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Numerical Prediction of Semi-Submersible Non-Linear Motions in Irregular Waves X. B. Chen and B. Molin Institut Fransa~s du Petrole Rueil-Malmaison, France Abstract A numerical model to predict non-linear vertical motions of a semi-submersible in regular and irregular waves is presented. Non-linear second-order forces are evaluated by using Molin's method which permits to obtain them without explicitely solving for the second-order potential. Non-linear low frequency motions being large in magnitude, variations of second-order forces with the motion of the body mean position are considered in the time simulation. As application cases, two floating bodies with small waterplane areas are taken into account. Comparisons of calcu- lations with experimental results show that this theory correctly predicts the low frequency motions. 1. Introduction The nonlinear behavior of floating offshore structures in regular and irregular waves is an interesting and important topic in ocean engineering. The problem of dynamic interaction of ocean waves and a floating structure is intrinsically nonlinear. Even though potential flow is assumed, direct solution to the three-dimensional problem has not been achieved yet. For small amplitude waves, the perturbation method is often used to sep- arate it into first- (linear) and second-order problems. The first- order problem can be solved by using a singularity method. The second-order problem is quite more complicated. The second- order problem is important as it yields the second-order loads whose frequencies, disjointed from those of the first-order, may be close to the resonant frequencies of the fluid-structure sys- tem. These exciting loads are not very large in magnitude but significant amplification may be induced due to the small system damping. The large low-frequency motions of floating bodies ex- cited by second-order low frequency loads are dealt with in this paper. In recent years, numerous contributions (Newman[02]i 1974, Pinkster and Huijsmans[03] 1982, Molin[04] 1983 etc.) have been published. Although horizontal low-frequency motions for a moored floating structure are often evaluated and in reason- able agreement with experiments, vertical resonant motions have been little studied and poorly predicted (Pinkster et DiJk[05] 1985~. A time simulation model based on separation of mean large slow motions of bodies and their first-order motions by two time scales is presented. Triantafyllou's[06] (1982) theory on the horizontal surge, sway and yaw motions is extended, in this model, to full six degrees low frequency motions. Specifically the variation of the second-order loads with body mean position is Reference number at end of paper 391 studied and some interesting remarks are deduced. Exact second-order loads are obtained by applying Mo- lin's[~] (1979) method and using two identities to transform Haskind integrals (Chen[~] 1989) for second-order potential con- tributions, besides first-order quantities contributions which are evaluated straightforwardly after solving the first-order problem. The system damping being an important factor is composed of radiation part and viscous part. At low frequencies as the radi- ation damping is negligible, the viscous part presented by linear and quadratic model is used in time simulation. The validation of numerical model is made by comparison with experiment results. Two floating bodies having low vertical resonant frequency idealizing characteristic offshore structures which have large submerged volumes with comparatively small waterplane areas, are used in experiments undertaken at the Ship Research Institute of Japan by Molin[~] (1982~. Comparisons of numerical calculations with experimental results show that this theory predicts correctly the large low frequency vertical motions. 2. Low Frequency Motions Theory A floating body is assumed to be submitted to bichro- matic waves of amplitudes al, a2 and frequencies we, w2. It re- sponds oscillatorilly with small amplitudes at the frequencies we and w2 around its mean postion which moves with large am- plitude at the low frequency twoway. As illustrated by figure 1, three reference systems are defined: (1) A three-dimensional Cartesian coordinate system Oo~oyozo fixed in espace with the z- axis vertically upwards. (2) System ozyz following the large low frequency motion which has its origin at the body gravity center G. and (3) Another coordinate system OXYZ tied to the body which coincides with ozyz in the absence of small high frequency motions. The three systems coincide at the initial time. Assuming a vector rO(t) with six components of surge, sway, heave, roll, pitch and yaw which designate the six degrees of freedom of the body: r0(t) = tzoft), yO(t), Soft), joft)'coft), joft)3T (1) Two small parameters ~ and ~ are introduced to express this vector into two parts: r0(t) = routs + ~ {(t, the) (2) Where a, a perturbation parameter, is proportional to the ratio of wave amplitude to wave length while ~ represents the ratio between the two time scales of motions: a high frequency mo-

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tion with small amplitude around its mean position and a low frequency motion of the body mean position of large amplitude because of low resonant frequencies of the system. (More strictly Figure 1 Reference Systems so ~YO z / \1 1 /` and o 'it 1~ IVY ~ we should take six ratios of time scales since the six low resonant frequencies are not the same). It was shown by Triantafyllou[06], in the inviscid, irro- tational and incompressible fluid, that the two potential prob- lems relating these two-scales motions can be separated and that the first-order problem is the same as if no low frequency mo- tions exist and high frequency first-order forces and motions are only parametrically dependent on the low frequency motions. We shall emphasize this 'parametrical' dependence by representing the low frequency forces as a function of the body mean position. The high frequency motions excited by first-order forces can easily be derived by classical means of using motion transfer function and those from second-order high frequency excitation are of order 52 and negligible. As these high frequency motions are supposed to be small as compared to the low frequency mo- tions, our efforts are concentrated on the evaluation of the large mean position motions: rG(t) = [xG(t), yG(t), zG(t), ~G(t), ~G(t), (G(t)]T (3) We shall first establish the low frequency motion equation, in the following, assuming the dependence on the mean position of the second-order low frequency forces and proceed the time simulation and frequential analyses. The complete second-order forces are evaluated in the last paragraph of this section. 