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Ship Wave-Resistance Computations G. Jensen Ingenieurkontor Lubeck, Germany V. Bertram and H. Soding Ibstitut Bur Schif~bau Hamburg, Germany Abstract A method for the numerical determination of the po- tential flow around a ship moving steadily at the free surface of an ideal fluid solves iteratively for the non- linear boundary condition at the free surface, the body boundary conditions and the equilibrium floating posi- tion. The method can also be applied to shallow water and hulls with transom stern. The radiation condi- tion and the open boundary condition are enforced by a special arrangement of the collocation points at the discretisation boundaries. This paper also describes a new simple and flexible panel method for satisfying the body boundary condition; the method could be used for other potential flow problems as well. 1. Symbols F B breadth of the hull D draught Do draught at rest frictional resistance coefficient wave resistance coefficient unit vector pointing in the direction of the tow- ing force pressure force on the body f panel area FA additional force on the body En = U2/ I, Froude number 9 acceleration of gravity G weight G = (O,O,G) G(p, ~ potential at point q due to unit source at p 593 n p p - q r R Ru S So s,t T T A U vn H water depth k point on tangential sphere lPp ship length between perpendiculars M source strength unit normal pointing into body point projection of a point on the body surface onto the tangential sphere pressure point ratio between the area of the projection and that of the original surface element radius of tangential sphere wave resistance wetted or panelized body surface wetted surface at rest vectors tangential to the body surface moment due to pressure on body additional moment on body free stream velocity velocity component normal to body surface vs. v' components of velocity tangential to body sur- face point on free surface x,y,z right-handed coordinate system; the x and y axes are on the undisturbed free surface, x points upstream, z vertically downward point of action of towing force center of gravity of ship's mass towing force xz XG z do - p submergence of dipole moment of dipole trim angle (bow down trim is positive) trim angle at rest velocity potential approximation of = ~ - Us correction of ~ density of water nondimensional sinkage at midship section; see (26)

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z F K nondimensional trim; see (26) z-coordinate of free surface approximation of suffices: 1,2,3 vector component in direction of the :~-, y- or z-ax~s resp. refers to point or source at the free surface refers to point n p q refers to point or source on the body surface component in direction of the normal n refers to point p refers to point q x, y, z partial derivatives 2. Introduction For the design of a ship hull and its power require- ment it is of great practical interest to know the flow and the resulting forces due to the steady motion at the surface of a calm ocean. Experimental techniques try to separate viscous from potential-flow forces to scale measured force coefficients from a model to the full- scale ship. Due to the difficulty of computing viscous flow forces for high Reynolds numbers, the same sepa- ration is used in numerical predictions as well. Serious efforts to compute the potential flow and the accompa- nied wave resistance of a ship started with the pioneer- ing work of Michell t1] in 1898. In spite of the great emphasis placed on this problem, numerical solutions are only now at the threshold of practical applicabil- ity. They neglect breaking waves, spray, bow vortices and other details of the flow, which may then be de- scribed by a potential satisfying Laplace's equation and boundary conditions at all fluid boundaries. The major difficulty in this problem is the nonlin- ear boundary condition at the unknown location of the free surface. To circumvent this difficulty, most known methods use, apart from the physical simplifications stated above, additional mathematical simplifications: Almost exclusively a linearized free-surface boundary condition is implied at the location of the undisturbed free surface, and in most cases the solution is described by a superposition of complicated singularities that meet this linearized free-surface condition identically. All linear methods show good agreement with mea- surements only for special hull forms or high Froude numbers En = U2/~, where U is ship speed, g acceleration of gravity, and Lpp the ship length be- tween perpendiculars. In 1978 Dawson t2] published a method using a dis- tribution of Rankine sources (potential = 1/distance) on the body surface and on a local part of the free surface around the body. The flow is imagined to be superimposed from the double-body flow, i.e. the flow which would result in case of a rigid free surface, and a correction A. The source strength distribution is de- termined from the body boundary condition and a so- called double-body linearization of the free-surface con- dition. This linearization neglects nonlinear terms in ~ and is applied at the plane, undisturbed water sur- face. The radiation condition which states that waves may occur only behind the ship is enforced by a one- sided finite-difference operator for the second deriva- tive of the potential in the direction of the double-body streamlines appearing in Dawson's free-surface bound- ary condition. This numerical method of satisfying the radiation condition has the disadvantage that the sur- face waves are damped a little and are slightly shorter than they should be. Several authors have tried to extend Dawson's method to an iterative