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A Hybrid Mode} for Calculating Wave-Making Resistance V. Aanesland Norwegian Marine Technology Research Institute Trondheim, Norway Abstract A hybrid method for calculating the wave-making resistance of ships has been developed. The method is based on three-dimensional potential theory, and the flow is assumed to be steady. The fluid is divided into an inner and outer domain by two vertical control surfaces. These surfaces are parallel to the free stream, extending to infinity and one at each side of the body. The inter- nal flow is matched to the external flow on the control surfaces in order to satisfy the radiation condition. A three-dimensional source-sink method is adopted in the numerical treatment of the problem, using a source function for an infinite fluid. A distribution of source density on the wetted part of the body, on a local part of the free surface and on the control surfaces has to satisfy the boundary conditions in the inner domain. The wave resistance can be calculated by two differ- ent methods. One is to calculate the pressure distribu- tion on the wetted hull surface and to integrate for the force in longitudinal direction. The other is to use the inner solution combined with a control surface integra- tion. The fluid velocity and wave elevation along the control surface are needed. Introduction For a ship in service it is important to have as low to- tal resistance as possible in realistic sea conditions. Both model tests and calculations have been used extensively in order to be able to estimate the resistance at the de- sign stage. Methods of varying sophistication have been developed, and many restrictions and assumptions are made in each case. A large number of contributions on the wave-making problem have been published. Concentrating on theo- retical and numerical solution methods which are inde- pendent of model tests, an important contribution was given by J. H. Michell, t1], who introduced thin-ship 657 theory where the resistance is given by the geometry of the hull. T. H. Havelock, [2], used a Green's function method instead of the Fourier-integral method used by Michell and confirmed the results of Michell. The reason for referring to these two authors is not only because they where among the first to attempt to solve the problem theoretically, but also because their results will be used in the present solution method. For a comprehensive review on the wave resistance problem, see for exam- ple Wehausen, [3~. Two other contributions will be of importance when relating the present work to already ex- isting theories. G. E. Gadd, t4i, and C. W. Dawson, [5i, introduced the idea of distributing discrete source panels on both the ship hull and the mean free surface. The wave-making part of the velocity potential is calculated as a perturbation to the double-body flow. Gadd used a nonlinear free-surface boundary condition, while Dawson linearized with respect to the double-body flow. In the former case an iterative procedure was adopted because the position of the free surface is unknown. It can be argued that the methods of Gadd and Daw- son need a considerable amount of computer power and time. If preliminary results are needed quickly, a number of thin ship and slender ship methods are available. On the other hand, the restrictions set by these methods can be troublesome. For the present author, the flexibility of the method is of greater importance than a fast run- ning program on a medium-size computer. The capacity of new computers is increasing rapidly, and the corre- sponding cost is decreasing. Indeed, the use of super- computers minimize this problem. The present program has been run on both VAX785 and CRAY X-MP. On the latter computer a speed up factor of about 100 is typical. In other words a computation of about 1 hour on the VAX machine is finished in half a minute on the CRAY. The present method incorporates some ideas from Gadd and Dawson, the thin ship theory and some new ideas. Results are presented for a single point source, the Wigley parabolic model and the Series 60 block 0.60 ship.