2.1 Low Frequency Motion Equation The complex notations are afterward admitted in follow- ing analytical expressions. For exemple, a real harmonic function H(t) of frequency w having an amplitude h and a phase ,(] is rep- resented by: H(t) = h ~ costed +,l]) and so: = h cos,l} costed)h sin l] singlet) =~{h~e~iWt} (4) h=hcos,Bi~hsin,0=h~e~i~ (5) where i = ~/=[ and ~{F) designates the real part of the complex function F. Assuming the mass-inertia matrix of system and added mass-inertia matrix M + m, damping matrix B. restoring ma- trix K and exciting low frequency forces F(ro, t), the motion equation can be written: (M + m) d`2 rG(t) + B do rG(t) + KrG(t) = F(rG, t) (6) In this equation, the exciting low frequency forces can be caused by wind, current and waves. For the sake of simplicity, F(rG, t) only includes the wave effects. Using Taylor's series, these low frequency forces can be expressed by: F(rG, t) = F(roG' t) + t~rGTOG) V] F(rOG, t) + (7) By denoting: Fott) = F(roG, t) (8) F:(t) = VF(rOG, t) (9) noting that F~(t) is a 6 * 6 matrix, the equation (7) can be then written as: F(rG, t) = Forth + F:(t) (rGTOG) + (10) Substituting the equation (10) in which the first two terms are used and the vector TOG, representing the origin position of the body, is taken as zero vector into the equation (6), the latter can be rewritten as: (M+m~d'2rG(t)+Bd~rG(t)+[KF:(t)]rG(t) = Fott) (11) This equation ressembles Mathieu's equation if Fat (t) is periodic. The solutions to this equation will be discussed in detail for the cases of regular, Dichromatic and irregular waves. Motion Id Regular Waves A regular wave with an amplitude a and a phase ,B of frequency w, in the direction of angle ~ with the axis x is repre- sented by the following form: 7~(t) = a cos~kx cos ~ + by sin ~atl]) with: = JR {a emit } (12) a = a . eik(= cOs e+y sin 8)-i~ (13) The low frequency forces contain only the steady forces which are function of wave frequency and mean position: F(ro, t) = Foe + Fat ro (14) in which Foe is the steady forces at the origin position and F.`: their first derivatives with regard to mean position displacement. They are expressed in complex form: and: Foe = ~ { 2 an F20(w,w) } (15) Fat = at{ {aa*F (w w)} (16) where a* is complex conjugate of a, the complex amplitude of wave at the original point COG, YG) and F20(wj,we) is the trans- fer function of the second-order low frequency forces in bichro- matic wave of frequencies wj and we (here we impose Wj = we = w). F2~(wj,we) is the gradient matrix of Frown,we). Intro- ducing these equations (1~15 16) into (11), we obtained a new formula to calculate the mean displacement of the body: 392

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roO = [K37 { 2 aa* F2l (w ~Cal) }] 37 { 2 aa* F20(w,w) } (17) instead of the classical formula: rGoc = K-i ~ { 2 aa F20(w,w)} (18) The formula (17) indicates that mean displacement of a floating body in regular wave is not strictely proportional to the square of the wave amplitude and that it depends also on the derivative of the steady forces. For example in the case of heave motion, the vertical steady force and its derivative being usually of the same sign, the mean displacement predicted by (17) may be much larger than (18) which is classically used. Another in- teresting remark can be obtained by considering the free motion of the system represented by the equations (ll). Tl~e natural res- onant frequency of heave motion, assumed to be decoupled from the other degrees of freedom, is given by: WRz = ~ (= ~{2 aa F2l:(~'w)} (19) This shows that the natural frequency in still water is diflfer- ent from that in a regular wave. The natural frequency may be smaller when wave amplitude increases and it depends in addi- tion on wave frequency. In fact, the derivatives of steady forces play a role like a supplement (or decrease) of the restoring forces in the system. Motion In Bichromatic Waves Free surface elevation of Dichromatic wave is represented, at first-order form, as: t7(t) = 9~(8,~t,4:,t)+772(8,~2,42,t) = ~ {at e~i~l'} + ~ {a2 e~i~t } (20, here al and a2 are complex amplitudes of waves including the directional angle ~ and their phases ~ and ,02, and corresponding frequencies w~ and w2. In this Dichromatic wave system, the low frequency forces are composed of a steady part and an oscillatory part which has a difference frequency (~:and: and they are written by: F(rG, t) = Fa(rG) + F2(rG, t) (21) Fa(rG) = F`o AFRO rG (22, F2(lo,t) = F20(t) + F21(t) lo (23) Introducing above identities into the equation (11), we have the motion equation as: (M + m)d~2rG(t) + Bd~rG(t) + [KFatF2~(~)]rG(t) = Fdo + F20(t) (24) in which the low frequency forces can be rewritten in complex form including their transfer functions: F&o = ~ { 2a~a;F20(~t,Am) + 2a2a2F20(~2,~2)3 (25) Fat --~{2a~a;F2~(~:,we) + 2a2a2F2~((~,w2)} (26) F20(t) = ~ {ala~F20(~l,W2) e-i(~l-W2)~} (27) F2~ (t) = ~ {at a2 Fat (ma,w2) e -i(~t ~W9)t } (28) The evaluation of the transfer function of second-order low fre- quency forces will be considered in the next paragraph (2.2~. It is assumed to be known now. The exact solution to the equation (2~1) can be obtained by time simulation method. We integrate this equation, supposing a starting point of rG(O) = `& rG(O) = 0, by a stantard fourth-order Runge-Kutta method. The motion simulate;] becomes stable af- ter some transient periods. The frequency domain