E L S E V I E R PII: 50141-1187(96)00005-3
Applieit Ocean Research 17 (1995) 349-362
Copyright © 1996 Elsevier Science L i m i t e d Printed in Great B r i t a i n . A l l rights reserved 0141-1 187/95/109.50
Nonlinear wave effect on the slow drift motion of a
T a k u m i O h y a m a
Centre for Water Research, The Utiiversity of Western Australia, Nedlands, WA 6907, Australia'
&
J o h n R . C . H s u
Department of Environmental Engineering, The University of Western Australia, Nedlands, WA 6907, Australia
(Received 22 January 1996)
The slow drift motion of a floating body in a two-dimensional wave field has been investigated using a time-domain, fully nonlinear numerical model with non-reflective open boundaries. Prehminary computations were conducted for incident bichromatic waves, in which wave theories with difi'erent orders were applied in generating the waves required. The results show that the use of low-order theories generates undesirable free waves, and that fourth-order terms contribute markedly to low-frequency input. The motion of a rectangular floating body in response to nonlinear bichromatic waves was computed. The numerical results for small-amplitude incident waves agree reasonably well with the second-order approximation for both the steady and difference-frequency (Ao") components in the body's motion. For relatively large waves, however, the 2Ao- component becomes predominant compared with the ACT component. The motion of the body in irregular waves with different wave parameters has also been presented in order to discuss the validity range of a second-order approximation. Copyright (gjElsevier Science Limited.
1 I N T R O D U C T I O N
It is well known that the slow d r i f t motion of a moored floating body is excited by low-frequency components locked to incident wave groups.''^ When the mooring stiffness is weak and the natural frequency of the body's horizontal modon is low, the magnitude o f such slow mo-tion may overwhelm the fast oscillamo-tion associated with primary wave frequencies.
I n the case of smah-amplitude waves, the magnitude o f the slow d r i f t motion is described by a square function of wave height, so a second-order perturbation theory has been commonly apphed to the analysis of this phe-nomenon.-'"^ Within the second-order theory, the d r i f t force in response to irregular waves consists o f a steady (mean) part and a slowly varying part associated with
fre-^ O n leave f r o m the Institute o f Technology, Shimizu Corporation, Etchujima 3-4-17, Koto-ku, Tokyo 135, Japan.
quency differences among first-order primary wave com-ponents. The former corresponds to Maruo's solution^ derived f r o m the momentum theory, and can be expressed by first-order quantities only Although the second-order potential contributes to the latter part, it requires a rel-atively complicated formulation to solve the complete second-order radiation-diffraction problem. Therefore, based on the assumption that the effect of diffraction on the second-order potential is negligible, Newman^ and Pinkster"^ proposed alternative methods to approximate the slowly varying part. The validity of this assump-tion has been investigated by comparison with the re-sults obtained f r o m the complete second-order theory By using a multiple-scale analysis, on the other hand, Agnon et a/."''^ successfully derived a closed-form so-lution without solving the second-order boundary value problem. They indicated that Newman's approximation is no longer adequate when the blockage coefficient of a floating body is so large that moderate resonance takes
T. Ohyama. J. R. C. Hsu
place.
Despite these attempts based on second-order theory, it should be emphasized that this theory provides only the lowest-order solution for slow d r i f t motion and, hence, its applications are limited to weakly nonlinear situations. In light of this, the time-domain boundary integral equa-tion method ( B I E M ) has been applied to the problem of the nonlinear interaction between waves and floating bodies'^-'•^ following the successful computations o f wave overturning by Longuet-Higgins and Cokelet.'* I n con-trast to the second-order models, the B I E M can provide fully nonlinear solutions for the wave-body interaction because the nonlinear boundary conditions on the free water surface and the body surface are incorporated di-rectly without approximation. However, the lack o f ap-propriate open boundary treatment has made h diflicult to utilize this model for practical purposes. Vinje and Bre-v i g " assumed spatial periodicity at both ends of a two-dimensional analytical domain, whereas Isaacson''' im-posed an incident (undisturbed) wave potential at open boundaries. These treatments are apparently inadequate for long-term simulations, so computations must be ter-minated when diffracted and radiated waves reach the open boundaries from a floating body. Dommermuth and Yue'5 attempted to match the nonlinear computational domain to a linear wave field at some distance f r o m the body. This technique is effective for three-dimensional forced-oscillation problems because the nonlinearity o f radiated waves may be weakened whh radial distance. However, it is impossible to apply this technique to gen-eral diffraction-radiation problems in which nonlinear in-teraction among incident, diffracted and radiated waves may take place even at open boundaries. Apart f r o m the work mentioned above, most numerical models based on the time-domain B I E M utilize a Sommerfeld-type radi-ation condition at open boundaries.'^^''' I n order to in-vestigate the combined effects of slow and fast motions of a floating body, it is essential to deal with wave fields composed of multiple (low and high) wave frequencies. However, the Sommerfeld-type condition, based on a sin-gle wave celerity, is obviously incapable of handling such wave fields.
