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Prediction of Radiation Forces on a Catamaran at High Froude Number M. Ohkusu (Kyushu University, Japan) O.M. Faltinsen (Norwegian Institute of Technology, Norway) ABSTRACT Practical approach is investigated to predict three dimensional hydrodynamic interaction between two hulls of a catamaran oscillating and running at forward speed. Chapman's approach is tried to solve the boundary value problem for the un- steady velocity potential around twin hulls' sec- tion contour with retaining all the terms of the full linear free surface condition including for- ward speed effect. Results show that predicted hydrodynamic forces agree generally well at high Froude number with the ~eas~red on a anode 1 situp and it-~leraction between twill hulls is cer- tainly weak at Ellis high speed. INTRODUCTION It may be a practical approach to generalize strip theories to predict motions of a catamaran ad- vancing in waves. Strip theories are certainly the most successful theories for practical purpose of predicting wave induced motions of ships at mod- erate forward speed. For applying strip theories two hulls of the catamaran are considered to be close to each other. Two dimensional theory is used to eval- uate flow around the contour of two sections os- cillating on the free surface. Experiments tell, however, that hydrodynamic interaction eRects between two hulls are not so strong as expected from ship's lengthwise summation of the 21) ef- fects (Ohkusu(1971~. This is more so for higher forward speed. If we assume two hulls are located away frown each other, theories to be applied must not be strip theories any more. We need a prac- tical theory to take into account correctly three dimensionality of hydrodynamic interaction be- tween two hulls of the catamaran. Strip theories account for the effect of for- war<1 speed in a simple way. The linear free sur- face condition including forward speed term is simplified such that the unsteady waves gener- ated by flee body motions propagate only in the breadthwise directions of the ship. A more com- plicated wave system must be considered if the three di~nensionality of the flow has to be consid- ered. 'three di~nensinal panel method at forward speed will be one alternative for taxis purpose. But even will, recent progress in co~nputatio~al scenes of the Green function (Ol~kusu and Iwashita(1989~), it can not be so practial as strip theories because of long computational tine. Be- sides we have no valid approach to treat with a line singularity appearing at the intersection o the surface piercing body and the free surface. Chapman's approach (Chapn~an(1976~) is a simplified, but still satisfying the full linear free surface condition, high speed theory for a verti- cal surface-piercing flat plate in unsteady yaw and sway motion. Daoud's theory (Daoud~ 1975~) for analyzing stationary flow around the bow of the ship is also a theory retaining all the terms of the linear free surface condition in the region close to the body and solving essentially 2D problem around section's contour. Ogilvie's description (Ogilvie (1977)) on these theories is full of in- sight into hydrodynamics and very informative. Furthermore one of the present author (Faltin- sentl983~) presented a theory for solving diffrac- tion problem at the bow of the ship along a sign tar line. All those are attractive ideas in the point that the -derivatives ifs tile free surface condition is completely retained and still i~urnerical work is almost on two dimensional problem. Recent comprehensive analysis by Yeug and Kim (1985) Nlak`'tc~ ()hkusu. Research Institute fair Plied \iecha~i~ i;' K~ ushu l'nix-.. Fuku`'ka Japan ()cl~-1 N1. Faltinsen. Nc:)rs~-egian Illstit~te off Technology-. Tr ~n~ll~eil:l~ .\;or~. s