The boundary value problem (vi) radiation conditions. The steady-state wave-making problem is formulated ; n a Cartesia n coordi nate-system x, y, z movi ng with the ship velocity. The x - y plane describes the undisturbed free surface with the x-coordinate positive towards the stern, see figure 1. Another coordinate-system x~,y~,z~ is fixed in the ship and coincides with x,y,z when the ship is in its equilibrium position and with no forward velocity. The fluid is assumed ideal, (i.e. inviscid, in- compressible and homogeneous), and its motion is irro- tational. Surface tension is neglected. Figu re 1: Coordi nate systems The problem is formulated as a potential flow prob- lem where the total fluid motion is described by the ve- locity potential ~(x, y, z). The following conditions have to be satisfied: (i) La place's equation in the fluid V2~=0. (1) (ii) the dynamic boundary condition on the free surface ~ (~2 + 4?2 + ~2 _ u2) = 0 on y = ((x, y). (2) where subscripts denotes partial differentiation. (iii) the kinematic boundary condition on the free surface ~,.~. + I,,,~ = 0 on y = ((x,y). (3) (iv) the kinematic boundary condition on the hull ~.,.h.~. ~ ~V + 4) h,, = 0 on SB (4) where y ~ h(:c~,z~) = 0 defines the hull surface (v) the kinematic boundary condition on the sea bottom A,, = 0 on z =d, (5) which in the infinite depth case is replaced by lily V. = U. (6) 658 The hull surface is defined in the body-fixed coordi- nate system. y1 = ~h(~1,zl) (7) The transformation from one coordinate system is easily performed by using the relations Hi = ACCOST(Zs~sinct yl = y (8) (9) z] = Using + (zaccost (10) (11) where ~ is the sin kage and car is the trim angle. The hydrodynamic forces in x and z direction and the trim moment are calculated by integration of the pressure over the wetted part of the hull. F'. = - ~ p(:c, y, z~n.,.dS (12) F =A p(~:c,y,z)n dS (13) . so M`' = / pax, y, z)tn'.(zz`,) - n (xxc-')]dS (14) The Bernoulli equation is used to evaluate the pres- sure, p = - Pt(V~2 - U24 - pgz (15) and the normal unit vector n is positive into the fluid. Solution method The fluid is divided into an inner and outer domain as shown in figure 2. The inner domain is bounded by the body surface, SB, the free surface, SF, two vertical control surfaces, So, and a surface at infinity, S.,. The outer domain consists of the rest of the fluid domain, which is two segments of a sphere, marked by the dotted lines in the figure. The outer domain is mainly used in order to find a boundary condition on the matching surfaces, while the main task is to find a solution in the inner domain. The use of Green's second identity in the outer domain gives the boundary condition. The derivation is described in appendix A. The reason for introducing the vertical control sur- faces is partly to restrict the computational domain and partly to obtain well defined radiation conditions. The numerical scheme will show that a disadvantage is that panels have to be distributed on a restricted part of the

control surfaces in addition to the hull surface and the free surface. On the other hand many elements can be excluded from the free surface compared to the method of Dawson and Gadd. It is assumed that the wave effect will decrease rapidly with depth and a large part of the control surface can be truncated. In addition to the two above-mentioned reasons for using two domains, the scheme also suggests to linearize the free-surface condition differently in the outer and in- ner domain. A low-speed linearization is adopted in the inner domain (similar to the one used by Dawson) and a free-stream linearization in the outer domain (similar to Kelvin's thin ship formulation). Assuming that the free stream linearization is satisfactory in the outer domain, other linear or nonlinear boundary conditions on the free surface in the inner domain may be used. Another pos- sibility is to use a totally different numerical solution method in the inner domain. A finite difference scheme may be of interest when a local flow phenomenon is studied, and the fluid can be described by more complex eq cations. In the present case the appropriate boundary condi- tion on the free surface is found by linearizing the equa- tions (2) to (3~. On the free surface in the inner domain, the double-body flow is used as the main flow upon which the wavy perturbation flow is superimposed. The total potential is divided into three parts. = Ux+¢ = ¢~+~] = Ux+¢O+¢>~ (16) where Up = free-stream potential ¢,' = double-body potential ¢0 = disturbance potential in double- body theory distu rba nce potential i n low- speed theory ¢> = disturbance potential in thin- ship theory. The free surface condition is simplified as described by Dawson even though Raven, [11i, has reported that the formulation is inconsistent. (Anyhow, the difference in calculated wave resistance is within a few percent.) The subscript ~ denotes differentiation with respect to the streamlines of the double-body solution. Neglect- ing quadratic and higher-order products of ¢~ and its derivative, the following equation is obtained. Hi- = - 9 t(~.'