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Numerical Appraisal of the New Slender Ship Formulation In Steady Motion H. Marco (University of California, Santa Barbara, USA) W.-S. Song (Shanghai liao Tong University, China) ABSTRACT A new formulation for the fluid motion around a slender ship is developed, on the basis of an asymptotic expression of the Kelvin-source around its track. The boundary value problem is expressed by an integral eq- ation which is much more simplified than the solution of the Neumann-Kelvin approxima- tion. In order to examine the validity of this theory, numerical computations are carri- ed out with respect to the pressure distribu- tion, wave pattern and wave resistance of several types of hull forms, i.e. the Wigley hull, a sailing yacht hull, and a Series 60 (Cg= 0.60) hull. The results are compared with experimental data. 1. INTRODUCTION The final goal of ship hydrodynamics is the theoretical determination of hydrodyna- mic forces acting on the ship hull within the accuracy of practical allowance. One of the most important in this respect is the computation of wave resistance in the steady forward motion. The pressure distri- bution over the hull surface becomes important when the boundary layer calculation is intend- ed. Because of the complex geometry of the ship hull, the solution of fully or partially nonlinear boundary value problem by means of the computational approach of numerical simulation has not achieved the practical feasibility yet. The analytical solution, on the other hand, has to depend on the per- turbation technique which leads to the linear- ization of the problem as the first approxi- mation. It is well known, that the classical thin ship perturbation has not provided result which shows a satisfactory agreement with measured data. Several versions of linearized theory have been proposed, such as the Neumann Kelvin approximation (1~. However most of them are rather inconsistent approach, lacking the rational basis in the sense of the pertur- bation analysis. The slender body theory is another possibility of rational approach of this problem. The first attempt of the application of the slender body theory to ships in steady forward motion appeared in 1962 - 1963 (2~3~4~5~. However the formul 239 ation of the wave resistance by this theory was found quite unsatisfactory, because the values computed according to this theory showed a remarkable deviation from measured values, and the agreement was even poorer than the result of the classical Michell theory (6~. No progress in this problem has been observed for as long as 20 years since that time. In 1982' one of the present authors developed a new approach to the slen- der ship in steady forward motion (7~. The difference of this theory from the former theory lies in the treatment of the singular- ity which represents the body. The original formulation has followed the method of pertur- bation analysis, which is employed in the slender body in the unbounded fluid. Then it assumes that the slender body is represent- ed by the source distribution along the longi- tudinal axis. The new theory, on the other hand~begins with the expansion of the Kelvin- source aroundits track. It is disclosed that it is not possible to represent the slender ship floating on the free surface by the source distribution along the longitudinal axis considered in the free surface. The boundary value problem is expressed by an integral equation on the hull surface, because the singularity representing the hull must be distributed over the surface. Then the solution is more complex than the original slender body theory. However the solution of the integral equation is much more simplified than the solution in the Neumann-kelvin approximation. The reason is first that the integral equation is of the Volterra type, so that the boundary value problem becomes parabolic. That means there is no contribution from the disturbance in the downstream to the boundary condition at the upstream section of the body. The marching procedure starting from the bow end is possible to solve the boundary value problem in each section. Secondly, the kernel function of the integral equation can be expressed by known functions, so that the high accuracy of the numerical work is achiev- ed. An analytical method of solution by means of the conformal mappi ng has been i ntend- ed, and several numeri Cal resul ts have been obtai ned i n 1983. However i t i s found that the accuracy of the computati on i s not sati s

factory (8~. Furthermore, the analytical method is not suitable for the numerical work, because the mapping of the transverse section to a unit circle needs much computer time. Then a numerical method of solution is developed. This method employs the source distribution to represent the hull, and the density of sources is determined numerically by the panel method. The program library is prepared for the computation of kernel functions. Three kinds of hull forms are employed for the numerical example. They are the Wigley hull, a sailing yacht hull, and the Series 60 (CB= 0.60) model. Items of the numerical work are the pressure distri- bution on the hull surface, wave resistance, the lateral force when moving obliquely, the wave profile alongside the hull, and the wave pattern around the hull. Some of the numerical results are compared with meas- ured data. 2. LINEARIZATION OF THE VELOCITY POTENTIAL It is assumed that the fluid is inviscid and incompressible, and the depth of water is infinite. Take the Cartesian coordinate system with the origin on the undisturbed free surface, x- and y-axes on the horizontal plane, and z-axis directing vertically down- wards. Consider a slender ship fixed in a uniform flow of velocity U in the direction of positive x. Assume the irrotational motion and write the velocity potential in the form like Ux+~. The field equation is the Laplace equation. [L] V2¢ = 0 (1) The boundary condition on the hull surface is ~ An= -U3x/3n = -Unx (2) where n is the normal drawn outwards on the hull surface, and ~ = al/3n. The kinematic condition on the free surface at z=` is [K] (U ~ ~x)~x+ ~y~y~ Liz= 0 (3) The dynamic condition on the free surface, that the pressure is constant, is [D] U¢x+ 2~¢X2+ ~y2+ ~Z2) _ 9` _ o (4) Since the depth of water is infinite, ~ = 0 at Zen-=, and ¢~0 at x,y-+~. One can eliminate between (3) and (4) such as [F] [(U+$x~aa +$yaay +¢za33tu~x+ 2~¢x2+$y2+lz2) _9z] = 0 (5) In order to express the velocity potential of the fluid motion around the hull, we assume Green's function G(P,Q) with P=(x,y,z), Q=(x', y',z'), and apply Green's theorem in the x',y'z'-space bounded by the hull surface S bellow the still waterline, a large surface Son surrounding S in the lower half space, and the horizontal plane SO between S and 00. |: S+Sm+so ~ ~ P. Q ) ~$ ~ Q )~ ] d SQ (6) We have assumed the analytic continuation of ~ to the entire space below the still water surface in the above equation. We will employ the Kelvin-source as Green's function, which satisfies the boundary condition, U2¢ - gaG = 0 (7) on the horizontal plane z=0. If the surface S. is taken at infinite distance, the integral on S vanishes. On the horizontal surface SO, we have the relation derived from the free surface condition (5) such as )=0= U ~ )Z=0+ ~ (X,y) (8) where t~ x, y ) = [ 2u($xTxx+$y~xy+¢zfxz )+2 ~¢x~y~xy+ +$xtz~xZ+¢y~z~yz ) - y ~ 2~¢ ~ 2 ~ +/ (U2¢xxz-g~zz)dz (9) Making use of relations (7) (8) in the inte- gral on SO, and integrating by parts with respect to x', we obtain 4~lSt ~ (P'Q)-~(Q) ~ idSQ 4nglL [G(P'Q) ~ - ¢(Q) ~ ~z'=odY' o + 4~9i; t~x',y')G(P,Q)dx'dy' (10) SO where Lo is the curve of intersection of the hull surface with the still water plane. Now let us assume that the ship is very slender and the beam to length ratio is a small fraction £<< 1. Then the slope of the hull surface to the longitudinal axis is small in the order of £, i.e. nx= 0~£). From the hull boundary condition (2), we have An ~x~x+ny~y+nztzc Uo(£) (11) where n , n , n are direction cosines of the outXwardY norZmal to the hull surface. Because of the slender body, n =O(1), nz=0~1). If U=O(1), we have ~ =0~£), ~ -YO(£). Since the hull surface are] is regaZrded as 0~£), the first term on the right hand side of (10) is O(£2). t~x,y) is at most 0~£2) in the near field in the area of 0~£) within S=, so that the third term on the right hand side is at most 0~£3 ). Omitting this term, we have 240