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gives more analytical foundation for Chapn~an's where approach. We present in the main text an application of Chaprnan-Oaoud-Faltinsen's approach for com puting hydrodynamic forces on the catamaran. First we invesigate the far-field effect of unsteady bow flow caused by heave and pitch motions of the ship. There we describe a set of assumptions leading to the full linear free surface condition retaining the ~ derivatives in the near-field. Nu merical solutions of the 21) boundary problems for the velocity potentials near the body is a ver sion of L)aoud's approach. Finally we compare computed and measured added mass and da~np ing of a catamaran at high Froude number and present wave elevation generated by the ship mm tion between two hulls of a catamaran as well as in the far-field. FAR-FIELD SOLUTION Velocity potential describing the flow around an oscillating ship at forward speed is expressed in the form of ~ = US + +5(~ y, a) ~ Army y, Z)ei~t (1) where the -axis directs astern from the origin at FP, the z axis vertically upward and the By plane coincides with the calm water surface (Fig.1~. U is steady ambient flow velocity in the direction of the x-axis (or the ship's speed), L the ship length, circular frequency of motion. Is is the steady part of the potential and Or the unsteady part. We start with a simple expression of the sot lution of Shr valid in the far-field of the ship's hull. The solution Or satisfying the full linear free sur face condition (iw + U `' ) Or + g `' = 0 on z = 0 (2) is expressed at My = 0(1) by the velocity pm tential of a line source Vein distributed on the londituidir~al axis of the ship hull since we are concerned with the symmetrical flow field. A Fourier transform type of expressions is given by 1 roe /~-(w + Uk-i,~2/g (3) 6 IL a (k) = ~(x)e-ik~d ~(4) o We assume high frequency of motions ~ = o(~-l/2) just as in Ogilvie and Tuck (1969) and we may proceed along their line. In Ogilvie and l'uck's analysis this integral is assumed to include only the contribution from up to ski = o(1) ( in their notation Ike = o(~~~/2~), because a*(k) drops off rapidly enough with large Ilk ~ . Reason of this is very smooth variation of ~(x) everywhere in x. When we assume flow variables vary steeply over the bow region, for instance, the x-derivative of ~ is of the order of o(~-~/2~) over the region close to the bow, then ~*(k) does not drop off so rapidly. With this assumption we extend the integral intervals at least until Ike = o(~-~/2) is included (In Ogilvie and 'ruck's notation this cor- responds to Ike = 0~-~. Then we obtain from equation (3) rk2 hid* (~/Uk)( 1 + k)~/U) fr = - ~ dk 1_o(~-1/2) >/~1 + k)4 - (k/Ty2 x exp[ik(~/U)x + iL'y)(1 + k)4 - (k/T)2 +I/z(l+k)2] ~k2 2~*~/Uk)(1 + k)2(~/U) - I dk Jk1 (k/T)2 _ (1 + k)4 x exp[ik(~/U)x-~y>/(k/~)2-(1 + k)4 + vz(1 + k)2] {O(E-1/2) k-2icr. (~/Uk)(l + k)2(~/U) . w v ~ Jk1 >/(1 + k)4 - (k/~)2 x exp[ik(w/U)x-i/+ k)4 - (k/T)2 + zJz(1 + k)2] + 0~) where (5) k1,2 = - 1 - (1/2T ~ \/1/r + 1/4r2) (6) ~ = WU/9' L, = (.,2/9 The method of stationary phase (Erdelyi (1956)) can be applied to evaluating the integrals because we are many wave lengths away from the sources at w/U = o(~~l/2) and ~ = o(~-l). Let

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assume (~-() >> y in this equation, then one stationary point exists within the interval of each integral, as far as the 1st and 3rd terms of equa- tion (5) are concerned. For large T stationary points Act for the 3rd and K2 for the 1st integral are approximately ~c~,:~ - 11~ ~ (7) Diary part of the righthand side of equa- tion (5) is then evaluated as follows. I fir ~ - 2 J a(t,)d: o r.~ .h) i I(o(~ _ ()2 x At - exp [-~-(a-A) 1 ko(~-(,)2Z 4 y2 ~ - ~ 4 ] . ~ , W ~ i(~(X _ ()2 _~ exp[-au(~-()+ 4 y 1 /(o(~ _ ()2Z I] - (r + 1/4)1/4 ~eXP[ik1 U (x-()] /2~lk2~(WiUj 1 exp[7k2u(~)]} (8) where To = g/U2. The last two terms of this equation come from the singularities at end points k = kit and k2 of the interval (Erdelyi (1957~) and are higher order than the first two. z de- pende~ce in the last two terms is ignored because exp(~z/U) is 1 ~ o(~il2) If x is away from the bow region such as ~ = C)(l) and ~(~) varies smoothly there, we can once again apply the stationary phase method to evaluate tile parts of the integrals (8) from = (/2) to a. We obtain ~o(~/2)
