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Numerical Simulation of Viscous Flow around Practical Hull Form A. Masuko and S. Ogiwara Ishikawajima-Harima Heavy Industries Co. Yokohama, Japan Abstract This paper deals with numerical simulations of viscous flow around ships having practical hull form under propeller operating condition. The Reynolds-averaged Navier-Stokes equations for three-dimensional flow are discretized by finite-difference approximation and solved with SIMPLE algorithm. The k- ~ turbulence model and the standard wall-function are adopted. A propeller is simulated by giving pressure jump at its position. In order to eliminate skew grid around practical hull form, which violates the computational result, adjustment of grid angle is applied to the grid generated by solving the elliptic partial differential equation. Computational examples for the cargo ship Series-60 model and the practical tanker models under propeller operating condition are presented and compared with experiments. 1. Introduction In the ship building industries, computational fluid dynamics ( CFD ) technique is being int roduced to design and develop the ship hull form under the concept of "numerical tank". Although quantitative accuracy of the prediction is not sufficientat at the present stage, qualitative prediction of flow field can be applied to evaluate the propulsive performance and to design a new hull form. It is considered to become a useful design tool from the point of efficiency and cost for the development. The authors have been developing the numerical code for calculation of viscous flow' around ship hull aiming at the practical use in ship design. The preliminary studies using mathematical ship models2 showed that the method has a sufficient robustness of convergence for iteration number and the variation of computational grid. The comparisons between calculation and experiment demonst rated that the method is effective to predict the wake distribution, viscous pressure drag and propeller effects on them, although some discrepancies, for instance in strength of bilge vortices, are found in detail. In this paper, this method is applied to simulation of the flow around the practical hull form with complicated shape. Computational grid becomes often extremely skew for such form and calculation by the author's method diverges due to skew grid. In order to stabilize the calculation, the computational grid generated by solving the partial differential equations is modified to eliminate the skew grid. The above flow calculation method is applied to the cargo ship Series-00 model and two types of tanker model, one is the ordinary type and the other has the IHI B.O. ( Bulbous Open ) stern4. The calculated wake and p ressure are compared with the experiments and propeller/hull interaction is numerically investigated. 2. Calculation method 2. ~ Basic equation The governing time-averaged equations for three-dimensional turbulent flows in Cartesian coo rdinates are : Pa (pZtj ) = 0 ( ~ ) axe (Pa; hi ) = - Bali ~ All t,4 ( Hi ~ AL hi (2) where u i is the velocity component, p is the fluid density and p is the pressure. ,tl~ is the effective turbulent viscosity and given by: ~ = JO ~ at,- = ~ ~ Cow where ~ is the laminar viscosity, ,u~ is the turbulent viscosity, CD is the constant, k is the turbulent kinetic energy and £ iS the dissipation rate of k. In the k- ~ model of turbulences, k and ate governed by the following equations: (3) a3-(P~ik) = a mak ~ pi pa aa (PORE) = 211 Alp ~ axj Car P; -k - C2 p ok ( ~ ~

