A p p l i e d O c e a n R e s e a r c h 4 4 ( 2 0 1 4 ) 71 - 8 1
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Applied Ocean Research
j o u r n a l h o m e p a g e : v v w w . o l s e v i e r . c o r n / l o c a t e / a p o r
O C E A N
R E S E A R C H
Fully nonlinear analysis of near-trapping phenomenon around an
oossMarkarray of cylinders
W. Bai^'*, X. Feng^ R. Eatock Taylor^'^, K.K. Ang^
'Department of Civil and Environmental Engineering, National University of Singapore, Kent Ridge, Singapore 117576, Singapore ''Department of Engineering Science, University of Oxford, Parks Road, Oxford 0X1 3Pf UK
A R T I C L E I N F O A B S T R A C T Article history: R e c e i v e d 27 F e b r u a r y 2 0 1 3 R e c e i v e d in r e v i s e d f o r m 2 0 A u g u s t 2 0 1 3 A c c e p t e d 10 N o v e m b e r 2 0 1 3 Keywords: A r r a y s o f c y l i n d e r s N e a r - t r a p p i n g F u l l y n o n l i n e a r H a r m o n i c a n a l y s i s W a v e d i f f r a c t i o n
The wave diffraction around an array of fixed vertical circular cylinders is simulated in a numerical wave tank by using a fully nonlinear model in the time domain. The emphasis of the paper lies in the insightful investigadon of the nonlinear properties of the near-trapping phenomenon associated w i t h the muldple cylinders. The numerical model is validated by analydcal soludons as well as experimental data for waves propagadng past two and four verdcal cylinders in certain arrangements. An array of four idendcal circular cylinders at the corners of a square w i t h an incident wave along the diagonal of the square is the main focus here for invesdgating the near-trapping phenomenon. When near-trapping occurs, the present study shows that an extremely high wave elevation near the cylinders can be observed. At the same time, the hydrodynamic forces on different cylinders are found to be either in phase or out of phase, leading to some characteristic force patterns acdng on the whole structure. Due to the nature of the numerical model adopted, nonlinearity at different orders can be captured using a harmonic analysis. In addition to first- and second-order near-trapping, the third-order (triple-frequency) nonlinear component is presented for the first time. For the configuration selected, i t is found that at one specific incident wave frequency and direction one trapped mode is excited by second-order effects, while a different trapped mode (having similar symmetries) is excited by the third harmonic of the incident wave frequency.
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1. I n t r o d u c t i o n
I t has been several decades since researchers started to w o r k o n the p r o b l e m o f w a v e s t r u c t u r e interactions. The h y d r o d y n a m i c p r o p -erties o f c o l u m n - s u p p o r t e d o f f s h o r e structures have attracted m u c h a t t e n t i o n w i t h g r o w i n g practical experience i n the o f f s h o r e i n d u s t r y of c o l u m n b a s e d p l a t f o r m s such as tension leg p l a t f o r m s and s e m i -submersibles o p e r a t i n g w e l l i n deep-water fields. In a d d i t i o n , one particular interest a m o n g other issues Is the h i g h wave elevation a n d s i g n i f i c a n t l y large w a v e force caused b y the w a v e d i f f r a c t i o n and r e -d i f f r a c t i o n -due to the presence o f m u l t i p l e cylin-ders. Damage to the l o w e s t deck o f those m u l t i - c o l u m n p l a t f o r m s has been reported, as f o r e x a m p l e discussed i n Swan et al. [1 ], a n d such cases c o u l d be due to u n d e r e s t i m a t i o n of the p r e v a i l i n g w a v e c l i m a t e , a n d / o r the unreliable p r e d i c t i o n o f the m a x i m u m w a v e elevation and u p w e l l i n g d u r i n g the design o f the p l a t f o r m s . I t is the latter o f these t h a t concerns us here. I t is l i n k e d to the p h e n o m e n o n k n o w n as ' n e a r - t r a p p i n g ' , w h i c h has been observed f o r m u l t i - c o l u m n arrays i n l a b o r a t o r y experiments, for e x a m p l e by Ohl et al. [ 2 ] and Kashiwagi and O h w a t a r i [ 3 ] . For some arrangements o f an array o f cylinders, f o r e x a m p l e w i t h certain s y m m e t r i e s , n e a r - t r a p p i n g occurs at c e r t a i n frequencies, k n o w n as
* C o r r e s p o n d i n g author.
E-mail address: w . b a i @ n u s . e d u . s g ( W . B a i ) .
0 1 4 1 - 1 1 8 7 / $ - s e e front m a t t e r © 2 0 1 3 E l s e v i e r Ltd. A l l r i g h t s r e s e r v e d . http://dx.doi.0rg/lO.l 0 1 6 / j . a p o r . 2 0 1 3 . 1 1 . 0 0 3
n e a r - t r a p p i n g frequencies, at w h i c h o n l y a small a m o u n t o f scattered w a v e energy Is radiated o u t w a r d s to the far field: the w a v e is t r a p p e d w i t h i n the local v i c i n i t y o f the cylinders, f o r m i n g a near s t a n d i n g w a v e w i t h m u c h larger a m p l i t u d e compared w i t h that at o t h e r frequencies. In the design o f such m u l t i - c o l u m n p l a t f o r m s , therefore, j t is crucial to develop a reliable t o o l to predict and describe the n e a r - t r a p p i n g p h e n o m e n o n .
For an array o f b o t t o m m o u n t e d vertical circular cylinders, p i -o n e e r i n g w -o r k w a s d-one b y Spring and M -o n k m e y e r [4], w h -o i n 1974 d e r i v e d the linear analytical s o l u t i o n f o r the scattered p o t e n t i a l a r o u n d c i r c u l a r cylinders. M c l v e r and Evans [ 5 ] presented an a p p r o x -i m a t -i o n f o r e s t -i m a t -i n g w a v e forces o n a g r o u p o f v e r t -i c a l cyl-inders w i t h large distances b e t w e e n each one, and Eatock T a y l o r and H u n g [6] investigated the m e a n w a v e d r i f t forces o n m u l t i - c o l u m n struc-tures. M e a n w h i l e , an accurate algebraic m e t h o d was d e v e l o p e d by Kagemoto and Yue [7] to p r e d i c t the h y d r o d y n a m i c p r o p e r t i e s o f a system o f m u l t i p l e t h r e e - d i m e n s i o n a l bodies i n w a t e r waves. Based on the m e t h o d i n Spring a n d M o n k m e y e r [ 4 ] , Linton a n d Evans [8] f u r t h e r d e r i v e d a considerably s i m p l i f i e d f o r m u l a f o r t h e p r e d i c t i o n o f first-order and m e a n forces on m u l t i p l e cylinders as w e l l as f o r the c a l c u l a t i o n o f the f r e e surface profile.