ff [a¢(Q)G(p Q'_~(Q)aG(P'0)]d ~ " S vilQ Q _ 4U J [G(P,Q) ~ _ ¢(Q)DG(P Q)]zlOd 0 (12) Next assume a velocity potential ¢' in the lower half space, which is harmonic inside S. and satisfies the boundary conditions, ¢'= ~ on S and U257T2- gang= 0 on z=0 Applying Green's theorem to $' and G(P,Q) in the domain bounded by S and the plane z=0, 0 = 44~; [a~niQ)G(P'Q)~¢'(Q)3G(P Q)]dSQ + 4nglL [G(P'Q) ~ ~¢ (Q.)- aX : ) ] Z , Body o ..... (13) where n' is the normal of S drawn inwards. Adding (12) and (13), and putting ~ than + an') we obtain ~ = ffS0(Q)G(P'0)ds + 9 fL o(Q)G(P'Q)nx~s ds (14) ..... (15) This is the basic equation of the Neumann- Kelvin approximation. If the slender body is assumed, nx and dy/ds are 0(£~. Then the inte- gral along the waterline Lo is 0(£3 ), and can be omitted. In consequence, the velocity pot- ential is given by = iiS0(Q~G(P,QjdS (16) Thus the fluid motion around the hull is exp- ressed by the distribution of Kelvin-sources over the hull surface. The Kelvin-source is given by the formula, G(P,Q) = G(x,y,z;x'y'z') = ~ r + r'+ G'(x,y~z) (17) G'(x,y,z) = Ko lode ~ exp(-kz+3kxcosK6+ikysin6) dk (18) where r =[(x-x' )2+(y_y')2+(Z_z' )2 ]2 r'=[(X-X' )2+(y_y' )2+(z+z' '2 ]2 x=x-x', y=y-y', z=z+z', Ko= g/U2. The integral with respect to k is taken along the real axis indented by a small semicircle in the lower side of the pole at k=KOsec20. Then the velocity potential is written in the form like (19) ¢1 -ifSo~x my LIZ )(r~ r,jdS (20, $2 ffS0( x , Y , Z JIG (x,y,z~dS (21) 3. ASYMPTOTIC EXPRESSION FOR THE KELVIN-SOURCE Let us consider the asymptotic behavior of the Kelvin-source near the x-axis. In order to find out the asymptotic expression of G'(x,y,z), we consider the following integral in the complex u-plane. I = ~ exp[-uz+iu(xcosO+ysinO)] du (22, -c 'C u - KOsec'd along a closed circuit C composed of the positive real axis indented by a small semi- circle in the lower side around the pole at u=K sec2D, and the positive or negative part of The imaginary axis together with a large quadrant arc connecting the ends of the axes. In the case of xcosO +ysinO~ 0, the closed circuit is taken in the first quadrant. Since the pole is inside the contour, Cauchy's theorem gives Ic= 2ni(Residue at u=K0sec20) If the radius of the large circle tends to infinity, the integral along it vanishes, so that ~ 0 I +r = 2niexp[-K0zsec20+iK0sec20(xcosO+ysinO)] and accordingly exp[-uZ+iu(xcKoSO+~inG)] du ~=tCostz-Kosec Using -t(XcosO+ysin0) J t + Ko sec ~ dt - 2ne KOzSeC ~sin[K0sec20(xcosO+ysinO)] In the case of xcos0+ysinG<0 on the other hand, the closed circuit is taken in the fourth quad- rant. Since the pole is outside the closed circuit, Cauchy's theorem gives I = 0, and we have the result, c r exp[-uz+iu(xcos0+ysinG)] )0 u - K0sec20 du tcostz-KOsec2~sintz - t2+ K02sec40 e-tlxcos~+ysin~ldt 241

Therefore the function 0'~x,y,z) is given by 2Ko 2 loos z-KOseC2 S intz see d | Liz+ K ~ sect ~ x e-t~xcosO+ysinO~dt -4Kof~ e~KOSec ~Zsin(KOxsecO)cos(KOysecOtanG) x sec2OdO (23, where 01 is an angle between -~/2 and n/2 such that tanOl= -x/~y;. The double integral is bounded, and expressed on the x-axis by known functions such as 11 sec2OdOi e~t~x~cosO ~ tdt ~ :tHl(Ko~x~) - Yl(Ko~x~) _ 1] (24) where H is the Struve function and Y is the Bessel Junction of the second kind `93. Next we consider the single integral of the second term on the right hand side of (23~. Changing the integration variable by secO = u' e KOzSeC ~sin(KOxsecO)cos(KOysecOtanO) |~ ) ~ e KoZU sin(KOxu~cosKOyu~u2-1 ) x sec2OdO _->DmJ~ e Kozu ~ U cos(KOyu~-cos(KOyu z x sin(KOxu~du so KodXJlCoS(KoXU)(~ - u~du 2 Yl(KOX) - c°S(Kox)/(KOx) The second integral is finite. (27) yiom~oe KoZU sin(KOxu~cosKOyu Mu = 0 z - O Kox (28) Then the integral (25) is expressed asymptoti cally as + 2 ~ e~v Zcos(v2y~sin(v ~ ~dv when x>O, = 0 when x<0 (29) Summarizing the above results, the asymptotic expression for G(x,y,z) near the x-axis is given by G(x,y,z) ~ -87--o: e v Zcos(v2y~sin(vi-Ox~dv x U dU when x>O, + ~KotHl(Kox)+3Yl(KoX)] + x ~ 2Ko ,~ rm e KoZU sin(KOxu~cos(KOyui5~) u du when x<0 (25) It has an essential singularity along x-axis. In order to isolate the singularity, let us consider the identity, ~ e KoZU sin(KOxu~cos(KOyui;~) u du = ~ e KoZU sin(KOxu~cos(KOyu2)du 0 1 - ~ e KoZ U s i n( Koxu ~ cos ( Kosu2 Mu + ~ e Kozu [ U cos(KOyui~-cos(KOyu2 ~ ] x sin(KOxu~du (26) The singularity is condensed in the first term on the right hand side. It is readily shown that the last integral is bounded and uniformly convergent on the x-axis. ~ - ~KotHl(Kox)-Yl(KoX)] + x -2Ko when x>0 (30) when x<0 (31) The integral in (30) is expressed by the Fres- nel function of complex argument such as E(x,y,z) = ~ e v Zcos(v2y~sin(vi~0x~dv = -]m e~i KoX /4Z: F[XiKo/(2nZ)] (32) where Z = y+iz, and F(x) = C(x) + iS(x) = ~ ei~U2/2du (33) o 242

