Fast computation of the transient motions of moving vessels in irregular ocean waves

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Applied Ocean Research 65 (2017) 23-34

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Applied Ocean Research

j o u r n a l hornepagerwwwreIi5evier.com7locate/apoT"

Fast computation ofthe transient motions of moving vessels in

irregular ocean waves

Okey G. Nwogu*, Robert F. Beck

Department of Naval Architecture and IVlarine Engineering, University of Michigan, United States

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CrossMark

A R T I C L E I N F O

-Artide history: Received 24 October 2016

Received in revised form S February 2017 Accepted 21 March 2017

Available online 2 April 2017

Keywords: Real-time Ship motions Panel method EFT acceleration A B S T R A C T

A fast t i m e - d o m a i n method is developed in this paper for the real-time prediction o f t h e six degree of freedom motions of a vessel traveling in an irregular s e a w a y in infinitely deep water. T h e fully coupled unsteady ship motion problem is solved by time-stepping the linearized boundary conditions on both the free surface a n d body surface. A velocity-based b o u n d a i ^ integral method is then used to solve the Laplace equation at every time step for the fluid kinematics, w h i l e a scalar integral equation is solved for the total fluid pressure. T h e boundary integral equations are applied to both the physical fluid d o m a i n outside the body and a flcririous fluid region inside the body, enabling use o f t h e fast Fourier transform m e t h o d to evaluate the free surface integrals. The computational efficiency o f t h e s c h e m e is further improved through use o f t h e method of images to eliminate source singularities on the free surface w h i l e retaining vortex/dipole singularities that decay more rapidly in space. T h e resulting n u m e r i c a l algorithm runs 2 - 3 times faster t h a n real t i m e on a standard desktop computer. N u m e r i c a l predictions are compared to prior p u b h s h e d results for the transient motions of a h e m i s p h e r e and laboratory m e a s u r e m e n t s o f t h e morions of a free r u n n i n g vessel in oblique w a v e s w i t h good agreement.

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1 . I n t r o d u c t i o n

Technological advances i n the r e a l - t i m e measurement o f ocean w a v e c o n d i t i o n s w i t h r e m o t e sensing devices have led to a g r o w t h i n t h e need f o r fast techniques to p r e d i c t s h i p / p l a t f o r m m o t i o n s i n r e a l - t i m e aboard ships and oil p l a t f o r m s [ 1 - 4 ]. T h e m o t i o n forecasts c o u l d be used to p r o v i d e s h i p / p l a t f o r m operators advance w a r n i n g o f dangerous m o t i o n s ; evaluate i n r e a l - t i m e t h e o p t i m a l p a t h o f m a n n e d / u n m a n n e d vessels t o m i t i g a t e vessel m o t i o n s ; and design r e a l - t i m e m o t i o n c o m p e n s a t i o n systems f o r ships, ocean p l a t f o r m s o r w a v e energy devices.

N u m e r i c a l methods based o n b o u n d a r y i n t e g r a l equations are o f t e n used t o s i m u l a t e t r a n s i e n t w a v e - b o d y i n t e r a c t i o n . A n early f o r m u l a t i o n b y Finkelstein [ 5 ] u t i l i z e d a time-dependent Green's f u n c t i o n t h a t satisfies the linearized f r e e surface b o u n d a r y c o n -d i t i o n s . The source strengths are t h e n -d e t e r m i n e -d t o satisfy the k i n e m a t i c b o u n d a r y c o n d i t i o n o n the b o d y surface. Practical t h r e e -d i m e n s i o n a l c o m p u t a t i o n o f s h i p m o t i o n s w i t h a time--depen-dent Green's f u n c t i o n w e r e later o b t a i n e d b y a n u m b e r o f investigators [ 6 - 1 0 ].

* Corresponding author.

E-mail address: onwogu@umich.edu (O.G. Nwogu).

http://dx.doi.Org/10.1016/j.apor.2017.03.010

0 1 4 1 - 1 1 8 7 / ® 2017 Elsevier Ltd. All rights reserved.

N o n l i n e a r w a v e effects o n ship m o t i o n s can be i n c l u d e d b y u t i -l i z i n g the s i m p -l e r f r e e space Green's f u n c t i o n (Rankine source) and time-stepping the b o u n d a r y c o n d i t i o n s o n the free surface. Several n u m e r i c a l m e t h o d s have been developed along this l i n e i n c l u d i n g Isaacson [ 1 1 ], Beck et aL [ 1 2 ], K r i n g et a t [ 1 3 ] and L i u et aL [ 1 4 ]. The Rankine panel m e t h o d t y p i c a l l y leads t o a large m a t r i x t h a t is c o m p u t a t i o n a l l y expensive to i n v e r t since t h e free surface has t o be discretized i n a d d i t i o n to the b o d y surface. Several accelerated m e t h o d s have been developed to speed u p t h e c o m p u t a t i o n s such as the m u l t i p o l e expansion m e t h o d [15] a n d the pre-corrected FFT m e t h o d [ 1 6 ].

M i x e d spectral-panel m e t h o d s have also been developed f o r solving t r a n s i e n t w a v e - b o d y i n t e r a c t i o n problems. Chapman [17,18] pioneered this approach b y c o m b i n i n g a discrete Fourier series representation o f the free surface v e l o c i t y p o t e n t i a l w i t h a panel m e t h o d f o r the b o d y m o t i o n p o t e n t i a l . The Green's f u n c -t i o n f o r -the b o d y p o -t e n -t i a l was chosen -to sa-tisfy -the e q u i p o -t e n -t i a l c o n d i t i o n o n t h e mean f r e e surface. Chapman [ 1 8 | r e p o r t e d good results f o r the h e a v i n g m o t i o n s o f a hemisphere a l t h o u g h use o f a p u r e spectral m e t h o d w i t h i m p l i c i t p e r i o d i c b o u n d a r y c o n d i -tions makes i t d i f f i c u l t to enforce t h e r a d i a t i o n b o u n d a r y c o n d i t i o n f o r t h e scattered waves. Chapman's approach was also restricted to s m a l l - a m p l i t u d e waves. A n alternative spectral f o r m u l a t i o n f o r w a v e b o d y i n t e r a c t i o n p r o b l e m s t h a t allows f o r h i g h e r o r d e r n o n

-2 4 O.G. Nwogu, R.F. Beck /Applied Ocean Research 65 (2017) 23-34

linear w a v e effects is based o n the pseudo-spectral approach. In contrast to p u r e spectral methods t h a t evaluate the g o v e r n i n g equations i n Fourier space, the pseudo-spectral m e t h o d evaluates linear terms i n Fourier space and nonlinear t e r m s i n physical space. Pseudospectral methods have successfully been used t o i n v e s t i -gate nonlinear w a v e i n t e r a c t i o n w i t h submerged bodies [ 1 9 - 2 1 ]. Another v a r i a n t o f the spectral-panel m e t h o d is the FFT-accelerated b o u n d a r y integral m e t h o d developed b y N w o g u [22] f o r nonlinear w a t e r w a v e propagation. For n o n - o v e r t u r n i n g waves, the kernels o f the free surface b o u n d a r y integrals are expanded i n a wave steepness parameter and e f f i c i e n t l y evaluated w i t h FFTs. N w o g u and Beck [23] extended this approach to surface-piercing bodies o f a r b i t r a r y shape b y a p p l y i n g the b o u n d a r y i n t e g r a l equa-tions to b o t h the physical f l u i d d o m a i n outside the b o d y and a f i c t i t i o u s f l u i d r e g i o n inside the body. The FFT m e t h o d is t h e n used t o evaluate the integrals over the free surface i n b o t h the i n t e -r i o -r and exte-rio-r -regions w h i l e the integ-rals ove-r the b o d y su-rface are evaluated u s i n g a panel m e t h o d . P r e l i m i n a r y results w e r e p r e -sented b y N w o g u and Beck [23] f o r linear w a v e d i f f r a c t i o n b y a v e r t i c a l circular cylinder.

I n this paper, the FFT-accelerated b o u n d a r y i n t e g r a l m e t h o d is f u r t h e r extended to simulate the 6 - d o f m o t i o n s o f m o v i n g vessels i n m u l t i d i r e c t i o n a l w a v e fields. The coupled equations o f m o t i o n f o r the free surface and b o d y are s i m u l t a n e o u s l y i n t e g r a t e d using a fractional-step m e t h o d . The velocity/pressure f o r m u l a t i o n s o f the equations o f f l u i d m o t i o n are used w i t h separate b o u n d a r y integral equations solved at each t i m e step f o r the f l u i d k i n e m a t i c s and t o t a l pressure. The m i x e d spectral-panel m e t h o d is i n i t i a l l y validated w i t h p r i o r p u b l i s h e d data o n the f o r c e d oscillations a n d transient m o t i o n s of a f l o a t i n g hemisphere. The n u m e r i c a l m o d e l is t h e n used t o p r e d i c t the t r a n s i e n t m o t i o n s o f a s e l f - p r o p e l l e d vessel i n oblique waves and c o m p a r e d to data f r o m l a b o r a t o r y e x p e r i m e n t s .