Trapped modes w e r e first i d e n t i f i e d i n an o p e n c h a n n e l b y Ursell [9] i n 1 9 5 1 . Callan et al. [10] proved the existence o f t r a p p e d modes in t w o - d i m e n s i o n a l waveguides using Ursell's m e t h o d . Subsequently,
7 2 W. Baiet al. I Applied Ocean Research 44 (2014) 71-81
Evans et a l . [11 ] t h e o r e t i c a l l y s h o w e d the existence o f t r a p p e d modes f o r all s y m m e t r i c c y l i n d r i c a l cross-sections. Thereafter, the t h e o r y o f L i n t o n and Evans [ 8 ] was used by Evans and Porter [ 1 2 ] t o study the firstorder n e a r t r a p p i n g f o r waves a r o u n d an array o f b o t t o m -m o u n t e d v e r t i c a l cylinders, w h i l e M a n i a r and N e w -m a n [ 1 3 ] observed s i m i l a r n e a r - r e s o n a n t modes b e t w e e n adjacent cylinders i n a l o n g fi-nite array at critical w a v e numbers. Consideration has also been given to the effects o f n e a r - t r a p p i n g b y such c o n f i g u r a t i o n s i n focused w a v e groups, e.g. Grlce et al. [ 1 4 ] . Malenica et al. [ 1 5 ] e x t e n d e d the i n -v e s t i g a t i o n o f n e a r - t r a p p i n g f o r an array o f equally spaced, i d e n t i c a l circular c y l i n d e r s to second-order in w a v e steepness. By s t u d y i n g the secondorder d i f f r a c t i o n o f m o n o c h r o m a t i c waves via a s e m i -analytical approach, t h e y solved f o r the waves due to the second-order p o t e n t i a l , and suggested t h a t there also exists a n e a r - t r a p p i n g p h e n o m e n o n f o r the secondorder w a v e a r o u n d an array o f c y l i n -ders. This second-order n e a r - t r a p p i n g occurs f o r an i n c i d e n t w a v e at h a l f the c o r r e s p o n d i n g first-order n e a r - t r a p p i n g f r e q u e n c y . W a n g and W u [ 1 6 ] u n d e r t o o k a second-order analysis o f n e a r - t r a p p i n g o f such an array i n the t i m e d o m a i n . Linear n e a r - t r a p p i n g by t r u n c a t e d cylinders has been considered b y S i d d o r n and Eatock Taylor [ 1 7 ] . For m o r e c o m p l e x geometries, such as m u l t i - c o l u m n g r a v i t y p l a t f o r m s , semisubmersibles or tension leg p l a t f o r m s , and o t h e r t r a p p i n g struc-tures, n u m e r i c a l d i f f r a c t i o n codes have been e m p l o y e d by several investigators.
For the p r o b l e m o f higher-order n o n l i n e a r w a v e d i f f r a c t i o n , Malenica and M o l i n [ 1 8 ] f o r m u l a t e d the t h e o r y f o r t h i r d - h a r m o n i c d i f f r a c t i o n b y a v e r t i c a l c y l i n d e r based o n the p e r t u r b a t i o n procedure. M e a n w h i l e , Faltinsen et al. [19] proposed an a p p r o x i m a t e t h e -o r y ( n -o w k n -o w n as t h e FNV a p p r -o x i m a t i -o n ) , by assuming t h a t the w a v e a m p l i t u d e and t h e c y l i n d e r radius are o f the same order. I n or-der to consior-der f u l l y n o n l i n e a r effects at d i f f e r e n t oror-ders i n the case o f steep waves, a f u l l y n o n l i n e a r time d o m a i n s i m u l a t i o n m a y be the m o s t a p p r o p r i a t e m e t h o d . This can consider all the n o n l i n e a r i t y o f the p r o b l e m w i t h o u t any a p p r o x i m a t i o n s such as Taylor series expansion or p e r t u r b a t i o n procedure. So far, various f u l l y n o n l i n e a r n u m e r i c a l models have been d e v e l o p e d b y d i f f e r e n t research groups. Ferrant [ 2 0 ] developed a f u l l y n o n l i n e a r wave t a n k to s t u d y the d i f f r a c t i o n p r o b l e m a n d f o r a single v e r t i c a l c y l i n d e r o b t a i n e d the h i g h e r - o r d e r h a r m o n i c s u p to seventh order. Subsequently, Huseby and Grue [21 ] p e r f o r m e d a large n u m b e r o f tests i n a l o n g w a v e t a n k a n d c o m -pared the h i g h e r - h a r m o n i c w a v e forces o n a v e r t i c a l c y l i n d e r against the n o n l i n e a r c o m p u t a t i o n s o f Ferrant [ 2 0 ] . Good agreement, even at higher harmonics, was s h o w n f r o m the comparisons o f the m e a s u r e -m e n t s and f u l l y n o n l i n e a r c o -m p u t a t i o n s . M a et al. [22,231 s t u d i e d the f u l l y n o n l i n e a r w a v e d i f f r a c t i o n a r o u n d a pair o f fixed cylinders i n a n u m e r i c a l w a v e tank based o n the finite e l e m e n t m o d e l (FEM). A p p l i -cations o f this approach to s i m u l a t e w a v e i n t e r a c t i o n w i t h a m o v i n g c y l i n d e r i n c l u d e W u a n d H u [ 2 4 ] and W a n g et al. [ 2 5 ] . T h e r e a f t e r Koo and K i m [ 2 6 ] , Bai and Eatock Taylor [27,28] and Z h o u et al. [ 2 9 ] used the b o u n d a r y e l e m e n t m e t h o d to simulate the f u l l y n o n l i n e a r w a v e i n t e r a c t i o n w i t h structures I n 2 D and 3D, respectively.
A n i n v e s t i g a t i o n o f n e a r - t r a p p i n g i n a l o n g array o f c y l i n d e r s was u n d e r t a k e n by W a n g and W u [30] using a f u l l y n o n l i n e a r m e t h o d based o n a finite e l e m e n t m o d e l , and the t i m e histories o f f o r c e and w a v e r u n - u p w e r e p r o v i d e d f o r d i f f e r e n t situations. H o w e v e r , to the best o f o u r k n o w l e d g e , a systematic i n v e s t i g a t i o n o f the n o n l i n e a r features associated w i t h the n e a r - t r a p p i n g p h e n o m e n o n has n o t been p u b l i s h e d . This paper a t t e m p t s to shed l i g h t on this t o p i c by e m p l o y -i n g the f u l l y n o n l -i n e a r t-ime d o m a -i n n u m e r -i c a l m o d e l d e v e l o p e d b y Bai and Eatock Taylor [27,28] to investigate w a v e d i f f r a c t i o n a r o u n d an array o f b o t t o m - m o u n t e d cylinders. The adopted n u m e r i c a l m o d e l is v a l i d a t e d b y c o n s i d e r i n g the cases o f a pair o f b o t t o m - m o u n t e d ver-tical c i r c u l a r cylinders and an array o f f o u r idenver-tical circular c y l i n d e r s s i t u a t e d at the corners o f a square. N u m e r i c a l results f o r t h e w a v e elevation, r u n - u p and forces are presented, w h i c h are f o u n d to agree w e l l w i t h the e x p e r i m e n t a l data. The n e a r - t r a p p i n g p h e n o m e n o n i n
W a v e m a k o r
1
S i 1 \ / \ Damping1
V7
\ j
SIM S n X s„. F i g . 1. Sl<etch o f tlie n u m e r i c a l w a v e t a n k .an array o f cylinders is t h e n studied, and results i n c l u d i n g the m o d e shape o f the free surface e l e v a t i o n at n e a r - t r a p p i n g , and t h e p a t t e r n o f forces o n the cylinders, are obtained. In a d d i t i o n , a h a r m o n i c a n a l -ysis o f the time h i s t o r y o f the f u l l y n o n l i n e a r results is p e r f o r m e d using the Fast Fourier T r a n s f o r m (FFT), f r o m w h i c h the I m p o r t a n c e of h i g h e r - o r d e r effects can be i d e n t i f i e d , especially In the case o f t h e second-order n e a r - t r a p p i n g . I n this paper, unless o t h e r w i s e specified, the t e r m s firstorder, secondorder a n d t h i r d o r d e r refer to the s i n -gle frequency, double f r e q u e n c y and t r i p l e f r e q u e n c y c o m p o n e n t s , respectively, o f the f u l l y n o n l i n e a r results o b t a i n e d by using an FFT analysis. Thus first h a r m o n i c t e r m s r e s u l t i n g f r o m the t h i r d order I n -t e r a c -t i o n o f firs-t and second order -t e r m s are n o -t considered -to a f f e c -t the overall conclusions d r a w n here a b o u t n o n l i n e a r behavior.
2. M a t h e m a t i c a l f o r m u l a t i o n
A n u m e r i c a l w a v e tank d e f i n e d In Fig. 1 is a d o p t e d to s i m u l a t e the a f o r e m e n t i o n e d w a v e d i f f r a c t i o n a r o u n d an array o f v e r t i c a l cir-cular cylinders a n d the associated n e a r - t r a p p i n g p h e n o m e n o n . The schematic figure involves a w a v e m a k e r at the l e f t b o u n d a r y o f the tank, an array o f b o t t o m - m o u n t e d v e r t i c a l circular cylinders i n the m i d d l e o f the t a n k and a d a m p i n g layer placed o n the w a t e r surface to a v o i d wave r e f l e c t i o n f r o m the far e n d o f the w a v e tank.
