Second order wave diffraction forces and runup by finite - infinite element method

(1)

Second order wave diffraction forces and runup by

finite - infinite element methodl

B. Y . L i * , S. L . L a u and C. O . N g

Department of Civil and Structural Engineering, Hong Kong Polytechnic, Hung Horn, Hong Kong

T h i s paper presents a finite-infinite element m e t h o d f o r s o l v i n g the second order wave d i f f r a c t i o n p r o b l e m . The f o r m u l a t i o n o f the p r o b l e m is based o n the inhomogeneous f a r field c o n d i t i o n a n d its c o r r e s p o n d i n g higher order a s y m p t o t i c solutions f o r the second order d i f f r a c t e d p o t e n t i a l suggested by L i ^ , and f o l l o w s the finite-infinite element m e t h o d as used b y L a u a n d Ji^^. T h e i n t e r p o l a t i o n i n the i n f i n i t e element a l o n g the r a d i a l d i r e c t i o n has a f a r field decay rate c o m p a t i b l e w i t h t h a t i n the higher order a s y m p t o t i c solutions. T h e p r o b l e m considered i n this paper is the d i f f r a c t i o n by a vertical surface-piercing c i r c u l a r cylinder. N u m e r i c a l solutions o f the second order d i f f r a c t e d p o t e n t i a l , satisfying a l l the g o v e r n i n g a n d b o u n d a r y c o n d i t i o n s as well as the inhomogeneous f a r field c o n d i t i o n , are o b t a i n a b l e w i t h the m e t h o d . N u m e r i c a l tests o f the m e t h o d give very g o o d results o f wave forces a n d r u n u p profiles w h e n compared w i t h those p u b l i s h e d by other researchers.

K e y "Words: N o n - l i n e a r wave d i f f r a c t i o n p r o b l e m , second o r d e r d i f f r a c t e d p o t e n t i a l , wave d i f f r a c t i o n force, wave r u n u p , finite-infinite element m e t h o d , inhomogeneous f a r field c o n d i t i o n , higher order a s y m p t o t i c solutions, s o m m e r f e l d r a d i a t i o n c o n d i t i o n .

L I N T R O D U C T I O N

1.1. The Diffraction Problem

The e v a l u a t i o n o f wave d i f f r a c t i o n l o a d o n offshore structures has been extensively studied i n the past few decades as the k n o w l e d g e is increasingly d e m a n d e d by the ocean a n d offshore engineering i n d u s t r y . The analysis o f first o r d e r d i f f r a c t i o n l o a d o n a vertical surface-piercing cylinder was first studied by Havelock^ a n d M a c C a m y and Fuchs^. The first order d i f f r a c t i o n p r o b l e m , w i t h the S o m m e r f e l d r a d i a t i o n c o n d i t i o n , is wellposed i n m a t h -ematical f o r m u l a t i o n . M a n y effective n u m e r i c a l tech-niques have already been developed.

O n the other h a n d , the second order d i f f r a c t i o n p r o b -l e m is f a r m o r e d i f f i c u -l t to f o r m u -l a t e and t o so-lve t h a n its first order c o u n t e r p a r t . G e n e r a l l y the h i g h l y non-linear wave d i f f r a c t i o n p r o b l e m is n o t solved directly, b u t is split by Stokes expansions i n t o a series o f linear problems. T h e Stokes expansions w i l l result i n inhomogeneous (i.e. o f non-zero r i g h t h a n d side) free surface c o n d i t i o n s f o r the second a n d higher order problems. T h e i n -homogeneous free surface b o u n d a r y c o n d i t i o n m a y be considered as a pressure disturbance o n the still water plane i n the Stokes wave f o r m u l a t i o n . T h e surface pres-sure disturbance i n the s o l u t i o n is o n l y possible t o account f o r if a n a p p r o p r i a t e f a r field b o u n d a r y c o n d i t i o n is used. T h e success o f a n o n - l i n e a r d i f f r a c t i o n m o d e l therefore counts very m u c h o n the f o r m u l a t i o n s o f the f a r field c o n d i t i o n s .

* P e r m a n e n t address: Researcli I n s t i t u t e o f E n g i n e e r i n g M e c h a n i c s , D a l i a n U n i v e r s i t y o f T e c h n o l o g y , D a l i a n , C h i n a

P a p e r a c c e p t e d J u n e 1990. D i s c u s s i o n closes S e p t e m b e r 1992.

270 Applied Ocean Research, 1991. Vol. 13, No. 6

1.2 Far Field Conditions for the Second Order Diffracted Potential

L e t us n o w consider o n l y the second o r d e r d i f f r a c t e d p o t e n t i a l « t j a n d review some o f the d i f f e r e n t f a r field b o u n d a r y c o n d i t i o n s t h a t have been p u t f o r w a r d b y various researchers.

(i) Y a m a g o u c h i and Tsuchiya^ used the f o r m 5

l i m r'"- ( , ; (1)

where k is the real r o o t o f the characteristic e q u a t i o n

k tanh(/<rf) = or ₍₂₎

I n e q u a t i o n (2), m is t h wave f r e q u e n c y a n d d is the w a t e r depth.

(ii) R a m a n a n d V e n k a t a n a r a s a i a h * used the f o r m

l i m r^l"-or

i)M 0 ₍₃₎

where X is the real r o o t o f t h e characteristic e q u a t i o n

9

X t a n h ( l r f ) ₍₄₎

(iii) C h e n and H u d s p e t h ' expressed <b\ as the s u m o f (t>'^2 a n d <!>Y, where cD^* is the d i f f r a c t e d p o t e n t i a l asso-ciated w i t h the inhomogeneous t e r m t h a t occurs i n the b o d y surface b o u n d a r y c o n d i t i o n , and is the d i f f r a c t e d p o t e n t i a l associated w i t h the i n h o m o -geneous t e r m that occurs i n the free surface b o u n d a r y c o n d i t i o n . T h e f a r field b o u n d a r y c o n d i t i o n s f o r the t w o separated potentials are

(2)

Second order wave diffraction forces and runup by firute ~ infinite element method: B. Y. Li et al. dr iX 2r $ f = 0 o f = 0 (5) (Ó) where X is the real r o o t o f e q u a t i o n (4), and g,,, is the r o o t o f f„{q,„a) = 0 (J„ is the Bessel f u n c t i o n of the first k i n d o f order n, and a is the radius o f the cylinder),

(iv) Molin*""^ considered that the i n f i n i t y a s y m p t o t i c b e h a v i o u r of the second order d i f f r a c t e d p o t e n t i a l c o m -prises t w o components:

ffi(0) cosh X(z + d)r~"^ e''-' 1) cosh X*{z + d) r

(7) g^{tJ) cosh /i*(z + d) r~"^ g - a i - r ^ u . „,

where lp represents free-waves and « t j i , is f o r waves l o c k e d t o the f i r s t order wave system. g^{9) is an u n k n o w n f u n c t i o n and 02(0) was given i n Refs 6, 7 and 8, k and X are the respective real roots of equations (2) and (4). A* is given b y

X* = kJ2 -f- 2 cos 0 ₍₉₎

(v) L i ' o b t a i n e d an inhomogeneous far field b o u n d a r y c o n d i t i o n by e x t e n d i n g the S o m m e r f e l d r a d i a t i o n c o n d i -t i o n :

l i m ,1/2 8^

dr iX^- F*iz, 9) e i*r(l+cos 0) (10) or

where F*(z,

iXl = F*(z, 0) r 1/2 g i M i + c o s 9 ) _|_ o ( r " i )

(11)

= {![/c(l + cos 0) - X-Jiico/g) cosh2(kd) X k\3 tanh\kd) - 1 - 2 cos 0 ] X A^Bo{9) cosh X*iz + d)]

/ [ A * sinh(A*rf) - ( 4 M V Ê / ) cosh{X*d)'] (12) I n e q u a t i o n (12), k and X are the real roots of equations (2) and (4) respectively, X* is given b y e q u a t i o n (9), d is the water depth, CÜ is the wave frequency, A a n d Bo{9) are k n o w n f r o m the first o r d e r far field solutions. L i ' also separated the f a r field s o l u t i o n of the second order d i f f r a c t i o n p r o b l e m i n t o t w o components. One is the free c o m p o n e n t $ 2 w h i c h satisfies the S o m m e r f e l d r a d i a t i o n c o n d i t i o n and the homogeneous free surface c o n d i t i o n . T h e other is the f o r c e d c o m p o n e n t 2 w h i c h satisfies a n inhomogeneous free surface c o n d i t i o n a n d has the same wave numbers a n d frequencies as those of the surface disturbances. T h e higher order a s y m p t o t i c solutions o f these t w o components are of the f o r m s :

e"-' cosh l ( z + rf) X B,{9)r 1/2-; (13)

+ e^''^ Y E,iz, 0)r

1 = 0

l / 2 - (

(14)

where F,(z, 0) and E,{z, 0) can be d e t e r m i n e d f r o m the first order f a r field potentials. L i ' w o r k e d o u t a recursion m e t h o d f o r solving F,(z, 0) a n d E,{z, 0) i n w h i c h they are given by solutions o f o r d i n a r y d i f f e r e n t i a l equations i n terms of t h e i r l o w e r order f o r m s . T h e solutions of the first t w o orders (/ = 0, 1) o f P",(z, 0) a n d E,(z, 0) were presented i n R e f 9. I t can be s h o w n t h a t w h e n the above higher order asymptotic solutions are reduced to the z e r o t h o r d e r (/ = 0), the results coincide w i t h those given by M o l i n « - « .

As expected, problems w i t h d i f f e r e n t f a r field b o u n d a r y c o n d i t i o n s w i l l produce d i f f e r e n t results, as different a s y m p t o t i c behaviour i n the f a r field is being represented. J u d g i n g f r o m the u n d e r l y i n g physical g r o u n d s o f the above conditions, we consider that the f a r field a s y m p t o t i c f o r m s by M o l i n ' ' " ^, a n d that b y L i ' are reasonable. These c o n d i t i o n s are capable of reflecting that there exists no o t h e r d i s t u r b i n g source at the i n f i n i t y t h a n that due to the inhomogeneous free surface c o n d i t i o n .