solution of the exact, nonlin- ear problem, but it was only quite recently that Ni t3] and Jensen t4] succeeded. Both methods show good agreement with resistance force measurements for the few cases compared so far. Ni's method uses source-panels on the wetted part of the body surface and on a local part of the free surface as computed in the previous iteration step. In each step the body boundary condition and lin- earized dynamic and kinematic free-surface conditions are used to derive a system of linear equations for the unknown source strengths and surface elevations at control points. After solving this system the body is repanelized automatically up to the computed wa- terline. The radiation condition is enforced by means of a one-sided finite-difference operator as in Dawson's method. Our method, on the other hand, employs Rankine point sources in a layer above the water surface, and in each iteration step it uses the same panelization of the body and a mirror image of it above the water surface. A linearized free-surface boundary condition for the flow potential is used in each step of the itera- tion together with the body boundary condition. After solving the resulting system of linear equations for the source strengths, the shape of the free surface is deter- mined from the dynamic free-surface condition. The radiation condition and the open boundary condition are enforced by adding an extra row of control points at the upstream end of the free surface mesh, and an extra row of source points at the downstream end. The effectiveness of this simple method was shown already in [4,5~. For the body boundary condition a new, flexi- 594

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ble panel method with simple numerical integration of the influence function is used. 3. Problem We want to compute the stationary flow of an incom- pressible, irrotational fluid around a laterally symmet- ric body at or near the free surface of an otherwise unbounded fluid. Far upstream the fluid has the uni- form velocity U opposite to the direction of the x-axis of our Cartesian x, y, z coordinate system with z point- ing downward. ~ ~=~-~ - The velocity potential ~ meets Laplace's equation: /\~= 0 (1) at points below the free-surface height ~ and outside of the body and, in case of limited water depth, above the bottom. the dynamic free-surface condition; for z = ~ outside. the body: I OCR for page 593
4. Free surface boundary condition The unknown surface height ~ can be eliminated from (4) and (5~; at z = (: 2VV (vi)2 _ 9~5z = 0. (13) This free-surface condition is nonlinear; it is valid at the unknown location of the water surface. Therefore we iterate solutions with a linearized condition which assures that - if convergence is reached - the solution fulfills (13~. To derive this condition, let ~ and Z be approxi- mations for ~ and (. We substitute ~ = 4} + ~ in (13), neglect terms nonlinear in derivatives of A, and obtain at z = (: V4}V (2 (V~2 + V4>V~) (14) +V~V (2 (Vamp - 9(~z + Liz) = 0 4} and ~ are developed into a Taylor series about Z. The series is truncated after the linear term. If prod- ucts of ~Z with derivatives of ~ are neglected, (14) becomes V4iV (2 (V~2 + V4iV~) + V~V (2 (V45~2) (15) - gaff + Adz) + t2V~V (Vet _ gee] (( _ Z) = 0 at z = Z. The index z designates partial derivatives. To decrease the number of unknowns, ~ is substituted by expressions involving Z. 4~(Z) and ~(Z) only. To this end (4) is developed into a Taylor series as well and linearized: = 29 [(Vim - U2]z=` 2i [(v~2 + 2V~V~ - Up 21 [V~2 + 2V~V(p +2VIV4>z (( - Z) - U2] z . This gives (16) ~ [(v4~2 + 2V4?V,o - U2] - gZ (17) with ~ and ~ being evaluated at z = Z. Inserting (17) into (15) and substituting ~ by ~4>, we obtain the linearized free surface condition 596 V~V [- (Vet + VIVA] + -Vow (Vet _ 9Z + t2v~v ~vq>~2 - god] (18) 2 [- (V~2 + 2V~V~ - U2] - 9Z O 9 - V~V~z at z = Z outside of the body. The denominator in the last term is zero if the vertical component of the parti- cle acceleration VIVID is equal to the acceleration of gravity g; this is the stability limit of the approximate flow Hi. 5. Radiation and open boundary condition ~ is approximated as the sum of the potentials of the uniform stream, a regular mesh of point sources in a layer above the free surface and the potential of singularities on the body surface. The point sources are generally located vertically above the collocation points. Only at the upstream end of the grid there is an additional row of collocation points, and at the downstream end there is an additional row of point sources. To validate this simple scheme to enforce the radi- ation condition and the open boundary condition, the flow around a submerged dipole with Kelvin condition at the free surface was computed and compared to an- alytical solutions given by Nakatake A. Figs. 1 and 2 show the resulting free surface deformation from the analytical and the numerical solution for different sub- mergence speed parameters F = gt/U2. ~ is the sub- mergence of the dipole. The numbers at the contour lines indicate values of (go<3/~4,rU,u), where ,u is the moment of the dipole. The agreement is excellent. 6. Body boundary condition The Neumann condition (2) at the body surface is fulfilled using a new panel method [4]. The commonly used method of Hess and Smith [7] uses plane quadri- lateral panels with constant source strength and control points in the (suitably defined) centre of each panel. The integral over the derivatives of the source poten- tial can be evaluated analytically for each panel. This results in complicated expressions containing transcen- dental functions. Such expressions are expensive to evaluate numerically.

