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Application of a 3-D Time Domain Panel Method to Ship Seakeenin~ Problems H. Yasukawa (Nagasaki Experimental Tank, Mitsubishi Heavy Industries, Japan) ABSTRACT This paper describes outline of a 3-D time domain panel method for ship seakeeping. The method in- cludes nonlinearity of hull and free-surface bound- ary conditions. To reduce the computational time and the memory size, we employ 2 types of the free- surface meshes: ship fitted mesh for near field and space fixed regular mesh for far field. This also accel- erates the computation of free-surface influence func- tion. Hydrodynamic forces, ship motions in waves and wave pressure on the hull surface are computed for several ships. The results are compared with ex- periments and the calculated results bar strip method. The present method is validated for the hydrody- namic forces, ship motions and wave pressures, and they are much better predicted than by strip method. 1. Introduction The most commonly used tools to determine seakeep- ing performances are based on strip theory. The . ~ . . . . ~ . . ~ such as actual container ships. In addition, Iwashita pointed out that the fre- quency domain approaches do not work so well in the region where At_ Uw/g) is smaller than 0.5[4][5], where U denotes ship speed, ~ the encounter fre- quency and 9 the acceleration gravity. This means that RAG is not obtained for a wide variety of wave direction. That may be serious problem at the stage of short-term prediction of the response. We will present here a time domain 3-D panel method including nonlinearity of hull and free-surface boundary conditions for ship seakeeping. Time do- main approach has no limitation about revalue and is applicable to many ship and offshore problems with large amplitude motions. Hydrodynamic force and ship motion analyses using the similar nonlinear time domain method have been made, for instance, by Maskew[6] [7] [8], Beck et al. [9] [10] and Scorpio et al.[11]. LAMP code is also well known for large am- plitude ship motion analysis method in wavest12] [13~. Generally, the time domain code requires consid- str~p metnoct approach Is cheap, last, and tor most erable computational time for obtaining significant cases also quite accurate. However, strip methods do solution. We employed some numerical techniques to not perform so well for high-speed ships, ships with reduce the computational time with keeping the ac- strong flare, and generally for low encounter frequen- curacy and the robustness. Outline of the method cies which typically occur in following seas. will be presented in this paper. The present method As new calculation methods to overcome the strip can be regarded as an extension of the method for un- methods, frequency domain 3-D Rankine panel moth- steady wash analysis of a high speed vessel[14]. After oafs were proposed. For a recent survey of Rankine that, we put a scheme of ship motion computation to singularity methods for forward-speed seakeeping, we the original method[15]. To validate the method, hy- refer to [1] and [2]. The methods include fully 3-D drodynamic forces, ship motions in waves and wave effects of the flow and forward speed effects where pressure on the hull surface are computed for several they are not taken into account in the strip method, ships. The results are compared with experiments and we confirmed that local pressures, especially for and the calculated results by strip methodt16]. We shorter waves, are much better predicted than by found that hydrodynamic forces, ship motions end lo- strip method[3]. The frequency domain methods deal cal pressures are much better predicted than by strip with linear ship motion problem based on the steady method. The present method is promising as a new flow. Therefore, the methods may not be applicable ship design tool for more accurate prediction of sea- to large amplitude motion for ships with strong flare keeping performances.

2. Basic Equations 2.1 Boundary value problem Let us consider a ship advancing in a towing tank as shown in Fig.1. The ship moves with speed U(t) which varies as the function of time t. The coordinate system fixed in the space is employed. The x-axis is defined as direction from ship stern to the bow, y- axis to port and z-axis vertically upward. The x y plane coincides with the still water surface. The motion displacements with respect to surge, sway and heave in the fixed coordinate system are defined as (1,(2,T,3 respectively, and the vector A. Euler angles with respect to roll, pitch and yaw are represented as A, §, ~ respectively, and the vector Q. Incident waves are generated by movement of flap angle /9 of wave maker attached to tank wall. Deep water is assumed. .~ ·~` ~ W. Fig.1: Coordinate system Incident Wave where The perturbation velocity potential due to ship moving in the tank is defined as oryx, y, z, t). Then, fib has to fulfill the following boundary conditions: ,9t 9; 45~2Z;2 (Vo) on z = o (1) 0L LIZ + ~Z2 ~~ C Zinc, 3 J) on z = 0 (2) ,)¢ = (v +w x r) r' on SH (3) 655 = vo on Sw (4) where V = (~/0x, 0/0y, 0/0z). ~ denotes wave ele- vation. SH and Sw mean hull and tank wall surfaces respectively. Eqs.(1) and (2) are dynamic and kinematic free- surface conditions respectively. For simplicity, free- surface conditions including the 2nd order terms with respect to o and ~ obtained by Taylor expansion at z = 0 are employedt144. Eq.(3) is hull surface condition, and has to be sat- isfied on actual wetted surface SH. In eq.(3), r de- notes coordinate of hull surface position, n the out- ward normal vector of the hull. Further, v = (a, v, w) means the ship velocity vector defined in the coor- dinate system fixed to ship, and co = (p,q,r) the angular velocity vector. It should be noted that v in- cludes the component of ship velocity U(t). Relations between v and A, and between w and Q are written as follows: v= LE(Q)~(, a) = tH(Q)]Q Here, means d/dt. The matrix E and matrix H are written as: Em, A, ¢) = feijI (7) eat = cos ~ cos em = cos ~ sin el3 =sin e2l = sin ~ sin ~ cos ~cos ~ sin e22 = sin ~ sin ~ sin ~ + cos ~ cos e23 = sin ~ cos em = cos ~ sin ~ cos ~ + sin ~ sin e32 = cos ~ sin ~ sin ~sin ~ cos e33 = cos ~ cos 0 sin ~ H(~, 8, A) = O cos ~ cos ~ sin ~ (8, sin ~ cos ~ cos ~ Eq.(4) is boundary condition of tank wall surface, and v0 means normal velocity on Sw. When gener- ating the waves in the tank, we have to give a proper value to vo. On the tank wall without wave maker, we set v0 = 0. The detailed explanation is described below. The velocity potential ~ is represented using source strength ~ as follows: 0(P) = /7 ~(Q~G(P;QjdS (9) SH+SF+SW G(P; Q) = )2 + ( _ yi)2 + (z _ zI)2 ( ~