where Pk is the production rate of k and given by: pa = pt Maui ( ~Xj ~ Wi ) Standard values for the constants in Eqs.(4) and (5) are as follows6: C., = 0. Og C] = 1.44 C2 = ~ . g2 c;r~ = 1 . O ~e = ~ .3 Eqs.(1), (2), (4) and (5) are represented in the following general form: pa (paid) = sax ( P~ flex ) ~ S' (8) where ~ is a general dependent variate le. =( l,u i ,k, c). When a general curvilinear coordinate system ($ I, $2, $3 )=($, A, () is introduced, Eq. (8) is transformed to the following equation: As `j(~i~) = Jar: ( P' A;~; ) ~ S' ' AP = ,~a'~j ( ~ Ant ) ~ S¢* as) where J is the Jacobian of transformation, So' is the transformed source term corresponding to Eq. (8), and S. * is the modified source term including So' and the cross derivatives in the diffusion terms. G I is the contravariant velocity components without the metric normalization, and defined as follows: Gi = a, ju; Ai j and ai j are the metric coefficients for transformation: a;%, = 9~ J DXJ Production term Pk in general curvilinear coordinate is given by: p ~ a, Du; ( ~ ale ~ a, ,'~uj ) (~3' Table 1 shows a, F. and S. * in Eq. (9). 2.2 Finite difference equation Variab les u i and p are set at staggered location to avoid spurious error7. Integrating Eq. (9) over a control volume (Fig. 1), the finite difference equations are given as fol lows: pop !t,];v ~ ~t,2 ~~ ~ ~ ~G2 Gas n ~~] ~3 ~ [pG3~J6~2 ~~ A:. and e ~~- ~~- ~ [~' Aims n at. At~ ~ [~ J43~;d~3 ~ At, J5~2 ~ By ~ }fit,] ~2 ~3 (6) Further discretization by the Hybrid schemer leads to the to flowing algebraic equation : AD ~ =A,' ~ BAD ~ CAN ~N HAS As CAT ~T +~ MOB USE (I, . . . ,~, . . ., etc. ) (15) where Su denotes a source term which involves the diagonal terms (~E, etc.) and off-diagonal terms (AN E ,etc.). Coefficients AE ~ AP are as follows : AE = [0, D20 ~0.5C10 ~ ~C10 Ad = [0, Did +0.5C~, Cede A.~, = Al = As = = and ~ 0 , D2 n ~ 0. 5C2 I7 ~ ~C? {0, D2S +0.5C25 ~ C25 ~ [0, D~-0.5C3e, -Calf [0, Dad +0.5C3~, C36] As ~ An ~ AN ~ As ~ Ar ~ ~ - Sp (16) _ Al ~ Ajar ^83 _ r' Jell [, 23 C2 = pG2~~dt3 (~7) JAI2 Aids At] at2 - JAIL A sign [,, ] in Eq.(16) means the maximum ( ~ O.) value in [ ]. Su and Sp are given by the following equation: SO ~ S.' UP = JS, ~ d(' d(~ ~3 (18) (11 ) The SIMPLE algorithms links the pressure to the velocity through the pressure- '2) correction equation. When pressure p is supposed to be summation of previous value p* and correction p', the pressure-correction equation is obtained from the continuity equation and momentum equations. The discretized pressure-correction equation has the same form of Eq.(15). In the present calculation, off-diagonal terms in Su of pressure-correction equation are assumed to be small compared with diagonal terms and neglected for saving computing time. In consequence, the sequence of solving the governing equations is as follows: (1) Solve the momentum equations with the initial or previous pressure field to obtain the intermediate velocity components u j * . (2) Calculate the intermediate contravariant ve] ocity components G j with u i * . (3) Solve the pressure-correction equation using G j (4) Obtain the new values of G j and u i which ( 14 ) satisfy the continuity equation using pressure 212

correction. (5) Correct the pressure by the pressure correction. (6) Solve the governing equations for k and c. (7) Iterate steps (1 ) to (6) until the solution converges. As a solving algorithm of algebraic equation, a "checker-board" method with SUR (Successive Under Relaxation method) ~ ~ is employed to vectorize the calculation on a supercomputer. 2.3 Computational domain In this paper, a body fixed Cartesian coordinates are adopted whose origin is settled at the bow on the still waterplane, x- axis in positive direction of uniform flow and z-axis downwards. In the body fitted coordinate system ( $, r7, ~ ), constant- ~ planes are chosen as correspond to constant-x planes, 71-axis in radial direction from hull surface and (-axis in girth direction. Calculation is carried out in the domain surrounded by the following boundaries ( see Fig.2 ): Hull Surface Center plane Water plane Upstream boundary Downst ream boundary Outer boundary : x= 0.0 ~ L .Y= Ye z= 0.0 ~ d : y= 0.0 : z= 0.0 : x=-0.5L : x= 2.0L : r= 0.5L where L, d and ye denote ship length, draft and half b readth of a ship respectively. 