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pressure of the second order on sections behind the bow region. FORMULATION OF THE SOLUTION Our general assumptions are: Al = 0(~), tI2,3 = 0~1), ~ = 0~-2) (10) ~3 = 0(`f~ ~ ~ Or = 0(~) ( 11 ) where n~,2,3 are the x,y and z components of a unit normal vector to the wetted part of the hull surface whose positive direction is into the fluid. is the slenderness ratio of the hull geometry and ~ characterize smallness of the oscillatory motions' amplitude. f is any flow variable caused by the body in some region near the hull. 'the unsteady part of flow jr is rather straightfowardly linerized based on ~ that is gen- erally independent of the hull geometry. The steady flow , and its interaction with the un- steady part are related closely to the hull geome- try and their linearization is strongly dependent Ott flow characteristics we assume. We have two alternatives in order to retain U0/~ term in the free surface condition: (i) up = o(~-~12), of/ = 0~1) (ii) U = 0~1), I/ = to-. (i) gives , = o(~3/2) and steady wave ele- vation , = 0~. The result is that we can not transfer the free surface condition to z = 0 be- cause of ~5/0Z = 0~14; the free surface condi- tion for , becomes non-linear. (ii) may be justified with some assumptions on the flow characteristics close to the bow as mentioned in the previous section. Recently Faltinsen and Zhao (1990) employed another al- ternative no = o(~/2) to analyze the same prob- lem. With this assumption, however, we once again end up with a non-linear free surface condi- tion for the steady flow 5. In this context jr ho to satisfy the free surface condition not on z = 0 but on the steady free surface displaced finitely from z = 0. Assume (ii) as well as (10) and (11), then we obtain , = o`~2y,<~ = ote3/2) with which we have no trouble to linearize the free surface con- ditio~ for As. If we retain tile lowest order terries with respect to ~ and it, tiled we reacts to tile following linearized free surface con OCR for page 5
free surface elevations equal to zero at x < 0. Rea- son of taxis assurnptio~ is that no upstrea~n waves are generated far in front of the bow because we are concerned with the case of T >> 1/4. \Ve assume here simply the lowest order terms retained in (16~. Some of results in which we keep ever the raj terns inco- sistently will be presented. Chapman (1975) presented a simplified high- speed theory satisfying the full linear free surface condition in the near field for a surface pierc- ing flat plate in yaw and sway motion problems. We apply a generalized and liniearized version of his idea to solve the boundary problems (12), (13), (14), (15) and (16~. In details we follow the approaches proposed by Ogilvie (1977) and Daoud (1975) in analyzing bow flow for the wave resistance problem. This method is economical in the computational effort because orate identical scheme can be applied for both the steady and the unsteady problems. lt isconvenientin the analysts to define new ,r= / ~(iQLr(J:;7l~l:~ Arm y Z) = ei(~/U)2~r(X y Z) <,r = 0 ~ Fir - 0 ate ~ 0 (22) We will now introduce an expression for the solution fir as suggested by Ogilvie (1977~. J L(2)+R(= ) 4~/~; d; J d'Er(~;71~),~) o L(~)+R(~) (18) too x / dw exp~w2(z + (, cosw2(y ~ t,)) o Without any simplification the boundary x sin To W(X-() value problems for both As and &6r then become Flee similar problems with the only difference in the body boundary conditions. For brevity we do not state the boundary value problems for 45 hereafter. 19 tiler + t9 fir = 0 (19) u2 jar + 9 '~ = 0 Off z = 0 (20) Alar ON I ei(~/U)X[iW1~3-U0N ~ ] for j = 3 ei(~/U)x[-in-L/2) U(-n3 + (A-L/2) ON Liz )] for j = 5 (21) on the body surface below z = 0. We set starting condition on fir and 0r/0x at x=0. We Essence the velocity potential and the 9 where (23) r, r' = >/(y _ 77)2 Jr (Z IF )2 L(x) and it(x) are the contours of the cata- ~naran's left and right sections at x ~ we ignore the contribution front L(x) for a single hull ship). We assu~ne<1 ilk tItis fortnulatio~ that each de'~i- hull of tile catamaran locates in the near field of each other. Equation (23) implies that tar at ~ is repre- sented by the potential caused by a distribution Yr of impulsive sources on that section