Second-order wave interaction with
two-dimensional floating bodies by a
time-domain method
Joseph Y . T . Ng & Michael Isaacson
Department of Civil Engineering, University of British Columbia, Vancouver, B.C. Canada, V6T 1Z4
I n this paper a time-domain method is used to simulate second-order wave interactions with a large floating body in two dimensions. I n the numerical scheme adopted, a boundary integral equation method based on Green's theorem is used to calculate the velocity potential of the resulting flow field at each time step, and the second-order free-surface boundary conditions and the radiation condition of the corresponding initial-boundary value problem are treated by a time-integration scheme to obtain the development of the flow. The equations of motion of the body are solved numerically by the fourth-order R u n g e - K u t t a algorithm. The method is applied to the case of a semi-submerged circular cylinder. Numerical calculations presented include the transient m o t i o n of a freelyfloating cyhnder with a specified initial displacement, and the d i f f r a c t i o n -radiation of Stokes second-order waves by a moored floating cylinder. Important second-order wave effects associated with the hydrodynamic forces and motions of the floating structures in regular waves are highlighted. I t is found that the present approach is both computationally efficient and stable.
Key words: wave-structure interactions, hydrodynamics, floating structures, second-order solution, time-domain, wave forces.
time-stepping procedure (e.g. V i n j e & Brevig,^ Isaac-son,^ a n d C o i n t e ' ) . A n alternative n u m e r i c a l m e t h o d recently developed by Isaacson & C h e u n g ' " m a y be considered as a h y b r i d o f these t w o approaches. T h e i r m e t h o d involves T a y l o r series expansions a n d the a p p l i c a t i o n o f the Stokes p e r t u r b a t i o n procedure, a n d a t i m e - i n t e g r a t i o n scheme is used t o o b t a i n the r e s u l t i n g flow development.
M o s t previous investigations o f the related n o n l i n e a r wave r a d i a t i o n p r o b l e m i n v o l v i n g f o r c e d m o t i o n s o f a r i g i d b o d y i n otherwise s t i l l water have been f o r m u l a t e d i n the f r e q u e n c y - d o m a i n , a n d i n m o s t cases t w o - d i m e n s i o n a l bodies have been considered (e.g. P o t a s h , ' ' P a p a n i k o l a o u & N o w a c k i , ' ^ a n d K y o z u k a ' ^ ) . B y a d o p t i n g the t i m e - d o m a i n m e t h o d , the secondorder wave r a d i a t i o n p r o b l e m f o r t w o -d i m e n s i o n a l structures o f a r b i t r a r y shape u n -d e r g o i n g sinusoidal f o r c e d m o t i o n s has been presented b y Isaacson & N g . ' ' * T h e Stokes p e r t u r b a t i o n p r o c e d u r e a n d T a y l o r series expansion a b o u t the m e a n b o d y p o s i t i o n can be used t o o b t a i n the c o r r e s p o n d i n g b o d y - s u r f a c e b o u n d a r y c o n d i t i o n t o second order. I n general, the m e t h o d has been f o u n d t o be accurate, c o m p u t a t i o n a l l y efficient, a n d n u m e r i c a l l y v e r y stable. 1 I N T R O D U C T I O N
P r e d i c t i o n s o f h y d r o d y n a m i c loads a n d responses f o r large floating structures have generally been o b t a i n e d o n the basis o f linear h y d r o d y n a m i c t h e o r y w h i c h is v a l i d f o r the case o f small-ampUtude sinusoidal waves. D e t a i l reviews o f the available a n a l y t i c a l methods have been given b y W e h a u s e n , ' M e i , ^ a n d Y e u n g . ^ O v e r the past decade, researchers have focused increasingly o n the study o f n o n l i n e a r w a v e - s t r u c t u r e interactions i n order t o extend the a p p l i c a b i l i t y o f the c o n v e n t i o n a l Hnear a p p r o a c h . C o m p l i c a t i o n s o f this nonhnear p r o b l e m are t h a t the s o l u t i o n is r e q u i r e d to satisfy the t w o n o n l i n e a r free-surface b o u n d a r y c o n d i t i o n s , a n d the f o r m u l a t i o n s h o u l d i n v o l v e a correct treatment o f the r a d i a t i o n ( f a r -field) c o n d i t i o n . A survey o f previous w o r k indicates t h a t the n o n l i n e a r free-surface flow p r o b l e m can be solved b y t w o categories o f m e t h o d : one a p p r o a c h is a second-order f r e q u e n c y - d o m a i n s o l u t i o n based o n the Stokes p e r t u r b a t i o n p r o c e d u r e , (e.g. M o l i n , ' ' E a t o c k T a y l o r & Jung,^ a n d K i m & Y u e ^ ) , a n d the other is a f u l l n o n l i n e a r s o l u t i o n t o the r e s u l t i n g wave field b y a Applied Ocean Research 0141-1187/93/$06.00
96 Joseph Y.T. Ng, Michael Isaacson
T h e extension o f the second-order d i f f r a c t i o n a n d r a d i a t i o n problems to t r e a t i n g wave interactions w i t h f l o a t i n g structures is o f considerable p r a c t i c a l engineer-i n g engineer-i m p o r t a n c e . O g engineer-i l v engineer-i e ' ^ has p r o v engineer-i d e d an extensengineer-ive survey o n this subject area. T r a d i t i o n a l l y , second-order forces acdng o n fixed o r floadng bodies have been evaluated by t w o t h e o r e t i c a l approaches. T h e f a r - f i e l d m e t h o d ( M a r u o , a n d F a l t i n s e n & M i c h e l s e n ' ^ ) equates the m e a n second-order forces to the t o t a l change o f fluid m o m e n t u m flux. T h i s a p p r o a c h is restricted t o the mean components o n l y a n d is considered t o be c o m p u t a ü o n a l l y i n e f i f c i e n t f o r the v e r d c a l plane forces a n d m o m e n t s . T h e alternative near-field a p p r o a c h ( P i n k s t e r ' ^ ) involves the direct i n t e g r a t i o n o f h y d r o -d y n a m i c pressure c o n t r i b u t i o n s over the instantaneous w e t t e d b o d y surface, a n d general expressions can be derived f o r a l l h y d r o d y n a m i c f o r c e components to second order. T h i s m e t h o d has the advantage o f p r o v i d i n g physical i n s i g h t i n t o the resulting forces a n d indicates the relative significance o f each c o m p o n e n t .
I n general, the emphasis o f previous research has been o n s o l v i n g the c o r r e s p o n d i n g wave d i f f r a c t i o n and r a d i a t i o n problems separately u p t o second order using the m o r e c o n v e n t i o n a l f r e q u e n c y - d o m a i n a p p r o a c h . T h e present paper considers the second-order wave d i f f r a c t i o n - r a d i a t i o n p r o b l e m f o r a floating structure i n t w o dimensions by the t i m e - d o m a i n m e t h o d . I n order to i l l u s t r a t e the m e t h o d , n u m e r i c a l examples presented relate t o the transient m o t i o n o f a f r e e l y - f l o a t i n g cylinder i n otherwise still water, a n d the i n t e r a c t i o n o f regular waves w i t h a m o o r e d floating cylinder. I m p o r t a n t features related to the secondorder h y d r o -d y n a m i c forces a n -d m o t i o n s o f floating structures i n regular waves are discussed. W i t h suitable m o d i f i c a -tions, i t is expected t h a t the present m e t h o d can readily be extended t o the three-dimensional p r o b l e m w i t h structures o f a r b i t r a r y shape.