analysis, in Dichromatic wave case, is possible if the following assumption is admitted: rG(t) = rGO + 37 [rG~e~i(~~~)~} + ~ {ro ze~i2(~l~w2)~} (29) Introducing above equation into the equation (24), each mode of motion can be approximatively obtained by below identities: rGO = [KF`~-~ F`o (30) rot = [ - (M ~ m)(w:waysiB(~w2) + KF,`~] ~ Alar [F20(~,w2) ~ F2i (at,W2)rGO] (31) rG2 = [ - 4(M + m)(w:waysi2B((~:w2) + KFat] 2 alas [Fat (at,w2)rG~] (32, The formula (30) for mean displacment is the same as (17~. The oscillatory amplitude of frequency (adw2~) iS determined by (31~. It depends not only on the low frequency forces but also on the mean displacement (rGo) and derivatives of steady forces (F`~. Since the second-order forces are proportional to the square of the wave amplitudes, the amplitude of the low frequency is thought to be the same too. It may not be true be- cause the 'supplement' stiffness (Fat) increases with the square of wave amplitudes. For instance, in case of heave motion, Fizz being positive, the motion amplitude does not increase paraboli- cally as wave amplitude increases. The motion amplitude should be less than proportional to wave amplitude square when the difference frequency is larger than that of resonance and more when the difference frequency is smaller. We keep the double dif- ference frequency 2(~1w2) term because this frequency may be closer to the resonant frequency of the system when the different frequency (wew2) is much lower. Motion In Irregular Waves The unidirectional irregular wave is often represented by a finite series of elementary Airy components whose amplitudes are deduced from the wave power spectrum S,7,7(W): N 7~(t) = ~ aj cos(kjx cos ~ + kjy Sill ~wjtdj) (33) j= and: aj = ~/2S,1,7 (A )tWj (34) where ~ is the wave direction angle. Phase angles Al (j = 1, 2, , N) are determined at random between (0, 27r) and {Wj is the jth discretization step of power spectrum frequency width. If we use the complex presentation, the wave system is rewitten by: 393

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t7(t) = 32 {I ad e } with: ad = `72S,7,7 (wj )~j . e$ki(~ cOs 8+y sin 8)-~J (36) If all discretization steps bwj(j = 1, ,11) are equal (= bw) and the largest low frequency considered am is: Wm < n bw n < N (37) we can then express the second-order steady part and the oscil- latory part of the low frequency forces by following simple and double summation forms: F`o = ~ { ~ 2 as a; F20 (wj,wj ) } (38) Fat = FIR { Am, 2 a} ad* Fat (wj ~no ) } (39) ~ n N ~ F20(t) = ~ ~ ~ ~ ajaj_eF20(wj,Wj_~) e~i(~j~~j-~)t t=t j=~+i ) (40) n N ~ Fat (t) = SIR }~ Am, Am, ajaj*_~F2~ (Wj,Wj_~) e~i(~j~~j~~)t 1~=1 j=l+1 ) (41) As we see the above expressions of oscillatory part of low frequency forces, two remarks have to be pointed out: one is that the number N of element wave frequencies should be large enough to represent correctly irregular waves and to evaluate the low frequency forces at the difference frequencies (WjWj_~) for = 1,.~.,n and j = 1,---,N (N is typically between 100 and 200). Another is that the double summations of low frequency forces at each time step need more computing tinge as the num- ber N is large. Prior to doing the simulation by same equation (24) as in Dichromatic wave case, two special files have to be built up. First file containing the second-order force transfer function at origin mean position can be obtained by running repeately a first-order diffraction-radiation code for a number (m) of fre- quencies (m may be equal to N/10 or N/5) which cover the whole frequency range and successively a second-order code for repeating m(m+1~/2 times The first- and second-order solutions for other frequencies can then be obtained by an interpolation method. Second file containing the derivatives of second-order force transfer function can be achieved by etablishing two files at two or three different mean positions and using the finite dif- ference method. 2.2 Low Frequency Forces The low frequency forces, obtained by integrating the second-order hydrodynamic pressure on the body wetted surface, consist of one part which depends only on first-order quantities (potential, velocity and motions etc.) and another part which depends on the second-order velocity potential. They can then be written as: Frown,w') = F21(Wj,w') + Fb2(Wj,w`) (42) The first part forces F2~(Wj,we) are formulated by integrat- ing the hydrodynamic pressure which corresponds to quadratic terms in Bernoulli's equation and corrective terms due to the in- tegration over the average surface instead of instantaneous sur- face (supposing a given mean position). These forces consisting of four terms are presented by: 2 Afro +P2~/ v~ji).v(~) NdS 2 ./. ./s.Co(MoMj~'v( )MoAl'~ivit)'Nds 2(Rj Fens + R' Fog) (43) where p is water density and 9 is the acceleration due to gravity. The indices j and e represent the first-order quantities corre- sponding the wave frequencies Hi and we and the sign * des- ignates the conjugate complex. N is the general normal vector of body surface towards the fluid. The first contribution is an integral of relative water elevation (difference of water eleva- tion ~ and vertical displacement () on the average waterline I'o. The second contribution is the integral of pressure due to the fluid quadratic velocity (~(~) being first-order velocity potential). The third one comes from the corrective term of the first-order pressure on the average surface instead of instantaneous surface (MoM being translation of one point Mo on body surface) and the fourth one is a correction of first-order inertia forces Fin due to the first-order rotations R. All these contributions can be evaluated directely once the first-order problem is solved. In fact, the first-orecr veloc- ity potential obtained by a singularity method is used to have the first-order excitation forces and motions by mechanical equa- tions. A source distribution which is kinemalica]ly equivalent to the body responding in waves can be obtained by considering first-order motions and used to evaluate first-order quantities (potential and its spatial