In this connection, Ohyama and Nadaoka^^ applied an alternative open boundary treatment^' to a BIEM-based fully nonlinear model, which utilizes a combination of a 'sponge' layer with the Sommerfeld radiation condition at the outer edge of the layer. A major advantage o f this technique is the insensitivity of its wave-absorption per-formance to wave frequency, which makes it possible to address irregular wave fields even in the nonlinear range. Fundamental numerical analyses verified the reliability of this open boundary treatment for nonlinear irregular waves with no signs of instability in prolonged computa-tions.2').22
A numerical model has been developed in this study to investigate the effect o f nonlinear waves on the slow d r i f t motion o f a floating body The basic theory and the
numerical formulation are described in Sections 2 and 3, where the equations of motion of a floating body are solved simultaneously with the boundary integral equa-tions at successive time steps. Prior to investigating the wave-body interaction, incident wave fields with two free-wave modes (referred to as 'bichromatic' free-waves, for sim-plicity) are examined in Section 4. I n order to eliminate higher-order parasitic free waves, it is essential to use an appropriate nonlinear wave theory in generating the waves. I n Section 4, therefore, comparison is made among wave fields produced by using wave theories with differ-ent orders of nonlinearity The slow drift motions of a rectangular floating body computed under bichromatic wave actions are presented in Section 5, where the present numerical results are compared with a second-order ap-proximation'^ in order to discuss nonlinear wave effects. Lastly, the motion of a body in irregular waves contain-ing 16 free-wave modes whh different significant wave heights is demonstrated in Section 6.
2 B A S I C T H E O R Y
2.1 Governing equation and boundary conditions for fluid motion
Figure 1 illustrates a wave-body interaction problem in a two-dimensional domain. A floating body of arbitrary cross-section oscillates in nonlinear irregular waves with three degrees o f freedom: sway, heave and roll.
A t each end of the domain, a wave-absorption fil-ter, composed of a simulated sponge layer with a Sommerfeld-type radiation boundary behind it, is posi-tioned for open boundary treatment. The basic concept of this filter is the same as that used by Israeli and Orszag.21 I t should be noted that most conventional time-domain computations applied to this problem have been conducted solely with a Sommerfeld-type radia-tion condiradia-tion.'^'^'2' However, this radiaradia-tion condiradia-tion, based on a single phase velocity, is inadequate for anal-yses o f multicomponent wave fields consisting o f both short and long waves with different phase velocities. I n contrast, the present filter has been found to provide satisfactory wave-absorption capacity in a wide range of wave frequencies, even in nonlinear wave fields.^"-jj^ addition to the filters, the fluid domain also incorporates a vertically distributed wave-making source (^s), initially introduced by Brorsen and Larsen.^'' The combination of this source with the wave-absorption filter achieves non-reflective wave generation, since waves reflected from the floating body propagate through the source without re-reflection and are absorbed by the filter behind it. Detailed descriptions have been given by Ohyama and Nadaoka^'* for the basic concept and performance of these open boundary treatments.
Assuming irrotational flow in an inviscid and incom-pressible fluid, the fluid motion can be described by the velocity potential 0 ( x , z, / ) , where {x, z) represents Cartesian coordinates (see Fig. 1) and / is time. Since
Nonlinear wave ejfect on the slow drift motion of a floating body 351 D i s t r i b u t i o n o f n Sponge •'2 !,!:: L a y e r 'onge ^ IS. i Layer ; H ^ 3J3
Fig. L Numerical wave tank model and coordinates.
the wave-making source is incorporated into the com-putational domain Q, the governing equation for fluid motion is given by the Poisson equation:
^^(P = U*{z,t)5(x-Xs). inn (1)
where U* is the flux density imposed on the wave-making source Ss, 5 is Dirac's delta function and Xs is the x-position of 6's (zero in the present study).
The boundary conditions imposed on the free surface
Sp, the impermeable horizontal bottom (z = - / ) ) , and
the side boundaries 5*2 and S4 are given as
3 0 dt + ^ ( V c / > ) ^ + ^ , + ^ c ^ > - | | 0 | ^ ^ ^ ^ d x = O, on Sp (2) 90 — = « . — , o n ^ F dn dt 3 0 3z = 0, at z = -/? (3) (4) 3 0 3x
1 ( 3 0 ^ f
3/i
^
fgn at J dx dx onSfJ
on ^ 2 (5) 3 0 3xJ_
3 0 dt .Ï4 f du + A . 0 - J - 0 dx onsr J on S4 (6) in which n is the water surface elevation f r o m the still wa-ter level, /J is the damping factor, n is the outward normal on the boundary, n^ is the z-component o f the outward unit normal, and g is the acceleration due to gravity. As shown in Fig. 1, the damping factor ^ is distributed lin-early in the layer in order to relieve the wave reflection on the leading side o f the layers.The kinematic condition on the body surface Sy is expressed as where 3 0 k=l (8) { n\, nj, « 3 } = { " . V , « r , nAz - ZG) - n^x - XG)} {Du D2. D3} = {XG, ZG, 0G} (9)
rix and are the x- and z-components of the outward
unit normal, ( Z Q , ZQ) represents the gravhational cen-ter of the floating body (see Fig. 1), 0Q is the angular displacement about ( X G , Z G ) , and an over-dot denotes a time derivative. The position of Sy and the normal vec-tor {nx, nr) change in the time domain, and are computed at each time step.
2.2 Equations of motion of floating body
I n the above two-dimensional case, the equations of mo-tion of the floating body may be written as
Mkbk + Rk = Fk + Wk ( / C = 1 , 2, 3), (10)
where
{Ml, M2, M^} = {m, m, 1}
{ Wu W2, PFj} = {0, -mg, 0} (11)
in which k = 1,2,3 correspond to the sway, heave and roll motions o f the body, is the restoring force due to the mooring system, is the hydraulic force, and m and
I are the body's mass and the inertia. The hydraulic force
term F^ can be evaluated by the Bernoulli equation:
pttkds ( / c = l , 2 3) where P=-P Sr 3 0 1 - d f ^ 2
. : ( M
ds + gz (12) (13)T. Ohyama, J. R. C. Hsu
in which p is the pressure including the hydrostatic com-ponent, p is the fluid density and s denotes the tangential direction on the surface.