P = (x, y, z) is filed point, and Q = (xl, Y1, zl) the respectively. These vectors and matrix are expressed singular point. SF denotes the free-surface position as follows: (z = 0~. Substituting eq.(9) to eqs.(3) and (4), the following equations are obtained: I2X O IF TIJ= O IYY O (17) !!SH+SF+SW (Q) Art dS = fH(P) (1l) IF O IZZ where I (P) _ ,{ (v +w x r) n for P on SH v0 for P on Sw (12) Eq.(11) represents the boundary conditions on the hull and tank wall surfaces. 2.2 Hydrodynamic forces and motion equa- tions Pressure on the ship hull surface is given from Bernoulli's equation as: p/p _ _ 0° _ ~ (V¢,)2 (13) where p is density of water. Hydrodynamic forces acting on the ship is obtained by integrating p over the hull surface. Here, we define that subscript 1, 2 and 3 denotes directions of surge, sway and heave respectively, and subscript 4, 5 and 6 the directions of roll, pitch and yaw respectively. Then, the hydrodynamic force/moment with respect to i-direction Fit is written as: / 9 sin ~ FG = m gcos~sin~ (18) -gcos~cos~ ~ qwrv FI = m | rupw (19) \ pvqu ~ (IzzIyy)qrI~zpq MI = (ItIzz)rp + I=Z(p2r2) (20) (IYYIx=)pq + I=zqr 2.3 Wave-maker and numerical absorbing beach Incident waves are generated by numerical wave mak- ers (vertical flaps) on tank wall. Here we deal with only regular waves. The flap starts to move from vertical position, and the angle is given as follows: O(t) = Go sin at (21) where (3(t) denotes the actual flap angle, Go the am- plitude of flap angle and ~ the frequency of flap mo- tion. Then, v0 in eq.(4) is represented ast174: /7su [9 ,'~ + 2 (Ho) ] rlidS (~4) v0 = (d1 + Z) do (22) where (rl~,~2,n3) al, and (n4,ns,n6) _ r x n. Solving the boundary value problem mentioned above under proper initial conditions, velocity potential can be obtained through the source strength. Calculating velocity components and pressure on the ship hull surface from the potential, unsteady hydrodynamic forces are obtained from eq.(14~. Motion equations of the ship are expressed as: me + F] = FG + F (15) LI] ~ + M] = M (16) Here m denotes the ship's mass and tI] the matrix with respect to moment of inertia of the hull. F and M are vectors of hydrodynamic force and moment respectively, defined as F = (F1,F2,F3) and M = (F4, F5, F6). FG denotes force vector due to the hull gravity force. F] and M] mean vectors of force and moment due to centrifugal force acting on the ship where do is the height of flap part (see Fig.1). For simplicity, here, nonlinear boundary condition is not adopted on the flap surface. Only v0 is given at ver- tical fixed flap position in the computations. As numerical absorbing beach technique, the method proposed by Cointe et al. (184 where a proper damping is added to free-surface conditions, is adopted at only face-side wall against the wave mak- ers. On tank-side, we do not impose any numerical beach and allow of wave reflections. 3. Numerical Procedure 3.1 Flow of numerical computation Numerical flow for calculating ship motions and hy- drodynamic forces is as follows. 1. Accelerations of ship motions (( ,Q ), and time derivatives of wave elevation and velocity

potential on free-surface ((k+l, Ok+ respec- 9. Hydrodynamic forces acting on hull are calcu- tively at (k + 1~-th time step are assumed using lated using eq.~14~. The ¢~ term appears in the values at k-th step. Here suffix t represents eq.~13) is evaluated as time derivative. 2. According to Newmark's ~ method, ship motion velocities and the displacements at (k+1~-th step 10. are estimated by: Ok+ Ok ^t (<k Ok+ ) Ok+ Ok At Ok (~2<k 2 ~ .. k + ~ ·- k · k+1 · k ^t ~ ·- k ·- ken) Qk+i = Qk + /\t Qk + (/\~2 ·- k +~/\t)2 (Qk+i Qk ) (26) (23) (24) 3. Using eqs.~5) and (6), ~ and Q are trans- formed to v and w. 4. Ship hull and free-surface panels are arranged using given ship speed and estimated motions. 5. According to Newmark's ,() method, wave ele- vation and velocity potential on free-surface at (k + Moth time step are estimated as: ~ = ~ + /\t (hi + (~ + ) /2 (27) Ok+ = ok + At (of + of ) /2 (28) where /\t is time increment. 6. Influence functions with respect to free-surface, ship hull and tank wall surfaces are calculated for constructing the base matrix where unknown variable is source strength. The base matrix is obtained by discretizing eqs.~9) and (11) using constant panels. -I. Solving the base matrix by SOR method, source strength on free-surface, hull and tank wall sur- faces is obtained. 8. Using eqs.~1) and (2), ~k+l and )+i are cal- culated. Then, velocity components on free- surface are analytically calculated using the source strength. The derivatives of ~ with re- spect to x and y are numerically obtained using finite differential technique. Ok+ = (¢k+l _ ok)/~t (29) Obtaining the hydrodynamic forces, v and ~ are calculated from eqs.~15) and (16~. Using eqs (5) and (6), v and ~ are transformed to ~ and Q. .. .. 12. (,Q, ~ and ¢~ obtained in step 8 and 11 are compared with those assumed in step 1. When the difference between both is sufficiently small, .. .. hi, Q. ~ and of are regarded as reaching to the convergence. Otherwise, returning to step 2, cal- culations are continued using the I, Q. ~ and ¢~ instead of old values until obtaining converged solution through this iteration. 13. We set k one time step ahead and return to step where /\t is time increment, and ,l? the accelera- 1 tion factor. As shown in eq.~29), 04/0t on SH is represented as backward differential of a. For exactly evaluating this term, additional boundary value problem with respect to do/0t has to be solvedt194. In the present method, however, this approach was not employed for saving the computational time. 3.2 Matrix solver The base matrix obtained by discretizing eqs.~9) and (11) can be written as follows: LA'J ~ aF ~ + :,B~J {AH ~ = f F (30 tC]{aF) + tDj~aH) = fH (31) Here subscript F means value on free-surface, and subscript H the value on body surface such as hull and tank wall. fA] iS the matrix which is composed of influence function induced by free-surface source with unit strength on the free-surface. Likewise, [B] is the matrix which is composed of influence function induced by the body source with unit strength on the free-surface, iC] the matrix which is composed of influence function induced by the free-surface source with unit strength on the body surface, and fD] the matrix which is composed of influence function in- duced by the body source with unit strength on the body surface. The JF and fH are constant terms. We slightly modify eqs.~30) and (31) as follows: ~LA] {OF ~ = JF ABE {aH ) (32) tD: {AH ~ = f HtC] {aF ~ (33 ~