4. BOUNDARY VALUE PROBLEM FOR THE SLENDER SHIP The velocity potential near the slender ship is simplified by the asymptotic expres- sion of the Kelvin-source given in the preced- ing section. We have divided the velocity potential into two parts ¢1 and ¢2 in (19~. |1 is expanded with respect to E . Omitting higher order terms, it is expressed near the hull by x ln'Y Y.) + 'Z Z.' ds ¢1 is(x) ~ ~ (y_y )2+ (z+z )2 (34) The expression for $2 near the hull is obtain- ed from the expression (30) and (31~. |2= idx iC`x ~ (x ,y ,z JIG (x-x ,y-y ,z+z ids ..... (35) where G (x,y,z) = -47-oE(x,y,z)~1 + sgn x) + ~KoH1(Ko~x~) + [nKoY1(Ko~xj)+ ?/~x ~ ~ x (1 + 2sgn x) - 2Ko (36) Then we can write ¢2= -4iro/~1 + son x~dx ic~x,'0(x ,y ,z ~ X E(x,y,z~ds + JH(x~dx Jc~x''0(x ,y ,z ids where H(x) = KoH1(Ko~x~) + [nKoYl(Ko~x~+2/~x~] (37) x (1 + 2sgn x) - 2Ko (38) Let us consider first, the case that the longitudinal axis of the ship is along the x-axis. The ship is moveing at zero drift angle, and the fluid motion is symmetric on both sides of the ship. The hull surface is given by the equation, y = +f~x,z) Then the boundary condition on the hull surface is written as 3¢ = -Ufx/il+fx2+fz2 (40) Because of the slender body assumption f - 0~£) f = C(l). Omitting terms of o(~2), the ~irec- t~on cosine of the normal can be written as axe -fX/i:--fz = VX nay +1/~ = v nor -fz/ ~ = v Then the hull surface condition is expressed as a;; ~ it;= -Unix where 3/3v = vy3/Dy + v 3/3z. Taking the representation ~ = ¢1+ $2 we can write 3$l 3~2 - = -us - ~ When the ship is placed obliquely to the uniform flow, with a drift angle A, the bound- ary condition on the hull surface becomes (43) = -(Ucos~nx+ Usin~ny) (44) Then the velocity potential can be divided into the symmetric part ~ and the antisymmet- ric part Ma such as s = Viscose + Casing (45) If ~ = 0~), the boundary condition on the hull surface for Us is given by (42), and that for is a atlas = -Uvy From the definition of 11 12 =200(x,y,z) (46) + J ~ ,~[ln'yy-yy, '2+ ~z+zl '20`x y z ids ..... (47) a 2= -8j~oJ dx ~ Ev~x,y,z)·~(x ,y ,z ids ..... (48) where x=xO gives the bow end, and Ev~x,Y,Z) = vy3E + vzaz ' Equations (47) (48) are substituted in (43), giving an integral equation for o~x,y,z). The expression (48) suggests that the integral `39' equation with respect to x is of the Volterra type, so that the boundary value problem is parabolic. This fact facilitates the solution to a great extent. The integral of (48) is determined by the source density in cross sections upstream. Then a¢~/3v is regarded as a known function at the section where the integral equation along the hull contour is solved. The solution begins at the bow end, and marches downstream. 5. WAVE PATTERN AND WAVE RESI STANCE The pressure on the hull is given by the Ber (41) noulli equation. p = p _ pO = p[-utcos~$x+sin~¢y) - 2~¢y2+lz2 ~ + 9Z] (50) (42) 243

is omitted because of higher order compared with ~ 2, ~ 2. The elevation of the free sur- face if obtained by ~ = -z = - -[U(cos~¢x+sin~ly) + 2(¢y2+¢z2)]Z O (51) The quadratic term ~ 2+¢ 2 may be omitted in the formula for the iaveZpattern, but it is better to include in the pressure distribution below the waterline. The wave resistance, the lateral force and the yaw moment are calculated bv the ore~- sure integral. F = -fisPn dS ~ -fdxfc( )Pv ds F = -JfsPn dS ~ -fdxfc( )Pv ds Nz= fisP(-nxy+nyx)dS ~ fxdxic(x'Pvyds (54) Substituting the expression for ~ in f50;, the pressure distribution is determined. The free surface elevation, forces and moment are calculated therefrom. The velocity poten- tial ¢2 includes a term which does not depend on z. It gives the pressure distribution ir- respective of the depth. However the numeric- al example for pressure distribution indicates that better agreement with measured results is obtained by taking account of the attenua- tion of this term by depth of water. The draftwise variation is related to DG/az. There is the relation by the free surface condition, (52) (53) DG = ~ a2G at z=0 (55) Therefore the variation of the kernel H(x), defined by (38), is given by (z/Ko)02H(x)/3x2. This problem appears when Koz is not small, or Kox is large. Then the asymptotic expression for H(x) may be employed. At large Kox, the asymptotic expression is determined by Y1(Kox). and we have the asymptotic relation 32Yl(Kox)/8x2~ -Ko2Y1(Kox) Therefore we can express the value of H(x) at depth z in the form like (1 - Koz)H(x) We make further simplification by taking the average of the attenuation factor t-hrough z. Then the factor 1 - 2K z is multiplied to the corresponding term in ~he free surface eleva- tion, and (1 - 2K Z)2 iS multiplied in the calculation of hy~rodynamic forces. 6. NUMERICAL METHOD The solution of the integral equation for the distribution of sources is calculated by means of the panel method. Since the hull is symmetric, one side of the hull sur- face under still waterline is divided in IXJ = M panel elements AS. , with I divisions in x and J divisions in ziJ The source density is defined at the center of the panel, over which the density is assumed uniform. The integral equation is discretized as J i-1 J k-1 ij( ) vxij-Qelki o(Qk)Mjj(Qk) (j = 1,2, J) for symmetric part (56) J i-1 J k-1 ij( ) vyjj-Qilki o(Qk)Mjj(Qk) (j = 1,2,...J) for antisymmetric part. (57) L (k) and Mj.(Qk) denote the normal velocity oiJ$~ and $2 Jby unit source at the control point respectively. The left hand side is the source density which is to be determined. The summation on the right hand side is determined by the source density along the cross section upstream. It is regarded as a known quantity, because the solution is carried out from the foremost section and proceeds backwards. Lj (k) is calculated analytically. The co~putation of the kernel matrix Mj (Qk) is more time-consuming. J Mjj(Qk) = vyjjEy(x,y,z) + vzjjEz(x,y,z) (58) We have to calculate the derivatives of ~ and ¢2 for the determineation of various quantiti- es. 3~1/3x is replaces by the finite differ- ence, a~l (¢l )i- ($l )i Dx Ax (59) Analytical expressions are employed for other derivatives. The most time-consuming is the computation of E, E , E , E . In order to facilitate this, theXfol~owinzg transform- ation is employed. Consider the integrals ECC = J e Ctcos(2a2t2)cos(bt)dt ECS = f e Ctcos(2a2t2)sin(bt)dt QSC = f e~ctt2sin(2a2t2)cos(bt)dt QSS = J e Ctt2sin(2a2t2)sin(bt)dt EQC = J e ctt 2sin(2a2t2)cos(bt)dt EQS = I e ctt 2sin(2a2t2)sin(bt)dt Put p = c+ib, and define A = ECC - iECS = J e Ptcos(2a 2 t 2) dt ~ B = QSC - iQSS = J e Ptt2sin(2a2t2)dt ~ (61) C = EQC - iEQS = J e~Ptt~2sin(2a2t2)dt J (~ (60) 244

