t o n a n d Evans [ 4 ] p r e s e n t e d t h e n e a r - t r a p p i n g p h e n o m e n o n a r o u n d a n a r r a y o f c i r c u l a r c y l i n d e r s . P o r t e r a n d Evans [ 5 ] s o l v e d w a v e s c a t t e r i n g b y v e r t i c a l b a r r i e r s , a n d M c l v e r [ 6 ] i n v e s t i g a t e d g e n e r a l t o r u s - l i k e s t r u c t u r e s . D i i e t o the c o m p l e x g e o m e t r y o f w a v e resonances i n t h e . g a p b e t w e e n t w o t h r e e d i m e n s i o n a l s i d e b y -side vessels, r e c e n t l y t h e p r o b l e m has b e e n m a i n l y s i m u l a t e d b y n u m e r i c a l m o d e l s based o n l i n e a r w a v e t h e o r y . E a r l i e r , N e w m a n a n d Sclavounos [ 7 ] m o d e l e d t w o close r e c t a n g u l a r barges b y a p a n e l m e t h o d a n d r e p o r t e d u n u s u a l l y h i g h w a v e e l e v a t i o n s i n t h e n a r r o w g a p as w e l l as large h y d r o d y n a m i c f o r c e s . Koo a n d K i m [ 8 ] i n v e s t i g a t e d t h e h y d r o d y n a m i c i n t e r a c t i o n s a n d c o u p l i n g e f f e c t s o f a m o o r e d FPSO-LNG s y s t e m w i t h h y d r o d y n a m i c c o e f f i c i e n t s o b t a i n e d f r o m t h e p a n e l p r o g r a m W A M I T . S u n e t a l . [ 9 ] u t i l i z e d t h e 3 D p r o g r a m DIFFRACT t o s i m u l a t e t w o a d j a c e n t barges a n d s u g g e s t e d t h a t b o t h first- a n d s e c o n d - o r d e r r e s o n a n c e s associated w i t h t h e g a p m a y e x i s t . H o w e v e r , a p p l i c a t i o n s o f l i n e a r p o t e n t i a l flow m o d e l s are r e p o r t e d t o p o t e n t i a l l y o v e r - p r e d i c t t h e w a v e responses, n a m e l y
120 X. Feng, W. Bai/Applied Ocean Researcli SO (2015) 119-129 RAOs (Response A m p l i t u d e O p e r a t o r s ) o f t h e f r e e s u r f a c e e l e v a -t i o n s i n -t h e gap a-t -t h e f r e q u e n c y o f -t h e p i s -t o n m o d e . C o n s e q u e n -t l y t h e m e a n d r i f t forces o n the h u l l s a n d s h i p m o t i o n s c o u l d v a r y m u c h f r o m t h e p r e d i c t i o n s . I n o r d e r to suppress t h e u n r e a l i s t i c w a v e ele-v a t i o n s p r o d u c e d f r o m l i n e a r m o d e l s , seele-veral m e t h o d s haele-ve b e e n d e v e l o p e d . H u i j s m a n s et a l . [ 1 0 ] a p p l i e d a r i g i d l i d o n t h e f r e e s u r -face b e t w e e n s i d e - b y - s i d e m o o r e d vessels. S u b s e q u e n t l y B u c h n e r e t a l . [ 1 1 ] n u m e r i c a l l y a n d 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 t h e f l o a t -i n g LNG s y s t e m based o n t h e ' l -i d m e t h o d ' a n d p a r t -i a l l y j u s t -i f -i e d t h e a p p r o a c h o f a d a m p i n g l i d i n a p p l i c a t i o n s . W h i l e t h i s ' l i d m e t h o d ' is able t o suppress u n r e a l i s t i c values, i t does n o t h o w e v e r m a k e p h y s -ical sense. N e w m a n [ 1 2 ] also m o d e l e d a d a m p i n g l i d o n t h e gap s u r f a c e a n d used a g e n e r a l i z e d m o d e t e c h n i q u e t o c o m p u t e t h e l i d m o t i o n s . M e a n w h i l e , Chen [ 1 3 ] i n t r o d u c e d a d a m p i n g f o r c e t e r m i n t o t h e f r e e s u r f a c e b o u n d a r y c o n d i t i o n s , w h i c h w a s e x p l a i n e d as e n e r g y d i s s i p a t i o n . The e f f i c i e n c y o f t h e l i n e a r d i s s i p a t i o n t e r m w a s t h e n p r e s e n t e d b y F o u r n i e r et a l . [ 14] w i t h c o m p a r i s o n s t o t h e l i n e a r p r o g r a m s W A M I T a n d HYDROSTAR, as, w e l l as e x p e r i m e n t a l data. T h e r e a f t e r , P a u w et a l . [ 1 5 ] u t i l i z e d a s i m i l a r t e c h n i q u e t o p r e d i c t t h e w a v e response w i t h i n t h e gap o f s i d e - b y - s i d e m o o r e d vessels, at t h e r e s o n a n t f r e q u e n c i e s o f p i s t o n a n d s l o s h i n g m o d e s o b t a i n e d b y t h e a p p r o x i m a t e d f o r m u l a t i o n s o f M o l i n [ 2 ] . P a u w et a l . [ 1 5 ] , h o w e v e r , d e m o n s t r a t e d t h a t t h e r e is n o a p r i o r i m e t h o d o f d e t e r -m i n i n g t h e c o e f f i c i e n t o f t h e d a -m p i n g t e r -m unless c a l i b r a t e d b y e x p e r i m e n t a l tests. R e c e n t l y t h e p i s t o n m o d e o f w a t e r c o l u m n m o t i o n i n t h e gap w a s i n v e s t i g a t e d b y K r i s t i a n s e n a n d F a l t i n s e n [ 1 6 ] , w h o a d o p t e d a 2 D n u m e r i c a l w a v e t a n k m o d e l a n d c a p t u r e d viscous e f f e c t s b y m e a n s o f a v o r t e x t r a c k i n g m e t h o d . M e a n w h i l e , Lu et a l . [ 1 7 ] e m p l o y e d a viscous fluid m o d e l to s t u d y the 2 0 w a v e i n t e r a c t i o n s w i t h c l o s e l y floaflng b o x e s . I t w a s r e p o r t e d t h a t i n t h e 2 D case gap e l e v a t i o n c a l c u l a t e d b y l i n e a r p o t e n t i a l t h e o r y c o u l d b e c o m e u n r e a s o n a b l y h i g h ( o v e r f o u r times h i g h e r t h a n m o d e l t e s t s ) i f t h e gap is s u f f i c i e n t l y n a r r o w . T w o - d i m e n s i o n a l m o d e l s , h o w e v e r , are s o m e w h a t l i m i t e d i n c a p t u r i n g real 3 D c h a r a c t e r i s t i c s o f t h e fluid i n t h e gap, e s p e c i a l l y i n r e p r e s e n t i n g h i g h e r - r e s o n a n t m o d e s . C o m p a r i s o n s b e t w e e n 3 D l i n e a r s i m u l a t i o n a n d e x p e r i m e n t a l tests i n M o l i n et a l . [ 1 ] s h o w t h a t i n d e e d , l i n e a r m o d e l s t e n d t o o v e r - p r e d i c t t h e r e s o n a n t w a v e r e s p o n s e ( a b o u t 40% l a r g e r at peaks), y e t l i n e a r results o f gap f r e e surface RAOs i n 3 D m o d e l s m a y n o t be as u n r e a l i s t i c as i n 2 D s i m u l a t i o n s .