2.4 Boundary conditions As for flow field around a fixed model in the uniform flow U. the boundary conditions are given as follows under neglect of free surface disturbance : On hu l l su rface : a, v, a, k, ~ = 0 On center plane: flu law ak v = 0, and and An, On water plane: flu 3D ak w = 0, an' an' fin' At infinity: = 0 = 0 ~ = U. v, a, p, k, ~ = 0 where a/an is normal derivative to th e boundary su rface. When the finite difference equation is solved in the computational domain shown in Fig.2, the boundary values subscribed with p are determined by the following manners considering the boundary conditions. On the hull surface: up =0, UP =0, ~P =0, ~P=O, EP=0 On the center plane: UP=~N ~ vP=0, w~=m~ kin Ok, eP =~ On the water plane: UP =ur, vP =~T, wP =0, kP =~T, EP =EN On the upst ream boundary : uP =U, P.- =0, On the downstream boundary: uP=O, wP=O, icP=O, EP=0 ~,~ =U`, VP =m, lo =W~, PP =PU . UP =iC~, EP =ES} On the outer boundary: (20) UP=U, VP=O, WP=O, ps=O, , iCP=O, EP=0 Since the standard k- £ model cannot be applied in the viscous sublayer and transition layer around the hull, the standard wall-functions6 are adopted. In the present calculation, the effective exchange coefficient rat is modified at the wall boundary so as to make the velocity profile fit to that from the log-law. Therefore the grid spaces in r'-direction adjacent to the hull surface have to be set to satisfy the following criterion. 20 ~ ye = d7Jm i r; pU+/p < ]00 (21) where IBM in denotes the minimum spacing of ~ direction ( distance from the hull surface to the nearest grid point ) and u+ means the frictional velocity. 2.5 Propeller model In order to simulate the propeller effect, the pressure jump model is employed in which a propeller is replaced by a accelerating disc'. The pressure jump is assumed uniform in the disc and its value is derived from the measured thrust of the self-propulsion test. Fig.3 shows the grid configuration at the section of the propeller. The grid is not Age fitted to the propeller disc and the uniform pressure jump is applied at the grid points indicated by circles. Fig.4 shows the pressure distribution calculated by this method for a propeller which is operating in open water. The calculated pressure connects smoothly with the given pressure jump at the propeller position. The velocity at the propeller position and far behind the propeller coincide with the values given by the momentum theory. 213

3. Computational grid for practical hull form In the present calculation, the computational grids are generated by solving the elliptic partial differential equations, however these grids do not always suitable for practical hull form. In general, computational grid for viscous flow calculation requires some characteristics such as orthogonality, smoothness, adequate concentration, configuration like streamlines and so on. Above method does not guarantee orthogonality condition of the grid and often provides the extremely skew grid. The present flow calculation method has a characteristic to be sensitive to the skew of the grid and the convergence of the calculation is violated by highly skew grid. Fig. 5 shows an examp le of the grid configuration including skew grids and the results of flow calculation which is violated by skew grid. The grid in Fig.5 is generated by Kodama's methods, which satisfy the above characteristics necessary for computational grid by geometrical manner. This is one of the sophisticated method to generate the grid for arbitrary hull form and the flow calculation method proposed by Kodama gives satisfactory results using these grids. However there are some skew grids near stern region because the grid line is chosen to correspond to the end profile and when these grids are applied to the flow calculation by the present method, the solution diverges. The computational results shown in Fig.5 is the velocity vector near hull surface just before calculation breaks down. It is found that the calcu ration is getting to be vio lated around the skew grid. The reason why this breakdown occurs is considered that the all off-diagonal terms are treated as the source term and they are ignored in the pressure-correction equation. Although it is possible to treat these terms more precisely, it brings enormous increase of computational time. Therefore, from the practical point of view, it is much convenient to generate the computational grids which restrain the breakdown of calculation. In order to restrain this breakdown, computational grid generated by Thompson's method is modified so as to improve the shape of the grid in the transverse plane. For convenience of expression of computational results, constant- ~ stations are chosen to correspond to the transverse section. 