derivatives on bocly's surface and the water elevation on the waterline) that are needed for the first part of the second-order forces. The second-order velocity potential ~(2)~,y,z,t) is as- sumed to have incident, diffraction and radiation parts in the same manner as the first-order potential: 4~(2)(X ye Z t) = ~ {all (2) + (2) + ~~(2)) e~~(Wj~~)~} (44) in which the incident potential (2) is known: 2 ~9 where: in which H is the waterdepth. y,, _ 1 g~kjkey tankk'~H)(WjW')2 cosh[l kjk' (A + H)] A- i(kj - k')(~ cos 8+y sin 8) (45) A- = j ~ kjk' t1 + tanh~kjH) tanh~k'H)] - [wj cosh2(kj0) we cosll2(k'H)] (46) The second-order radiation potential is supposed to sat- isfy the same equations as the first-order radiation potential. The only difference corresponds to low frequency (wj~c) instead of the frequencies Wj or we. The solution by using the same sin- gularity method gives us the potential damping and added mass for one low frequency. 394 All nonhomogeneous properties of the second-order prob-

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lem are tied with the second-order diffraction potential which satisfies not only Laplace's equation, the sea bed condition equa- tion and a proper radiation condition but also the nonhomoge- neous free surface condition and the nonhomogeneous body sur- face condition. These two latter conditions are written as: on the free surface and: on the body surface. GodW'y2D2) + 9,' (2) = a(2) (47) ~ IDA) =~ (2) + a(2) (48) The nonhomogeneous terms a(2) arid a(2) being functions of the first-order potentials are expressed by: a(2) = it W`~(Vit) V( ) + Vest Vpj ) _u~ [j~)~-w' 5~(~)*+g5 (~)*) integral in fluid domain including the Laplace operator on ODE) and (l into boundary surface integrals. By analysing asymptotic expressions of the velocity potentials and using the stationary phase theorem for the surface integral at infirmity, Molill[~] (1979) has shown that the following identity is true: I ISCO D ||SCO [ knit + c ] jIdS 9 | .l o aL D . bids (54' where: 1 = 1, 2, , 6 Introducing this identity in the expression (51), we have the com- plete formula for the Ith component of the low frequency forces: F221(Wj,Wl) = F2II(Wj,WI) + F2CI(Wj,WC) + F231(Wj,W') (55) in which we name the incident integral: +(Aid I +Oz2~' )Pj] F2,,(~j,w') =it'sw'Jpll off Mods (56) and: + 2 ~ [() (_Wj2 By ( j) + g 5z2 ( )) +~-wj2,9 (j)+g5 2~(~j))~(~)*] (49) a(2) = (VEjVji))R'* nO + (VE'!V(~) JRj nO (MoMj V)V(~)*nO(MoM' V)Vji)nO (50) in which () is the first-order incident potential and (~) is the sum of the first-order diffraction and radiation potentials. VE is the velocity of the point Me on the body surface. It is important to note in the above expressions the dou- ble spatial derivatives of the first-order velocity potential. For a three dimensional body of arbitrary form, because of the sin- gularity of Green function as shown by t09], the double spatial derivatives near the body surface are not possible to be evaluated accurately by using a numerical method. The nonhomogeneous terms can not be then obtained directely. Belt we are going to try to calculate the integrals containing Close terms. by: The second part of the low frequency forces are written F22(h~j,Ad) = i~h~jW.C)p|| (~(2) + fD2))NdS (51) sea The contribution of the incident potential is easily obtained as the second-order incident potential is known (see equation 45~. In order to evaluate the contribution of the diffraction potential, additional potentials ',61 (I = 1,,6) are introduced. These ad- ditional potentials satisfy Laplace's equation, the sea bed and the radiation conditions and the two following boundary conditions: (Wjw')2?~' + g,90 ~,b1 = 0 z = 0 (52) ' = Nl on ScO (53) 1= 1,2,.~.,6 these equations being the same as in the first-order radiation problem, these additional potentials are obtained in the same way. Green's second identity is applied to transform the volume 395 the Haskind integral on the body surface: F2CI(~j'Ad) = i(WjW')P Jl [~( )a( ] ?/)ldS (57) and the Haskind integral on the free surface: F231~j,me) = it~e) | /, aLDIdS (58) With these three integrals including only terms function of the first-order velocity potential, the complete low frequency forces may be evaluated without explicitely knowing the second-order velocity potential which is more difficult to have, on condition that the difficulty of the double derivatives of the first-order ve- locity potential can be overcome. An identity deduced from Stokes' theorem can be derived by making the vectorial analysis: | | (G V)V~ no dS = /(V~ ~ G) dI' + | IS [(V~ V)G + Vat div(G>)] no dS (59) By using this identity into (57) and supposing G = ~.MoM, the Haskind integral on the body surface is transformed into integrals which don't contain any more the double spatial derivatives and can so be evaluated accurately. In order to evaluate the Haskind integral on the free sur- face, we divide the unlimited free surface into two regions: an in- ner region Sinn which is limited by the waterline and a boundary circle line some distance far away from the body and an outer region SOut which is the surface extending from the boundary circle line to infinity. In the inner region, since the Haskind integral contains the double spatial derivative of the first-order potential, another identity derived from Riemann's theorem is developped as: s. LIZ Jar [0Y 02 ] ~/IS [02 62~+ ,~yG- Gyp] dS (60) Using this identity, the Haskind integral on the inner region can be transfomed into integrals in which double derivative does not appear any more and may be so calculated numerically by using