Based on the a s s u m p t i o n t h a t the f l u i d is incompressible and i n -viscid, and the flow i r r o t a t i o n a l w i t h i n the fluid d o m a i n , p o t e n t i a l flow t h e o r y can be used to describe the w a v e d i f f r a c t i o n p r o b l e m , w h e r e a velocity p o t e n t i a l <j){x, y, z, f ) d e f i n e d i n a global c o o r d i n a t e system Oxyz satisfies the Laplace e q u a t i o n .
v'-d) = 0 .
(1)
On the f r e e w a t e r surface Sf, the k i n e m a t i c and d y n a m i c w a v e c o n d i -tions i n the Lagrangian d e s c r i p t i o n areD X Df D 0 D f -Vcf) • ( 2 ) ( 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 denotes t h e p o s i t i o n o f w a t e r particles on the free w a t e r surface and g is the g r a v i t a t i o n a l acceler-ation. As the bodies are b o t t o m - m o u n t e d and t h e r i g i d boundaries should be i m p e r m e a b l e , the k i n e m a t i c c o n d i t i o n o n the body sur-faces, b o t t o m and sldewalls o f the t a n k is
dn
0, ( 4 )
w h e r e n is the n o r m a l u n i t vector p o i n t i n g o u t o f the 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 taken as
: 0 , z = 0 a t t = 0 . ( 5 )
The 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 t o solve the m i x e d b o u n d a r y value p r o b l e m described above at each t i m e step, w h e r e the surface over w h i c h the 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
W. Baiet al. I Applied Ocean Researcli 44(2014)71-81 7 3
(•>.) 0';
F i g . 2 . E x a m p l e m e s l i e s for w a v e p a s t a n a r r a y of v e r t i c a l c y l i n d e r s at a c e r t a i n t i m e i n s t a n t : ( a ) t w o c y l i n d e r s ; ( b ) four c y l i n d e r s .
by quadratic isoparametric elements. In the present m e t h o d , struc-t u r e d 8-node quadrilastruc-teral meshes are d i s struc-t r i b u struc-t e d on struc-the solid sur-faces i n c l u d i n g the body surface Sb, w a v e m a k e r S^ju and t a n k w a l l s Sw. On the f r e e surface Sp, u n s t r u c t u r e d 6-node triangular meshes are generated by using the Delaunay t r i a n g u l a t i o n m e t h o d . Fig. 2 shows snapshots o f the meshes f o r waves a r o u n d t w o and f o u r c y l i n -ders respectively at a certain t i m e instant. Once the velocity p o t e n t i a l has been f o u n d , the h y d r o d y n a m i c forces can be predicted by using a m e t h o d i n w h i c h a separate boundary value p r o b l e m is r e q u i r e d to solve f o r some a u x i l i a r y f u n c t i o n s , instead o f p r e d i c t i n g the time derivative o f the potential directly; this m e t h o d has been p r o v e n to be m o r e stable.
A t the end o f each time step, the free surface g e o m e t r y and p o t e n -tial are updated by the standard 4 t h - o r d e r R u n g e - K u t t a scheme, and a cosine r a m p f u n c t i o n Is used to m o d u l a t e the b o u n d a r y c o n d i t i o n on the w a v e m a k e r d u r i n g t h e i n i t i a l time steps. In order to avoid the 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 that may occur i n the wave p r o f i l e a f t e r a s u f f i c i e n t l y l o n g time, mesh regeneration o n the free surface is i m p l e m e n t e d by a d o p t i n g the Laplacian s m o o t h i n g technique to obtain the n e w nodes on the free surface. I n t e r p o l a t i o n is t h e n used to update the variables at each n e w node. M o r e details can be f o u n d In Bai and Eatock Taylor [27,28], w h e r e several v a l i d a t i o n studies are conducted f o r simple geometries.
I n the n e x t f e w sections, the f u l l y n o n l i n e a r n u m e r i c a l m o d e l is e m p l o y e d to investigate the w a v e f i e l d a r o u n d an array o f circular cylinders. The present results are f i r s t c o m p a r e d w i t h e x p e r i m e n t a l data, as w e l l as w i t h n u m e r i c a l results o b t a i n e d by other researchers, for b o t h t w o a n d f o u r vertical circular cylinders o f equal size I n d i f f e r -ent arrangem-ents. The near-trapped mode In an array o f f o u r cylinders is t h e n h i g h l i g h t e d , and an I n v e s t i g a t i o n is carried o u t i n t o various aspects i n c l u d i n g the mode shape, the w a v e p r o f i l e and r u n - u p at a n e a r - t r a p p i n g frequency and the harmonics o f d i f f e r e n t orders. I n all the f o l l o w i n g studies, a Is the c y l i n d e r radius, / the distance be-t w e e n be-the cenbe-ters o f be-t w o adjacenbe-t cylinders, and d be-the w a be-t e r d e p be-t h . Other notations f o r various w a v e properties include k the w a v e n u m -ber, w the w a v e frequency, A the w a v e a m p l i t u d e and ?; the surface elevation.
3. W a v e diffraction a r o u n d two cylinders
The f i r s t case concerns t w o b o t t o m - m o u n t e d vertical c i r c u l a r cylinders aligned i n the i n c i d e n t w a v e d i r e c t i o n at the centeriine o f the tank. I n order to compare w i t h the p u b l i s h e d results, the f o l -l o w i n g parameters are se-lected: a = 0.1416, w = 1.6748, A = 0.004 and kA = 0.0113. This corresponds to a w a v e o f l o w steepness, to facilitate c o m p a r i s o n w i t h the linear analytical s o l u t i o n . Unless o t h -erwise specified, the w a t e r d e p t h d, the g r a v i t a t i o n a l acceleration g and the fluid density p are taken to be unity, l e n g t h scales are n o n -d i m e n s i o n a l i z e -d by the w a t e r -d e p t h -d an-d t e m p o r a l parameters are thereby also n o n - d i m e n s i o n a l i z e d . The n u m e r i c a l w a v e tank is 12.0 units l o n g and 2.0 units w i d e , and the center o f the upstream c y l i n d e r is located a distance 7.0 units f r o m the w a v e maker. There are six
b o u n d a r y elements i n the vertical d i r e c t i o n o n the body surface and on the sidewall, and the n u m b e r o f elements around the c i r c u m f e r -ence o f each c y l i n d e r is 24. Results are o b t a i n e d f o r a range o f spacings / b e t w e e n the axes o f the cylinders, at the above m e n t i o n e d wave f r e -quency. In a typical mesh f o r kl = 2.0, f o r example. I n c l u d i n g 3 8 7 4 elements on the w a t e r surface, there are 13,511 nodes d i s t r i b u t e d o n the w h o l e c o m p u t a t i o n a l boundary.