Then we have ECC = ~A, ECS = -~A QSC = ;~8, QSS = -SUB} (62) EQC = Ji:C, EQS = _aac Applying the Laplace transform, A = p-1+ jrr2a2p~3/2e~a/Perf~ia2p-2y(63) B 2p-2 j~2p-5/2~2p-aye-a/perf~ia2p 2) (64) C = -i rr2p~5/2e~a/Perf~ ja2p2'(65) where erf(Z) is the error function defined by erf(Z) = - Erf(Z) = -J. e dt ~ ~0 There is the expansion for the error function about Z=0 as n_O41 e~Z2> 2nZ2n+](67) where (2n+1~!! = (2n+1~2n-1~2n-3 ~5.3.1 (-1~!! = 1 On the other hand, the error function has an asymptotic expansion at CZAR such as Erf(Z) = v; e~Z2) (mu) (22-~ Now we put Erf(Z) ~ -e~Z ~ (-a) (2n-1~!! + ~' ~z' (69) (68) The first term on the right hand side is the asymptotic expansion. If an denotes the n-th term of the asymptotic expansion, and£gives the accuracy, we have N(Z) 2 for ~an~_£ (70) J)N(Z) = EN ntO ni(2n-2N+l) for ~an~>£ (71) where N is the integral part of ~z2 Me When N=0, the first term of (69) becomes zero, and Erf(Z) = $0(Z). Then (69) coincides with (67~. If Z is pure imaginary, Z = iy, Erf~iy) = iiYe~tdt = jfyet2dt O O Then it is pure imaginary. Put Erf~iy) = i[eY ~ (2nn+~2n+l] + ~N(iy) (73) (iy) is given by (72) A)N(iY) = 0 for |an| £ (74) N(iY) it EN nt0 n!~2n-2N+~] for an>£ (75) Sample calculations of the Fresnal integral, to which the existing program library is avail- able, confirm excellent accuracy of the above method. Putting a = Kox2/2, b = y, C = z > 0, (76) we obtain E (x y z) = 2~oECC = 2;140^A~ Ey~x,y,z) = -2QSS = 2 ~ B| `77' (66) Ez~x,y,z) = -2QSC = -CAB E(x,y,z) = 2EQC = ARC In order to check the computer time, the wave Pattern of a point source is calculated. The computation by means of the above expression takes 2.04sec.CPU by HITAC M-240H, while it takes 2min.-7sec.CPU, if E is calculated by Simpson's rule. x 7. RESULTS OF THE WIGLEY HULL equation The hull surface geometry is given by the y = bt1 _ (Q)23~1 _ (Z'2 ~ (78) where Q = L/2 is half length, b = B/2 is half breadth, and d is the draft at still waterline A model of dimensions L = 2.000m, B = 0.200m, d = 0.125m is employed for experiments in the towing tank of Yokohama National University. The panel division for the numerical work is 40~1ength) x 10(draft) = 400, in equal intervals. The hull form with panel division is illustrated in Fig.1. First of all, the source distribution over the hull surface is determined. A few sample results are illustrated in Figs.2,3. Comparison is made with the source distribu- tion of the double model. Remarkable differ- ence is observed near the free surface. This fact suggests the inadequacy of the origi- nal form of the slender ship theory, which employs the double body potential as the near field solution. Fig.4 shows the pressure distribution on the hull surface at Froude number 0.267. The result of computation is compared with measured results with 6m- model published by Namimatsu et al (10~. Generally speaking, good agreement is observ- ed between computed and measured results. Slight deviation at the stern region may be attributed to the boundary layer displace- ment effect. 243

Present Flethod X U x lo2 ~Doub le Hodel Fig. 1 Coordinate system (Wigley hull) -Present Metl~od SC'~E Dolible Model u 1 2 3 v-Xl02 X/1--O - 975 i: x/~- -o. B75 x/~=-O. 825 X/, -0. 1 75 X/~--O. 1 25 0 - ~ 47/ X/~- 0.825 X/1- 0. 675 X/1= 0. 92S X/~ 0. 975 Fig. 2 Source distribution at each cross section F = 0.267 ' n 246 o. , o.o -o., , F.P. Fig. 4 Pressure distribution on the hull surface at F= 0.267 n .n's ~=~-~_~`Q3363~\ 1' .P. r2 -o s ~,Z/~:-0000-.0125 ~ ~=~-t 2 AP. 11\ / W;-~, r2 0;~'' -U.S~' \_ Z/L:.JO00-.0l2s 1, I, F" = 0 . 3 1 6 F.P. Fi g. 3 Long i tud i na 1 d i stri buti on of sources at uppermost pane 1 s c - P ~ P° P ~ u 2 \e \~ O O F =0 267 Present Hethod n . ° Measured (6m Model) , ~Z/~-0. 015 0 9/ ~ ~:~ NNbW _ ~/e-o,~,~ _ , Z/l=0 . 08S ~ v''° ° °~° -~ , '~s: 0),% . z/~- 0 . I 05 ~ 1