A s t r a i g h t f o r w a r d e x p l a n a t i o n f o r t h e d i s c r e p a n c y is t h a t t h e gap s u r f a c e e l e v a t i o n s i n l i n e a r t h e o r y are o v e r - p r e d i c t e d d u e t o t h e n e g l e c t o f fluid v i s c o s i t y i n t h e p o t e n t i a l flow m o d e l , i.e. v o r t e x s h e d d i n g a n d flow s e p a r a t i o n at s h a r p edges a n d c o r n e r s . O n t h e o t h e r h a n d , i t is k n o w n t h a t w i t h i n t h e f r a m e w o r k o f p o t e n t i a l flow t h e o r y t h e f r e e s u r f a c e b o u n d a r y c o n d i t i o n s are n o n l i n e a r , w h i c h are s i m p l i f i e d i n t h e l i n e a r a p p r o x i m a t i o n . T h e r e f o r e , b o t h w a v e n o n l i n e a r i t y a n d fluid v i s c o s i t y m a y c o n t r i b u t e t o t h e d i s c r e p a n c y b e t w e e n l i n e a r r e s u l t s a n d m e a s u r e m e n t s . Some research w o r k has b e e n d o n e to t h r o w l i g h t o n t h e i n f l u e n c e o f viscous e f f e c t s a n d n o n l i n e a r e f f e c t s o f t h e f r e e s u r f a c e . The v o r t e x - s h e d d i n g e f f e c t s w e r e e v a l u a t e d i n F a l t i n s e n e t a l . [ 1 8 ] b y a d i s c r e t e - v o r t e x m e t h o d f o r a s i m p l e case, i.e. a 2 D m o o n p o o l f o r m e d b y t w o r e c t a n g u l a r h u l l s u n d e r g o i n g heave m o t i o n s . C o m p a r i s o n s w i t h e x p e r i m e n t s d e m o n s t r a t e d t h a t t h e a g r e e m e n t o f r e s o n a n t f r e -quencies is reasonable f o r s m a l l f o r c i n g a m p l i t u d e s , w h i l e t h e d i s c r e p a n c y increases f o r l a r g e r e x c i t a t i o n s a n d a w i d e r m o o n p o o l . A d d i n g v o r t e x - s h e d d i n g e f f e c t s i n t h o s e cases does n o t suppress the d i s c r e p a n c y . The r e a s o n m i g h t be t h e r e l a t i v e l y s m a l l f o r c i n g a m p l i t u d e s used i n t h e i r e x p e r i m e n t s a n d t h e q u a d r a t i c v e l o c i t y d e p e n d e n c e o f the v o r t e x - i n d u c e d forces, as e x p l a i n e d b y F a l t i n s e n e t a l . [ 1 8 ] . It is o f i n t e r e s t t h a t h i g h e r h a r m o n i c s i n time h i s t o r i e s o f w a v e e l e v a t i o n s w e r e c a p t u r e d i n t h e i r m e a s u r e m e n t s , w h i c h h i g h -l i g h t e d possib-le effects o f f r e e surface n o n -l i n e a r i t i e s . I n K r i s t i a n s e n a n d Faltinsen's [ 1 9 ] v o r t e x t r a c k i n g analysis, i t w a s f o u n d t h a t flow
s e p a r a t i o n m a i n l y accounts f o r the d i s c r e p a n c y o f t h e gap surface a m p l i f i c a t i o n b e t w e e n l i n e a r results a n d m e a s u r e m e n t s , a n d n o n -l i n e a r f r e e s u r f a c e b o u n d a r y c o n d i t i o n s are o f m i n o r i m p o r t a n c e . H o w e v e r , i t s h o u l d be n o t e d t h a t t h e p r o p a g a t i n g w a v e s i n t h e i r m o d e l tests are o f r e l a t i v e l y l o w w a v e steepness, I<A a p p r o x i m a t e l y f r o m 0.3% to 1.0% {k is w a v e n u m b e r a n d A w a v e a m p l i t u d e ) . T h e r e -f o r e , n o n l i n e a r -f r e e s u r -f a c e e -f -f e c t s m a y n o t be s i g n i -f i c a n t i n t h e i r cases.
I n a r e c e n t s t u d y o f K r i s t i a n s e n a n d Faltinsen [ 2 0 ] , t h e y u t i l i z e d a d o m a i n - d e c o m p o s i t i o n a p p r o a c h , w h i c h c o m b i n e s p o t e n t i a l flow t h e o r y a n d CFD, t o analyze t h e 2 D gap resonance p r o b l e m . They a g a i n c o n c l u d e d t h a t flow s e p a r a t i o n at b a r g e / s h i p b i l g e s e x p l a i n s t h e d i s c r e p a n c y o f peak r e s o n a n t response b e t w e e n l i n e a r p o t e n t i a l flow m o d e l a n d e x p e r i m e n t s . I n t h e t h r e e d i m e n s i o n a l , e x p e r i -m e n t a l a n d n u -m e r i c a l i n v e s t i g a t i o n i n M o l i n et a l . [ 1 ] , barges w i t h b o t h r o u n d e d a n d s q u a r e bilges w e r e s i m u l a t e d a n d e x p e r i m e n t a l r e s u l t s suggested t h a t t h e d i s c r e p a n c y is m o s t l y d u e t o t h e flow s e p a r a t i o n at the b a r g e b i l g e s . N o t m u c h w o r k , h o w e v e r , has b e e n p u b l i s h e d c l o s e l y i n v e s t i -g a t i n -g the n o n l i n e a r e f f e c t s o f f r e e surface o n t h e -gap resonance. This is one o f o u r m a j o r i n t e r e s t s h e r e i n m o d e l i n g t h e g a p res-onance, w h e r e t h e gap s u r f a c e behaves as a m a s s - s p r i n g s y s t e m (e.g. i n its p i s t o n m o d e ) . T h e o r e t i c a l analysis o f s i m i l a r n o n l i n e a r m a s s s p r i n g s y s t e m s can be f o u n d i n V i n j e [ 2 1 ] f o r a n a r r o w m o o n -p o o l and M i l e s [ 2 2 ] f o r a c i r c u l a r w e l l , as w e l l as m o r e d e t a i l s i n F a l t i n s e n a n d T i m o k h a [ 2 3 ] f o r s l o s h i n g . I n o r d e r to assess t h e n o n -l i n e a r effects o f t h e f r e e surface i n t h r e e - d i m e n s i o n a -l s i t u a t i o n s , the s a m e t w o barges w i t h square b i l g e s c o n s i d e r e d as i n M o l i n et al. [ 1 ] are m o d e l e d i n t h i s s t u d y , w i t h w a v e steepness v a r y i n g f r o m 0.34% to 6.7%. Calculations w i t h a s m a l l f r e q u e n c y step near the p i s t o n m o d e r e s o n a n t f r e q u e n c y are p e r f o r m e d , w i t h s u f f i c i e n t l y l o n g time s i m u l a t i o n s i n o r d e r t o achieve s t e a d y state. C a r e f u l p l o t s o f gap peak response near t h e resonance s h o w t h a t t h e res-o n a n t f r e q u e n c y s l i g h t l y s h i f t s t res-o h i g h e r values as w a v e steepness increases, w h i l e t h e peak responses are n o t s i g n i f i c a n t i y r e d u c e d . This s t i f f s p r i n g b e h a v i o r is also o b s e r v e d i n t h e d r i f t f o r c e s . W i t h t h e change o f barge d r a f t , t h e s t i f f s p r i n g b e h a v i o r w i l l t u r n t o a s o f t s p r i n g b e h a v i o r w h e n t h e b a r g e d r a f t over l e n g t h r a t i o is s u f f i c i e n t l y large.
T h e a i m o f t h i s p a p e r is t o s t u d y t h e e f f e c t s o f f r e e s u r f a c e n o n -l i n e a r i t y o n t h e w a v e resonance i n a gap b e t w e e n t w o barges. To achieve this, a f u l l y n o n l i n e a r p o t e n t i a l flow m o d e l capable o f r u n -n i -n g l o -n g time s i m u l a t i o -n s e f f i c i e -n t l y is d e v e l o p e d b y s e p a r a t i -n g t h e t o t a l w a v e i n t o a p r e s c r i b e d i n c o m i n g w a v e a n d a n u n k n o w n s c a t t e r e d w a v e . The p r e s e n t n u m e r i c a l m o d e l is a f u r t h e r e x t e n s i o n o f t h e n u m e r i c a l w a v e t a n k ( N W T ) d e v e l o p e d b y Bai a n d Eatock T a y l o r [ 2 4 , 2 5 ] . I n t h e p r e s e n t m o d e l , the c o m p u t a t i o n a l d o m a i n is c i r c u l a r a n d a d a m p i n g z o n e is p l a c e d o n t h e f r e e s u r f a c e near t h e t a n k w a l l t o a b s o r b t h e o u t g o i n g s c a t t e r e d w a v e s u c h t h a t the t a n k w a l l e f f e c t can be e l i m i n a t e d . T h e flow field i n t h e n o n l i n e a r i n c i d e n t w a v e is s p e c i f i e d e x p l i c i t l y , t h u s o n l y t h e s c a t t e r e d w a v e needs t o be s o l v e d . F e r r a n t e t a l . [ 2 6 ] a p p l i e d a s i m i l a r a p p r o a c h i n t h e i r n o n l i n e a r timedomain m o d e l t o s i m u l a t e t h e w a v e d i f f r a c -t i o n a r o u n d a v e r -t i c a l c y l i n d e r , w h i c h d e m o n s -t r a -t e d a n u m b e r of p r a c t i c a l advantages i n t e r m s o f accuracy a n d c o m p u t a t i o n a l e f f i -c i e n -c y . I n t h e i r m o d e l , t h e s e p a r a t i o n o f t h e i n -c i d e n t w a v e a n d the s c a t t e r e d w a v e is i m p l e m e n t e d i n a s e m i - L a g r a n g i a n f o r m u l a t i o n of t h e n o n l i n e a r f r e e surface b o u n d a r y c o n d i t i o n s , w h e r e t h e h o r i z o n -t a l m o -t i o n s o f -t h e f r e e s u r f a c e p o i n -t s are i n h i b i -t e d a n d -t h e v e r -t i c a l c o o r d i n a t e s b e c o m e s i n g l e - v a l u e d as z = f ; ( x , y, f ) . This a p p r o a c h c o u l d lead t o d i f f i c u l t i e s i n s i m u l a t i n g m o v i n g s t r u c t u r e s w h e r e i n t e r s e c t i o n lines b e t w e e n t h e fluid a n d t h e s t r u c t u r e s are n o t h o r -i z o n t a l l y f-ixed. I n t h -i s s t u d y w e p r e s e n t f o r m u l a t -i o n s -i n a f u l l y L a g r a n g i a n d e s c r i p t i o n o f t h e f r e e surface b o u n d a r y c o n d i t i o n s , w h i c h w e suggest is m o r e r o b u s t f o r w a t e r w a v e - b o d y i n t e r a c t i o n p r o b l e m s .