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classical quadratic method. In the outer region, the asymptotical form of the non- homogeneous term a(2), in the cylindrical reference (r,E,z), is derived by using the developpment of Kocl~in's function in the presentation of the first-order velocity potential. This asymptotic expression of a(2) as form of a product of Fourier series in ~ and oscillatory radial functions in r and the similar expression for FEZ are introduced into the Haskind integral. This integral is then separated into a product of two line integrals: first integral in which is not difficult to be calculated by applying the orthogo- nality properties of Fourier series product and another integral in the radial distance r which is easily obtained by analytical results of Fresnel integral. The value of the Haskind integral on the free surface is the sum of integral results on the inner region and on the outer region. This value varies oscillatorily with the radial distance of the boundary circle line which separates Else inner and outer regions. As shown in [10i, the oscillatory value converges much more rapidly than that without the outer region integral and the final result is very close to the average value of oscillation. In a test case of a vertical cylinder (height/radius=5.) for which we have analytical results, the inner region is discretized by 36 points in circumference direction and 10 points in radial direction per composite wave length Ac which is defined lay: 2,T = c kjk' + kj' where kj' satisfies: (61) Gus _ w')2 = gkj' tanh~kj'H) (62) A good agreement between numerical and analytical results is obtained when the radial distance is more than three Ac. The average value of integral results for the radial distances between 4Ac and 5Ac is taken as the final result of flee lIaskind integral. The error between the numerical result and the analytical result is less than five per cent. The computing time for the low frequency forces is dom- inated by the evaluation of the Haskind integral on flee free sur- face, since other contributions are just in form of simple addi- tions of the first-order quantities which are obtained once the first-order problem is solved completely. In flee same case of a vertical cylinder whose one quarter surface is divided into 108 panels, the first-order solution costs about 3 ~ni~utes on Vax8700 computer while the free surface Haskind integral is obtained after 10 minutes calculus for one frequency. Ashen the system responds to only some frequencies, it is necessary to evaluate completely the low frequency forces for these frequencies. But it is also rea- sonable, when the frequencies considered are numerous, to take the approximation which consists of neglecting purely and sim- ply the free surface integral. Allis economical approximation is adequate when the diffraction effects are weal;. For in.stance, flee submersible or semi-submersible with small waterplane areas are the cases where the approximation can lie used. This is shown by Matsui[~] who considered the ITTC semi-subn~ersil~le platform. The work in [09] has the same conclusion. 3. Model Test Presentation The model tests were carried out at the Ocean Basin of the Ship Research Institute of Japan, while the second author was on sabbatical leave in 1981-1982. Two l~ottle-sl~aped models were considered, with different neck diameters. Cable 1 presents some characteristics of both models. Table 1 Characteristics of the Models Designation Model No.1 Model No.2 R1 Neck diameter 11.5 cm 21.6 cm HI Neck height 20.0 cm 20.0 cm R2 Bottle diameter 63.0 cm 63.0 cm H2 Bottle height 70.0 cm 70.0 cm M Displacement 220.3 Kg 225.5 Kg Cg Center of gravity 33.8 cm 34.7 cm Cb Center of body 35.4 cm 36.4 cm Kz Vertical Stiffness 113.3 N/m 370.6 N/m The test set-up is discribed on figure 2. The mooring sys- tem was constituted by two linear springs. The ballasts of the models were adjusted to ensure large natural periods in roll and pitch. In all tests it appeared that the pitcl, response was neg- ligibly small. The heave motion of the models was measured by a potentiometer. It was checked during the tests that the surge motion was always small enough to ensure an accurate measure- ment. Figure 2 Test set-up ,// ~ ~ r ~ . Potentiometer T R1 1 wave ~ R2 Fir . . 11 H ~ ~ A/ ~ ' ~~ ~1 ~ ~ / r Tests in regular waves were first undertaken for tile pur- pose to evaluate the vertical steady force. The mean vertical dis- placements were measured for wave periods ranging from 0.6 to 2.0 seconds and double wave amplitudes of 4. and 6. centimeters. Extinction tests were made in still water and in regular waves, in order to estimate the heave damping, which is an im- portant quantity in the evaluation of low frequency motion. In this series of tests, the model was given a vertical downward dis- placement (10 to 20 cm) and released. The leave damping divas classically estimated from the record of the decaying motion. Tl~e tests showed that the damping increases antler the effect of wave superimposition. Another feature observed in regular waves, is that the zero down-crossing intervals decreased as tl,' heave mo- tion decayed to zero. This variation of ~~at~'ral periods is tal;en into account by the equation (19~. A large proportion of the tests undert akel1 were in bichro- 396

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matic waves. In these experiments, two regular waves of equal amplitudes were superimposed. One pulsation was kept constant and different beat frequencies were achieved by varying the other. Another kind of tests was aslo made at constant beat frequency and varying wave amplitude. Tests in irregular waves plus cur- rent were aslo undertaken. In next section, the tests results and analysis are pre- sented in the comparison with the numerical predictions. 0.40 - 4. Numerical Predictions and Comparisons - For the purpose of the numerical solution to the diffrac- tion-radiation problems that is needed to evaluate the low fre- quency forces, the body wetted surface is n~csl~ed by quadri- lateral panels. The principle of discretization is that the panels close to the free surface and near the corners of body surface are finer than the others. Figure 3 presents the mesh used in the full numerical calculus. The total number of panels is 880. Figure 3 Mesh of the body surface The numerical accuracy depends directly on the repre- sentation of the body form, that is, the size of the discretiza- tion. A mesh giving an accurate solution to the first-order prob- lem may not be refined enough for the second-order problem. The convergence with discretisation was so investigated first. Model No.1 is chosen to evaluate the first-order vertical force and the second-order steady force in a regular wave of frequency (w = 6.283 rad./sec.) and the low frequency vertical force in Dichromatic waves of frequencies (~ = 7.222 rad./sec. and w2 = 6.283 rad./sec.