L3 Solving the Second Order Diffraction Problem

T h e second order d i f f r a c t i o n p r o b l e m can be solved i n t w o approaches. One a p p r o a c h is t o a v o i d finding the second order d i f f r a c t e d p o t e n t i a l , b u t to solve directly f o r the second order quantities f r o m the k n o w l e d g e of the first order potentials (Refs 6, 7, 8, 10 11, 12, 13). T h i s is accomphshed by using the Green's i d e n t i t y t o t r a n s f o r m the b o d y surface integral i n v o l v i n g the second order d i f f r a c t e d p o t e n t i a l i n t o an i n t e g r a l over the free surface i n v o l v i n g o n l y the first order potentials. H o w e v e r this m e t h o d still requires the f o r m u l a t i o n of the f a r field b o u n d a r y c o n d i t i o n f o r the second o r d e r d i f f r a c t e d p o t e n -t i a l . A n u m b e r of researchers w o r k e d o n -this a p p r o a c h , b u t few o f t h e m c o u l d give consistent results. I t is o n l y recently t h a t M o l i n • ' • ^ and E a t o c k T a y l o r a n d H u n g ' ° , w h o w o r k e d independently, c o u l d give results agreeing w i t h each other. W e consider t h a t t h e i r results are acceptable i n the sense that they have apphed reasonable a s y m p t o t i c solutions i n the free surface integrals. Recent-ly, E a t o c k T a y l o r , H u n g a n d C h a u ' \ have f u r t h e r developed the w o r k b y E a t o c k T a y l o r and H u n g ^ ° , a n d d e m o n s t r a t e d that the second order pressure d i s t r i b u t i o n o n the submerged surface o f a b o d y m a y be derived w i t h o u t h a v i n g to solve the b o u n d a r y value p r o b l e m f o r the second order p o t e n t i a l . T h e y presented, i n p a r t i c u l a r , an a n a l y t i c a l s o l u t i o n f o r a vertical cylinder. T h e i r m e t h o d is theoretically sound. H o w e v e r , it appears t h a t i n the case o f a r b i t r a r y objects, the m e t h o d is l i k e l y to be i n f l i c t e d by h a v i n g to solve f o r the h y p o t h e t i c a l p o t e n t i a l 1// a g o o d m a n y times.

T h e second a p p r o a c h is t o determine the second order d i f f r a c t e d p o t e n t i a l first, a n d next t o o b t a i n various second order variables w i t h the k n o w n potentials (Refs 3, 4, 5, 14, 15, 16, 17). T h e m a j o r d i f f i c u l t i e s i n this a p p r o a c h are the use of an a p p r o p r i a t e f a r field b o u n d a r y c o n d i t i o n a n d the p r o p e r treatment o f the c o r r e s p o n d i n g i n f i n i t y integral. O w i n g to differences i n these aspects, analyses i n this a p p r o a c h also lacked agreement w i t h one another i n their results. Recently, K i m a n d Y u e ' ' ' have managed t o show b y a n u m e r i c a l example that t h e i r results of wave forces are consistent w i t h those b y M o l i n ' ' ^ a n d E a t o c k T a y l o r a n d H u n g ' ° .

(3)

Second order wave diffraction forces and runup by finite - infinite element method: B. Y. Li et al. L4 The Present Study

I n this paper, we shall present an i n f i n i t e element m e t h o d f o r solving the second order wave d i f f r a c t i o n p r o b l e m i n the second approach. T h e m e t h o d employs the f a r field c o n d i t i o n (10), a n d the c o r r e s p o n d i n g higher o r d e r a s y m p t o t i c expansions (13) and (14) suggested b y L i ' , a n d f o l l o w s the finite-infinite element technique as a p p l i e d b y L a u a n d Ji'**. W e shall discuss at l e n g t h i n the f o l l o w i n g sections a b o u t the details o f the m e t h o d , i n c l u d i n g the d e r i v a t i o n s o f the far field c o n d i t i o n s a n d the finite-infinite element technique. T h e a i m o f this study is to c o m p u t e second order wave forces a n d r u n u p profiles a r o u n d a vertical surface-piercing c y l i n d e r i n regular waves. I n o u r m o d e l , the w a t e r region is reasonably d i v i d e d i n t o an i n n e r d o m a i n and a n i n f i n i t e outer d o m a i n , w h i c h are a p p r o x i m a t e d w i t h finite a n d i n f i n i t e elements respectively. T h e finite a n d i n f i n i t e elements are c o n f o r m i n g at the i n t e r b o u n d a r y between the t w o d o -mains. W e also compare o u r c o m p u t a t i o n a l results w i t h those p u b l i s h e d b y E a t o c k T a y l o r and H u n g ' ° , a n d w i t h the experimental results given by C h a k r a b a r t i a n d b y I s a a c s o n ' ' . I t w i l l be seen that o u r results are very satisfactory.

2. T H E N O N - L I N E A R D I F F R A C T I O N P R O B L E M 2.1 The Non-linear Problem

A s s u m p t i o n s o f ideal, incompressible fluid a n d i r r o t a -t i o n a l flow a l l o w us -t o express -the n o n - l i n e a r d i f f r a c -t i o n p r o b l e m as f o f l o w s : / V2(p = 0

it

+ 2 + ^ " (I) ( f l u i d d o m a i n ) (15) (z =

n)

(16) d}i d(p df] 1 d(p 5// ^ = 0 dz dn active d i s t u r b i n g source dji dz = 0 (z = ,/) (17) (z = - d ) (18) (19) (r ^ CO) (20)

where cp is the velocity p o t e n t i a l , g the g r a v i t a t i o n a l acceleration, d the water d e p t h , )/ the w a t e r surface elevation, T{t) an i n t e g r a t i o n constant, the b o d y surface, and (r, 6, z) the c y l i n d r i c a l coordinates. W e m a y view that there exists a d i s t u r b i n g source at i n f i n i t y w h i c h is responsible f o r generating waves p r o p a g a t i n g i n t o the r e g i o n o f o u r concern. F o r s i m p h c i t y , we assume that the d i s t u r b i n g source has the f o U o w i n g p r o p e r t i e s : (a) i t exists at X — 0 0 a n d generates o n l y plane disturbances; (b) i t is i n sinusoidal m o t i o n w i t h a single f r e q u e n c y ro; (c) its strength is k n o w n a n d finite. W e shall denote f o r b r e v i t y i n the ensuing b o n d a r y value p r o b l e m s a n active d i s t u r b -i n g source sat-isfy-ing the above propert-ies as :

d i s t u r b i n g source , (20)' (x ^ — 0 0 : plane, ro, finite) ^ ^

W e are g o i n g to develop the second o r d e r d i f f r a c t i o n p r o b l e m f o r m the non-linear b o u n d a r y value p r o b l e m (I).

F o r completeness, we shall give a f u l l account o f the procedure, t h o u g h readers m a y already be f a m i l i a r w i t h the problems f o r the first a n d second order i n c i d e n t potentials w h i c h w i l l appear i n the course o f p r o b l e m decompositions. W e also urge the readers to pay a t t e n t i o n to the far field b o u n d a r y c o n d i t i o n we specify i n each o f the f o l l o w i n g b o u n d a r y value p r o b l e m .

2.2 Decomposition of the NoJi-Iinear Problem T h r o u g h the Stokes expansions, P r o b l e m (I) m a y be decomposed i n t o a series o f linear p r o b l e m s o f d i f f e r e n t order, a n d the free surface c o n d i t i o n s are a p p r o x i m a t e d at z = 0. F o r the f o l l o w i n g problems, the water r e g i o n — d ^ z ^ O i s denoted by Q. T h e p e r t u r b a t i o n parameter £ is i d e n t i f i e d w i t h the wave steepness kH/2, where k is the wave n u m b e r a n d H is the wave height. W e w r i t e

(21) (22) (23) (p = BCPi £ (Pj + •• • )] = eiu + + ••• T = eT^+ £^T2 + . . .

a n d i d e n t i f y the first t w o decomposed p r o b l e m s as f o f l o w s : (1) F i r s t O r d e r P r o b l e m / V V l = 0 d^(p^ dcp^ dT^ (II) dt^ dcpi dz dn 9 dz = 9 dt (Q) (z = 0) (24) (25) 0 d i s t u r b i n g source

(x -» — (X): plane, ro, finite)

(z = - r f ) (26)

(S,) (27)

(r^co) (28)

T h e first order free surface displacement is then expressed as 'h _{g dt g} (2) Second O r d e r P r o b l e m / V > 2 = 0 (z = 0) (29) ( I I I ) / dt^ dz S<P2 dn 9 dz dT2 dt = 0 (Q) (z = 0) (z = (30) (31) -rf) (32) (33) n o d i s t u r b i n g source other t h a n f { ( p i ) ( r ^ o D ) (34) where I d(p^ d fd^ipi dcpi ct (V<Pi)' (35) T h e second order free surface displacement is

(4)

18(P2 CJ dt

Second order wave diffraction forces and runup by finite ~ infinite element method: B. Y. Li et al.