X. Feng, W. Bai/Applied Ocean Researdi 50 (2015) 119-129 121
2. M a t h e m a t i c a l f o r m u l a t i o n
2.1. Separation of incident and scattered waves
A c i r c u l a r n u m e r i c a l w a v e t a n k ( N W T ) is d e f i n e d t o i n v e s t i g a t e the w a v e - b o d y i n t e r a c t i o n p r o b l e m . Fig. 1(a) presents a p l a n v i e w o f t h e N W T m o d e l , i n c l u d i n g a c i r c u l a r tank, t w o r e c t a n g u l a r barges, a f r e e w a t e r surface a n d a d a m p i n g z o n e o n t h e f r e e surface near t h e t a n k side w a l l . The o r i g i n o f a c o o r d i n a t e s y s t e m Oxyz is p l a c e d at t h e c e n t e r o f t h e gap o n t h e c a l m w a t e r surface, w i t h zaxis p o i n t -ing u p w a r d s . The d i r e c t i o n o f t h e i n c i d e n t w a v e is d e n o t e d b y p m e a s u r e d f r o m t h e p o s i t i v e x d i r e c t i o n , a n d t h e d a m p i n g z o n e has a w i d t h o f o n e w a v e l e n g t h . Before w e p e r f o r m t h e s e p a r a t i o n o f an i n c o m i n g flow a n d a s c a t t e r e d flow, w e p r e s e n t b r i e f l y t h e g o v -e r n i n g -e q u a t i o n s f o r f u l l y n o n l i n -e a r p o t -e n t i a l flow. F o l l o w i n g t h -e a s s u m p t i o n s o f p o t e n t i a l flow t h e o r y , i.e. t h e fluid is i n c o m p r e s s -ible, i n v i s c i d a n d flow i r r o t a t i o n a l w i t h i n t h e f l u i d d o m a i n , t h e flow v e l o c i t y p o t e n t i a l ({){x, y, z, t) satisfies t h e Laplace e q u a t i o n : V^(j) = 0 ( 1 ) On t h e f r e e w a t e r surface Sp, t h e k i n e m a t i c a n d d y n a m i c w a v e c o n d i t i o n s i n t h e Lagrangian d e s c r i p t i o n are: ^ = _ g z + i v 0 . V 0 , ( 3 ) w h e r e D / D f is the m a t e r i a l d e r i v a t i v e , X d e n o t e s t h e p o s i t i o n o f w a t e r p a r t i c l e s o n t h e f r e e w a t e r surface, t is t h e time a n d g is t h e g r a v i t a t i o n a l a c c e l e r a t i o n . N o - f l u x c o n d i t i o n s o n t h e fixed s o l i d b o u n d a r i e s Sr are a p p l i e d ón (4) w h e r e n = (nx, Uy, nz) is t h e n o r m a l u n i t v e c t o r p o i n t i n g o u t o f t h e fluid d o m a i n . The i n i t i a l c o n d i t i o n is t a k e n as:
(p = 0 o n z = 0 a t t = 0 ( 5 )
I t is b o t h p r a c t i c a l a h d c o m p u t a t i o n a l l y e c o n o m i c a l to sepa-r a t e t h e t o t a l flow i n t o an i n c i d e n t a n d s c a t t e sepa-r e d flow. This has b e e n w i d e l y a p p l i e d i n t h e l i n e a r d i f f r a c t i o n / r a d i a t i o n w a v e t h e o r y . W h e n w e r e w r i t e t h e t o t a l v e l o c i t y p o t e n t i a l as ( 6s a n d t h e p o s i t i o n o f w a t e r particles o n t h e f r e e w a t e r surface as X = X i + X s , t h e g o v e r n i n g Laplace e q u a t i o n r e m a i n s l i n e a r a n d t h e s c a t t e r e d p o t e n t i a l 0 s s a t i s f i e s : :0. (6) S u b s t i t u t i n g the t o t a l p o t e n t i a l a n d p o s i t i o n i n t o t h e c o r r e -s p o n d i n g b o u n d a r y c o n d i t i o n -s i n Eq-s. ( 2 ) - ( 4 ) lead-s t o t h e f o l l o w i n g b o u n d a r y c o n d i t i o n s : D X i DXs (7) V(/) onSp, (8) (9) w h e r e t h e s u b s c r i p t s '1' a n d 'S' d e n o t e t h e q u a n t i t i e s f o r i n c o m i n g a n d s c a t t e r e d flows r e s p e c t i v e l y . I f t h e p r e s c r i b e d i n c o m i n g flow satisfies t h e f r e e surface b o u n d a r y c o n d i t i o n s , D X | / D t = V 0 / a n d D 0 i / D t = g z i + ( l / 2 ) V i ^ i i • V<^i o n t h e f r e e w a t e r surface, t h e c o r -r e s p o n d i n g n e w b o u n d a -r y c o n d i t i o n s f o -r t h e s c a t t e -r e d p o t e n t i a l can be e x p r e s s e d as: D f = Vcj)-y<j)\ onSp, ^ = -g2s + i v < ^ . V < / . - i v < ^ , . V < / . , 9ii dn onSp, (11) (12) w h e r e b o t h t h e t o t a l flow v e l o c i t y V 0 f r o m t h e t o t a l p o t e n t i a l 0 a n d t h e v e l o c i t y c o m p o n e n t V*^; d u e t o the i n c i d e n t w a v e p o t e n t i a l (pi are c a l c u l a t e d o n t h e real t i m e t o t a l f r e e w a t e r surface Sp. The s a m e c o n c e p t has also b e e n used i n Ferrant et a l . [ 2 6 ] , e x c e p t f o r t h e i r use o f t h e s e m i - L a g r a n g i a n d e s c r i p t i o n . The n o r m a l v e l o c i t y o f t h e i n c o m i n g w a v e o n t h e s o l i d s u r f a c e SR can be o b t a i n e d e x p l i c i t i y .