The an and AH can be obtained by solving eqs.~32) and (33) alternately until the solution is reached to the convergence. To quickly solve the matrices tA] and tD], we consider the use of SOR method which is fast and simple. Diagonal term of iD] is theoretically larger than other terms, so we can solve this matrix easily using SOR method. For tA], on the contrary, the diagonal term is not always larger. This means that we do not always obtain the solution. To certainly get the solution, we place restrictions on free-surface panel geometry with rectangular panels of the same size. In this case, slightly larger diagonal can be achieved in [A], and can solve the matrix by SOR method. For faster computations, suitable relaxation factor for each eqs.~32) and (33) should be chosen. For solving eqs.~32), under-relaxation factor is effective. 3.3 Treatment of Resurface panels Actually, we can not keep the same geometry and the same size of the free-surface panels in the proximity of ship hull because the hull has certain breadth. To settle the problem, we employ the following treat- ment: . 2 different free-surface panels are used as shown in Fig.2: rectangular panels fixed to the tank (tank fixed panels) and arbitrary shaped panels moving with the ship (moving panels). · In the tank fixed panels, ignoring the presence of the ship, regular rectangular panels are ar- ranged. The free-surface panels are often ar- ranged inside the hull on z = 0 plane. · The moving panels are restricted within a small region outside the ship's water line. The panel geometry automatically changes so as to match with boundary of the fixed free-surface panels. The moving panels are treated as additional un- known panels with source strength aF. Then, the panels within the moving panel region are double arranged. To avoid the trouble comes from the double arrangement, we impose the condition where source strength of the fixed pan- els becomes zerot204. In this treatment, we have to solve a new matrix with respect to aF. However, increment of unknown vari- ables is about several hundred, and the small-sized matrix can be quickly solved by Gaussian elimina- tion. That is no big difficulty. The free-surface panel arrangement using regular rectangular panels is also useful for efficient construc- tion of matrix tA]. In this case, evaluation of influ- ,Tank Fixed F/S Panels Fig.2: Treatment of free-surface panels ence function with respect to (A] depends on only dis- tance between 2 panels because the panel geometry is all the same. Then, tA] can be constructed using the influence functions induced by a row of panels on the whole free-surface panels. We do not need so large memory size for matrix [A] when employing this treatment. Remarkable saving of the memory size was achieved. 3.4 Ship's acceleration from the rest In the present calculations, not forward thrust force is given to surge force term but forward velocity is forced at the motion equation of surge. This is corre- sponding to simulating not self-propulsion test, but resistance test in waves. Then, ship speed from the rest U(t) is assumed to be given as follows: ~ Uo j6(t/to)5 - 15(t/to)4 + l°(t/to)3) U(t) = ~ for t < to Uo for t ~ to (34) Here, Uo denotes the target steady speed and to the acceleration time until steady state. As to, we em- ploy value of to~/377=15.7 to avoid the long period oscillation of hydrodynamic forces appears in case of rapid accelerationt214. In acceleration period until reaching to the steady ship speed, large damping is artificially added to the force term of the motion equation. This roughly con- strains the ship motion in the computations like a cramp in the experiment. The treatment is useful for reducing the computational time until reaching to steady state of the ship motion.

4. Computations of Hydrodynamic Forces 4.1 A modified Wigley hull with longitudinal motions First, the present method is validated for hydrody- namic forces with respect to heave and pitch. The hy- drodynamic forces acting on the hull with forced os- cillation (radiation forces) and the restrained hull ad- vancing in regular head waves (wave exciting forces) are evaluated for a modified Wigley hull. The hull is a mathematical form expressed as follows: A' = ¢1X,2~1 _ z'2y where x' = x/(L/2), y' = y/(B/2), Z = Z/d with L/B = 10.0 and B/d =1.6, where L, B and d de- note the ship length, breadth and draft respectively. Fig.3 shows panel arrangement of the hull surface. Number of the hull panels is about 800 for a half body. We employ the free-surface panel region with 12.5L for the length and 1L for the half width. The number of the free-surface panels is 5,000~= 250 x 20~. Cal- culated results are compared with the experiments conducted in Delft University of Technologyt224. Oscillation frequency and V the ship's volume. The time until t>/~=15.7 is acceleration zone and the heave motion is given so as to gradually increase. Where the non-dimensional time is up to 20, the calculated forces look stable and regular sinusoidal forces are obtained. Time averaged values of CF~ and CF3 are negative. This means that steady resis- tance and sinkage forces are acting on the hull. The averaged value of CF5 is almost zero. By analyzing the time histories of hydrodynamic forces, added mass and damping are obtained. Fig.5 shows comparison of added mass and damping coeffi- cients. The air and bit mean added mass and damp- ing respectively, with respect to i-th mode induced by j-th motion, where i/j=3 means heave and i/j=5 the pitch. Horizontal axis in the figures means non- ~ .o _ ~ Fn=0.3 / ~ . ~ o.o~ 10 of . . . . . modified Wialev - Forced Heave Mode w{(Ug)=3.86: . . . i - 30t T(g/L) , . . . . , . .] Fig.3: Panel arrangement of modified Wigley hull. This figure includes hull panels above the still water surface. Fig.4 shows time histories of hydrodynamic forces excluding restoring force component for forced heave with w~/~75=3.86. Froude number based on the ship length En is 0.3. Amplitudes of the oscillation (As, A5) are assumed to be 0.01L for heave and pitch respectively. The hydrodynamic forces induced by i-th motion are normalized as: o.o, , J - cO o -0.01 o t T(g/L) ! ~ t T(g/L) u.u~ . . I ~ . . . ~ . . . . ~ 002 ~~. -0.04 0 10 20 30 10 20 tT(g/L) F F3 F5 Fig.4: Computed time histories of ship speed CFl'CF3= pVAc,,2' CF5= pVA w2l (36) (U/Uo), surge force(CFl), heave force(CF3), pitch moment(CF5) and forced heave (63/L) where F1,F3 and F5 mean surge force, heave force and pitch moment acting the hull respectively, ~ the

