Applied Ocean Research 41 (2013) 57-64
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Applied Ocean Research
j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / a p o rThird-order interactions, wave run-up and hydrodynamic loading ( J )
crossMarkon a vertical plate in an infinite wave field
I.K. Chatjigeorgiou^'', B. Molin''
'School of Naval Architecture and IVIarine Engineering, National Technical University of Athens (NTUA), 9 Hereon Polytechniou Avenue, Athens 15773, Greece ''École Centrale Marseille & Institut de Recherche sur les Phénomênes Hors Equilibre (IRPHE), 13451 tViarseille Cedex 20, France
A R T I C L E I N F O A B S T R A C T
Article history: The nonlinear w a v e interaction problem w i t h a vertical plate of finite length is considered. Reference is made Received 31 January 2013 fo previous e x p e r i m e n t a l and n u m e r i c a l studies reported in M o l i n et al. [ 1 - 3 ] , w h e r e it w a s s h o w n that Accepted 5 March 2013 observed r u n - u p p h e n o m e n a are due to third-order (or tertiary) interactions b e t w e e n the incoming and
reflected w a v e systems. In this paper a n e w numerical model is proposed w h e r e the presence of lateral walls is relaxed. R u n - u p computations, w i t h and without confinement effects, are c o m p a r e d . It is found that, in the model tests reported in M o l i n et al. [3], the effect of c o n f i n e m e n t w a s relatively smafl. T h e time-varying and steady w a v e loads w h i c h are exerted on the plate are also investigated. T h e dedicated n u m e r i c a l predictions s h o w that as the w a v e steepness is increased the response amplitude operators of the t i m e - v a r y i n g loads first increase, reach a m a x i m u m and then decrease dramatically, due to p h a s i n g effects.
® 2 0 1 3 E l s e v i e r Ltd. M l rights reserved.
1. Introduction
The present paper c o m p l e m e n t s previous w o r k s b y the second a u t h o r and his g r o u p o n r u n - u p phenomena o n the w e a t h e r side o f reflective structures [ 1 3 ] , I n the q u o t e d references i t is d e m o n -strated, t h r o u g h e x p e r i m e n t a l , theoretical and n u m e r i c a l investiga-dons t h a t the observed r u n - u p effects are associated w i t h n o n l i n e a r ( t h i r d - o r d e r ) interactions b e t w e e n the i n c o m i n g w a v e system a n d the r e f l e c t e d w a v e system b y the structure: t h e reflected waves " s l o w d o w n " t h e i n c o m i n g waves, alike a shoal w o u l d do, i n d u c i n g energy f o c u s i n g effects.
Several e x p e r i m e n t a l campaigns w e r e carried o u t i n the BGO-FIRST w a v e - t a n k ( l e n g t h 4 0 m x w i d t h 16 m ) i n La Seyne-sur-Mer (France) and i n the Ship Dynamics Laboratory ( l e n g t h 150 m x w i d t h 30 m ) o f CEHIPAR (Spain), w i t h s i m i l a r set-ups: a r i g i d plate o f large d r a f t stuck against one o f the lateral walls o f the basins ( t h e r e b y d o u b l i n g the plate a n d basin w i d t h s t h r o u g h m i r r o r i n g e f f e c t ) . The lengths o f the plates, f r o m the w a l l s to t h e i r edges, v a r i e d f r o m 1.2 m [1 ] to 3 m [ 2 ] and 5 m [ 3 ] . Typical wavelengths w e r e i n the range o f 1-3 m w h i l e w a v e steepnesses H/L ranged f r o m 2% u p t o 6%. I n the BGO-FIRST tests, the plates w e r e located at about 20 m f r o m the w a v e m a k e r , m e a n i n g a rather s h o r t i n t e r a c t i o n area b e t w e e n the i n -c o m i n g and the refle-cted w a v e systems; i n the CEHIPAR tests the plate was located at 100 m f r o m the wavemaker, m e a n i n g a m u c h longer i n t e r a c t i o n area.
T w o n u m e r i c a l models w e r e devised to reproduce the observed
• Corresponding author. Tel.:+30 2107721105; fax:+30 2107721412. E-mail addresses: chatEi#naval.ntua.gr, cliatji@central.ntua.gr (I.K. Chatjigeorgiou).
0141-1187/$ - see front matter ® 2013 Elsevier Ltd. All rights reserved. http://dx.doi.Org/10.1016/j.apoi-.2013.03.001
r u n - u p phenomena, b o t h based on p o t e n t i a l f l o w t h e o r y . The f i r s t m o d e l , described i n [ 1 ] , looks f o r a steady state s o l u t i o n i n the f r e -quency d o m a i n . The reflected w a v e system b y the plate is d e r i v e d via linearized p o t e n t i a l f l o w t h e o r y and use o f e i g e n f u n c t i o n expan-sions i n the t w o sub-domains at either side o f the plate, b o u n d e d by the basin walls. The reflected w a v e f i e l d is locally i d e n t i f i e d w i t h a plane w a v e o f a m p l i t u d e AR(^X, y) and d i r e c t i o n o f p r o p a g a t i o n p{x,
y). Use is then m a d e o f the m o d i f i c a t i o n o f t h e w a v e n u m b e r o f the
i n c o m i n g w a v e g i v e n by Longuet-Higgins and Phillips [ 4 ] to derive a parabolic equation t h a t describes the space e v o l u t i o n o f the a m p l i -tude o f the i n c o m i n g waves as t h e y propagate t o w a r d the plate. The parabolic equation is integrated i n the i n c o m i n g w a v e d i r e c t i o n (x) s t a r t i n g f r o m some distance / ahead o f the plate, w i t h / b e i n g taken as the distance f r o m the w a v e m a k e r to the plate i n t h e comparisons w i t h the experiments. The i n c i d e n t and the r e f l e c t e d w a v e fields are successively updated u n t i l a converged s o l u t i o n is f o u n d . I t has been observed t h a t i n m a n y cases w h e n the w a v e steepness ( o r the i n t e r -action l e n g t h ) exceeds some t h r e s h o l d value, no steady state s o l u t i o n can be f o u n d no m a t t e r h o w m u c h r e l a x a t i o n is i n t r o d u c e d i n the i t e r a t i v e scheme.