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lath ~ a 4~ `' ( -~ ~ \~\v~/} ~ ~~-~--` ~ \ it ~ ~ ~ \  ~ J I\ _ _ ~ \ \ _ Fig.1. Contour lines of surface elevation due to a submerged dipole for F = 0.5 The upper half shows the analytical solution due to Nakatake t6], the lower half shows the numerical solution I / ~ / ~ 5~ =' / ,- ~ / /; /, ~ ,,~:~411~ hi' A\;;' OCR for page 593
Here we show a possibility to evaluate these inte- grals with simple numerical integration by eliminating the singularity of the integrand at the control point on the panel: f (id) = ~ M(fi) G(`p, ~ dSp, (~19) s ~ is the potential at a point q induced by a source distribution M on the body surface S; G(p,~ is the potential of the unit source at p: G = - (dip-. The normal velocity on the body surface S is vat id) = no Vq(~) = (20) ( M(O no) VQG(p, A) dSp9 My) s If the normal velocity is prescribed by the boundary condition, the velocity on S in two different tangential directions t is to be determined only: vt = toy) Vq~) = ~ M( OCR for page 593
Table 1: Error depending on discretization N number of elements I N 11 1 1 4 1 16 1 64 1 256 1 1024 1 error 11 13.5 1 5 62 1 3 19 1 1 58 1 0 78 1 0 39 1 This shows that the method (like that of Hess&Smith) is of first order; the error decreases pro- portional to the mesh spacing. As in most panel methods a smooth body surface is required. Ships often have sharp stems or sterns. The body boundary condition will always be violated near such a corner. The panel method may still be applicable for practical purposes if the overall solution is not disturbed. Figure 3 validates this property by showing the computed deviation from the parallel flow for a Wigley double model with different panelizations. Fig.3. View of the forward lower half of a Wigley double model in direction of the Taxis. The arrows show the deviation from the parallel flow for 8, 32, 128 und 512 panels on one eighths of the hull surface and for a grid with 208 panels, which is the 128 panel grid with local refinement near the stem. I; 599

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7. Body at the free surface For determining the stationary potential flow about a body sailing steadily at a free surface of an ocean of finite depth, the methods described in the preceding sections are combined as follows. Source panels are used on the wetted starboard hull surface, and . . . on a part of the above-water surface up to a plane above the uppermost point of the assumed water- line at the hull. Further source panels of equal source strength as those below this plane are arranged on a mirror image of the hull surface above this plane. The reason for this is that, on one hand, our special panel method requires the source panels to con- stitute a closed surface, and, on the other hand, this mirror image reduces the necessary width of the grid of further sources arranged along the free water surface. Here it is sufficient to use point sources above the free water surface instead of source panels on the free surface. We use point sources located one grid spacing above the free surface. Further, all these sources are mirrored both at the symmetry plane of the ship and - in case of shallow water - at the bottom to satisfy the respective boundary conditions. The source strengths are determined from the following conditions: On the ship surface up to the plane somewhat above the water surface, the condition of vanish- ing flow velocity normal to the surface is satisfied. For the part of the hull surface above the water, this condition is applied for two reasons: On one hand, we do not know in advance where the ac- tual water surface is, and, on the other hand, the continuous source strength along the contin- uous ship surface and its mirror image above the horizontal plane helps to obtain a smooth poten- tial at the intersection between body and water- line. The same argument could and has been proposed for a continuation of the free-surface source panels into the body. However, because we use sources not at, but above the water sur- face, at the free surface the potential due to these sources is smooth anyway. . The transom stern is also covered by source pan- els. However, if the water flow is assumed to separate smoothly at the transom such that the transom face is not wetted, the usual condition of vanishing normal velocity is not applied on the transom. Instead, at