**Suggested Citation:**"Grid Generation and Flow Computation for Practical Ship Hull Forms and Propellers Using the Geometrical Method and the IAF Scheme." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Grid Generation and Flow Computation for Practical Ship Hull Forms and Propellers Using the Geometrical Method and the IAF Scheme." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Grid Generation and Flow Computation for Practical Ship Hull Forms and Propellers Using the Geometrical Method and the IAF Scheme." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Grid Generation and Flow Computation for Practical Ship Hull Forms and Propellers Using the Geometrical Method and the IAF Scheme." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Grid Generation and Flow Computation for Practical Ship Hull Forms and Propellers Using the Geometrical Method and the IAF Scheme." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Grid Generation and Flow Computation for Practical Ship Hull Forms and Propellers Using the Geometrical Method and the IAF Scheme." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Grid Generation and Flow Computation for Practical Ship Hull Forms and Propellers Using the Geometrical Method and the IAF Scheme." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Grid Generation and Flow Computation for Practical Ship Hull Forms and Propellers Using the Geometrical Method and the IAF Scheme." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Grid Generation and Flow Computation for Practical Ship Hull Forms and Propellers Using the Geometrical Method and the IAF Scheme." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Grid Generation and Flow Computation for Practical Ship Hull Forms and Propellers Using the Geometrical Method and the IAF Scheme." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Grid Generation and Flow Computation for Practical Ship Hull Forms and Propellers Using the Geometrical Method and the IAF Scheme." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Grid Generation and Flow Computation for Practical Ship Hull Forms and Propellers Using the Geometrical Method and the IAF Scheme." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Grid Generation and Flow Computation for Practical Ship Hull Forms and Propellers Using the Geometrical Method and the IAF Scheme." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Grid Generation and Flow Computation for Practical Ship Hull Forms and Propellers Using the Geometrical Method and the IAF Scheme." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Grid Generation and Flow Computation for Practical Ship Hull Forms and Propellers Using the Geometrical Method and the IAF Scheme." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Grid Generation and Flow Computation for Practical Ship Hull Forms and Propellers Using the Geometrical Method and the IAF Scheme." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

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Grict Generation and Flow Computation for Practical Ship Hull Forms and Propellers Using the Geometrical Method ant! the lAF Scheme Y. Kodama Ship Research Institute Tokyo, Japan Abstract Grid generation for ship hulls and a propeller blade was made using the geometrical method, where an ini- tial grid is modified iteratively under several geomet- rical requirements such as orthogonality, smoothness, and clustering. Using the generated grid, the computation of the incompressible Navier-Stokes equations was made for flows past a flat plate and four different ship hull forms using the IAF scheme and the Baldwin-Lomax zeros equation turbulence model. The hull forms chosen are a Wigley hull and Series 60 (Cb=0.6, 0.7, 0.8) hulls. Prior to the ship flow computation, the flow around a flat plate was computed, where the agreement of the com- puted result with experiment was very good in terms of the wall shear stress, displacement thickness, shape factor, and velocity distribution. The computed flow past a Wigley hull, which is not very different from the flat plate result because of its fine hull form, showed good agreement with experiments. The computed flows past Series 60 (Cb=0.6, 0.7, and 0.8) hull forms showed systematic change with the Cb block coefficient values. The computed Cb=0.6 result showed reasonable agree- ment with experiments. However, the agreement be- came poorer with increase in the Cb values (=0.7, 0.8~. 1. Introduction Prediction of flow past a ship hull has been an im- portant subject in ship hydrodynamics because of its practical importance. If one succeeds in accurate pre- diction of the flow, one can get a resistance value to be used in powering, and the wake field at the propeller plane to be used in propeller performance estimation. A classical approach to the above problem is the bo~ndary-layer method t14. But recently, aided by the rapid development of computer hardware, methods called NS solvers, in which the governing equations of 71 the flow are discretized and computed, are becoming increasingly popular t2],(3],[43. The author previously computed the flow past a Wigley hull using the non- conservation form of the NS solver t53. The present work is an extension of the previous work. The major change is in the use of the conservation form. The scheme is called the IAF scheme [6], which is widely used for computing compressible flows. Pseudo-compressil>ility is introduced in the continuity equation of the incom- pressible flow, in order to make the system of equations hyperbolic. A conservative 2nd-order central differenc- ings are used for convection and diffusion terms, and the 4-th order numerical dissipation terms are e~cplcitly added to the equations to damp out high frequency wig- gles. The degree of accuracy of computed results of NS solvers is affected by the quality of the grid used. There- fore, it is important to be able to generate grids of high quality, in order to obtain good computed results. In the present paper, a grid generation method called the geometrical method is used. It is an extension of the method used by the author previously t73. Using the method, the grid on a propeller blade and the grids around ship hulls are generated. Although the use of NS solvers greatly reduces the need for experimental informations, a turbulence model is still needed for computing high Reynolds number flows such as those past ship hulls. The turbulence mocl- els widely used today in engineering applications are the Cebeci-Smith (CS) zero-equation turbulence model t8], and the k-e two-equation model t93. In the present paper the Baldwin-Loma.x (BL) zero-equation t~rbu- lence model, derived by making a modification to the CS model, is used. There, in contrast to the original CS model, the necessity for finding the edge of flee boundary-layer is avoided. The model is widely vised for computing compressble flows in the field of aerody- namics, and is said to be accurate for ~~nseparater1 floes s,