The second m o d e l [5,6], f u l l y nonlinear, solves t h e p r o b l e m i n the time d o m a i n , alike a n u m e r i c a l w a v e t a n k . I t makes use o f the so-called extended Boussinesq equations as described, e.g., i n B i n g h a m et al. [ 7 ] . Good agreement b e t w e e n p r e d i c t e d and m e a s u r e d time histories o f the free surface elevations a l o n g the plates have been r e p o r t e d i n Jamois et al. [5] and M o l i n et al. [ 2 , 3 ] .
Both n u m e r i c a l models and the physical tanks share a c o m m o n feature t h a t the d o m a i n s are b o u n d e d l a t e r a l l y . This means t h a t the reflected w a v e system by the plate is c o n f i n e d and thus its a m p l i t u d e , averaged over the basin w i d t h , remains m o r e o r less c o n s t a n t whereas
58 I.K. Chatjigeorgiou, B. Molin/Applied Ocean Research 41 (2013) 57-64
Control area of width b
F i g . l . Definition sketch.
i t w o u l d decay as o p g ^ ocean w i t h R the h o r i z o n t a l distance f r o m the plate. I t m a y thus be i n f e r r e d t h a t the n o n l i n e a r interactions b e t w e e n i n c o m i n g and reflected w a v e systems, as achieved i n the tanks, are exaggerated as compared to w h a t t h e y w o u l d be w i t h o u t lateral walls, p a r t i c u l a r l y i n the cases w h e n the distance f r o m the w a v e m a k e r to the plate is large as related to the t a n k w i d t h .
This issue has m o t i v a t e d the present s t u d y w h e r e , i n t h e parabolic m o d e l , the presence o f lateral walls is relaxed i n the d e t e r m i n a t i o n o f the reflected w a v e system. This is achieved by a s s i m i l a t i n g the plate w i t h a degenerated elliptical cylinder w i t h nearly zero s e m i - m i n o r axis and solving the d i f f r a c t i o n p r o b l e m i n elliptic coordinates i n an i n f i n i t e ocean.
The f o r m u l a t i o n o f the n e w l y developed n u m e r i c a l m o d e l is out-lined i n detail i n Section 2. A p p l i c a t i o n is t h e n made to some o f the CEHIPAR tests [ 3 ] w i t h the 5 m plate i n regular waves o f 1.01 s period. The parameters w h i c h are varied are the i n c o m i n g w a v e steepness
H/L, the i n t e r a c t i o n l e n g t h / and the w i d t h o f the tank.
In all o f the previous w o r k s , the quantities o f interest w e r e the free surface elevations along the plates. Obviously free surface elevations p r o p e r l y r e f l e c t the local loads b u t the integrated loads along the plates are another matter. In this paper w e s h o w n u m e r i c a l results, as o b t a i n e d w i t h the t w o models ( c o n f i n e d and n o t c o n f i n e d ) , f o r the f i r s t - h a r m o n i c loads ( t a k i n g place at the w a v e f r e q u e n c y ) , and f o r the time-averaged loads (or d r i f t forces). It is obtained t h a t albeit the RAOs of the local elevations tend to increase w i t h the w a v e steepness, more p a r t i c u l a r l y b y the plate - w a l l corner, the RAOs o f the f i r s t - h a r m o n i c loads do n o t necessarily increase w i t h the w a v e steepness, because of phase compensations.
2. Numerical model
A n e l l i p t i c a l c y l i n d e r a p p r o x i m a t i n g a finite length v e r t i c a l plate is considered. To this e n d w e let e 1 w h e r e e = ^ 1 ~ [a'/af denotes the e l l i p t i c eccentricity. I t is noted t h a t a and a' represent the s e m i -m a j o r and the s e -m i - -m i n o r axes, respectively g i v e n by a = c cosh UQ and a' = c sinh U Q w h e r e c is the half distance b e t w e e n the t w o foci and Uo is the elliptical b o u n d a r y o f the c y l i n d e r w h i c h is o b t a i n e d by U Q =
a t a n h ( a 7 a ) . The c y l i n d e r is exposed to the action o f m o n o c h r o m a t i c
beam seas o f linear a m p l i t u d e A\ and f r e q u e n c y o), p r o p a g a t i n g along the X d i r e c t i o n as indicated by the d e f i n i t i o n sketch depicted i n Fig. I . The z coordinate is fixed on the free surface p o i n t i n g vertically u p w a r d s and the w a t e r d e p t h is assumed to be i n f i n i t e .
It s h o u l d be n o t e d that the finite w i d t h b o f the d o m a i n that ap-pears i n the flgure only has the purpose o f b o u n d i n g the d o m a i n w i t h i n w h i c h the parabolic equation, describing the space e v o l u t i o n of the i n c o m i n g w a v e system, is b e i n g solved.
The linearized v e l o c i t y p o t e n t i a l reads
<P [u, V, z; t ) = Re {4> ( u , v, z) e-'^^) ( 1 )
w h e r e u = const and v = const denote o r t h o g o n a l l y intersecting f a m -ilies o f confocal ellipses and hyperbolae, respectively. The o r i g i n o f the elliptic coordinate system coincides w i t h the body fixed center o f the ( x , y ) Cartesian system s h o w n i n Fig. 1. According to the n o t a t i o n s of Fig. 1, the s e m i - m a j o r and the s e m i - m i n o r axes o f the e l l i p t i c a l "plate" are along t h e y and x coordinates respectively.
2.1. Linear diffraction problem
I n elliptic coordinates, the i n c i d e n t w a v e c o m p o n e n t o f 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 is w r i t t e n as [ 8 ] 0j - 2 i ^ e ^ 0) m=0
J2
' " ' M c , ( 1 ) i i ; g ) c e „ , ( v ; g ) c e m ( a ; ( j ) (2) + J2i'"MsO\u;q)se,niv;q)se„,{a;q) m=lIn Eq. (2) k is the w a v e n u m b e r , a is the angle o f p r o p a g a t i o n w h i c h h e r e i n w i l l be taken equal to 90= and q is the M a t h i e u parameter g i v e n by
M o r e o v e r ce„, and se,n denote the even and the odd periodic M a t h i e u f u n c t i o n s o f order m w h i l e Mcj,]' and MsJ,!' p r o v i d e the equal order even and odd m o d i f i e d M a t h i e u f u n c t i o n s o f the first k i n d .