contours am the effect of source distributional at all sec- tions upstream of a. One may use Daoud's ex- pressio~ (1975) for the velocity potential instead of equation (23~. He distributed sources and nor- r~al dipoles based on snore systematic derivation. But we can transform his expression into that of only the source distributions without difficulty by considering another boundary value problem on the flow interior the body. So there is no reason that the expression (23) is not appropriate. Contribution from the line singularities at the intersection of the body and the free surfaces is consi(le~re(1 to be of higher order for slender hull form based on Daoud's arguement (1975~. We re- ~nark that if we had to include really this term, one Galore condition such as the least singularities

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of flow in addition to the body boundary condi- tion should be imposed in order to determine the strength of this line singularities. We do not know what correction is physically correct. The last term of equation (23) is the com- plicated triple integral including the unkown Or in its integrand. But this term gives the effect only from the sections upstream of the section and does not depend on the source density at ~ _ ~ where the integral equation is to be solved, be- cause of the term sin I/ wax-Hi. Starting from x=O, we can solve this equa- tion step by step. Details of numerical computa- tions are described in the following section. When we let y ~ oo on the equation (23), a far field approximation of the near field solution is to be obtained as follows. blur = e i(~/U)r A' ~ +; d(S(~;~)~Ko X expt-it W ~,~ _ ~ To (x-(~2 1 It'o(~-(Liz + 4 y2 +i~ dust; x expt-it w ~y + ~ Ifo~x _ {~2 1 I(o~x-(~2Z + 4 where ~ = lKo(z-() (25) This leads to Sin; ~) = / di Arty; 9' ()e~Z cos Fly-0) R(~) / d[~r(~;77,() for y ~ oo R(~) (26) Comparison of expressions (8) and (24) sug- gests that ~) describing the far field solution is determined from E(x) by the following relation. U ~4 ~ /R(~) ~ ; 77' (I) (27) Dynamic pressure PeiWt linear to the ampli- tude of motion os will be derived by liernoulli's formula as follows. P = -ph exp(-iw/Ux)U 9~ ajar -pt exp(-iw/Ux)t ,~ ,~ + ~9 ~ ] (28) in this equation we did not include the static pressure component induced on the hull surface by the displacement of the body in the non- uniform steady pressure field. This component will be measured independently of the dynamic pressure in our experiments. Once again there are different order terms in the right hand side of equation (28~. The lowest of them is the first line. The second line is ci/2 (24) higher order than the first and should not be in cluded in the computation of forces on the body, while we include this terns in some cases. In our computation an identical scheme is employed to solve the unsteady problem as well as the steady problem and , is always available when Or iS cal culated. inclusion of the second term in the com putation of the forces does not need more com putational effort. Pressure integrated on the body surface gives the forces Fjtj = 3, 5) on the ship. Fj = - | d OCR for page 5
NUMERICAL COMPUTATION 'lihe source density must be determined such that the body boundary condition (21) is satis- fied. We rewrite the second term of equation (23) into the discrete source density form: we divide each section contour of the catamaran into M pieces of segments on each of which the density is assumed constant Er~x;j). After integrating analytically the constant source density ore each segment, the normal derivative is taken. Then we have-an integral equation ON IN /~(~)+R(~) r ~2N 7rIfoJ d.