Samples of computation of the wave pro- file alongside the model are illustrated in Fig.5. The result at Froude number 0.267 is compared with the experiment of 2m-model at YNU towing tank, and the result at Froude number 0.316 is compared with the measurement of 6m-model mentioned before. Slight discre- pancies are observed at the wave trough. They may be attributed to the nonlinear effect because the phase of the wave is in good agreement. The computed wave pattern around the hull at Froude number 0.267 is illustrated in Fig.6, and the corresponding measurement is illustrated in Fig.7. Similar configura- tions in the crest and trough of waves are observed between computeation and measurement. Fig.8 shows the result of computation of the wave resistance coefficient. The computed values given by white spots are compared with the values obtained by the longitudinal cut wave survey given by black spots. Good agreement in the position of humps and hollows is observed. The computed wave resistance is slightly higher than the wave-pattern resistance. It may be a common trend, that the wave-pattern resistance shows a little lower value. The dotted line gives the wave resistance obtained by the subtraction of viscous resistance defined by CF(1+K), where OF is the Schoenherr friction coefficient and K = 0.15, the form factor, from the total resistance coefficient. This curve fits well with the computed values. The full line gives the Michell resistance, which shows a great deviation from the measurement. Computations are also carried out with respect to the Wigley hull at finite drift angles, ~ = 5°, 10°, 15°, 20°, as the example of the asymmetric flow. Fig.9 shows the computed wave profile alongside the model at ~ =10°, Froude number 0.267. The result of measurement with 2m-model at Yokohama National University is also shown. Fairly good agreement except near the bow on the back side (leeway side), at which the leading edge separation may be present. The computed and measured wave pattern around the model is illustrated in Fig.10 and Fig.ll respect- ively. In spite of good agreement in the wave profile' some deviation is observed in the diverging wave pattern. It may be attributed to the distortion of the base flow due to the hull, while the computation does not take account of it. Longitudinal and lateral components of the force and the moment about the vertical axis are given in Fig.12, Fig.13 and Fig.14 respectively. The nonlinearity of curves at large angle may be due to the inclusion of the term of velocity squared. There are no data available for comparison with measurement. The computation does not include the viscous force which must be present, though the flow separation at the leading edge is observed clearly. Therefore the computation under the condition of continuous flow of a perfect fluid may not hold in the actual condition. O.04: 2~/L ~o 0.02 :\ J. ~ O.00 -OjI on ~` n no Present Method pleasured (Y..~.U.) - Pleasured (em Node j Fig. 5 Wave profile alongside the model at En= 0.267, 0.316 0.6 ).5 0.4 03 0a O.' Michell ° Present Method Towing - t (Krause · Wave Analysis , ~ O ~ ~ 1<'- ~ ,: 0.15 a23 n ,~ 0~0 n ~ EN Fig.8 Wave resistance coefficient of Wigley hull Fig. 6 Computed wave pattern at F = 0.267 Fig. 7 Measured wave pattern at F = 0.267 n 247

0.06r 2~/L 0.04 _ 0.0_ _n n: n n n n: o.n -O.0~ ~. Fi 9. 9 Computed and measured wave profi le alongside the hul l o~ = 10°, Fn= 0.267 Fi 9. 10 Computed wave pattern ~ .6 o ~= 10°, Fn= 0.267 o.~ PRESENT METHOD : · MEASURED .CX~to3 ,-__ t~ __ Rx fn-o.3le ro u L - ~e~ o . I ~ ~_ ~ _ ~ fn-0 . 204 O O I .' ~ . ~° s° ~o° a 15° 20° Fn-0 . 2nd F i 9. 1 2 Computed l ong i tud i na l force coeff i c i ent 1.8 -Ci ~_~In' side ~a9-c~ Fig. 11 Measured wave pattern ~= 10°, Fn=0.267 /~ 0.0( ~' ~1 1 '0 50 10° a ls° 20° 248 `~,,~ Fig. 13 Computed Lateral force coeffi ci ent

0.6 n 5~ - - - 0.4 0.3 0.2 O . 1 - CURIO: // o.~( i, , or so I Go if/ Fig. 14 Computed yaw moment coefficient 8. RESULTS OF THE SAILING YACHT HULL Cu=< 1~ , , a 15° 20° The hull form has been designed by NCAC following the rule of 12 metre class yacht. The hull form with panel division is illust- rated in Fig.15. The length is divided into 40 segments in equal intervals. The radial cut is employed in the draftwise division. The angle of the radial cut from the waterline j 0° 8° 1 6° 24° 32°,40°,50°,70°, 80°, 90° . The panel division on the solid keel is 11X8, but small panels are combined, resulting 70 panels. A model with a detachable solid keel and a rudder is made for towing tank experiment. The resistance test at zero drift angle is carried out at Yokohama National University tank. Experiments at the finite drift angle are conducted by Akishima Labora- tory of Mitsui Shipbuilding Co. _~ FP Fig. 15 Sailing yacht hull form Fig.16 gives the longitudinal distribu- tion of the symmetric part of sources on the canoe body with and without keel, and Fig.17 gives that of the antisymmetric part of sources at Froude number 0.269. A remarkable effect of the vertical keel to the source distribution on the canoe body at finite drift angle is observed. Fig.18 gives the wave profile alongside the hull, and Fig.19 illustrates the wave pattern around the hull by the wave contour, at the drift angle 0°, 4°, 8°, at Froude number 0.269. Because of small length to beam ratio and the flat stern, the wave pattern behind the stern becomes very complex. No measured data are available for comparison, but a particular feature of this hull form is observed in these figures. 4 O,xIO 4 8_ on_ B° 2 ~2 Of ~O -2 PRtSENT HtTHO0 lUltH Kttt ) --- PRESIFT HEtHOD 4 'al THOUT KEEL ) ----- DOUBLE HOtEL IHITH KtEL) 2~--_--O.S FP 9= 9° - 16° L-2 ~4 ~ 2 UP 4 O,x10 §=16° - 24° ·RISE'r HETHOØ INtTH KEEL, ___ FRESEtll HETHO8 4' tH1 tHOUt KEEL ) ount HOTEL IN l TH KEEL ) l_2 4- O,XIO A_ PRESENT HETHe0 `~11H KEEL ) ___ PRESENT HETHO0 IRITHOUT KEtL) ----- DOUI'E NOtEL tH1 TH KEEL ~ =50° - 60° =60° - 70° L_2 UP ~4 L 2 O --2 -2 FP 249 Fig. 16 Longitudinal distribution of symmetric part of sources Fn= 0.269