Using an analogous expansion, the d i f f r a c t i o n c o m p o n e n t o f the t o t a l velocity p o t e n t i a l is w r i t t e n as - 2 i ^ e ^ ^ (I) m=0 .(3), u ; g ) c e m ( v ; q ) (4)
w h e r e Mcm and MsL"?^ are the m t h order even and odd m o d i f i e d M a t h ieu f u n c t i o n s o f the t h i r d k i n d whereas Am and Bm denote the u n k n o w n expansion coefficients w h i c h are obtained t h r o u g h t h e f u l f i l l -m e n t o f the n o - f l o w c o n d i t i o n at the cylinder's w a l l
dtpo i)d>i
— = - — a t t t = Uo (5)
Use o f Eq. (5) together w i t h the relations o f o r t h o g o n a l i t y o f even and odd periodic M a t h i e u f u n c t i o n s [9] i m m e d i a t e l y renders
M c ' ( ] ' ( u o ; q M c ' t ^ ' ( U o ; q ) M 4 ! ' ( » o ; q ) Msf,\uo:q) ce„,{a:q) sem(a;q) (6) (7)
w h e r e the primes denote d i f f e r e n t i a t i o n w i t h respect to t h e argu-m e n t . The free surface elevation along t h e c y l i n d e r is t h e n obtained f r o m
g
Pi +'PD) (8) a t z = 0 and u = UQ.
A f t e r s u b s t i t u t i n g Eqs. (2) and (4) i n t o Eq. (8) and using Eqs. (6) and (7) the f o l l o w i n g is obtained
A, 4i m=0 , c e n , ( v ; q ) c e n i ( g : g ) Mc''„^'(Uo;q) s e „ , ( v ; ( j ) s e m ( t ï ; q ) ] (9)
I.K. Chatjigeorgiou, B. Molin/Applied Ocean Research 41 (2013) 57-64 59
For the d e r i v a t i o n o f Eq. (9) the f o l l o w i n g W r o n s k i a n relations w e r e e m p l o y e d [10]
Wc=McS,l
)(Uo
;(j)Mc'S
^'(uo;q)
2i ws = M s L l ' ( u o ; g ) M s V ( u o ; g ) M s ' ( J ) ( u o ; ( j ) M s J ^ ' ( u o ; g ) ( 3 ) / 2/ (10)(11)
2.2. Third-order interactions between the incoming and the reflected
wave fields
Solving the linear d i f f r a c t i o n p r o b l e m constitutes the first stage o f the t h i r d o r d e r i n t e r a c t i o n p r o b l e m . The r e f l e c t e d w a v e field i n t e r -acts w i t h the i n c o m i n g w a v e field, c o m p l e t e l y changing its pattern. Reference is made to Longuet-Higgins and Phillips [ 4 ] w h o tackled the case o f t w o plane waves, one p r o p a g a t i n g along the x-axis and the o t h e r at an angle c o m p o s i n g a v e l o c i t y field t h a t is described by the f o l l o w i n g v e l o c i t y p o t e n t i a l
* = M e X Z s i n ( ; « - « t ) + — e * ^ ^
s i n ( / « c o s / i + ky sin p - cot)
(12)
A c c o r d i n g to the t h i r d - o r d e r analysis o f Longuet-Higgins a n d Phillips [4] the w a v e n u m b e r o f the first c o m p o n e n t is m o d i f i e d by a q u a n t i t y kf^ that is g i v e n b y / < i = / < 3 A 2 / ( / ? ) - f c 3 A 2 ( 1 3 ) w h e r e 1 f { P ) = - (1 - c o s ^ ) V 2 - | - 2 c o s / ? + 2cos/( +-sir? p 2 ( 1 - c o s fl) — \ + , ^ 1 + c o S y S - F V2 + 2cos,8 y 2 + 2 c o s / 5 - 4 V " '^J ( 1 4 )
I n order to e m p l o y the result o f Longuet-Higgins and Phillips [ 4 ] , M o l i n et al. [ 1 ] idealized locally the reflected w a v e field as a plane w a v e o f a m p l i t u d e AR a n d d i r e c t i o n p. The angle o f p r o p a g a t i o n is taken equal to the d i r e c t i o n o f steepest v a r i a t i o n o f the reflected f i e l d . The a m p l i t u d e o f the i n c o m i n g w a v e field A{x,y] is rendered c o m p l e x and s l o w l y v a r y i n g w i t h respect to the space coordinates x a n d y , and is s h o w n to obey the parabolic e q u a t i o n [1]
2ikA^ + Ayy + 2lé [Alf{p)+Aj - ||A||2j A = 0 ( 1 5 )
This e q u a t i o n is solved s t a r t i n g f r o m a distance / ahead o f the plate w h e r e A is taken equal to A;, over a d o m a i n o f finite w i d t h b. Details on f u r t h e r processing o f Eq. ( 1 5 ) can be f o u n d i n M o l i n et al. [ 1 ] w h e r e the interested reader is referred. A t variance w i t h the m e t h o d adopted i n that paper, w h i c h e m p l o y e d an a r t i f i c i a l second-order accurate finite differences i m p l i c i t scheme, here the parabolic system represented b y Eq. ( 1 5 ) is integrated n u m e r i c a l l y b y the R u n g e - K u t t a m e t h o d .
The i n t e r a c t i o n o f the reflected and i n c o m i n g waves is a c o n t i n u -ous process t h a t results i n the constant m o d i f i c a t i o n o f the c o m b i n e d w a v e field. The empirical f e e l i n g i m p l i e s t h a t this process m u s t finally be stabilized and a steady state be attained. To achieve a relevant theo-retical f o r m u l a t i o n , the i n t e r a c t i o n process is t u r n e d i n t o an iterative procedure whereby, at each step, the r e f l e c t e d and i n c o m i n g w a v e systems are updated. The n u m b e r o f iterations r e q u i r e d to o b t a i n a steady s o l u t i o n is a r e f l e c t i o n o f t h e d u r a t i o n o f the transient stage i n the physical tank.