2.3 Boundary integral equations
Applying Green's theorem to the fluid domain Q, the Poisson equation is transformed into the following inte-gral equation: a {P)<PiP) + ƒ ds+ju*G ds = 0 (14) where GiP. e ) = - l n ( / - ) - l n ( / ) , = yj(xp - XQ)^ + {zp - ZQ)^, >•' = yjixp - x e ) 2 + (zp + ZQ + 2hy (15)
in which S represents the closed boundary surface con-taining n, P(xp, Zp) and Q{XQ, ZQ) denote arbhrary po-sitions along the surface S, and G is a Green function. The coefficient a*{P) is defined by a*(P) = a(P) {zp ^
~h) and a*{P) = 2a(P) (zp = - / ; ) , where C((P) is the
internal angle of the boundary at position P. The Green function G is selected as the sum of the fundamental so-lution of the two-dimensional Laplace equation and its mirror image about the horizontal bottom, which makes it possible to exclude the bottom surface f r o m S.
Substituting the boundary conditions, eqns (3)-(8) into eqn (14) results in <x*{P)4>iP)+ f ds- \,u^G ds J dn J - dt 3
f
-X\F>k Gnk ds/
^2 \ ' d4>f
X2 1r I
30 ( du ' " ^ J h ^ ^ ^ ^ - J a ^ ' ^ l o n . . Gds + U*Gds = 0 (16) SsBy using the weighted residual method, on the other hand, the dynamic condition on ^ F , eqn (2), can be trans-formed into an integral equation:
d^
dx 0 onsp dA- ds=0 (17)
where eqn (3) is substituted into the second term in eqn (2), and co represents a weighting factor which is a func-tion o f t h e tangential coordinate s. Equafunc-tion (17) is ap-plied to the free surfaces on both sides o f the floating body.
3 NUIVIERICAL F O R M U L A T I O N
3.1 Spatial discretization of the integral equations The integral equations, eqns (16) and (17), are discretized into a finite number of elements (see Fig. 2), over which linear distributions of 0 , rj and their time derivatives are assumed. Equation (16) is then rewritten as
« * 0 / + A f ( f ) + A^n, + A j ' 0 , + A j ' D , = A f U *
(i= l,2,...,N) (18) 4>^ = {01, 02, . . . , 0 / v }
0 ^ = { 0 0 / a O i , O0/9O;^*+i O0/3r);v} nf = {(af?/30i O;7/30A'P,+I, (3/]/30;y*+i,
(9^7/90;v*, + i}
D f =
{Z)i,
1 ) 2 , Z)3}U * ' ' = { t / ; , % „ IJ^,2 (19) I n eqn (18), A/, (/ = 1, 2 , . . 5 ) denote coefficient
vec-tors which vary with the time and with the location of the /th control point.
On the other hand, collecting for nodal values of the weighting factor, eqn (17) is transformed into the follow-ing f o r m :
A'FI + I
7 = 1
/VR*, + 1
(20)
where contains the unknown variables 0y, r]j,
{d4>ldt)j and {dr}jdt)j. The arbitrariness o f the
weight-ing factor leads to y ; = o
(7 = 1 7VFI + 1 , A ^ ^ + 1 N^2 + \) (21)
Similar discretization using linear elements is also ap-plied to eqn (13) i n order to evaluate hydraulic pressure on the body's surface. A more detailed description of the discretization procedure can be seen in Ohyama and Nadaoka.^"
3.2 Time-stepping procedure
Considering the finite-amphtude motions of both the wa-ter surface elevation and the floating body, the discretized
Nonlinear wave effect on the slow drift motion of a floating body 353
Fig. 2. Discretization o f boundary surfaces.
, Free Surface Sf [n + l)-step (n)-step (n + l)-step Body Surface Sv
Fig. 3. Movement of nodal points on % and Sy during the time increment A / .
integral equations, eqns (18) and (21), and the equations of motion of a floating body, eqn (10), are solved si-multaneously at successive time steps. In order to assign an identical spatial interval between neighboring nodal points, nodal points located on ,S'F and Sy are consid-ered to move in arbitrary directions with an advancing time step, as shown in Fig. 3. I n the present method, the surface elevation, velocity potential and velocity compo-nents of the body's motion at the {n -I- l ) t h time step, fj("+i), 0("+i) and are given as
,("+i)
ri = q"" + A r ] ' " ' on Sp
0(« + l ) = ^(n) ^ ^ ^ ( « )
i)(''>+AZ)<'" / c = l , 2, 3
(22)
where A t ] ' " ' , A0<") and A i ) [ " ' are the increments o f each variable during the time interval AZ.The displacement components of the floating body at the («+l)thstep,Z)["+'> {k= 1, 2, 3), can be obtained by
D ("+i) («) -Df + 0 { ( A r ) 3 }
' 4
+ 0{{At?] A: = 1 , 2 3
= 4 ' " + Atbi
i 4 " ' - f
AZ)["-'>'-(23)which prescribes the locadon of the body's surface Sy at the {n+ l ) t h time step.
Using Taylor expansions around the correspond-ing values at the («)th time step, {dqjdtY"*'^^ and
(d4>/dt)'-"^^^ are expressed as . dt («+i) 2Ar] («) Az A x " " At + 0{iAtf} \ d t ) '\dx in) + A \ d x ) («) (24) d^\ dt j ( « + i ) 2A0 («) At A x ' " ' d4> ^ 'di At At I [dz
+o{m^}
HS)""-(f)1
/ \dz J (25) in which Az<"' is identical to A / j ' " ' for nodal points onSp (see Fig. 3). Once the locations of the body's surface
and the free surface at the (n -H l ) t h step are calculated, { A x " " , A z " " } at the nodal points on Sy and A x " " at those on Sp can be prescribed accordingly. I n eqns (24) and (25), the spatial derivatives of 0 and rj are obtained f r o m the kinematic conditions on Sp and Sy, and their increments, A{dri/dx)<-"\ A ( 3 0 / a x ) " " and A O ^ / B z ) " " , are evaluated in an iteration procedure described below.