W e first examine the convergence o f the c o m p u t a t i o n w i t h d i f f e r -ent meshes. Besides the a b o v e m e n t i o n e d mesh t h a t is denoted "Mesh b", t w o other mesh systems, d e f i n e d as "Mesh a" a n d "Mesh c", are also tested. "Mesh a" is a coarser one I n c l u d i n g 16 elements a r o u n d the cylinder, w h i l e the finer "Mesh c" involves 32 elements a r o u n d the cylinder. The n u m b e r s o f quadratic elements o n the free surface per first order w a v e l e n g t h are a p p r o x i m a t e l y 1 3 , 1 8 and 24 f o r "Mesh a", "Mesh b" and "Mesh c" respectively. Fig. 3(a) and (b) shows t h e m a x i m u m wave forces on the upstream and d o w n s t r e a m cylinders (these may be compared w i t h the a m p l i t u d e s o f the linear forces u n -der sinusoidal f o r c i n g ) . Our c o m p u t e d results are g i v e n f o r the three meshes for three d i f f e r e n t c y l i n d e r spacings. It should be noted t h a t in o r d e r to evaluate the i n t e r a c t i o n b e t w e e n the t w o cylinders, the forces are n o n d i m e n s i o n a l l z e d by the force Fisoiated t h a t acts o n a s i n -gle isolated c y l i n d e r under the same c o n d i t i o n s : i t can be seen t h a t the c o m p u t a t i o n converges fast. I n fact, i t is hard to d i s t i n g u i s h the results o b t a i n e d w i t h "Mesli a", "Mesh b" and "Mesh c". The linear analytical results o f Linton and M c l v e r [31 ] are also s h o w n f o r c o m p a r i s o n . I n a d d i t i o n , the time h i s t o r y o f w a v e r u n - u p at the u p w a v e face o f the upstream cylinder at /</ = 4.0 o n these three meshes Is s h o w n i n Fig. 3(c), also i n d i c a t i n g that the c o m p u t a t i o n w i t h "Mesh b" is convergent for the case of t w o cylinders. One can o n l y observe s m a l l differences (less t h a n 1%) at the crest and t r o u g h o f the w a v e elevations f o r these three meshes i n Fig. 3(c). In the latter p a r t o f the paper, a s i m i l a r mesh density is adopted f o r the i n v e s t i g a t i o n o f f o u r cylinders.
Fig. 4 shows the v a r i a t i o n o f the h o r i z o n t a l w a v e forces Fx on the upstream and d o w n s t r e a m cylinders versus the dimenslonless spac-i n g kl b e t w e e n these t w o cylspac-inders. For the purposes o f c o m p a r spac-i s o n , results f r o m the analytical solutions o f Spring and M o n k m e y e r [4] and L i n t o n and M c l v e r [31 ], based o n linear w a v e theory, are also i n c l u d e d in this figure. It can be seen t h a t agreement is generally achieved f o r the forces on b o t h the upstream a n d d o w n s t r e a m cylinders. The s m a l l discrepancies (less t h a n 5%) b e t w e e n our n u m e r i c a l a n d the analytical results are assumed to be due to n o n l i n e a r effects. Nevertheless, i t is clear that f o r this s m a l l steepness w a v e the behavior established f r o m linear analyses o f this c o n f i g u r a t i o n are r e p r o d u c e d : the effects o f i n -teractions between the cylinders are pronounced, leading to s t r o n g variations w i t h f r e q u e n c y o f the forces o n the t w o bodies. The effects o f nonlinearlties are considered later.
To f u r t h e r validate the c u r r e n t f u l l y nonlinear n u m e r i c a l m o d e l in s i m u l a t i n g m u l t i p l e bodies, the time histories o f w a v e forces are also compared w i t h the results in M a et al. [23], w h o used the FEM to s t u d y the wave a r o u n d t w o cylinders w i t h a spacing o f / d = 2.0. As can be seen f r o m Fig. 5, our results agree q u i t e w e l l w i t h those o f Ma et al. [ 2 3 ] a f t e r the steady state has been achieved at a r o u n d f = 35. N o t e t h a t before the steady state is reached, the present results are s l i g h t l y smaller because o f the a p p l i c a t i o n o f a r a m p f u n c t i o n d u r i n g the early stages o f the s i m u l a t i o n (Bai and Eatock Taylor [27]). The purpose o f the r a m p f u n c t i o n is to e l i m i n a t e the i m p u l s e e f f e c t w h e n a sudden velocity is assigned to the wave m a k e r at the first t i m e step; i t leads to a s m o o t h t r a n s i t i o n f r o m the still w a t e r to the f u l l y developed w a v e field.
The preceding cases d e m o n s t r a t e that the f u l l y n o n l i n e a r s o l u t i o n is capable o f r e p r o d u c i n g p u b l i s h e d linear results w h e n the w a v e steepness is l o w . The behavior under steeper waves is considered i n the next section.
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A f t e r considering tvyo cylinders, i n this section w e focus o n an ar-ray o f f o u r vertical circular cylinders i n the t w o c o n f i g u r a t i o n s s h o w n i n Fig. 6. Fig. 6(a) shows a g r o u p o f f o u r c y l i n d e r s standing i n the cor-ners o f a square and the i n c i d e n t w a v e is p r o p a g a t i n g along one side o f the square: this is r e f e r r e d to as the 0= heading. Fig. 6(b) is a s i m i l a r a r r a n g e m e n t o f the cylinders except the I n c i d e n t w a v e is propagating along the diagonal o f the square, w h i c h is d e f i n e d as the 45= heading. In o r d e r to compate w i t h the e x p e r i m e n t s c a r r i e d o u t i n a w a v e basin by O h l et al. [2], an I n c i d e n t w a v e o f 0.8 Hz and a m p l i t u d e 0.049 m is first considered f o r the b o t h 0° and 4 5 ° headings. The same c y l i n d e r as t h a t used In the e x p e r i m e n t s is a d o p t e d here, w h i c h has a radius
o f a = 0.203 m situated i n a w a t e r d e p t h o f d = 2.0 m w i t h a spacing b e t w e e n c y l i n d e r s o f / = 4a. The corresponding d i m e n s l o n l e s s w a v e n u m b e r is ka = 0.524. The c o m p u t a t i o n s are p e r f o r m e d i n a n u m e r i c a l w a v e tank 1 1 m l o n g and 2.6 m w i d e , and the o r i g i n o f the space-fixed global coordinates is set at the center o f the c y l i n d e r g r o u p , w h i c h is located 7.0 m f r o m the w a v e maker on the c e n t e r i i n e o f t h e tank. I n this case, the w a v e steepness corresponds to l<A = 0.126, so t h a t a considerably steeper i n c i d e n t w a v e is investigated.
Fig. 7 s h o w s the dimenslonless m a x i m u m w a v e e l e v a t i o n along the X-axis In the centeriine o f the tank f o r the 0° heading, and its c o m p a r i s o n w i t h the linear analytical s o l u t i o n o b t a i n e d by L i n t o n and Evans [ 8 ] . The present f u l l y nonlinear results are g e n e r a l l y close
W. Baiet al./Applied Ocean Research 44 (2014) 71-81 7 5 3.0 A - ' + kl (b) kl F i g . 4 . H o r i z o n t a l w a v e forces o n a p a i r of c y l i n d e r s w i t h d i f f e r e n t d i s t a n c e s s i t u a t e d o n the c e n t e r i i n e o f the tanlc at a) = 1.6748 a n d A = 0 . 0 0 4 : ( a ) f o r c e o n u p s t r e a m c y l i n d e r ; (b) force on d o w n s t r e a m c y l i n d e r . (•,,1 • • • - . M f v u l
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5 10 (S ja IS V> 55 >« ' 5 SO • I F i g . 5. C o m p a r i s o n of t i m e h i s t o r y of h o r i z o n t a l w a v e forces on a p a i r o f c y l i n d e r s s i t u a t e d o n the c e n t e r i i n e o f the tanl< at M = 1 . 6 7 4 8 , /\ = 0 . 0 0 4 a n d W = 2.0: ( a ) force o n u p s t r e a m c y l i n d e r ; ( b ) force o n d o w n s t r e a m c y l i n d e r .i
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CyiiMdM J F i g . 6. A r r a n g e m e n t s for a n a r r a y o f four c y l i n d e r s : ( a ) 0 h e a d i n g ; (b) 4 5 h e a d i n g .to the e x p e r i m e n t a l results, w h i l e the linear solutions tend to under-estimate the w a v e p r o f i l e . Particularly at the peaks near x = -0.8 m and X = 0.0 m , the linear a p p r o x i m a t i o n s could have as large as 40% errors, whereas the present f u l l y n o n l i n e a r results agree m u c h better w i t h the tests. I n a d d i t i o n , the linear result seems to be s y m m e t r i c about the m i d d l e p o i n t b e t w e e n the t w o peaks at a r o u n d x = - 0 . 4 m , b u t the n o n l i n e a r result is o b v i o u s l y a s y m m e t r i c . This suggests t h a t i n this case the n o n l i n e a r i t y plays a s i g n i f i c a n t role and the linear t h e o r y no longer provides a valid m o d e l f o r p r e d i c t i n g the m a x i m u m w a v e elevation. In Fig. 8 the dimenslonless mean w a v e elevation is presented and compared w i t h the physical e x p e r i m e n t . The overall trends are in good agreement, t h o u g h the value at x = -0.2 m is s l i g h t l y over-predicted. It s h o u l d be borne i n m i n d that the mean w a v e elevation is o f second order a n d Is m u c h smaller and m o r e sensitive t h a n the wave a m p l i t u d e .