the collocation points on the transom we assume a normal velocity through the transom of a size which satisfies the dynamic (pressure) condition at the boundary between the transom and the rest of the ship surface, assum- ing the normal velocity to be constant over the whole area of the panel. This results in a nor- mal velocity greater than the ship's speed U if the edge of the transom is below the undisturbed water surface, and a normal velocity smaller than U if the transom edge is above the undisturbed water surface. Only if the edge of the transom is at or above the stagnation height level, the condition of vanishing normal velocity is applied. Whether the transom is wetted or not, has to be guessed in advance. At the free water surface the linearized free sur- face condition (18) is imposed. These boundary conditions are satisfied at, typi- cally, 1000 collocation points on the body and on the free surface. When the source strengths have been de- termined, the free surface elevation is computed from the dynamic boundary condition. The collocation points on the free surface are moved to this height. Then the process is repeated to get an iterative solu- tion of the nonlinear free surface conditions. On the water surface, the collocation points form a regular grid outside of the body up to a distance which was determined by test calculations and which depends on the Froude number. Usually, point sources are located about one grid length above each colloca- tion point. However, if the distance between source and collocation point is smaller than about 3 grid lengths, instead of one collocation point a pattern of 4 points is used, and the average error of the boundary condition at those 4 points is considered. It may be questioned whether our numerical meth- ods applied to solve the potential-flow problem are cor- rect for derivatives up to 3rd order. Without theoret- ically investigating this matter, we simply found that the method usually converges to a solution in which the errors of the nonlinear free-surface conditions are indeed extremely small at all collocation points, usually below 1/lOOOth of the ship speed if they are expressed in form of a velocity. Due to the smoothness of the potential, errors are relatively small also between col- location points. To obtain such an accuracy, it was necessary to apply a normalization of the equations 600

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which led to approximately equal values on the main diagonal of the coefficient matrix. To decrease the cases of divergence, we found it helpful to determine the maximum error of the free- surface condition at the newly determined positions of the free surface both with the previous and with the current source strength, and to use intermediate source strengths if the maximum error was larger with the new than with the old source strengths. In the iteration to solve the nonlinear free-surface condition the pressure distribution is integrated on the actually wetted part of the hull to obtain the resulting forces and moments. Together with corrections for the pitch moment of the viscous resistance and the towing or propeller force, a correction of the equilibrium float- ing position is determined, and the panel grid of the body is shifted and rotated correspondingly before the next iteration step is performed. The linear system of equations obtained during each iteration step is solved by a combination of elimination steps with a Gaul3-Seidel iteration: At first, only those elements below the main diagonal the absolute value of which exceeds a certain limit are eliminated to im- prove the condition number of the matrix. If the fol- lowing Gaul3-Seidel iteration indicates no convergence, further incomplete eliminations using a smaller limit are performed. This method constitutes a completely safe and quick solution scheme. 8. Applications 8.1 Series 60 with CB = 0.6 in deep water The Series-60 hull shape with a block coefficient CB = 0.6 was chosen because extensive and careful experiments and resistance evaluations have been per- formed for this form by Ogiwara A. The principle relations of his model are: breadth B = 7P5; draught at rest Do = ~~P75, wetted hull area at rest So = 5 86~5. The hull surface was panelized up to a height 0.3125Do above the floatation line at rest. There were 453 panels used on one half of the body. As in Ogi- wara's experiments the horizontal towing force was applied at 0.485lPP from the aft perpendicular and 0.461Do under the floatation line at rest. The vis- cous resistance coefficient CF = Rf /~0.5pU2So) was es- timated to be 3.5-10-3 for all speeds. For the speed range investigated here it had only a small influence on the trim. The height of the centre of gravity which is not given by Ogiwara was assumed in the floatation line at rest. 5.0j 2.5~ 1O3cw 0 linear . non-linear measured .