3.1 Grid points on the hull surface The coordinates of grid points on the hull surface are given by interpolation technique from the offset data. The method of a circular arc approximations is adopted for the interpolation. The procedure of the interpolation is as follows ( see Fig.6 ). First, an angle of tangent of each data point ( P i ) is determined as follows. Using four data points near the point P i, make three circular arcs ( arc Pi-2P- Pi, arc PA PiP:+, and arc P i P j + ~ P i + 2 ). Each arc has an ang le of tangent at point Pi( (,), (5i)2 and (hi) ), and then a mean of these three angles is taken for an angle of tangent of point P I. Next make a triangle PiPi+i P' using a segment PiPi+~, and tangents at Pi and Pi ~ i, where P' is a cross point of two tangents. An interpolated point Q is taken at a inner center of this triangle. By treating an interpolated point as a new data point, these process are repeated until a length of the segment becomes sufficiently short. These process are carried out along the waterlines and frame lines. The coordinates of grid points on the hull surface are determined by choosing a point from these sequential points. 3.2 Generation of grid The coordinates of grid points in the computational domain are generated by Thompson's method. They are solutions of the following Poisson equation: Up, (22) Exchanging the independent variables and the dep endent variate les in Eq. (22), fo 1 lowing partial differential equation is obtained: hi, ~~ '~: ~ J~Pj 3~' = 0 (23) It is not necessary to solve the equation for x, in Eq.(23), because the constant-$ stations are chosen to correspond to the transverse sections. This method has some problems when generating grid for arbitrary hull form. Since this method does not guarantee the orthogonality of the grid, extremely skew grid is often generated depending on the hull form. When sufficient number of grid points cannot be taken for the limitation of computer storage, grid lines often break into the ship hull. Fig.7(a) shows a typical example of such case. This is the grid at the section of bulbous bow which has extremely convex configuration. At the side of bulb, high] y skew grids are found and some grid lines break into the bulb. Since the stability of the solution by the present flow calculation method is very sensitive to the skew of the grid, a method of grid modification which adjusts the angle between grid lines is adopted. Fig.8 shows the way of this adjustment. The two segments on constant- ~ lines which close to the hull surface are adjusted normal to the constant-a lines. For the other segments on constant- ~ lines, the direction is adjusted to the angle between 45 deg. and 135 deg. When this adjustment is carried out directory, the change of the grid direction is too large and the computational grid breaks down. So the above adjustment is carried out iteratively using relaxation method. After obtaining constant-r' coo rdinates, smoothing of constant- r~ line is carried out by Lagrange 214