D u r i n g a c e r t a i n p e r i o d at t h e b e g i n n i n g o f t h e s i m u l a t i o n , a r a m p f u n c r i o n is i m p o s e d o n t h e s p e c i f i e d i n c i d e n t w a v e fleld to ensure t h a t t h e i n c i d e n t flow s m o o t h l y d e v e l o p s f r o m the c a l m w a t e r s u r f a c e t o a f u l l y p e r i o d i c w a v e . The s a m e cosine r a m p i n g f u n c t i o n as i n Bai a n d Eatock T a y l o r [ 2 4 ] is u t i l i z e d a n d t h e r a m p d u r a t i o n is c h o s e n as t w o t i m e s t h e i n c i d e n t w a v e p e r i o d . A t t h e d a m p i n g zone near t h e t a n k w a l l o n l y t h e s c a t t e r e d w a v e w i l l be d a m p e d o u t . T h e r e f o r e , t h e f r e e surface b o u n d a r y c o n d i t i o n s i n Eqs. (10) a n d ( 1 1 ) are m o d i f i e d t o i n c l u d e t h e a r t i f i c i a l d a m p i n g i n t h e d a m p i n g area, a n d b e c o m e DXs Dt = V 0 - V 0 1 - v ( r ) X s onSp, ^ = - g z s + l v 0 . V 0 - l v < ^ , . V 0 , - v ( r ) 0 s onSp, (13) (14) (10) w h e r e v ( r ) is a d a m p i n g c o e f f i c i e n t c a l c u l a t e d b y Eq. ( 3 0 ) i n Bai a n d Eatock T a y l o r [ 2 4 ] . The e f f i c i e n c y has b e e n d e m o n s t r a t e d i n [ 2 4 , 2 6 ] . A t a g i v e n t i m e step, t h e c o m p u t a t i o n a l m e s h is g e n e r a t e d o n t h e t o t a l f r e e w a t e r surface, o n w h i c h t h e s c a t t e r e d p o t e n t i a l is k n o w n f r o m t h e p r e v i o u s time step. By s o l v i n g t h e b o u n d a r y v a l u e p r o b l e m f o r t h e s c a t t e r e d w a v e , t h e n o r m a l d e r i v a t i v e o f t h e scat-t e r e d p o scat-t e n scat-t i a l can be o b scat-t a i n e d . Based o n scat-t h i s o b scat-t a i n e d d e r i v a scat-t i v e , t h e t o t a l flow v e l o c i t y is c a l c u l a t e d o n t h e t o t a l f r e e s u r f a c e . I n t h e c a l c u l a t i o n o f t o t a l flow v e l o c i t y t h e i n c i d e n t w a v e v e l o c i t y a n d p o t e n t i a l are r e q u i r e d , b o t h o f w h i c h are e v a l u a t e d o n t h e t o t a l f r e e w a t e r surface. By t h e t i m e i n t e g r a t i o n o f Eqs. ( 1 0 ) a n d ( 1 1 ) , t h e p o s i t i o n a n d p o t e n t i a l o f t h e scattered w a v e c a n be u p d a t e d . C o n -s e q u e n t l y t h e n e w t o t a l f r e e w a t e r -s u r f a c e can be d e t e r m i n e d b y s u p e r i m p o s i n g 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 n d t h e c o m p u t a t i o n can p r o c e e d t o t h e n e x t time step.
To solve t h e b o u n d a r y i n t e g r a l e q u a t i o n f o r t h e s c a t t e r e d w a v e , t h e h i g h e r - o r d e r b o u n d a r y e l e m e n t m e t h o d is e m p l o y e d , w h e r e t h e s u r f a c e o v e r w h i c h t h e i n t e g r a l is p e r f o r m e d is d i s c r e t i z e d b y q u a d r a t i c i s o p a r a m e t r i c e l e m e n t s . I n p a r t i c u l a r , s t r u c t u r e d 8 - n o d e q u a d r i l a t e r a l e l e m e n t s are d i s t r i b u t e d o n t h e v e r t i c a l w a l l s such as t h e t a n k w a l l a n d b o d y v e r t i c a l surfaces. O n t h e f r e e s u r f a c e Sp as w e l l as o n t h e b a r g e b o t t o m s , u n s t r u c t u r e d 6 n o d e t r i a n g u l a r e l e -m e n t s are g e n e r a t e d b y u s i n g t h e D e l a u n a y t r i a n g u l a t i o n -m e t h o d . U n s t r u c t u r e d t r i a n g u l a r m e s h e s have p r o v e n t o be m o r e r o b u s t t o t h e c h a n g e o f b o u n d a r y shapes t h a n q u a d r i l a t e r a l meshes. M e s h d e n s i t y is c o n t r o l l e d b y t h e m e s h size at t h e b o u n d a r i e s . The w h o l e b o u n d a r y o f t h e d o m a i n is d i v i d e d i n t o several p a t c h e s o n w h i c h t h e m e s h e s c a n be a d j u s t e d a c c o r d i n g l y s u c h t h a t w e are able t o m a n u a l l y c o n t r o l the m e s h e s o n each p a t c h . Near t h e s t r u c t u r e s w h e r e h i g h m e s h r e s o l u t i o n is r e q u i r e d , f o r i n s t a n c e o n t h e g a p surface, m o r e e l e m e n t s can be d i s t r i b u t e d ; w h i l e o n t h e f r e e s u r -face n e a r t h e t a n k w a l l w h e r e t h e s c a t t e r e d w a v e w i l l be d a m p e d out, m e s h size can be large to m a x i m i z e c o m p u t a t i o n a l e f f i c i e n c y . A n e x a m p l e o f a 3 D m e s h f o r a w a v e p a s s i n g s i d e - b y - s i d e barges is s h o w n i n Fig. 1(b), w h e r e i t is seen t h a t a h i g h d e n s i t y o f e l e m e n t s is d i s t r i b u t e d n e a r t h e t w o barges. Once t h e s c a t t e r e d w a v e p o t e n -t i a l a n d v e l o c i -t y are o b -t a i n e d b y s o l v i n g a se-t o f algebraic l i n e a r e q u a t i o n s , t h e h y d r o d y n a m i c forces a c t i n g o n t h e s t r u c t u r e s c a n be c o m p u t e d b y i n t e g r a t i n g t h e pressure o v e r t h e b o d y surfaces.
122 X. Feng, W. Bai/Applied Ocean Researdi 50 (2015) 119-129 T h e Standard 4 t h - o r d e r R u n g e - K u t t a s c h e m e is e m p l o y e d i n t h e p r e s e n t m o d e l to u p d a t e t h e p o t e n t i a l a n d p o s i t i o n o f t h e s c a t t e r e d w a v e . I n o r d e r t o achieve a l o n g - t i m e s i m u l a t i o n , m e s h r e g e n e r a t i o n o n t h e f r e e surface is r e q u i r e d t o m i t i g a t e t h e s a w -t o o -t h n u m e r i c a l i n s -t a b i l i -t y , a n d -t h i s is i m p l e m e n -t e d b y a d o p -t i n g t h e L a p l a c i a n s m o o t h i n g t e c h n i q u e t o o b t a i n t h e n e w n o d e s o n t h e f r e e s u r f a c e . The v a r i a b l e s at the n e w nodes are o b t a i n e d f r o m i n t e r p o l a t i o n w i t h those o f t h e o l d ones. D e t a i l e d n u m e r i c a l i m p l e -m e n t a t i o n s are p r e s e n t e d i n Bai and E a t o c k T a y l o r [ 2 4 , 2 5 ] , w h e r e s e v e r a l v a l i d a t i o n studies w e r e c o n d u c t e d f o r s i m p l e g e o m e t r i e s .
2.2. Prescribed incident wave models
T h e a b o v e r e f o r m u l a t e d b o u n d a r y v a l u e p r o b l e m f o r s o l v i n g t h e s c a t t e r e d w a v e is e q u i v a l e n t t o t h a t f o r t h e o r i g i n a l t o t a l f l o w i f t h e i n c i d e n t f l o w satisfies t h e Laplace e q u a t i o n a n d a l l t h e f u l l y n o n l i n -ear b o u n d a r y c o n d i t i o n s . The c o n v e n i e n c e o f r e p r e s e n t i n g d i f f e r e n t sea states b y p r e s c r i b i n g s p e c i f i c i n c i d e n t w a v e s is o n e o f t h e i m p o r -t a n -t f e a -t u r e s o f -t h e p r e s e n -t m o d e l . P r a c -t i c a l l y , -t h e i n c i d e n -t w a v e s can be d e s c r i b e d b y a n y w a v e m o d e l . For w e a k l y n o n l i n e a r w a v e s , a Stokes m o d e l ( 5 t h o r d e r f o r i n s t a n c e ) m i g h t be u t i l i z e d . The t r a d e -o f f is t h a t t h e t -o t a l f l -o w is t h e n a p p r -o x i m a t e d t -o t h e e x t e n t t h a t t h e i n c i d e n t flow is a p p r o x i m a t e d .