2 M A T H E M A T I C A L F O R M U L A T I O N
Statement of boundary value problem
T h e f u f l n o n l i n e a r w a v e - s t r u c t u r e i n t e r a c t i o n p r o b l e m d e f i n i n g the fluid m o t i o n is first considered. T h e t w o -d i m e n s i o n a l p r o b l e m , as i l l u s t r a t e -d i n F i g . 1, is -define-d w i t h respect t o t w o r i g h t - h a n d e d cartesian c o o r d i n a t e systems: one is the i n e r t i a l (space-fixed) c o o r d i n a t e system (x, z); the second is the b o d y - f i x e d c o o r d i n a t e system ( x , z ) . W h e n the b o d y is i n its e q u i l i b r i u m p o s i t i o n , the t w o sets o f c o o r d i n a t e systems are p a r a l l e l a n d the centre o f mass is located at (A'g,Zg) w i t h respect t o the i n e r t i a l c o o r d i n a t e system. T h e seabed is assumed impermeable a n d h o r i z o n t a l a l o n g the plane z = —d. Based o n the assumptions o f a homogeneous, i n v i s c i d a n d incompressible fluid, a n d an i r r o t a t i o n a l flow, the fluid m o t i o n can be described by a v e l o c i t y p o t e n t i a l 4>
y / / } / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / ^ ^ ^ ^ Fig. 1. Definition sketch.
w h i c h satisfies the Laplace e q u a t i o n w i t h i n the fluid d o m a i n , together w i t h b o u n d a r y c o n d i t i o n s o n the seabed, the instantaneous wetted b o d y surface 5^^, a n d the free surface 5*5 at z = 77, where 77 denotes the f r e e surface e l e v a t i o n above the s t i l l water level. F i n a l l y , f o r a well-posed b o u n d a r y value p r o b l e m , a suitable b o u n d a r y c o n d i t i o n o n a c o n t r o l surface S^, w h i c h is located at a s u f f i c i e n t l y f a r distance f r o m the b o d y is a p p l i e d t o ensure t h a t a l l waves scattered a n d r a d i a t e d b y the s t r u c t u r e are p r o p a g a t i n g i n the o u t g o i n g d i r e c t i o n . The m a j o r difficulties o f the r e s u l t i n g m a t h e m a t i c a l p r o b l e m are associated w i t h the s o l u t i o n being r e q u i r e d t o satisfy the t w o n o n l i n e a r free-surface b o u n d a r y c o n d i t i o n s a n d the k i n e m a t i c b o d y -surface b o u n d a r y c o n d i t i o n at the f r e e -surface a n d the w e t t e d - b o d y surface, respectively, b o t h o f w h i c h are u n k n o w n a priori, as w e l l as a p o o r l y d e f i n e d r a d i a t i o n c o n d i t i o n i n the f a r - f i e l d .
C o n v e n t i o n a l l y , the m o t i o n s o f the r i g i d t w o -d i m e n s i o n a l b o -d y are -define-d i n terms o f the t r a n s l a t i o n a l m o t i o n s i n the h o r i z o n t a l a n d v e r t i c a l directions and the r o t a t i o n a l m o t i o n a b o u t the centre o f mass, namely the sway, heave a n d r o l l m o t i o n s w h i c h are d e n o t e d by f , ( a n d a, respectively. Since r o t a t i o n a l m o t i o n is considered here, the displacement vector x a n d the instantaneous u n i t n o r m a l vector N = {N^,N2) at each p o i n t [x, z) o n the b o d y surface are given b y
X = {i + x{cosa - 1) - l - z s i n a ,
C — x s i n a - f z ( c o s a — 1)) (1) N = («^. cos a + n^ sin a, —n^ sin a -\- n^ cos a) (2)
where n = ( ; j ^ , 7 ? 2 ) is the u n i t n o r m a l vector directed o u t w a r d l y f r o m the fluid region w h e n the b o d y is at rest. T h e n o r m a l v e l o c i t y o f the b o d y surface is s i m p l y given b y X • N , where a n o v e r - d o t indicates a t i m e d e r i v a t i v e . I n s t u d y i n g the m o t i o n response o f floating structures, the h y d r o d y n a m i c loads i n d u c e d by the time-dependent flow field associated w i t h wave disturbances give rise t o the structure m o t i o n s . T h e i n t e r a c t i o n involves the k i n e m a t i c b o d y surface b o u n d a r y c o n d i t i o n r e l a t i n g the fluid velocities and structure m o t i o n s as w e l l as the equations o f m o t i o n o f the structure. T h e latter are
given f o r the three modes o f m o t i o n as:
(3) where ^ = , ( , a) represents the displacement vector o f the floating b o d y , M is the mass m a t r i x , K is the s t r u c t u r a l stiffness m a t r i x , and ¥* = {F^,F^,M^) is a vector c o n t a i n i n g the t o t a l h y d r o d y n a m i c forces, w i t h F^ a n d F^ d e n o t i n g the f o r c e c o m p o n e n t s i n the x and z direcdons, respectively, a n d My d e n o t i n g the m o m e n t c o m p o n e n t i n the y d i r e c t i o n , where y is the axis perpendicular t o the v e r d c a l plane o f the t w o - d i m e n s i o n a l p r o b l e m . Since the o r i g i n o f the b o d y - f i x e d axes is c o i n c i d e n t w i t h the centre o f mass o f the b o d y , the mass m a t r i x is d i a g o n a l , w i t h the first a n d second terms given as the mass o f the structure a n d the t h i r d t e r m g i v e n b y the mass m o m e n t o f i n e r t i a a b o u t the j^-axis. T h e c o m p o n e n t s o f the sdffness m a t r i x i n c l u d e c o n t r i b u t i o n s due to the m o o r i n g system. I t is n o t e d t h a t the drag-induced viscous d a m p i n g is neglected here. F u r t h e r m o r e , and i n contrast t o the f r e q u e n c y - d o m a i n a p p r o a c h , the c o n t r i b u t i o n s o f the h y d r o d y n a m i c added mass and d a m p i n g associated w i t h the m o t i o n s o f the s t r u c t u r e , a n d o f the h y d r o s t a t i c stiffness are i n c l u d e d w i t h i n the f o r c i n g vector o n the r i g h t - h a n d side o f eqn (3).