interpolation. At the same time, the minimum spacing of the grid Ar7m in iS set constant and determined using Of of the Prandtl-Schlichting's formula at the aft end of equivalent flat plate as follows: 30~ 30~ Hum; rat pus ~/~ p T~ = pU2 Of 2 PU2O.455(!Og~N )~2 SS (1 4~10gR2 ~ (25) Fig.7(b) shows the modified result of Fig.7(a) by this method. Fig.9 and Fig.10 show the computational grids for tanker forms with normal stern and IHI B.O. stern respectively. In every case, orthogonality condition is almost satisfied. 4. Convergence property Convergence histories of pressure at three monitoring points, midship, after perpendicular (A.P.) and in the wake, close to the keel line, are shown in Fig.11 in the case of Series-60 model (Cb=0.6) at the Reynolds number based on ship length Rn=9.22x106. The number of grid points is 94x25x21 and the iterative calculation is repeated 300 times. The calculation seems to converge at about 100 times. Fig. 12 shows the convergence histories of variate les u, w, p, k and ~ near the keel line at A.P. The values are non-dimensionalized by the differences between maximum and minimum values of each variable. Convergence of variables excepting pressure is considerably slow. This may be attributed to the lack of grid smoothness. Fig. 13 shows the variation of mass- imbalance with the number of iteration. SSUM means a sum of mass-imbalance for all the grid points normalized by inlet mass flow rate. SSUM decreases rapidly with the number of iteration and this verifies that the equation of continuity is satisfied. From the above results, 200 iteration steps is chosen for practical use. It takes about 20 minutes of CPU time to calculate 200 steps using the supercomputer FACOM VP-50. In the following calculations, the iteration is stopped at 200 steps in every case. 5. Computational results and discussions 5.1 Wigley model The Wigley hull is defined by the following parabolic equation: ye = BL1 ~ ( r2X_ 1 )2 ~ [] ~ ( d A] (26) where B is the ship breadth. Calculations are carried out using 93x25x19 grid with L=~.Om, B=0.6m, d=0.375m, U=0.85m/s and RN =4. 5X1 0G, and compa red with two kinds of experimental data. One is obtained in the towing tank of Ishikawaj ima-Harima Heavy Industries Co., Ltd. (IHI) with em length model, where resistance and hull surface pressure are measured at RN=4.2X1O69 and the other data is from the wind tunnel test by Sarda' ~ using a double model at RN=4.5X1O6. Fig. 14 shows the iso-wake contours at the (24 ) aft end station compared with Sarda's results. The calculated result shows qualitative agreement with the experimental results, however, it is a little diffusive due to the numerical diffusion of the Hybrid scheme. Fig.15 shows the comparison of hull surface pressure distributions, and reveals good agreement between the calculation and experiment. In this figure, inviscid results obtained by Hess-Smith method are also shown. The results of the present method agree with the inviscid results in the fore part and the displacement effect of boundary layer is simulated near the aft end. Fig. 16 shows the comparison of the local skin-friction coefficient Cf. Calculated skin- friction is given by the following equation: ply /2 _ 1 ~cqC~ i ~ ~ kit 2 (27) Up /2 l ~t( E,oC3 ~ 4 k~ ~ d7?m; n /~) where K iS Karman constant and E is constant The figure shows good agreement between the calculation and experiment except slight discrepancy at the station x=5.9m close to the aft end. Agreement of the turbulent kinetic energy between calculation and experiment is poor as shown in Fig.17. Calculation does not simulate the sharp peek of the turbulent kinetic energy in the region close to the hull near the after end. As the same tendency appears in Sarda's calculation which also adopts k- E model, this seems to be due to the defect of the turbulence model. The pressure resistance and the frictional resistance are calculated by integrating the hull surface pressure and local skin-friction respectively on the hull surface. The calculated results are as follows compared with the experimental values given by th ree dimensional analysis of the resistance tests. Calculation r Total resist. (rT) Fric. resist. (rF) Press. resist. (rp ) _ ( RN =4.2X1O6 EN =0.1043 ) 12.6xlO- 3 10.8xlO- 3 1.8xlO-3 215 Experiment Total resist. (rT) Fric. resist. (rF0) Residual resist. (rig) 13.2xlO- 3 12.7xlO- 3 (Schoenherr) 0.5xl 0- 3