)/i ^ T2 ^^^^ Likewise, we m a y express 1 2g g dtdz

+

_(I (36)

2.3 The First Order Problem

T o f u r t h e r decompose the linear p r o b l e m i n t o the incidence and d i f f r a c t i o n problems, we express

(37) (38) (39) Tl = t [ +

and o b t a i n the f o l l o w i n g first order p r o b l e m s (1) F i r s t O r d e r Incidence P r o b l e m / v > l = 0 5 > i ( I V ) [I d(p[ dT[ 0 dt (fi) (z = 0) (40) (41) vz = 0 [ z = - d ) (42) d i s t u r b i n g source

(A- ^ - CO: plane, co, finite)

and the free surface displacement is expressed as (43)

a

5t

g

(z = 0) (44) (2) F i r s t O r d e r D i f f r a c t i o n P r o b l e m (V) 0 > i , d<p\ dt^ 9 dT\ dt ( f i ) (z = 0) (45) (46) dcp\ {z= - d ) (47) dn dcp\ dn no d i s t u r b i n g source ( ; • - CO (48) (49) and the free surface displacement is expressed as

.s

(z = 0) (50) g dt g

W e f u r t h e r assume t h a t the mean water level f a r away f r o m the b o d y is at z = 0. I t can be s h o w n that

T{ = t I = 0 everywhere (51) The s o l u t i o n o f the first order incident p o t e n t i a l is

le sinh(/<rf)

cosh k{z + d) sm{kx - mt) (52) where the wave n u m b e r k is given b y the real r o o t o f e q u a t i o n (2). T o separate the p o t e n t i a l f r o m the t i m e variable, e q u a t i o n (52) is r e w r i t t e n i n the f o r m 1 where ^ [ is the c o m p l e x p o t e n t i a l — ico a.; k^ smh{kd) cosh k{z + d) e"-' (53) (54) <p'i ((P^e- (55)

and recast P r o b l e m (V) i n terms o f its complex p o t e n t i a l «tl as / V ' O i = 0 ( V I ) 5 $ 2 dz d(!>l d<^\ dn or (Df = 0 ( f i ) (56) (z = 0) (57) { z = - d ) (58) V -l i m r'-l^ d<^\ 'dn' d(t>l (S,) ('• CO (59) (60)

where the S o m m e r f e l d r a d i a t i o n c o n d i t i o n (60) is em-p l o y e d as the far field c o n d i t i o n , as the o n l y d i s t u r b i n g source at the r i g h t side o f e q u a t i o n (59) radiates o u t - g o i n g waves f r o m the b o d y .

2.4 The Second Order Problem

I n P r o b l e m ( I I I ) , the free surface i n h o m o g e n e o u s t e r m f(q>i) m a y be decomposed i n t o three components accord-ing to the d e c o m p o s i t i o n o f cp^: fWx) = f'\<p\) + f"{(p[, + f'Hvl) (61) where

f'W

_{g dt dz\ dt^ ^ c}1 d(p[ d (8\p\ d(p[ (Vcp (62) g dt dz \ dt dz 1 dcp\ d (d'cp[ dcp\ g dt d z \ dt 2j^{v<p[-y<p\) (63) g dt dz\ dt 1 d(p\ d (d^(p\ (J dcp\ (V<p^)^ (64) at

T o o b t a i n the second order incidence a n d d i f f r a c t i o n problems, we first w r i t e 'I2 = '?2 + '?2 Ti + Tl (65) (66) (67) w h i c h a l l o w the p r o b l e m t o be decomposed as f o l l o w s : (1) Second O r d e r Incidence P r o b l e m

(5)

Second order wave diffraction forces and runup by finite - infinite element method: B. Y. Li et al. / V V ' = 0 I 1 (fi) ( V I I ) / + . ^ ^ r V j + . ^ ( . ^ 0 ) (69) dep'. dz 0 no d i s t u r b i n g source other t h a n ƒ (z = - d ) (70) ( r ^ c x D ) (71)

T h e c o r r e s p o n d i n g free surface displacement is \ djp\ 1 g dt Ti

'A

_{g dtdz}

+

(2) Second O r d e r D i f f r a c t i o n P r o b l e m (z = 0) (72) / V V 2 = 0 d ^ l , . d^l ( f i ) (73) dt^ +^ dz + ƒ ' ' ( < ? i) + g ( V I I I ) / dT\ ~dt 5)1 an n o d i s t u r b i n g source other \ t h a n f & n d f ' ' (z = 0) (74) { z = - d ) (75) (S,) (76) ( r ^ c o ) (77)

T h e c o r r e s p o n d i n g free surface displacement is 1

1 dq>\

g 5 f ö z 0 (z = 0) (78)

O n the a s s u m p t i o n t h a t the mean displacement i n water surface at i n f i n i t y is zero, we m a y determine that

n =

-= 0

2k sinh(2/frf) (79) (80) W i t h (p[ given by equations (52), P r o b l e m ( V I I ) has the s o l u t i o n cp'2 3(0 cosh 2k{z + d) sm2{kx - (at) (81) 8/(^ sinh-'ikd) Its c o r r e s p o n d i n g c o m p l e x p o t e n t i a l is 3icü l^i

S/c^ sinh''(/(rf) cosh 2/<(z + d)e^ (82)

S i m i l a r l y we express f'^ a n d f^^ as f o l l o w s :

f i s j ^ f s s ^ y i(F" + F''')e~^"" + ( f " + F^^)e2''°'] + gG (83) where 2g ^ d z \ d z g d (d<b\ 100 (S>, 2g ' d z \ dz + ^ ( V c D ' , . V ( D ^ )

g

or d fd\ dz im ^ d fd(b 4g dz \ dz

g J g

$ 1 S . 2 (84) (85) ioo - d ( d ^ , H f t I — — -Ag d z \ dz (86)

A c c o r d i n g t o e q u a t i o n (83), we m a y express the second o r d e r d i f f r a c t e d p o t e n t i a l (p\ as ^ \ e so that P r o b l e m ( V I I I ) is f u r t h e r decomposed i n t o 0 d<^% 4cö'

g

O", = + F^ (fi) (z = 0) (87) (88) (89) : )

( ^

an a n d (X) no d i s t u r b i n g source other t h a n and F^^ a ^ a ^ a57 = 0 0 (z = - r f ) (90) (S,) (91) (;• ^ CX3) (92) ( f i ) (93) (z = 0) (94) (z = - r f ) (95) no d i s t u r b i n g source other t h a n G (S,) (r ^ O ) ) (96) (97) 2.5 Remark

A t this p o i n t , we have reached the p r o b l e m f o r the second order d i f f r a c t i o n p o t e n t i a l . Before g o i n g any f u r t h e r , we wish to m a k e one r e m a r k here. T h e w e l l -k n o w n first and second o r d e r i n c i d e n t potentials given by equations (52) a n d (81) respectively carry

(6)

Second order wave diffraction forces and runup by finite ~ infinite element method: B. Y. Li et al. the significance o f the far field physical c o n d i t i o n s

we have prescribed at the beginning. I t is a c o m m o n oversight that m a n y people use d i r e c t l y the incident potentials, and the S o m m e r f e l d r a d i a t i o n c o n d i t i o n f o r the first order d i f f r a c t e d p o t e n t i a l w i t h o u t p a y i n g due regard to their relationships w i t h the c o r r e s p o n d i n g far field physical backgrounds. I n the f o l l o w i n g sections, we shall f u r t h e r develop the b o u n d a r y c o n d i t i o n s and the a s y m p t o t i c solutions i n the far field f o r the d i f f r a c t i o n problems. / V ^ c t f = 0 (XI) (

dm

dz

a

pis ^ pSS 0 ( - r f ^ z ^ 0, )• ^ R) (100) (z = 0, )• 5= R) (101) { z = - d , r ^ R ) (102) n o d i s t u r b i n g source other t h a n F'^ a n d f (/• -> co) (103) 3. FAIR F I E L D B O U N D A R Y C O N D I T I O N S A N D F A R F I E L D S O L U T I O N S

W e have seen i n Section 2 that the decompositions of the n o n - l i n e a r p r o b l e m (I) to the second order w i l l result in three problems yet t o be solved: the first order d i f f r a c t i o n p r o b l e m ( V I ) , the second order d i f f r a c t i o n p r o b l e m ( I X ) , and an a d d i t i o n a l p r o b l e m (X). L e t us first discuss a b o u t the far field b o u n d a r y c o n d i t i o n s and a s y m p t o t i c solutions f o r the t w o d i f f r a c t i o n problems. 3.1 The First Order Diffraction Problem (VI)

T h e f a r field b o u n d a r y c o n d i t i o n f o r P r o b l e m ( V I ) is the w e l l - k n o w n S o m m e r f e l d r a d i a t i o n c o n d i t i o n (60). T h e far field s o l u t i o n f o r the p r o b l e m can be w r i t t e n as

0\ = cosh k{z + d) H\l^ (/«-)Ki sin(«6

CO CO

+ C0S(,7Ö)] + X I COS /c„,(z + rf)

11 = 0 111 = 2

X ^„(/<„i'')Ca sin(nö) + /?„,„ cos(n0)] (98)

where k is the real r o o t of e q u a t i o n (2), ik,„ is an i m a g i n a r y r o o t o f e q u a t i o n (2), H , ' / ' is the H a n k e l f u n c t i o n o f the first k i n d of order n, and K „ is the m o d i f i e d Bessel f u n c t i o n o f t h e second k i n d o f order n. A n u m b e r of s i m p l i f i c a t i o n s m a y be made t o e q u a t i o n (98).

F i r s t , we m a y ignore the w h o l e t e r m i n d o u b l e sum-m a t i o n at the r i g h t side of e q u a t i o n (98) i n v i e w oïK„{k,„r) w h i c h decays i n the r a d i a l d i r e c t i o n a c c o r d i n g t o e"*™'' and therefore has l i t t l e significance at the i n f i n i t y .

I f the H a n k e l f u n c t i o n is replaced b y its a s y m p t o t i c expansions, e q u a t i o n (98) is f u r t h e r s i m p l i f i e d t o :

^ \ = A cosh k[z + d)e""- ^ B,{9)r~' 12-1 ₍₉₉₎

P r o b l e m ( X I ) does n o t have a b o u n d a r y c o n d i t i o n specified o n the inner surface r = R, so its s o l u t i o n is n o t unique. Suppose the s o l u t i o n o f P r o b l e m ( I X ) is given by $ 2 and a p a r t i c u l a r s o l u t i o n o f P r o b l e m ( X I ) is

difference between these solutions, as denoted b y :

0)^" = <s>l

has to satisfy the f o l l o w i n g homogeneous p r o b l e m / V ^ d ) ^ " = 0

dm

dz dz

a

0

m =

( - r f =^ z ^ 0, )• ^ R) 0 (z = 0, /• ^ R) -rf, r > R) n o d i s t u r b i n g source (f ^ c o ) (104) (105) (106) (107) (108) T h e f a r field c o n d i t i o n (108) o f P r o b l e m ( X I I ) is given by the Sommerfeld c o n d i t i o n : l i m } .1/2

dm

_ixm

0 (109) w h i c h , by equation (104), can be expanded i n t o :

l i m r ' / 2

dm

/ l d ) ' l i m ,.1/2

dm,

dr

im.