Second-order expansion
A s s u m i n g t h a t the amplitudes o f the b o d y m o t i o n s are smaU c o m p a r e d t o a p r i n c i p a l b o d y d i m e n s i o n a n d t h a t the wave height is m o d e r a t e , the body-surface a n d free-surface b o u n d a r y c o n d i t i o n s , o r i g i n a l l y defined o n the instantaneous surfaces, m a y be expanded a b o u t the c o r r e s p o n d i n g m e a n positions b y the use o f T a y l o r series expansions: ( V ^ + X - V ( V , / . ) dz 'dt dr] d(j) drj •).N-d X - ^ o n 5b 'dz d(j) drj d(j) dr] 0 o n 5„ (4) (5) •gv-1 'dz d<p^ dl + ''' • m i 0 o n 5„ (6)
where t denotes time a n d g is the g r a v i t a t i o n a l constant. T h e b o u n d a r y c o n d i t i o n s m a y n o w be i m p o s e d o n the surface o f a time-independent geometry w h i c h includes the m e a n b o d y surface 5b a n d the still water surface Sg. F o l l o w i n g the Stokes expansion p r o c e d u r e , quantities at first a n d second order are separated b y i n t r o d u c i n g a
p e r t u r b a t i o n series f o r ^ , </> and T]: (/) = e(pi + e^(j)-^ r] = erii+ + ••• (7) + ••• (8) (9) where e is a p e r t u r b a t i o n parameter related to the wave steepness w h i c h is s m a l l , a n d the subscripts 1 and 2 indicate, respectively, components at first a n d second order. T h e first a n d second-order p o t e n t i a l s a n d the f r e e surface e l e v a t i o n are f u r t h e r decomposed i n t o i n c i d e n t wave c o m p o n e n t s a n d wave disturbance c o m p o n e n t s due to c o m b i n e d wave d i f f r a c t i o n - r a d i a t i o n effects:
0 = e(<^r + 0 f ) + £ ^ ( < / . ^ + 0 f ) + V = <v7 + v f ) + e'{7]^'+v!) + \v
(10) (11) T h e first-order p o t e n t i a l (pf corresponds s i m p l y t o a s u p e r p o s i t i o n o f solutions t o the separate linear wave d i f f r a c t i o n a n d r a d i a t i o n p r o b l e m s , whereas the second-order p o t e n t i a l comprises n o n l i n e a r wave c o m p o n e n t s due to self- a n d cross-interactions o f the first-order i n c i d e n t a n d d i s t u r b e d wave p o t e n t i a l s . I n the c o n -v e n t i o n a l f r e q u e n c y - d o m a i n m e t h o d , this gi-ves rise to a n u m b e r o f sub-problems w h i c h are considered to be algebraically tedious. H o w e v e r , i n the present a p p r o a c h all the second-order wave c o m p o n e n t s due t o wave disturbances are collectively represented b y the wave p o t e n t i a l 0 f .
U p o n s u b s t i t u t i n g the Stokes p e r t u r b a t i o n expansions f o r ^, cf) a n d r] i n t o the g o v e r n i n g L a p l a c e e q u a t i o n a n d the c o r r e s p o n d i n g b o u n d a r y c o n d i t i o n s e x p a n d e d a b o u t their m e a n positions, a n d c o l l e c t i n g terms o f equal order, the c o r r e s p o n d i n g b o u n d a r y value p r o b l e m s f o r the e a n d terms i n the p o w e r series expansions m a y be developed.
Solution to second order
T h e b o u n d a r y value p r o b l e m at each o r d e r is n o w linear a n d is f o r m u l a t e d w i t h respect to a time-independent fluid d o m a i n D b o u n d e d b y the seabed, the b o d y surface i n its e q u i l i b r i u m p o s i t i o n 5b, the still w a t e r surface a n d the c o n t r o l surface S^. I n the /cth o r d e r wave d i f f r a c t i o n - r a d i a t i o n p r o b l e m ( w i t h ^ = 1, 2 i n t u r n ) , the disturbance p o t e n t i a l satisfies the Laplace e q u a t i o n i n D
V'<P^ = 0 inD (12) a n d is subject to the b o u n d a r y c o n d i t i o n s a p p h e d o n the
seabed, the m e a n b o d y surface a n d the s t i f l w a t e r surface, given respectively as
9 ^
dz 0 a t z -d
o n Su
(13)
98 Joseph Y. T. Ng, Michael Isaacson
wave p r o p a g a t i o n (e.g. Orlanslci,'^ L e & L e o n a r d , ^ " a n d
d4
dri ,
dt grik^fk o n 5 „
(15)
(16) T h e terms f ^ , fl- a n d fl/ are given respectively as
w / i = - ^ + (^1 + + (Cl - a i ' ^ ) « z (17a) w f l dn + (il + ó-2z)n^ + ( 4 - a2x)n. dn 2 dn dsdn 2 "X dsdn (Cl - «1'^') d<p\ d(j)i dn ds + ainA Cl - ^ n , . - - ^ ? 7 . . a,n^[i:i-—n.+—n^ 7 dv'^\ ^dcPxdrji ^ ö V i dz dt r dx dx "^^'dz^ fl'=0 f 2 = -dzdt (17b) (18a) (18b) (19a) (19b)
H e r e n denotes distance i n the d i r e c t i o n o f the u n i t n o r m a l vector n, s is the t a n g e n t i a l d i r e c t i o n o n the m e a n b o d y surface as s h o w n i n F i g . 1, a n d the subscript /c indicates the p e r t u r b a t i o n order. T o account f o r the structure m o d o n s , the three equations o f m o t i o n at each order m a y be expressed i n m a t r i x f o r m as
Mék+Mk = n (20)
I n the present context, the second-order h y d r o d y n a m i c f o r c e is made u p o f the steady d r i f t f o r c e a n d o s c i l l a t o r y f o r c e at twice the first-order i n c i d e n t wave f r e q u e n c y . T h e q u a d r a t i c terms i n the second-order b o d y surface b o u n d a r y c o n d i t i o n can be i n t e r p r e t e d as corrections t o the time-dependent m o t i o n s o f the fluid-structure interface. Each secondorder f o r c i n g t e r m o n the r i g h t -h a n d side o f eqns ( 1 7 ) - ( 1 9 ) i n p r i n c i p l e can be calculated either f r o m the first-order s o l u t i o n or f r o m the Stokes second-order i n c i d e n t wave components, once the m o t i o n s o f the b o d y are o b t a i n e d f r o m the d y n a m i c analysis procedure.
W i t h respect to a p p l y i n g the b o u n d a r y c o n d i t i o n o n the c o n t r o l surface S^., 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 m a y be m o d i f i e d to treat the case o f unsteady
Isaacson & C h e u n g ' " ) , a n d is a p p r o x i m a t e d as:
9<Pk
dt + c dn 0 o n 5"^ (21) where c is the time-dependent celerity o f the p e r t u r b e d waves at the c o n t r o l surface. T h i s c o n d i t i o n p e r m i t s wave disturbances due to d i f f r a c t i o n - r a d i a t i o n to be t r a n s m i t t e d o u t o f the a r t i f i c i a l c o m p u t a t i o n a l d o m a i n w i t h o u t r e f l e c t i o n . B y u t i l i z a t i o n a suitable t i m e -dependent celerity o n S^, e q n (21) can then be a p p l i e d t o calculate the p o t e n t i a l at each t i m e instant o n the c o n t r o l surface.