~. Figure 4 presents the numerical results with respect to the panel number in which the first-order verti- cal force Few) with the markers ~ ~ ~ ~ is adimensionalized by pgLLa (here the reference length L = 1 m), the second- order steady force F2z~w,w) with the markers (-. ~ ~ by (pgLaa/2) and the low frequency force Fame,w2) with the markers ~ ~ ~ ) by (pgLa~a2~. From the figure 4 we see that the first-order forces have converged with 100 panels, while the second-order forces need over 800 panels. If the criterion of panel size for the first-order is one sixth of wave length, that of the second-order should be one fifteenth of the average wave length (about 800 panels on the whole surface). In order to achieve a good accuracy for the low frequency forces, we have then chosen the discretisation of 880 panels presented by the figure 3. The computer times on the series of Vax8000 computer, for one frequency, are about 9 minutes for the first-order problem and 30 minutes for the evaluation of the low frequency forces. Figure 4 Convergence with discretisation In o 0.30 ._ IS _ ~ 0 400 800 1200 1600 Panel Number In the regular waves, the second-order steady forces of model No.1 are evaluated for 16 frequencies from 2 rad./sec. to 14 rad./sec. at three vertical position of the body: the origin po- sition (neck height H1=20 cm), the higher position (neck height lI1=18 cm) and the lower position (neck height H1=22 cm). The results are presented by the figure 5 in which the curve ~ ~ describes the steady forces as a function of the wave frequencies at the origin position, the curve ~ jet the higher position and the curve (- - - -) at the lower position. Figure 5 Vertical drift forces at three reference positions 0.30 - cn ~ 0.20 - .= 0.10 - . at' i.~\ ~'~ 0.0 3.0 6.0 9.0 Frequency (Hz) 12.0 15.0 As the first-order vertical body motions are small, the drift forces dominated by the contribution of the quadratic term of the fluid velocity (see the equation 43) are always positive upward because the fluid motion is larger above the model than underneath. For the same reason the drift forces are higher at the higher position and lower at the lower position, as compared to the origin position, and they have a peak for a frequency equal to about 6.5 rad./sec. Figure 6 presents the derivatives of vertical drift forces with respect to the mean vertical position. These derivatives are obtained by finite differences. The adimenionalized vertical drift forces Fdo: at the origin mean position are drawn by the curve ~ ~ and their first derivatives Fd1z by the curve ~ ~ which are adimensionalized by (`pgLaa/2 * L/3cm). The second derivatives adimensionalized by (`pgLaa/2 * 2L2/9cm2) are also presented by the curve (- - - -I. 397

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Figure 6 Derivatives of the vertical drift forces 0.40 0.30 u' a) o 0.20 0.10 0.00 l"\ o.o 3.0 6.0 9.0 12.0 15.0 Frequency (Hz) The derivatives shown by the figure have similar shapes (a peak for a frequency equal to 6.5 rad./sec.) as the drift forces at the origin position. The values of the first derivatives are smaller than those of the drift forces and the second derivatives are much smaller than the first derivatives (with the adimensionalizations given above). So it is legitimate to use only the first term in the Taylor series of the steady forces (see the equation 7~. Using the vertical drift forces and their first derivatives evaluated numerically, the vertical mean displacements of the model in regular waves are obtained by the equation (24~. Figure 7 presents this result as the curve ~ ~ and the comparison with the model tests undertaken at the Ship Research Institute of Japan (markers ~ ~ ~ ~ and with the results calculated by the classical method of the equation (18) (the curve- - - -I. Figure 7Vertical mean displacements 0.201 1 0.15 c`i *E : cot ~ 0.10 c' - o 0.05 0.0 3.0 6.0 9.0 Frequency (Hz) 12.0 1 5.0 The vertical mean displacements are adimensionalized by the square of the wave amplitude. The comparison from the fig- ure 7 shows a good agreement between the experiments and the numerical predictions. But the classical method underestimates the vertical mean displacement. In the Dichromatic waves tests for model No.2 the second wave frequency ~2 = 5.712 rad./sec. is kept constant while the first frequency we varies from 6.283 rad./sec. t.o 7.140 rad./sec., which covers the whole frequency range used in the model tests. The second-order low frequency forces at the origin mean po- sition are presented on figure 8. The top curve describes the amplitude of total low frequency forces which consist of the first 398 part (see equation 42) shown in the middle, flee seco~-~d-order in- cident contribution and the Haskind integral on the body surface shown underneath and the contribution of the IIaskind integral over the free surface (bottom curve). Figure 8 Low frequency forces cat 0.40- ~L o 0.20 F2s 1 0.50 0.75 1.00 1.25 1.50 Low Frequency (Hz) The second-order low frequency forces are dominated by the first part. But it is not correct to take al ly this part as an ap- proximation of the low frequency forces because flee contribution of second-order potential (set-down effect) is important and in- creases with the difference frequency. Nevertheless the contribu- tion of the Haskind free surface integral is very small ~ compared to the other contributions. The approximation which consists in neglecting this small but expensive contribution is justified. The steady forces are evaluated as the sum of the steady forces in two regular waves of two different frequencies. The first derivatives of the low frequency forces and Else steady forces are obtained by the same procedure as those of flee steady forces in regular waves, applying the finite difference Clod to the results at three different mean positions. The obtained low frequency forces and the steady forces and their first derivatives are used in equation (24) for the motion simulation. Figure 9 presents one of the Dichromatic wave which was used ill Else model tests (co: = 6.756 rad./sec. and w2 = 5.712 r(l~l./.sec.) with a beat amplitude equal to 4 cm fat = a2 = 2cm.