a33lp V 0.80 06 -O. C -I--` 0.20 O ECal. (Delict) - Strip Method o{(U5) 0.00 2 3 i 5 a351p VL 0.04 0.02 .. . . . . . . . ... . . . .. -0 02 ~ - ~ O _ - -0.06 ~ '' ...... - ~0.10 2 3 4 5w[(U' ) v.vv 2 3 4 5~(Ug) a53/pVL b /pV~L 0.1C n53n- 0.0E 0.06 0.04 0.0z O.OC .0~ ~ no b~3/p VEIN 1 .00 0.80 0.60 0.40 _ 6 . 0.20 . ~ --- 0.00 2 3 4 ~ w{(U 1) b351 p Vm L 0.10 0.08 - Strip Method 0.06 0.04 0.02 rid on , . . O Cal. ....... · Exp.(Delit) Strip Method 3 u- ~. · Exp.(Delit) · ".,, ,2 . ~ .. _ hA I , , , , 0.00 _ -0.02 -0.04 -0.06 -0.08 ~ 1^ _ ~ ~ .. ~ 1 o cad. 1 ............... · Exp.(Delit) . ; Strip Method _ ~ 2 3 4 5~1~(U~ it u 2 3 4 5w[(U~ I) asgp VL 0.04 'D 0.03 non _ . _ 0.01 _ w[(Ug) b551 p VL2w . 002 'a' 001~ 849 0.00 2 3 4 5 0OO 2 3 4 5m{(Ug Fig.5: Comparison of added mass and damping coef- ficients for a modified Wigley hull (F~=0.3) dimensional frequency of forced oscillation. Agree- ment with experiments is satisfactory as a whole. Es- pecially, calculated results of a33, b33, a35, a53 and b55 show good agreement with experiments. How- ever, small discrepancy is observed in terms of b35, b53 and a55. On the other hand, calculation accuracy by strip method is clearly worse than by the present method. Fig.6 shows comparison of wave exciting forces Ei~i=3:heave force, ~=5:pitch moment). As the flap angle of numerical wave maker, O0=0.4deg were se- lected. Then, amplitude of incident waves ha was about 0.0024-0.0030L. Calculated amplitude and phase angle of heave force and pitch moment agree well with experiments as a whole. In the region where )/L is smaller than 1.0, however, pitch moment is larger predicted. The reason may be due to lack of free-surface panel density to capture the shorter waves. The calculated results by strip method agree well with experiments. 15.C _ ~10.0 . Q 5.0 0.q modified WiqleY . . . . . . . . . . . . . . . . modified WiqleY .L . . . · ' ! ' 1 ~ ,0'~.'~- ..~. L°f- ... ,.... ---.. -] | O Cal. | · Exp.(Delh) - - Strip Method 8.5 i 1;5 A /L 2 Fig.6: Comparison of wave exciting forces for a mod- ified Wigley hull (F~=0.3) 4.2 A container ship with lateral motions Next, the present method is validated for hydrody- namic forces on the lateral motions such as sway, roll and yaw. Calculations are carried out for a container ship with Lpp/B = 6.72 and B/d = 2.74 as shown in Table 1, and are compared with experimental data. To obtain the hydrodynamic force coefficients on the lateral motions, 3 forced oscillation tests such as pure swaying, pure rolling and pure yawing tests were conducted at towing tank, Nagasaki R & D cen- ter, MHIt234. The size of the towing tank is 120m in length, 6.1m in width and 3.65m in water depth. The ship model was 3.0m in length, and has no bilge keels and no rudder. Froude number based on Lpp was 0.20. The amplitudes of forced oscillation in pure swaying, pure rolling and pure yawing test were 0.0448B, 7.5deg and 2deg respectively. The forced motions were given to center of gravity of the ship model. The height of center of gravity was adjusted so as to coincide with still water level. Fig.7 shows panel arrangement of the hull. 990 panels for ship hull, 8,000~= 200 x 40) panels for free- surface and 2,880 panels for tank walls were used in the computations. The size of the numerical towing tank (free-surface panel region) is 8Lpp in length and 2.03Lpp in width. The width and the water depth was coincided with those of the actual towing tank. Table 1: Principal dimensions of a container ship Ship length Lpp Breadth B draft d Displacement V _ Model 3.000m 0.446m l 0.163m _ 121.77kg _ Ship l 175.0m 26.0m 9.5m 24,776ton

Fig.7: Panel arrangement of a container ship. This figure includes hull panels above the still water sur- face. 1 As an example, wave patterns generated by the ship with pure swaying motion are shown in Fig.8. Non-dimensional frequency is A' = 0.683 where A' _ w~,/B/~2g), and ~ = 0.501. Solid line means posi- tive and dotted line the negative in wave elevations. We see that the wave component generated by forced sway motion laterally moves and reflects at the tank- side. The unsymmetrical waves periodically appear and are superimposed over the steady wave compo- nent. Figs.9-11 show comparison of added mass and damping coefficients on the lateral motions. The aij Fig.8: Computed wave patterns around a container ship with pure swaying motion (Fn=0.2, T=0.501' ~'=0.683~. The phase angles against the swaying mo- tion are 0, 45, 90 and 135deg from the top. and bij mean added mass and damping with respect to i-th mode induced by j-th motion, where i/j=2 means sway, i/j=4 the roll and i/j=6 the yaw. They are non-dimensionalized as follows: a22 = a22/(pV), a42 = a42/(pVB), a62 = a62/(pVB), a24 = a24/(pVB), a44 = a44/(pVB2), a64 = a64/(pVB2), a26 = a26/(pVLpp), a46 = a46/(pVL2p), a66 = a66/(pVL2p), b22 = b22 ~7~/(pV) b42 = b42~/(pVB) b62 = b62~7~/(pvB) b24 = b24~7~/(PVB) b44 = b44~7~/(pVB2) b64 = b64~7~/(pVB2) b26 = b26~7~/(pvLpp) b46 = b46~7~/(pvL2p) b66 = b66~7~/(PVL2p) Diagonal terms for the pure sway mode such as a22 and b22 are in good agreement with experiments, although some discrepancy is observed in coupling terms b42 and b62. Added mass coefficients with re- spect to pure yawing motion are also in good agree- ment with experiments. However the damping coef- ficients in yawing motion are uniformly smaller than those in experiments. This may be due to viscous flow effect on the hydrodynamic forces acting on the hull. The present method treats the ship hull as non-lifting body. To improve the accuracy of yaw damping, we should include the lifting body effect due to change of attack angle of the hull against the flow. Bertram et al. calculated the ship motions including the lift- ing body effect in frequency domain Rankine panel method t244. In the pure rolling mode, calculation accuracy becomes worse than that in other modes as over all tendencies. Diagonal terms such as a44 and b44 are not in good agreement with experiments. This is also due to viscous flow effect on the hydrody- namic forces. To accurately simulate the lateral ship motions, we need some empirical correction for b44 as usually employed in strip method. The results by strip method do not agree with ex- periments except a22. In some coupling terms, ten- dency versus forced frequency is quite different from the experiments. The calculated accuracy by strip method is clearly worse than by the present method. 5. Ship Motions in Heacl Waves for S- 175 Container Ship Next, the present method is validated for ship mo- tions in regular head waves. Calculations are carried out for S-175 container ship as shown in Table 2, and are compared with experimental data. The experi- ments were conducted at seakeeping & maneuvering basin, Nagasaki R & D center, MHI. The size of the tank is l90m in length, 30m in width and 3.5m in