( U O )

T h e above e q u a t i o n gives an i n d i c a t i o n of h o w the far field c o n d i t i o n f o r the second order d i f f r a c t e d p o t e n t i a l cDj m a y be determined. As l o n g as we can find a p a r t i c u l a r s o l u t i o n f o r P r o b l e m ( X I ) , the r i g h t h a n d side of e q u a t i o n (110) wiU be k n o w n . O n the other h a n d , e q u a t i o n (104) suggests a basis o f separating the second order d i f f r a c t e d p o t e n t i a l i n t o t w o c o m p o n e n t s : one is the s o l u t i o n o f P r o b l e m ( X I I ) a n d the other is a p a r t i c u l a r s o l u t i o n o f P r o b l e m ( X I ) . I n v i e w o f the h o m o g e n e i t y o f the problems, these c o m p o n e n t s are called the free and the f o r c e d s o l u t i o n respectively. L e t us consider these solutions i n succession.

W h e n R is s u f f i c i e n t l y large, the s o l u t i o n o f P r o b l e m ( X I I ) m a y be expressed as:

I t s h o u l d be n o t e d t h a t the s o l u t i o n (99) satisfies o n l y the b o u n d a r y c o n d i t i o n s b u t n o t the Laplace e q u a t i o n . 3.2 The Second Order Diffraction Problem ( I X )

W e first consider the f o l l o w i n g p r o b l e m o f w h i c h the d o m a i n lies outside a c y l i n d r i c a l surface o f radius R enclosing the d i f f r a c t i n g b o d y :

m = cosh Aiz + rf) X (''''•)[a„i sin(n0) 11 = 0

CO CO

+ /5„i cos(n0)] + X Z cos 2,„(z + rf) 11 = 0 m = 2

X K„(2,„r)[a,„„ sin()70) + p,„„ c o s ( « 0 ) ] (111) or

(7)

Second order wave diffraction for C6S clud vuuup by fjuite

(I) cosh X{z + d)é'' Z C , ( 0 ) r - i 12-1

(112)

infinite element method: B. Y. Li et al.

where X is the real r o o t o f e q u a t i o n (4), a n d ü,,, is an i m a g i n a r y r o o t o f e q u a t i o n (4).

O n the other h a n d , w h e n R is sufficiently large, the expressions f o r the f i r s t order i n c i d e n t a n d d i f f r a c t e d potentials are:

(D'l = A cosh k{z + ^ f ) e * — "

(b\=A cosh k{z + d)e"" B,{9)r'' 12-1 (113) (114) where A and B , are k n o w n parameters. S u b s t i t u t i n g equations (113) a n d (114) i n t o equations (84) and (85), we have •1/2-/ pis ^ — cosh\kd)e'''^'^''°"'^ y G,(0)r 9 1 = 0 pss = ^ 2 ! ^ cosh2(/crf)e2'^^ X H,{e)r-'-9 1 = 0 where G,(0) = k\3 tanh^ikd) - 1 - 2 cos 0)B,(0) / I \ + 2ik - - / c o s 0 B , _ i ( 0 ) \ 2 / (115) (116) 2ik sin 9 dB,_,{9) d9 (117) and j = 0 3fc" 2 cosh2(/crf) 5;(Ö)5,_,.( + 7 / c ( - - / + 7 B , ( 0 ) - i ? , _ , _ i ( 0 ) + ik^--jjBj^,{9)B,^j{9) + •(^\-l+J^Bj_,i9)B,_j_,i9)

+

dBj_,i9) dB,_j_,{9) dO dB (118)

T o m a t c h the f o r m s of the f o r c i n g terms F^^ and F^^, i t is reasonable to let P r o b l e m ( X I ) have a s o l u t i o n (bl = A^'^cosh\kd)f ikr( \ + C O S 9)

1 = 0

I2~l

i = 0

(119) w h i c h has the same wave numbers as those o f the disturbances F'^ and F^^. Readers m a y find that e q u a t i o n (119) satisfies the far field physical c o n d i t i o n o f P r o b l e m s ( X I ) . S h o u l d there exist d i s t u r b i n g sources other t h a n F'^ and F^^ i n the far field, a free wave o f wave n u m b e r X must be present there. H o w e v e r , the f o r m o f e q u a t i o n (119) eliminates this possibility since a s o l u t i o n f o r free wave has n o t been i n t r o d u c e d i n t o the e q u a t i o n .

W e next p u t equations (115), (116) and (119) i n t o P r o b l e m ( X I ) and o b t a i n the f o l l o w i n g problems f o r the f u n c t i o n s F,(z, 0) and £ , ( z . Ö) (/ = 0, 1, . . . ) :

/

o ^ I dz' (X*yF, = l2ik([ + C O S 0 ) ( / - 1) + ik cos 0 ] f , . ( X I I I ) 7 + 2ik sin dF,_, 8'F,. 90 89' ( - d ^ z ^ O ) 8F, 4co' 8z a

V

8F, 8z F, = G, (2 = 0) V 0 (z = - d ) 'Z = 2 , 7 < ( 2 / - ! ) £ , _ ! - ( / - 1 ) 2 £ , _ 2 (120) (121) (122) ( X I V ) / 8z g E, = H, ( - r f ^ z ^ O ) (123) (z = 0) (124) dE, ~8^ = 0 [ z = - d ) (125)

where E _ i = E _ 2 = E - i = E - 2 = Ö - 1 ^ the above equa-tions, the f u n c t i o n s i n the 0-derivatives are o n l y o f (/ — l ) t h or (/ — 2)th order. The equations m a y therefore be regarded as o r d i n a r y d i f f e r e n t i a l equations i n z a n d can easily be solved in a recursion manner. T h e s o l u t i o n s f o r the first t w o orders (/ = 0, 1) are given below. F o r / = 0:

Fo(z, 0) = feo(0) cosh X*{z + d) (126) E^{z, 9) = do{9) cosh 2/c(z + d) (127)

where

bo{9) = Go(0)/[;.* sinh {X*d) - {4u>'/g) cosh (A*rf)] (128) d^{9) = Ho(0)/[2/< sinh (2/d) - {4oj'/g) cosh (2/crf)]

(129) F o r / = 1:

F,iz, 9) = ai(0) cosh A*(z + d) + a2(0)(z + d) X sinh X*(z + d) + a3(0)(z + df X cosh X*{z + d) (130) E i ( z , 0) = /3i(0) cosh 2/c(z + d) + (z + d) X sinh 2/<(z + d) (131) where «2(0) = Öi(0)/2A* - b,m2X*r (132) « 3 ( 0 ) = Ö2(0)/4A* (133)

(8)

Second order wave diffraction forces and runup by finite - infinite element method: B. Y. Li et al.

ai(ö) = { G i ( 0 ) - a2(0)[sinh {X*d) + X*d cosh (A*rf)

- {Aoj'lg)d s i n h (/l*d)] - a^ü-d cosh + X^d' sinh (A*rf) - {Aa)'lg)d' cosh

/ [ A * sinh {X*d) ~ {Aa>'lg) cosh (134) P2{0) = mdJ4li (135) ^^(0) = { H i ( 0 ) - i92(ö)[sinh (2/crf) + 2kd cosh (2/crf)

- {4oj'/g)d sinh (2/crf)]}

/[2fc s i n h (2kd) - {4(x>'/g) cosh (2]<d)] (136) and

i)i(0) = (7( cos 0bo(9) + 2/7c sin 0 ;: rf0

^2(0) = 2ik sin 0 bo{9)

dX{e)*

dB

We have n o w c o m p l e t e d the d e r i v a t i o n o f the f a r f i e l d s o l u t i o n f o r the second o r d e r d i f f r a c t e d p o t e n t i a l O j - I n s u m m a r y , we have considered O j as composed o f t w o c o m p o n e n t s : one is the free c o m p o n e n t w h i c h satisfies the homogeneous free surface c o n d i t i o n a n d the S o m -m e r f e l d r a d i a t i o n c o n d i t i o n ; the other is the f o r c e d c o m p o n e n t w h i c h satisfies an i n h o m o g e n e o u s free surface c o n d i t i o n a n d possesses the same wave n u m b e r s as those o f F'^ a n d F^^. T h e f o r c e d c o m p o n e n t can be d e t e r m i n e d i n the w a y as s h o w n above, whereas the coefficients o f the free c o m p o n e n t are left to be d e t e r m i n e d w h e n the second order d i f f r a c t i o n p r o b l e m is solved. F i n a l l y , we m a y develop the i n h o m o g e n e o u s f a r field c o n d i t i o n (10) or (11) b y using the expression (119) f o r the f o r c e d c o m p o n e n t t o o b t a i n

Hm ( .1/2

dm

ixm F*{z, 0) e ifcr( 1 + cos 9) _(103)'

where the expression (12) f o r F*(z, 0) is o b t a i n a b l e by t a k i n g i n t o account the equations (126), (128), (117), (113) a n d (114). W i t h equations (103)' and (110), the f a r field c o n d i t i o n s (10) a n d (11) are developed.

3.3 The Additional Problem ( X )

Suppose the first o r d e r s o l u t i o n i can be w r i t t e n i n the separable f o r m

$ 1 = Z(z) [/()•, 0) (139) where Z(z) is real a n d U{r, 0) is complex. W e then have

the i n h o m o g e n e o u s t e r m G ( $ J given b y e q u a t i o n (86) i d e n t i c a l l y equal t o zero. I n view o f this, we m a y d r a w t w o conclusions f o r P r o b l e m ( X ) :

(i) I f the b o d y is v e r t i c a l and p r i s m a t i c , the first o r d e r s o l u t i o n is expressible by e q u a t i o n (139). G{<b^) is therefore zero. P r o b l e m (X) has n o other d i s t u r b i n g source, so it can o n l y have a zero s o l u t i o n .

(ii) I n the general case, the first order s o l u t i o n i n the f a r field is also expressible b y e q u a t i o n (139). G(<Di) therefore diminishes at distance f a r away f r o m the b o d y .