Nevertheless, i t is e v i d e n t t h a t the d i f f r a c t i o n p r o b l e m m u s t be treated i n d i v i d u a l l y at a l l iterations. Hence assuming a r a n d o m i t e r -a t i o n i n w h i c h the c o m p l e x " -a m p l i t u d e " h-as become A(x, y) or, b y
analogy, A{u, v) i n elliptic coordinates, the w a v e field t h a t impacts the plate can be n o w w r i t t e n as
^ i " ' M c S , ; j ( u ; ( ? ) c e m ( v ; ( j ) c e „ ( a ; q )
m=0 (16)
+ 22 ""MsLlJ (u; q) sem (v; q) se,„ («; q) m=l
The d i f f r a c t i o n c o m p o n e n t is n o w taken i n the f o r m
CO ^ i ' " A , „ M c | „ ^ ^ ( u ; q ) c e „ , ( v ; q ) m=0 + f ] / ' " B „ M s S „ ^ ) ( u ; q ) s e , „ ( v ; q ) <PD = -2i^e^ (17)
and the u n k n o w n expansion coefficients are o b t a i n e d by a p p l y i n g the n o - f l o w c o n d i t i o n o n the elliptical "plate" at u = UQ (see Eq. (5)). These are An = 271 1 - . M c ' ( ^ ) ( u o ; q )
/
•• m=o M c V ' ( i i o ; q ) A ( u o , v ) c e m ( v , q ) c e „ ( v , q ) dv cem(a;q) M c f ) ( u o i q ) (18) sem(a;q)f
J 0 Bn A ( u o , v ) s e m ( v , g ) c e n ( v , q ) d v ^ E ' - - " ? ? ^ c e . ( . : q ) m=0 M s f ( u o i q ) A ( u o , v ) c e m ( v , q ) s e n ( v , q ) dv l g . . _ „ M s| ) ( . o: q ) Ms'(^)(Uo;q) ( 1 9 ) 2,T A ( u o , v ) s e m ( v , q ) s e „ ( v , q ) dvƒ
The d i f f r a c t i o n p o t e n t i a l is obtained b y s u b s t i t u t i n g the expansion c o e f f i c i e n t s f r o m Eqs. (18) and (19) i n t o Eq. (17), the t o t a l v e l o c i t y p o t e n t i a l is taken b y <pi + 0 D . w h e r e 0; is g i v e n b y Eq. ( 1 6 ) , w h i l e the w a v e e l e v a t i o n is derived t h r o u g h Eq. (8). The final r e l a t i o n t h a t provides t h e w a v e elevation as a f u n c t i o n o f the plane variables (x, y ) o r (u, v ) is q u i t e l e n g t h y and its details are o m i t t e d .
3. Wave run-up calculations
W e focus on the CEHIPAR tests w i t h the 5 m l o n g plate. The basin is 30 m w i d e , 5 m deep and 150 m long. D u r i n g t h e tests t h e plate was stuck against one o f the w a l l s at 100 m f r o m t h e w a v e m a k e r ; this set-up is e q u i v a l e n t to a 10 m long plate i n the m i d d l e o f a 60 m w i d e tank. I n M o l i n et al. [3] detailed comparisons are s h o w n b e t w e e n calculated a n d measured free surface elevations a l o n g the plate. W e r e f e r the reader to this paper and w e w i l l s h o w o n l y n u m e r i c a l results here.
W e consider the tests at the l o w e s t w a v e p e r i o d o f 1.01 s. This means a w a v e l e n g t h o f 1.6 m that is about one t h i r d o f t h e plate l e n g t h , and the distance f r o m the w a v e m a k e r to the p l a t e represents about 63 w a v e l e n g t h s .
First w e consider a l o w steepness case H/L = 0.025 and w e v a r y the i n t e r a c t i o n l e n g t h and the tank w i d t h . The results s h o w n i n t h e figures are the transfer f u n c t i o n s (or RAOs) o f the free surface e l e v a t i o n along the plate, f r o m the w a l l (y = 0) to the edge (y = 5 m ) , as o b t a i n e d f r o m b o t h n u m e r i c a l models. Results obtained w i t h the m o d e l o f M o l i n et al. [ 1 ] are labeled as CWF ( c o n f i n e d w a v e field), w h i l e those o b t a i n e d w i t h the n e w m o d e l are labeled as IWF ( i n f i n i t e w a v e field). In the first m o d e l (CWF) the finite w i d t h o f the t a n k is accounted f o r b o t h i n
60 U(. Chatjigeorgiou, B. Molin /Applied Ocean Research 41 (2013) 57-64
Fig. 2. Comparisons between infinite and confined wave field calculations. H / t = 2.5%
and 7 = 1.01 s. The depicted results correspond to the end of iterations after conver-gence. The waves are generated at x = - 5 0 m ahead of the plate.
Fig. 4. Compar isons between infinite and confined wave field calculations. H/L = 2.0% and T = 1.01 s. The depicted results correspond to the end of iterations after conver-gence. Width of the control area b = 30 m.
Fig. 3 . Comparisons between infinite and confined wave field calculations. H/L = 2.5% and T = 1.01 s. The depicted results correspond to the end of itei'ations after conver-gence. The waves are generated atx = - 1 0 0 m ahead of the plate.
Fig. 5. Comparisons between infinite and confined wave field calculations. H/L = 2.0% and r = 1.01 s. The depicted results corr espond to the end of iterations after conver-gence. Width of the contr ol area b = 30 m.
the r e s o l u t i o n o f the linear d i f f r a c t i o n p r o b l e m , and i n the n u m e r i c a l r e s o l u t i o n o f the parabolic equation. I n the second m o d e l ( I W F ) the linear d i f f r a c t i o n p r o b l e m is solved i n an u n b o u n d e d d o m a i n b u t the parabolic e q u a t i o n is still solved over a d o m a i n o f f i n i t e w i d t h .