2 . M a t h e m a t i c a l f o r m u l a t i o n

2.1. Equations of body motion

Consider a vessel t r a v e l i n g w i t h speed U{t) i n a m u l d d i r e c t i o n a l w a v e f i e l d i n i n f i n i t e l y deep w a t e r The body m o t i o n s are d e f i n e d i n t e r m s o f three t r a n s l a t i o n a l (surge, sway, heave) and three angular rotations ( r o l l , p i t c h , y a w ) about a b o d y - f i x e d c o o r d i n a t e system w i t h o r i g i n at its center of g r a v i t y G and o r i e n t e d along t h e p r i n c i p a l axes o f i n e r t i a o f the body. The six degrees o f f r e e d o m equations f o r r i g i d b o d y m o t i o n can be w r i t t e n as:

[m]{Vi] = F (1) [ l ] ( f t f c } = M (2) w h e r e [ m ] is the mass m a t r i x , [/] is m o m e n t o f i n e r t i a m a t r i x , V,,

is the t r a n s l a t i o n a l velocity i n the b o d y - f i x e d (x,y,z) directions and Sit is the angular v e l o c i t y vector i n the b o d y axis. The i n v i s -c i d h y d r o d y n a m l -c for-ce F and m o m e n t M ve-ctors are o b t a i n e d by i n t e g r a t i n g the f l u i d pressure p over the w e t t e d b o d y surface:

fl

X xn

dS (3)

w h e r e n ( x , y , z) is an i n w a r d u n i t n o r m a l vector o n the b o d y surface i n the b o d y - f i x e d coordinate system.

2.2. Equations of fluid motion

The equations g o v e r n i n g the f l u i d m o t i o n are d e f i n e d i n a r i g h t -handed Oxyz coordinate system t r a v e l i n g w i t h the ship w i t h its o r i g i n at the s t i l l w a t e r level v e r t i c a l l y i n line w i t h the ship's center o f gravity, G. The x-axis is assumed t o p o i n t t o w a r d the ship's b o w w h i l e the z-axis is measured v e r t i c a l l y u p w a r d s as s h o w n i n Fig. 1.

Fig. 1. Definition sl<etcii.

The f l u i d is assumed to be i n v i s c i d and incompressible w i t h the fluid m o t i o n i r r o t a t i o n a l . The g o v e r n i n g equations f o r the fluid m o t i o n are the c o n t i n u i t y e q u a t i o n f o r the conservation o f mass and Euler's equations o f m o t i o n f o r the conservation o f m o m e n t u m :

V • u = 0 ( 4 ) ( 5 ) -1-VP = 0 9ii dt w h e r e V = (9/9x, 9/3y, 9/9z), u = (u, v, w ) is t h e t h r e e - d i m e n s i o n a l v e l o c i t y field, P=plp + u-ul2+gz is the t o t a l pressure, p is t h e local fluid pressure, p is the fluid density and g is the g r a v i t a -t i o n a l accelera-tion. W e assume s m a l l - a m p l i -t u d e i n c i d e n -t waves and decompose the t o t a l pressure and v e l o c i t y fields i n t o c o m p o -nents associated w i t h t h e i n c i d e n t and scattered waves, r e s u l t i n g

p = p(') + p M ; u = U + u<-'1 + u^'^ (6) The e f f e c t o f the f o r w a r d speed o f the vessel is i n c l u d e d i n the v e l o c i t y field as an i m p o s e d c u r r e n t U = (-17, 0, 0).

2.2.J. Boundary value problem for pressure

The g o v e r n i n g e q u a t i o n f o r the t o t a l scattered pressure over the fluid d o m a i n is the Laplace e q u a t i o n w h i c h is obtained b y t a k i n g the divergence o f the Euler e q u a t i o n (Eq. ( 5 )) :

V 2 p ( S ) ^ Q

₍₇₎

The t o t a l scattered pressure field has to satisfy b o u n d a r y c o n d i -tions o n the free surface S f , b o d y surface SB a n d f a r - f i e l d b o u n d a r y Soo. Given t h a t p = 0 o n t h e free surface z = ï ) , t h e linear b o u n d a r y c o n d i t i o n f o r the t o t a l scattered pressure o n Sp can be w r i t t e n as:

p W = g , , + U . „(S) (8)

The pressure b o u n d a r y c o n d i t i o n o n the b o d y surface is obtained b y t a k i n g the d o t p r o d u c t o f the Euler e q u a t i o n (Eq. (5)) w i t h a u n i t n o r m a l vector o n the b o d y surface:

„ . V P ( S ) . - „ . | H ' ^ '

d t

( 9 )

w h e r e n(x, t) is a u n i t i n w a r d n o r m a l vector o n the body surface i n the h y d r o d y n a m l c reference f r a m e . Since the fluid pressure needs to be calculated at p o i n t s fixed o n a m o v i n g b o d y surface, i t is m o r e convenient to evaluate the t i m e d e r i v a t i v e i n the b o d y - f i x e d coordinate system:

, . = „ . - n . m t -f fib X X ) . V)u(^)]

^ ( i f ^ _ „ ( 5 ) . | _ „ . [ ( ( v , + ft,xS).V)u(«] ( 1 0 )

O.G. Nwogu, R.F. Beck/Applied Ocean Research 65(2017)23-34 ₂₅ Given thiat d n / d f = ( f t j x n ) , t l i e b o u n d a r y c o n d i t i o n f o r the t o t a l

pressure on the b o d y surface can be f u r t h e r reduced t o :

n • VP(^) = -d(ti' ( S ) ,

dt + i j ( ^ ' - ( i 2 b x n )

+ n . [ ( ( l / b + flj,xx).V)iiM] ₍₁₁₎

The n o r m a l v e l o c i t y o f the scattered w a v e f i e l d is g i v e n b y the k i n e m a t i c b o u n d a r y c o n d i t i o n o n the b o d y surface SB:

u^^^ • n = (Vb + fli X x ) • n - U • II - u " ) • i i (12)

S u b s t i t u t i n g Eq. ( 1 2 ) i n t o Eq. ( 1 1 ) and n e g l e c t i n g h i g h e r - o r d e r t e r m s i n v o l v i n g p r o d u c t s o f t h e o s c i l l a t o r y f l u i d and b o d y v e l o c i -ties leads t o the f i n a l f o r m o f the b o d y b o u n d a r y c o n d i t i o n f o r the scattered pressure:

j = i

(13)

w h e r e a n d tij represent the generalized d i s p l a c e m e n t and n o r m a l f o r the j t h m o d e o f m o t i o n respectively w i t h :

(14) ( ^ 4 , ^ 5 , ^ 6 ) = a i ,

( n ] , n 2 , n 3 ) = n ( n 4 , n 5 , n 6 ) = S x n

The irij t e r m s arise f r o m the last t w o t e r m s o n t h e r i g h t h a n d side o f Eq. (11) and represent c o u p l i n g b e t w e e n t h e steady f l o w a n d unsteady r o t a t i o n o f the b o d y . They w e r e o r i g i n a l l y d e r i v e d by T i m m a n and N e w m a n [ 3 6 ] as a c o r r e c t i o n t o e v a l u a t i n g the k i n e -m a t i c b o u n d a r y c o n d i t i o n at the -m e a n p o s i t i o n o f t h e b o d y instead o f its instantaneous p o s i t i o n . For slender bodies, the d o m i n a n t c o n -t r i b u -t i o n -to -the -trij -t e r m s is due -t o U • ( f i j x n ) :

( m i , m 2 , m a ) = ( 0 , 0 , 0 ) ( m 4 , m s , m e ) = (0, Un^, -Utiy

(15)

W e note t h a t f o r b l u n t bodies, s i g n i f i c a n t spatial variations occur i n the local steady f l o w f i e l d and a d d i t i o n a l c o n t r i b u t i o n s to the mj t e r m s due t o the spatial g r a d i e n t t e r m i n Eq. (11) s h o u l d be retained.