Integral equation
U s i n g Green's second i d e n t i t y , the p o t e n t i a l at /cth o r d e r o n the surface o f the s o l u t i o n d o m a i n m a y be expressed b y the f o l l o w i n g b o u n d a r y i n t e g r a l e q u a t i o n :
i ? ( x )
T T j s ' dn
^ f ( x ' ) g ( x , x ' ) ] d 5 (22)
Here x ' represents a p o i n t ( x , z ) o n the surface S' over w h i c h the i n t e g r a t i o n is p e r f o r m e d , G is a Green's f u n c t i o n a n d n is measured f r o m the p o i n t x ' . I n the present context the surface S' w o u l d comprise the m e a n b o d y surface b e l o w the still water level, the s t i l l w a t e r surface So, the c o n t r o l surface S^ s u r r o u n d i n g the b o d y a n d the seabed. H o w e v e r , f o r the case o f constant w a t e r d e p t h , the seabed can be discarded f r o m S' b y c h o o s i n g a Green's f u n c t i o n c o r r e s p o n d i n g t o the f u n d a m e n t a l s o l u t i o n o f the Laplace e q u a t i o n a n d its image a b o u t the seabed. T h i s is
G ( x , x ' ) - £ l n ( r , ) (23) k = \
where , \ is the distance between the p o i n t s x a n d x ^ . T h e source p o i n t s x^ are defined such t h a t x'l = {x',z) is a p o i n t o n S'; a n d •s!2 — {x,-{z' + 2d)) is the r e f l e c t i n g image o f x'l a b o u t the h o r i z o n t a l seabed.
A c c o r d i n g t o the b o u n d a r y i n t e g r a l e q u a t i o n f o r m u l a t i o n , either the p o t e n t i a l or its n o r m a l derivative o n each p o r t i o n o f the b o u n d a r y S' is given f r o m the c o r r e s p o n d i n g b o u n d a r y c o n d i t i o n s , the s o l u t i o n to the i n t e g r a l eqn (22) can n o w be solved b y a n u m e r i c a l procedure f r o m w h i c h the r e m a i n i n g b o u n d a r y variables can be evaluated. I n the present a p p l i c a t i o n o f a floating structure, the c o u p l e d fluid-structure equations can be solved b y a t i m e - i n t e g r a t i o n scheme i n v o l v i n g a d i s c r e t i z a t i o n o f the surface S' a n d a suitable iterative p r o c e d u r e .
Free surface elevation and hydrodynamic forces
A f t e r s o l v i n g f o r the v e l o c i t y p o t e n t i a l , the free surface elevation to second o r d e r is calculated e x p l i c i t l y b y the d y n a m i c free surface b o u n d a r y c o n d i t i o n given b y eqn (16). F o r regular wave e x c i t a t i o n , the free surface elevation T] t o second o r d e r contains three c o m p o n e n t s
rjx, ?7o a n d 772, w h i c h are, respectively, the f i r s t - o r d e r
o s c i l l a t o r y c o m p o n e n t at the wave f r e q u e n c y , the order steady c o m p o n e n t a n d the second-o r d e r second-o s c i l l a t second-o r y c second-o m p second-o n e n t at twice the wave f r e q u e n c y . T h e expressions f o r rjx, rjo a n d 772 are given respectively as: V i = -1 fd(l>i Vo V2 Vl
dzdt I
a 9'
where ( ) denotes a time average. T h e c o m p o n e n t s 770,772 are associated, respectively, w i t h a steady set-up o r set-down o f the m e a n w a t e r level due t o the non-zero m e a n o f the free surface b o u n d a r y c o n d i t i o n s at second order, a n d an o s c i l l a t o r y v a r i a t i o n o f the free surface e l e v a t i o n at second o r d e r due t o b o t h forces a n d free waves.
T h e h y d r o d y n a m i c forces a n d m o m e n t s exerted o n the b o d y can be calculated b y c a r r y i n g o u t a direct i n t e g r a t i o n o f the d y n a m i c pressure over the i n s t a n -taneous w e t t e d b o d y surface 5 ^ . I n the present a p p l i c a t i o n , the pressure i n the fluid can be d e t e r m i n e d b y the unsteady B e r n o u l h e q u a t i o n :
0 0 1 , ,
(27)
where p is the fluid density. T o account f o r the b o d y m o t i o n s , the pressure a c t i n g o n the instantaneous b o d y surface can consistentiy be expanded a b o u t the m e a n b o d y surface b y a T a y l o r series expansion, f o l l o w e d b y the s u b s t i t u t i o n o f the p e r t u r b a t i o n series o f 0 a n d C. R e t a i n i n g terms to second order, t h i s gives
d4> 1 2
- pg[ C2 - Q2X - - a x z ] - p{Cx + oixz)
- p(Ci -
axx)
o n 5h (28)
By a direct pressure i n t e g r a t i o n , the r e s u l t i n g f o r c e a n d m o m e n t vectors are given respectively b y the i n t e g r a l s
¥ = p pNdS
My = p\ p[{z - z^)N, - [x - x^)N,]&S\
(29)
(30)
where j is the u n i t v e c t o r i n the y d i r e c t i o n . N o t e t h a t the m o m e n t is t a k e n a b o u t the centre o f mass o f t h e b o d y . T h e t o t a l h y d r o d y n a m i c forces F t o second o r d e r consists o f three f o r c e components: F j , Fg a n d F 2 , w h i c h are, respectively, the first-order o s c i l l a t o r y f o r c e at the i n c i d e n t wave f r e q u e n c y , the second-order steady f o r c e , a n d the second-order o s c i l l a t o r y f o r c e at t w i c e the wave f r e q u e n c y ; a n d are given respectively as:
— - n dS - /3g^„(Ci - a i X f j k Sh 01 (31) Fo = - ^ p ( | | ^ j V 0 i p n d 5 ^ + i p g (77, - C ] - | - a i x ) ^ n d M ' [ ( 6 + a . s (Cl - ai^-) n d ^ n d S 5, ^ " ".vk)d5j> - ^ pgA,,Zg(a?)k 1 (32) Fa = - P d 5 - p g ^ w ( C 2 - a 2 X f ) k - \ p \ \V<t>i?n&S + l-pg 2 2 t . 5 (771 - Cl +axxfndw (Cl - aix) d (I dndt wdS ndS - P ^ " 1 ("zi - « . x k ) d 5 - ^pgA,^Zga]k - F 0 (33) where ^ „ is the m e a n w a t e r p l a n e area, Xf is the x c o o r d i n a t e o f the centre o f flotation w i t h respect t o the i n e r t i a l c o o r d i n a t e system, a n d i, k denote u n i t vectors i n the X a n d z d i r e c t i o n s , respectively. F u r t h e r m o r e , the c o r r e s p o n d i n g m o m e n t c o m p o n e n t s Mxy, M^y a n d M2y
100 Joseph Y. T. Ng, Michael Isaacson
can be o b t a i n e d i n a similar m a n n e r , as s h o w n b y Isaacson & N g . ' ' * I t is m e n t i o n e d that there are alternative expressions f o r the f o r c e a n d m o m e n t c o m p o n e n t s p r o v i d e d i n the l i t e r a t u r e , a n d t h e i r differences are m a i n l y associated w i t h the d e f i n i t i o n o f the t w o f r a m e s o f c o o r d i n a t e reference a n d the assumed p o s i t i o n o f the centre o f mass.
3 N U M E R I C A L P R O C E D U R E
T h e s o l u t i o n m e t h o d o l o g y o f t r e a t i n g the b o u n d a r y integral eqn (22) a n d m a t h e m a t i c a l f o r m u l a s used i n the time-stepping scheme have been presented i n detail b y Isaacson & C h e u n g ' " i n the c o n t e x t o f the w a v e d i f f r a c t i o n p r o b l e m , a n d therefore o n l y a b r i e f s u m m a r y o f the m e t h o d is o u t l i n e d here.