Fig. 2 shows the o b t a i n e d RAOs f o r an i n t e r a c t i o n length o f 50 m a n d three d i f f e r e n t basin w i d t h s o f 20 m , 30 m and 40 m . The RAO f r o m linear t h e o r y (zero steepness) is also s h o w n f o r reference. W h a t e v e r the basin w i d t h , t h e RAOs delivered by the IWF m o d e l are identical. The CWF RAOs agree w i t h the IWF RAOs at 30 m a n d 4 0 m w i d t h s , discrepancies o n l y appear i n the 20 m w i d t h case. Fig. 3 shows the c o r r e s p o n d i n g results w i t h a n i n t e r a c t i o n l e n g t h o f 100 m t h a t is the actual distance f r o m the w a v e m a k e r to the plate i n the e x p e r i m e n t s ( e x p e r i m e n t a l results f o r this case are s h o w n i n Fig. 9 o f Ref. [3]). Again the IWF RAOs r e m a i n identical w h a t e v e r the basin w i d t h : this proves t h a t the choice o f d o m a i n w i d t h does n o t m a t t e r i n the r e s o l u t i o n o f the parabolic equation. But the CWF RAOs are all d i f f e r e n t and d i f f e r e n t f r o m the IWF RAOs, p r o v i n g t h a t c o n f i n e m e n t effects, a f f e c t i n g the r e f l e c t e d w a v e fleld f r o m the plate, come i n t o play.
F r o m n o w on t h e tank w i d t h is set to its actual value o f 30 m and the w a v e steepness and i n t e r a c t i o n l e n g t h are varied. Figs. 4 a n d 5 address the 2% steepness case, w i t h i n t e r a c t i o n lengths o f 25 m , 50
m , 100 m (Fig. 4, see also Fig. 7 i n Ref [ 3 ] f o r e x p e r i m e n t a l results), 2 0 0 m and 3 0 0 m (Fig. 5). The IWF a n d CWF RAOs r e m a i n v e r y s i m i l a r u p to i n t e r a c t i o n lengths o f 100 m b u t s t r o n g differences appear i n the 200 m and 300 m cases. A t 300 m , i n particular, the CWF RAO has a q u i t e d i f f e r e n t shape by the edge o f the plate. It is remarkable t h a t the IWF RAOs f o r 200 m and 300 m are n e a r l y i d e n t i c a l ; this can be related to the fact that the IWF r e f l e c t e d w a v e - f i e l d has become v e r y w e a k past 2 0 0 m f r o m the plate, so b e y o n d some distance i d e n t i c a l results are obtained.
Fig. 6 shows the o b t a i n e d RAOs i n the 2.5% steepness case, f o r i n t e r a c t i o n lengths o f 100 m and 2 0 0 m . A t 100 m l e n g t h the CWF and I W F RAOs are very close. A t 2 0 0 m t h e CWF RAOs n o t a b l y d i v e r g e f r o m the IWF ones. There are some s m a l l d i f f e r e n c e s b e t w e e n the t w o IWF curves, suggesting that 100 m is a too s h o r t distance f o r the i n t e r a c t i o n p h e n o m e n a to have f u l l y developed. U n f o r t u n a t e l y c o m p u t a t i o n s fail a t distances larger than 2 0 0 m w h e r e convergence o f the i t e r a t i v e scheme cannot be reached. Likewise, i n the 3% steepness case (Fig. 7), t h e i t e r a t i v e scheme fails b e y o n d 100 m a n d i n the 3.5% steepness case (Fig. 8) b e y o n d 75 m . As w r i t t e n above, t h e f a i l u r e o f the i t e r a t i v e scheme suggests that no steady state c o n d i t i o n can be reached i n the physical tank; this is actually w h a t happened i n the CEHIPAR tests.
I.K. Chatjigeorgiou, B. Molin /Applied Ocean Research 41 (2013)57-64 61 3.5 c S 3 f 2.5
1
1.5 1 0.5 0, L I N E A R C W F , x=-100m C W F , x = - 2 0 0 m - B - I W F , x = - 1 0 0 m —*— IWF, x ^ 2 0 0 m L I N E A R C W F , x=-100m C W F , x = - 2 0 0 m - B - I W F , x = - 1 0 0 m —*— IWF, x ^ 2 0 0 m\ I
\ .
. ' ^ a - . * . ^ M \ ; — , i ^- IFig. 6. Comparisons between infinite and confined wave field calculations. H/L = 2.5% and r = 1.01 s. The depicted results correspond to the end of iterations after conver-gence. Width of the control area h = 30 m.
Fig. 8. Comparisons between infinite and confined wave field calculations. H/L = 3.5% and T = 1.01 s. The depicted results correspond to the end of iterations after conver-gence. Width of the control area b = 30 m.
5 4.5 4 3.5 I 3 2.5 2 1.5 1 0.5 L I N E A R C W F , x=-25m C W F , x=-50m C W F , x = - 1 0 0 m - a - I W F , x = - 2 5 m — t - - I W F , x = - 5 0 m I W F , x = - 1 0 0 m
Fig. 7. Comparisons between infinite and confined wave field calculations. H/L = 3.0% and T = 1.01 s.The depicted results correspond to the end of iterations after conver-gence. Width of the conti'ol area b = 30 m.
Fig. 9. Amplitude of the wave field AR(x.y)//\| that is reflected by the plate at the 1st iteration (linear diffraction). Wave steepness H/L = 2.0%;" the regular waves are generated atx= - 1 0 0 m; T = 1.01 s; width of control area i = 30 m.
w h e r e no steady state could be attained i n the 3.5% a n d 4% steepness cases (see Figs. 11 and 12 i n Ref. [ 3 ] ) .