T h e b o d y we consider i n this paper is a v e r t i c a l c i r c u l a r cylinder. W e m a y therefore ignore P r o b l e m ( X ) .

X

(138) pjg J Geometry definition sketch

4. T H E F I N I T E - I N F I N I T E E L E M E N T M E T H O D T h e c o m p u t a t i o n a l regions are depicted i n F i g . 1, the symbols b e i n g explained as f o l l o w s : a, = radius a n d surface o f the c y l i n d e r ; Q , , fi^ = i n n e r a n d outer water d o m a i n s ; b, Sj = radius and surface o f t h e i n t e r - b o u n d a r y between the i n n e r and the outer d o m a i n s ; Spj, Sp^; = free water surface i n the t w o d o m a i n s ; S^, S^p = b o t t o m surface i n the t w o d o m a i n s ; = surface o f a c y l i n d e r whose radius tends to be i n f i n i t e .

4.1 Functionals for the Diffracted Potentials W e m o d e l the i n n e r d o m a i n w i t h finite elements a n d the outer d o m a i n w i t h i n f i n i t e elements. T h e f u n c t i o n a l s Hj^ and n 2 o f the first and the second o r d e r d i f f r a c t i o n p r o b l e m s are decomposed i n t o c o m p o n e n t s i n the i n n e r a n d the outer d o m a i n s .

F i r s t order:

n, =

u,j + u,p

Second o r d e r :

n 2 = n 2 , + u^p These f u n c t i o n a l s are given b y

1 ' (140) (141) n „ ( < 5 ? ) 2 ,

+

1 2 (V(I)^)Mv ^ an ( V ( D i ) M v CO' 2 g J CO 2g [m.fds (142)

{^\Yds

Sp, ik ~2 {m^fds (143) n 2 ; = (va)^)Mv -203'

m

+ F ' + F^^)(D^c/s + 1 7 2 rf^S (144)

(9)

Second order wave diffraction forces and runup by finite - infinite element method: B. Y. Li et al. + + F'' W,ds 2üj' iX 0 . _|_ ^ * g ; * r ( l + c o s 9)^.-1/2 j ^ S (145) B y the f a r field r e l a t i o n s h i p O^ = O j ' + we f u r t h e r decompose T12E{^1) i n t o three c o m p o n e n t s

^2E{^1) = n 2 H i ( 0 f ) + U2,2im;, O)^) + U^p^i^,) (146) where 2oj' ' 1 2 (V<D^ydv -n„ g (147) ^2Ei^\) 2m' iA

1

r r {Vm )Mv - ^ g i'^iYds J S p , (1>2?rfs ' ' ' ' C ' * 2 ' S " . (153) T h e f u n c t i o n a l s (142), (143), (144) a n d (153) f o r m the basis f o r the finite-infinite element m o d e l . F o l l o w i n g the a p p r o a c h i n R e f 18, we shall first develop the f o r m u l a -t i o n s o f -t h e fini-te and -the i n f i n i -t e elemen-ts f o r -the second o r d e r p r o b l e m , a n d m a y then derive c o r r e s p o n d i n g f o r m u l a t i o n s f o r the first o r d e r p r o b g l e m by degenerating the second order results.

4.2 Modelling the Second Order Diffraction Problem I n the inner d o m a i n , the s t a n d a r d finite element m e t h o d is employed. T h e second o r d e r d i f f r a c t e d p o t e n -t i a l i n an elemen-t is (VO^ ) • ( V O 2 )dv E rr 4m' m +F' + F^'

w

:ds

^2EZ{'^1 + F'"

\m

ds -m l + F*e''"-<'+'°^'" r - " ' ) ^ f ds (148) S E E \ 3 iX ( V O ^ ' ) M v O^' + F' + F* e ' M i + c o s e ) , . - 1 / 2 U S ' (149)

As discussed i n Section 3, the f o r c e d c o m p o n e n t O 2 is already given by a p a r t i c u l a r s o l u t i o n , so the v a r i a t i o n a l o f n 2 £ 3 ( < l ' 2 ) m u s t be zero. T h e t h i r d f u n c t i o n a l can therefore be i g n o r e d . T h e first i n t e g r a l i n e q u a t i o n (148) m a y be expanded by the Green's i d e n t i t y : (Vd)^") • ( V D f )dv = (V^cDf)(D^" dv - I -

_m

ds (150) S u b s t i t u t i n g e q u a t i o n (150) back t o e q u a t i o n (148) a n d using the c o n d i t i o n s (100) to (102) a n d (103)', we o b t a i n n 2 £ 2 ( « > 2 ' ' , ^ 2 ) = (151) 5 * 2 S" ^ * 2 ds (152) A d d i n g e q u a t i o n (152) to e q u a t i o n (147) gives 1>2 = E N,<bl i= 1 (154)

where iV,. is the standard shape f u n c t i o n , O j ; are the n o d a l values, a n d n is the n u m b e r o f nodes i n the element. W i t h this i n t e r p o l a t i o n a n d e q u a t i n g the v a r i a t i o n a l o f the f u n c t i o n a l (144) to zero, we have the element coefficient m a t r i x a n d the right-side vector as f o l l o w s :

(VN,)-(ViVj.)dv 4m' g NiNjds N f d s + ( / , ; • = 1,2, . . . , H ) (155) (F'' + F'')N,ds 'e (r = l , 2 , . . . , ; ; ) (156) T h e outer i n f i n i t e d o m a i n £1^. is a p p r o x i m a t e d w i t h i n f i n i t e elements. A n i n f i n i t e element has its nodes located on the b o u n d a r y surface Sj a n d m e e t i n g the e x t e r i o r nodes of the c o r r e s p o n d i n g finite element. T h e i n t e r p o l a t i o n i n an i n f i n i t e element is c a r r i e d o u t b y t w o steps:

(i) O 2 is assumed o f the f o l l o w i n g f o r m a l o n g the r a d i a l d i r e c t i o n

{^iz,

e){r

/br'/'

+

D{z, e ) [ ( r / è ) - ' / 2 (157)

T h i s i n t e r p o l a t i o n f u n c t i o n corresponds to the first t w o terms i n the f a r field s o l u t i o n (112). T w o par-ameters, <bi,{z, 9) a n d D{z, 9), are i n t r o d u c e d so as to enhance the flexibility o f the m e t h o d . S u b s t i t u t i n g e q u a t i o n (157) i n t o e q u a t i o n (153), we have

(10)

Second order wave diffraction forces and runup by finite - infinite element method: B. Y. Li et al. 8^,80 80 [ c , m . \ c j + 2 ^ ' \8d + c 32 a o , 8D 1 Am' g 8z 82 \8z a o V d z ) \ds + C32(D,(0, 0)D(O, 0) + ^ C33D^(0. 0)1 dl 5 * 2 where 4 2^,2^ (158) (159) (160) C i 3 = -iX/2-ia-2X'b)Q^ ^3

+

X'b' + AiXb)Q2 b + 3iXb' 0 3 + 7 ^ ' 6 4 C21 = ^ ' 0 4 = bQ, - b'Q, C23 = 02 - 2^03 + b'Q, C31 = b'Q, C32 = ^'01 - b'Q, C33 = and e „ , = 2 I 2feQ, + ^ ^ 0 2 g 2 i A ( r - b ) j , - m J j . (161) (162) (163) (164) (165) (166) (167) (m = 1, 2, 3, 4) (168) (ii) W e next c a r r y o u t i n t e r p o l a t i o n s i n the z- and

0-directions. T o ensure c o m p a t i b i l i t y o f values on the b o u n d a r y surface Sj given by the f i n i t e elements a n d the i n f i n i t e elements, we w o r k as f o l l o w s . F r o m e q u a t i o n (157),

[ * 2 % = . = 3>.(z, d) (169) b y w h i c h we can w r i t e

(170) Hence the i n t e r p o l a t i o n can be applied to ['ï>2]r=6:

(171)

The 2 - D shape f u n c t i o n is given by the degener-a t i o n o f the 3 - D shdegener-ape f u n c t i o n of the i n n e r finite element to the b o u n d a r y surface Sj. I n this way, consistent values o f O j are considered o n Sj. The f u n c t i o n s ( t . and D are n o w of the f o r m s

©.(z, 0) = X M , * !

-D{z, 0) = X M A

(172)

(173)

P u t t i n g equations (172) and (173) i n t o e q u a t i o n (158) and e q u a t i n g the v a r i a t i o n a l of to zero, we get the e q u a t i o n f o r the i n f i n i t e element:

LK2rl [ X 2 2 ] . [ f f ]

[•B2] The n o d a l vectors are:

CP s i r 2m} T (174) (175) (176) [ C ] = {3>21<1>22 [ Z ) J = {D,D,--- D,„r

Elements i n the coefficient m a t r i x are (i,j = 1 , 2 , . . , m): 4m' Kl UJ ^ i i A j + C 21 ^eij + hij g Kl2iJ — K.2lji Am' C , 2 f j + C22l„,,+ C:,,! 22 ' 3 2 ^ z i j C,2JiJ

K-22ij — ÎZîj + C 2^1 Bij + C^Îîj

(177)

(178)

Am'

g ^33-^ ij (179) The right-side elements are (/' = 1, 2, . . . , m):

Am' g ^ 3 1 0 1 , - + g2i (180) Ao? C320U I n equations (177) to (181), J i j -M,Mjds 8M, 8Mj 1)6 ^ 8M, 8M: ds ds M.Mjdl MiQ>2 ds

K =

8M,

dm

1)6 ^ ds (181) (182) (183) (184) (185) (186) (187)

(11)

Second order wave diffraction forces and runup by finite - infinite element method: B. Y. Li et al. 'J^ dz ' J z ' ' ^ ' gu = 92i = M , $ f dl

dm

M , ds (188) (189) (190)

I n the above equations f o r the i n f i n i t e element, whenever the f o r c e d p o t e n t i a l O f is r e q u i r e d , we calculate i t b y e q u a t i o n (119) b u t o n l y u p t o the first t w o r r d e r s (/ = 0, 1). T h e accuracy is acceptable because the t r u n c a t i o n e r r o r is 0{r~^''), w h i c h is the same as i n the $2" i n t e r p o l a t i o n f u n c t i o n (157). 4.3 Modelling the First Order Diffraction Problem

F o r m u l a e developed above f o r s o l v i n g the second o r d e r d i f f r a c t i o n p r o b l e m can easily be reduced t o the first o r d e r case.