~s~s24~s¢~ss] on z = 0. (17) 6s9 t Figure 2: Inner and outer domains This is the same result as Dawson obtained in his equa- tion (14), [5~. When matching is performed on the vertical control surfaces, 0 from the outer solution has to equal ¢0 + At. Equation (17) can be written in the form 0 + - ffr2l,9¢sS2~04sO`Iss] = ~~2~2l~¢,~Sx - (O`~(Ux~s~s] on z = 0. (18) The problem is solved with respect to source strength which automatically gives the velocity components i.e. the first derivatives of the velocity potential. A numerical operator identical to the one obtained by Dawson, t5i, is adopted in order to estimate the second derivative with respect to s. The vertical control surfaces are assumed to be so far from the body that the waves will satisfy the linear free-surface condition. It is then appropriate to use the Kelvin source function, Go, to describe the outer flow. It is shown in appendix A that the boundary condition on the vertical control surfaces is v= 27r //s, G~;o,7dS on y= sob (19) and the corresponding boundary condition on the hull is ¢~7 =U.- on z = 0. (20) The radiation condition in the inner domain is sat- isfied by using an upstream differential operator when satisfying the free-surface condition. The operator in- sures that waves are only present behind the ship. On the downstream boundary, an artificial damping is ap- plied to the free surface condition. The inner solution is only affected a short distance upstream of the damping

Numerical tests of a single point source F., 1 0.10 1 0.20 1 0.30 1 0.40 1 0.50 1 H,~/L 1 003 1 013 10.29 1 0.511 0.79 1 Table .1: Minimum depth of the vertical control surfaces as function of Froude number. area. The calculation of the wave resistance by control surface integration is obtained from the undisturbed part of the inner domain. A far-field solution may be applied instead as a matching condition on a downstream trans- verse boundary. The necessary depth of the vertical control surfaces can be estimated by use of the fundamental wave length, A, defined by the Froude number, F,i, and the length of the ship at waterline, L. Of = 2~F,~L. (21 ) As seen from the formula, the wave length increases as the square of the Froude number. Using the assump- tion that an elementary wave disturbs to a depth of half the wave length, the minimum depth of the control sur- faces, His, is given by Hit- = Jo = OFF (22) Table .1 indicates the necessary depth of the ver- tical control surfaces. At the higher Froude numbers, however, the results obtained using depths much smaller than indicated by the table are good. On the other hand, it is necessary to have a depth about twice the draft of the hull, which indicates that the near-field flow about the hull is dominating the wave making. While running the program, the contribution to the wave re- sistance from the first vertical row, the last vertical row and the bottom horizontal row of panels on the vertical control surface is checked in order to control the exten- sion of the surface. The depth of the surface can then be within the limits of confidence without actually satisfying the numbers in table .1. The evaluation of the Kelvin Green's function, Go;, is only needed on the vertical control surfaces. Until recently the calculation of this function has been rather time consuming. In the present work, nondimensional values have been tabulated and linear interpolation has been used. New, fast algorithms are now available, for example Newman [6i, and can easily be included in the progra m. Verification of a computer program is very important, t10~. The present code has been tested using single point sources and single point dipoles situated below the free surface. The results have been compared with results obtained by Nakatake [9] in the case where the double- body linearization in the inner domain is replaced by the common free stream linearization. Figure 3 shows the panel distribution in the case of 8 longitudinal rows of elements on the free surface. Figure 3: Panel distribution on one fourth of the surfaces in the case of a single point source situated at F = 1. b is the distance between the center plane and the vertical control surfaces. Different aspects were important to investigate in these single point tests. The first