The f a r - f i e l d b o u n d a r y Soc encompasses the l a t e r a l boundaries o f the c o m p u t a t i o n a l d o m a i n and i n f i n i t e w a t e r d e p t h . The lat-eral boundaries are t r e a t e d as p e r i o d i c boundaries w i t h o u t w a r d p r o p a g a t i n g waves absorbed i n d a m p i n g layers placed n e x t to the boundaries. The i m p u l s i v e and m e m o r y c o m p o n e n t s o f the t o t a l pressure t h u s vanish o n 5^, since the fluid m o t i o n s are e i t h e r d a m p e d o u t or decay t o zero a s z - > - o o :

p t s ) - ^ 0 onSoo ( 1 6 ) One p o t e n t i a l d i f f i c u l t y w i t h u s i n g Eq. ( 1 3 ) to solve the b o u n d a r y

value p r o b l e m f o r the t o t a l fluid pressure is t h a t the b o d y b o u n d a r y c o n d i t i o n depends o n t h e instantaneous acceleration w h i c h needs t o be d e t e r m i n e d as p a r t o f the s o l u t i o n o f t h e e q u a t i o n o f m o t i o n s f o r the b o d y . To m i t i g a t e the d e s t a b i l i z i n g e f f e c t o f the c o u p l i n g b e t w e e n t h e ship and fluid m o t i o n s , w e a d o p t t h e approach o f C u m m i n s [ 2 4 ] and decompose t h e t o t a l scattered pressure i n t o an i m p u l s i v e c o m p o n e n t P(<^) due t o t h e instantaneous m o t i o n o f t h e body, and a m e m o r y c o m p o n e n t P'-'^l associated w i t h the radiated f r e e surface waves a f t e r the i m p u l s i v e fluid m o t i o n :

piS)^p{oc)^p(M) ^^^^ The i m p u l s i v e pressure occurs at a v e r y s m a l l t i m e scale c o m

-pared to g r a v i t a t i o n a l effects and can consequently be neglected o n

the free surface. The free surface and b o d y b o u n d a r y c o n d i t i o n s f o r the i m p u l s i v e and m e m o r y c o m p o n e n t s o f the t o t a l pressure are thus decomposed as:

P<°°' = 0 o n S f _ 6 6 p ( M ) = g , , + [ / . u ( ^ ) o n S f „ . V p W = | ( u ( ' ) . n ) onSB (18) (19) (20) (21)

2.2.2. Initial/boundary value problem for fluid kinematics The g o v e r n i n g e q u a t i o n f o r the v e l o c i t y field o f the scattered waves over the fluid d o m a i n is o b t a i n e d b y t a k i n g the g r a d i e n t o f the c o n t i n u i t y e q u a t i o n (Eq. (4)), r e s u l t i n g i n the v e c t o r f o r m o f t h e Laplace e q u a t i o n :

(22) The fluid velocities have to satisfy the Laplace e q u a t i o n over the fluid d o m a i n subject to b o u n d a r y c o n d i t i o n s o n the free surface Sp, body surface SB and f a r - f i e l d b o u n d a r y S^. The linearized f r e e surface b o u n d a r y c o n d i t i o n s can be expressed as a set o f e v o l u t i o n equations f o r the f r e e surface e l e v a t i o n n and h o r i z o n t a l velocities ( u j f ' , 1/J;') at the s t i l l w a t e r level ( z = 0 ) : dul dt dt ' 3y l^bVo (23) (24) (25)

w h e r e w f f ' is the v e r t i c a l v e l o c i t y at the s t i l l w a t e r level, (W,(x, y) is a d a m p i n g c o e f f i c i e n t t h a t is a p p l i e d over t h e i n t e r i o r o f t h e b o d y to suppress resonant i n t e r i o r fluid m o t i o n s and Ati)(x,y) is t h e d a m p i n g c o e f f i c i e n t o f a n u m e r i c a l beach t h a t is used to absorb o u t w a r d p r o p a g a t i n g waves. The k i n e m a t i c b o u n d a r y c o n d i t i o n on the b o d y surface SB is g i v e n b y Eq. (13). The fluid velocities v a n i s h o n S^o since the fluid m o t i o n s are e i t h e r d a m p e d o u t or decay to zero as Z ^ - o o .

The scattered v e l o c i t y field is also decomposed i n t o an i m p u l s i v e c o m p o n e n t due t o the instantaneous m o t i o n of t h e b o d y and a m e m o r y c o m p o n e n t , u ^ , associated w i t h the r a d i a t e d free surface waves a f t e r the i m p u l s i v e fluid m o t i o n :

( 2 6 ) The i m p u l s i v e m o t i o n o f t h e b o d y introduces a v e r t i c a l flux at the s t i l l w a t e r level t h a t radiates o u t as free surface waves. The free sur-face and b o d y b o u n d a r y c o n d i t i o n s f o r the i m p u l s i v e and m e m o r y c o m p o n e n t s o f the scattered fluid velocities are thus decomposed as: u[,°°' = 0: v[,°°' = 0 o n S f ^ = v v t ' ^ ' + w r + U ^ - ( M b + M,)'7 onSp _dt dr, dx ^ = - | [ g , - t ; u r i - / x . u ^ - ) o n s , dr, dt dy - / X f c v ' ^ ' onSp ( 2 7 ) ( 2 8 ) ( 2 9 ) ( 3 0 )

26 O.G. Nwogu, R.F. Beck/Applied Ocean Research 65 (2017)23-3'! 6 6 u('^).n = J2^jnj + J2^jmj on SB (31) j = i J=i u ( ' ^ ' . n = - l l - n - u ( " n ODSB (32) 3. N u m e r i c a l s o l u t i o n

3.1. Time integration of equations of motion

The transient w a v e - b o d y i n t e r a c t i o n p r o b l e m is solved over a d o u b l y - p e r i o d i c c o m p u t a t i o n a l d o m a i n o f size Lx, Ly w i t h u n i f o r m g r i d sizes A x and A y i n the x and y directions respectively. The free surface variables are d e f i n e d at the g r i d p o i n t s i n a staggered m a n n e r w i t h t h e surface elevation d e f i n e d at the g r i d nodes w h i l e the h o r i z o n t a l velocities are d e f i n e d at half a g r i d p o i n t on either side o f the elevation g r i d points.

The coupled equations of m o r i o n f o r the fluid and b o d y are i n t e -grated s i m u l t a n e o u s l y w i t h a f r a c t i o n a l step m e t h o d . A t any instant of t i m e t, a f o u r t h - o r d e r R u n g e - K u t t a scheme is used to integrate the free surface e v o l u r i o n equations f r o m t t o t + A t w i t h t h e n o r m a l b o d y v e l o c i t y assumed to be constant over t h a t time i n t e r v a l and equal to its value at t + A t / 2 . A d y n a m i c boundary-value p r o b l e m is solved f o r t h e fluid pressure and i n t e g r a t e d over the i m m e r s e d b o d y surface t o y i e l d t h e e x c i t i n g forces o n the ship. The 6-dof equations o f b o d y m o t i o n are t h e n i n t e g r a t e d w i t h a f o u r t h - o r d e r R u n g e - K u t t a scheme f r o m f + A t / 2 to t + 3 A t / 2 t o d e t e r m i n e the b o d y m o t i o n s and b o u n d a r y conditions f o r t h e n e x t fluid m o t i o n t i m e step.

?ÈÉ!l- [ o - t j . ( x ' ) V G * ( x ; x ' ) - n ( x ) dS = mj f o r ) = 1 6 ( 3 8 )

2 JSB

Eqs. (37) and (38) are solved n u m e r i c a l l y using the constant panel m e t h o d [ 2 5 ]. The i m m e r s e d body surface is discretized i n t o a finite n u m b e r o f quadrilateral panels w i t h the source s t r e n g t h d i s t r i b u t i o n assumed to be u n i f o r m over each panel. The b o u n d -ary c o n d i t i o n is t h e n applied at the c e n t r o i d o f each panel t o y i e l d a m a t r i x equation f o r the u n k n o w n source strengths. Once the source strengths have been d e t e r m i n e d , the fluid pressure is evaluated using Eq. (6) and integrated over the b o d y surface to y i e l d the i m p u l s i v e h y d r o d y n a m l c forces. The forces/moments are p r o p o r -tional t o the body acceleration and v e l o c i t y and can be expressed i n terms o f a m a t r i x o f i n f i n i t e - f r e q u e n c y added mass and d a m p i n g terms:

Fij = ki I Paj"idS + / p,,jn,dS = - A , j ( c » ) t j - B^{oc)tj (39) JSB JSB

The i n f i n i t e f r e q u e n c y added mass and d a m p i n g terms are i n d e -p e n d e n t o f time a n d transferred to the l e f t hand side o f the rigid body equations o f m o t i o n (Eqs. (1) and ( 2 ) ) . The final f o r m o f t h e equations o f b o d y m o t i o n i n terms o f the generalized modes o f m o t i o n is given by:

6

j^mij+Aij{^mj+[B|,P+B^ioomj+Bf%\kj

+lCij + qj]^j = Fi(tl i = l , 2 , . . . , 6 (40)

3.2. Boundary integral solution for fluid pressure

The s o l u t i o n o f Laplace equation f o r i m p u l s i v e c o m p o n e n t o f t h e t o t a l pressure subject to boundary c o n d i t i o n s (18) and (19) can be obtained using a b o u n d a r y i n t e g r a l o f sources over the i m m e r s e d b o d y surface:

P ' ° ° ' ( x , t ) = / a(x',t)G*{x;x') dS

JSB

(33)

w h e r e a{x', t) are the u n k n o w n source strengths and G * {x;x') is a Green's f u n c t i o n that satisfies the U p l a c e e q u a t i o n and b o u n d a r y conditions at the free surface (Eq. (18)) and r a d i a t i o n b o u n d a r y Soo. It is g i v e n b y t h e m e t h o d o f images as:

G * ( x ; x ' ) =

" 4 7 r | x - x ' | 4n\x

(34)

w h e r e x'i = (x', y ' , - z ' ) are the i m a g e source locations. S u b s t i t u t i n g Eq. (33) i n t o t h e d y n a m i c b o d y b o u n d a r y c o n d i t i o n (Eq. (19)) yields an integral equation f o r the source strengths:

6 6

/ a{x',t)VG''{x:x')-n{x) dS = ^ I j - n ^ + ^ ^ JT HJ (35)

JSB j = l j = l It is m o r e convenient t o separate o u t the e x p l i c i t t i m e depen-dence o f t h e source s t r e n g t h o n t h e velocities and accelerations f o r the d i f f e r e n t modes o f m o t i o n :

(36) a{x', t) = f ^ a a j i x ' Ü t ) + Y.'^bjW^j

; = l j = l

The i n t e g r a l equations f o r t h e d i f f e r e n t modes o f m o t i o n thus become:

^"-'^^^ - I a „ j ( x ' ) V G * ( x ; * ' ) • n[x) dS = Uj f o r j = 1 , . . . , 6 (37) 2 JSB

w h e r e Cy is the linearized h y d r o s t a t i c r e s t o r i n g force c o e f f i c i e n t and f i is the e x c i t i n g force t h a t includes c o n t r i b u t i o n s f r o m the i n c i d e n t waves a n d the m e m o r y c o m p o n e n t o f t h e h y d r o d y n a m l c force due to w a v e d i f f r a c t i o n and m o t i o n o f the body. W e have also i n c l u d e d e m p i r i c a l linear B'-}'' and quadratic B [ ^ ' viscous d a m p i n g coefficients t h a t are critical f o r r o l l and h o r i z o n t a l plane m o t i o n s (surge, sway and y a w ) , and s o f t s p r i n g constants qj to stabilize the h o r i z o n t a l plane m o t i o n s .

The m e m o r y c o m p o n e n t o f the t o t a l pressure has to satisfy the Laplace equation a n d b o u n d a r y conditions on the f r e e surface, b o d y surface and f a r - f i e l d b o u n d a r y . It is g i v e n at any p o i n t i n t h e fluid d o m a i n b y the scalar f o r m o f Green's second i d e n t i t y as:

P('^)(x) = (f P " ^ ' ( x ' ) V G • n ( x ' ) - G(x; x ' j V p C ^ ' • f i ( x ' ) dS (41)

w h e r e G ( x ; x ' ) = - l / ( 4 7 r ] x - x ' | ) is the free-space Green's f u n c t i o n . W e also consider fictitious fluid m o t i o n s inside t h e b o d y governed by the Laplace e q u a t i o n and free surface b o u n d a r y c o n d i t i o n (Eq. (20)). The b o u n d a r y c o n d i t i o n s along t h e i n t e r i o r b o d y surface can be chosen a r b i t r a r i l y . A p p l y i n g c o n t i n u i t y o f the n o r m a l d e r i v a t i v e o f the t o t a l pressure across the body surface yields a d i p o l e - o n l y d i s t r i b u t i o n w h i l e c o n t i n u i t y o f the t o t a l pressure yields a source-o n l y d i s t r i b u t i source-o n . Tsource-o ensure c source-o n t i n u i t y source-o f the pressure d i s t r i b u t i source-o n across the b o d y surface, w e adopt a source-only d i s t r i b u t i o n and c o m b i n e the t o t a l pressure o n the i n t e r i o r and e x t e r i o r b o d y sur-faces t o o b t a i n :

P t ^ " ( x , t ) = / p ( ' ^ ' ( x ' , t ) V G - n ( x ' ) d S - / G(x;

ls,+sf

x')VpW(^x',t)-n{x')dS+ I crpix',t)G(x;x')dS ( 4 2 )

w h e r e S^'"'' is t h e i n t e r i o r f r e e surface and crp(x', t) represents the j u m p i n the n o r m a l derivative o f the t o t a l pressure across t h e b o d y

surface. W e n e x t use the m e t h o d o f images to e l i m i n a t e the source d i s t r i b u t i o n over the free surface, r e s u l t i n g i n :

p W ( x , t ) = P " ' ^ ( x , t ) + [ ap{x',t)G*{x:H')dS ( 4 3 )

w h e r e pC^ is the t o t a l pressure i n d u c e d on the body surface by the free surface dipoles:

O.G. Nwogu, R.F. Beck/Applied Ocean Research 65 (2017)23-34

e x t e r i o r problems t o o b t a i n :

2/

X t ) = - /

Js, •s,+^m ° 27T\X-X'\3

dS (44)

The source strengths ap are d e t e r m i n e d b y s o l v i n g an integral equation derived f r o m the d y n a m i c body b o u n d a r y c o n d i t i o n (Eq. (21)):

M", t ) f ,

/ ap{x',t)VG*{x:x').n{x)dS JSB

= - g ^ ( u « . n ) + VpP(x, t ) . n ( x )

The constant panel m e t h o d is also used to evaluate the integral over the body surface w i t h the r e s u l t i n g m a t r i x equation i d e n t i c a l to t h a t obtained f o r t h e i m p u l s i v e pressure. The r i g h t - h a n d side of Eq. (45) requires evaluation o f the spatial g r a d i e n t o f the free surface c o m p o n e n t o f the m e m o r y - f l o w pressure. A p p l i c a t i o n o f the gradient operator to Eq. ( 4 4 ) leads t o a h y p e r - s i n g u l a r integral w h i c h is d i f f i c u l t to evaluate n u m e r i c a l l y . Hence, w e use Eq. (5) t o derive a m o r e c o n v e n i e n t f o r m o f 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 n u m e r i c a l evaluation as: a-p(x, t ) - / < T p ( x ' , t ) V C * ( x ; JSB x ' ) . n ( x ) d S = - | - ( u ( ' ^ ) . » i - F u « . n ) at (46)

w h e r e uC^ is t h e f r e e surface c o m p o n e n t o f the f l u i d v e l o c i t y t h a t w i l l be d e r i v e d as p a r t o f t h e b o u n d a r y integral s o l u t i o n o f t h e k i n e m a t i c p r o b l e m .

3.3. Boundary integral solution for fluid kinematics

The i m p u l s i v e c o m p o n e n t o f t h e f l u i d v e l o c i t y is obtained f r o m a b o u n d a r y i n t e g r a l o f sources over the i m m e r s e d b o d y surface as:

° ' ( x , t ) = [ ai'°\x',t)VG*(x:x') dS

JSB

(47)

Given t h a t duldt=- V P, the i m p u l s i v e v e l o c i t y source strengths are u n i q u e l y d e t e r m i n e d f r o m t h e b o u n d a r y i n t e g r a l s o l u t i o n f o r the i m p u l s i v e pressure (Eqs. ( 3 7 ) and (38)), i.e.

6 6

<rlr\x; t ) = -J^^ajWtjit) - Y,crtj(x')^j{t) ( 4 8 )

J=i j = i

The m e m o r y - f l o w c o m p o n e n t o f t h e f l u i d velocities can be obtained f r o m a d i s t r i b u t i o n o f sources a n d vortices along t h e b o u n d a r y u s i n g the v e c t o r f o r m o f Green's second i d e n t i t y :

u " ^ ' ( x , t ) = ƒ ( u ( ' ^ ' ( « ' , t) X n{x'))

- ƒ ( u ( ' " ' ( x ' , t ) . n ( x ' ) ) V G ( x ; x ' ) d S

x V G ( x ; x ' ) d S ₍₄₉₎

w h e r e the source and v o r t e x strengths are respectively the n o r m a l and t a n g e n t i a l velocities a l o n g the b o u n d a r y . W e also consider f i c -titious f l u i d m o t i o n s i n s i d e the b o d y and c o m b i n e the i n t e r i o r and

Sf+S^""' u^'^Hx,t)= f ( u ( ' « ) ( x ' , f ) x n ( x ' ) ) x V G ( x ; x ' ) d S - /

( u " ^ V , t ) . n ( x ' ) ) V G ( x ; « ' ) d S - F / ' ( x ' , t ) V G ( x : x ' ) d S ( 5 0 ) JSB

w h e r e a'-^\x', t) represents the j u m p i n the n o r m a l v e l o c i t y across the b o d y surface. To ensure consistency b e t w e e n the k i n e m a t i c and dynamics problems, the m e t h o d o f images is used to e l i m i n a t e t h e source d i s t r i b u t i o n over the free surface, r e s u l t i n g i n :

u ( « ) ( x , 0 = u ' ^ ' ( x , t ) + / a f ' ( x ' , t ) V G * ( x ; x ' ) dS ( 5 1 ) JSB

w h e r e is the v e l o c i t y i n d u c e d at any p o i n t i n the f l u i d d o m a i n by free surface vortices:

(45) " ' ' ' ' ( ' ' . t ) = 2 / {u('^)(x', t ) x n{x')) X VG(x; x') dS ( 5 2 )

./Sp+sj,'""

For small a m p l i t u d e waves w i t h o u t w a r d n o r m a l vector n = (0, 0 , 1 ) at the s t i l l w a t e r level, Eq. ( 5 2 ) can be w r i t t e n i n t e r m s o f t h e h o r i z o n t a l a n d v e r t i c a l velocities at z = 0 as:

- u j , " V l « - » ' l ^

-^"''z/\x-x'\^ \ dx'dy' [(u'^"ix-x') + vf^{y-y))/\x~x'\' j

(53)

The u n k n o w n source strengths e r f ' are d e t e r m i n e d f r o m Eq. (51) b y a p p l y i n g the k i n e m a t i c b o d y b o u n d a r y c o n d i t i o n (Eq. (32)), y i e l d i n g the f o l l o w i n g i n t e g r a l e q u a t i o n :

^ - / . n « ' . t ) V G . ( x ; x ' )

JSB JSB

•nix)dS = d^l • n + u ' ' ' • n - Un^ ₍₅₄₎

W e note t h a t the time d e r i v a t i v e o f the above i n t e g r a l e q u a t i o n is s i m i l a r to the i n t e g r a l e q u a t i o n f o r the m e m o r y pressure source strengths (Eq. (46)). The m e m o r y pressure source strengths are thus derived f r o m the k i n e m a t i c source strengths u s i n g Op = 9 o - f /dt.

The free surface e v o l u t i o n e q u a t i o n (Eq. (28)) requires k n o w l -edge o f t h e v e r t i c a l v e l o c i t y at the s t i l l w a t e r level. The i m p u l s i v e c o m p o n e n t o f t h e v e r t i c a l v e l o c i t y is g i v e n b y Eq. ( 4 7 ) w h i l e t h e m e m o r y c o m p o n e n t i n Eq. ( 5 1 ) can be e x p l i c i t l y w r i t t e n as:

w n - , y , t ) = - r r u f \ K ' , y ' , t )

/

OO poo , / v f ' ( x ' , y , t ) - J ^ oo7-oo 2n\x-x>\^ + / a f ' ( ^ ' , Js„ dx'dy' 27t\X-X'\ dS (55)

The free surface integrals i n t h e f o r e g o i n g e q u a t i o n are convo-l u t i o n integraconvo-ls i n space a n d are evaconvo-luated w i t h t h e fast Fourier t r a n s f o r m m e t h o d : w f ' ( x , y , t ) = F - i

+ / 4^'(x',t)

JSB

- f ( | F ( u r ) + | f ( . r

dS 2n\x-x'\^ (56)

2 8 O.G. Nwogu, R.F, Beck/Applied Ocean Research 65 (2017)23-34

w h e r e the Fourier operators are d e f i n e d as:

Flë{^,y)]=Èik.,ky)=^ / g{x,y]e-<ix^^^'<yy^ dxdy ( 5 7 ) J - o o J - 0 0

F''mx,ky)]=g{x,y].

J —00 J -o

g{kx,ky)e'^>""'+''yy'> dkxdky (58) and k = ^/i<[+l^ w i t h h , ky the w a v e n u m b e r s i n the x a n d y direc-tions respectively. H i g h e r - o r d e r closure equadirec-tions f o r the v e r t i c a l velocity are presented i n N w o g u and Beck (2010).

3.4. Computational considerations

The p r e d o m i n a n t c o m p u t a t i o n a l cost i n m o s t t i m e - d o m a i n Rankine panel ship m o t i o n codes is associated w i t h s o l v i n g the U p l a c e equation every t i m e step. Iterative m a t r i x solvers t y p i c a l l y have costs o f 0 [ ( N f + N B ) ^ ], w h e r e N B is the n u m b e r o f panels o n the body surface and N f is the n u m b e r o f grid points o n the free surface. The c o m p u t a t i o n a l costs can be d i v i d e d i n t o three c o m p o n e n t s : the free surface c o m p o n e n t [0{Nj)], t h e body c o m p o n e n t |0(N|)1, and the b o d y / f r e e surface i n t e r a c t i o n t e r m [ 0 ( 2 N B N F ) ] .

The present FFTaccelerated b o u n d a r y i n t e g r a l m e t h o d i n c o r p o -rates several features t h a t s i g n i f i c a n t l y speed up the c o m p u t a t i o n s relative to t r a d i t i o n a l Rankine panel methods. Use o f t h e FFT m e t h o d to evaluate the free surface integrals reduces the cost o f the free surface c o m p o n e n t f r o m 0{Nj) t o O ( N f l o g N f ) . The cost o f the m a t r i x i n v e r s i o n f o r the b o d y source strengths is s t i l l o f 0 ( N | ) , b u t is done once p r i o r to the start o f the t i m e - s t e p p i n g scheme. The c o m p u t a t i o n a l cost o f i n t e r a c t i o n terms is reduced by o p t i m a l l y selecting singularities t h a t decay r a p i d l y i n space ( 0 [ z / | x - x ' p ] ) , and r e s t r i c t i n g t h e e v a l u a t i o n o f t h e velocities/pressure to a near-field region a p p r o x i m a t e l y one characteristic w a v e l e n g t h away f r o m the s i n g u l a r i t y . This decreases the cost f r o m 0{NBNF) to O ( N B M P ) , w h e r e is the n u m b e r o f free surface points i n the near-field region. The t o t a l c o m p u t a t i o n a l cost f o r s o l v i n g the b o u n d a r y value p r o b l e m at each t i m e step is thus o f 0 ( 3 N F logNp + S M ^ N B ) . For t y p i c a l c o m p u t a t i o n s w i t h N p = 3 2 , 7 6 8 , i W , = 6 4 and NB = 500, the FFT-accelerated scheme is a p p r o x i m a t e l y 1800 t i m e s faster t h a n an i t e r a t i v e solver o f 0 [ ( N f + N B ) 2 ] .

3.5. Specification of incident wave field

The i n c i d e n t m u l t i d i r e c t i o n a l w a v e field is specified as a linear s u p e r p o s i t i o n o f a finite n u m b e r o f w a v e components w i t h d i f f e r e n t amplitudes, w a v e n u m b e r s and d i r e c t i o n s :

7j(x,y, t) = Re y^An exp{i{k„xcosp„ + k„y s i n P „ - o)nt)] n=l

(59)

w h e r e An is the c o m p l e x w a v e a m p l i t u d e , k„ is the w a v e n u m b e r , is t h e w a v e p r o p a g a t i o n d i r e c t i o n relative to the ship head-ing, and cOn is t h e encounter w a v e f r e q u e n c y t h a t is related to the w a v e n u m b e r a n d ship speed t h r o u g h the dispersion r e l a t i o n :

- Ukn cos p„ (60)

The i n c i d e n t w a v e velocities and pressure can be obtained f r o m Eq. ( 5 9 ) u s i n g l i n e a r w a v e t h e o r y .

3.6. Wave absorbing layers

Use o f t h e F F T m e t h o d t o accelerate e v a l u a t i o n o f t h e f r e e surface b o u n d a r y integrals imposes d o u b l y - p e r i o d i c b o u n d a r y conditions at the ends o f the c o m p u t a t i o n a l d o m a i n . To prevent t h e o u t -going waves f r o m w r a p p i n g a r o u n d and propagating back i n t o

the c o m p u t a d o n a l d o m a i n , w e place d a m p i n g regions a r o u n d the p e r i m e t e r o f t h e c o m p u t a t i o n a l d o m a i n t o absorb o u t g o i n g waves. A Gaussian-type d i s t r i b u t i o n is adopted f o r the spatial v a r i a t i o n o f the d a m p i n g t e r m i n Eq. (23) w i t h i n each d a m p i n g layer, e.g.:

M b ( x , y ) = / x o e x p [ - 2 j r ( x / L d ) ^ l f o r 0 < x < L d , 0<y<Ly (61) w h e r e and /Xo are respectively the w i d t h and s t r e n g t h o f the d a m p i n g layer. N u m e r i c a l tests showed t h a t waves c o u l d be effec-tively d a m p e d o u t i n a layer one w a v e l e n g t h w i d e using d a m p i n g s t r e n g t h IIQ =

3.7. Suppression of resonant interior fluid motions

One o f t h e challenges w i t h a p p l y i n g the b o u n d a r y i n t e g r a l equa-t i o n m e equa-t h o d equa-t o surface-piercing bodies is selecequa-ting a p p r o p r i a equa-t e b o u n d a r y c o n d i t i o n s f o r the fictitious fluid m o t i o n s inside t h e body. The FFT-accelerated b o u n d a r y i n t e g r a l m e t h o d requires c o n t i n u i t y o f t h e pressure b e t w e e n the e x t e r i o r and i n t e r i o r regions, lead-ing t o the e l i m i n a t i o n o f d i p o l e slead-ingularities o n the b o d y surface. This h o w e v e r leads t o resonant oscillations at t h e eigenfrequencies o f t h e i n t e r i o r D i r i c h l e t p r o b l e m i f i d e n t i c a l f r e e surface b o u n d -ary c o n d i t i o n s are a p p l i e d i n b o t h the i n t e r i o r and e x t e r i o r regions [26]. Several techniques have been proposed t o suppress reso-nant m o t i o n s at "irregular" frequencies such as i m p o s i n g a r i g i d l i d o n the i n t e r i o r free surface [27], d a m p i n g the w a v e e l e v a t i o n i n the i n t e r i o r r e g i o n [ 2 8 ], p l a c i n g a d d i t i o n a l m u l t i p o l e singularities inside the b o d y to absorb energy o f t h e resonant modes [29], o r using a m o d i f i e d i n t e g r a l e q u a t i o n w i t h sources and dipoles o n t h e body surface [ 3 0 ].