Figs. 9 and 10 show the dimenslonless m a x i m u m and m e a n wave
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0 . 5 F i g . 8 . M e a n w a v e e l e v a r i o n a l o n g the x - a x i s for 0 h e a d i n g at ƒ = 0 . 8 0 0 H z . A = 0 . 0 4 9 m a n d / = 0.812 m .elevations along the diagonal BC f r o m the d o w n s t r e a m C y l i n d e r 1 to the u p s t r e a m Cylinder 3 (see Fig. 6) f o r the 45= heading. A l t h o u g h some discrepancies can be observed near the u p s t r e a m Cylinder 3, the present results are generally able to describe the global trends o f the w a v e p r o f i l e . A n o t h e r reason f o r the discrepancies m a y be that possible e x p e r i m e n t a l errors could result f r o m the presence o f d i s t u r -bances i n d u c e d b y the wave probes i n the w a v e basin, as c o m m e n t e d by Ohl et al. [ 2 ] In the discussion o f t h e i r e x p e r i m e n t s .
N o w t h a t the present n u m e r i c a l m o d e l has been v a l i d a t e d by c o m -p a r i n g w i t h b o t h n u m e r i c a l and e x -p e r i m e n t a l data f r o m the literature, the effects o f nonlinearlties are considered. First, the c o m p l e t e wave elevation, w a v e force and r u n - u p a r o u n d the c y l i n d e r s is presented f o r the 45= h e a d i n g c o n f i g u r a t i o n ( s h o w n i n Fig. 6b), at a d i m e n -slonless w a v e n u m b e r ka = 0.468. The c o m p u t a t i o n a l c o n d i t i o n s are s i m i l a r to those presented above f o r the same c o n f i g u r a t i o n , except that a smaller w a v e w i t h A = 0.02 i n s h a l l o w e r w a t e r at d = 1.0 is considered, w h i c h corresponds to a w a v e o f steepness l<A = 0.047.
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0.2 0.4F i g . 9. I V I a x i m u m vyave e l e v a t i o n a l o n g tlie x-axis for 4 5 h e a d i n g a t / = 0 . 8 0 0 H z , A = 0 . 0 4 9 m a n d / = 0 . 8 1 2 m . 2.0 1 . 5 - ^ 1.0
1-0 . 1-0 - 0 . 5 - 1 . 0 X Experiment -"— Present - 0 , 4 - 0 , 2 0 , 0x(m)
0 . 2 0 . 4 F i g . 1 0 . M e a n w a v e e l e v a t i o n a l o n g t h e x - a x i s for 4 5 h e a d i n g at ƒ = 0 . 8 0 0 H z , A = 0 . 0 4 9 m a n d / = 0 . 8 1 2 m .This case is also considered b e l o w f o r a steeper w a v e than above, i n the c o n t e x t o f n o n l i n e a r n e a r - t r a p p i n g phenomena. Fig. 11 shows the f i r s t 20 periods o f the surface e l e v a t i o n r u n - u p at the upstream and t h e d o w n s t r e a m faces o f Cylinders 1 and 3 respectively. As expected, the r u n u p at Points B and C i n Fig. 5(b) o n the inner faces o f t h e c y l i n -ders are larger t h a n those at Points A and D o n the o u t e r faces, due to the s u p e r p o s i t i o n o f the i n c i d e n t w a v e w i t h m u l t i p l y scattered waves f r o m the o t h e r cylinders. E x a m i n a t i o n o f these figures suggests t h a t n o n l i n e a r effects appear i n the results at Points B a n d C, even f o r this l o w e r steepness w a v e , b u t the characteristics o f the n o n l i n e a r i t y at these t w o points are rather d i f f e r e n t . Such n o n l i n e a r behavior w i l l be f u r t h e r i n v e s t i g a t e d i n Section 5.3.
Fig. 12 plots the r u n u p on the cylinders against the c o u n t e r c l o c k -w i s e angle y3 measured f r o m the back o f the c y l i n d e r (see Fig. 6). Due to the s y m m e t r y o f the c o n f i g u r a t i o n , the r u n - u p o n Cylinder 4 is n o t presented, w h i c h Is i d e n t i c a l to t h a t o n Cylinder 2. The r u n - u p o n b o t h C y l i n d e r 1 and Cylinder 3 are a p p r o x i m a t e l y s y m m e t r i c a b o u t yS = 1 8 0 ° as expected, t h o u g h this Is n o t the case f o r Cylinder 2.
5. Near-trapping p h e n o m e n o n
In this section, w e use the array o f f o u r b o t t o m - m o u n t e d cylinders w i t h the c o n f i g u r a t i o n s h o w n i n Fig. 6 ( b ) i n order to investigate the n e a r - t r a p p i n g p h e n o m e n o n using the f u l l y n o n l i n e a r time d o m a i n n u m e r i c a l m e t h o d . The a i m is to relate the n o n l i n e a r results to a w e l l - s t u d i e d e x a m p l e f o r w h i c h linear results w e r e g i v e n b y Evans and Porter [12], and second o r d e r results by Malenica et al. [15] and W a n g and W u [ 1 6 ] , In o r d e r to m a k e the compansons, at this stage i t is convenient to express all o f the quantities o n l y i n dimenslonless f o r m . The spacing ratio is again / = 4a, and w e use the same w a t e r d e p t h as i n [15], d = 3a.
5.1. Near-trapped mode shape
In this arrangement, the n e a r - t r a p p i n g f r e q u e n c y was f o u n d b y Evans and Porter [121 using linear t h e o r y to be ka = 1.66. In order to consider nonlinear effects, i n the present s t u d y the w a v e steepness w e adopt is equal to l<A = 0.157, w h i c h w a s also used by W a n g and W u [ 16] i n t h e i r t i m e d o m a i n second-order analysis. The contours of w a v e elevation predicted by o u r m o d e l are s h o w n i n Fig. 13. Fig. 13(a) and (b) shows the w a v e e l e v a t i o n d i s t r i b u t i o n s at the t i m e instants t h a t b o t h the d o w n s t r e a m r u n - u p on C y l i n d e r 1 and the upstream r u n - u p on Cylinder 3 have t h e i r m a x i m u m (crest) a n d m i n i m u m ( t r o u g h ) values, respectively. The near s t a n d i n g w a v e p a t t e r n w i t h i n the array can be clearly i d e n t i f i e d . The w a v e w i t h i n the array is oscillating be-t w e e n be-the sbe-tabe-tes o f Fig. 13(a) and (b) w h i l e be-the w a v e oube-tside be-the array is passing by i n the w a v e d i r e c t i o n . Fig. 13(c) shows the m a x i m u m w a v e elevation at every l o c a t i o n near t h e array f o r any instant i n the w a v e cycle. It can be seen t h a t the highest elevations are f o u n d near the inside faces o f the cylinders, w h e r e the m a x i m u m w a v e eleva-tions o f the near standing w a v e occur i n Fig. 13(a) and (b). A l o n g the t w o axes o f s y m m e t r y , y = ±x, the m a x i m u m w a v e e l e v a t i o n is close to t h a t outside the surface array, w h i c h is a p p r o x i m a t e l y the I n c i d e n t w a v e a m p l i t u d e .