was to check the ra- diation condition on the vertical control surfaces. The closer to the center plane it was possible to position the surfaces, the better. Then a very limited part of the free surface was needed to be panelized. The effect of moving the control surfaces was checked by investigat- ing the wave elevation along the innermost row of panels compared to the wave elevation obtained by using the Greens function definition, equation 25, in appendix A. The plots presented in figures 4 and 5 show the case when the nondimensional vertical position of the point source is F = '`(z) = g/U2(z) = 1.0. Figure 4 includes the results as presented by Nakatake. Secondly, the number of elements needed to dis- cretize the inner domain was checked. It was found that a number of about 20 elements pr wave length was needed to obtain a stable solution. By increasing the number of elements beyond that limit, nearly no differ- ence was observed in the case of 8 longitudinal rows of elements on the free surface. In the case of 3 rows of elements on the free surface however, even a higher num- ber of elements would increase the accuracy of the wave elevation as seen in figure 5 where three different grid sizes in longitudinal direction is presented. Also included are the curve when the control surfaces are removed. It is obvious that this solution is totally wrong both with 660

-1 0 2 ~ ~~\~\~"o>~°: Figure 4: Contour plot of the wave elevation due to a single point source situated at F = 1. The upper part of the figure is from Nakatake. Kelvin source 8 rows, Max - Q25 3 rows, Max - 0.40 .- 3rows,6yx=025 3rows,6~x=0.15 - 3 rows, no control surface respect to the wave length and wave amplitude. In addi- tion the result for the Kelvin Green's function evaluated at the center plane is given in figure 5. A third aspect of importance is the behaviour of the numerical differential operator. Two-, three- and four- point upwind operators where tested and the last one was selected. As can be seen from the curves in figure 5, a reduction of the first wave length of about 5 percent is observed. But additional computations have shown that the wave length is very good further downstream. This might be caused by the boundary condition applied on the control surfaces. Finally, it is important to check to influence of com- bining a double-body linearization in the inner domain and a free-stream linearization in the outer domain. New tests where carried out for the single point source using these conditions. The wave elevation changed, as ex- pected, somewhat compared to the tests with only free- stream linearization in the inner domain, but the effect of using the vertical control surfaces where similiar. Tests have also been carried out for single point dipoles which confirm the results. The Wigley hull The hull surface of the Wigley parabolic hull is de- fined by the equation Y' = 2 t1~ L')~1 - ~ L' )~] (23) whereL/2 ~ xi < L/2 and -H < z, < 0. The main parameters are given in table .2, and the body plan in figure 6, which has smooth lines arm fore- aft symmetry. A lot of numerical and experimental data Figure 5: Wave elevation along the innermost row of panels. The results for different positions of the verti- Aloe \Vigley hull cal control surfaces are compared with results using the centre plane source function. it\ \ :7 '''-I -'' TTI- r ,~ TO I Ill Wigley B/L,,,, 0.1000 H/L,,,, O. 0625 Series 60 | 0.1333 0.0625 L/L,),, 1.0000 1.0167 Ca 0.444 1 0.600 Cal 0.661 1 0.710 To ble .2: Main parameters of the hulls . I:;~-;lt1 ~ ~ 1 ~ i' , . T ., l _~/ / /' ~ The Series 60 hull Figure 6: Body plans of the different ship hulls. 661

exists which makes the hull well suited for comparisons. Data was taken from the 1.st and 2.nd Workshops on Wave Resistance Computations, [12] and [13i, and plot- ted in figure 7 as the envelope of residual resistance coef- ficients. In addition, the mean values of the experiments of the two workshops are plotted separately in the case of a restrained ship and a ship free to sink and trim. In [7i, the results of a great number of test cases are listed, varying the number of elements on the hull, free surface and control surfaces. Also the position and ex- tent of the vertical control surfaces were changed. Two configurations will be presented. The first is a reference case, where the control surfaces are positioned at a suf- ficient distance from the hull surface so that reflections as described by Dawson are avoided. The solution is essentially the same as Dawson's solution. Computer programs, essentially based on the same 5- o 4_ x an <a ~ 3 Us o Is 2- V' V, CLI 1 - _ ·MEAN OF EXPERIMENTS INCLUDING SINKAGE AND TRIM MEAN OF EXPERIMENTS FIXED MODEL (FROM 2.W0RKSHOPJ 0.0- . . 