> ~ - o 0.25 0.35 Fig.4. Computed wave resistance coefficients for Series-60 with CB = 0.60 and values measured by Ogiwara t8] In Fig. 4 our computed wave resistance coefficients are compared to Ogiwara's measurement evaluations. For comparison also the results of the Neumann-Kelvin problem are included. The results were obtained with the same program (1. iteration step) and the same grids; trim and sinkage were suppressed and the still- water line used as an integration limit for the pressure integration. Especially for higher Froude numbers non- linear results agree much better with experiments. Figures 5 and 6 show the nondimensional sinkage and trim defined by 2 DoD Or = . ~ _ F=2 L ' F2 (A8) (26) n as functions of the Froude number For sinkage the agreement is good. The curve for the computed trim is similar to the corresponding measurements, but ap- pears to be shifted somewhat to aft trim. 0.2 0.25 0.3 0.0 . computed measured Fn Fig.5. Computed and measured t8] nondimensional sinkage ~ at midship section of a Series-60 model with CB = 0.60 601

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0.1~ o.o- . computed . / measured / Fn .,,,,,t,,,,,.,~r,,,,~ 0.25 _~ ~ 0.4 ~ ~ Fig.6. Computed and measured [9] nondimensional trim ~ for a Series-60 model with CB = 0.60 / Aim \: ~r: /:J': r:3J ~ / ~/~L ! ~ ~ / ~~/~-: / Ad/ V ////~- d' Fig. 7 shows the computed elevation of the free surface for the solution obtained with the linearized free-surface condition (Kelvin condition) U2~2 - 9z = 0 at z = 0 (27) and the solution with the exact nonlinear condition (13) for F7, = 0.25. Fig. 7. Contour lines of deformed water surface for a Series-60 models CB = 0.60' at F7, = 0.25. Left side: linear solution; right side: nonlinear solution. The vertical distance between contour lines is 10-3Lpp. 602

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Table 3: Influence of grid spacing at water surface 8.2 Containership in shallow water The following results apply to a containership the principal data of which are shown in the following table. Table 2: Principal data of investigated containership length between perpendiculars 161.44 breadth 28.40 draught at U=0 10.00 trim at U=0 0 block coefficient 0.68 designed load waterline coefficient 0.82 Dynamic sinkage and trim of ships on shallow water are of interest not only because of possible grounding. Forward trim usually encountered on shallow water can reduce yaw stability so severely that ships may loose their ability to keep course. Figure 8 shows results for the draught at the for- ward perpendicular divided by the stagnation height U2/2g depending on depth Froude number F72h and length Froude number An. The theoretically most in- teresting region around depth Froude numbers of 1 can be investigated only in case of relatively large length Froude numbers because otherwise the ship touches the ground. Limits of ground-touching are indicated in Figure 9. This is the reason for investigating also high length Froude numbers which are unrealistic for a containership. As can be seen, near to a depth Froude number of 1 large changes of the squat are experienced. This is known also from model experiments and small boats. To determine the accuracy of these results, model experiments are being performed but not yet finished. Instead, Figure 10 shows results of approximation for- mulae according to Barras t9] which were established on the basis of model experiments and measurements aboard ships, and results of the slender-body theory of Tuck t104 applied to our ship. The nearly exact coin- cidence with Tuck's formula is striking; however, near to the critical depth Froude number 1 this formula is not applicable. Possible errors of our method were investigated also on the basis of numerical experiments with different grids on the body and on the free surface. In the fol- lowing tables ~ denotes the difference in sinkage to the