a'22 2.0 . , i l ' 1.0 ~ ~ · · · Exp. . ~ Stnp Method on _ I . . . I . i ~0 0.2 0.4 0.6 0.8 ~,~~ 1 biz 1.0 _ . , . , . , . , . O Cal. · Exp. ~ ~; ,. - - - Strip Method ~ .~ . . . · . .. ' . , · .~..- :"""', . . . _.~)( ) 0.2 0.4 0.6 0~8 w' 0.5 r. n O. 0.c ~ 1 0.1 0.0 ~ ~ ~ ·' -01 . i . , . . . , . , _~' ~. 0 0.2 0.4 0.6 0.8 ~,.~ 1 0 0.2 0.4 0.6 0.8 ~,~ a,42 a'62 0 10 _ O 0 2 0 0.00 _ ~ ~ ~ ~ -0 2 _ . , ; --,--- , --- -0.4 .05 `0 6 0 0.2 0.4 0.6 0.8 c`' 1 0 0.2 0.4 0.6 0.8 `~, 1 bl62 1.0 . , I ' ' . O O , O.5 . O u.u~ O.O , j . i 0 0.2 0.4 0.6 0.8 `,' 1 0 0.2 0.4 0.6 0.8 `o 1 U.U~ 0.0d o.or ~.o, Fig.9: Added mass and damping coefficients in pure swaying mode (Fn=0.2' 62a/B=0.0448) a'26 a'46 a'66 0.2 1 1 ,- , 0.000 · 1 1 1 1 0.20 ,, . 1 ~ ' ' ': . · ~ ~ ~ -0.001 (~~ '; 0. 15 .. - O : -0.002 _ .. 0.10 ~ :---O ~ ) ~ O Cal. I . -0.003 . ................. ' 0.05 ~ ~ e ~. · - StriP Method | - . : _ , -0 004 . 1 . 1 . 1 . 1 . o on . 1 . 1 . 1 1 O 0.2 0.4 0.6 0.8 w~ 1 0 0.2 0.4 0.6 0.8 ~~~ 1 0 0.2 0.4 0.6 0.8 ~~~ 1 b,26 bl46 0.2 . 1 1 1 1 ~ 0.001 · ·3 ~ . - - - - ~ - 1 ° °°°t 0 ... ~ -0. ''''''-0-----°- -"'. l t bl66 1 010 1 0.08p | O Cal. | 0.06~-| - - Stnp Method 1 0.o4t 1 o~o2t I o.oot 1 -0.020 0.2 0.4 0.6 0.8 `~' 1 Fig.10: Added mass and damping coefficients in pure yawing mode (Fn=0.2, ~a=2de=1 at24 0.1 0 E ° 0.05p - - -- - --- - ---- ----- - --- --- - -- --- - ~ ·-. O 0.00F · ~ O O ~ .; ; '1'~''''''''''','~""' _ -O. 0R 0 0.2 0.4 0.6 0.8 ~,~~ 1 bl24 0.03 0.02p 0.01 g --- -- --- -- -- --- - i--- ·-- o.oo~ -._~ !.~ .o 0 01 t -002 0 0.2 0.4 0.6 0.8 `~' at44 ] 0.041 3 0.02~--- 3 o.ooF...... 3 o 02h 3 -o.o4k 1 ~ ~~~21 ......... . o I 11 | · Exp. 14 | - - - Strip Method | l - -u.ucO 0.2 0.4 0.6 0.8 <~' 1 bl44 0.006L 0.004L ] 0.002t I ^ n~rl al64 °-°t . I . I . I . I . ~ 0.1t - - C, ~.2t .-. ~ . 1 . 1 . 1 . 1 . ~ 0 0.2 0.4 0.6 0.8 `~, 1 bt64 0.O5L I I ! I ~ ao ~ o.ooF -~ .;;;..;.~; e. ............1 ~ _0.05: O cat .. ·-3] t .. i - u uou~ ~ ~~ - - - ~ -010i - --- ~ --- - - -'- i ~ ~ 1 0 0.2 0.4 0.6 0.8 `~, 1 0 0.2 0.4 0.6 0.8 <~ 1 Fig.11: Added mass and damping coefficients in pure rolling mode (Fn=0.2, ~a=7.5deg)

Table 2: Principal dimensions of S-175 container ship Ship length Lpp Breadth B draft d Displacement V Model 3.500m 0.508m O.l90m . 193.57kg Ship 75.0m 25.4m 9.5m 24,801ton Table 3: Region of numerical towing tank and the number of free-surface panels in the computations Froude number Range of A/Lpp Tank Length Tank Width Number of Panels 0.15 0.7-1.3 1.5-2.0 8Lpp 8Lpp 2Lpp 3lpp 5,000 5,000 Fig.12: Panel arrangement of S-175 container ship. This figure includes hull panels above the still water surface. water depth. The ship model used in the test was 3.5m in length and the radius of gyration key was 0.25Lpp. The tests were carried out at 2 Froude numbers Fn = 0.15 and 0.25. There is no restoring term with respect to surge. For certainly measur- ing the surge motion in the tank test, we artificially added the restoring force to the ship model using coil springs. In the calculations, we put the restoring force to the motion equation so as to coincide with the tank test condition. Fig.12 shows panel arrangement of the hull used in the computations. The number of the panels is about 800 for a half body. Table 3 shows the region of numerical towing tank and number of the panels used in the computations. The length of the numerical tank is 8Lpp for Fn = 0.15 and lOLpp for Fn = 0.25. 5,000~= 250 x 20) panels were used for free-surface. This number may be insufficient for capturing shorter waves less than A/L = 0.5. For relatively long waves, we confirmed no problem in view of practical uset264. Flap angle of the wave maker was selected as OO=l.Odeg so as to coincide with the amplitude of incident waves in the tank test. The calculated am- plitude was about 2-3m in fullscale. No numerical wave absorbing beach was used at the tank-side and we allowed the wave reflection in the computations. We checked the tank-side effect on the ship motion in waves by referring to Kashiwagi et al.~254. Based on their study, we made a diagram showing the minimum wave length for disappearing the tank-side effect versus tank width (see Appendix). From the diagram, we selected the tank width disap- peering the tank-side effect. lOLpp 2lpp 5,000 S-175 Container Ship ~ n _ I I I I L I ~= ~1 ~ r- `0 20 30 40 50 60 t {(g/L' a ~P .005 Fn=0.15 A /L=1.0 3 -0.011 ~ ~ . , , 1 20 30 40 50 60 70 80 t {(g/L) O,Oc - _ C ~D _O.Oc 20 30 40 50 60 70 80 t {(g/L) l 20 30 40 50 60 70 80 t {(g/L) Fig.13: Computed time histories of ship speed (U/UO), surge ((l/l), heave (63/L), pitch (~) and wave elevation (~/L) in regular head waves

Fig.13 shows the calculated time histories of ship speed, ship motions and wave elevation at Fn = 0.15 and A/L = 1.0. The ship starts at non-dimensional time T = 25 and reaches to given speed at around T = 40. The ship motions gradually grow and at around T = 50 reach to steady state. We see 2 com- ponents of oscillation in (~/L: one is long period os- cillation due to artificial restoring force term and the other the surge motion oscillating with encounter fre- quency. Such the behavior can be often seen in the actual tank test results. Regular periodic oscillations are obtained in 63/L and 8. Some fluctuation is ob- served in amplitude of incident waves (~/L), however this is negligible in view of practical purpose. Fig.14 shows the computed wave patterns around the ship in regular head waves. The wave component generated by ship motion appears at the ship fore part, and moves rearward and reflects at the tank- side. Incident waves and steady waves components are superimposed to the radiated waves. At present, there is no validation data for wave computations, but those look plausible. Fig.15 shows snap shots of the ship attitude change and wave profile at Fn = 0.15 and A/L = 1.0. Deck wetness appears in the computations, however, Frl 0.15' A/L Lo ~~ ~ hydrodynamic effect due to the deck wetness is ex- cluded. No trouble occurred in such a large ship motion case and it was confirmed that the present method is robust. By analyzing the time histories of the ship motions, we obtained the amplitude and phase against the in- cident wave. Figs.16 and 17 show the comparison of amplitudes and phases of surge, heave and pitch at Fn = 0.15 and 0.25. The amplitudes of surge and heave, and pitch are non-dimensionalized by the am- plitude of incident wave ha and the wave slope hat respectively. The calculated results by strip method are also plotted. In case of Fn = 0.15, the calculated accuracy is satisfactory for surge. However, ampli- tudes of heave and pitch are overestimated and are rather close to the results by strip method. In case of Fn = 0.25, calculations of amplitude and phase agree well with experiments. The calculated accuracy by strip method is not satisfactory for surge and heave. Particularly, the heave amplitude by strip method is too large in the range from 1.0 to 1.5 of A/Lpp. Fr'0.15' A/L - 1.0 Fig.14: Computed wave patterns around S-175 con- tainer ship advancing in head waves. The phase an- gles against incident wave are -9Odeg, -45deg, 0 and 45deg from the top. Fig.15: Snap shots of ship attitude change and wave profile in regular waves (one cycle from-9Odeg to - 90deg in phase angle)