( Er(~;j) O j=1 x Re [ - (ny + it ) exp(-id ) W(Zj+1 ) Y ~ VIZ + j+1 I + i(y - q) ~ 1 1 ~ ~ ~ + (ny-inz ) exp(iC~; ) X (- W(Zj+l) VIZ +i+ll - i(y - ~j+l) W(Zj ) )] VIZ + j l - i(y - Rj) (30) where (17j,j) and (~j+1,j+1) are the coordinates of the end points of the j-th segment, Re denotes the real part and Z - 1 -) (31) J 2~1z+jl+i~y-0j) CXj = tan-1 Rj+1-71: (32) j+1 - j where Zj is the complex conjugate of Zj. wit) is Error function for complex arguements defined as wit) = em erfc~-iz) (33) In the derivation of (30) we assumed wall sided hull form and the segment closest to the free surface is vertical. This leads to w('Z: )( ) ~ ( it, z, j = 0 (34) This singularity may not cause serious prob- lem in solving the integral equation (30) numer- ically. Efficient program to compute Liz) at accuracy of 12 digits is available (Iwashita and Ohkusu (1989)) in which several different expres- sions, continued fractions, asymptotic expansions and finite series approximations, are combined the most efficient way. mj, that we need to compute when we retain the second order terms in the body boundary con- ditions will not be so seriously singular close to the free surface, provided the hull form is wall sided. It is not difficult to show 0,'3 = 0~1), z,,0 3 = 0~1) (35) Wave elevation Or in the region near to the hull surface is computed with numerical -derivatives of the solution Or onz = 0. Or = -- (him + U ~9 ~texp~-i U x~fr]z=o 1 ,~ 4'r(X + Az)-jr(X) --expel-Mix) Ax Both hulls of a catamaran are considered to be within the near field of each other, the source density On both contours must be symmetrical with respect to fez plane. Each section contour is divided into a number of segments on which the source strength is assumed constant and the inte- gral equation (30) is solved such that the bound- ary condition is satisfied on the midpoint of each segment. '['his implies that upstream effect rep- resented by the second term of (30) is evaluated on the midpoint. 'lihe ship length is divided too into a num- ber of strips with thicknes of Ax. Assuming the source is concentrated at the center of /`x we step the integral equation (30) to next section. Of course only the sources located upstream have ef- fect on the current section; the sources from O to x-/\x have the effects. 11

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\Ve tested our method using only the i~npul- sive sources ageist results of Daoud (1975) and Faltinsen ( 1983 ) on stationary waves around a wedge. Daoud solved the integral equation on the velocity potential on the contour of each sec- tion instead of the source strength. laltinsen ex- pressed the velocity potential around a section with the fundamental sources distributed on the free surface as well as on the section contour. The conditions are imposed on the free surface, on the contour and at infinity. He solved the integral equation on the velocity potential and step the free surface condition to next section with using the dynamic arid the kinematic free surface con- ditions. Wave elevations around a wedge for a half angle 7.5 degree are compared in Fig.2. Results of our method agree well with their results. In our method magnitude of /\x will have essential effect on the accuracy of results because we employ the concentrated sources in the x-wise and we are assuming fast variation of the solu- tion in the x direction. Added mass coefficient for heave and added moment of inertia for pitch of a hull form (one of twin hulls of a catamaran model later described) were computed with in- creasing the number of strips N frown 20 up to 60 as shower in Figs.3 and 4. Results of compu- tation without raj seem to converge consistently and N = '2() gives accurate enough results, while results with mj, added moment of inertia in par- ticular, are slow in convergence. This may be due to the smallness of Ass value vulnerable to singularity in mj close to the intersection of the section contours and the free surface. We need a smaller signet size there to have more consis- tent results. Hydrodynamic forces computed with the present method for a mathematical hull form with se~ni-circle section contour and water plane form y = (0.3/L)x(L-x) are compared with those computed by Newman's unified theory (Newman (1978~) in Figs. 5 to 8. Results are obtained with ignoring the terms including (, in the body boundary condition (21) and retaining only the first term in the pressure (28~. Actually in pitch Node the term-Un3 in (21) is retained, though it is of higher order. In order to evaluate the first terra of (28) we need to dilferetiate numerically fir ill the x direction