I n t h e present study w e i m p r o v e the discussion o n the actual phe-n o m e phe-n o phe-n addressiphe-ng aspects w h i c h w e r e phe-n o t cophe-nsidered by M o l i phe-n et al. [ 1 , 3 ] . I n the f o l l o w i n g w e e x a m i n e the details o f the e n t i r e w a v e f i e l d and this is p e r f o r m e d u s i n g the v i s u a l representadons o f the c o n t r o l area given i n Figs. 9 and 10 w h i c h correspond to the 2% steep-ness case. In b o t h depicdons t h e waves are generated at 100 m ahead o f the plate. Fig. 9 depicts t h e reflected w a v e field i n a b = 3 0 m w i d e c o n t r o l area at the first i t e r a t i o n ( l i n e a r p r o b l e m ) w h i c h a c t u -ally represents the i n i t i a l i m p a c t . Moreover, Fig. 10 shows the same field at the e n d o f iterations a f t e r the s t a b i l i z a t i o n o f the w a v e field and t h e a t t a i n m e n t o f the steady c o n d i t i o n . It is n o t e d t h a t the i t e r -ative calculations are the n u m e r i c a l i n t e r p r e t a t i o n o f the c o n t i n u o u s i n t e r a c t i o n s b e t w e e n the i n c i d e n t and t h e r e f l e c t e d w a v e fields. A c -c o r d i n g l y , the i n i t i a l i m p a -c t (Fig. 9) results i n s m a l l m a g n i t u d e s i n the r e f l e c t e d field a p p r o x i m a t e l y 1-1.5A;, whereas the disturbances o c c u p y a r e l a t i v e l y large area ahead o f the plate. W h e n the steady state c o n d i t i o n is finally attained, at the e n d o f i t e r a t i o n s (Fig. 10), the r e f l e c t e d w a v e field depicts p r o f o u n d disturbances a l o n g the x-axis.
The m a x i m u m w a v e elevation reaches up to 2.5Aj a n d is extended l e n g t h w i s e t o a p p r o x i m a t e l y 20 m ahead o f the plate. It is also i n -teresting to observe t h a t the flow lines bend t o w a r d t h e x-axis and decay progressively t o w a r d the wavemaker. The reflections appear to be i n s i g n i f i c a n t w i d t h w i s e a f t e r the edge o f the plate.
4. Wave loads
In this section w e present n u m e r i c a l results f o r t h e w a v e loads on the plates, and o n h o w they evolve w h e n the w a v e steepness i n -creases. I n a l l o f the e x p e r i m e n t a l campaigns t h a t w e r e p e r f o r m e d a n d are r e p o r t e d i n M o l i n et al. [ 1 - 3 j , o n l y the f r e e surface elevations along the plates w e r e measured. Free surface elevations p r o p e r l y r e -flect the local w a v e loads, b u t integrated loads a l o n g the plate are another m a t t e r since the phases come i n t o play.
Here w e address b o t h the time-varying loads t a k i n g place at the w a v e f r e q u e n c y , and the time-averaged loads, o r d r i f t forces.
62 I.K. Chatjigeorgiou, B. Molin /Applied Ocean Research 41 (2013) 57-64
Tig. 10. Amplitude of the wave fleld Ag(x,y)/Ai that is reflected by the plate at the
15th iteration (end of iterations). Wave steepness H/L = 2.0%; the regular waves are generated atx = - 1 0 0 m; r = 1.01 s; width of control area b = 30 m.
1.1 1 0.9 0.8 0,7 0 0,5 - s - L / d = 1 , 0 - - » - L / d = 1 . 5 ^ - - L / d = 2 . 0 - s - L / d = 1 , 0 - - » - L / d = 1 . 5 ^ - - L / d = 2 . 0
\ \ \ ^
- * - L / d = 2 . 5\
\ ; • ; . p w . . . ; . , 0.01 0,02 0,03 0.04 0.05 0.06 0.07 0.08 2A/LFig. 11. Plate 1.2 m long. First-order force ratio for 1 = 0.88 s [L/d = 1), 1.07 s (L/d -¬ 1.5), 1.24 s (t/d = 2), and 1.39 s (L/d = 2.5).
4.1. Short plate case
First w e consider the early experiments at BGO-FIRST [1] w i t h the 1.2 m l o n g plate located at 19.3 m f r o m the wavemaker. This distance is taken as the i n t e r a c t i o n l e n g t h i n the calculations, w h i l e the w i d t h o f the n u m e r i c a l t a n k is taken equal t o the actual w i d t h of BGO-FIRST t h a t is 16 m . U n d e r these conditions i t is expected t h a t c o n f i n e m e n t effects do n o t come i n t o play and t h a t quasi-identical results are p r o v i d e d by b o t h n u m e n c a l models.
The tests w e r e r u n at w a v e periods o f 0.88, 0 . 9 8 , 1 . 0 7 , 1 . 1 6 , 1 . 2 4 , 1.32 and 1.39 s, m e a n i n g wavelengths f r o m 1.2 m up to 3 m , a n d at steepnesses H / L o f 2%, 3%, 4%, 5% and 6%. Calculations w e r e first r u n w i t h the c o n f i n e d m o d e l , at w a v e periods o f 0 . 8 8 , 1 . 0 7 , 1 . 2 4 and 1.39 s, f o r w a v e steepnesses f r o m zero (linear case) to 8% ( i t s h o u l d be noted that this h i g h steepness value is s o m e w h a t unrealistic at the l o w e r periods since b r e a k i n g was observed t o take place i n the experiments f r o m smaller H/l values). Fig. 11 shows the e v o l u t i o n o f the RAOs | f < ' ) | / ( p g A/d2) o f the " f i r s t - o r d e r " loads ( t a k i n g place at the wave f r e q u e n c y ) w i t h the increasing steepness, w i t h d the l e n g t h o f the plate (1.2 m ) . I n the calculations the pressure a c t i n g o n b o t h sides of the plate w a s accounted for. I n the figures the RAOs are n o r m a l i z e d w i t h t h e i r values at zero steepness t h a t is f r o m linearized t h e o r y (these values being, respectively, 0.318, 0.483, 0.617 and 0.860 f o r the 4 periods). It is s t r i k i n g that, except f o r the longest w a v e l e n g t h , the n o r m a l i z e d RAOs first increase w i t h increasing w a v e steepness, reach a m a x i m u m , and t h e n decrease d r a m a t i c a l l y b e l o w t h e i r linear values. This behavior is due to phase cancelation effects, as can be made clear f r o m the figures t h a t f o l l o w : Figs. 12 and 13 show, i n t h e 0.88s w a v e p e r i o d case, the RAOs o f the free surface elevations (Fig. 12) and t h e i r phases (Fig. 13), along the plate, f o r steepnesses o f 0% (linear case), 2%, 4%, 6% and 8%. Even t h o u g h the w i d t h - a v e r a g e d f r e e surface RAOs do increase w i t h increasing wave steepness, the phase differences along the plate also increase, leading to the local loads n o t s t r i c t i y a d d i n g up. I n M o l i n et al. [2] some c o m p a r a t i v e results are s h o w n b e t w e e n calculated and measured phase angles, w i t h excellent agreement ( w i t h a plate 3 m long), so w e are c o n f i d e n t that the phase angles are p r o p e r l y rendered b y the n u m e r i c a l m o d e l .