,-175 Container Ship o8t. 0.E - 0.4 ~ SURGE Fn-0.15, X =1 80deg 0 Cal. . · Exp. ~ Strip Method 0.2 0. .5 1 ~ . 18C O O ~ ,~ a--~--0--~-0~ ) ~ <5 .~4, 0 .~................................................................................................. ~ ·e -18 .5 i 1 5 A/L . i .... 1 ~ 2/1 2 S-175 Container Ship S-175 Container Shi . . . . , . . . . , .P. . . HEAVE Fn=0.15, x=180deg 1.5 . 0 Cal. · Exp. ....... Strip Method _ 1.0 0.5 ~ -.,,,e,.'', . . °8 5 1 1.5 A/L . 180 : 44~ o , ·,· ~ ~ ~ 5-~-~ ~ ( 188 5 1 1.5 A/L / 1 .5 PITCH 1 .0 ............ s - _ 05 -- - io Fn=O.15, x=180deg ~ . .. 0 Cal. G. · Exp. ~ ........ Strip Method O [. ~ . . . , . , . . . . . .5 1 1 .5 A /L 2 360 a' =180 : 000_,,_,,0,,,,,,,_____ ___. O ..- ~ .- -- ---- -- ............ O"e li ,,,,3.~ ~ IJ 05 1 - 7/L Fig.16: Comparison of amplitudes and phases of surge, heave and pitch in regular head waves for S-175 container ship (Fn = 0.15) S-175 Container Ship ~ ....,....,.-... SURGE - 0.8 - - too. ~ 0.t O.` Fn=0.25, x=180deg . 0 Cal. · Exp. Strip Method °8 ; .5 1 1 .5 A /L 2 180 ~ O O Q ~ ~ ......... ,~,, O .................................................................................................... .' : ~ : ' : -1 8Q u.5 1 1.5 A /L 2 S-175 Container Shin _ S-175 Container Shin 1 .£ _ 1.t co ~P . .. _ 0.5 C' 0 Cal. . · Exp. ~ · ~ Strip Method on' ~---..~.., ....i.... _ v.5 1 1.5 A /L 2 180 . . . . . . . . . O' ......................................................................................... =, ; . ~ c~) r~ !..~ 44e U _ . .e,,,,,~,., - ' ~ 1 8n v.5 1 1.5 A /L 2 v.5 . ~ · '----- . .~ O - .i; c Fn=0 25 x=180dea , .~ 1 .C - y - ~D O.c fe ........ ~ .......... . . ,,.e O - -- --- ---r'. -° -- F'1=0~25, x~ =1 80deg ~ Cal. ~i · Exp. 1 ~ ~ ~ I Strip Method|4 V.vx.5 1 1.5 A /L 2 360, - . . . . . a' m18C t n~ ;~ e-~--~---~------ -------1 1 1.5 A/L 2 Fig.17: Comparison of amplitudes and phases of surge, heave and pitch in regular head waves for S-175 container ship (Fn = 0.25)

6. Wave Pressures of a Large Container Ship Next, the present method is validated for wave pres- sure in regular head waves. Calculations were carried out for a 4,900TEU large container ship as shown in Table 4, and were compared with experimental data. The experiments were also conducted at seakeeping & maneuvering basin, Nagasaki R & D center, MHI. The pressure fluctuation was measured at 2 hull sec- tions of square station (S.S.) 7 1/4 and 5 1/2 using load-cell typed pickup. The pickup was installed to the model on 4, 7, 10 and 13m water lines in fullscale. Here, pi., P2, pa and p4 denote the pressures at 13, 10, 7 and 4m water lines respectively. In the test, ampli- tude of the incident waves was corresponding to 2m in fullscale. Radius of gyration of the model key was 0.25Lpp. The detailed test procedure and the ship model used in the test were described in Ref.~274. Table 4: Principal dimensions of a 4,900TEU large container ship Ship length Lpp Breadth B Depth D draft d ship speed U Model 4.200m 0.652m 0.396m 0.204m 1.38m/s . Ship 258.0m 40.0m 24.3m 12.5m _ 21kn In computations, about 700 panels for a half body were used. The length of the numerical tank is l OLpp and free-surface panels were 5,000 (= 250 x 20~. Flap angle of the wave maker was selected so as to coincide with the amplitude of incident waves in the tank test. Fig.18 shows comparison of amplitudes of heave and pitch in regular waves. The calculated ampli- tudes are larger than the experiments as a whole, al- though the present calculation is much closer to the experiment than by strip method. F. =0.215, X =1 80deg n ...... 10 HEAVE . ~ --- ~10 : ,,.4e, . ~ ,. ~ it- '' A..' ~ . . . . , . . . . , . . no 0.8 , Fn.0.215' X e1 80deg .. .. I · ' ' · ! PITCH ....... -- - - .~" · ' O ...... .... .. .. . .. . .. ... . . ... .... . ... .... .' ~ 0 Cal. if · Exp. ~ Stnp Method I 5 1 1.5 A IL u.~ i ~ . j . 1.5 A /L Fig.18: Comparison of amplitudes of heave and pitch in regular head waves for a large container ship S.S.7 1/4 Cal. -------- Exp. ~ 1 1 1 1 1, 1 ! 1 mm _ j~ ~ ~ ~ ~ ~ _ Q o ~ ~ ~ -i Id/ -~ ~ . ; ; ~ -2 60 61 62 63 64 65 66 67 68 t ~g/Lj70 _= 60 61 62 63 64 65 66 67 68 t ~ /L)70 , , ~ , , ~ ~ .. ~ , 60 61 62 63 64 65 66 67 68 69 70 t Jr(g/L) 2r , , , , , , , I 1 1 Ct s Q C ~1 2 60 61 62 63 64 65 66 67 68 tin /L)70 Fig.19: Comparison of time histories of hull pressures on a large container ship (Fn = 0.215, A/Lpp = 1.1) Fig.19 shows time histories of hull pressures at S.S.7 1/4. Here, the pressures are set to be zero when ship model is just floating without forward speed in tank test. The calculated pressures are adjusted to coincide with the test condition. Time history of Pi has flat bottoms which occur due to exposure of the pickup in the air. The P2 behaves the similar to Pi. The present method can capture the nonlinear be- havior due to the pickup exposure. Almost the sinu- soidal history is obtained for pa. The history of p4 has round for the top and sharp for the bottom, and the present method captures well such the behavior. The calculated results are about 20% larger than the experiments in absolute value of pressure peaks. This copes with the situation that calculated ship motion is larger than the experiment. Fig.20 shows the comparison of pressure amplitude distribution. In the figure, lip defines the hull surface location of the given section: lip = 0 means the ship bottom and lip = +9Odeg the hull sides of still wa- ter line. The amplitudes were obtained by taking