just like in equation (36~. Unified theory?s results we have shown here are too without A. The present method and unified theory predict hydrodynamic forces al most similar in magnitude, while agreement is not consistent. Damping of pitch computed by the present method is much smaller in lovrer fre- quency. We need experiments to decide which is better in such a high I)oude number region. We did forced heave and pitch motion test for a catamaran whose demihull has a form shown in Fig.9. Length of the model is lm, the breadth 0.086m, the draft 0.043m and the distance be- tween the centers of both hulls 0.30m. Heave or pitch of appropriate magnitude was given on tile towed model with other Reties suppressed. Measured force record was analyzed into several terns of harmonics. Length of records rneasurec1 after stationary state is reached is averagely 10 cycles of motion. In-phase force measured contain the force that would act when the ship were displaced from the original position at w = 0. Ilere we mea- sured it by giving some displacement to the model towed on calm water. Assuming magnitude of this force is approximately linear to amount of the ~lisplacetnent, we get a spring constant as illustrated in Fig.14. This component was ex- cluded frown the measured in-phase forces to ob- tain added mass. In the frame work of the present theory, however, difference of this component at non-forward spee(l and at forward speed in the non-uniform steady flow field must be considered to be of the second order. Results of experiments are plotted in Figs.10 to 1~3. A few of black circles depicted in Fig.11 have dash mark. At these points noise level was so high with reasons we do not know at present (magnitude of higher harmonics was as large as 5() percent of the fundamental one) that we are not confident of their accuracy. (:on~puted hydrodynamic force with the present method are shown in those figures. Re- sults depicted as without rat, which were ob- tained with ignoring consistently the effects of Us in equations (16) and (28), predict well the mea- sured ones at high Froude number 0.5 to 0.89. One reason of a little dicrepancy will be that the analysis is not valid at the ends of the ship. In the case of transom stern like our model we ex- pect that the flow leaves the stern tangentially in the downstream direction so that there is ato~n- ospheric pressure at the last section. This means that the suns of the added mass forces and hydro- static restoring force on the last section is zero. The same is true with the damping force on the last section. In our analysis the hydrodynamic 2

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behaviour at the last section is dependent on the upstream effect. There is no eRect frown down- stream. '['his means we have no information that there should be atmospheric pressure at the last section. 'the consequence is a wrong prediction of hydrodynamic forces at the last section. lnclu- sion of the interaction with steady flow at high Froude number ~ with end ~ does not seem to i~n- prove flee correlation between the predicted and the measured. Wave elevation between two hulls of a cata- rr~aran will cause occurence of impact pressure or the bridge structure connecting two hulls of a catamaran. We need a practical prediction method of it. It is well known that strip the- ory in which we consider two hulls are in the near field of each other predicts too strong interaction between them (Ohkusu (1971~) and the predic- tion of wave elevation is not realistic. Figs.15 and 16 are examples of wave elevation computed by the present method. '['hey are for the cases of pitch Notion at Fn = 0.5 and En = 0.89 with w2L/g = 20. The far left of Figs.15 and 16 is the midpoint of the distance between two hulls. It is clear that wave elevation is almost symrnetri- cal around each hull. This implies that there is almost no interaction of two hulls if we are close to the hull. Wave elevation in these examples is certainly high at I'7n = 0.5 than at I'n = 0.89 . We can compute the velocity potential and wave elevation in the far-field generated with a line source Next determined by