- onto 8- ~ - ~ '- ~ r ~--~ v- -o.s o.o ~ e.s \ .` 1 re'3'Hr NETHER tHl TH HEEL ) --- FR`SEH' HtTH00 lu I TNOUt REEL ) ----- lOUlLE ttOt'L (~1 TH HEEL ) 2 O -2 PREsEHr NETHOD IH1 TH HIEt ) --- "ESEHT NETHOD '~ 1 You r KEEL ) ----- lOUJ~£ HOVEL tH 1 TH KEEL ) 6 =2 (a-3 O . O2x 1 0 2 4 - 2 \~-2 ~4 l-2 UP ~ =5~-6~ ~2 FR6SEH! METRO: IH 1 TH KEEL ~ _ PRESEH! nETHOD . IN 1 THOU r KEEL I --~- lOUlLE HOVEL tH I TN KEEL ) UP PtESEHr NETHOD IUlTH KEEL ~ --- rtE5EH' NEtHUD 4_ t~1 THOUr KEEL ) ----- GUILE noDEL INtTH KEEL 2 ~' O 0 ~=-in 2 - 2 FP L_2 UP Fig. 17 Longitudinal distribution of antisymmetric part of sources En= 0.269 Fig.20 shows the wave resistance coeffi- cient of the canoe body without keel. Towing test results are given by the total resistance coefficient CT, the residual resistance coef- ficient CR based on the Schoenherr friction coefficient, and the wave resistance coeffi- cient derived by the assumption of the form factor K = 0.29. The curve of residual resis- tance fits the computed values approximately. The deviation at higher Froude numbers may be attributed to the change of wetted hull geo- metry due to sinkage and trim, together with the bow wave elevation. Fig.21 shows the wave resistance coefficient of the hull with keel. The effect of the keel to the computed wave resistance is mainly due to the differ- ence in the source distribution near the stern. The difference in CT between results with and without keel is remarkable. The conventional method of the form factor for viscous resistance gives 0.54, which seems to be too large. The computed points are not parallel to the experimental curve. This deviation may be partly due to the change of trim, which is more remarkable than in the case without keel. The axial and lateral forces when the model is moving at finite drift angle are calculated. The axial force coefficient C is shown in Figs.22,,23. The computation does not include the induced drag of the solid keel. Therefore the difference between the results with and without keel is not remarkable. Experimental data obtained by the Akishima laboratory are shown in white spots. Since the model is free to heel and trim in the experiment, there is a considerable difference between experiment and computation. Then the comparison is only for reference. The lateral force coefficient Cy is shown in Figs. 24, 25. The computation does not include the lift of the solid keel. In order to calculate the lift of the keel, the lifting surface computation such as the vortex lattice method is necessary. However the result of computation by the theory of large aspect ratio is added for simplicity. The result is shown in dotted lines. The measured data are shown by white spots. However comparison is difficult, because of the difference in conditions mentioned before. The computed yaw moment coefficient C is shown in Figs.26, 27. Different from the lateral force, difference between the moment with and without keel is remarkable. This means that the theory includes the moment of the keel. One may conclude from the above results, that the present theory is applicable to the wave resistance of the canoe body of the yacht, in spite of the small length to beam ratio, while other theories such as the vortex lattice computation is necessary for the prediction of hydrodynamic forces acting on the hull with vertical keel at finite drift angle. 250

- 2 t/L _ . _ 0.02 O. 00 -U.1 fP -0 . 02 _ Fn=0 .269 F ATTACK ANCLE=ee ----- AITACK ANCLE=4° A I TACK AIdGLE _0° FAtE 5 I DE ,, , =3 I /~\ ,~t ,~/r~oA/P o.o2t O.OC -O. O~ ~ ~/ 2X/L _ f~ - Fn_O. 269 F i ~ . 1 8-a Computed wave prof i 1 e alongside the hul 1 without keel Fn= 0.269 .~ Q~ lIlifK A'lr.l E=4° [n=0.269 ~ ~r.l .F =R~ Fn :0 . 269 Fi 9. 1 9-a Computed wave contour wi shout keel Fn= 0 269 0.06F 0.04L 0. 02~- 0.0c 1 2 (/L A 1 T ACK AllCLE = e° ----- AITACH ANCLE=4° Al TACH AtICLE=0° FACE SIDE Fn=0.269 -0.0 _ 0.04 2t/L lACK SIDE ~0.02~1 ' ~,~ F i 9. 1 8-b Computed wave prof i 1 e alongside the hul 1 with keel Fn= 0.269 A'TArx AN0.t E=48 Fn=0.269 ~? ~AT TACK AllGLE-1~° Fn ~0. 269 lr I I I Fi 9. 1 9-b Computed wave contour with keel Fn= 0.269 251

X 1 O - 3 6 _ 5 _ 4 C- R - rOU2L2 TOHINC TEST C' . C~(K=0.29) ~CALCULI`TED CU 2 I ~CI ~ 74 CFO _ 0.3 0.4 Fn 0.5 Fig.20 Computed wave resistance coefficient compared with test results of yacht model without keel s.o~ 4.0 _ 3.0 _ 2.0 _ 1.0 _ c, Fig. 21 Computed axial force coefficient of yacht model F = 0.269 n C= ~ reNl'JG JEST Cl. CR. CuIK=0.54 ~ ~ CALCULATED CU 4~t _CXXl03 5 0 Cx= ~ r Fn =0 . 269 C' MEASURED B). n I TSU I (WITH KEEL ~ RUDDER ) - CALCULATED {HITHOUT ItEEL ~ RUDDER) CALCULA TED t)J I TH KEEL ) (~AYE-MAKING FORCE ONLT ) c, 1 n 252 /CI / CR , / ///cH / // ~ ~ /,/ Fig. 21 Computed wave resistance coefficient compared with test results of yacht model with keel Cxxlo3 4.0 _ 3.O _ 2.0 _ CX= ~ Fn =0 .359 nEAsuRED BT n I TSU I (UITI.I KEEL ~ RUDDER) ~- CALCULA TED {HITHIJUT KEEL ~ RUDDER) CALCULA TED (HITH KEEL ) (II4VE_MAK ING FORCE ONLY ) -cFn I Fig. 23 Computed axial force coefficient of yacht model Fn= 0 359

15.0 10.0~ s.o~ 4.0 1 3.0 _ 2.0 _ 1.D :) . U - Crxlo3 lS.0 crXlo3 Fn=~.269 EASURED B] H 1 TSU I (HItH KEEP ~ RUDD£R ) CALCULA TED (HItHCUT KEEL ~ RUDDER) CACCULA TED {~1 TH KEEP ) (~AVE-MAKING fCRCE ONLY' CALCULAtED HlTH KEEL ) (HAVE-dAK ING fORCE* LlfT OF KEEL ) ~10.0 C7=~ fn=~.359 MEASURED BY H 1 TSU I (N I tH KEEL ~ RUDDER ) ----- CALCULA TED ~ ~ l rHcu r KEEL ~ RUD DER ) - CALCULA TED (~1 TH KEEL ) (~AVE-MAKING fCRCE ONLY ) ---------- CALCULATED (H I TH KEEL ~ (HAVE -MAK I NG fURCE. L I FT 3F KEEP ) ~,, ~-oO 20 4° 6° 8a a Fi 9. 24 Computed 1 ateral force coefficient of yacht model F = 0.269 _CMX 104 CM= .~'U: L] Fn =~.263 ----- CALCULATED (HITHCUt KEEL 4 RUDDER) 3.o CALCULAtED (~1 TH KEEP ) (HAVE-MAKING FORCE 3NLT 24 40 Fi 9. 25 Computed 1 ateral force coefficient of yacht model Fn= 0.359 4 ~ CMX10 __. , , 6° 80 a CM= - fn =~ .359 --- CALCULAtED {HITHOUT KEEP 4 RUDDER ) CALCULA TED (HITH KEEP ) / {~4VE-MAK ING fORCE CNcT ) 60 80 Fig.26 Computed yaw moment n n a coefficient of yacht model Fig. 27 Computed yaw moment __ ,_ a coef f i c i ent of yacht mode 1 Fn= 0 359 253 ,_