most accurate computation in To of sinkage at forward perpendicular FP. The results indicate that for the small length Froude number of 0.15 (6 m/s) our results are doubtful. U (mls) 6 6 6 10 10 H _ (m) _ 14 14 14 14 14 _ ., . ,grla spawn, 4.5m x 4.5m 5m x 5m em x 6m 6m x 6m 8m x 8m _ at AP -20% To To atFP | 5 To 16~o -3% m m m Table 4: Influence of grid size behind aft perpendicular m AP and sideways _ U (mls) 6 6 10 10 _ H (m) 14 14 14 14 grid size aft side -135m 93m -70m 70m -19lm 101m _ -135m 85m atAP | atFP | -1% 18~o 2~o -ho Table 5: Influence of height of source points over still water plane U (mls) 6 6 12 12 12 . H (m) 14 14 18 18 18 height 4m 6m 8m 10m _ 12m at AP | at FP 25% -14~o -2% 0% -3% 2% Table 6: Influence of distance limit for taking 4 collo- cation points instead of 1 U (m/s) 6 6 6 l H T (m) 14 14 14 distance limit 2.0xgrid spacing 3.3 x grid spacing 5.5xgrid spacing 1 atAP | atFP | -do 1 To -ho 1 0% The reason for this is held to be the fact that, due to limitations of the maximum number of grid points, the grid length is not small compared to the 23 m length of the transverse surface waves and the even shorter di- agonal waves generated at this small Froude number. Fortunately, in practice the squat is hardly relevant at such small speeds. On the other hand, for Froude num- bers of 0.25 and more, the table seems to indicate that errors due to the discretization are small. 603 Computing times are about 10 minutes for each it- eration step on a VAX 8550 in case of about 1000 collo- cation points. The necessary number of iteration steps to obtain an accuracy of 1 cm for the squat ranges typ- ically from 2 to 5, depending mainly on depth Froude number.

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1.0- _ 0.5~ O - sinkage 0.15 0.2 U2/29 ~ /. ' \ i'; i, Fn=0.15 l ~ ~ O '0.25X a" 0.2 o , ~ 0.3 ~ ~ . ~ . ~ 0.25 0 3 1 I \ 0,.4 \ . \ ,, F7lh 0.5 \1.0-i`9 Fig. 8. Nondimensional sinkage at FP Dotted lines are limits of ground-touching em 4m 2m _ \\ . ._ ~.~_ _ . I- O lOkn 20kn U Fig.9. Distance between ship bottom and sea bottom at FP depending on speed for 3 water depths 10] 0.5~ O - sinkage U2/2g Barras Tuck / 0 15 ~ -0 ~ 0.,S / ,~ / / / Fig. 10. Nondimensional sinkage at FP Fnh l according to Barras [9] and Tuck [10] 604 8.3 SWATH ship For an research SWATH ship of the German navy we used our method for wave resistance prediction. We estimated frictional and viscous pressure resistance as in Salvesen et. al. [11]. The rather ununsual shape of the cross section caused some difficulties in the non- linear computation. Therefore, in each iteration step the free-surface collocation points were not only shifted in height but also in horizontal direction according to the current waterline. For a demihull this modifica- tion ensured rapid convergence. The error in the free surface was reduced by a factor of 10 in each iteration step. Each demihull induces a slightly oblique flow at the other demihull. Due to limitations in time, we did not incorporate a Kutta condition at the rear of each strut which would be necessary to take this effect prop- erly into account. We felt justified in this decision by the results of Bai et al. t12] who found for another SWATH ship that including a Kutta condition had no significant effect on the wave resistance. Bai et al. re- ported only some differences in the local velocity field near the trailing edges of the struts. Figure 11 shows a comparison of computational results with experiments of the Hamburg Towing Tank HSVA. Unlike the com- putational model the experimental model was equipped with rudders, fins and a propeller guard. 1 O3CT O O ~ 5 ~ o 0 ~ Ago ~ 0 linear non-linear measured Fn , ~ 0.1 0.2 0.3 0.4 0.5 0.6 Fig. 11. Computed total resistance coefficients for SWATH ship and values measured by HSVA Only in a few cases a nonlinear solution succeeded. The agreement with experiments is worse than for conventional ships. For the medium Froude-number range we noticed a considerable phase shift between the waves on the inside and the outside of each demi- hull before the computation breaks down. The point with the highest error in the free-surface condition and also the highest vertical acceleration was at the end of the strut. This seems to indicate that cross-flow effects afterall might be important for nonlinear solu- tions despite Bai's et al. findings. For high Froude numbers the computation breaks down at a point be- hind the SWATH ship at the plane of symmetry. Two wave crests starting from the trailing edges of the struts superimpose resulting in a splash. This phenomenon