rat 90 - 60 2.OI 1.5 Q 1.0 n. . o.o _ _ 1 1 ,, .,.,, 0 ECap. 1 .... -- Stri ~ Method | _ ,, ,,, ,,,.,,,,, .,, ._. _ ma. - ...., ,=_ ~ _ ) 0 0 0 0 0 Ol)Ooc~oo, -30 0 30 60 90 ~ p(deg) S.S.7 1/2 l/L=O.9 . , ~ . 0, , . o' ~ ~LCL.~,''O~ C?- - - ~ or . , +., . ,,, ~ ,. . ', 90 -60 , ,, ,, . . . _ . . _, , -30 0 30 60 90 ~ p(deg) ~ .R .s 1/d ~ It =n ~ S.S.5 1/4 l/L=1.1 S.S.5 1/4 AIL=1.3 I, 1 1 2.0 1.5 Q. 1.0 0.5 0.0 2.( -a Q 1.t 0.! O.` 5 ~ _ I I 1.5 . .; -- . . . | Strip Method | Q 1.0 ...... 5 _ Oo°c oco°° ; 0 0 ]° O 0 0 ~~ ~~ n r. -90 -60 -30 0 30 60 90 6 p(deg) S.S.7 1/2 A/L=1.1 1 _ , . i _ , ~ L b ~ ~ ~ l - --I- C) i ., ...,..,.,,.~.,, .,,, .,, +,......... ~0 - ·.O..C., ° , ... , , ., . .,. ... , ... .,,, ... .,, ,, ;, . o .. . .. .. ... . .... ... C O _ . ' ., -90 -60 -30 0 30 60 90 ~ p(deg) _ _ _ ' ' ' "'amp- Cr CJ 0~15 ~ ~~ - oO D -90 So -30 0 30 60 90 ~ p(deg) S.S.7 1/2 A/L=1.3 2.0 . 1.5 ._ Q 1.0 0.5 0.0 _< 10 1 ! 4~ IQO~~ IC-''''--'''-'',j-..,_ _o ~ ,- -''t' 1 1 '1-'---'------'--~---- - --------1-,_,,_,,----t' 1 1 1 ' 1- 90 ._,,. _. I' °= o .. _,. _.~2 ' ',.'.' ' ?. .oOoOOOc ~ L o - ~n ~n 0 30 60 (deg) Fig.20: Comparions of hull pressure amplitude distribution for a large container ship (Fn = 0.215) the 1st order solution of Fourier analysis result of the time histories. In the case of appearance of nonlin- ear behavior as shown in the top of Fig.l9, the same procedure was adopted for analysis. We also plot- ted the results by strip method with correction for considering the nonlinear behavior of pressure~285. In the calculated distribution at S.S.5 1/4, notice- able hump appears near Bp = +75deg. At A/L = 1.3 the calculated results agree well with experiments be- cause the hump is clearly observed in the experiment. At A/L = 0.9 and 1.1, however, such the hump is not so remarkable in the experiments. Comparing with the results by strip method, the present method im- proves the prediction accuracy. At S.S.7 1/2, the calculated pressure amplitude turns down at the sides of the hull due to the nonlin- ear effect as shown in Fig.l9. The calculated ampli- tudes are about 10-20~o larger than the experiments. As mentioned above, this comes from the larger ship motions in the calculations. Strip method is overpre- dicted as same as the present method. In summary, the present method is better predicted in amplitude of the pressure fluctuation at midship section than strip method, and can capture the non- linear behavior of the pressure appearing near the ship hull sides at fore part. 7. Concluding Remarks by means of the time domain code. All the results except the wave patterns were compared with exper- iments and the calculated results by strip method. As a result, we found that hydrodynamic forces, ship mo- tions and wave pressures are much better predicted than by strip method. The present method is promis- ing as a new ship design tool for more accurate pre- diction of seakeeping performances. However, the calculated accuracy of hydrodynamic forces on lateral motions is not satisfactory. This is due to viscous flow effect and we need improvement of the method. For this purpose, simple empirical correction may be useful as commonly employed in strip method. As the next step of the work, we will extend the present method to large amplitude ship motion prob- lem such as slamming. Acknowledgement We are grateful for the support of younger colleagues Mr. S. Mizokami and 'Cowboy' Tanaka in prepara- tion of the paper. References ill Bertram, V. and Yasukawa, H., " Rankine Source Methods for Seakeeping Problems", Jahrbuch der Schiffbautechnischen Gesellschaft, In this paper, a 3-D time domain panel method Springer, 1996, pp.411-425. for seakeeping has been presented. Hydrodynamic forces, ship motions in head waves end wave pressure t24 Bertram, V., "Numerical Investigation of on the hull surface were computed for several ships Steady Flow Effects in 3-d Seakeeping Compu-