9. RESULTS OF SERIES 60 HULL As an example of the conventionalhull form, the well-known Series 60 model (C =0.60) is employed for computation. The bodily plan is shown in Fig.28. The panel division is 40~1ength~x 8(draft) in equal interval. The towing test of 3m-model is conducted at Yoko- hama national University tank. The wave profile alongside the model at Froude number 0.28, 0.30, 0.32, 0.34 are illustrated in Figs.29 ~ 32. Full lines give the measured results and dotted lines give the computation. Fairly good agreement between computed and measured wave profiles is observed throughout the results. The slight difference may be attributed to effects of the nonlinearity and the boundary layer displacement. The wave patern around the model at Froude number 0.30 is illustrated by contour courves in Fig.33. Similarity between computed and measured wave patterns is observed. In Fig.34, the computed wave resistance coefficient is compared with the residual resistance coefficient based on the Schoenherr friction coefficient and the wave pattern resistance by the longitudinal cut method. The computed value is slightly higher than the residual resistance. This may correspond to the tendency of lower wave height at the stern in the computed wave profile. Since the curvature of the hull surface at the stern is much higher than the case of other models, the number of panel division in this area may not be sufficient for good accuracy. The position of humps and hollows is in good agreement in theoretical and experimental curves. L~38 Bso.lOOs Cb=.8 1 .. _ l\ V\'\\\\\<- In\\ \\:L~\ Ah ~\1 ~1 id. Fig. 28 Body plan of Series 60 CB= 0.60 0.03~ :N 0.0; ~ of f -1.0 <.l 0.~/ O.] ~ J^" 0 ' n at I ~.ml Fig. 29 Computed and measured wave profile alongside Series 60 model Fn= 0.28 A ivy . 0.0 odor ~_._. ~0 01 1 ~ - a - a - flu {2~, ,~o.:e - x - ~ - - . -06t ^4 -0.2 t ~ ~2 ~0;`' 4.05 ^04 Fig. 30 Computed and measured wave profile alongside Series 60 model Fn= 0 30 To: .' '`\ 0.03 j o ml 0.01: dry. -1.0 -0.8 -0.6 ~.',0. ~ -0.2 0 0 ~ -0.01 ~ ~ {., -0.03 -O. O. - X - X - Con 1 ~ ; ~Q;5, 8~ 1,0~ Fig. 31 Computed and measured wave profile alongside Series 60 model Fn= 0.32 2h'L 1 ~ I; \ 6 i: o.m ,. `: ;~ -1.0 ^e ^l \,(0~` ~L2 0. V. Mel ~` ^02 ^04 , _._ 0.03 n hi -° - .o- ~ <2.) r_~.~` -x.- x - con it. my; ~ : ?~-^ a 0.2 tar. ,,~'o.e 1.0 f.~' /.'' ,~,i /,, Fig. 32 Computed and measured wave profile alongside Series 60 model Fn= 0~34 254

o ,4 ~_ 844: ~ Tar 10. Fig. 33 Computed and measured wave contour of Series 60 model En= 0 30 9.0' , B.04 i 7.~. , 6.Q. 5.04 4~0t ~ no 2 D - Cw T11£0RETICAL RESISTANCE COEFF. Cr IESIOUAI" lESISTA#CE COEFF. nAr.l7~089 (Schoenherr) V.T.-I4.O'C VAVE FATTERN RESISTANCE COEFF. x CUP Fig. 34 Computed wave resistance coefficient compared with test results Series 60 model 10. CONCLUSIONS The validity of the new slender ship formulation is examined by the computation of the wave pattern and wave resistance of the Wigley hull, the sailing yacht hull and the Series 60 model. Satisfactory agreement is obtained in pressure distribution on the hull surface, the wave profile alongside the hull, and the wave resistance, between computed and measured values with respect to the Wigley hull. The computed wave resistance of the canoe body of the sailing yacht hull without keel shows good agreement with towing test results, in spite of small length to beam ratio. The effect of the solid keel is not fully accounted for by the present theory. The hydrodynamic forces, when the hull is at a finite drift angle, may be calculated by the present theory supplemented by the lifting surface computation. Although a good agreement between the computed wave profile alongside the Series 60 hull and the measurement is obtained, a slight devia- tion is observed in the computed wave resis- tance from the towing test result. This fact may suggest that fine panel division is required for the prediction of wave resis- tance of conventional hull forms such as Series 60 model, because of the large curva- ture of the hull surface, especially at the stern area. As the conclusion, the new formulation of slender ship approximation has achieved a remarkable improvement in the theoretical computation of the wave pattern and wave resistance. The result seems to confirm the usefulness of this theory in the predic- tion of hydrodynamic characteristics of prac- tical hull forms in steady forward motion. ACKNOWLEDGMENTS The authors express their thanks to Prof. M.Ikehata and staffs of Marine Hydro- dynamic Laboratory of Yokohama National Uni- versity for their cooperation in the experi- mental work. Their thanks are also to NCAC and the Akishima Laboratory of Mitsui Ship- building Co. for their generous permission for publishing the data concerning the sailing yacht hull, It is noted that the numerical work has been carried out by the use of HITAC M28D computer of Yokohama National University Information Processing Center. REFERENCES: . Brard, R., "The Representation of a Given Ship Form by Singularity Distributions when the Boundary Condition on the Free Surface is Linearized." Journ. Ship Res. Vol.16 (1972) . Vossers, G., "Some Applications of the Slender Body Theory in Ship Hydrodynamics." Thesis, Delft Technological University, (1962) . Maruo, H., "Calculation of the Wave Resis- tance of Ships, the Draught of which is as small as the Beam." Journ. Soc. Naval Arch. Japan, Vol.112 (1962) 4. Tuck, E.O., "The Steady Motion of a Slender Ship." Thesis, University of Cambridge, (1963) 5. Joosen, W.P.A., "Velocity Potential and Wave Resistance Arising from the Motion of a Slender Ship." Internationnal Seminar on Theo- retical Wave-Resistance, Ann Arbor, (1963) 6. Lewison, G.R.G., "Determination of the Wave-Resistance of a Partly Immersed Axisym- metric Body." International Seminar on Theo- -~;~3 u~v~-Resistance' Ann Arbor (1963) 255