I n all t h e cases s t u d i e d i n this paper, t h e i n c i d e n t w a v e s are p r o p a g a t i n g i n r e l a t i v e l y d e e p w a t e r a n d w a v e steepness is n o t v e r y h i g h . To s o m e w h a t s i m p l i f y t h e m o d e l , a 5 t h - o r d e r Stokes w a v e is i m p o s e d i n t h e fluid d o m a i n as the n o n l i n e a r i n c i d e n t flow,
3 . V a l i d a t i o n a n d m e s h c o n v e r g e n c e W e first i n v e s t i g a t e t h e w e a k l y n o n l i n e a r case o f s e c o n d - o r d e r n e a r - t r a p p i n g associated w i t h an a r r a y o f f o u r c y l i n d e r s . T h i s is t o v a l i d a t e t h e n e w l y d e v e l o p e d n u m e r i c a l m o d e l , w h e r e h i g h e r h a r -m o n i c s are o f i n t e r e s t a n d p l e n t y o f a n a l y t i c a l a n d e x p e r i -m e n t a l r e s u l t s are a v a i l a b l e f o r c o m p a r i s o n . S u c h n e a r t r a p p i n g is a r e s -o n a n t p h e n -o m e n -o n s i m i l a r t -o t h e gap res-onance. I n t h i s case, t h e f o u r b o t t o m - m o u n t e d c i r c u l a r c y l i n d e r s are o f r a d i u s a = 0.203 m , s t a n d i n g at t h e corners o f a square 4a x 4a, i n a w a t e r d e p t h d=3a. The s e c o n d o r d e r a n a l y t i c a l s o l u t i o n i n M a l e n i c a et al. [ 2 7 ] d e m o n -s t r a t e d a -s e c o n d - o r d e r n e a r - t r a p p e d m o d e at t h e i n c i d e n t w a v e f r e q u e n c y to = 0.468 i n a 4 5 ° h e a d i n g w a v e . This is t h e case t e s t e d i n t h e e x p e r i m e n t s i n O h l et al. [ 2 8 ] w i t h a n i n c i d e n t w a v e o f steepness /</l = 0.126, a n d h e r e is s i m u l a t e d f o r c o m p a r i s o n . M o r e i n f o r m a -t i o n o n -t h e n e a r - -t r a p p i n g a n d -t h e c o n f i g u r a -t i o n s o f -t h i s case can be f o u n d i n Bai e t al. [ 2 9 ] . I n t h e i r analysis, w h e r e a r e c t a n g u l a r n u m e r i c a l w a v e t a n k w a s a d o p t e d a n d t h e p r o p a g a t i n g m o n o c h r o -m a t i c w a v e w a s g e n e r a t e d b y a p i s t o n - l i k e w a v e -maker, a l o n g ^ 0 — Baie/al. (2014) Present — Baie/al. (2014) Present
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Fig. 2. Time liistory of wave elevation at the center of cylinder array with ka = 0,468 in 45° heading wave, s i m u l a t i o n time w a s r e q u i r e d b e f o r e t h e s t e a d y - s t a t e r e g i m e w a s a c h i e v e d . As e x p e c t e d , i t is f o u n d t h a t the n e w m o d e l d e v e l o p e d i n t h e p r e s e n t p a p e r c a n have a s m a l l e r c o m p u t a t i o n a l d o m a i n , a n d p r o v i d e s t e a d y state results m o r e q u i c k l y . Fig. 2 p l o t s t h e n o n - d i m e n s i o n a l t i m e h i s t o r y o f w a v e eleva-t i o n aeleva-t eleva-t h e c e n eleva-t e r o f eleva-t h e array. I eleva-t can be seen eleva-t h a eleva-t eleva-t h e p r e s e n eleva-t r e s u l t s are v e r y close t o those i n Bai et a l . [ 2 9 ] ( n o t e t h a t t h e y h a v e a phase s h i f t as e x p e c t e d ) a f t e r t h e s t e a d y states have b e e n reached, y e t a s m a l l e r n u m b e r o f w a v e p e r i o d s is r e q u i r e d f o r t h e p r e s e n t m o d e l t o a c h i e v e the flnal steady state. A s i m i l a r c o m p a r -i s o n -is m a d e -i n F-ig. 3 f o r t h e d -i m e n s -i o n l e s s h o r -i z o n t a l w a v e force o n the u p s t r e a m c y l i n d e r . A l i k e l y e x p l a n a t i o n f o r t h e s m a l l d i f -f e r e n c e o b s e r v e d i n Figs. 2 a n d 3 m a y be t h e side w a l l e -f -f e c t s i n t h e n u m e r i c a l m o d e l o f Bai et a l . [ 2 9 ] , w h i l e t h e c u r r e n t s i m u l a -t i o n is m o r e s u i -t a b l e f o r o p e n sea c o n d i -t i o n s . As i n [ 2 9 ] , w e have u s e d FFT t e c h n i q u e s t o e x t r a c t f r o m t h e time h i s t o r y o f w a v e ele-v a t i o n seele-veral h i g h e r h a r m o n i c s , as i n d i c a t e d i n Fig. 4 . This s h o w s ( s u i t a b l y n o n - d i m e n s i o n a l i s e d ) d i f f e r e n t h a r m o n i c c o m p o n e n t s o f w a v e e l e v a t i o n a l o n g t h e d i a g o n a l w i t h i n t h e c y l i n d e r a r r a y at the s e c o n d - o r d e r n e a r - t r a p p i n g f r e q u e n c y c o r r e s p o n d i n g t o ka = 0.468. T h e first, second a n d t h i r d h a r m o n i c s are t h e c o m p o n e n t s o f f u l l y n o n l i n e a r r e s u l t s at s i n g l e - , d o u b l e - , a n d t r i p l e t h e f r e q u e n c y o f , 5 : 8 Bai elar (2014) 8 Present 4
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0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 t/TFig. 5. Time history of wave elevation at x = 0,366 m with ka = 0,468 for three differ-ent meshes. w h i c h are n o n - d i m e n s i o n a l i s e d s i m i l a r l y ) , t h e o v e r a l l a g r e e m e n t is f a i r l y g o o d . I n o r d e r t o d e m o n s t r a t e t h e c o n v e r g e n c e o f t h e p r e s e n t f u l l y n o n l i n e a r m o d e l w i t h c o m p u t a t i o n a l meshes, e s p e c i a l l y f o r h i g h e r h a r m o n i c c o m p o n e n t s , a n o t h e r t w o m e s h c o n f i g u r a t i o n s , coarse ' M e s h a' a n d fine ' M e s h c', are a d o p t e d f o r t h e a b o v e case, w h i l e ' M e s h b ' c o r r e s p o n d s t o t h e r e f e r e n c e m e s h d e n s i t y used f o r r e s u l t s i n Fig. 4. The e l e m e n t sizes o n t h e f r e e s u r f a c e n e a r t h e c y l i n d e r a r r a y i n cases o f ' M e s h a', ' M e s h b ' a n d ' M e s h c' are a p p r o x i m a t e l y o n e t h i r t i e t h , o n e fiftieth a n d o n e s e v e n t i e t h o f t h e l i n e a r w a v e l e n g t h , r e s p e c t i v e l y . Fig. 5 displays t h e t i m e h i s t o r y o f w a v e eleva-t i o n aeleva-t X = 0.366 m , eleva-t h e u p w a v e face o f eleva-t h e d o w n s eleva-t r e a m c y l i n d e r f o r these t h r e e m e s h c o n d i t i o n s . T h e y are a l m o s t i d e n t i c a l e x c e p t at t h e m a x i m a w h e r e discrepancies appear w i t h i n a v e r y s m a l l range (less t h a n 3%). Because h i g h e r - o r d e r e f f e c t s are o f i n t e r e s t i n t h e p r e s e n t s t u d y , second a n d t h i r d h a r m o n i c s e x t r a c t e d f r o m f u l l y n o n l i n e a r r e s u l t s , o f f r e e surface e l e v a t i o n a l o n g t h e d i a g o n a l ( e q u i v a l e n t t o those i n Fig. 4 ) are i l l u s t r a t e d i n Fig. 6 f o r the t h r e e m e s h c o n f i g u -r a t i o n s . One c a n n o t i c e t h a t s e c o n d - o -r d e -r c o m p o n e n t s i n Fig. 6(a) are v e r y close a m o n g d i f f e r e n t m e s h densities, a n d o n l y m i n o r v a r i -ances are o b s e r v e d f o r t h e t h i r d h a r m o n i c s i n Fig. 6 ( b ) . The above c o m p a r i s o n s suggest t h a t t h e c u r r e n t n u m e r i c a l m o d e l converges v e r y f a s t w i t h c o m p u t a t i o n a l meshes, a n d i n t e r m s o f accuracy, i t is n o t v e r y s e n s i t i v e t o m e s h d e n s i t y or m e s h size as l o n g as t h e c o m -p u t a t i o n a l m e s h is n o t t o o coarse. I n a l l t h e f o l l o w i n g s i m u l a t i o n s , s i m i l a r m e