where r=R/,oU2 V2 ~ 3, R is the resistance and is the disp lacement of the model. Calculated value of the p ressure resistance is larger than residual resistance which does not include wave resistance because of very low Froude number. This may result from the fact that the calculated pressure of after part is a little lower than measured one as shown in Fig.15 observing in detail. However the order of the total resistance comparatively agrees with experiment. 5.2 Series-60 model (Cb=0.6) The first application to a practical hull form is made for prediction of flow field around Series-60 model (Cb=0.6) under propeller operation. The calculation is carried out with L=7.0m and U=1.5m/s, corresponding Reynolds number is RN=9.22xlO6. In the calculation of propeller operating condition, pressure jump Ap=412.47N/m2 which is equivalent to the measured thrust T=22.119N ( propeller diameter is .2613 m ) is set on the propeller disc. Total number of grid points is 94x25x21. Fig.18 shows the hull surface pressure distribution, where measured data is from VEB towing tank using 5m model at U=1.54m/s (RN=7.7X108)] 2. Calculated and measured patterns of pressure contour without propeller resemble each other. In the propeller operating condition, suction effect of propeller is appeared in the stern region. Fig. 19 shows a comparison of wake patterns at three different transverse sections. Experimental data is obtained in IHI towing tank using 7m model. The unit of calculated vector is twice of the measured one in order to make clear the direction of the flow. The calculated contour of wx=O.l is a little diffusive compared with the measured one, however, the pattern of the iso-wake contour is well simulated as a whole. Fig.20 shows the effect of propeller on the iso-wake contours. It can be seen that the effect of propeller is restricted in the propeller disc. The measured iso-wake contours with propeller at A.P. section (just abaft the propeller) show the asymmetrical feature about center line due to the rotating flow ( Fig.21 ). As the present method does not deal with the rotating flow, calculated results are compared with the measured contour taking the mean of the contours in starboard and port side. It is found that the present method can simulate well the propeller effect on the contour. 5.3 Practical tanker form 'I'he second example of the application to practical hull form is the simulation of the flow around two kinds of tanker form shown in Fig.10 and Fig.11. Fi g.22 shows the comparison of iso-wake contour of ordinary tanker form ( Fig.10 ) at propeller position in towing condition at R~ =7.8xl 0~-; . The vortical motion can be simulated, however, it is smaller than that of experiment and an island-like contour of vortical motion is not found in the computed resu its. The flow around tanker form with IHI B.O. stern ( Fig.11 ) is calculated at RN=4.94X106. IHI B.O. stern is developed aiming at both merits of wake gain by bulbous stern and low thrust deduction by open stern. The configuration of B.O. stern is so complicated that the flow calculation does not succeed by the o rdinary method of grid generation. However the present method of grid modification makes the flow calculation possible. Fig. 23 and Fig. 24 are the comparison of hull surface pressure distribution and wake pattern in towing condition respectively. Calculated pressure distribution agrees well with the experiment except near the stern end where calculation gives lower pressure. B.O. stern gives uniform wake in propeller disc compared with ordinary stern shape. Present calculation simulates this feature of wake pattern, however the correspondence with measured results is not good because the bilge vortex is not simulated well. In order to improve the accuracy of the prediction, further examinations are necessary for finite- difference scheme, grid generation, adoption of wall-function, turbulence model and so forth. Fig.25 shows the velocity vectors near the hull surface for both cases of with and without propeller. In the propeller operating condition, the pressure jump of 691.67N/m2 which corresponds to measured thrust of 17.6N is set on the propeller disc of