The equations of motion o f the body, eqn (10), are solved at the [n -F ( l / 2 ) ] t h step, i.e.
AD («) R [" + ( 1 / 2 ) ] At ik = 1, 2, 3) where ^ [ « + ( 1 / 2 ) ] ^ _ p " ^ [ « + ( l / 2 ) ] „ [ « + Sc "'^ds+Wk (26) « + ( 1 / 2 ) ] n + ( l / 2 ) ] ^ [ H + ( l / 2 ) ]
T. Ohyama, J. R. C. Hsu 2\\ds [«+(1/2)]-) 2 [« + (1/2)1 (27) and D ^[«+(1/2)] _ n(«) (") OUAt)^} ^ [ « + ( l / 2 ) ] ^ ^ ( » ) + ^ ^ ^ ( « ) + ^ ^ ( ^ ^ ) 2 j \ dt J [«+(1/2)1 A 0 M At - A r l t e ) ^ 2 ^ ( a ! ) At { [ d z ]
^1
\ d z ) + 0{{Atf} («) (28)I n eqn (27), O0/a5)["+('/2)l is calculated using
0[«+( 1/2)1 at neighboring nodal points.
Substituting eqns (22), (24) and (25) into eqns (18) and (21), and truncating after 0{At), a set of linear algebraic and eqn (26) simultaneously, the values of A4>j ( j equations can be established. By solving these equations
1 N), Anf (7 = 1 A^Fi + 1, iVi? + 1 iVF*2 +1) and Abk (k = 1, 2, 3) can be obtained. A t this stage, however, the coefficient matrix is associated with the boundary surface profile at the (ti + l ) t h step, which is not known after the calculadon at the «th step. Thus, in order to obtain the successful solutions associated with the corresponding profile, an iteration procedure is em-ployed using the profile at the previous time step as the inidal input. 4 I N C I D E N T WAVE F I E L D S 1.0 0.8 i.O c e 0.6 0.4 1 1 - ; — (Ist-order theory) 1 O O O 0 O Numerical results O A a^+Ao • Aa A 2Aa •
• *
i^J £ _ fi Ö d J—* U i A O A-1— 0 — A — 15.0 x/h 15.0 x/h 15.0 x/h • ( b ) o n I I I I-\
(2nd.order theory) 0 U O u O -Aa (2nd-order theory) • r . . . . O . .•
9 -A A— A A J A Uk 1 4JL_ —k— 1 • ( c ) I I I I 0 O n O u O " (3rd-order theory) A AOj+Aa (3rd-order theory)
. . . . A A
-•
•
^ . • . . . . a . . . . ^ . . . . * . . . . f . = Ao (3rd-order theory) A 1 A A A I I I IThe unintentional generation of free waves at a higher order along with bound waves when first-order wave the-ory is applied to generate large waves is known as the triad interaction phenomenon.^^-^^ The phase mismatch between the free and bound waves results in spatial mod-ulations of harmonic amphtudes, i.e. the wave spectrum. Therefore, for example, in testing the slow d r i f t motion of a floating body by either physical or numerical means, triad interaction can cause a significant problem in that the results obtained vary with the test location. I f the test model of a floating body is set at a location where free and bound long waves have inverse phases and cancel each other, the magnitude of the slow drift motion may be underestimated.
These parasitic free waves may be eliminated by utiliz-ing an appropriate nonlinear wave theory in wave gener-ation. For physical wave flumes, Barthel et al?'^ and Sand and Donslund^* have made attempts to suppress such free long waves in random wave trains by applying
second-0.4 0.2
( d )
^ Ö Ö O (4th-order theory)
a +Ao (4th-order theory)
Aa (4th-order theory)
A • ' • I
-/
2Aa (4th-order theory)
* A * Ï "
• » a^ « . 15.0
yjh
Fig. 4. Spatial evolution o f harmonic amplitudes o f (0-2), ( A c ) , ( o i + Acr) and (2A<T)-components in bichromatic
waves (aih/g = 0.5312, Aa/ao = 1/8 and lailh =
2fl2//' = 0.125). Wave theories with the following orders were applied in the generating waves: (a) first-order the-ory applied; (b) second-order thethe-ory applied; (c) third-order theory applied; (d) fourth-third-order theory applied.
Nonlinear wave effect on the slow drift motion of a floating body 3 5 5 + Numerical result Ist-order theory
32.0 34.0 36.0 38.0 40.0 42.0 44.0
+ Numerical result 4th-order theory
1.5 - 1 ' ' 1 ' 1 ' 1 1 1 ' ' -•n 1.0 L (b) . i
h h
k Ah
f{ ~^ 0.5\ l\
nh l\
f t f: 0.0 T t \ 1 \ 1 ^ i t t \ i' -0.5 . yV V y y
-1.0 -1.5 - r , , , . J . , , 1 , , , 1 , , . -32.0 34.0 36.0 38.0 40.0 42.0 44.0Fig. 5. Comparison o f bichromatic wave profiles between the numerical computation and the corresponding wave theory (al^lg = 0.5312, Aa/ao = 1/8 and 2a\lli = Zaj/li = 0.125): (a) first-order theory applied in generating
waves; (b) fourth-order theory applied in generating waves.
order control signals to the wave board. A similar inves-tigation using a second-order theory was made by Mad-sen and SorenMad-sen^' in numerical experiments. I t should be noted, however, that a second-order theory provides only the lowest-order solution for bound long waves so that free waves at much higher orders may appear i n wave trains with large amplitudes.