s h densities t o t h a t i n ' M e s h b ' are a d o p t e d , i n o r d e r t o a c h i e v e a balance b e t w e e n accuracy a n d c o m p u t a t i o n a l e f f o r t , I n s u m m a r y , t h e p r e s e n t f u l l y n o n l i n e a r p o t e n t i a l m o d e l is capable o f c a p t u r i n g t h e h i g h e r o r d e r e f f e c t s associated w i t h n o n -l i n e a r f r e e surface b o u n d a r y c o n d i t i o n s a n d i m p r o v e s t h e e f f i c i e n c y t h e i n c i d e n t w a v e , r e s p e c t i v e l y . A n a l y t i c a l s o l u t i o n s b y the s e c o n d -o r d e r p -o t e n t i a l t h e -o r y i n M a l e n i c a et a l . [ 2 7 ] , n u m e r i c a l r e s u l t s i n Bai e t a l . [ 2 9 ] a n d e x p e r i m e n t a l d a t a f r o m O h l et al. [ 2 8 ] are i n c l u d e d i n t h e c o m p a r i s o n s . The first-order results i n Fig. 4 ( a ) are g e n e r a l l y v e r y close. M o r e s t r i k i n g is t h e n o n - d i m e n s i o n a l i s e 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 i n Fig. 4 ( b ) , w h e r e r e s u l t s o f b o t h Bai et al. [ 2 9 ] a n d t h e p r e s e n t m o d e l agree w e l l w i t h t h e e x p e r i m e n t s . The a n a l y t i -cal s o l u t i o n o f M a l e n i c a et a l . [ 2 7 ] , h o w e v e r , t e n d s t o o v e r - p r e d i c t t h e s e c o n d o r d e r e l e v a t i o n n e a r x = 0,366 m u p w a v e o f t h e d o w n -s t r e a m c y l i n d e r . A po-s-sible r e a -s o n i-s t h a t a l l r e -s u l t -s o t h e r t h a n t h e a n a l y t i c a l s o l u t i o n are t h e s e c o n d h a r m o n i c c o m p o n e n t s e x t r a c t e d f r o m n o n l i n e a r t i m e series a n d e f f e c t s f r o m h i g h e r orders, p o t e n -t i a l l y h i g h e r r e s o n a n -t m o d e s , c a n be presen-t, w h i l e -t h e a n a l y -t i c a l s o l u t i o n o f M a l e n i c a e t a l . [ 2 7 ] is l i m i t e d t o first a n d second o r d e r o n l y . One e x a m p l e o f t h i s m a y be f o u n d i n Fig. 4 ( d ) , w h i c h i l l u s -t r a -t e s -t h e -t h i r d h a r m o n i c s o f w a v e e l e v a -t i o n f r o m Bai e-t a l . [ 2 9 ] a n d t h e p r e s e n t m o d e l . B o t h r e s u l t s s h o w a v e r y s i m i l a r o v e r a l l t r e n d , w i t h v e r y h i g h v a l u e s at s o m e p o s i t i o n s . These h i g h t h i r d -h a r m o n i c e l e v a t i o n s m i g -h t be e x p l a i n e d b y a t -h i r d - o r d e r r e s o n a n t m o d e e x c i t e d i n t h e c u r r e n t c o n f l g u r a t i o n . C o m p a r i s o n s o f t h e s e c o n d - o r d e r m e a n e l e v a t i o n are p l o t t e d i n Fig. 4 ( c ) . C o n s i d e r i n g t h e s m a l l range (e.g. c o m p a r e d w i t h the second h a r m o n i c results
( a ) 20 15 10
^5
A Mesh a - - -•- - Mesh b 0 Mesh c A -0.4 -0,3 -0.2 -0,1 ( b ) 0.0 X 0,1 0,2 40 30 20 a 10 A Mesh a - -m-- Mesh b o Mesh c A y / 1'7
A -0,4 -0,3 -0.2 -0.1 0,0 X 0.1 0.2 0.3 0 4 0.3 0.4 Fig. 6. a t t a =Mesh convergence of higher harmonics of wave elevation along the diagonal 0.468: (a) second harmonic and (b) third harmonic.
124 X. Feng, W. Bai/Applied Ocean Research 50 (2015) 119-129 against t l i e p r e v i o u s m o d e l i n Bai et a l . [ 2 9 ] i n t e r m s o f c o m p u t a -t i o n a l e f f o r -t . M o r e o v e r , -t h e n a -t u r e o f -t h e p r e s e n -t m o d e l enables a l o n g - t i m e s i m u l a t i o n t o be p e r f o r m e d w i t h o u t t h e c o n c e r n a b o u t w a v e r e f l e c t i o n f r o m t h e t a n k w a l l ; t h i s is v e r y i m p o r t a n t i f a large m a r i n e s y s t e m is c o n s i d e r e d a n d a l o n g t i m e is r e q u i r e d f o r t h e s t e a d y state t o be a c h i e v e d . I n the i n v e s t i g a t i o n o f gap r e s o n a n c e s b e t w e e n s i d e - b y - s i d e barges w h e r e t h e t r a n s i e n t r e g i m e m i g h t b e l o n g , these a d v a n t a g e s b e c o m e e x t r e m e l y u s e f u l .
4. Gap r e s o n a n c e s
In t h i s s e c t i o n , w e i n v e s t i g a t e the r e s o n a n t m o d e s i n a n a r r o w gap b e t w e e n s i d e - b y - s i d e barges. The same t w o f i x e d r e c t a n g u l a r barges w i t h square bilges as i n M o l i n et al. [ 1 ] are s i m u l a t e d . The c o n f i g u r a t i o n o f the s i d e - b y - s i d e barges at m o d e l scale is as f o l l o w s : b a r g e l e n g t h is 2.47 m , w i d t h 0.6 m , d r a f t 0.18 m a n d gap w i d t h 0.12 m . T h e w a t e r d e p t h is set as 3 m ( t h e s a m e as i n t h e tests), a n d t h e t a n k r a d i u s f o r the p r e s e n t s i m u l a t i o n s is e i t h e r 5 m or f o u r rimes the i n c i d e n t w a v e l e n g t h , w h i c h e v e r is larger. The i n c i d e n t w a v e h e a d i n g c o n s i d e r e d h e r e is 90 degrees, i.e. b e a m sea, i n w h i c h t h e resonances are m o r e c r i t i c a l . The c o o r d i n a t e s y s t e m is s h o w n i n Fig. 1(a). A n e x a m p l e o f a m e s h used f o r c a l c u l a t i o n s is d i s p l a y e d i n Fig. 1(b), w i t h 4 8 8 4 p o i n t s a n d 1606 e l e m e n t s o n t h e barges, a n d 11,137 p o i n t s a n d 5 3 8 3 e l e m e n t s o n t h e f r e e w a t e r surface.
To closely i n v e s t i g a t e the r e s o n a n t f r e q u e n c i e s , response peaks a n d m o d e shapes, w e s i m u l a t e t h e barge s y s t e m i n r e g u l a r b e a m seas. As s t a t e d i n S e c t i o n 2.2, a r e g u l a r 5 t h - o r d e r Stokes w a v e is s p e c i f i e d as t h e i n c o m i n g w a v e . F i r s t l y w e u t i l i z e a r e l a t i v e l y s m a l l w a v e steepness /o4 = 0.0034 f o r a l l w a v e f r e q u e n c i e s . The n o n l i n -e a r i t y o f t h -e p o t -e n r i a l f l o w m o d -e l w i l l b-e c o n s i d -e r -e d i n S -e c t i o n 5. W i t h s u c h l o w steepness, w e e x p e c t t h a t t h e results s h o u l d b e c o n v e r g e n t t o the l i n e a r s o l u t i o n s . C o m p a r i s o n s o f f r e e s u r f a c e e l e -v a t i o n against l i n e a r t h e o r y a n d e x p e r i m e n t a l data i n M o l i n et al. [ 1 j are p l o t t e d o v e r t h e f r e q u e n c i e s 5 . 0 - 9 . 0 rad/s i n Fig. 7. Fig. 7 ( a ) - ( d )
s h o w s f r e e surface Response A m p l i t u d e O p e r a t o r s (RAOs) i n the gap at f o u r p o s i t i o n s : x = 0.0 m , x = 0 . 3 m , x = 0.6 m , x = 0 . 9 m , respec-t i v e l y , w h i c h are f r o m m i d s h i p respec-t o w a r d respec-t h e barge e n d . The o v e r a l l a g r e e m e n t is g o o d a n d t h e RAOs are c h a r a c t e r i z e d b y t h r e e peaks o v e r t h i s r a n g e o f f r e q u e n c i e s . C a r e f u l i n s p e c t i o n finds t h a t the p r e s e n t results w i t h l o w w a v e steepness a l m o s t c o i n c i d e w i t h the l i n e a r c a l c u l a t i o n s as e x p e c t e d , e v e n n e a r t h e peaks. H o w e v e r , dis-crepancies are o b s e r v e d b e t w e e n t h e 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. It is f o u n d t h a t t h e p o t e n t i a l flow m o d e l s , b o t h the l i n e a r cal-c u l a t i o n s a n d f u l l y n o n l i n e a r s i m u l a t i o n s w i t h l o w w a v e steepness, o v e r - p r e d i c t the w a v e e l e v a t i o n s near t h e r e s o n a n t m o d e s .