T h e c o r r e s p o n d i n g inner finite element f o r m u l a e f o r the first o r d e r d i f f r a c t e d p o t e n t i a l are o b t a i n e d f r o m equations (155) a n d (156) by

(i) r e p l a c i n g 4m'/g i n e q u a t i o n (155) w i t h m'/g, (ii) r e p l a c i n g dm/Sn i n the first i n t e g r a l i n e q u a t i o n (156)

w i t h dm/dn, a n d

(iii) r e m o v i n g the second i n t e g r a l i n e q u a t i o n (156). T h e c o r r e s p o n d i n g outer i n f i n i t e element f o r m u l a e f o r the first o r d e r d i f f r a c t e d p o t e n t i a l are o b t a i n e d f r o m equations (159) to (168) a n d f r o m equations (174) to (190) by

(i) r e p l a c i n g A i n equations (159) to (168) w i t h k, (ii) setting the r i g h t h a n d sides o f equations (168) t o (190)

to zero, thus m a k i n g the vectors [ B J a n d [B,'] zero, a n d

(iii) r e p l a c i n g 4coVfif i n equations (177) to (179) w i t h o?/g.

5. C O M P U T A T I O N S O F W A V E F O R C E S A N D W A V E R U N U P

5.1 Wave Forces T h e fluid pressure is

P= -pgz-p^~^{Wcpr + pT{t) (191)

B y Stokes expansions a n d i n t e g r a t i n g the pressure over the b o d y surface, we get the wave f o r c e c o m p o n e n t s :

f = fo +A+f2 + 0{E') where — pgz ri dz 8(Pi , p " ds (192) (193) (194) f2 p -r^ n ds d(p2 ^ , s dt «. 2g\dt {Vcpifn ds pT^n ds (195)

So is the b o d y surface b e l o w z = 0, 1^ is the perimeter o f the b o d y section at z = 0, « is the i n w a r d n o r m a l vector, a n d T[ is given by e q u a t i o n J 7 9 ) . f , is a c t u a l l y the h y d r o s t a t i c force. E x p r e s s / ^ a n d i n complex f o r m s :

f , = Re{F, e-'""} (196) / ; = Re{(F22 + F 7 ) e - 2 . - - ) (197) where E J = —ipas - lipojs' C&iH ds <b.iï ds (198) (199) ^ 2 1 - 4 {SJ<b,fn ds f20 = .2 r (<I)i)'« ds (200) (V<I)i)-(V<I)i)n ds 2„2 PCO E + P^'T[[t) <b,^fi ds w n ds (201)

F o r the sake o f the discussions o n the n u m e r i c a l results, we define t h a t :

(a) i T = / i + / 2 (202) (b) w h e n a force variable is s t r i p p e d o f the vector

n o t a t i o n , i t refers t o the x - c o m p o n e n t o f the force, (c) the dimensionless force coefficients are

{CF,, CF22, CF,,, C f , „ Cf^)

= {F„ E22, E21, f2o,fj)/pgHad 5.2 Wave Runup

T h e water surface elevation above z = 0 is \ dw (7 „ , ' / = - - - ^ - f ( V ^ ) ^ + T ( r ) / , B y Stokes expansions where 'h = (203) (204) (205) (206) 1 g " dtdz 2k sinh(2/crf) (207) Express i], a n d i], i n c o m p l e x f o r m s :

1],= Re{Ei e~""} (208)

)/2 = Re{(S22 + S„)e-""'} + ,,20 (209) where

^,=-m

(210)

(12)

Second order wave diffraction forces and runup by finite - infinite element method: B. Y. Li et a l .

Table 1 Computed wave forces for various values o f k b (ka = / , H/a = 2, d/a = 1)

kb n x n x n r Ö z F 1 tN) F 21 (N) 20 F 22 (N) CF 22 kb n x n x n r Ö z 20 Re Im Re Im Re Im MD AG

1 . 01 4x15x2 1 155E4 -3 090E4 -9 205E3 -6 746E3 1 .037E4 1 018E3 -9 169E2 6 815E-2 -0 7332 1 .25 8x15x2 1 155E4 -3 090E4 -9 203E3 -6 727E3 1 .037E4 1 782E3 - 1 190E4 5 985E-1 -1 4220 1 .50 8x15x2 1 155E4 -3 090E4 -9 205E3 -6 724E3 1 037E4 7 057E3 - 1 498E4 8 236E-1 -1 1280 1 75 8x20x2 1 155E4 ~3 090E4 -9 205E3 -6 724E3 1 037E4 1 003E4 -1 340E4 8 326E-1 -0 9283 2 00 8x20x2 1 155E4 -3 090E4 -9 206E3 -6 725E3 1 037E4 1 003E4 -1 153E4 7 601E-1 -0 8549 2 25 8x20x2 1 155E4 -3 090E4 -9 206E3 -6 725E3 1 037E4 9 040E3 - 1 111E4 7 124E-1 -0 8878 2 50 8x28x2 1 155E4 -3 090E4 -9 206E3 -6 725E3 1 037E4 8 548E3 - 1 164E4 7 183E-1 -0 9374 2 75 8x28x2 1 155E4 -3 090E4 -9 206E3 -6 725E3 1 037E4 8 781E3 - 1 204E4 7 412E-1 -0 9407 3 00 8x28x2 1 155E4 -3 090E4 -9 206E3 -6 725E3 1 037E4 9 102E3 - 1 196E4 7 476E-1 -0 9203 3 25 8x28x2 1 155E4 -3 090E4 -9 206E3 -6 725E3 1 037E4 9 088E3 - 1 170E4 7 369E-1 -0 9104 3 50 8x28x2 1 155E4 -3 090E4 -9 205E3 -6 725E3 1 037E4 8 855E3 - 1 166E4 7 283E-1 -0 9213 3 75 8x28x2 1 155E4 -3 090E4 -9 205E3 -6 725E3 1 037E4 8 743E3 - 1 186E4 7 329E-1 -0 9355 4 00 12x36x2 1 155E4 -3 090E4 -9 206E3 -6 725E3 1 037E4 8 886E3 -1 206E4 7 451E-1 -0 9358 4 50 12x36x2 1 155E4 -3 090E4 -9 205E3 -6 725E3 1 037E4 9 192E3 -1 176E4 7 424E-1 -0 9074 5 00 12x36x2 1 155E4 -3 090E4 -9 205E3 -6 725E3 1 037E4 8 817E3 -1 166E4 7 271E-1 -0 9234 5 50 12x36x2 1 155E4 -3 090E4 -9 205E3 -6 725E3 1 037E4 8 901E3 - 1 200E4 7 432E-1 -0 9326 6 00 12x36x2 1 155E4 -3 090E4 -9 204E3 -6 725E3 1 037E4 9 095E3 -1 180E4 7 410E-1 -0 9141 6 50 12x36x2 1 155E4 -3. 090E4 -9 203E3 -6 724E3 1 038E4 8 919E3 -1 177E4 7 345E-1 -0 9223 7 00 12x36x2 1 155E4 -3. 090E4 -9 203E3 -6 724E3 1 038E4 8 961E3 -1 185E4 7 390E-1 -0. 9233 7 50 12x36x2 1. 155E4 -3. Q90E4 -9 202E3 -6. 724E3 1 038E4 8 968E3 - 1 . 181E4 7 376E-1 -0. 9213 8 00 12x36x2 1. 155E4 -3. 090E4 -9. 200E3 -6 724E3 1 038E4 8 968E3 -1 180E4 7 372E-1 -0 9209 Ref. 10 1 . 155E4 -3. 090E4 -9. 207E3 -6 724E3 1 037E4 8 937E3 -1 183E4 7 375E-1 -0. 9238

2!ft) O , (211) j _ 4 ^ i?ianh\kd) - ( V ^ i ) ' 2 ^ " ^ ^ * ^ ' ^ ' ^ ^ ^ ^ ^ '?20 1 ^ ^ k\aXi\v\kd) ^ -4i/ 2g T h e r u n u p a r o u n d the c y h n d e r is g i v e n b y R(0) = max(C)/(c = a, 6, t)) 5.3 Numericcd Examples (215) 1 2k s i n h {2kd) (213)

W e define the dimensionless surface e l e v a t i o n coefficients as

( C S i , CS22, C H 2 1 , C/I20, Cij)

= {eE„ s'E,„ s'E,„ shuo,'iVH (214)

W e have w o r k e d o u t three n u m e r i c a l examples i n o r d e r to give results o f wave forces a n d w a v e r u n u p p r o f i l e s t o c o m p a r e w i t h those p u b l i s h e d b y o t h e r researchers. W e e m p l o y e d 20-node i s o p a r a m e t r i c finite elements a n d 8-node i s o p a r a m e t r i c i n f i n i t e elements. B y s y m m e t r y o f the p r o b l e m , we considered o n l y h a l f o f the d i f f r a c t i o n d o m a i n i n the c o m p u t a t i o n s . I n the d a t a we s h o w b e l o w , we d e n o t e the n u m b e r s o f finite elements a l o n g the r-, 6-, a n d z-directions i n the i n n e r d o m a i n b y n,. x n^ x n^. T h e mesh is u n i f o r m i n each d i r e c t i o n . 0 . 9 0 - | 1 1 1 1 1 1 1 3 5 7 kb

(13)

Second order wave diffraction forces and runup by finite - infinite element method: B. Y.