Field solution
T h e surfaces S<a, a n d S^, are dlscretized i n t o finite n u m b e r s o f facets m a d e u p o f straight-line segments, a n d the c o r r e s p o n d i n g values o f 0 ^ a n d dcj^/dn are assumed constant over each facet a n d a p p h e d at the facet c e n t r o i d . T h i s enables the p r o b l e m to be f o r m u l a t e d as a m a t r i x e q u a t i o n f o r the u n k n o w n values o f a n d dcpf/dn, w i t h m a t r i x coefficients t h a t are f u n c t i o n s o f geometry o n l y , a n d a r i g h t - h a n d side i n p u t vector w h i c h contains time-dependent b o u n d a r y variables. T h u s , a m a t r i x i n v e r s i o n a n d m u l t i p l i c a t i o n is r e q u i r e d o n l y once t o o b t a i n the s o l u t i o n t o the system o f hnear algebraic equations, a n d subsequently a m u l t i p l i c a t i o n o f the r e s u l t i n g m a t r i x a n d the i n p u t vector at each time step t h e n gives the flow development. T h e i n p u t vector contains d(j)f /dn o n the b o d y surface 5),, a n d o n the c o n t r o l surface a n d the still water surface S,,- These are evaluated respectively b y the b o d y surface b o u n d a r y c o n d i t i o n s o n 5b a p p l i e d i n c o n j u n c t i o n w i t h the equations o f m o t i o n , a n d b y time-stepping procedures applied to the r a d i a t i o n c o n d i t i o n o n a n d the free surface b o u n d a r y c o n d i t i o n s o n
S^-Time-stepping scheme
T h e r a d i a t i o n c o n d i t i o n a n d the t w o free-surface b o u n d a r y c o n d i t i o n s are treated i n the m a n n e r described i n detail b y Isaacson & C h e u n g . ' " A t i m e -i n t e g r a t -i o n procedure -is appl-ied t o the a p p r o x -i m a t e r a d i a t i o n c o n d i t i o n given b y eqn (21) i n order to o b t a i n (pf o n the c o n t r o l surface S^. f r o m a n u m e r i c a l e x t r a p o l a t i o n based o n values o f (pf near 5^ a n d inside the d o m a i n at previous time steps. A d i f f e r e n t t i m e - i n t e g r a t i o n procedure is a p p l i e d t o the free-surface b o u n d a r y c o n d i t i o n s to evaluate the free-free-surface elevation qf a n d p o t e n t i a l (pf at a new time step / I -i n terms o f the k n o w n s o l u t -i o n u p t o t -i m e t. T h e f-irst-
first-o r d e r A d a m s B a s h f first-o r t h e q u a t i first-o n is a p p l i e d t first-o the k i n e m a t i c free-surface b o u n d a r y c o n d i t i o n t o o b t a i n ?/f, a n d the first-order A d a m s M o u l t o n e q u a t i o n is applied t o the d y n a m i c free-surface b o u n d a r y c o n d i t i o n to o b t a i n ( p f . F i n a l l y , the n o r m a l derivative d(pf/dn o n the e q u i l i b r i u m b o d y surface 5b is given b y the b o d y -surface b o u n d a r y c o n d i t i o n , e q n (14), i n terms o f the b o d y displacements a n d velocities, and these are o b t a i n e d b y s o l v i n g the equations o f m o t i o n e m p l o y i n g a s t a n d a r d f o u r t h - o r d e r R u n g e - K u t t a m e t h o d .
T h e wave field at the advanced t i m e t + At can n o w be o b t a i n e d b y s o l v i n g the m a t r i x e q u a t i o n w i t h a k n o w n r i g h t - h a n d side vector. I n f a c t , a n iterative process is needed t o treat the free-surface b o u n d a r y c o n d i t i o n s . T h i s is because the d y n a m i c c o n d i t i o n used t o o b t a i n (pf requires rif, a n d the k i n e m a t i c c o n d i t i o n used to o b t a i n rjf requires d(pf/dn, w h i c h is itself o b t a i n e d as a s o l u t i o n to the field e q u a t i o n . T h u s , an i n i t i a l c a l c u l a t i o n o f a l l the b o u n d a r y variables at the advanced time is c a r r i e d o u t , a n d r e f i n e d values o f 'qf are o b t a i n e d using the calculated values o f d(pf/dn. T h e successive steps can t h e n be repeated i d e n t i c a l l y w h i c h lead t o m o r e accurate values o f (pj^ o n 5b f o r h y d r o d y n a m i c f o r c e calculations. Previous c o m -p u t a t i o n a l ex-perience indicates t h a t o n l y a f e w iterations are r e q u i r e d t o achieve r a p i d n u m e r i c a l convergence a n d the c o m p u t a t i o n can t h e n proceed t o the n e x t t i m e step.
4 R E S U L T S A N D D I S C U S S I O N
Computational considerations
I n o r d e r t o d e m o n s t r a t e the present m e t h o d , n u m e r i c a l results are presented f o r a semi-submerged c i r c u l a r cylinder i n deep water w i t h its centre o f mass a t the intersection o f the still water level a n d the v e r t i c a l plane o f s y m m e t r y . Examples considered here relate to the transient m o t i o n o f a f r e e l y - f l o a t i n g cylinder i n otherwise still water, a n d the i n t e r a c t i o n o f Stokes second-order waves w i t h a m o o r e d floating cylinder.
T y p i c a l l y , a b o u t 4 0 0 - 6 0 0 facets m a d e up o f straight-line segments are used f o r the discretization o f the b o d y surface, the c o n t r o l surface a n d the s t i l l water surface. T h e influence coefficients i n the dlscretized b o u n d a r y i n t e g r a l e q u a t i o n are calculated b y a f o u r - p o i n t Gaussian i n t e g r a t i o n a n d d o u b l e p r e c i s i o n is used t h r o u g h o u t the c o m p u t a t i o n . T h e m a x i m u m t i m e step size f o r a given d i s c r e t i z a t i o n can be d e t e r m i n e d b y the C o u r a n t c r i t e r i o n . W i t h the Green's f u n c t i o n chosen t o satisfy the b o u n d a r y c o n d i t i o n o n the seabed, the seabed can be excluded f r o m the surface d i s c r e t i z a t i o n scheme leading to a substantial saving i n c o m p u t a t i o n a l costs i n terms o f the r e q u i r e d C P U t i m e a n d v i r t u a l m e m o r y storage. T o describe the m o r e r a p i d v a r i a t i o n s o f the p o t e n t i a l near the b o d y surface a n d t o speed u p the
convergence o f the n u m e r i c a l calculations w i t h o u t increasing the facet density everywhere, smaller facets are used on the b o d y surface a n d o n the nearby free surface t h a n those f u r t h e r away f r o m the b o d y surface. T h e c o m p u t a t i o n s r e p o r t e d here were p e r f o r m e d o n an I B M 3090/150S c o m p u t e r at the U n i v e r s i t y o f B r i d s h C o l u m b i a . A s a n i n d i c a t i o n o f the c o m p u t a t i o n a l e f f o r t r e q u i r e d , a p r o b l e m i n v o l v i n g 500 facets t y p i c a l l y requires a C P U t i m e o f a b o u t 40 seconds f o r setting u p a n d s o l v i n g the m a t r i x e q u a t i o n , and a b o u t 10 seconds f o r the flow c o m p u t a t i o n over 100 t i m e steps.