The peaks a p p e a r i n g i n t h e f r e e s u r f a c e RAOs c o r r e s p o n d to the resonances i n t h e w a v e m o t i o n s . I n t h i s case, these t h r e e resonan-ces o c c u r near t h e f r e q u e n c i e s 5.75 rad/s, 6.85 rad/s a n d 8.0 rad/s, r e s p e c t i v e l y . They c o r r e s p o n d t o m o d e 1, m o d e 3 a n d m o d e 5, as o n l y s y m m e t r i c s l o s h i n g m o d e s a p p e a r i n t h e case o f a b e a m sea, w h i l e t h e r e are i n t e r m e d i a t e a n t i - s y m m e t r i c s l o s h i n g modes at i n c i d e n c e s o t h e r t h a n 9 0 degrees. T h e first r e s o n a n t m o d e or p i s t o n m o d e , w i t h large w a v e responses, is g e n e r a l l y m o r e c r i t i c a l t h a n the o t h e r , l o n g i t u d i n a l s l o s h i n g m o d e s i n t h e gap. To have a d i r e c t i l l u s t r a t i o n o f t h e r e s o n a n t m o d e s , w e p l o t t h e c o n t o u r s of m a x i m u m f r e e surface e l e v a t i o n s n o r m a l i z e d b y t h e i n c i d e n t w a v e a m p l i t u d e i n Fig. 8 w h e r e t h e i n c i d e n t w a v e p r o p a g a t e s v e r t i c a l l y d o w n w a r d s . Fig. 8(a) c o r r e s p o n d s to t h e p i s t o n m o d e at 5.75 rad/s, w h e r e a n e a r - s t a n d i n g w a v e p a t t e r n i n f r o n t o f t h e u p w a v e barge f o r m s . The n e a r - s t a n d i n g w a v e p a t t e r n i n f r o n t o f t h e u p w a v e barge b e c o m e s m o r e c o m p l i c a t e d at h i g h e r r e s o n a n t m o d e s i n Fig. 8(b) a n d (c). R e f l e c t i o n s i n f r o n t o f t h e u p w a v e barge s e e m m o r e e v i d e n t at m o d e 5 i n Fig. 8(c) t h a n t h o s e i n m o d e s 1 a n d 3. Three f o c u s i n g r e g i o n s are f o r m e d i n f r o n t o f t h e u p w a v e b a r g e a n d m o r e are f o u n d f u r t h e r u p s t r e a m . U n l i k e t h e decrease o f m a x i m u m w a v e e l e v a t i o n f r o m m i d s h i p t o the gap e n d at the first m o d e , t h r e e a n d five l o w -a m p l i t u d e pe-aks -are p r e s e n t -a n d u n i f o r m l y d i s t r i b u t e d -at m o d e 3 a n d m o d e 5 r e s p e c t i v e l y .
X. Feng, W. Bai/Applied Ocean Research 50 (2015) 119-129 125
Fig. 8. Contours of maximum free surface elevation near the barges at near-resonant modes in beam sea: (a) (i) = 5.75 rad/s; ( b ) « = 6.85 rad/s and (c) CÜ = 8.0 rad/s.
F o c u s i n g o n t l i e gap surface, Fig. 9 p r e s e n t s t l i e m a x i m u m w a v e e l e v a t i o n s a l o n g t h e gap i n these t h r e e m o d e s . T h e y are all s y m m e t -ric w i t h respect to t h e m i d s h i p due to t h e s y m m e t r y i n a 9 0 ° h e a d i n g w a v e . S p e c i f i c a l l y , t h e m a x i m u m e l e v a t i o n near m i d s h i p i n t h e gap at t h e f i r s t m o d e is o v e r 6 t i m e s t h e i n c i d e n t w a v e a m p l i t u d e a n d i t decays t o n e a r i y t h e i n c i d e n t w a v e a m p l i t u d e at t h e e n d o f t h e gap. It is v i s i b l e f r o m t h e m o v e m e n t o f t h e f l u i d t h a t the f r e e s u r f a c e i n t h e gap oscillates as a f l e x i b l e p l a t e . A t m o d e 3, t h r e e peaks, o n e at m i d s h i p a n d t w o near t h e g a p ends, are o b s e r v e d i n t h e gap. T h e peak e l e v a t i o n at m o d e 3 b e c o m e s a b o u t h a l f o f t h a t at t h e f i r s t m o d e a n d e v e n l o w e r at m o d e 5. To d e m o n s t r a t e t h e m o v e m e n t o f f l u i d i n t h e gap, w e p l o t i n Fig. 10 t h e i n s t a n t a n e o u s f r e e surface i n o n e cycle ( 5 0 t i m e steps i n one w a v e p e r i o d ) at m o d e 3 w i t h t h e w a v e f r e q u e n c y 6.85 rad/s. I n t h e steady state, t h e w a v e s o s c i l l a t e
I 3 1 •g 2 mode 1 mode 3 '''^ T^s^^ mode 5
\ /
-1.2 -0.8 -0.4 0.0 X 0.4 0.8 1.2 w i t h i n t h e e n v e l o p e . I t is p r e d i c t a b l e t h a t at e v e n h i g h e r m o d e s ( r e s o n a n t e l e v a t i o n s are v e r y l o w a n d o f less i n t e r e s t ) analogous m o d e shapes w o u l d also occur.The h y d r o d y n a m i c forces o n t h e barges d e m o n s t r a t e a n o t h e r aspect o f t h e resonances. I n t h i s c o n f i g u r a t i o n w i t h t h e barges i n b e a m seas, s w a y forces are o f m o r e p r a c t i c a l i n t e r e s t t h a n f o r c e s i n o t h e r d i r e c t i o n s . Fig. 11 s h o w s t h e m a x i m u m s w a y f o r c e s o n t h e u p w a v e a n d leeside barges, as w e l l as o n t h e e n t i r e t w o - b a r g e s y s t e m . Forces are n o n - d i m e n s i o n a l i s e d b y pgALD, w h e r e L is t h e l e n g t h a n d D t h e d r a f t o f t h e barge. I t is f o u n d t h a t s w a y forces o n b o t h t h e u p w a v e a n d leeside barges are s i g n i f i c a n t l y a m p l i f i e d near t h e p i s t o n m o d e f r e q u e n c y 5.75 rad/s. N e v e r t h e l e s s , o n l y a m i l d h u m p appears near m o d e 3 f r e q u e n c y 6.85 rad/s a n d n o a m p l i f i c a -t i o n is o b s e r v e d near m o d e 5 f r e q u e n c y 8.0 rad/s. M o r e o v e r , w h e n t h e i n c i d e n t w a v e f r e q u e n c y is b e y o n d t h e p i s t o n m o d e r e g i o n ( o v e r a b o u t 6.0 rad/s), t h e s w a y f o r c e o n t h e u p w a v e barge d o m i n a t e s a n d s w a y f o r c e o n t h e leeside barge b e c o m e s v e r y s m a l l a n d e v e n t u a l l y close t o z e r o . This is p h y s i c a l l y r e a s o n a b l e because t h e s h i e l d i n g
Fig. 9. Maximum free surface elevation along the gap between two barges in beam sea at modes 1,3 and 5.
Fig. 10. Envelope of wave elevation along the gap at resonant mode a> = 6.85 rad/s in beam sea.
126 X. Feng. W. Bai/AppHed Ocean Researdi 50 (2015) ï 19-129
e f f e c t b e c o m e s m o r e p r o n o u n c e d at h i g h e r w a v e f r e q u e n c i e s a n d r e f l e c t i o n b y t h e u p w a v e barge b e c o m e s m o r e s i g n i f l c a n t . This has b e e n d e m o n s t r a t e d i n t h e c o n t o u r s o f w a v e e l e v a t i o n i n Fig. 8. I n t e r e s t i n g l y , t h e t o t a l s w a y f o r c e s h o w n i n Fig. 11 r e m a i n s a l m o s t c o n s t a n t ( s l i g h t l y d r o p p i n g ) o v e r t h e b r o a d r a n g e o f f r e q u e n c y , e v e n near t h e p i s t o n m o d e at w h i c h s w a y forces o n u p w a v e a n d l e e -side barges are m u c h h i g h e r . This suggests t h a t t h e gap resonances h a v e g r e a t i n f l u e n c e o n e a c h o f t h e barges, y e t n o r e s o n a n c e occurs o n t h e w h o l e s i d e - b y - s i d e barge s y s t e m . I n a d d i t i o n , t h e m e a n d r i f t forces i n s w a y o n each o f t h e barges at d i f f e r e n t f r e q u e n c i e s are p l o t t e d i n Fig. 12, w h i c h again i l l u s t r a t e s the g a p resonances. D r i f t forces i n Fig. 12 are n o r m a l i z e d b y pgA^L The d i r e c t i o n s o f t h e d r i f t forces i n d i c a t e t h a t t h e s t a n d i n g w a v e s i n t h e g a p are s e p a r a t i n g t h e s i d e - b y - s i d e barges n e a r t h e resonances.