diameter 0.18m. Applying this propeller model, the accelerated flow afore and abaft the propeller can be simulated as well as decelerated flow just above the propeller. The present method predicts the boundary layer flow into the propeller around such a complicated stern form. 6. Conclusions Present studies are summarized as follows: (1) In the present flow calculation method, the off-diagonal terms in source term in pressure-correction equation are ignored as small quantities in order to save the computational time. It is found, however, that this leads to the breakdown of computational results when there are skew grids in computational domain around practical hull form. (2) In order to stabilize the calculation for practical hull form without increase of computational time, improvement of the grid shape generated by solving the elliptic partial differential equation is carried out by adjustment of grid angle. (3) Using this grid, the calculations of viscous flow around practical hull form (Series-6O and tanker forms) under propeller operating condition are carried out and the 216

results are compared with experimental results. This method is applicable for hull form examination at the initial design stage. In order to improve the accuracy of the prediction, further examinations are necessary for finite-difference scheme, grid generation, turbulence model, adoption of wall-function and so forth. The final goal of the present study is to build a design code which can evaluate self propulsion factor of a ship taking account rudder effect. Acknowledgements The authors are indebted to Dr.Y.Kodama of Ship Research Institute for providing us the computational grid of Series-60 generated by his own method. They also express their thanks to Dr.Y.Ando, Mr.M.Kawai, Dr.T.Tsutsumi, Dr.R.Sato and Mr.Y.Shirose of Ishikawajima- Harima Heavy Industries Co.,Ltd. for their support and advice, and all the staff of IHI towing tank for their help with the experiments. References 1. Ando,Y., Kawai,M., Sato,Y. and Toh,H., "Prediction of three-dimensional turbulent flows in a dump diffuser", AIM 2Cth Aerospace Sciences Meeting, Reno, Nevada, (1988). 2. Masuko,A., Shirose,Y. and Ishida,S., "numerical simulations of the viscous flow around ships including bilge vortices", Proceedings of the 17th ONR Symposium on Naval Hydrodynamics, The Hague, (1988). 3. Thompson,J.F., Warsi,Z.U.A. and Mastin,C.W., "Numerical grid generation, foundation and applications", North-Holland, New York, Amsterdam and Oxford, (1985). 4. Koshiba,Y. and Mori,M., "How to design the stern form"( in Japanese ), Ishikawajima- Harima Engineering Review, Vol.27, No.5, pp288-pp293, (1987). 5. Launder,B.E. and Spalding,D.B., "Mathematical models of turbulence", Academic Press, London and New york, (1972). G. Launder,B.E. and Spalding,D.B., "The numerical computation of turbulent flow", Computer Method in Applied Mechanics and Engineering, Vol.3, (1974). 7. Patanker,S.V., "Numerical heat transfer and fluid flow", Hemisphere Publishing, (1980). 8. O rli, S. and Haraguchi,M., "Efficient implementation of a fluid simulation algorithm on the FACOM-VP 100/200-( in Japanese ), The 28th National Convention of ISPJ, Information Processing Society of Japan, (1984). 9. Kodama,Y., "Three-dimensional grid generation around a ship hull using the geometrical method", Journal of the Society of Naval Architects of Japan, Vol.164, pp.9-16, (1988). 10. Oki,Y., Ochi,M. and Ohgane,E., "Ship lines design system"( in Japanese ), IshikawaJima- Harima Engineering Review, Vol.21, No.5, pp .422-427, ( 1 98 1 ) . 11. Sarda,O.P., "Turbulent flow past ship hulls -- An experimental and computational study --", Ph.D. Thesis, the University of Iowa, (1980). 12. "Flow examination on a model of Series 60 with Cb=0.60 Model No.675", VEB Report, (1983, Applied to the ITTC Cooperated Experimental Program). Table 1 Effective exchange coefficients and source terms of equation (9) 1 1 r~ ~I s, 0 11 0 u; ~ ~ I _ ai ~P t ~ a,, auk ~ ~ mAfk3~t f k J9t, J3tk k ~ ~^ , ~ 1 ~ 'CJ' _ pi _ pe ~ ~ ~ ~Aj~3k C] Pk k~ - C2 p k ~ ~ ~ ~Aj'3E ~ - ~ J3t,, ~E J3tk T ( ~ Cont ro I Vo I ume | B ~ Fig. 1 Grid points and a control volume 217