Our approach to suppress resonant fluid m o t i o n s inside the b o d y is to m o d i f y 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 on the i n t e r i o r free surface S^'"'' to i n c l u d e a d a m p i n g t e r m (Eq. (23)). W e note t h a t t h e i n t e r i o r d a m p i n g t e r m is n o t a p p l i e d 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 s (Eqs. (24) a n d (25)) t o a v o i d i m p a c t i n g the pressure inside t h e body. The spatial d i s t r i b u -tion o f t h e d a m p i n g c o e f f i c i e n t over the body's w a t e r p l a n e area Awp is chosen to be o f Gaussian f o r m :

iU.i(x.y) = Moexp.^ - 2 ; r

_(=)'-(l)'

f o r ( x , y ) e Avvp ( 6 2 ) w h e r e a and b are respectively the h a l f - l e n g t h and h a l f - w i d t h o f a rectangle enclosing t h e w a t e r p l a n e area. The d a m p i n g c o e f f i c i e n t thus decays t o zero a r o u n d the w a t e r l i n e , ensuring c o n t i n u i t y o f t h e free surface e l e v a t i o n betiween the i n t e r i o r and e x t e r i o r regions.

The o p t i m a l d a m p i n g s t r e n g t h is d e t e r m i n e d by e x a m i n i n g t h e dispersion r e l a t i o n o f the w a v e p r o p a g a t i o n equations i n the i n t e -rior d a m p i n g layer (Eqs. ( 2 3 ) - ( 2 5 )) :

a) + U k c o s ^ = i ^ ±

J g k - ( f

)

(63) For /Xi > 2^/gk, the apparent w a v e f r e q u e n c y i n t h e d a m p i n g layer becomes p u r e l y i m a g i n a r y c o r r e s p o n d i n g to an o v e r d a m p e d system. The s t r e n g t h o f t h e i n t e r i o r d a m p i n g layer was t h u s chosen as /xo = 2 y ^ .

4 . N u m e r i c a l r e s u l t s

4.1. Forced and transient motions of a floating sphere

The n u m e r i c a l m e t h o d w a s i n i t i a l l y used to investigate t h e f o r c e d oscillations o f a h a l f - i m m e r s e d floating sphere f o r w h i c h a s e m i - a n a l y t i c a l s o l u t i o n is available [ 3 1 ]. A h e m i s p h e r e o f radius R w a s discretized i n t o 324 flat q u a d r i l a t e r a l panels u s i n g a u n i -f o r m polar and a z i m u t h a l r e s o l u t i o n o -f 10° as i l l u s t r a t e d i n Fig. 2.

O.G. Nwogu, R.F. Beck/Applied Ocean Research 65 (2017) 23-34 ₂₉

Fig. 4. Wavefield generated by a surging motions of a hemisphere (id? = 1).

The hemisphere was forced t o undergo p e r i o d i c oscillations at a given f r e q u e n c y (o thereby generating f r e e surface waves w i t h w a v e l e n g t h X^JngjaP-. N u m e r i c a l s i m u l a t i o n s w e r e p e r f o r m e d b y i n t e g r a t i n g t h e free surface e v o l u t i o n equations over a square g r i d (8A X 8A) w i t h r e s o l u t i o n A x = A y = A , / 3 2 u s i n g time step size A t = O . M y ^ K / g . Snapshots o f the instantaneous w a v e f i e l d gener-ated b y the heave ( z - d i r e c t i o n ) and surge ( x - d i r e c t i o n ) oscillations at n o n - d i m e n s i o n a l w a v e n u m b e r kR = 1 are s h o w n i n Figs. 3 and 4 respectively. The p u l s a t i n g sources o n t h e panelized b o d y surface e f f e c t i v e l y act as submerged wavemakers, generating waves t h a t radiate o u t b o t h inside and outside the body. The waves converge at t h e center o f t h e hemisphere f o r t h e heave m o d e o f m o t i o n and o n the p e r i m e t e r f o r the surge m o d e . The f l u i d m o t i o n s thus correspond to a s t a n d i n g w a v e inside the b o d y . The free surface ele-v a t i o n is obserele-ved t o be continuous across t h e i n t e r i o r a n d e x t e r i o r regions. 0.8 0.4 0,2 Hulme(1982) Present Method

\

' ' ' ' I ' ' ' ' I I ' ' ' I 2 3 4 5 kR

Fig. 7. Comparison of the predicted surge added mass coefficient with the semi-analytical solution of Hulme [31].

The added mass and d a m p i n g coefficients w e r e o b t a i n e d f r o m a Fourier analysis o f t h e predicted force time histories and are c o m -pared to the semi-analytical s o l u t i o n o f H u l m e [ 3 1 ] i n Figs. 5-8. Reasonably good agreement is o b t a i n e d b e t w e e n the n u m e r i c a l and analytical results f o r b o t h modes o f m o t i o n s over a b r o a d f r e q u e n c y range i n c l u d i n g the f i r s t irregular frequencies {k^R = 2.56 f o r heave,

3U O.G. Nwogu, RJ. Beck/Applied Ocean Research 65 (2017) 23-34

I • I I • I I . I r t . I • ' I

0 50 100 150 200 250 300 350

Fig. 11. Free surface elevation of Gaussian wave packet.

Fig, 8. Comparison of tlie predicted surge damping coefficient with the semi-analytical solution of Hulme | 3 1 ].

20

-20

- w i t h Inl Damp Without Inl Damp

10

Fig. 9. Interior free surface elevation generated at center of body by hemisphere oscillating vertically at irregular wavenumber JcR=2.5.

10 20 30 40 50 60 70 80

Fig. 10. Comparison of heave radiation forces with/without interior damping layer for hemisphere oscillating at irregular wavenumber kR - 2.5.

k l R = 3.92 for surge). The effectiveness of the i n t e r i o r d a m p i n g layer i n suppressing resonant f l u i d m o r i o n s at i r r e g u l a r frequencies was f u r t h e r evaluated b y repeating t h e s i m u l a t i o n o f the forced heave oscillarion at kR=2.5 w i t h o u t an i n t e r i o r d a m p i n g layer over the sphere's w a t e r p l a n e area. The f r e e surface elevation at the cen-ter o f t h e hemisphere ( n o n - d i m e n s i o n a l i z e d b y the a m p l i t u d e o f oscillation A) is p l o t t e d i n Fig. 9. The w a v e e l e v a t i o n inside the hemisphere is observed to resonantly g r o w w i t h rime w h e n f o r c e d near t h e irregular f r e q u e n c y i n the absence o f i n t e r i o r d a m p i n g . Our technique o f a p p l y i n g a d a m p i n g t e r m t 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 prevents the resonant g r o w t h b u t does not d a m p o u t n o n - p r o p a g a t i n g waves inside t h e body. The d a m p i n g t e r m also has negligible i n f l u e n c e on the h y d r o d y n a m l c pressure as can be seen i n Fig. 10 w h e r e t h e predicted forces are c o m p a r e d to t h e analytical s o l u t i o n .

We n e x t consider the response o f a h a l f - i m m e r s e d f l o a t i n g sphere w i t h its center o f g r a v i t y at its geometric center t o t r a n -sient w a v e packet [32]. The tran-sient w a v e t r a i n is characterized by a f i n i t e n u m b e r o f w a v e components (Eq. (59)) w i t h the phases u n i q u e l y selected t o converge at the b o d y at a specific time. A Gaus-sian d i s t r i b u t i o n is adopted i n t h i s paper f o r t h e shape o f t h e w a v e a m p l i t u d e s p e c t r u m ;

^2

A(c6>) = /lo V27r f {(ü-cOpf\ (64)

-0.5^ 1 0.5 -0.5 -1 - h

ft

-' • ' 1 1 • ' ' 1 1 t l - I -. 1 . . . . 1 , , 1 , ., 1 1 50 100 150 200 250 300 350 0 50 100 150 200 250 30O 350 ^g/Rf''

Fig. 12. Surge and heave motion of hemisphere induced by Gaussian wave packet.