Fig. 14 shows the time histories o f h o r i z o n t a l forces o n the three cylinders over five wave periods. Note t h a t the solid line a n d the dash-dot line are the l o n g i t u d i n a l forces Fx o n Cylinder 1 and Cylinder 3 respectively, and the dashed l i n e is the transversal force Fy o n Cylinder 2. As can be seen, s t r o n g n o n l i n e a r i t y appears and i t s i g n i f i c a n t l y affects the m a x i m u m forces as w e l l as the m e a n forces. It is i n t e r e s t i n g to find t h a t Fx on Cylinder 1 is i n phase w i t h Fy o n Cylinder 2, b u t i t is o u t o f phase w i t h Fx o n Cylinder 3. This indicates t h a t w h e n the l o n g i t u d i n a l forces o n Cylinder 1 and Cylinder 3 reach t h e i r m a x i m a ( i n opposite directions), the transverse force o n Cylinder 2 also has a local m a x i m u m value. Likewise, w h e n the l o n g i t u d i n a l force o n Cylinder 2 ( n o t p l o t t e d ) is at its m a x i m u m , the l o n g i t u d i n a l forces o n Cylinder 1 and Cylinder 3 are equal to zero. The transversal forces Fy on Cylinder 1 and Cylinder 3 are always f o u n d to be v e r y close to zero due to the s y m m e t r y o f the g e o m e t r y considered.
As m e n t i o n e d i n [ 12], this k i n d o f behavior can be characterized b y various c o m b i n a t i o n s o f f e r e e s o n the f o u r cylinders. Fig. 15 sketches the f o u r possible force patterns w i t h the forces b e i n g either at the m a x i m u m or equal to zero. Fig. 15(a) is t h e s i t u a t i o n w h e r e the l o n -g i t u d i n a l forces o n Cylinders 1 and 3 are p o i n t i n -g i n w a r d s and the transverse forces o n Cylinders 2 and 4 are p o i n t i n g o u t w a r d s . Fig. 15(c) shows a s i m i l a r p a t t e r n b u t w i t h the c o r r e s p o n d i n g forces i n the opposite directions. Fig. 15(b) and ( d ) are the patterns w h e r e the forces o n Cylinders 1 and 3 are zero a n d the l o n g i t u d i n a l forces o n Cylinders 2 and 4 are i n the same or opposite d i r e c t i o n to the i n c i d e n t w a v e respectively. Globally, the w h o l e s t r u c t u r e w i l l subsequentiy experience the force c o m b i n a t i o n s c y c l i n g f r o m one p a t t e r n to a n -other.
A f t e r calculating the m e a n force o n each o f the cylinders, w e find t h a t all the mean forces p o i n t t o w a r d s the center o f the array, as
5 ; O
W. Baiet al./Applied Ocean Researcli 44 (2014) 71-81
2 , 7 7 ( a ) 10 12 14 16 18 20
//r
10 12 14 16 18 20l/T
F i g . 1 1 . T i m e tiistoiy of w a v e e l e v a t i o n at four d i f f e r e n t l o c a t i o n s for 4 5 treading a t / I = 0 . 0 2 , w = 1.44 a n d / = 0.8: ( a ) P o i n t A ; ( b ) Point B; ( c ) Point C; ( d ) P o i n t D.
1 8 1.7 1.6 i , 5 1.4 1.3 1.2 1.) 1.0 0 . 9 0, 0.7 1¬ 0 . 6 0 • C y l i n d e r I - C y l i n d e r 2 C y l i n d e r 3 9 0 180 2 7 0 36'! F i g . 1 2 . M a x i m u m w a v e r u n - u p a r o u n d t h e c y l i n d e r s for 4 5 h e a d i n g a t / 1 = 0.02, a) = 1.44 a n d 1 = 0.8. 1 • F i g . 1 3 . M o d e s h a p e s for t h e n e a r - t r a p p e d m o d e w i t h a n a r r a y of f o u r c y l i n d e r s a t to = 1.66: ( a ) w a v e e l e v a U o n w h e n d o w n s t r e a m of C y l i n d e r 1 a n d u p s t r e a m o f C y l i n d e r 3 a t the m a x i m u m ; ( b ) w a v e e l e v a t i o n w h e n d o w n s t r e a m of C y l i n d e r 1 a n d u p s t r e a m of C y l i n d e r 3 at t h e m i n i m u m ; ( c ) m a x i m u m w a v e e l e v a t i o n n e a r t h e c y l i n d e r a r r a y . s h o w n i n Fig. 16 ( i t should be n o t e d h o w e v e r t h a t t h e m e a n l o n g i t u -d i n a l forces o n Cylin-ders 2 an-d 4 are v e r y small, so are i g n o r e -d here). The m e a n forces tend to push t h e c y l i n d r i c a l c o m p o n e n t s i n w a r d s , y e t the t o t a l mean d r i f t force o n the entire system is actually rather s m a l l .
5.2. Maximum run-up at near-trappetl mode
In o r d e r to understand better 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 and its i n f l u e n c e o n surface elevation a r o u n d t h e cylinders, t w o m o r e cases w i t h ka = 0.754 and ka = 0.468 respectively are investigated. These t w o ka values correspond to t w o t h i r d s a n d the h a l f o f the near-t r a p p i n g f r e q u e n c y a near-t ka = 1.66 respecnear-tively f o r near-t h e same arrange-m e n t o f an array o f f o u r cylinders s h o w n i n Fig. 6(b). These cases w e r e
F i g . 1 4 . T i m e h i s t o r y o f h o r i z o n t a l f o r c e s o n the c y l i n d e r s at the n e a r - t r a p p i n g w h e n ka = 1.66.
W. Bai etal./Applied Ocean Research 44 (2014) 71-81 7 8
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(a) W (c) (Ó) F i g . 1 5 . P o s s i b l e force p a t t e r n s of the n e a r - t r a p p e d m o d e a t / ( a = 1.66: ( a ) p a t t e r n 1; ( b ) p a t t e r n 2; ( c ) p a t t e r n 3 ; ( d ) p a t t e r n 4.F i g . 1 6 . M e a n forces o n the a r r a y o f f o u r c y l i n d e r s at the n e a r - t r a p p i n g w h e n ka = 1.66. 8 0 1 1 I 1 1 1 t I . I . I , I I 1 -0.4 -0.3 -0.2 -0.1 0:0 0.1 0.2 0.3 0.4 X F i g . 1 7 . M a x i m u m w a v e e l e v a t i o n a l o n g the x - a x i s for 4 5 - h e a d i n g at a = 0.2, / = 4 a , d = 3a a n d k/l = 0.157.
Fig. 17 shows the m a x i m u m w a v e elevation along the diagonal BC f o r the three d i f f e r e n t to numbers. It can be seen t h a t the m a x i m u m w a v e elevations near the d o w n s t r e a m face o f Cylinder 1 and the u p s t r e a m face o f Cylinder 3 are e x t r e m e l y h i g h w h e n n e a r - t r a p p i n g occurs at to = 1.66: they are m o r e t h a n 6 times the i n c i d e n t w a v e a m p l i t u d e at Point B and about 7 times at Point C. F u r t h e r m o r e , the m a x i m u m wave e l e v a t i o n at the n e a r - t r a p p i n g frequency decreases v e r y q u i c k l y f r o m o v e r 6 a t x = ± 0 . 3 7 to less t h a n 2 a t x = ± 0 . 2 , and subsequently there appears another small peak at x = 0.0. For the other t w o to numbers, the m a x i m u m w a v e elevation along the diagonal BC is gen-erally less t h a n t w i c e the i n c i d e n t w a v e a m p l i t u d e and the v a r i a t i o n o f the m a g n i t u d e is l i m i t e d to a relatively small range.
The same effects associated w i t h the n e a r - t r a p p i n g p h e n o m e n o n are f o u n d f o r the m a x i m u m w a v e r u n - u p o n the cylinders as s h o w n in Fig. 18. From the positions o f the m a x i m u m w a v e r u n - u p at the n e a r - t r a p p i n g f r e q u e n c y o f to = 1.66, i.e. 0° f o r Cylinder 1, 270= f o r
Cylinder 2 (or 90= f o r Cylinder 4) and 180° f o r Cylinder 3, w e can also c o n f i r m the presence o f t h e near s t a n d i n g wave i n the v i c i n i t y o f t h e array center.