equation (27~. In- formation on the wave field will make it possible to evaluate added resistance of high speed ships in waves. L)a~nping is also computed from energy flux in the far field. A line doublet as well as a line source will be required to describe correctly the wave field around a catamaran. Considering low interaction effect between two hulls in the near field, two line sources distributed on both hulls may be enough to give wave elevation correctly. Short wave components of the waves gener- ated by a point source will be dominant at high speed and at position close to the source. Ac- tually waves by a point source vary dreadfully, while waves by a line source does not because of integration of the effects from all sources on a line. But realizing this smoothing effect correctly on numerical computation requires very accurate evaluations of wave elevation generated by each source. New scheme to evaluate the Green func- tio~ for ship -notion at forward speed proposed by one of the present author (Ohkusu and Iwashita (1989~) will be usefull for this purpose. We computed Or by solving the integral equation at 2() strips along the ship length and then interpolated a at 2()0 positions. Effects from 2()0 point sources are surnamed to represent the effect of a line source. Figs.17 and 18 are corn- puted examples of the wave elevation at t = 0 for the heaving rno OCR for page 5
REFEREN CES (l)(:hapman, R.B.~1976~. Free Surface Effects for Yawed Surface-Piercing Plate. J.Ship Research 20 (2)Daoud, N. (1975~. Potential Flow Near to a Fine Ship's Bow, Rep. No.177, Dep. Nav. Archt. Marine long., Univ. of Michigan. (3)Erdelyi, A. (1956~. Asymptotic Expansions, Dover Publications, New York. (4)Faltinsen, 0.~1983~. Bow Flow and Added Re sistance of Slender Ships at High Froude Number and Low Wave Lengths. J. Ship Research 27 (5)Faltinsen, O. and Zhao, R. (1990) . Numeri cal Predictions of a Ship Motion at High Forward Speed, the Meeting oil the Dynamics of Ships, the Royal Society of London . (6)Iwashita,H. and Ohkusu, M. (1989~. Hydrodynamic Foeces on a Ship Moving with Forward Speed in Waves, Vol.166, J. Society Naval Architects of Japan (7)Newman, J.N.~1978~. The theory of Ship Mo tions, Advance in Applied Mechanics, Vol.18, Academic Press. (8~0gilvie, T.F. (1977~. Singular-Perturbation ' Problems in Ship Hydrodynamics, Advances in o.e Applied Mechanics, Vol.17, Academic Press, New 06 York .. . (9~0gilvie, T. F. and Tuck, E.O. (1969~. A Ratim nal Strip Ttheory for Ship Motions, Part 1, Rep. 0.2 _ No.013, Dep. Nav. Archit. Marine Eng. Univ. o , ~' ~' ~^ '~ of Michigan (lO)Ohkusu, M. and Iwashita, H. (1989~. Evalu ation of the Green Function for Ship Motions at Forward Speed and Application to Radiation and DifEraction Problems, 4th International Work shop on Water Waves and Floating Bodies, Nor way (ll)Ohkusu, M. (1971) On the MoLio~ of Twin Hull Ship in Waves, Vol.129 J. Society Naval Ar chitects of Japan (12)Yeung, R.W. and Kim, S.H. (1985) A New Development in the Theory of Oscillating and 'l'ranslating Slender Ships, 15th Symposium of Naval Hydrodynamics 1.OI /.6 /.4 /.2 Fig.1 Coordinate Systen~ ~ / ~ :~2.0 //\W x~v.l O \ U. ~U.~ U.D U.0 1. V y ye~X) H Dooud Fo/tinsen Pre s en t me tho d F~ ' T~T ' 0 5 Xlr H F~ Fig.'2 Waves around a Wedge FN=0.8D, ~9 L=10.0 Without mj A55 0.1 A33 0. 5 L_ ~40.05 O ^ 3 4 5 -2 xlo 1 2 Fig.3 Convergence with Increase of Number of Strips 4

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Ass o.os 0.5 A33 1 O j FN=O.89. ~g L=lo.o A33 - O 1 2 3 O 4 5 xlO Fig.4 Convergence with Increase of Number of Strips 1~ 0.4 f, ~ 0.1 _ I L I 0.4: - Present Method(without mj) -- Unified theory FN=0-5 0.3h LFN=1.O l l 0 5 10 15 20 2 25 wg L Fig.5 Added Mass (Heaving) of a Single Hull Ship ~1 ~1 |~ Present Method(without mj) > - -- Unified theory c~ 1.0 _ .~ IFN=0.5 ~ V 5 10 rFN=o.so 1 1 1 15 20 ,~ 25 Fig.6 Damping (Heaving) of a Single Hull Ship 0.1 Present Method(without mj) --- Unified theory 1 , , , 5 10 15 20 2 25 W T L 9 Fig.7 Added Mt. of Inertia (Pitching) of a Single Hull Ship 1 c~ \\~\\' o ~ `/ FN=0~5 ~_ OS / ~_4FN=1.O _ ~~ - ___ Present Method(without mj) --- Unified theory Fig.8 Damping (Pitching) of a Single Hull Ship W2 L -- Fig.9 Body Plan of a Catan~aral1 9 15