1.5 Ptesenl Melhod Oairisoo(1975) _ 1 1 L _ l — ! — 1 - J — i — I . I I I I I i . I I < I

i

£ 0 I I I I M • I I I I I I 2.5 0.5 1 1.5 2

Fig. 13. Amplitude and phase of surge transfer function for a floating hemisphere.

w h e r e cop is t h e peak frequency, a is a spectral w i d t h parameter and To is the d u r a t i o n o f the w a v e t r a i n . The transient w a v e p r o -f i l e was s i m u l a t e d using 96 equally-spaced -f r e q u e n c y c o m p o n e n t s w i t h dp = ^/g/R. a- = ^/Ö^gjR and To = 355-y/R/g. The result-ing w a v e p r o f i l e is s h o w n i n Fig. 11 w i t h t h e correspondresult-ing surge and heave m o t i o n s s h o w n i n Fig. 12. Transfer f u n c t i o n s r e l a t i n g t h e i n c i d e n t w a v e p r o f i l e t o the sphere m o t i o n s w e r e o b t a i n e d f r o m a cross-spectral analysis o f the s i m u l a t e d time histories. The a m p l i t u d e and phase o f the surge and heave transfer f u n c t i o n s are compared i n Figs. 13 and 14 respectively t o results o b t a i n e d b y Gar-rison [33] using a f r e q u e n c y - d o m a i n b o u n d a r y i n t e g r a l m e t h o d . Excellent agreement is observed b e t w e e n t h e t i m e and f r e q u e n c y d o m a i n results over a w i d e range o f frequencies.

O.G. Nwogu, R.F. Beck / Applied Ocean Research 65(2017)23-34 31

2

Fig. 14. Amplitude and pliase of heave transfer function for a floating hemisphere.

4.2. Transient motions of a vessel with forward speed

In this section, w e apply t h e n u m e r i c a l m e t h o d to t h e m o r e general p r o b l e m o f the six degree o f f r e e d o m m o t i o n s o f a ves-sel advancing at f o r w a r d speed i n irregular ocean w a v e f i e l d s and present comparisons w i t h data f r o m l a b o r a t o r y e x p e r i m e n t s [ 3 4 ]. The e x p e r i m e n t s w e r e carried o u t i n the seakeeping basin at the U.S. Naval Surface W a r f a r e Center, Carderock w h i c h is 1 1 0 m l o n g by 73 m w i d e and 6 m deep. T w o banks o f p n e u m a t i c wavemak¬ ers arranged i n an L - c o n f i g u r a t i o n w e r e used t o generate i r r e g u l a r u n i d i r e c t i o n a l waves. A 1 1 5 m - l o n g bridge spans the basin t h a t supports a t o w i n g carriage.

A 1:23 fiberglass m o d e l o f t h e oceanographic research vessel R/V M e l v i l l e was b u i l t for the laboratory experiments. The p r i n c i p a l particulars o f the ship are s u m m a r i z e d i n Table 1. The m o d e l w a s f i t -ted w i t h t w o podded p r o p u l s i o n u n i t s . D u r i n g the experiments, the m o d e l was loosely t e t h e r e d t o t h e carriage and m a n u a l l y c o n t r o l l e d by t w o pilots. One p i l o t a d j u s t e d t h e t h r o t t i e t o m a t c h t h e carriage speed w h i l e the o t h e r a d j u s t e d t h e steering angle o f the pods t o m a i n t a i n a constant heading. A n i n d o o r global p o s i t i o n i n g system was used to track the vessel's p o s i t i o n i n the basin. The ship's trans-l a t i o n a trans-l m o t i o n s w e r e measured w i t h a 3-axis accetrans-lerometer at the vessel's center of g r a v i t y w h i l e t h e angular m o t i o n s w e r e measured w i t h a 3-axis angular rate gyroscope. The w a v e c o n d i t i o n s i n the basin w e r e measured i n b o t h an e a r t h - f i x e d f r a m e o f reference w i t h eight ultrasonic probes m o u n t e d o n t h e bridge and i n t h e ship's f r a m e o f reference w i t h f o u r w a v e probes attached t o the carriage. The seakeeping tests w e r e p e r f o r m e d for three sea states, f i v e head-ings and three ship speeds corresponding to Froude n u m b e r s F„ = 0, 0.15 and 0.22.

Table 1

Main characteristics of oceanographic vessel RA' Melville.

Length between perpendiculars (m) 77.4 Maximum beam (m) 14.0 Draft at FP(m) 4.89 Draft at AP(m) 5.01 Displacement volume (ni') 2910 Vertical center of gravity above BL (m) 6.22 Longitudinal center of gravity from FP 36.9 Roll radius of gyration around (m) 5.29 Pitch radius of gyration around (m) 1 9 3 Yaw radius of gyradon around (m) 19.3

0,1

Mid-Cycle Roll AmpUuide £,

Fig. 15. Variation of log-decrement with roll amplitude.

Fig. 16. 3D panelization of R/V Melville ship hull.

20

T i m e ( s )

Fig. 17. Comparison of measured and simulated roll decay time histories (F„ = 0.15).

43.1. Calibration of roll damping

M i n n i c k et al. [34] analyzed the measured c a l m - w a t e r r o l l decay t i m e series t o o b t a i n the l o g a r i t h m i c d e c r e m e n t o f consecutive r o l l peaks. The l o g - d e c r e m e n t values are p l o t t e d i n Fig. 15 f o r the three ship speeds. I t can be seen t h a t the r o l l d a m p i n g ratio exhibits a s t r o n g dependence on r o l l a m p l i t u d e and s h i p speed. There is also significant v a r i a b i l i t y at smaller r o l l a m p l i t u d e s (<4°).

The r o l l d a m p i n g m o m e n t was parameterized w i t h a linear c o m p o n e n t t h a t is p r o p o r t i o n a l the r o l l v e l o c i t y and a quadratic c o m p o n e n t t h a t is p r o p o r t i o n a l t o the square o f the r o l l velocity. Linear regression analysis was t h e n used to o b t a i n the b e s t - f i t linear and quadratic d a m p i n g coefficients over t h e range (4° < I4 < 15°).

4.2.2. Roll decay simulations in calm water

The n u m e r i c a l m o d e l was setup t o reproduce t h e r o l l decay experiments. The R/V M e l v i l l e ship h u l l was discretized w i t h 322 quadrilateral panels b e l o w the w a t e r i i n e as s h o w n i n Fig. 16. A v i r -t u a l appendage -t o vessel s-tern as sugges-ted b y Couser e-t al. [ 3 7 ] -t o o b t a i n stable n u m e r i c a l c o m p u t a t i o n s . The s i m u l a t i o n s w e r e per-f o r m e d o n a rectangular g r i d w i t h spatial r e s o l u t i o n A x = A y = 7.5 m a n d time step size A t = 0.35 s. A n i n i t i a l r o l l angle was applied t o t h e ship a f t e r a t t a i n i n g a steady w a k e p a t t e r n . Fig. 17 shows a c o m

-32 O.G. Nwogu, R.F. Beck/Applied Ocean Research 65 (2017)23-34

Fig. 18. Comparison of measured and predicted time tiistories of ttie vertical plane motions in bow-quartering seas (F„ = 0.15).

100 150 200 250 300 Time (s)

Fig. 19. Comparison of measured and predicted time histories of the horizontal plane morions in bow-quartering seas (F„ = 0.15).

parison o f the measured and p r e d i c t e d r o l l decay t i m e h i s t o r y w i t h the b e s t - f i t d a m p i n g coefficients f o r a test r u n w i t h Fn =0.15. The quadratic d a m p i n g parameterization is able to accurately describe the r o l l decay f r o m an i n i t i a l angle o f 16° t o ~ 2 ° .

4.2.3. Six-degree of freedom motions in bow-quartering seas W e n e x t consider six degree o f f r e e d o m ship m o t i o n s i n b o w -q u a r t e r i n g waves (;6 = 2 2 5 ° ) f o r one o f t h e test runs w i t h a ship speed o f 4.11 m/s (Fn = 0.15) i n sea state 4 conditions characterized b y a Bretschneider spectrum w i t h Hs = 1.9 m and Tp = 8.8 s. N u m e r -ical simulations w e r e p e r f o r m e d i n a f r a m e o f reference m o v i n g w i t h t h e vessel. The i n c i d e n t w a v e conditions w e r e d e r i v e d f r o m a Fourier analysis o f the measured w a v e elevation t i m e histories on t h e carriage probe 145 m u p w a v e o f t h e vessel. The equations of f l u i d m o t i o n w e r e integrated o n a rectangular g r i d (Ix = 2 5 6 A x , L;, = 1 2 8 A y ) w i t h spatial r e s o l u t i o n A x = A y = A.p/16. The t i m e step size was selected to y i e l d a Courant n u m b e r o f 0.7. A s o f t spring system w i t h a n a t u r a l period o f 50 s was u t i l i z e d to restrain the l o w - f r e q u e n c y m o t i o n s i n the h o r i z o n t a l plane.