5.3. Harmonic analysis
To f u r t h e r investigate t l i e n o n l i n e a r n e a r - t r a p p i n g p h e n o m e n o n , the Fast Fourier T r a n s f o r m (FFT) is adopted to carry o u t the a m p l i t u d e spectral analysis o f the t i m e series obtained f r o m the present f u l l y nonlinear time d o m a i n s i m u l a t i o n s . For each o f t h e time series to be processed, and to m i n i m i z e the l e n g t h o f the c o m p u t a t i o n s , w e extract 5 w a v e periods o f the observed steady state signals and e x t e n d the signals t o 20 w a v e periods by repeating these 5 w a v e p e r i o d signals f o u r times. By e x t e n d i n g the signals i n this w a y , w e are able to increase the f r e q u e n c y r e s o l u t i o n i n the spectral analysis, w i t h o u t m u c h a d d i t i o n a l c o m p u t a t i o n a l e f f o r t or loss o f accuracy.
First w e consider the w a v e e l e v a t i o n at the d o w n s t r e a m face o f Cylinder 3 i n the case o f ka = 0.468 and l<A = 0.157, f o r the purpose o f d e m o n s t r a t i o n . Fig. 19 shows the f r e q u e n c y s p e c t r u m o f the w a v e elevation f o r this case based o n the t i m e h i s t o r y over 5 w a v e periods. This t i m e h i s t o r y is displayed b y the solid line i n Fig. 20. As can be seen f r o m Fig. 19, the peaks appear at the Incident f r e q u e n c y wu d o u -ble f r e q u e n c y 2a)u and t r i p l e f r e q u e n c y 3öJj, w h i c h are r e f e r r e d to as the first, second and t h i r d o r d e r harmonics respectively. This f i g -ure indicates t h a t the f i r s t - o r d e r h a r m o n i c is p r e d o m i n a n t , however, the second- and t h i r d - o r d e r harmonics also play an i m p o r t a n t role In the w a v e elevation. In a d d i t i o n , w e can even notice the f o u r t h - o r d e r h a r m o n i c i n the figure, w h i c h h i g h l i g h t s again t h a t n o n l i n e a r i t y is sig-n i f i c a sig-n t i sig-n this p r o b l e m ; isig-ndeed this is the m o t i v a t i o sig-n f o r the presesig-nt study.
The n o n l i n e a r i t y can also be I l l u s t r a t e d by p e r f o r m i n g inverse FFTs o f the spectra, f i l t e r e d to isolate each o f the f i r s t three peaks. The time d o m a i n a p p r o x i m a t i o n s t o the f i r s t - , second- and t h i r d - o r d e r eleva-tions may t h e r e b y be calculated and c o m p a r e d w i t h the o r i g i n a l f u l l y n o n l i n e a r results, and these comparisons are p l o t t e d i n Fig. 20 (here second-order elevation indicates the total second o r d e r e l e v a t i o n i n a Stokes expansion, a n d s i m i l a r l y f o r t h i r d - o r d e r ) . The f i r s t - o r d e r ap-p r o x i m a t i o n is f o u n d to give a relative e r r o r o f about 30% at the w a v e t r o u g h . The second-order s o l u t i o n s i g n l f i c a n t i y i m p r o v e s the accu-racy i n general, however, one can observe s i g n i f i c a n t discrepancies i f one examines the concave curve near the w a v e crest. Finally, the t h i r d o r d e r a p p r o x i m a t i o n is able to capture every d e t a i l and r e p -resent accurately the o r i g i n a l f u l l y n o n l i n e a r results. The f r e q u e n c y analysis discussed above suggests t h a t under the c o n d i t i o n s e x a m i n e d the f i r s t o r d e r harmonics can o n l y give an overall v i e w o f the p r o b -l e m , and p o t e n t i a -l -l y i m p o r t a n t detai-led effects are m i s s i n g . A d d i t i o n o f the second-order h a r m o n i c c o m p o n e n t provides an i m p r o v e d ap-p r o x i m a t i o n as exap-pected, b u t one needs the t h i r d - o r d e r c o m ap-p o n e n t t o achieve a g o o d m a t c h t o the f u l l y n o n l i n e a r t i m e h i s t o t y i n this case.
A s i m i l a r f r e q u e n c y analysis can be conducted at any p o i n t i n w h i c h w e are interested. W e have used this approach to decompose the first-, second- and t h i r d - o r d e r harmonics o f the w a v e e l e v a t i o n i n the v i c i n i t y o f the c y l i n d e r array, a n d thereby to Investigate the p o t e n t i a l f o r d i f f e r e n t neartrapped modes to be excited by d i f f e r -e n t harmonics. IVlal-enica -e t al. [15] d -e m o n s t r a 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 f o r second-order waves f o r this array o f cylinders. Their second order analysis i n c l u d e d results f o r waves at i n c i d e n t f r e q u e n -cies corresponding to ka = 1.66 and ka = 0.468. The first o f these corresponds to a trapped m o d e frequency, and as discussed above i t leads to large elevations f o r waves at 45°. The i n c i d e n t w a v e f r e -q u e n c y ka = 0.468 corresponds ( f o r the w a t e r d e p t h considered here) to h a l f this t r a p p e d m o d e frequency. Therefore one m a y a n t i c i p a t e ( a n d Malenica e t a l . [15] c o n f i r m e d ) t h a t second o r d e r effects can ex-cite this same t r a p p e d mode. Here w e re-examine this s i t u a t i o n u s i n g the f u l l y n o n l i n e a r code, a n d make a p r e l i m i n a r y a t t e m p t to e x t e n d
W. Baiet al I Applied Ocean Research 44(2014) 71-81 7 9 - k i i = 1.66 k a = 0 . 7 5 4 - k a = 0 . 4 6 8 46 90 135 180 2 2 5 270 3 1 5 3 6 0 ( a ) 6 . - ki\ = 1,66 kü ^ 0 . 7 5 4 - k;i = 0 . 4 6 8 9 0 135 180 2 2 5 2 7 0 316 360
( b )
0 46 90 136 180 2 2 5 270 316 360 F i g . 1 8 . M a x i m u m w a v e r u n - u p o n ttie c y l i n d e r s a t a = 0.2,1 = 4 a , d = 3 a a n d icA = 0.157: ( a ) C y l i n d e r 1: ( b ) C y l i n d e r 2; ( c ) C y l i n d e r 3 . F i g . 1 9 . F r e q u e n c y s p e c t r a o f w a v e e l e v a t i o n at the d o w n s t r e a m o f C y l i n d e r 3 w h e n l<a = 0 . 4 6 8 . F i g . 2 0 . H i g h e r - o r d e r a p p r o x i m a t i o n s of w a v e e l e v a d o n at the d o w n s t r e a m of C y l i n d e r 3 w h e n l<a = 0 . 4 6 8 .8 0 W. Baiet al./Applied Ocean Research 44 (2014) 71-81
al. [2], w h o r e p o r t e d results f o r ka = 0.465. Ohl et al. measured results f o r t w o steepnesses: liA = 0.135 and kA = 0.261, and the d i m e n s i o n -less elevations f r o m t h e i r h a r m o n i c analyses d i f f e r e d s o m e w h a t at the t w o steepnesses. They suggested that p a r t o f the reason f o r this was b r e a k i n g o f the i n c i d e n t plus scattered waves i n the higher steepness case. Our results are calculated f o r this w a v e n u m b e r at a steepness /<7l = 0.157.