y/L -0.5 0.0 FP n n- 0.5 3~ X/L AP l.o 1.5 2.0 -0.5 0.0 0 FP . _ l ~ ~ = = ~ ~ ~ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ , .5 1.0 1.5 2.0 X/ L AP Fig.2 Computational domain and grid configuration ~: O Grid point where pressure jump is imposed Fig.3 Grid configuration at the propeller position cp = P 0.6 0.4 0.2 o.o , -1 .o 0.2 -0.4 0.6 x(m) _ - I 1 UU VI'V2 Prediction with momentum theory U | Propeller position 0.4 _ Vl ~ /i x(m) 0.0 , ~/ 1 1 1 -1 n o.o l.o 1.0 2.0 Fig.4 Computational examination of pressure jump model 218

Fig.5 Influence of skew grid (a) Before modification (b) After modification Fig.7 Modification of the grid 219 x - \\W (~: P,: X;" -K,""" to, ig.6 Generation of interpolated point i=] 2 (Hull Surface) I\\ \ \ \ ~ \ id,'- \ *: Angle adjusted to be between 45°~135° Fig.8 Adj ustment of grid angle

Bow Part Stern Part Stern Section Fig.9 Generated computational grid ( Tanker form with normal stern ) Row Fort _~ Stern Part Stern Section Fig.10 Generated computational grid ( Tanker form with IHI B.O. Stern ) 220

P (Njm2) 50C Hi 500 Max SSUM Fig.11 Convergence history of pressure / / / ~ , , w 300 ISTEP 150/ 200 ~_ u - Cal. - Exp. ~) (Sarda) ~ Fig. 14 Calculated and measured wake pattern ( Wigley model, RN=4.5X1O6 ) Cp - PU2/2 ' Fig. 12 Convergence history of u, w, p, 0.2 k, £ ~ at A.P. ~o 10-'1( io-t 1n-. 10-' ~ ~\\ - - - 10-' O SO 1W lSO 200 250 3W ISTEP Fig. 13 Convergence history of mass-imbalance 0 1~ 0.0 ; z/d - 0.2 iC, .. . .$ . , . FP .. ~ D - ,. _ · Cal. O Exp. (IHI, R,,-~.2X106) Inviscid (Calculation) .e .. . :. , ~ ,. _ . . ~ ~ , ~ I 5 ° AP . .~ Fig. 15 HU11 surface pressure distribution ( Wigley model, RN=4.5X1O6 ) 221

5.0 C'X103 x =3.0m (Midship) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ao ~ ~e ' ' z/d 0 0 0.5 1.0 . Cat. 5 . 0 x = 5 . 4 m 2 ( y-yO)/B 0 Exp. (Sarda) (S. S. 1 ) O x-5.7m x-6.0m x -6.6m x -9.Om ~ 1.0 - (S.S. 1/2) 1.0 - (A.P) 1.0 1.0 C'X103 ~ ~ ~ ~d.~- 0.8~ 0.8~ 0.8t 0.8 0.6~ 0.6~ 0.65 0.6~ 1 ~ z/d 0 4 Oo 0 4 - Oo e 0 4 °O. 0.4 - 0 e 0 0 0.5 0.2 .2 0.2 ~ (S.5~/2) . ~°~ C, X103 1 ~ o · °'° k/U7XIO' 5'00'0 k/U2 103 5'00'0 / ~ , 5.00.0 5,0 · ~ ~O · · Fig.17 Comparison of turbulent kinetic energy l 1;0 z/d ( Wigley model, RN =4.5xlO6, z/d=O.O ) 0.0 0.5 5.0 x =5.9m - · Cal. C~ X103 - 0 Exp.(Sarda) ° · · ·o · · - ~ I ' ' z/d 0.0 0.5 1.0 Fig.16 Comparison of local skin-friction coefficient ( Wigley model, RN=4.5X1O6 ) ~ ~ j ~ ~ ~ Cal. (without propeller) Cal. (with propeller) C) ~_ . .. , . ... .... .. .. . _ Fig.18 Hull surface pressure distribution ( Series-60 Cb=0.6 ) 222 2_-0~075 ~=

x - ~ ~ ~ ~ ~ ~ - ~ o ooo o o o o ~ ::: S-~:~ S.S.1 Cal. 1 ~ ~ u, x . . ~ ~ 0 0 0 0 /` `~` `: i_ ~'NJ''~ / N/' N/' //~//~: \ ~ ~ / \ ~ ~ \ ~ ~/1 \ \ ~ \ 1 1 ~ ~ \ ~ ., -.~I Exp. O.SU . . . O o 0 / ~ / _/ _ )` ~ ~ (` ~ ~ ~ ~ -;'/~/`J~ 1' . t\~/~0.5~U:' Cal. Exp. ~/ n su t / \1 t 1 /1i' .,., ~ A.P. Cal. Exp. 1 1 Fig.19 Calculated and measured wake patterns ( Series-60 Cb=0.6, RN=9.22X1O6 ) 11 x ~ r~ ~ Ln _ . . . . 5 o o o o ~ \N without propeller -- with propeller S.S. 1/4 Cal. Exp. 0.1 '~ without propeller ---------- with propeller A.P. Cal. Exp. Fig.20 The effect of propeller on the calculated and measured wake patterns ( Series-GO Cb=0.6, RN=9.22X1O6 ) o. without propeller ---------- with propeller Fig.21 The effect of propeller on the measured wake pattern ( Series-60 Cb=0.6, RN=9.22X1O6 ) 223

- - / / / ~~/\~ Cal. ~,,~J ~ -/ - - · , /~ ~ \ , ,/1("~1/\ Am)' \ K'\4 1\ /\~N ~ \ '\ ''_)t / 1 1 ~ t t ~ ~ ~ 0 5U Prop. Position Exp. 1 ' )' Fig.22 Calculated and measured wake patterns ( Tanker form with normal stern, RN=7.8X1O6 ) ~ '\~"'\"\ V'\'\\~ - / ~ _11 -~11 Exp. Fig.23 Hull surface pressure distribution ( Tanker form with IHI B.O. Stern, RN=4.94X1O6 ) ~.~\ Prop. Position Exp. Fig.24 Calculated and measured wake pattern Cal. 1 ~~ 224 Fig.25 The effect of propeller on the velocity vector near hull surface