T w o error measures are i n t r o d u c e d t o evaluate t h e f i d e l i t y o f the predictions, the phase c o r r e l a t i o n c o e f f i c i e n t :

y =

E ? ' " ( t ) ? p ( t )

and t h e m a g n i t u d e error factor:

X / Ë ¥ W - \ / Ë M ( Ö

(65)

(66)

w h e r e ^ „ ( t ) and | p ( t ) are t h e measured and p r e d i c t e d m o t i o n s . The p r e d i c t e d t i m e histories o f the translational m o t i o n s are c o m -pared t o measured data i n Fig. 18, w h i l e the r o t a t i o n a l m o t i o n s compared i n Fig. 19. A l l six modes of m o t i o n s have c o r r e l a t i o n coef-ficients greater t h a n 0.88 w i t h t h e highest phase correlations f o r the heave and p i t c h m o t i o n s at 0.95.The m a g n i t u d e error, however, has m o r e v a r i a b i l i t y r a n g i n g f r o m 4% ( y a w ) to 21% ( r o l l ) . W e note t h a t the r o l l a m p l i t u d e s are relative small [ 0 ( 2 ° ) ] and t h e differences c o u l d be due to t h e p a r a m e t e r i z a t i o n o f viscous r o l l d a m p i n g at smaller angles and/or the a p p l i c a t i o n o f the b e s t - f i t d a m p i n g coef-ficients f r o m the r o l l decay tests i n calm w a t e r t o w a v e conditions.

300

SO 100 150 200 250 300 Time (s)

Fig. 20. Comparison of measured and predicted time histories ofthe vertical plane motions in stern-quartering seas (F„ = 0.15).

4.2.4. Six-degree of freedom motions in stem-quartering seas Accurate m o d e l i n g of ship m o t i o n s i n f o l l o w i n g or stern quarter-i n g seas quarter-is crquarter-itquarter-ical f o r shquarter-ip s t a b quarter-i l quarter-i t y studquarter-ies due t o the p o t e n t quarter-i a l loss of transverse s t a b i l i t y o n w a v e crests at l o w encounter f r e q u e n -cies. Simulations w e r e p e r f o r m e d f o r one o f the s t e r n - q u a r t e r i n g (/S = 4 5 ° ) m o d e l test runs f o r a ship speed o f 4.11 m/s (Fn = 0.15) i n a sea state w i t h H j = 1.9 m and Tp = 8.8 s. The translational m o t i o n s are compared to the measured data i n Fig. 20 w h i l e the r o t a t i o n a l m o t i o n s are compared i n Fig. 2 1. Overall, the agreement b e t w e e n the measured and p r e d i c t e d m o t i o n s is considered t o be excellent. The phase c o r r e l a t i o n values f o r a l l six modes o f m o t i o n s exceeded 0.9 w h i l e the m a g n i t u d e errors ranged f r o m 2.5% (heave) to 16% ( r o l l ) . The r o l l m o d e o f m o t i o n appears to be the m o s t challeng-i n g m o t challeng-i o n to s challeng-i m u l a t e due to uncertachalleng-intchalleng-ies challeng-i n t h e vchalleng-iscous d a m p challeng-i n g and s e n s i t i v i t y t o unsteady variations i n the ship speed/heading o f a self-propelled vessel.

O.G. Nwogu, R.F. Beck/Applied Ocean Research 65 (2017)23-34 ₃₃

-2

r ! I ' I i I I ' r I I J r I I I r I I I I I I I i I I I I I

50 IDQ 150 200 250 300

2c — ,

0 50 100 150 200 250 300 Time (s)

Fig. 21. Comparison of measured and predirted time histories of the horizontal plane motions in stem-quartering seas (F„ = 0.15).

5. Concluding remarlcs

A t i m e - d o m a i n t h r e e - d i m e n s i o n a l n u m e r i c a l m e t h o d has been d e v e l o p e d t o s i m u l a t e the t r a n s i e n t m o t i o n s o f a s h i p t r a v e l i n g i n i r r e g u l a r ocean waves. The c o u p l e d eciuations g o v e r n i n g t h e m o t i o n s o f b o t h the f r e e and b o d y surfaces w e r e s i m u l t a n e o u s l y i n t e g r a t e d u s i n g a f r a c t i o n a l - s t e p a l g o r i t h m . The f l u i d k i n e m a t i c s at each t i m e step w e r e o b t a i n e d b y s o l v i n g a v e l o c i t y b a s e d b o u n d -a r y i n t e g r -a l m e t h o d w h i l e t h e f l u i d pressure w -a s o b t -a i n e d f r o m t h e b o u n d a r y value p r o b l e m f o r t h e t o t a l pressure. The c o m p u t a -t i o n s w e r e speeded u p -t h r o u g h use o f -t h e f a s -t Fourier -t r a n s f o r m m e t h o d t o evaluate t h e f r e e surface c o n v o l u t i o n integrals a n d a n e a r f i e l d s p a d a l e v a l u a t i o n o f the v e l o c i t i e s a n d pressure i n d u c e d b y free-surface v o r t e x / d i p o l e s i n g u l a r i t i e s .

The n u m e r i c a l m o d e l w a s i n i t i a l l y v a l i d a t e d w i t h s e m i -a n -a l y t i c -a l d-at-a f o r t h e f o r c e d m o t i o n s o f -a h -a l f - i m m e r s e d f l o -a t i n g sphere w i t h excellent a g r e e m e n t . The s i m u l a t i o n s also d e m o n -s t r a t e d t h e effectivene-s-s o f t h e i n t e r i o r d a m p i n g layer t e c h n i q u e t o suppress resonant m o t i o n s at i r r e g u l a r frequencies. The a b i l i t y o f t h e m o d e l to p r e d i c t t i m e s y n c h r o n i z e d s h i p m o t i o n s was e v a l -u a t e d -u s i n g data f r o m l a b o r a t o r y e x p e r i m e n t s w i t h a f r e e r -u n n i n g vessel. Reasonably g o o d c o m p a r i s o n s w e r e o b t a i n e d f o r a l l six rigid b o d y m o d e s o f m o t i o n s i n b o t h b o w a n d s t e r n q u a r t e r i n g waves despite t h e v a r i a b i l i t y i n ship speed a n d h e a d i n g f o r a s e l f - p r o p e l l e d vessel. Accurate p r e d i c t i o n o f r o l l s t i l l r e m a i n s a c h a l l e n g i n g task d u e t o t h e need t o i n c o r p o r a t e e m p i r i c a l l y t u n e d viscous d a m p i n g c o e f f i c i e n t s i n an i n v i s c i d code.

The c o u p l e d t h r e e - d i m e n s i o n a l s p e c t r a l / p a n e l m e t h o d is an o r d e r o f m a g n i t u d e f a s t e r t h a n n o n - a c c e l e r a t e d b o u n d a r y i n t e g r a l m e t h o d s . For i l l u s t r a t i v e purposes, the c o m p u t a t i o n o f t h e f o r c e d h e a v i n g m o t i o n o f a h e m i s p h e r e w i t h 16,384 panels o n t h e f r e e surface a n d 108 panels o n t h e b o d y surface r e q u i r e d 3 0 s o f CPU time f o r 513 time steps (20 o s c i l l a t i o n p e r i o d s ) o n a 2 GHz l a p t o p c o m p u t e r w i t h 8GB o f R A M . This is a p p r o x i m a t e l y 600 t i m e s f a s t e r t h a n t h e 0 ( 5 h ) CPU time f o r a s i m i l a r c o m p u t a t i o n b y Zhang a n d Beck [ 3 5 ] w i t h a 3-D Rankine p a n e l m e t h o d . The s i m u l a t i o n s o f t h e 6 d o f m o t i o n s o f a vessel i n o b l i q u e w a v e s r a n 2 - 3 times f a s t e r t h a n real time o n the l a p t o p c o m p u t e r . P a r a l l e h z a t i o n o f the code is b e i n g u n d e r t a k e n t o a l l o w f o r l o n g e r t e r m [ 0 ( 1 m i n ) ] forecasts o f s h i p m o t i o n s aboard ships a n d o f f s h o r e p l a t f o r m s w i t h r e m o t e l y sensed w a v e c o n d i t i o n s .

Adtnowledgements

This p r o j e c t w a s s u p p o r t e d b y t h e E n v i r o n m e n t a l a n d Ship M o t i o n Forecasting (ESMF) P r o g r a m o f the Office o f Naval Research t h r o u g h Contract No. N 0 0 0 1 4 - 1 1 - D - 0 3 7 0 t o t h e U n i v e r s i t y o f M i c h i g a n . The a u t h o r s g r a t e f u l l y a c k n o w l e d g e t h e guidance o f t h e p r o g r a m m a n a g e r D r Paul Hess, Wi.—

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