0 15 0,20 0.25 1 /,1 V: 1 .' 1 1: 0.30 0.35 0.40 0.45 0.50 FR0U0E NUMBER Figure 7: Wave-resistance coefficients calculated from experiments using the longitudinal cut method. O `, x A - ~ 3- o Z 2- V, - 1- 0~. DAWSON'S METHOD FEN'S METHOD O PRESENT METHOD 1 1 / /K _~ ~ ~- _ _ . . . 0,15 0.20 0,25 0,30 0~5 0. FR0UDE NUMBER O ~ 7 . .- . _ 0.45 0,S0 Figure 8: Wave-resistance coefficients calculated by com- puter programs. 5 1 - o `_ PRESENT METHOD O HULL PRESURE INTEGR. \7 WAN INTEGRATION O HULL PRESURE, FREE MODEL WALL I~GR" WEE MODE' 0~- 0,15 0.20 0.25 1 1 t~1 .~ 030 OJ5 0,~ 0.45 0. FR0U0E NUMBER Figure 9: Wave-resistance coefficients calculated by the present method using both hull-pressure integration and control-surface integration. .20 .25 .30 .35 .40 .45 .50 1 i 1 1 1 1 ~3 experimental data I D calculations I anti 0.02- nnR- a ~ Figure 10: Nondimensional sinkage of the Wigley hull t 2.0 - 1.B - 1.2 - 0.8 - 0.4 - 0.0 - ~3 experimental data calculations Figure 11: Nondimensional trim of the Wigley hull. 662

theory, have often produced results with a large spread- ing. In figure 8 the numerical results of both Dawson, [5], and Xia Fei, t14], have been plotted as reference to the present data. Sinkage and trim have been neglected. The discrepancy is small compared to the known limits of experimental data. Note that the plotted curves are the experimental values of the wave-resistance coefficients estimated using the longitudinal-cut method. The wave-resistance coefficients plotted in figure 9 are results of the second test case where the vertical control surfaces are inserted at y/l = +0.07. Remem- bering that the half-breadth of the hull is 0.05L it is obvious that the control surfaces are very close to the hull surface. This configuration is presented because it gives satisfactory results even though the matching sur- face is closer than what would be a reasonable distance were linear waves assumed. The wave resistance coeffi- cients are in good agreement with the results obtained in the first test case. The effect of including sinkage and trim is obvious at the higher Froude numbers. The resistance obtained from both pressure integration over the hull surface and over the control surface is plotted. Some differences are observed between the two methods which can be explained by errors when truncating con- trol surfaces or local effects close to the body which are eliminated in the far field. case is tested in the present paper and the main param- eters are given in table .2. Experimental data, from Kim and Jenkins (1981), [16i, are plotted together with nu- merical results in figure 12. Mewis and Heinke (1984), t17], have published almost the same results, which con- firm the difference between the ship fixed and free to un- dergo sink and trim. The experimental wave-resistance coefficients are obtained by using the longitudinal-cut method. Two numerical test cases are considered where the difference is basically the number of panels used. The position of the vertical control surfaces and the upstream and downstream truncation boundaries are kept con- stant. The computer time for solving the equation sys- ~ O i- The sin kage, nondimensionalized with respect to U2/29 Froud8 no.: 0.2SC and trim, nondimensionalized with respect to the length are plotted in figure 10 and 11. The Series 60 hull (CB = 0.60) The Serie 60 is a collection of ship forms with block coefficients varying from 0.60 to 0.80. The C,~ = 0.60 of x _ 2.5_ - ~ 2.0- MU ,,, 1.S- - L~ a: ~ .0 0.5 - O FREE 1400" 3400" O FREE IdOO~ · nXED ~ FREE NOEL ~ Kl~ AND _ FIXED hdOOEL. ~ ~~ (1~1) 1 10< . . ~ } C7E~ 1 _ ~ } CABE 2 ~ ] ~ ' it. O 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36 FRO=E NUMBER Figure 12: Wave-resistance coefficients calculated by the present method and experimental results from Kim and Jenkins, 116~. . . . .. u~'no' - L ax - ~~m x~ froud8 no.: 0.320 Figure 13: Wave elevation along the hull for Serie 60 ship. 663