In this connection, 'bichromatic' waves, consisting of two free-wave modes and corresponding bound compo-nents were generated using wave theories with different orders of nonlinearhy The first-order free waves consid-ered had identical amplitudes {2a\ih = lailh = 0.125) and different angular frequencies, cri and cr2 (CTI < ai). The normalized mean frequency crlhlg was 0.5312, and the frequency difference Ao" ( = era - cri) was (TO/8, where (To = {a\ + (Ti)l2. The computational channel had a con-stant depth A, and was equipped with wave-absorpdon filters at both ends. The filter and the channel lengths, including the filters, were set at LQ and 6Lo, respectively, where £ 0 is the wavelength corresponding to Oo. The wave-making source, to which the wave theories were ap-plied, was located at the leading side o f the left-hand filter. In the computations, the time increment, At, and the horizontal projection of distance between the sur-face nodes on ^ F , A^^-;,, were ro/32 (7b = 2n/cro) and
1 , 0 / 3 0 , respectively. The inhial condition was zero surface displacement and velocity throughout the computational domain.
The numerical results, based on incident wave theo-ries up to fourth order,^" are presented in Figs 4 ( a ) ^ ( d ) . In these figures, spatial evolutions for the amplitudes o f the (0-2),
(Aa),
(cr2 + ACT) and (2Acr) components are plotted, together with the corresponding analytical solu-tions. The lowest orders of these components are the first, second, third and fourth orders, respectively. I n the case where the first-order theory was applied [Fig. 4(a)], i t is clearly seen that, along with wave propagation, a part o f the energy in the primary wave mode is gradually trans-ferred to higher-order components, such as the (Aa) and( a 2 + Aa) components. As shown in Fig. 4(b), the use o f
the second-order theory is effective to suppress such en-ergy transfer f r o m the primary to the bound waves. How-ever, the amplitude of the ( a 2 -I-Aa) component, of which the lowest order is the third, still increases with wave propagation, resultant from the generation o f the corre-sponding free waves. The third-order theory prescribes a relatively large amplitude for the ( a 2 -I- Aa) component at the wave-making source, which may cause further wave interactions [Fig. 4(c)]. I n this case, the amphtude of the
T. Ohyama, J. R. C. Hsu 0.30 0.25 0.20 0.15 0.10 0.05 h
a=Ao {4th-order theory)
a=2^a (4th-order theory)
t- a=Ao (2nd-order theory)
0.00
0.05 0.10 2 a i / h
0.15
Fig. 6. Amplitudes o f low-frequency bound components in bichromatic waves, obtained f r o m the second- and fourth-order theories (alhlg = 0.5312, Aa/ao = 1/8
and ai = ai).
0.00 0.05 0.10 2a,/h
0.15
Fig. 7. Amplitudes of the steady and low-frequency com-ponents in sway motion o f a rectangular body (alli/g = 0.5312, Aa/ao = 1/8, 6//i = 2.0, ? = 0.5 undK^/pgh =
0.05).
(2A(T) component. Figure 4(d), on the other hand, shows that spatial modulations of these harmonic amplitudes are practically eliminated by using the fourth-order the-ory. However, it should be noted that much higher order theories, which unfortunately have not yet been derived for random waves, are required to generate larger ampli-tude waves whhout parasitic free waves.
I n addition to the harmonic amphtudes, the corre-sponding wave profiles at x/h = 28.0 are given in Figs 5(a) and (b) for the cases where the first-order and fourth-order theories were applied in the wave generation. I n the former case [Fig. 5(a)], it is found that triad inter-action causes significant discrepancies in amplitude and phase between the computational and first-order solu-tions. I n contrast, the use of the fourth-order theory pro-vides satisfactory agreement [Fig. 5(b)], indicating that the magnitudes of the spurious free waves become negli-gible. Moreover, Figs 4(d) and 5(b) show that the outgo-ing wave trains containoutgo-ing both high- and low-frequency components can be favorably absorbed by the present open boundary treatment without significant reflection.
Comparison between Figs 4(b) and (d) further indi-cates that the magnitude of the (Ao") component ob-tained f r o m the fourth-order theory is smaller than that f r o m the second-order theory. Figure 6 shows variations of the normalized amplitudes of the (Aa) and (2Acr) components whh the first-order amplitude 2ai //?. It was found that the (ACT) component at the fourth order has an inverse phase to that at the second order. This results in the cancellation of the corresponding magnitude at 2 a i / / j = 0.134. Conversely, since the (2Ao-) component is produced only at the fourth order, \n2Ao-\/a\ indicates monotonous increase with 2ai/h. Therefore, the fourth-order terms contribute significantly to the low-frequency part o f t h e incident waves, which apparently influence the slow d r i f t motion of a floating body.
I n accordance whh the results presented here, the sub-sequent computations utilize the fourth-order theory^" in generating incident waves.
5 C O M P A R I S O N W I T H T H E SECONDORDER A P
P R O X I M A T I O N FOR B I C H R O M A T I C WAVE I N C I -D E N C E
Among the various approaches based on second-order theory, Agnon et al}'^ successfully derived a closed-form solution. Although its applications are restricted to only a narrow frequency band of incident wave trains and a rectangular body with sway motion, comparison with the present numerical results may reveal nonlinear wave ef-fects on slow d r i f t motion. For this comparison, numeri-cal computations were carried out for a rectangular body with a linear mooring system, which is allowed only to sway under the action o f bichromatic waves. The width of the body, b, and the elastic constant o f the mooring,
Kx, were set to bjh = 2.0 and K^/pgh = 0.05,
respec-tively. The incident wave conditions and the discretiza-tion parameters are identical to those given in the pre-vious section, i.e. a^h/g = 0.5312, ACT = cro/8, ai = « 2 ,
At = To/32 and ASf^ = Lo/30.