7. Maruo, H., "New Approach to the Theory of Slender Ships with Forward Velocity." Bulletin Faculty of Eng. Yokohama National University Vol.31 (1982) 8. Maruo, H., Ikehata, M., "An Application of New Slender Ship Theory to Series 60, Cb= 0.60" The Second Workshop on Ship Wave Resistance Computations, DTNSRDC Bethesda (1983) 9.Havelock, T.H., "Ship Waves: the Calculation of Wave Profiles." Proc. Royal Soc. A Vol. 135 (1932) 10. Namimatsu, M., Ogiwara, S., Tanaka, H., Hinatsu, M., KaJitani, H., "An Evaluation of Resistance Components on Wigley Geosim Models, 3. An analysis and Application of Hull surface Pressure Measurement." Journ. Kansai Soc. of Naval Architects Japan No.197 (1985) . . . 11.i Maruo, H., Ikehata, M., Takizawa, Y., Masuya, T., "Computation of Ship Wave Pattern by the Slender Body Approximation." Journ. Soc. Naval Arch. Japan, Vol.154 (1983) 12.i Song, W.-S., Ikehata, M., Suzuki, K., Computation of Wave Resistance and Ship Wave Pattern by the Slender Body Approximation." Journ. Kansai Soc. Naval Arch. Japan No.209 (1988) 13.i Song, W.-S., "Wave-making Hydrodynamic Forces Acting on a Ship with Drift Angle and Wave Pattern in her Neighborhood." Journ. Kansai Soc. Naval Arch. Japan No.211 (1989) 14.~ Song, W.-S., Ikehata, M., Suzuki, K., "On Wave-making Hydrodynamic Forces and Wave Pattern of a Sailing Yacht." Journ. Soc. Naval Arch. Japan, Vol.166 (1989) 15.2 Maruo, H., "Evolution of the Theory of Slender Ships." Ship Technology Research, Vol. 36, No.3 (1989) Results in this paper are in part reported in these articles. 2 The theory is discussed from the perturba- tion point of view. 256

DISCUSSION Ronald Yeung University of California at Berkeley, USA In spite of the nonlinear conditions stated in the beginning of section 2, it seems clear that the starting point of this work remains the same as the Neumann-Kelvin problem. This is evident from the representation, Eq. (12) of the paper. What followed from there is essentially an approximation to the N-K solution, and one should not expect the present calculations can do any better than the 3-D Neumann-Kelvin solution. In terms of the slender-body anoroximation carried out here. or in Prof. Maruo's 1982 work, I don't feel that it is rationally based, at least not completely. I will point out 2 objections. (1) The neglect of the line integral in Eq. (15) cannot be justified simply on the basis of traditional infinite-fluid slender-body theory. Sources on the free surface exert much stronger influence than submerged distribution. It is well established these days that the waterline integral in the N-K problem yields a significant contribution. This contribution is taken into account in the matched asymptotic theory of Yeung & Kim (1984, 15th ONR Symposium). This leads to the 2nd point. (2) In our work, which Prof. Marno might not be aware of, we showed that the near field approximation, Eq. (25), of paper is more elaborate than an impulsive 2-D source and a function, say, F. that depends only on the axial distance 'x'. Using matched asymptotics, we showed that the transverse wave-field is contained in F(x,y,z), with explicit DISCUSSION expressions given in Yeung & Kim (1984). Prof. Marno's F-function corresponds to setting y-z-0 in ours. It is clear that your analysis eventually lead to a rather ad-hoc afire in the paragraph following Eq. (55). It appears this deficiency can be corrected in the manner that we have derived from the matched field. I don't think such development should be done as a matter of convenience, rather, it should be rationally based. I would like Prof. Marno to comment on these two issues. err ~ AUTHORS' REPLY The slender body theory is based on the rational perturbation analysis. The fundamental technique is the series expansion of the complete solution of the fully nonlinear boundary value problem, its existence being assumed, with respect to the slenderness ratio a, as a small perturbation parameter. The lowest order of the expansion gives the linearized solution, which is discussed in this paper. The rigorous derivation of the result by the perturbation technique is not employed in this paper, because it has been Riven in another - literature. The general form of the linearized solution in the near field is given by the velocity potential of the form ~ = HAD' + g(x) where Cited' is the solution of the two-dimensional Laplace equation q}~D',,~, + q}~2D'zZ = 0 and satisfies the boundary conditions on the body surface and on the free surface. g(x) is a function of x only, which is determined by matching with the far field solution. Both of these functions are o(~2~. It is readily proved that the line distribution of sources along the waterline b is 0(63), SO that it must be deleted from the linearized scheme. The function F(x,y,z) referred in the discussion is derived from the linearized far field potential, which is not correct in the near field. In order to obtain the consistent approximation in the near field, one must expand it with respect to the transverse coordinates y, z, and detain only the term of the lowest order. It is reduced to the limit at y = z = 0, if the function is bounded. It corresponds to the function g(x). It should be emphasized that the higher order terms with respect to ~ should not be detained, because they are subject to the nonlinear portion of the boundary conditions which is not taken into account in the theory. The present discussion is concerning the higher order terms only, and does not make sense accordingly. It seems that the argument presented by Prof. Yeung is noting but the consequence of the lack of knowledge about the rational perturbation analysis of the complex nonlinear problem. DISCUSSION Hongbo Xu Massachusetts Institute of Technology, USA (China) Prof. Marno, the results you have are impressive. My question is about the theoretical results for Wigley hull at an angle of attack. Have you compared the lateral force and yaw moment coefficients with experimental data? The angle of attack or used in your computation appears to be rather large (up to 20°). As we know, the stall angle for a wing is about 12° to 15°. It may be important to find the approximate range for ~ in which the slender ship theory is valid. AUTHORS' REPI,Y The example for the Wigley hull in finite drift angle is rather an academic aspect because the leading edge separation must be present by the form with a sharp edge, though the theory does not take account of it. Therefore, the forces and moment computed by the theory may not represent the actual value except at small angles, less than 5° say. However, the comparison of the wave profile shown in Fig. 9 indicates that the theory can predict the behavior of the free- surface flow fairly well, even at 10°, which is not so small. Kazu-hiro Mori Hiroshima University, Japan You explained that your results agree with the measured Fairly wells I don't think so, but the agreement is strikingly well! In your introductory remarks, you disclosed your negative opinion to the direct numerical method. Although it takes much computing time, it has a potentiality; e.g., the viscosity can be taken into account. The methods may be complimentary. I hope you may not have such a negative opinion to the numerical method, for you are so influential. AUTHORS' REPLY The computational fluid dynamics depends on the capacity of computers. It has achieved a great success in various fields in hydrodynamics, such as in compressible aerodynamics. Unfortunately, the present stage of the application of CFD to the free-surface flow around the hull does not seem to reach the level of feasibility as a useful tool to resolve problems in the practical field of shipbuilding. However, the recent progress of the computer capacity is remarkable, so that there is much prospect that CFD will become a powerful tool with practical feasibility in the field of the full form research in future. Please, never be afraid. 257