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--- Present Method(without mj) Present Method(with mj) O Measured(FN=0.89) ~Measured(FN =0 50) 1.0 0.5 --Present Method(without mj) -Present Method(with mj) Measured(FN =0.89) Measured(FN=0 50) O O o orF~=0.89 1 ~O. ~ F~=O.SO 0; 5 10 15 w2I 20 ol l l I I _ 5 10 15 w~L 20 9 9 Fig.13 DampiIlg (Pitchillg) of a Catamaran Fig.10 Added Mass (Heaving) of a Catamaran 2.0H I.0~- Present Method(without m;) Present Method(with mj) O MeaSUred(FN =0~89) Meneured(FN=o.5o) -W== ~ FN=0.89 ~ _ 0 ~ - - _ ~== ~ - - _ L FN=0 50 5 10 15 Fig.ll Damping (Heaving) of a Catamaran ~1 C33 pgL B2/2 0.5 ~N=05 /'/4FN= 0.89 / //~FN=0 ~15 //; ~ 1 ~/~ _ ~ ~0.5 15mm Heave Fig.14a Variation of Restoring Force (Heave) Cse ---- Present Method(without mj) r Present Method(with mj) P9 L2B2/2 l!vieasured(F~ =0.89) Measured(FN =0.50) 0.2~ O. I _ ~ FN=0 50 o o 0_/ 0 0 FN=O _ ~ F~=O.89 1 , , I 0 5 10 15 W2L20 ~3 / g //, Fig.12 Added Mt. of Inertia (Pitching) of a Cata naran ', ~-0.2 _ ~> / 3 deg. / Pitch -FN ~0.89 -FN =0.5 Fig.14b Variation of Restoring Force (Pitch) 6

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- - /L ~'` oo -0.5 5/L For ~ 0.5. At L .20, Pitching ,' X-0.6 _ closet " ~= ~ - A_ _ ~ - ~ ~ _ _ _ NO ~ \ 0.6 \~N I As ~ X-0.6 0 4 ~, / ' t - _ _ \ /1' 1 '. /~1 - - J I; I ^- ~~~5 0.6 / 0.8/ Fig.15 Wave Elevation TV For 0.89, ~ L-20, Pitching /'X-0.8 / / /0.6 / \` X-O.B cos ~ t 0.6 ' At' 0.~ 1 0.2> ~ a\ ~ 0.4 0.4 Fig.16 Wave Elevation 17

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Fn = 0. 500 T = 3. 162 Y/L = 0.500 _ Y/L = 0. 4 17 Y/L = 0.333 Y/L = 0.251/' . =/ Y/L = 0. 16 7~ Y/L = 0.084 1 \/~. , . . -L O L F.P A.P Fig.1/ Wave Elevation in the Far Field Y/L Y/L = 0. 333 Y/L = 0. 251 Y/L = 0. 167 -L ~ _ / ~_~y \-,~ /~A, _ _ ~ ~ ~ I A ~ ~ ~ ~ ~ ~ ~ ~; ~ ~ A ~ A _ ~ ~ ~- ~V' V - ~ ~' V ~ V ~ ~ ~ ~ -\f~^-J W-'`~--^~-- ~ \ ~_ A~ V-~ -J~' ~ ,~J=~ _ _ ~ ~ ,\ ~ ~ ~ ~ ~ ~ ~ .1 . . ~ AA~ _ . A~ _AA_ ,.... r V V ~V - ~r ] ~ V%r - ~ ~r ~ 2L 3L 4L 5L ~__ ,/ / \ ~ . ,/~\,/> \ ~^J ~ ~ r ~ A ^~ ~- ~ /' - ~ / \~, ~ ~_ \ ~ - ~ ^~ - - - ~ A ~ --~V5\,~ ~ ~ r~ ~ ~ \J \J ~ - v \] ;--~-^~ ,- ~,'t.,~q, `-~ 2L 3L 4L 5L Fig.18 VVave Elevation in the Far Field 18

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DISCUSSION Ronald W. Yeung University of California at Berkeley, USA The approach taken here is one already taken up by Yeung & Kim (1981, 3rd Numerical Ship Hydrodynamic Conference) for a single body. The present calculations have qualitative features, in terms of agreement with expt., not so different from the above work which the authors are apparently unaware of. Indeed, our experience, as I recall, was that the terms associated with the change in waterline width need to be accounted for. In our calculations of those days, the mj terms involving the steady-state body potential was not included, but improvement over strip approximation was evident. The numerical approximation over the artificial time variables (x-O is particularly important when (x-O is large and (yet) is small. Accuracy and precision can be achieved if the ~ variable is integrated analytically. The procedure is described in more detail in Yeung (1982, J.Engrg.Math). The role played by the transverse waves was analysed in Ref [12] but I suspect they are not important as the Froude numbers being considered here. AUTHORS' REPLY I appreciate Prof. Yeung for attracting our attention to his extensive works done before. I understand the terms associated with the change in waterline width indicates a line integral on the intersection of the free surface and the body surface. This term is of higher order in our analysis, that is, of the same order as mj. Our results do not show any essential difference from the results obtained with solving the near field problem by distributing the fundamental sources on the free surface as well as the body surface. This seems to show that inclusion of the line integral has the secondary effect on the results. 19

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