Figs. 21 and 22 show the first-, second-, t h i r d - o r d e r components and the m e a n w a v e elevation along the diagonal BC (see Fig. 6b) at these t w o w a v e n u m b e r s , respectively, and the c o m p a r i s o n w i t h the results o b t a i n e d b y the second-order w a v e t h e o r y f r o m Malenica et al. [ 1 5 ] a n d the e x p e r i m e n t a l data f r o m O h l et al. [ 2 ] at the l o w e r steepness they considered. The second order results m o d e l the array in the o p e n sea, whereas the experiments w e r e conducted i n a square tank; and f o r practical reasons our n u m e r i c a l m o d e l was based on a l o n g tank, o f w i d t h 14a. It is n o t expected t h a t reflections f r o m the tank w a l l s w o u l d s i g n i f i c a n t l y a f f e c t the higher harmonics, associated w i t h slower w a v e speeds. The first, second and t h i r d harmonics are n o n -d i m e n s i o n a l i z e -d by A, l<A^, an-d li^A^ respectively. Results f o r three meshes f o r the w a v e n u m b e r ka = 0.468 are i n c l u d e d i n Fig. 22, to d e m o n s t r a t e convergence o f the h i g h e r - o r d e r h a r m o n i c s . Mesh 1, the coarsest, has elements around the c y l i n d e r w i t h a m a x i m u m size o f about one t h i r t i e t h o f the first order w a v e l e n g t h . In m e s h 2 these elements are about one s i x t i e t h o f the first order w a v e l e n g t h , and i n m e s h 3, the finest, t h e y are about one n i n e t i e t h o f the first order w a v e l e n g t h . It can be seen i n Fig. 2 2 ( a ) - ( c ) t h a t b o t h t h e first- and second-order components o f the i n t e r m e d i a t e and fine mesh cases are essentially Identical. Small discrepancies are observed i n the t h i r d -order c o m p o n e n t s i n Fig. 22(d), nevertheless, the overall t r e n d still can be considered as convergent. The above compansons p r o v i d e a s i m p l e d e m o n s t r a t i o n o f the convergence o f the h i g h e r - o r d e r harmonics, extracted b y the FFT, up to t h i r d order i n the present n u m e r i c a l m o d e l o f the array.,
For the w a v e n u m b e r corresponding to linear n e a r - t r a p p i n g , the present first-order results i n Fig. 21(a) generally agree w i t h those f r o m Malenica et al. [15] except near the c y l i n d e r edges at Points B and C w h e r e the present results are higher. For the w a v e n u m b e r causing n e a r - t r a p p i n g at second-order, the first-order c o m p o n e n t i n Fig. 22(a) is m u c h smaller than t h a t In the first-order near t r a p p i n g case, and t h e first order results f r o m b o t h Malenica et al. [15] and the present study are close to the e x p e r i m e n t a l data. H o w e v e r , the dimenslonless second-order c o m p o n e n t i n this case becomes m u c h larger c o m p a r e d w i t h t h a t at the wave f r e q u e n c y causing linear near-t r a p p i n g . O u r n u m e r i c a l resulnear-ts snear-till agree q u i near-t e w e l l w i near-t h near-the ex-p e r i m e n t a l data w h i l e the values near the inside faces o f C y l i n d e r 1 and C y l i n d e r 3 f r o m Malenica et al. [15] are s i g n i f i c a n t i y larger. It should be m e n t i o n e d t h a t the results o f Malenica et al. [15] have been c o n f i r m e d by another independent d i f f r a c t i o n n u m e r i c a l p r o g r a m i n N e w m a n [ 3 2 ] . One possible explanation f o r this discrepancy b e t w e e n the f r e q u e n c y - d o m a i n second order results on the one hand, and the e x p e r i m e n t a l and f u l l y nonlinear time-domain results o n the other, is the d i f f e r e n c e b e t w e e n the i n c i d e n t w a v e n o n l i n e a r characteristics. Differences arise at second and higher orders because o f the d i f f e r e n t b o u n d waves generated b y a w a v e m a k e r (as In the e x p e r i m e n t s and the n o n l i n e a r m o d e l ) and those i n a Stokes expansion f o r waves i n the o p e n ocean. The mean components o f e l e v a t i o n are s h o w n i n Figs. 21(c) and 22(c). As expected, the mean e l e v a t i o n at ka = 1.66 is v e r y m u c h larger t h a n at ka = 0.468, because the mean q u a n t i t i e s i n the Stokes expansion to second order depend o n l y o n first o r d e r effects, and these are strongly i n f l u e n c e d by n e a r - t r a p p i n g at ka = 1.66.
The dimenslonless t h i r d order components at the t w o I n c i d e n t w a v e n u m b e r s are s h o w n i n Figs. 2 1 ( d ) a n d 2 2 ( d ) . It is s t r i k i n g t h a t the t h i r d o r d e r c o m p o n e n t f o r the case ka = 0.468 is a l m o s t an order o f m a g n i t u d e larger t h a n t h a t f o r ka = 1.66. A l i k e l y e x p l a n a t i o n w o u l d seem to be t h a t this i n c i d e n t wave excites a d i f f e r e n t trapped m o d e at t h i r d order. I t m a y easily be s h o w n , using a linear analytical
6 4 T ? - 2 O - O - Present Malenica p - cf' 1 ( 1 -0.4 -0..'! -0.2 -0.1 0.0 0.1 0.2 0.3 0 . 4
-7^
10 -1 - - Prescnl 1 Cr--'^ O - , f> . <:^ --0.4 -0.3 -0.2 -0.1 ( i l ) 0.0 0.1 X 0.2 0.3 (1.4 F i g . 2 1 . 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 of w a v e e l e v a t i o n a l o n g the x - a x i s f o r the firstorder n e a r t r a p p e d m o d e a t ka = 1.66: ( a ) firstorder c o m p o n e n t ; ( b ) s e c o n d -o r d e r c -o m p -o n e n t ; ( c ) m e a n v a l u e : ( d ) t h i r d - -o r d e r c -o m p -o n e n ts o l u t i o n based o n [ 12|, t h a t this c o n f i g u r a t i o n exhibits a n e a r - t r a p p e d m o d e excited by waves f r o m 4 5 ° at ka = 3.734. Such a m o d e w o u l d be excited i n the w a t e r d e p t h considered here ( d / a = 3 ) b y t h i r d harmonics due t o an i n c i d e n t w a v e h a v i n g w a v e n u m b e r ka = 0.468. This leads to the r e m a r k a b l e s i t u a t i o n t h a t i n this case, f o r an i n c i d e n t w a v e at ka = 0.468 at 4 5 ° , one trapped m o d e is excited b y second order effects, a n d a d i f f e r e n t t r a p p e d mode by t h i r d harmonics.
6. Conclusions
Fully n o n l i n e a r w a v e d i f f r a c t i o n a r o u n d an array o f fixed v e r t i -cal circular cylinders has been s i m u l a t e d by means o f a 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 o d e l i n a n u m e r i c a l w a v e tank. The n e a r - t r a p p i n g p h e n o m e n o n i n the array has been Investigated by the present time d o m a i n s i m u l a t i o n . The n u m e r i c a l m o d e l has c o n f i r m e d and e x t e n d e d the findings o f Malenica e t a l . [ 1 5 ] that higher o r d e r n o n l i n e a r effects can excite near-trapped modes. For a p a r t i c u l a r case o f a square a r r a y o f f o u r circular cylinders, Malenica et al. s h o w e d t h a t f o r a n i n c i -d e n t w a v e at h a l f the f r e q u e n c y o f the w a v e t h a t excites a p a r t i c u l a r near-trapped m o d e t h r o u g h linear interactions, the same m o d e is ex-c i t e d by seex-cond o r d e r effeex-cts. A h a r m o n i ex-c analysis has been used here to decompose linear, double f r e q u e n c y and t r i p l e f r e q u e n c y c o m p o -nents i n the time histories o b t a i n e d f r o m the f u l l y n o n l i n e a r m o d e l . These have s h o w n s i m i l a r behavior to t h a t f o u n d by M a l e n i c a e t al. w i t h their second order analysis. W e have also p r o v i d e d p r e l i m i n a r y evidence o f t h i r d order e x c i t a t i o n o f a trapped mode. For t h e par-ticular c o n f i g u r a t i o n considered here, w e observed the r e m a r k a b l e