Table 2 Comparison with wave forces of Eatoclc Taylor & Hung'"

Li et al. ka Ref. 10 _(N) F^^ (N) f (N) F^^ (N) CF 2 2 ka . , n ,xn xn kb r 9 z _{Re Im Re Im}2 0 _{Re Im MD AG}

0. 125 Ref. 10 y.837El -7.957E3 -2.339E1 -1.936E3 1.193E2 3.9nE4 -7.755E5 3.862E1 -1.520 0. 125

3.0| 15x24x1 9.711E1 -7.907E3 -2.367EI -1.871E3 1. 191E2 3.875E4 -7.711E5 3.840E1 -1.521 0.25 Ref. 10 7.856E2 -1.590E4 -1.769E2 -3.722E3 9.0O8E2 7,144E4 -3.649E5 1.849E1 -1.377 0.25

3.0| 15x24x1 7.844E2 -1.589E4 -1.774E2 -3.699E3 9.001E2 7.183E4 -3.653E5 1.852E1 -1.377 0.50 Ref. 10 5.232E3 -2.880E4 -1.313E3 -5.483E3 5.321E3 4.477E4 -9.992E4 5.446E0 -1.150 0.50

6.0| 15x30x1 5.225E3 -2.878E4 -1.314E3 -6.464E3 5.324E3 4.480E4 -9.995E4 5.448E0 -1.149 0.75 Ref. 10 1.057E4 -3.305E4 -4.284E3 -7.535E3 9.796E3 1.140E4 -3.300E4 1.737E0 -1.238 0.75

6,0| 15x30x1 1.056E4 -3.305E4 -4.281E3 -7.530E3 9.799E3 1.140E4 -3.318E4 1.7C5E0 -1.240 1. 00 Ref. 10 1.155E4 -3.090E4 -9.207E3 -6.724E3 1.037E4 8.937E3 -1.183E4 7.375E-1 -0.9238 1. 00

6.0| 12x36x2 1.155E4 -3.090E4 -9.204E3 -6.725E3 1.037E4 9.095E3 -1.180E4 7.410E-1 -0.9141 1.25 Ref. 10 8.8S9E3 -2.717E4 -1.459E4 -4.477E3 8.921E3 1.804E4 -3.038E3 9.100E-1 -0.1668 1.25

6.0| 12x36x2 8.888E3 -2.717E4 -1.459E4 -4.484E3 8.922E3 1.812E4 -2.760E3 9.117E-1 -0.1512 1. 50 Ref. 10 5.009E3 -2.354E4 -1.796E4 -1,314E3 7.843E3 2.687E4 -1.116E3 1.338E0 -0.04151 1. 50

6,0| 12x36x2 5.009E3 -2.354E4 -1.796E4 -1.323E3 7,840E3 2.691E4 -7.291E2 1.339E0 -0.02709 1.75 Ref. 10 1.239E3 -2.017E4 -1.735E4 2.668E3 7.481E3 2.888E4 -5.433E3 1.462E0 -0,1859 1.75

6.0| 12x36x2 1.239E3 -2.017E4 -1.735E4 2.658E3 7.470E3 2.895E4 -5.073E3 1,462E0 -0,1734 2.00 Ref. 10 -1.939E3 -1.696E4 -1.270E4 7.310E3 7.229E3 2.262E4 -1.531E4 1,359E0 -0,5950 2.00

6.0[ 12x36x2 -1.939E3 -1.696E4 -1.271E4 7.299E3 7.206E3 2.284E4 -1.502E4 1,360E0 -0,5817 2. 25 Ref. 10 -4.404E3 -1.390E4 5.580E3 1.172E4 6.903E3 1.022E4 -2.753E4 1,461E0 -1,215 2. 25

6.0| 12x36x2 -4.402E3 -1,390E4 5.611E3 1.171E4 6.863E3 1.066E4 -2.730E4 1.458E0 -1.199 2. 50 Ref. 10 -6.167E3 -1.097E4 -1.835E3 1.449E4 6.683E3 -4.647E3 -3.683E4 1,846E0 -1.696 2. 50

6.0( 12x36x2 -6.166E3 -1.097E4 -1.801E3 1.447E4 6.615E3 -4.089E3 -3.676E4 1.840E0 -1,682 2.75 Ref. 10 -7.289E3 -8.207E3 -7.728E3 1.421E4 6.604E3 -1.873E4 -3.790E4 2,109E0 -2.028 6.0| 12x36x2 -7.288E3 -8.206E3 -7.706E3 1,419E4 6.499E3 -1.847E4 -3.781E4 2.093E0 -2.023 3. 00 Ref. 10 -7.847E3 -5.650E3 1.124E4 1.034E4 6.551E3 -2.995E4 -2.811E4 2.043E0 -2.388 6.0| 12x36x2 -7,846E3 -5.643E3I 1.124E4 1.036E4 6.404E3 -3.033E4 -2.740E4 2.033E0 -2.407

iJ Wave Forces at Various kb and ka

E a t o c k T a y l o r a n d H u n g " presented results o f c o m -p u t e d forces a c t i n g o n a c y l i n d e r o f r a d i u s a = I m b y incidence wave o f h e i g h t ƒƒ = 2 m i n v a r i o u s w a t e r d e p t h d.

F o r c o m p a r i s o n purposes, we have generated results o f forces w i t h H/a = 2 a n d dja = \ . F i r s t l y we w i s h t o examine h o w the results m a y be affected b y the choice o f t h e d i m e n s i o n o f the i n n e r d o m a i n , w h i c h is represented b y the r a d i u s b. I n T a b l e 1 we present forces we o b t a i n e d at v a r i o u s values o f the dimensionless p a r a m e t e r kb w i t h ka fixed at one. A t the end o f the table, we present also the resuhs given b y E a t o c k T a y l o r and H u n g ' ° . I t can be seen fiom T a b l e 1 t h a t the first o r d e r f o r c e F , a n d the second o r d e r forces a r i s i n g f r o m the first o r d e r

p o t e n t i a l F , , a n d f , ^ have very shght dependence o n kb. H o w e v e r the o t h e r second o r d e r f o r c e F , , oscillates w h e n kb is s m a l l . I n F i g . 2 we s h o w the v a r i a t i o n s w i t h kb o f the m o d u l i | CF,, \ a n d | CS221. I t is evident t h a t c o n -vergence o f these q u a n t i t i e s is p r a c t i c a l l y achieved w h e n kb is greater t h a n 6.

W e present i n T a b l e 2 the forces we c o m p u t e d at d i f f e r e n t ka values a n d the c o r r e s p o n d i n g results given b y R e f 10. I t is seen t h a t the agreement between these t w o sets o f results is excellent, the difference b e i n g less t h a n one percent.

ii) Wave Runup Profiles

I s a a c s o n ' ' c a r r i e d o u t tests t o measure wave r u n u p p r o f i l e s a r o u n d a c y l i n d e r i n the s h a l l o w water range.

1 . 4 1 . 3 1.2 1. ! 1 0.9 0. 8 0. 7 0.6 0.5 N O N - L I N E A R (kd - 0 . 5 ) K O N - L I N E A R (kd - 0 . 3 3 ) T E S T B Y I S A A C S O N " L I N E A R 180

Fig. 3 Wave runup profile around cylinder (ka = 0.4, h = 1.45)

(14)

and checked his results w i t h the predictions b y the c n o i d a l wave d i f f r a c t i o n theory. O v e r the range of c o n d i t i o n s covered i n his experiments, o n l y t w o parameters are o f i m p o r t a n c e : ka and A = H/{k'd^). I n the paper R e f 19, f o u r graphs are p r o d u c e d t o indicate the measured r u n u p profiles f o r c o n d i t i o n s : (a) ka = 0.40, A = 1,45; (b) ka = 1.00, A = 1.45,(c)/ca = 0.52, A = 2.14;(d)/<fl = 1.30, A = 2.14.

W e c o m p u t e d wave r u n u p profiles c o r r e s p o n d i n g t o the above f o u r sets o f c o n d i t i o n s . I n each case we consider t w o values o f the d e p t h parameter, kd = 0.33 a n d 0.5, w h i c h are w i t h i n the range o f kd i n the tests. O u r results together w i t h the measured profiles a n d the predictions by the linear t h e o r y are s h o w n i n Figs 3 t o 6. As seen

f r o m the figures, o u r c o m p u t e d profiles agree very w e l l w i t h the measured profiles, and certainly far better t h a n the hnear theory predictions do.

iii) Maximum Wave Forces

C h a k r a b a r t f gave a few experimental data o n m a x i -m u -m wave force f o r d/a{ = 1.16 and H/d = 1/4 ~ 1/5. W e c o m p u t e d the wave forces f o r d/a = 1 . 1 6 a n d H/d = 0.5 X (1/4 + 1/5) = 0.225. T h e results are given i n T a b l e 3. T h e m a x i m u m dimensionless t o t a l force coeffi-cients I C / j . I are p l o t t e d against ka i n F i g . 7, so as to compare w i t h C h a k r a b a r t i ' s d a t a a n d w i t h the linear t h e o r y predictions. A g a i n o u r results agree w e l l w i t h the experimental data.

(15)

Table 3 Computed wave forces for various values o f k a (d/a = 1.16, H/d = 0.225, kb = 6.0) ka HD AG 1.417 - 1 . 3 9 2 1.643 -1.261 1.526 - 1 . 2 1 3 1.295 - 1 . 2 5 5 1.072 -1.361 0.890 - 1 , 5 1 0 0.745 - 1 . 6 8 5 2 1 MD AG MD 0.4255 AG -1.116 Cf 20 ( C f , ) MD AG ( C f ) , T « l n MD AG ICf^l AG 0.50 0.75 1.00 1 .25 l . S O 1.75 2.00 10x31x1 12x36x1 12x36x1 12x36x2 12x36x2 12x36x2 12x36x2 0.0389 - 1 . 7 5 6 0.0523 - 2 . 0 3 6 0.0668 -2.451 0,0876 - 2 , 8 1 2 0,1031 - 3 , 0 5 7 0.1007 - 2 , 9 9 2 0.0842 - 2 , 6 2 4 0,1204 -1.111 0.0674 - 0 . 3 1 6 0.1227 0.081 0.1733 0,039 0,1858 - 0 . 1 6 0 0.1732 - 0 . 5 8 0 0.0291 0.0525 0.0546 0.0466 0,0410 0.0393 0,0382 1.650 - 0 , 9 4 9 1.782 - 1 . 1 1 5 1.616 - 1 . 1 4 7 1.323 -1.191 1.053 - 1 . 2 8 8 0.850 - 1 . 4 2 3 0.708 - 1 . 5 2 4 - 1 . 6 1 8 - 1 . 5 5 3 - 1 . 4 4 3 -1.271 - 1 . 0 6 9 - 0 . 9 3 3 -0.791 1.329 1.687 1.850 1.832 1.734 1.593 1.392 1.650 - 0 . 9 4 9 1.782 - 1 . 1 1 5 1.616 - 1 . 1 4 7 1.323 -1.191 1.096 1.734 0 . 9 3 3 1.593 0.791 1.392 6. C O N C L U S I O N S

element m o d e l t o c o m p u t e second o r d e r d i f f r a c t i o n forces I n the f o r e g o i n g , we have presented a f i n i t e - i n f i n i t e a n d r u n u p profiles o n a vertical cylinder i n regular waves.

k a Fig. 7 Maximum total wave force versus ka

(16)

O u r approach o f a n a l y z i n g the wave d i f f r a c t i o n p r o b l e m t o the second order is also discussed i n detail. W e have utilized the inhomogeneous f a r f i e l d b o u n d a r y c o n d i t i o n s and the higher order a s y m p t o t i c solutions f o r the second order d i i f r a c t e d p o t e n t i a l , as suggested by L i ' . The i n f i n i t e elements we have used are capable o f representing the decay rate o f the d i f f r a c t e d potentials i n the f a r field. T h e integrals i n the i n f i n i t e element coefficient m a t r i x are a l l expressible by explicit f o r m u l a e .