In Fig. 7, the computed amplitudes of the steady and low-frequency components of the sway motion, \Xa-\ (cr = 0, ACT, 2Ao-), are shown for ^ = 0.5 (^ is the ratio of the body's draft to the water depth), together whh the corresponding second-order solutions given by Agnon et
alP Within 2a\/h < 0.05, the present numerical results
for both \Xo\ and I I A O - I agree reasonably well w h h the second-order solutions and the magnitude o f | X 2 A O - | is negligible. For \Xo\, the deviation f r o m the second-order solution is still negligible for 2a\/h < OA. However, for
2ai/h > 0.05, the distinction in low-frequency input
be-tween the second-order and fourth-order theories (see Fig. 6) resuhs in a significant difference in the corre-sponding response of the body. The numerical result for
\XAO-\ shows smaller magnitude than the second-order
solution, and I X A O - I overwhelms \X^cr \ for 2a[/h > 0.11. The time histories of the horizontal displacement of the body X (s XG - [XG]Q, [XG]O. the initial location o f t h e gravitational center of the body) and its low-frequency part Xs (0 < or < ao/2) are given in Figs 8 and 9
Nonlinear wave ejfect on the slow drift motion of a floating body 357
32.0 34.0 36.0 38.0 40.0 42.0 44.0 46.0 48.0
Numerical result Agnon et al, (1988)
Fig. 9. Sway motion ( X ) and its low-frequency part (As) o f a rectangular body f o r 2a\lh = 2ailli = 0.15 {allilg = 0.5312, A t r / f f o = 1/8, bjh = 2.0, q = 0.5 and K^lpgli = 0.05).
for 2ai/h = 0.05 and 0.15, respectively. For the former smaller-wave case, both the amplitude and phase of the computed low-frequency response agree favorably with those of the second-order solution, indicating the valid-ity of the present numerical model in a small-amplitude range. Associated with the fourth-order effects on inci-dent waves, on the other hand, the (2Ac7")-component be-comes predominant for 2a\/h = 0.15. This resuhs in a significant deviation from the second-order profile.
I n Figs 10(a), (b) and (c), the numerical results for \Xo\, \XAO- \ and IXZAO-I are plotted versus the draft ratio o f the body q for different values oï2ai/h. I n these figures, all
amplitudes are normalized using the square o f ai so that the corresponding second-order solutions for the steady (o" = 0) and Acr-components do not vary w h h 2a\lh. For the wave condition examined, the second-order so-lution of |Xol is nearly constant for the range 0.3 < 9 < 0.9, and is in good agreement with the numerical results for 2a\ih = 0.05 and 0.1 [Fig. 10(a)]. The difference for
2a\lh = 0.15 may be primarily due to the deviation in
estimating the wave momentum, which determines the steady d r i f t force. I n contrast to |Ao|, resonant peaks ap-pear in the oscillatory parts [Figs 10(b) and (c)], at which the natural frequency o f the sway motion coincides with
h | X o l (2a,)^ h X Ao (2a,) 10.0 8.0 6.0 4.0 2.0 0.0
- Agnon et al. (1988) A 2aj/h=0.1 O 2aj/h=0.OS • 2aj/h=0.15 i * * i • 4 • • t * * _ t • j) A 0 Ó 6 ^ 6 — é — ê — 6 ó ö ó è - é — ó — ^ - 4 0.3 0.4 0.5 0.6 0.7 0.8 0.9 25.0 20.0 15.0 10.0 5.0 0.0 '-- — 1
-yfV
^ A...4...A - - ~ ; i A A A ^ <• « « « ^ ^ A A ' • • • 1 i > r1
A i 0.3 0.4 0.5 0.6 q 0.7 0.8 0.9 h X 2 Aal 20.0 (2a,)2 15.0 10.0 5.0 0.0 L . _ -..•....< 0 t. ~ • 1 1 . ^ • i> A A - A 1 A ^ - r ^ - 1 A -O ° i -O Q • * ~i\
" ^
'
• • . > 0 O < > « : \ A ? 0.3 0.4 0.5 0.6 0.7 0.8 0.9 qFig. 10. Variation in amplitudes o f the steady and low-frequency motion with body draft [crlhlg = 0.5312, Aa/ao 1/8, b/h = 2.0 and K^lpgh = 0.05).
Aa or 2Aa. Favorable agreement between the
computa-tion and the theory for 2ai/h = 0.05 shows the applica-bility o f the second-order approximation for such small-amplitude waves. I n the numerical results for 2ai/h = 0.15, on the other hand, a significant resonant peak of the
{2Aa) component emerges at q = 0.38, whereas IfjAo-l
becomes much smaller than the second-order solution in an entire range of q. These facts may indicate the impor-tance of higher-order nonlinear wave effects on both the incident waves and the interaction between waves and a floating body
6 S L O W D R I F T M O T I O N I N I R R E G U L A R WAVES I n addition to bichromatic wave incidence, the motion of a rectangular floating body in irregular waves was com-puted by the present numerical model.