Fig. 14, also i n the 0.88 s w a v e period case, shows the firstorder force ratios vs. the w a v e steepness, as o b t a i n e d f r o m the c o n -fined w a v e - f i e l d (CWF) m o d e l a n d f r o m the i n f i n i t e w a v e - f i e l d (IWF) m o d e l . The t w o curves are very close, c o n f i r m i n g t h a t c o n f i n e m e n t effects are negligible i n this case o f large basin w i d t h and short i n t e r -action distance. The calculations that correspond to the i n f i n i t e w a v e
(I i , i .. \ 1
0 0.2 0.4 0.6 0.8 1 1.2 y
Fig. 1 2 . Plate 1,2 m long, RAO of the free surface elevation along the plate f o r r = 0.88
y
Fig. 1 3 . Plate 1.2 m long Phase (in radians) of the free surface elevation along the plate for r = 0.88 s.
I.K. Chatjigeorgiou, B. Molin / Applied Ocean Research 41 (2013)57-64 63
Table 1.
Long plate (5 m) in 1.01 s wave; first-order for ce ratio.
2.0% 2.5% 3.0% 3.5%
ll CWF IWF CWF IWF CWF IWF CWF IWF
25 m 0.850 0.848 0.643 0.650 0.482 0.492 0.363 0.376 50 m 0.747 0.748 0.551 0.556 0.389 0.396 0.222 0.225 75 m 0.732 0.731 0.542 0.545 0.394 0.399 0.239 0.235 100 m 0,727 0.727 0.541 0.549 0.405 0.413 200 m 0.732 0.719 0.565 0.553 300 m 0.664 0713 Table 2.
Long plate (5 m) in 1.01 s wave; drift force ratio.
2% 2.5% 3% 3.5% 25 m 0.997 0.975 0.974 0.985 50 m 1.163 1.166 1.175 1.215 75 m 1.281 1.239 1.199 1.141 100 m 1.341 1.257 1.172 200 m 1.403 1.272 300 m 1.534 0.01 0.02 0.03 0,04 0.05 0.06 0.07 0.08 2MI 1 1,7 1,6 1,5 1.4 1,3 1,2 1,1 1 - & - L / d = 1 . 0 : ! 1 i 1 1 . - « - L / d = 1 . 5 - ^ L / d = 2 . 0 - * - L / d = 2 . 5 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 2A/L
Fig. 14. Plate 1.2 m long. First-order force ratio for 7 = 0.88 s. Confined wave field vs. infinite wave field.
Fig. 15. Plate 1.2 m long Drift force rario for T = 0,88 s (L/d = 1), 1.07 s {L/d = 1.5). 1.24 s (L/d = 2), and 1.39 s (L/d = 2.5).
field approach have b e e n o b t a i n e d b y i n t e g r a t i n g the h y d r o d y n a m i c pressure o n the degenerated e l l i p t i c a l c y l i n d e r u s i n g e l l i p d c c o o r d i -nates. The expression t h a t provides the h y d r o d y n a m i c e x c i d n g forces, n o r m a l i z e d b y pgAitP is w r i t t e n as ƒ ( ! ) 0 -F kd ^ i ' " M c S , ! ' ( U o ; q ) c e „ ( « ; q ) m=0 71(Uo,V)ceni(V, q ) s i n v d v 2n (20) Y,i'"Ms'-m\uo;q)sem{ci:q) f A ( u o , v ) s e m ( v , g ) s i n v d v m=l ° -•j-:Ti"'BmMs'i^'>iuo;q) / s e m ( v , q ) s i n v d v '^^ m=i Jo
The t w o branches i n the above r e l a t i o n denote respectively the F r o u d e - K r y l o v and the d i f f r a c t i o n components.
Finally Fig. 15 shows the d r i f t force ratio vs. the w a v e steepness, at the same w a v e p e n o d s as i n Fig. 1 1 . The d r i f t force is calculated t h r o u g h d i r e c t i n t e g r a t i o n o f the h y d r o d y n a m i c pressure a c t i n g o n
b o t h sides o f the plate t h a t is w i t h the f o r m u l a
~ p j dz ƒ ^ d y { ( 0 - , y, z) - V0V</>' ( 0 + , y , z ) }
(21)
As i n Fig. 1 1 , w h a t is s h o w n is the n o n - d i m e n s i o n a l d r i f t force values Fd/{pgAjd) at the considered w a v e steepness n o r m a l i z e d b y t h e i r values at zero steepness ( f r o m l i n e a r theory, t h e values being, respectively, 0 . 9 9 3 , 1 . 0 0 3 , 0 . 9 6 4 and 1.070 f o r the 4 p e r i o d s ) . I t can be seen t h a t the d r i f t force ratios first increase, reach a m a x i m u m , and t h e n decrease as the w a v e steepness increases f r o m zero. I t is s t r i k i n g that, i n the 0.88 s case, at a steepness H/L o f 5%, the m e a n d r i f t force r a t i o has increased by a l m o s t 50%, w h i l e , f r o m Fig. 1 1 , t h e first-order f o r c e RAO has decreased by a b o u t 10%.
4.2. Long plate case
W e come back to the 5 m l o n g plate case t h a t was tested i n CE-HIPAR m o d e l basin. W e o n l y consider t h e 1.01 s w a v e p e r i o d case, and v a r y the w a v e steepness and the i n t e r a c t i o n l e n g t h .
64 I.K. Chatjigeorgiou, fi. M o / i i i /Applied Ocean Research 41 (2013)57-64
First w e present, i n Table 1, the f i r s t - o r d e r force RAOs, as o b t a i n e d f r o m the CWF a n d IWF n u m e r i c a l models. The i n t e r a c t i o n l e n g t h varies f r o m 25 m u p to 300 m , and the w a v e steepness f r o m 2% up t o 3.5%. RAOs o f the f r e e surface elevadon along the w e a t h e r side o f the plate have been s h o w n i n Figs. 2 - 8 . Empty boxes i n the table are cases w h e r e the i t e r a t i v e scheme d i d n o t converge, and w h e r e w e believe no steady state s o l u t i o n exists.