I n o r d e r to m i n i m i z e adverse transient effects related t o an a b r u p t i n i d a l e o n d i d o n a n d aUow a g r a d u a l d e v e l o p m e n t o f the r e s u l t i n g wave field, the n o r m a l d e r i v a t i v e o f the i n c i d e n t wave p o t e n t i a l o n the r i g h t -h a n d side o f eqn (14), w -h i c -h corresponds t o t-he b o d y surface b o u n d a r y c o n d i t i o n , a n d the h y d r o d y n a m i c pressure forces exerted o n the b o d y surface are m u l t i -p l i e d b y a m o d u l a t i o n f u n c t i o n w h i c h increases g r a d u a l l y f r o m zero to u n i t y over a specified m o d u l a -t i o n -t i m e (Isaacson & C h e u n g ' " ) . I n effec-t, -the i m m e r s e d b o d y surface g r a d u a l l y materiahzes a n d the d y n a m i c pressure o f the s u r r o u n d i n g fluid is s l o w l y established over the m o d u l a t i o n t i m e . I n the results r e p o r t e d here, a m o d u l a t i o n time T^ — 2T, where T denotes the incident wave p e r i o d , is a d o p t e d t o p r o v i d e a stable and s m o o t h development o f the steady state s o l u t i o n a f t e r the d u r a t i o n o f the m o d u l a t i o n .
Transient motion of a freely-floating cylinder
T h e unsteady free surface flow p r o b l e m r e l a t i n g t o the transient heave m o t i o n o f a f r e e l y floating c y l i n d e r w i t h a specified i n i t i a l displacement has been the subject o f a n u m b e r o f a n a l y t i c a l investigations (e.g. M a s k e f l & U r s e f l , ^ ' Yeung^^ a n d Lee & Leonard^") based o n a l i n e a r i z e d p o t e n t i a l flow t r e a t m e n t . T h i s transient p r o b l e m extended t o second order has been used as a p r e l i m i n a r y test case o f the present t i m e - d o m a i n a p p r o a c h p r i o r t o t r e a t i n g the m o r e general case o f a floating b o d y i n waves. Since the r a d i a t e d waves generated by the b o d y m o t i o n s are unsteady, the celerity o f the o u t w a r d p r o p a g a t i n g waves at the c o n t r o l surface is determined n u m e r i c a l l y at each t i m e step. F u r t h e r m o r e , f o r this p r o b l e m the m o d u l a t i o n f u n c t i o n is n o t needed, a n d the i n i t i a l c o n d i t i o n is t a k e n t o c o r r e s p o n d t o zero v e l o c i t y p o t e n t i a l everywhere, a specified i n i t i a l v e r t i c a l b o d y displacement, a n d zero i n i t i a l v e l o c i t y .
F i g u r e 2 shows calculated time histories o f the heave m o t i o n a n d the vertical h y d r o d y n a m i c f o r c e c o m p o n e n t s o n the c y l i n d e r f o r case c o r r e s p o n d i n g to 1^1/0 = 0 - 1 , rf/fl=10-0 and Ljd=A-0, where \A\, a a n d are respectively the i n i t i a l v e r t i c a l displacement o f the c y l i n d e r , the radius o f the c y l i n d e r , a n d the h o r i z o n t a l length o f the c o m p u t a t i o n a l d o m a i n . N o t e t h a t t h e c y l i n d e r axis is situated at x/L = 2-0. F o r _]0 I 1 1 1 \ . 1 1 I I I 1 1 0.0 1.0 2.0 3.0 4.0 B.0 6.0 -1.5 0.0 1.0 2.0 3.0 4.0 5.0 6.0 t / T
Fig. 2. Development with time of vertical displacement and vertical hydrodynamic force components for a semi-submerged circular cylinder undergoing transient heave motions with \A\/a = OT and d/a = 10-0. , solution to first order; , solution to second order; , second-order
component.
convenience, t i m e / is n o r m a h z e d w i t h respect t o y V ö / g w h i c h is associated w i t h the n a t u r a l p e r i o d o f the c y l i n d e r i n heave. T h e present first-order s o l u t i o n o f the displacement gives excellent agreement w i t h the c o r r e s p o n d i n g results o f M a s k e l l & U r s e l l , ^ ' such t h a t t h e i r results o v e r l a p f u l l y w i t h the first-order p l o t i n the figure a n d hence are n o t s h o w n . N o t e t h a t the t i m e between each successive zero upcrossing o f the heave m o t i o n a n d the c o r r e s p o n d i n g h y d r o d y n a m i c force varies shghtly, i n accordance w i t h the unsteady features o f the m o t i o n . T h e figure also indicates t h a t differences between the first a n d second-order solutions are h a r d l y distinguishable f o r this case, so t h a t errors caused b y neglecting the second-order s o l u t i o n t o the i n i t i a l - b o u n d a r y value p r o b l e m w o u l d be m i n i m a l .
T h e development w i t h t i m e o f the c o r r e s p o n d i n g free-surface p r o f i l e s t o second order is s h o w n i n F i g . 3. T h e unsteady r a d i a t i o n w a v e m o t i o n s c o n t i n u o u s l y t r a n s m i t energy away f r o m the c y l i n d e r a n d , i n t u r n , d a m p o u t the h a r m o n i c b o d y m o t i o n s . I n contrast t o the results f o r the case o f a c y l i n d e r u n d e r g o i n g f o r c e d sinusoidal m o t i o n s r e p o r t e d earlier b y Isaacson & N g , ' ' ' i n w h i c h substantial second-order effects can be observed, the c o n t r i b u t i o n o f the second-order wave c o m p o n e n t s t o the free surface elevation, i n the present case, is f o u n d to be i n s i g n i f i c a n t .
Wave interaction with a moored floating cylinder
N u m e r i c a l results i n c l u d i n g h y d r o d y n a m i c forces a n d m o t i o n responses t o second o r d e r are presented f o r a m o o r e d semi-submerged c i r c u l a r c y l i n d e r subject t o deep water regular waves. T o the a u t h o r s ' k n o w l e d g e , such results have n o t yet been r e p o r t e d . H o w e v e r , the c o r r e s p o n d i n g first a n d second-order results o b t a i n e d b y the time-domain m e t h o d f o r wave d i f f r a c t i o n (Isaacson & C h e u n g ' " ) a n d w a v e r a d i a t i o n (Isaacson & N g ' ' ' ) have recently been c o m p a r e d w i t h p u b l i s h e d t h e o r e t i c a l a n d e x p e r i m e n t a l data a n d i n d i c a t e excellent agreement. F o r
102 Joseph Y.T. Ng, Michael Isaacson t/T = 7.4 t/T = 2.6 t/T = 1.0 t/T = 0.2 0 1 2 3 4 x / d
Fig. 3. Development w i t h time o f free surface profiles to second order for a semi-submerged circular cylinder undergoing transient heave motions with |.4|/a = 0T and c?/a=10-0.