O u r n u m e r i c a l examples have yielded very g o o d results when c o m p a r e d w i t h other c o m p u t a t i o n a l results and w i t h some measured data. The v a l i d i t y o f o u r m o d e l is therefore demonstrated. H o p e f u l l y , o u r w o r k w i l l a d d t o the knowledge o f solving the second order wave d i f f r a c -t i o n p r o b l e m .

A p p l i c a t i o n has been made t o h o r i z o n t a l forces o n a vertical cylinder, b u t the m e t h o d can easily be generalized to c o m p u t e wave forces o n any other three-dimensional structures, i n c l u d i n g the case o f a m o o r e d floating body.

R E F E R E N C E S

1 H a v e l o c k , T . H . T h e p r e s s u r e o f w a t e r waves o n a fixed o b s t a c l e , Proc. Royal Soc, L o n d o n , Series A , 1940, 175, 409

2 M a c C a m y , R. C. a n d F u c h s , R, A . W a v e f o r c e s o n p i l e s : a d i f f r a c t i o n t h e o r y , Tech. memo 69, B e a c h E r o s i o n B o a r d , 1954 3 Y a m a g o u c h i , M . a n d T s u c h i y a , Y . N o n l i n e a r effect o f w a v e s o n

w a v e pressure a n d w a v e f o r c e o n a l a r g e c y l i n d r i c a l pile, Proc. Civil Engineering Soc. in Japan, T o k y o , 1974, N o . 229

4 R a m a n , H . a n d V e n k a t a n a r a s a i a h , P. F o r c e s d u e t o n o n l i n e a r w a v e s o n v e r t i c a l c y l i n d e r s , / . Waterways, Harbors and Coastal Engineering Division. ASCE 1976, 102, 301

5 C h e n , M . C. a n d H u d s p e t h , R. T . N o n l i n e a r d i f f r a c t i o n b y e i g e n f u n c t i o n e x p a n s i o n s , J. Waterway, Port, Coastal and Ocean Division, ASCE m i , 108, 306

6 M o l i n , B . S e c o n d - o r d e r d i f f r a c t i o n l o a d s u p o n t h r e e - d i m e n s i o n a l b o d i e s , Applied Ocean Researcli, 1979, 1(4), 197

7 M o l i n , B . a n d M a r i o n , A . E t u d e a n d i e u x i e m e o r d r e d u e c o m p o r t e m e n t des c o r p s flottants a n h o u l e r e g u l i e r e , Institut Fracais du Petrole, Report No. 8334008004707501, 1985 8 M o l i n , B . a n d M a r i o n , A . S e c o n d - o r d e r l o a d s a n d m o t i o n s f o r

floating b o d i e s i n r e g u l a r waves, Sth OMAESymp., T o k y o , 1986 9 L i , B . Y . Stokes w a v e d i f f r a c t i o n p r o b l e m - its f a r field b o u n d a r y c o n d i t i o n s a n d its s o l u t i o n b y a finite-infinite e l e m e n t m e t h o d , PliD thesis (in Chinese), D a l i a n U n i v e r s i t y o f T e c h n o l o g y , 1989

10 E a t o c k T a y l o r , R. a n d H u n g , S. M . S e c o n d o r d e r d i f f r a c t i o n f o r c e s o n a v e r t i c a l c y l i n d e r i n r e g u l a r waves, Applied Ocean Research, 1987, 9(1), 19

11 E a t o c k T a y l o r , R., H u n g , S. M . a n d C h a u , F . P . O n t h e d i s t r i b u t i o n o f s e c o n d o r d e r pressure o n a v e r t i c a l c i r c u l a r c y l i n d e r , Applied Ocean Research, 1989, 11(4), 183

12 L i g h t h i l l , J. W a v e a n d h y d r o d y n a m i c l o a d i n g , Proc. Second Int. Conf. on the Behaviour of Offshore Structip-es, B H R A F l u i d E n g i n e e r i n g , 1979, 1, 1

13 R a h m a n , M . W a v e d i f f r a c t i o n b y l a r g e o f f s h o r e s t r u c t u r e s : a n exact s e c o n d t h e o r y . Applied Ocean Research, 1984, 6(2), 90 14 C h a k r a b a r t i , S. K . N o n l i n e a r w a v e f o r c e s o n v e r t i c a l c y l i n d e r ,

J . Hydraulic Division, ASCE 1972, 98, 1895

15 C h a k r a b a r t i , S. K . S e c o n d - o r d e r w a v e f o r c e o n l a r g e v e r t i c a l c y l i n d e r , J. Waterways, Harbors and Coastal Engineering Divi-sion, ASCE 1975, 101, 311

16 H u n t , J. N . a n d B a d d o u r , R. E . T h e d i f f r a c t i o n o f n o n l i n e a r p r o g r e s s i v e w a v e s b y a v e r t i c a l c y l i n d e r . Quart. J. Mechanics and Applied Mathematics 1 9 8 1 , 34(1), 69

17 K i m , M . H . a n d Y u e , D . K . P . T h e c o m p l e t e s e c o n d - o r d e r d i f f r a c t i o n w a v e s a r o u n d a n a x i s y m m e t r i c b o d y , P a r t 1: m o n o -c h r o m a t i -c i n -c i d e n t waves, J. Fluid Me-chani-cs, 1989, 200, 235 18 L a u , S. L . a n d J i , Z . A n e f f i c i e n t 3 - D i n f i n i t e e l e m e n t f o r w a t e r

w a v e d i f f r a c t i o n p r o b l e m s , Int. J. Num. Methods in Engineering, 1989, 28, 1371

19 I s a a c s o n , M . de St. Q . W a v e r u n u p a r o u n d l a r g e c i r c u l a r c y l i n d e r , J. Waterway, Port, Coastal and Ocean Division, ASCE 1978, 104, 69

N O M E N C L A T U R E i) Symbols

a = cylinder radius; b = inner d o m a i n radius;

D = extra n o d a l parameter i n i n f i n i t e element; d = water d e p t h ;

E,, F, = parameters i n the a s y m p t o t i c s o l u t i o n o f the second o r d e r forced d i f f r a c t e d p o t e n t i a l ; F = wave force i n complex f o r m ;

ƒ = wave force;

CF, Cf= dimensionless force coefficients; g = g r a v i t a t i o n a l acceleration; H = incident v/ave height;

H J / ' = H a n k e l f u n c t i o n o f the first k i n d o f order n; i = c o m p l e x n u m b e r y / — 1;

J„ = Bessel f u n c t i o n o f the first k i n d o f o r d e r n; K„ = m o d i f i e d Bessel f u n c t i o n o f the second k i n d o f

order n; k = i n c i d e n t wave n u m b e r ; M , = standard shape f u n c t i o n o f 2 - D i s o p a r a m e t r i c element; A'; = s t a n d a r d shape f u n c t i o n o f 3 - D i s o p a r a m e t r i c element; fi = i n w a r d n o r m a l vector to a surface; p = d y n a m i c wave pressure; 9, z = c y l i n d r i c a l coordinates; = b o d y surface; = b o t t o m surface; Sp = w a t e r free surface;

Sj = i n t e r - b o u n d a r y surface between i n n e r a n d outer d o m a i n s ;

S „ = outer d o m a i n b o u n d a r y surface at i n f i n i t y f r o m the b o d y ;

t = t i m e ;

£ = p e r t u r b a t i o n parameter;

= wave p o t e n t i a l i n complex f o r m ; 1] = water surface elevation;

^ = the time-independent c o m p o n e n t o f the second order d i f f r a c t e d p o t e n t i a l ;

A = H/{k'dy,

X = wave n u m b e r o f second o r d e r free w a v e ; X* = wave n u m b e r given by e q u a t i o n (9); n = f u n c t i o n a l ; p = water density; (p = wave p o t e n t i a l ; n = water d o m a i n —d^z^O; CO = i n c i d e n t wave angular f r e q u e n c y ; S = water elevation i n complex f o r m ; a n d CH, C ; ; = dimensionless w a t e r elevation coefficients.

ii) Superscripts I = i n c i d e n t value; S = d i f f r a c t e d value; S' = f o r c e d c o m p o n e n t o f d i f f r a c t e d p o t e n t i a l ; a n d S" = free c o m p o n e n t o f d i f f r a c t e d p o t e n t i a l . iii) Subscripts 1 = first order v a l u e ; 2 = second order value;

22 = second order value arising f r o m the second o r d e r wave;

= second o r d e r value arising f r o m the first o r d e r wave;

(17)

20 = second o r d e r time-independent value; / = n o d a l value' I = m n e r d o m a m ; , ^ T • E = outer d o m a i n 1 = element^domam; and

T = t o t a l value.

Second order wave diffraction forces and runup by finite - infinite element method