tem in test case number 2 is about four times the time in case 1. The values plotted by filled markers in figure 12 include the effects of sinkage and trim. The numerical results are in good agreement with experimental data, and the convergence of the method is reasonable. Fig- ure 13 shows the wave elevation along the hull plotted for three different Froude numbers. The main observa- tion is that the calculations underpredict the wave crests and troughs, while the wave length is well predicted. Conclusions The present method has proved to be successful in solving the linear wave-making problem. Results for the single point source show that the combination of an inner and outer domain is working well and that the prediction of the wave elevation is good. Numerical tests have been carried out in order to verify the computer code. In specific the Kelvin Green's function, the numerical differential operators and the panel sizes have been tested. Results such as wave-making resistance, wave eleva- tion, sinkage and trim are generally in good agreement [6] with experimental data. Only limited number of ship forms have been tested in the present work. Further development of the method should include a number of additional test cases. The high Froude number range (F., > 0.5 ~ is especially im- portant. The knowledge thereby obtained, will be useful in refinements of the numerical method. The theory has many possibilities for further develop- ment. By means of moderate changes in the program, both the velocity field in the fluid and the streamlines and equipotential lines on the body can be calculated. Further, the method should be well suited for ships in channels or when wall effects is investigated. References [1] MICHELL, J. H. 1898. The Wave Resistance of a Ship. Phil. May. The Collected Mathematical Works of J. PI. and A. G. M. Michell, Noordhoff, Groningen, 1964, pp. 124-141. [2] HAVELOCK, T. H. 1923. Studies in Wave Resis- tance The Collected Papers of Sir Thomas Have- loc1: on Hydrodynamics, published by the Office of Naval Research, Washington D.C. 1966, pp. 30-38. t3] WEHAUSEN, J. V. 1973. The Wave Resistance of Ships Advances in Applied Mechanics No. 13, pp 93-245. [4] GADD, G. E. 1975. A Method of Computing the Flow and Surface Wave Pattern Around Full Forms. RINA, pp. 377-392. [5] DAWSON, C. W. 1977. A Practical Computer Method for Solving Ship-Wave Problems. Second International Conference on l!lumerical Ship Hy- drodynamic~, pp. 30-38. NEWMAN, J. N. 1987. Evaluation of the Wave- Resistance Green Function: Part 1 - The Double Integral. Journal of Ship Research Vol. 31, No. 2, June, pp 79-90. AANESLAND, V. 1986. A Theoretical and Numeri- cal Study of Ship Wave Resistance. Dr. ing. thesis, Report UR-86-48. Norwegian University of Tech- nology, Division of Marine lIydrodynamics. [8] AANESLAND, V. 1988. Hydrodynamic Properties of Floating Bodies Calculated by Supercomputer. CADMO 88. Southampton, 20-22 September. t9] NAKATAKE, K. 1966. On the Wave Pattern Cre- ated by Singular Points of Ship Wave Resistance. Jo urnal of the Society of Naval Architects of West Japan. No. 31 pp 1-19. The present method can be extended to include non- [10] SCLAVOUNOS, P. D., NAKOS D. E. 1988. Sta- 1'near free-surface conditions In the Inner domain. The .. . ' a,. Zloty Analysis of Panel Methods for Free-Surface matching boundaries have to be situated at a scent Flows with Forward Speed. 17th Symposium on distance from the hull so that the waves In the outer Naval Hydrodynamics The Hague po.29-48 domain can be treated by linear theory. v , Acknowledgements The work has been carried out as a dr. ing. study at the Norwegian Institute of Technology, and I wish to express my sincere thanks to my advisor Prof. Odd M. Faltinsen. The study was financed by a scholarship from the Norwegian Institute of Technology and the computer costs paid for by the Norwegian Council for Scientific and Industrial Research (NTNF). t11] RAVEN, H. C., 1988. Variations on Theme by Daw- son. 17. th Symposium on Naval Hydrodynamics The Hague, pp.9-28. 664 [12] PROCEEDINGS of The Workshop on Ship Wave- Resista nce Computations. David Taylor Naval Ship Research and Development Center, 13.-14. November 1979. [13] PROCEEDINGS of The Second Workshop on Ship Wave-Resistance Computations. David Tay- lor Naval Ship Research and Development Cen- ter, 16.-17. November 1983.

t14] FEI, X. 1984. Calculation of Potential Flow with a Free Surface. Report no. 65 ISSN 009-112X, Chalmers University of Technology, Department where of Ship Hydrodynamics, Sweden. [15] NEWMAN, J. N. 1976. Linearized Wave Resistance Theory International Seminar on Wave Re~i~- tance. pp 33-43. [16] KIM, Y. H. and JENKINS, D. 1981. Trim and Sink- age Effects on Wave Resistance with Series 60, CB=0.60. David Taylor Naval Ship Research and Development Center. Report DTNSRDC/SPD- 1013-01. t17] MEWIS, F. and HEINKE, H. J. 1984. Untersuchun- gen der Umstromung eines Modells der "Serie 60" mit CB=0.60. Schiffbaufor~chung, 23 3/1984, pp. 148-154. FRY, X. and KIM, Y. H. 1984. Bow Flow Field of Surface Ships. Fifteenth Symposium on Naval Hydrodyn amics. Boundary condition on the vertical control sur- face Consider the vol u me bou nded by So , SFa nd S.