Successive profiles are at times 0-4 T apart.
s i m p l i c i t y , the c y l i n d e r m o t i o n is c o n s t r a i n e d i n the x d i r e c t i o n b y a p a i r o f hnear h o r i z o n t a l springs l o c a t e d at the s t i l l w a t e r level a n d w i t h a c o m b i n e d e q u i v a l e n t stiffness k = {•K/2)pga. W i t h this p a r d c u l a r c o n f i g u r a -d o n , r o t a -d o n a l m o -d o n s i n -d u c e -d b y the m o m e n t c o m p o n e n t s v a n i s h and are therefore n o t considered i n the c o m p u t a t i o n . T h e i n c i d e n t waves c o r r e s p o n d t o a Stokes secondorder wave t r a i n inside the c o m -p u t a d o n a l d o m a i n , and the m o d u l a t i o n f u n c t i o n is used t o p r o v i d e the i n i t i a l c o n d i t i o n s a n d the i n i t i a l d e v e l o p m e n t o f the f l o w .
F i g u r e 4 shows the d e v e l o p m e n t w i t h time o f the free surface p r o f i l e s t o f i r s t a n d second o r d e r f o r
i/a/g = 0-4, A/a = 0-3 a n d deep w a t e r , where u> a n d A are, respectively, the angular f r e q u e n c y a n d a m p l i t u d e
o f the first-order i n c i d e n t wave t r a i n . T h e c y l i n d e r is located at the centre o f the d o m a i n w i t h h o r i z o n t a l l e n g t h o f AL, where L is the w a v e l e n g t h o f the i n c i d e n t wave t r a i n . I n the figure, a steady state is a t t a i n e d f o r the f u l l y developed wave field a f e w cycles a f t e r the m o d u l a t i o n t i m e , and the secondorder wave c o m -ponents associated w i t h the f o r c e d a n d free waves are observed to be c o n c e n t r a t e d near the c y l i n d e r . I n a d d i t i o n , the u p w a v e r e g i o n i n w h i c h i n c i d e n t a n d p e r t u r b e d waves p r o p a g a t e i n opposite directions is characterized b y the presence o f s t r o n g second-order wave effects. Since the i n c i d e n t a n d p e r t u r b e d waves i n the d o w n w a v e r e g i o n are o u t o f phase a n d p r o p a g a t e i n the same d i r e c d o n , the r e s u l t i n g wave a m p l i t u d e a n d the related second-order c o m p o n e n t s are relatively s m a l l . I n the deep water case considered here, the free wave
' i ' :
^ ^ t / T = 3 — y
1 . 1 , 1 .
1 — ' 1 — >
1 . 1 , 1 .
Fig. 4. Development with time of free surface profiles f o r a moored semi-submerged circular cylinder in waves f o r
iJa/g = 0-4, A/a = 0-3 and deep water. , solution to
first order; , solution to second order.
system is expected to be the d o m i n a n t second-order wave c o m p o n e n t .
F i g u r e 5 shows d e v e l o p m e n t w i t h t i m e o f the h o r i z o n t a l a n d v e r t i c a l m o t i o n s (denoted b y ^ a n d C respectively) a n d the c o r r e s p o n d i n g h y d r o d y n a m i c f o r c e c o m p o n e n t s ( d e n o t e d b y a n d F^ respectively) t o first a n d second o r d e r f o r the same c o n d i t i o n s , u/a/g = 0-4,
A/a = 0-3 a n d deep water. A s i n d i c a t e d i n the figure, the
b o d y m o t i o n s a n d the h y d r o d y n a m i c forces are m o d u l a t e d t o suppress the transient effects associated w i t h the i n i t i a l c o n d i t i o n s , a n d a second-order steady state s o l u t i o n is g r a d u a l l y developed over a reasonably s h o r t d u r a t i o n a f t e r the f u l l i m p o s i t i o n o f the b o d y -surface b o u n d a r y c o n d i t i o n . I n this p a r t i c u l a r case, the second-order c o m p o n e n t s c o n t r i b u t e s i g n i f i c a n t l y t o the t o t a l v e r t i c a l h y d r o d y n a m i c f o r c e a n d the c o r r e s p o n d i n g
Fig. 5. Development with time of the motion response and hydrodynamic force components f o r a moored semi-sub-merged circular cylinder i n waves for u/a/g = 0-4, A/a = 0-3 and deep water. , solution to first order; , solution
heave m o t i o n . O n the other h a n d , the amplitudes o f the second-order force components i n the x d i r e c t i o n an the r e s u l t i n g sway responses are relatively s m a l l c o m p a r e d w i t h those o f the first-order s o l u t i o n . I t is seen t h a t the h o r i z o n t a l d r i f t f o r c e a n d the r e s u l t i n g steady m o t i o n are i n the d i r e c t i o n o f wave p r o p a g a t i o n as expected, w h i l e their v e r t i c a l counterparts are i n the negative z d i r e c t i o n .
F i g u r e 6 shows the a m p l i t u d e a n d phase angle o f the first-order o s c i l l a t o r y f o r c e i n the h o r i z o n t a l a n d v e r t i c a l directions as f u n c t i o n s o f dimensionless wave f r e q u e n c y
uP'alg
f o r a semi-submerged c i r c u l a r c y l i n d e r i n deep water. I n order t o c o n f i r m these results, c o r r e s p o n d i n g first-order results have also been o b t a i n e d f r o m a f r e q u e n c y - d o m a i n s o l u t i o n o f the equations o f m o t i o n using added masses, d a m p i n g coefficients a n d wave-exciting forces o b t a i n e d f r o m the l i t e r a t u r e (e.g. D e a n & Ursell,^^ Vugts,^'' and Nestegard & Sclavounos^^). These results are i d e n t i c a l to those o f F i g . 6 a n d hence are n o t i n c l u d e d i n the figure. T h e first-order h o r i z o n t a l h y d r o d y n a m i c f o r c e is seen t o a p p r o a c h zero w h e n the dimensionless frequency u^ajg = TO, c o r r e s p o n d i n g t o the surge n a t u r a l frequency (defined w i t h the added mass excluded). T h e c o r r e s p o n d i n g phase angle t h e n undergoes a s h i f t o f ir. These features m a y readily be explained b y considering the e q u a t i o n o f m o t i o n i n the h o r i z o n t a l d i r e c t i o n , b e a r i n g i n m i n d t h a t wave r a d i a t i o n d a m p i n g is i n c l u d e d i n the h y d r o d y n a m i c f o r c e a n d t h a t the s t r u c t u r a l stiffness is p r o v i d e d solely b y the h o r i z o n t a l m o o r i n g lines. I n a s i m i l a r manner, the first-order vertical h y d r o d y n a m i c f o r c e approaches the2.5
. g o I : ^ ' 1 ' 1 ' ' ' '
0.0 0.4 O.B 1.2 1.6 2.0 ua/g
Fig. 6. Amplitude and phase angle of first-order oscillatory forces on the cylinder as functions of u/a/g. • , horizontazl
component; O , vertical component.
h y d r o s t a t i c r e s t o r i n g f o r c e at the heave n a t u r a l f r e q u e n c y . F o r s m a l l values o f o?a/g, i t is i n d i c a t e d t h a t the first-order v e r t i c a l h y d r o d y n a m i c force is o u t o f phase w i t h the free surface elevation.