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can also be observed in reality. This violates one of the fundamental assumptions of our method. More research will be necessary before breaking waves can be included. Similar agreement with experiments was found for linkage, t13~. Trim was suppressed both in computations and model tests. 9. Final remarks For many practical hull forms the present method can be used to compute the potential flow with nonlin- ear free-surface conditions. The wave resistance is pre- dicted quite well, although we do not yet achieve the accuracy of experiments. More comparisons to mea- surements are required to gain experience. The squat of the container ship seems to have been determined accurately by our panel method for Froude numbers above 0.20 in case of depth Froude numbers below 0.9 and perhaps also for critical and over-critical depth Froude numbers. However, the extremely simple slender-body formula of Tuck gives the same results for sub-critical depth Froude number, incuding small length Froude numbers. 10. References MICHELL, J. H.: "The Wave Resistance of a Ship", Phil.Mag. Vol 45, 1898. t2] DAWSON, C. W.: "A Practical Computer Method for Solving Ship-Wave Problems", Sec- ond International Conference on Numerical Ship Hydrodynamics, University of California, Berke- ley 1977. [34 NI, S.-Y.: "Higher Order Panel Methods for Po- tential Flows with Linear or Nonlinear Free Sur- face Boundary Conditions", Chalmers University of Technology, Goteborg, 1987. - ~~ - ~ '#~ ~ ~ [5] [6] [7] [8] [9] [4] JENSEN, G.: "Berechnung der stationaren Po- tentialstromung urn ein Schiff unter Berucksich- tigung der nichtlinearen Randbedingung an der freien Wasseroberflache", Institut fur SchifEbau Hamburg, Report No. 484, Juli 1988. JENSEN, G., MI, Z.-X., SODING, H.: "Rank- ine Source Methods for Numerical Solutions of the Steady Wave Resistance Problem", Sixteenth Symposium on Naval Hydrodynamics, University of Califorinia, Berkeley, 1986. NAKATAKE, K.: "On the Wave Pattern Created by Singular Points", Journal of Seibu Zosen Kai West Japan, No. 31, 1966 HESS, J. L., SMITH, A. M. O.: "Calculation of Non-Lifting Potential Flow about Arbitrary Three-Dimensional Bodies", Douglas Aircraft Di- vision Report No. E.S.40622, 1962. OGIWARA, S.: "Tank Experiments and Numer- ical Works on Series 60 Model in IHI Ship Model Basin", Report to the Cooperative Experiment Program of 18th ITTC, Kobe, Japan, October 1987. BARRAS, C.B.: "A Unified Approach to Squat Calculations for Ships", Bulletin of the PIANC 1, No. 32, 1979 [10] TUCK, E.O.: "Shallow-water Flow past Slender Bodies", J. Fluid Mech. 26, Part 1, 1966 t11] SALVESEN, N., von KERCZEK, C.H., SCRAGG, C.A., CRESSY, C.P., MEINHOLD, M.J.: "Hydro-Numeric Design of SWATH- Ships", SNAME Trans., Vol. 93, 1985 t12] BAI, K.J., KIM, J.W., KIM, J.W.: "The Cross Flow Effect on the Force and Moment acting on a SWATH Ship" Seventeenth Symposium on Naval Hydrodynamics, The Hague, 1988 (13; BERTRAM, V.: "Numerische Widerstandspro- gnose fur SWATH-Schiffe", Schiffstechnik Vol. 35, No. 3, 1988 605

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DISCUSSION by A. Musker Have you tried the non-linear calculation of Fig.4 with the ship f ixed and compared with the data compiled by ITTC on the experiment of Kim and Jenkins? The novel treatment of the radiation condition deserves f urther study to see how it behaves with high resolution surface grids. Author's Reply We did not try the non-linear calculation with fixed trim and sinkage for the Series 60. Fig.4 shows the result for the first and the final step of the interaction for the same computation. We believe that our treatment of the radiation condition will also work in the limit of very small grid spacing. The derivative are taken analytically and the free surface boundary condition is symmetric. What is needed to get the desired solution is a numerical stimulation for an asymmetric solution. If the source distribution is in a layer above the free surface, as in our method, the vertical distance has to be decreased with the mesh size. We performed trial computations with 50 points per wave length for the 2-d case and did not have any problems. 606