=, i n figure 14. An arbitrary point is located on the surface So at the position where the boundary condition is wanted. The point is enclosed by a hemisphere Sc with a small radius. It is assumed that the two functions ~ and ~ together with their first and second derivatives, are finite and single valued in the closed volume. It is then possible to use Green's second identity // (¢ ~¢~ ~¢ )dS = i| j; (¢V2¢ - (V2o)dV (24) whereS=S~ +S~+S,x, The surface So is located at y = b, Sx on a seg- ment of a sphere at infinity and SF is the free surface. Replacing ~ with the Kelvin Green's function, G.;, and using ~ as the perturbation potential Figure 14: Outer solution 665 r + r' + No(r ) + W(r') (25) = T(X - ()2 + (Y - r/)2 + (Z _ ()'24-2 (26) _ r r' [(xa) + (y _ r1)2 + (z + ()2]-2 (27) NO(r') = | d~pv| dke'(~-° Or 0 0 cos[k(x - ) cos ] cost (y - ) sin ; Is - k cos2 ~ (28) W(r') = 4~c i; 2 dd seC2 cet;( +0 sect ~ (29) Sill[~;(X - () see b] cos[~(y - A) sec2 ~ sin if] and ~ = g/U: and see ~ = 1/ cos §. Both ~ and Gal satisfy the Laplace equation, V:2¢ = O and V2G,` = 0, and equation (24) becomes /J~(¢ On -Go;, )dS=0. (30) Separating the integral into three parts, the contri- bution from each surface can be investigated. Remem- bering that it is a limited part of So that is needed as a control surface, STI-, the following integrals are consid- ered. The surface at infinity I==ll (¢ i," -Go- )dS (31) When (x - if) ~ 0 the potential ~ and its derivative (3o/~3n are of O(1/r) and O(l/r2) respectively. Simi- larly, when (x() < 0, G.`- and dG,;/0n are of O(1/r) and O(1/r'). Consequently the contribution will vanish when r ~ Do. The free surface The boundary condition on the free surface is applied to both the potential and the Green's function. By lin- earizing equation (2) and (3) and combining them, the condition becomes o + o.~..~. = 0 on z = 0. (32) IF = i/ (I on G.; A,,, ids U // (~0 GO - Hi ~ ¢2)dS 9 its' ,9~(V ,'~ -G]-0 )dS U ~ (CAGE _ Gil ' jdy (33)

The surface integral is now reduced to a line integral along the intersection of the free surface and both the vertical control surface and the surface at infinity. The integration along C,, will vanish when r ~ oo by a similar argument as for the surface at infinity. The y- position of the vertical control surface is constant, which gives no contribution, and the total integral from the free surface vanishes. The vertical control surface By differentiating equations (26), (27), (29) and (30), dG/~/0n on the vertical control surface is zero when x 76 ~ or z 76 (. The interpretation is that only the source point creates a normal velocity to the surface. For all other points on the control surface, only a com- ponent in the plane itself exists. The contribution from the source itself is found by enclosing the source point by a small hemisphere with radius c. The concept of an inner and outer domain is espe- cially attractive since it makes use of the matching sur- faces as a part of the control surface, S.`- = So- + S.=, as described in figure 2. Er is the intersection between the free surface and the Sir. In the case where the ver- tical control surfaces are extended to infinity, only So will ~i'!e a contribution to the integral because Vet and vanish at infinity. Equation (36) then simplifies to RI\- = 2P J! ¢~.¢,7dS (37) Here continuity has been used, (.~.ts`-. 0,,dS = 0~. In the computer program only a restricted part, So,, of the vertical control surfaces So- is used. Consequently two transverse control surfaces, STIR and STII, have to be inserted in order to enclose the control volume. The corresponding part of Er is E~ and E+. Figure 15 describes the configuration. When z < 0 and r ~ O. ~Gu/0y - ~ (y - b)/r:3 Finally the equation becomes I1 = || ~ ''dS-G~` ¢'dS a, on dry /./s~ ~ all, dS- /is G~;,,¢'dS = li ti dS- G dS '' - 0-~.5'. ~ 071 ·//S~ [ion r r An = 2~¢- |/s Gl\ rods (34) Finely IX + IF + I1 = 0 which gives 27r.//s\ on y= bib (35) Wave resistance calculated from control-surface integration Wave resistance can be calculated by pressure in- tegration over the hull. An alternative procedure is to calculate the change of momentum in a closed volume. Newman t15] derived an alternative expression based on the fluid velocity and wave elevation at an arbitrary con- trol surface. Using the perturbation potential ¢, the formula is: P //.. ~2(¢' + a''+ 0~-)7~. - ¢).,o,'JdS 2P9 .~(, ~ Y (36) RI, = 2P J; ¢,.¢,,dS !/S,,.~+.S+.~12(~' + BY + '>~)n'¢I'J''l~dS (38) The normal vector n'. of the elements on ST! T and Sit 7 is st 1, and the disturbance potential and the wave elevation vanish rapidly upstream. Consequently the computational domain will have contributions only from the vertical control surfaces and the transverse control surface behind the ship. , \ 1 \ , s;, 1 So x Figure 15: Control surfaces for calculating wave resis- tance. 666