F i g u r e 7 presents the second-order h o r i z o n t a l a n d vertical d r i f t forces as f u n c t i o n s o f u^a/g f o r a semi-submerged c i r c u l a r cylinder i n deep water. I t is evident t h a t the h o r i z o n t a l d r i f t f o r c e always acts i n the d i r e c t i o n o f wave p r o p a g a t i o n , as first s h o w n b y M a r u o . F o r the floating cylinder considered here, the v e r t i c a l d r i f t f o r c e acts d o w n w a r d s and is p r i m a r i l y due t o negative d y n a m i c pressure associated w i t h the v e l o c i t y squared t e r m i n the B e r n o u l h e q u a t i o n . F u r t h e r m o r e , its m a g n i t u d e increases r a p i d l y w i t h w a v e f r e q u e n c y to a peak and then reduces to a f a i r l y m o d e s t value at h i g h wave frequencies. N o t e t h a t the h o r i z o n t a l a n d v e r t i c a l d r i f t f o r c e components are counteracted, respectively, by the m o o r i n g line r e s t o r i n g f o r c e and the second-order f o r c e due t o the h y d r o s t a t i c stiffness, a n d these together give rise t o the c o r r e s p o n d i n g steady d r i f t m o t i o n s .
T h e a m p l i t u d e a n d phase angle o f the c o r r e s p o n d i n g second-order o s c i l l a t o r y force i n the h o r i z o n t a l a n d v e r t i c a l directions as f u n c t i o n s o f u?a/g are presented i n F i g . 8. I n this case, the h o r i z o n t a l second-order o s c i l l a t o r y f o r c e approaches zero at Lo^a/g = 0-25, c o r r e s p o n d i n g t o a wave f r e q u e n c y o f h a l f the n a t u r a l f r e q u e n c y , a n d associated w i t h the second-order f o r c i n g o c c u r r i n g at twice the i n c i d e n t wave f r e q u e n c y . F o r higher w a v e frequencies, b o t h the h o r i z o n t a l a n d v e r t i c a l second-order o s c i l l a t o r y components increase steadily w i t h u?a/g.
C o r r e s p o n d i n g results r e l a t i n g t o the c y l i n d e r m o t i o n s are s h o w n i n Figs 9 - 1 1 . F i g u r e 9 shows the a m p h t u d e a n d phase angle o f the first-order o s c i l l a t o r y sway a n d heave response as f u n c t i o n s o f u^a/g. T h e heave m o t i o n exhibits v i r t u a l l y n o d y n a m i c a m p l i t u d e at s m a l l values o f u?a/g a n d is then i n phase w i t h the i n c i d e n t w a v e c o m p o n e n t , as expected. A s i n d i c a t e d i n the figure, the a m p l i t u d e s o f the sway a n d heave m o t i o n s , b o t h o f w h i c h e x h i b i t a s i m i l a r t r e n d , increase w i t h increasing
1.0 I ^ r
0.0 0.4 O.a 1.2 1.6 2.0
Fig. 7. Magnitude of second-order drift forces on the cylinder as functions of u?alg. • , horizontal component; O , vertical
104 Joseph Y. T. Ng, Michael Isaacson
0.0 0.4 1.6 2.0
Fig. 8. Amplitude and phase angle of second-order oscillatory forces on the cylinder as functions o f u/a/g. # , horizontal
component; O , vertical component.
Fig. 10. Magnitude o f second-order d r i f t motion response o f the cyhnder as functions of i^/a/g. 9, sway; O , heave. values o f uP'a/g to d i f f e r e n t peak values a n d d r o p o f f steadily at h i g h wave frequencies. Once again, cor-r e s p o n d i n g cor-results have been o b t a i n e d o n the basis o f a f r e q u e n c y - d o m a i n a p p r o a c h . These are i d e n t i c a l w i t h those o f F i g . 9 a n d hence are n o t i n c l u d e d i n the figure.
F i g u r e 10 shows the c o r r e s p o n d i n g m a g n i t u d e o f the second-order d r i f t sway a n d heave response as f u n c t i o n s
ofo/a/g. T h e results indicate t h a t the second-order d r i f t
m o d o n s e x h i b i t s i m i l a r trends to those o f the cor-r e s p o n d i n g d cor-r i f t f o cor-r c e c o m p o n e n t s , as expected. T h e m e a n sway response is caused by the h o r i z o n t a l d r i f t f o r c e a n d corresponds t o a constant offset i n the p o s i d v e
X d i r e c t i o n , whereas the mean heave response is i n d u c e d
set-down o f the cylinder. F i n a l l y , F i g . 11 shows the a m p l i t u d e a n d phase angle o f the c o r r e s p o n d i n g second-order oscillatory sway and heave response. I n the f i g u r e , the amplitudes o f the second-order o s c i l l a t o r y sway a n d heave response e x h i b i t considerable v a r i a d o n s over the f r e q u e n c y range o f interest. F u r t h e r m o r e , the t r e n d o f the second-harmonic heave response appears t o be s i m i l a r to that o f v e r t i c a l second-order o s c i l l a t o r y f o r c e f o r relatively l o w wave frequencies.
5 C O N C L U S I O N S
A t i m e - d o m a i n second-order m e t h o d has been extended to study n o n l i n e a r wave interactions w i t h a large floating structure o f a r b i t r a r y shape i n t w o d i m e n -sions. The resulting flow development a n d structure m o d o n s t o second order are o b t a i n e d by a time-stepping scheme, i n w h i c h the field s o l u t i o n at each t i m e step is calculated b y a b o u n d a r y i n t e g r a l e q u a t i o n m e t h o d based o n Green's t h e o r e m . Steady state solutions are a t t a i n e d over a r e l a t i v e l y short d u r a t i o n t h r o u g h a g r a d u a l i m p o s i t i o n o f the b o d y surface b o u n d a r y c o n d i t i o n . The present t i m e - d o m a i n m e t h o d provides a relatively algebraically s t r a i g h t f o r w a r d a n d c o m -p u t a t i o n a l l y effective n u m e r i c a l a l g o r i t h m f o r t r e a t i n g the second-order d i f f r a c t i o n - r a d i a t i o n p r o b l e m i n r e l a t i o n t o c o n v e n t i o n a l f r e q u e n c y - d o m a i n m e t h o d s .
T h e m e t h o d is a p p l i e d to the case o f a semi-submerged circular c y l i n d e r i n deep water, b o t h f o r the transient m o t i o n o f a f r e e l y - f l o a t i n g c y l i n d e r i n otherwise still water, as w e l l as the i n t e r a c t i o n o f Stokes second-order waves w i t h a m o o r e d c y l i n d e r . T h e results indicate t h a t n o substantial second-order w a v e effects are present i n the predicted f r e e b o d y m o t i o n s . I n the general case o f a m o o r e d floating b o d y i n regular waves, s i g n i f i c a n t second-order f o r c e a n d m o t i o n components are developed at c e r t a i n wave frequencies. The a p p l i c a t i o n o f the m e t h o d to three-d i m e n s i o n a l p r o b l e m s m a y r e a three-d i l y be achievethree-d o n the same basis.
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