The irregular wave trains considered here are composed
of 16 free-wave modes with the corresponding bound components up to the fourth order.^° Based on the JON-SWAP spectra^' for Tp^/gTh = 8.8 and 9.9 (Tp is the peak period), the first-order free waves with random ini-tial phases were generated within the frequency range o f 0.5 < (j/o-p < 2.0 (cTp = 2n/Tp). I n both cases, the angu-lar frequency interval between the neighboring free-wave components Acr was 0. ItTp, corresponding to the lowest frequency of the bound components. The width and draft of the body, and the elastic constant of the mooring were
b/h = 2.0, q = 0.5 and Kxipgh = 0.03, respectively. I n
the computations, the length o f the wave-absorption fil-ters was set at a typical wave length, Lp, obtained f r o m the hnear dispersion relation with TJ,. The time increment
At was rp/32 for both cases, and AS^x was Lp/32 and
Z,p/36 for Tp^gjh = 8.8 and 9.9, respectively
Computations were first conducted for incident wave fields where the floating body does not exist. Figures 11(a) and (b) show the incident wave profiles for H / j / h = 0.3,
Nonlinear wave ejfect on the slow drift motion of a floating body 359 1/3 0.0 0.6 O.'? 0.2 0.0 -0.6 -0.8 24.0
+ Numerical result •4th-order theory , 1 r 1 • • [ (a) * 1 1 ' 1 ' . l . l , 1 1 1 26.0 28.0 30.0 32.0 34.0 36.0 1/3 24.0 36.0
Fig. 11. Incident wave profile measured at the location corresponding to the gravitational center o f a rectangular body {xlh = 7.0): (a) Txi^fgUi = 8.8 and //i/s/Zi = 0.3; (b) Ji/sVgM = 9.9 and H\i^llt = 0.3.
T. Ohyama, J. R. C. Hsu
measured at xjh = 7.0 corresponding to the inhial loca-tion of the gravhaloca-tional center of the body. The computed wave profiles for both cases are in favorable agreement with the fourth-order solutions, indicating the sufficient reliability of the present open boundary treatment i n a wide frequency range.
In Figs 12 and 13, the simulated horizontal motion o f the floating body is shown for Tpy/gjh = 8.8 and 9.9 with different incident wave heights, together with their low-frequency parts A's within a frequency range o f 0 < cr/0-p<O.5.
I n the case o f Tp^/gjh = 8.8 with Hip/h = 0.1 [Fig. 12(a)], while the slow d r i f t motion may not be noticeable as compared to the fast motion, it becomes conspicuous for Hip/h = 0.3 [Fig. 12(b)], resulting in a maximum
X/H\/i more than double that f o r / A = 0.1. A similar
tendency can be observed in the resuhs for T p f g f h = 9.9 (Fig. 13). I n this case, however, the difference in the phase o f t h e slow modon profile (Xs/Hus) is also disdnct between the results for H\i^/h = 0.1 and 0.3.
The slow d r i f t motions normalized using {Hm)^ are plotted in Fig. 14. I n both cases, the computed pro-files for Hi/3/h = 0.1 agree reasonably well with those for Hiji/h = 0.05. This fact implies that the slow mo-tion Xs can be favorably described by a square func-don of the wave height and, hence, a second-order the-ory may be applicable in this small-amplitude range. I n the case o f Tp^/gjh = 8.8, the deviation o f the profile
for Hi/j/h = 0.3 may not be conspicuous f r o m that for
Hi/i/h = 0.05, but its normalized maximum
displace-ment becomes 24% larger [Fig. 14(a)]. I n general, the magnitudes o f the higher-order components o f the inci-dent waves become greater for longer wave periods. This leads to a more distinct variation of the correspond-ing response w h h wave height for the case o f Tp.JgJh = 9.9 [Fig. 14(b)], where both the amplitude and phase o f
hXs/iHip)^ for Hi/i/h = 0.3 are totally different f r o m
those for the smaller waves. I n this case, the second-order theory may overestimate the maximum displacement of the floating body.
7 C O N C L U S I O N S
(1) A previously developed numerical model,^" based on a time-domain B I E M with nonreflective open boundaries, has been extended and applied to the problem o f the slow d r i f t motion o f a two-dimensional floating body I n contrast to the Sommerfeld-type radiation condhions utilized in most conventional models, the open boundary treatment in the present model provides satisfactory wave-absorption performance over a wide range of wave frequencies. This makes it possible to perform prolonged simulations o f wave-body interaction in nonlinear irregular waves.
Nonlinear wave ejfect on the slow drift motion of a floating body 361
(2) Prior to investigating the slow drift body modon, wave-wave interaction in incident wave fields with two free-wave modes has been discussed. Wave the-ories with different orders of nonlinearity (up to fourth order) were applied in generating waves, and the resultant wave fields were examined. The results indicate that the use of low-order wave theories for relatively large-amplitude waves can lead to the gen-eration of undesirable parasitic free waves, and that the fourth-order terms markedly modify the magni-tudes of the low-frequency input.
(3) Comparison has been made with second-order ap-proximations'^ for the slow sway motion of a rect-angular body in response to bichromatic wave in-cidence. Within 2ailh = lailh < 0.05 (ai, aj are the first-order amplitudes o f the incident waves), the present numerical results of the steady (cr = 0) and slowly varying (cr = ACT, A I T is the difference in angular frequency between the first-order com-ponents) motions agree well with the second-order solutions, indicating the validity of the present nu-merical model and the applicability o f the second-order approximation in a small-amplitude range. For larger-amplitude waves, the numerical results show that the (2Acr) component becomes predominant in the body's motion as compared to the (Acr) compo-nent.
(4) The body's motion in irregular waves, involving 16
free-wave modes and corresponding bound compo-nents up to the fourth order, were computed for six cases o f two different peak frequencies w h h three different wave heights. The slow drift motion Xs can be well represented by a square function of Hip within Hip/h < 0.1 {H\/3 is the significant wave height obtained f r o m a first-order input), indicat-ing the applicabhity of a second-order theory in this range. However, particularly in the case of longer waves, the normalized displacement, hXs/(Hi/s)'^, for Hi/3/h = 0.3 exhibits significant deviation f r o m those for Hi,3/h = 0.05 and 0.1.