~. The low frequency motion simulation is shown on figure 10. In the equation (17), the added mass is fallen as the nu- merical result at the frequency equal to 1.044 ~ ad./sec. and the linear heave damping is derived from the extinction tests. The simulation is started at an initial point such that the displace- ment and the velocity are zero. The simulate<] Notion is stable after less than ten periods. The low frequency motion a~npli- tude and the mean displacement are measured from the stable simulation. In the same Dichromatic wave the simulations are worked out for double beat amplitudes from 2 cm to 8 cm. The results are shown by the curve ~ ~ in figure 11. The markers ~ ~ ~ ~ are the experimental points and the curve (- - - -) designates the classical calculus without the first derivatives of the second-order low frequency forces. According to the frequency domain analysis in tile above section 2, the amplitude curve of classical method is parabolic because the second-order motions are simply proportional to the square of the wave amplitude. But the motion silllulation which takes account of the variation of the second-order forces with respect to the mean position is not the same. When the beat frequency is larger than the natural frequency the simulation

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Figure 9Bichron~atic wave 0 10 20 30 40 50 60 70 80 90 100 It0 120 130 140 150 Time {second) Figure 10Low frequency motion 0.10 0.05 - a, -O. 05 0 10 20 30 40 50 60 amplitude is less than a square function of the wave amplitude. The difference between the two increases with the wave ampli- tude. The comparison shows that the simulation amplitude curve is much closer to the experimental results than that from using the classical method. Figure 11 Heave amplitude as a function of wave amplitude 20-1 1 16 c' cat - 12 8 I /' 0 2 4 6 8 10 Bich. Wave Hight (cm) The whole low frequency motions are obtained for the frequency range of the resonance by the same way. The figure 12 presents the results of the double heave aml>lit~lde with respect 399 70 80 Time (second) 90 100 110 to the beat frequency. The markers ( ~ ~ ) are the pOilltS of the model tests results. The continuous line () is derived from the motion simulation while the dotte(l line is obtained by using the classical method. Figure 12Heave amplitude of model No.2 12.0- * I 9.0- ~ 6.0 - ce a) 3.0 - 0.50 0.75 1.00 1.25 1.50 Low Frequency (Hz) The heave amplitude of the motion sillllllatioll is in good agreement with the experimental results. The classical method underestimates the heave motion for frequencies lower than that of resonance while it over-estimates the heave motion at larger

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frequencies. Moreover the motion simulation predicts correctly the resonant frequency of the system. For model No.1 in dichromatic waves, the numerical com- putations are carried out for frequencies w~ varying from 6.347 rad./sec. to 7.222 rad./sec. and fixed w2 = 6.283 rad./sec. The double beat amplitude is 5.7 cm (a, = a2 = 2.85 cm). Same pro- cedures are used as for the model No.2. The results are presented on figure 13. Figure 13 Heave amplitude of model No.1 ~ 20 .o o cats 10 - ~D I ___,__ 0.00 0.20 0.40 0.60 0.80 1.00 Low Frequency (Hz) Again the classical method does not predict well the low frequency heave motions. The motion simulation predicts cor- rectly the resonant frequency of the body in Dichromatic waves. Table 2 Resonant frequency prediction ~ hi_ ~ S ~ ~ Experiment I Model No.1 0.57 rad/sec 0.47 rad/sec 0.45 rad/sec Model No.2 1.04 rad/sec 1.00 rad/sec 1.00 rad/sec Although the numerical prediction gives the correct fre- quency of resonance, the agreement with the experimental results for model No.1 is not excellent. It is necessary to note that the heave damping in the simulation using the equation (24) takes the form as: B = Bo + Bi | d'z0(t)| in which the linear damping Be is derived from to extinction tests and the quadratic damping Be is computed by the following formula: Bi = ~ pSCa where S is the bottom area and the drag coefficient C`` is taken equal to 3. It is possible that the unsatisfactory comparison with the model tests arises from the unclear determination of the heave damping. Further investigation on the viscous damping of the system is necessary. f. Conclusions Using the Haskind integral relations, the second-order low frequency forces can be completely evaluated by Else aid of two transformation identities, without explicitely solving for Else second-order potential. The numerical results show that for float- ing semi-submersible bodies, the Haskind integral on flee free surface can be negligible. A simulation model for the prediction of low frequency motions taking account of the variation of the second-order low frequency forces with regard to the mean position has been pre- sented. Even though satisfactory comparisons with the model tests results are obtained for the heave motions in Dichromatic waves, the system damping for low frequency notion is needed for further investigations. Reference 1. B. Molin 1979 "Second order diffraction loads upon three- dimensional bodies" Applied Ocean Research. Vol:l No:4 ppl97- 202. 2. J.N. Newman 1974 "Second-order, slowly-varing forces on vessels in irregular waves" Proceedings of International Sym- posium on the Dynamics of Marine Vehicles and Structures in Waves. ppl9~197. 3. J.A. Pinkster and R.H.M. Huusmans 1982 "The low fre- quency motions of a semi-submersible in waves" Proceeding of Conference on Behaviour of Offshore Structure. pp447-466. 4. B. Molin 1983 "Three-year experience in flee numerical pre- diction of the slow-drift motion of noosed tankers" Proceeding of 15th OTC. ppl87-191. 