5. N o n l i n e a r e f f e c t s o n g a p r e s o n a n c e s
5.1. Nonlinear behaviors
It has b e e n r e p o r t e d i n S e c t i o n 4 t h a t l i n e a r p o t e n t i a l flow m o d -els t e n d t o o v e r - p r e d i c t w a v e response i n t h e gap i n t h e p i s t o n m o d e . S e m i - e m p i r i c a l t r e a t m e n t s b y i n t r o d u c i n g e i t h e r l i n e a r o r n o n l i n e a r t e r m s i n t o the l i n e a r i z e d f r e e surface b o u n d a r y c o n d i -tions as a n a r t i f i c i a l ' d a m p i n g t e r m ' have b e e n d e v e l o p e d i n several r e s e a r c h g r o u p s ( N e w m a n [12], Chen [ 1 3 ] a n d M o l i n et al. [ 1 ]). It has b e e n r e c e n t i y d e m o n s t r a t e d i n K r i s t i a n s e n a n d F a l t i n s e n [ 2 0 ] t h a t flow s e p a r a t i o n at t h e barge bilges m a i n l y accounts f o r t h e d i s c r e p -a n c y b e t w e e n l i n e -a r p r e d i c t i o n s -a n d m e -a s u r e m e n t s -at t h e p i s t o n m o d e f r e q u e n c y , a n d n o n l i n e a r i t y d u e t o t h e f r e e surface is o f m i n o r i m p o r t a n c e i n m o d i f y i n g t h e h i g h r e s o n a n t response. H o w e v e r t h e n o n l i n e a r e f f e c t s o n t h e gap resonances have n o t p r e v i o u s l y b e e n m u c h discussed. F r o m t h e p o i n t v i e w o f a n o n l i n e a r m a s s - s p r i n g
2.5
1 5 1— . — I — . — , — . — , — . — I —.— , —.— , —.—I
5.0 5.5 6.0 6.5 7.0 7.5 8.0 8,5
Frequency rad/s
Fig. 12. Mean drift force in sway on eacli barge in beam sea.
s y s t e m a t the p i s t o n m o d e , t h e n o n l i n e a r i t y m a y s l i g h t l y s h i f t t h e r e s o n a n t f r e q u e n c y , w h i c h s h o w s a ' s t i f f / s o f t s p r i n g ' or D u f f i n g - l i k e b e h a v i o r ( F a l t i n s e n et a l . [ 1 8 ] ) . By p e r f o r m i n g s i m u l a t i o n s w i t h a s m a l l f r e q u e n c y step a r o u n d t h e peak a n d i n c r e a s i n g w a v e s t e e p -ness, w e discuss i n t h i s s e c t i o n t h e n o n l i n e a r i n f l u e n c e o f t h e f r e e s u r f a c e o n t h e gap resonances. I t s h o u l d be m e n t i o n e d t h a t i t is h a r d t o f u r t h e r increase t h e w a v e steepness i n t h i s case, as t h e c a l -c u l a t i o n s t e n d t o b r e a k d o w n b e f o r e t h e final steady state -c a n be a c h i e v e d . C a r e f u l i n s p e c t i o n s h o w s t h a t local u n s m o o t h s u r f a c e e l e -v a t i o n s n e a r t h e gap o p e n i n g s cause t h e crash o f s i m u l a t i o n s , w h i c h m i g h t be d u e t o t h e s a w - t o o t h n u m e r i c a l i n s t a b i l i t y .
The results o f s i m u l a t i o n s w i t h l o w w a v e steepness /oA = 0 . 0 0 3 4 are v e r y close t o l i n e a r c a l c u l a t i o n s as discussed i n S e c t i o n 4 . S i m -u l a t i o n s w i t h i n c r e a s i n g w a v e steepness w e r e n e x t p e r f o r m e d , k e e p i n g a l l t h e c o n f l g u r a t i o n s t h e same e x c e p t t h e i n c i d e n t w a v e a m p l i t u d e . A g a i n , o n l y t h e b e a m sea case w a s c o n s i d e r e d . O f t h e i n t e r e s t h e r e is t h e p i s t o n m o d e , w h e r e t h e r e s o n a n t f r e q u e n c y a c c o r d i n g t o t h e l i n e a r c a l c u l a t i o n is n e a r 5.75 rad/s. Gap s u r f a c e RAOs at m i d s h i p close t o t h e p i s t o n m o d e f r e q u e n c y w i t h v a r i o u s i n c i d e n t w a v e steepnesses (;<A = 0.0034, 0.034, 0.055, a n d 0.067), w i t h a fine f r e q u e n c y r e s o l u t i o n a r o u n d t h e peak, are p r e s e n t e d i n Fig. 13. I t can be seen t h a t t h e r e s p o n s e c u r v e s h i f t s s l i g h t l y t o a h i g h e r f r e q u e n c y r e g i o n as t h e w a v e steepness increases, l e a d i n g to t h e s h i f t o f r e s o n a n t f r e q u e n c y b y a b o u t 1% w h e n !<A increases f r o m 0.0034 t o 0.067. A t t h e s a m e t i m e , t h e RAO p e a k d r o p s b y a b o u t 5% w i t h t h e increase o f w a v e steepness o v e r t h e r a n g e c o n -s i d e r e d here. F r o m t h e figure w e can -see t h a t -steeper w a v e -s lead to s l i g h t l y s m a l l e r peak r e s o n a n t responses, h o w e v e r , t h e p r e s e n t f u l l y n o n l i n e a r p o t e n t i a l flow m o d e l s t i l l o v e r - p r e d i c t s t h e peak r e s o n a n t response. This i n d i c a t e s t h a t t h e f r e e surface n o n l i n e a r i t y plays a m i n o r r o l e i n d a m p i n g t h e h i g h r e s o n a n t response, c o n -flrming t h e w e l l a c k n o w l e d g e d u n d e r s t a n d i n g t h a t flow s e p a r a t i o n is t h e m a i n r e a s o n f o r t h e d i s c r e p a n c y b e t w e e n l i n e a r s o l u t i o n s a n d m e a s u r e m e n t s .
Fig. 14 s u m m a r i z e s t h i s s h i f t o f r e s o n a n t f r e q u e n c y as w e l l as t h e c h a n g e o f peak response w i t h i n c r e a s i n g w a v e steepness l<A. The h o r i z o n t a l axis r e p r e s e n t s t h e d i f f e r e n t w a v e steepnesses fc4, w h i l e t h e l e f t v e r t i c a l axis is t h e r e s o n a n t f r e q u e n c y at the d i f f e r e n t w a v e steepnesses, n o r m a l i z e d b y cog = 5.75 rad/s, t h e l i n e a r r e s u l t f o r the r e s o n a n t f r e q u e n c y . The right v e r t i c a l axis is t h e c o r r e s p o n d i n g peak R A O , n o r m a l i z e d b y t h e l i n e a r p e a k R A OQ. The s h i f t o f r e s o n a n t f r e q u e n c y t o h i g h e r values as f r e e s u r f a c e n o n l i n e a r i t y increases at t h e p i s t o n m o d e i l l u s t r a t e s a ' s t i f f s p r i n g ' b e h a v i o r o f s u c h a n o n -l i n e a r m a s s - s p r i n g s y s t e m , w h e r e a w a t e r c o -l u m n is p u m p e d i n a n a r r o w g a p f o r m e d b y s i d e b y s i d e barges. Some t h e o r e t i c a l a n a l -ysis o n t h e ' s t i f f / s o f t s p r i n g ' b e h a v i o r o f a n o n l i n e a r m e c h a n i c a l s y s t e m can be f o u n d i n F a l t i n s e n a n d T i m o k h a [ 2 3 ] f o r t h e s l o s h i n g p r o b l e m . E x p e r i m e n t a l s t u d i e s i n Fults [ 3 0 ] f o r the s t a n d i n g w a v e p r o b l e m also r e v e a l e d t h i s n o n l i n e a r b e h a v i o r associated w i t h a