R E F E R E N C E S
1. Hsu, F H . & Blenkam, K. A., Analysis of peak mooring forces caused by slow vessel drift oscillafions in random seas. Proc Offshore Tech. Conf, Paper 1159, Houston, TX, USA, 1970.
2. Pinkster, J. A., Low-frequency phenomena associated with vessels moored at sea. Soc Pet. Eng. Jour, AIME, SPE No. 4839 (1975) 487-94.
3. Faltinsen, O. M . & Loken, A. E., Slow drift oscillations of a ship in irregular waves. Appl. Ocean Res., 1, (1979) 21-31. 4. Matsui, T., Analysis of slowly varying wave drift forces on compliant structures. Proa 5th Offshore Mech. and Arctic
T. Ohyama. J. R. C. Hsu
5. Sdavounos, P. D., Radiation and diffraction of second-order surface waves by floating bodies. J. Fluid Mech., 196 (1988) 65-91.
6. Report of the Ocean Engineering Committee. Proc 19th
Int. Towing Tank Conf., Madrid, Spain, 1990, pp. 499-509.
7. Drimer, N . & Agnon, Y., A hybrid boundary element method for second-order wave-body interaction. Appl
Ocean Res., U {\99A) 21^5.
8. Maruo, H., The drift of a body floadng on waves. J. Ship i?ej., 4 (1960) 1-10.
9. Newman, J. N., Second-order slowly-varying forces on ves-sels in irregular waves. Proc hit. Symp. on the Dynamics of
Marine Vehicles and Structures in Waves, 1974, pp. 193-7.
10. Pinkster, J. A., Mean and low frequency wave drifting forces on floating structures. Ocean Engng, 6 (1979) 593-615. 11. Agnon, Y. & Mei C. C , Slow drift modon of a
two-dimensional block in beam seas. J. Fluid Mech., 151 (1985) 279-94.
12. Agnon, Y , Choi, H . S. & Mei C. C , Slow drift of a floating cylinder in narrow-banded beam seas. J. Fluid Mech., 190 (1988) 141-63.
13. Vinje, T. & Brevig, R, Nonlinear ship modons, Proc 3rd
Int. Conf Numerical Ship Hydrodynatnics, Paris, France,
1981, pp 257-68.
14. Isaacson, M . , Nonlinear-wave effects on fixed and floadng bodies. J. Fluid Meek, 120 (1982) 267-81.
15. Sen, D., Pawlowski, J. S., Lever, J. & Hinchey, M . J., Two-dimensional numerical modelling of large modons of float-ing bodies in waves. Proa 5th bit. Conf Numerical Ship
Hydrodynamics, Hiroshima, Japan, 1989, pp. 351-73.
16. Yang, C, Liu, Y. Z. & Takagi, N . , Time-domain calcula-don of the nonlinear hydrodynamics of wave-body interac-tion. Proa 5th Int. Conf Numerical Ship Hydrodynamics, Hiroshima, Japan, 1989 pp. 341-50.
17. Brorsen, M . & Bundgaard, H. I . , Numerical model of the nonlinear interaction of waves and floating bodies. Proa
22nd Int. Conf Coastal Engng. ASCE, 1990, pp. 3140-51.
18. Longuet-Higgins, M . S. & Cokelet, E. D., The deformation of steep waves on water. I . A numerical method of compu-tation. Proa R. Soc Lond, A 350 (1976) 1-26.
19. Dommermuth, D. G. & Yue, D. K. P., Numerical simula-tions of iionlinear axisymmetric flows with a free surface.
I Fluid Mech.. 178 (1987) 195-219.
20. Ohyama, T. & Nadaoka, K., Development of a numerical wave tank for analysis of nonlinear and irregular wave field.
Fluid Dyn. Res., 8 (1991) 231-51.
21. Israeli, M . & Orszag, S. A., Approximation of radiation boundary conditions. J. Comp. Phys., 41 (1981) 115-35. 22. Ohyama, T , Beji, S., Nadaoka, K. & Battjes, J A.,
Ex-perimental verification of a numerical model for nonhnear wave evolutions. / Waterway. Port. Coastal and Ocean
En-gng. ASCE, 120 (1994) 637^4.
23. Ng, J Y. T. & Isaacson, M . , Second-order wave interac-tion with two-dimensional floating bodies by a time-domain method, Appl Ocean Res., 15 (1993) 95-105.
24. Brorsen, M . & Larsen, J, Source generation of nonlinear gravity waves with the boundary integral equation method.
Coastal Engng, 11 (1987) 93-113.
25. Mei, C. C. & Ünlüata,Ü., Harmonic generation in shallow water waves. In Waves on Beaches, ed. R. E. Meyer. Aca-demic, New York, USA, 1972, pp 181-202.
26. Bryant, P. J, Periodic waves in shallow water. J. Fluid Mech.,
59 (1973) 625^4.
27. Barthel, V , Mansard, E. R D., Sand, S. E. & Vis, F. C , Group bounded long waves in physical models. Ocean g/!^, 10 (1983) 261-94.
28. Sand, S. E. & Donslund, B., Influence of wave board type on bounded long waves, J. Hydraulic Res, 23 (1985) 147¬ 63.
29. Madsen, P. A. & Sorensen, O. R., Bound waves and triad interactions in shallow water. Ocean Engng, 20 (1993) 359¬ 88.
30. Ohyama, T , Jeng, D. S. & Hsu, J R. C , Fourth-order the-ory for multiple-wave interaction. Coastal Etigng, 25 (1994) 43-63.
31. Hasselmann, K., Measurements of wind-wave growth and swell decay during the Joint North Sea Project (JON-SWAP). Report No. 12, Deut. Hydrogr. Inst., Hamburg, 1993.