tations",22nd Symp. on Naval Hydrodynamics, 1998, Washington D.C. t3] Bertram, V. and Yasukawa, H., "Investigation of Global and Local Flow Details by a Fully Three- dimensional Seakeeping Method", 23rd Symp. on Naval Hydrodynamics, 2000, Val de Reuil. t44 Iwashita, H.:, "Influence of the Steady Flow in Seakeeping of a Blunt Ship through the Free Surface Condition", 13th International Work- shop on Water Waves and Floating Bodies, 1998, Netherlands, pp.89-92. t54 Iwashita, H. and Ito, A., "Seakeeping Compu- tations of a Blunt Ship Capturing the Influence of the Steady Flow", Ship Technology Research, 1998, Vol.45, No.4, pp.159-171. t64 Maskew, B., "A Nonlinear Numerical Method for Transient Wave/Hull Problems on Arbi- trary Vessel", Trans. of SNA ME, Vol.99, 1991, pp.299-318. (13] Lin, W. M., Shin, Y. S., Chung, J. S., Zhang, S. and Salvesen, N., "Nonlinear Predictions of Ship Motions and Wave Loads for Structural Analy- sis", 16th International Conference on OMAE, 1997, Yokohama. t144 Yasukawa, H., "Unsteady Wash Computation for a High Speed Vessel", Numerical Towing Tank Symp.(NuTTS'99J, 1999, Rome. t15] Yasukawa, H., " Unsteady Wash Generated by a High Speed Vessel", 16th International Work- shop on Water Waves and Floating Bodies, 2001, Hiroshima. t164 Salvesen, N., Tuck, E. O. and Faltinsen O., "Ship Motion and Sea Loads", Trans. of SNA ME, Vol.78, 1970, pp250-287. t174 Shinkai, A. and Iseki, T., "Boundary Element Analysis of Non-Linear Water Wave Problems", Trans. of the West-Japan Society of Naval Ar- chitects, No.70, 1985, pp.67-76 (in Japanese). t74 Maskew, B., "Prediction of Nonlinear (184 Cointe, R., Geyer, P., King, B., Molin, B. and Wave/Hull Interactions of Complex Ves- Tramoni, M., "Nonlinear and Linear Motions sel", 19th Symp. on Naval Hydrodynamics, of a Rectangular Barge in Perfect Fluid", 18th 1992, Seoul. Symp. on Naval Hydrodynamics, 1990, Ann Ar- t84 Maskew, B., Tidd, D. M. and Frase, J. S., "Pre- bor. diction of Nonlinear Hydrodynamic Characteris- tics of Complex Vessels using a Numerical Time- Domain Approach", 6th International Confer- ence on Numer~cal Ship Hydrodynamics, 1993, Iowa City, pp.591-609. t94 Beck, R. F., Cao, Y., and Lee, T.-H., "Fully Nonlinear Water Wave Computations using the Desingularized Method", 6th Int. Conf. on Nu- merical Ship Hydrodynamics, 1993, Iowa City, pp.3-20. t10] Beck, R. F., Cao, Y., Scorpio, S. M. and Schultz, W. W., "Nonlinear Ship Motion Com- putations using the Desingularized Method", 20th Symp. on Naval Hydrodynamics, 1994, Santa Barbara. t114 Scorpio, S. M., Beck, R. F. and Korsmeyer, F. T., "Nonlinear Water Wave Computations using a Multipole Accelerated Desingularized Method", 21st Symp. on Naval Hydrodynamics, 1996, Tronheim. t12] Lin, W. M., Meinhold, M. J., Salvesen, N. and Yue, D. K. P., "Large-Amplitude Motions and Waves Loads for Ship Design", 20th Symp. on Naval Hydrodynamics, 1994, Santa Barbara. t194 Tanizawa, K., "A Nonlinear Simulation method of 3-D Body Motions in Waves", J. of the Soci- ety of Naval Architects of Japan, Vol.178, 1995, pp.179-191. t204 Nakatake, K. and Ando, J., "Rankine Source Method using Rectangular Panels on Water Sur- face", 11th International Workshop on Water Waves and Floating Bodies, 1996, Hamburg. t214 Yasukawa, H., "Time Domain Analysis of Ship Motions in Waves using BEM (1st Report: Computation of Hydrodynamic Forces)", Trans. Of the West-Japan Society of Naval Architects, No.100, 2000, pp.83-98 (in Japanese). t22] Journee, J. M. J., "Experiments and Calcula- tions on Four Wigley Hull Forms", Report No. 909, Ship Hydrodynamic Laboratory, Delft Uni- versity of Technology, 1992, Netherlands. t23] Fujii, H. and Takahashi, T., "Study on Lateral Motions of a Ship in Waves by Forced Oscilla- tion Test", Mitsubishi Technical Bulletin, No.87, 1973, MHI. t244 Bertram, V. and Thiart, G., "A Kutta Condi- tion for Ship Seakeeping Computations with a

Rankine Panel Method", Ship Technology Re- Appendix: Diagram for Tank-Side Ef- search Vol.45, 1998, pp.54-63. feet on Ship Motions in Waves t254 Kashiwagi, M., Ohkusu, M. and Inada, M., Kashiwagiet al. studded the tank-side effect on the "Side-Wall Effects on Radiation and Diffraction ship motions in waves by applying the unified the- Forces on a Ship Advancing in Waves", J. of the Society of Naval Architects of Japan, Vol.168, 1990, pp.227-242 (in Japanese). t26: Yasukawa, H., "Time Domain Analysis of Ship Motions in Waves using BEM (2nd Report: Mo- tions in regular head waves)", Trans. of the West-Japan Society of Naval Architects, No.101, 2001, pp.27-36 (in Japanese). t27] Mizokami, S., Yasukawa, H., Kurolwa, T. Sueoka, H., Nishimura, S. and Miyazaki, S., " Wave Load on a Container Ship in Rough Seas", J. of the Society of Naval Architects of Japan, Vol. 189, 2001, pp. 181-192 (in Japanese). t28] Toki, N., Fukushima, Y., Tozawa, S. and Wada, Y., "On the Characteristics and Long-term Pre- diction Procedure of Wave-induced Pressure Fluctuation on a VLCC Hull", J. of the Soci- ety of Naval Architects of Japan, Vol.176, 1994, pp.375-385 (in Japanese). Ory, and indicated the range of wave number dis- appearing the tank-side effectt254. Based on their achievement, we made a diagram showing the min- imum wave length for disappearing the tank-side ef- fect versus tank width as shown in Fig.21. We de- cided the minimum width of the numerical tank from the diagram. , Fn=0.2,5,/ ,, ----'.''' - " ~ Fn=0.20 1 O. . i . I 0 1 2 ., ., ., 1 3 4 5 Fig.21: Minimum wave length for disappearing the tank-side effect versus tank width

DISCUSSION V. Bertram ENSIETA, France I enjoyed the paper very much. I congratulate you to the significant progress achieved over the last years. For clarification: Eqs. (1) and (2) are formulated at z-0. In a time-domain method, we should be able to formulate them at the actual free surface position. The good agreement of the computations seem to indicate that for the investigated cases, this simplification is justified. Nevertheless, is this simplification in principle necessary? Do you see major problems in extending the approach to include a non- linearized free-surface condition? The new test case of a large container ship described in chapter 6 is a very useful addition to our suite of ship seakeeping test cases. Can I encourage you to follow the good tradition of MHI and have an English report on the experiments with sufficient data given for the world-wide community to use it as a validation case? AUTHOR'S REPLY Our main purpose of the study is to apply 3D time domain panel method to actual ship design. In view of this, we gave precedence to minimizing the computational time and keeping the robustness of the code rather than the full- nonlinear computations. The employment of the 2n~ order free-surface conditions on z=0 is useful for dramatically reducing the computational time because of no re-paneling the free-surface and no re-constructing the base matrix with respect to large number of free-surface panels. The calculated accuracy is satisfactory for practical use as mentioned in the paper. Now we are planning another tank test to capture nonlinear behavior of wave loads and wave pressures for other modern container ship. This test includes the cases of oblique waves and irregular waves. These data may be useful as validation data for various seakeeping prediction methods. We will open the measured results together with hull form data in near future.