As i n Fig. 11, t h e q u a n t i t y s h o w n i n the table is t h e ratio o f the f i r s t o r d e r force RAO {F^^^/ipgAicP) at the considered steepness n o r -m a l i z e d b y the value (0.1017) p r o v i d e d b y linear t h e o r y .
I t can be seen that, except i n the 300 m l e n g t h case, the n u m e n c a l results p r o v i d e d b y the C W F and IWF models are r a t h e r close. A t a g i v e n i n t e r a c t i o n l e n g t h /, w h e n the steepness increases, the first-order force ratio decreases r a p i d l y . A t a g i v e n w a v e steepness, w h e n the i n t e r a c t i o n l e n g t h increases, the force ratio first decreases, reaches a m i n i m u m , and t h e n increases. The m o s t m i n i m u m value is a t t a i n e d at the highest steepness 3.5%, f o r an i n t e r a c t i o n l e n g t h o f 50 m : t h e force RAO is o n l y a b o u t 23% o f the linear t h e o r y RAO.
Table 2 shows t h e d r i f t force ratios obtained w i t h t h e CWF m o d e l . As i n Fig. 15, the q u a n t i t y t h a t is given i n the table is the ratio o f the n o r m a l i z e d d r i f t force Fii/{1/2pgAjd) at the said steepness b y its value (1.0027) at zero steepness (given b y linear t h e o r y ) .
5. Conclusions
The present paper o n the t h i r d o r d e r w a v e i n t e r a c t i o n w i t h v e r tical plates p e r f o r m s a dual role: first i t supplements previous c o n -t r i b u -t i o n s r e p o r -t e d by M o l i n e-t al. [1,3] w h i c h deal-t e x p l i c i -t l y w i -t h the r u n - u p o n the plates and secondly, presents f o r the first time ever n u m e r i c a l p r e d i c t i o n s o n the h y d r o d y n a m i c l o a d i n g ( f i r s t - h a r m o n i c and d r i f t forces) exerted o n the plates. The n o v e l p a r t o f the s t u d y relies o n t h e s o l u t i o n o f the t h i r d - o r d e r i n t e r a c t i o n p r o b l e m assum-i n g the plate b e assum-i n g exposed t o the actassum-ion o f m o n o c h r o m a t assum-i c waves i n an i n f i n i t e w a v e field. To this end, the plate was assimilated as a degenerated e l l i p t i c a l c y l i n d e r w i t h nearly zero s e m i - m i n o r axis.
The purpose o f the s t u d y was to capture the global effects o f t h e waves r e f l e c t e d b y the w a l l s t h a t confine t h e w a v e field i n the en-v i r o n m e n t o f an e x p e r i m e n t a l basin. In f a c t the same approach w a s taken b y M o l i n et al. [ 1,3] i n the n u m e r i c a l m o d e l t h e y developed and presented. In the BGO-FIRST experiments [ 1 ] , the plate (1.2 m l o n g ) was r e l a t i v e l y s h o r t c o m p a r e d to the w i d t h o f the basin (16 m ) . On the contrary, i n CEHIPAR e x p e r i m e n t s presented by M o l i n et al. [3] t h e plate (5 m ) w a s longer c o m p a r e d to basin's w i d t h (30 m ) , and the d i s -tance f r o m the w a v e m a k e r to the plate m u c h longer, r e n d e r i n g m o r e possible the d e t e r i o r a t i o n o f the wave field due to the reflections f r o m the lateral w a l l s .
The results o b t a i n e d f r o m the comparisons o f the t w o n u m e r i c a l models, n a m e l y t h e c o n f l n e d w a v e field and the i n f i n i t e w a v e field approaches i n d e e d c o n f i r m e d the r e f l e c t i n g effects to t h e global w a v e
field and accordingly to the w a v e r u n u p o n the vertical plate. H o w -ever, i t was noted t h a t i n m o s t o f the cases considered, the i m p a c t o f the lateral walls w a s negligible unless the w i d t h o f the basin is r e -duced or the distance f r o m t h e w a v e m a k e r is increased. Differences w e r e observed i n the n u m e r i c a l tests that s i m u l a t e d the CEHIPAR ar-rangement. It was f o u n d the 100 m t r a v e l i n g distance o f the i n c o m i n g waves f r o m the w a v e m a k e r t o w a r d the plate is quite short and i n -capable to a l l o w f o r t h i r d - o r d e r interactions b e t w e e n the i n c o m i n g and the reflected w a v e fields to f u l l y develop. Significant differences w e r e observed f o r a t r a v e l l i n g distance equal to 2 0 0 m and above. A s i m i l a r feature was n o t i c e d i n the BGO-FIRST tests f o r the smaller w a v e p e r i o d o f 0.88s (see Fig. 26 i n Ref. [ 1 ] ) .
Further, n e w results have been presented on the w a v e i n d u c e d loads t a k i n g place at the w a v e f r e q u e n c y and time-averaged. Some-w h a t unexpectedly, at variance Some-w i t h the m a x i m u m r u n - u p elevation, i t has been f o u n d that the response a m p l i t u d e operators (RAOs) o f the w a v e - f r e q u e n c y loads, at a g i v e n w a v e frequency, d o not increase steadily w i t h the w a v e steepness: they first increase, reach a m a x i -m u -m a n d then decrease to values possibly l o w e r t h a n p r e d i c t e d b y linearized theory. This feature is p a r t l y associated w i t h phase c o m -pensations along the plate. As f o r the n o r m a l i z e d d r i f t forces ( d i v i d e d by the w a v e a m p l i t u d e squared), they f o l l o w a s i m i l a r t r e n d s h o w -i n g local m a x -i m a t h a t s h -i f t to h-igher w a v e steepnesses f o r -increas-ing wavelengths.
Acknowledgement
The first author g r e a t i y appreciates the s u p p o r t p r o v i d e d b y É c o l e Centrale Marseille d u r i n g his sabbatical stay i n France.
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