5. J.A. Pinkster and A.W. van DUk 1982 and 1985 "Low frequency behavior of semi-submersibles" and "Wave drift forces on large semi-submersibles" Joint Industry Projects NSMB. 6. M.S. ~i~tafyllou 1982 "A consistent hydrodynamic theory for moored and positioned vessels" Journal of Ship Research. Vol:26 No:2 pp97-105. 7. B. Molin 1989 "Comportements no~-lineaires des plate- formes semi-submersibles" Actes des 2emes Jo~rnees do l'Hydro- dynamiques. ppl2~140. 8. T. Matsui 1987 "Second-order hydrody~.~mic forces on n~oor- ed vessels in random waves" IUTAM symposium pp292-300. 9. X.B. Chen 1988 "Etude des reponses du second-ordre d'une structure soumise a une houle aleatoire" These de Doctorat de ENSM. 10. X.B. Chen 1989 "Evaluation des efforts de basse frequence sur une structure soumise a une houle irregl~licre: con~parai- son de differences approximations" Actes des Scenes Journees de l'Hydrodynamique. ppl41-156. 11. B. Molin 1982 "Some experiments on the low-frequency heave Notion of floating bodies with sill<`ll waterl~la~le areas" IFP report. Reference number: 30 166. 400

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DISCUSSION by R. H. Huijsmans I like to congratulate the authors with their fine paper and their treatment of the mean position dependency of the second order forces, which resembles the way the wavedrift damping concept has been introduced. I have a question regarding the low- frequency drift forces depending on the second order potential F22. In using Lighthill's transformation to the Haskind integral we showed [A1] that the simplified analysis of this integral as was proposed by Pinkster [A2] was valid for a wide frequency range. This simplified analysis has the benefit of small computational efforts. Would the authors comment on this? A next question concerns the Fig.8 of your paper. It seems to me that for practical irregular waves notably JONSWAP type wave spectra, these only exists a small frequency band where the envelope spectrum has some significance. Therefore larger difference frequencies need not to be regarded extensively. Can the authors comment on this. [A1] A. Bens chop, A.J. Hermans, R.H.M. Huijsmans, "Second order diffraction forces on a ship in irregular waves", Applied Ocean Research, 9, 1987. [A2] J.A. Pinkster, "The low frequency excitation forces on ships", PhD Thesis, University Delft, 1980. Author's Reply 1. The contribution of the second-order diffraction potential to the second-order loads consists of two Haskind integrals: one on the free surface that involves the second- order correction to the free surface equation, the other over the hull that involves the second-order correction of the body surface equation. For difference frequency problems it is generally accepted that the free-surface integral is small, which has been confirmed both by the discusser's results [A1] and by ours. However, if the body is allowed to respond to waves, the body surface integral is not small, even for the difference frequency case. Thus Pinkster's approximate method, which only takes account of the second-order incident potential on the body surface, may not be valid. 2. We are thankful to the discusser that this second comment has made us aware that all figures in the paper are incorrectly labeled. The low frequencies on the horizontal axes are not expressed in Hertz but in radian per second (as it is written in the text). We apologize for that error. DISCUSSION l by R. Eatock Taylor It appears that the second order correction due to the body motions involves the large low frequency motions (eq.7) and the small wave frequency motions (eq.43). This presumably results from the two time scale approach, although the details are not given. Is this consistent with including terms from the second order potential for the low frequency forces? My memory of Triantafyllou's argument is that the corresponding forces are of third order, because of the small parameter which results from taking the derivative of the potential with respect to the slow time [i.e., the factor ( ~j-~L)]. This leads to a second question. The expression for the second order incident potential given in eq.45 does not appear to be valid-at very small difference frequencies. I suggested this at the Water Waves Workshop in Norway earlier this year, and I would like to ask the authors whether their formulation leads to a discontinuity in the vertical force as ~j+WL ; i.e. do they reach the regular wave result in the limit? Author's Reply We agree that our method can be rigorously justified only in the case when the time scale of the low frequency motion is very long compared to the wave motion time scale. Obviously this is not the case for the vertical motion of more traditional semi- submersible platform. but still we contend that variation of the low frequency exciting forces with the low frequency vertical motion should be account for, just as the low frequency horizontal velocity affects the low frequency horizontal loads. The discrepancy between the vertical drift force and the ~j ' AL limit of difference frequency vertical force had already been noted in [9]. In the present case it does not appear as our numerical model is restricted to the infinite water depth case. 401

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DISCUSSION by R. Zhao (1) Discuss about vertical drift force by using direct pressure integration or based on momentum and energy relations. Author's Reply In an earlier paper[A3] of the author computed the vertical drift forces on the same models, using both the momentum method and the direct pressure integral method. Numerical results showed excellent agreements. This success however may have been partially due to the fact that the diffraction-radiation problem was solved with a fluid finite elements technique, which allows for an accurate evaluation of the fluid particle velocities. [A3] B. Molin and J. P. Hairault, "On Second- Order Motion and Vertical Drift Forces for Three-Dimensional Bodies in Regular Waves", Proc. Int. Workshop on Ship and Platform Motions, Berkeley, 1983 402