lout poor for separated flows. One of the objectives of the present work is to test the validity of the tubulence model for incompressible flows past ship hulls, and to find the range of applicability of the model by com- puting flows past ship hulls with various degree of full- ness. Although further modification to the BL model was published recently t10], the original BE model is used here. In Chapter 2, the NS solver is described, with dis- where cussions on the conservation property, boundary con- ~ F = IF + (YG + (ZH ditions, and the turbulence model. In Chapter 3, the ~ J J J geometrical grid generation method is described, where ~ G = 92 F + ~9 G + my H the use of bi-cubic splines for representing a body sur- ~ J J J face geometry is explained. In Chapter 4, computed re- sults of flows past a flat plate, a Wigley hull, and Series 60 Cb=0.6, 0.7, and 0.8 hulls are shown. In Chapter 5, conclusions are drawn. The flow past a propeller, which was originally planned, is not included in the present paper. 2. NS Solver 2.1 Governing Equations The governing equations are the combination of the incompressible Navier-Stokes equations and the conti- nuity equation. They are, in conservation form, q' + Fm + Gy + Hz = 0 (2.1a) where q= ~ ~ F= U +p-~q. UV-~y ~ uvAn G = v + pMY uw~xz vwTyz flu Jv ~ uwId VWfyz w + pfizz jaw where all the subscrits except those with ~ denote par- tial derivatives. The 1st, 2nd, and 3rd components are x-, y-, and z- momentum equations. In the 4th compo- nent, which is the continuity equation, bp/0t is artifi- cially added to give pseudo-compressibility,thus making the sytem of equations hyperbolic. ,B in the equation is a positive constant. The shear-stress terms ~ are ex- pressed as follows. ~m = (R + ut)2u~, boxy = (R + ~~) (fly + vm) My = (R +~2vy ~ me = (R +~) two +uz) (2.2) ''ZZ = (R Jr 7~2wz ~ adz = ('R + ut) (vz + wy) where u~ is the kinematic eddy viscosity. 72 Coordinate transformation from the (x,y,z) Carte- sian coordinate system to the body-fitted ((,r~,() coordi- nate system is made to the governing equation eq.~2. ] ). The transformation is assumed time-independent. The resulting equation is, again in conservation form Hi ~ J ~ + Fit + Go + H; = 0 (2.3~) (2.3b) tH=~JF+(JYG+(JZH J is the Jacobian, and all the x, y, and z derivatives of g", A, and ¢ are expanded using the chain rule. Numerical dissipation terms are added to the above equation to enhance numerical stability. ~ ~ J )+F~ +Gn +H: +~(q6466 +q0000+q<~¢) = 0 (2.~) where cat is a positive constant. In order that computed results obtained by solving eq.~2.4) satisfy the origi- nal equation with accuracy, these added terms must be small. 2.2 Discretization First, the time derivative is replaced by the time differencing. The Pade time differencing form is used here. ~q 1 _ = qn Bt fit l + c4/\ where i\qn En+ _ qn (2 5) where qn denotes q at timestep n. ~ is a constant which (2.1.b) takes the value of either 0 (Euler explicit), 0.5 (Trape- zoidal), or 1.0 (Euler implicit). Here ~1.0 is adopted. The nonlinear flux terms OF, AG, /\H are locally lin- earized into the form, for example, i\F = A/\q + A6/\qf. + A~\q,, + A¢~\q¢ (~2.6a) where A = ~' , At = ~ , An = ~ , At = `9 (2.66) Then the governing equation becomes /\qn+67\tt J~(` Ai\q + A6/\q~ + A'7/\qr~ + A¢Aq¢) + J~,,(~ Bi\q + BfAq~. + Br77\q,7 + B</\q¢) + J~¢( C~q + C6Aq~ + C~Aq~ + C¢Aq¢) + w(< Aqua< + i\qnrlnn + /\q¢¢¢~] =/\tpJ(Ft + G., + H¢) + w{<q~ . + q,, + q¢¢¢~]n(2 7)

The mixed derivative terms such as 0(A~^q~/~: are lagged to the timestep (n-1) and evaluated time- explicitly. Then the above equation can be approxi- mately factored into A-, A-, and ¢-sweeps. (-sweep {I + 9 At [J ,~: (A + Af ~ ~ + a, ~ in } A* =At [ J(F~ + Gn + H: ~ + w f qua + q + qua< )]n §^tJ[(A~^q~ + A(Aq`~4 + (B(Aq~ + B<Aq`~ +`C(^q~ + Cu~q '<]n1 (2.8) -sweep {I+~^t[J,7 (B+B~0 )+w~ 4]n}Aq** = Aq* (2.9) ¢-sweep {I+~^t[J0¢(C+C<~¢~+~¢4]n}Aqn = Aq** (2.10) /\q* and Aq** are intermediate variables defined as the above two equations. I is a (4x4) unit matrix. After the above three equations are solved, the variable qn is updated using eq.~2.5~. That is qn+l = qn + /<qn (2.11) In order to solve eqs. (2.8) to (~).10), spatial differ- enciation has to be approximated by spatial differenc- ing. Here the following 2nd-order central differencings are used. ~ ~ = J 06 l 054 = E6-2 - 4E6-1 + 6E`° - 4E<+1 + E`+2 _E<-2 +E~+2 (2.12) E<+"' is a shift operator in (-direction defined as Ed qi,j,k qi+7n,j,k (~.13) where (i,j,k) are the numberings in A-, A, and ¢- direc- tions respectively. Singular central differencing operators are vised in ~ and ¢ directions. The non-staggered grid system is used, i.e. velocity, pressure, and eddy viscos- ity values are stored at the grid points. Substituting the di~erencing operators into eq.~2.~) results in Ii i\q*i_o + LAq*i-: + JI^q*i + N~q is + O^q i+2 Ii = HI = THIS of (~).~] (2.14~) L = 9/\ttJ~2Ai_~/o +.~_~/~)4wI] Al = I + 9AtfJ~.iAi-ilo + ,.4i+~/o + Az-~/2 + AZ+~/2) +6wI] (2. 14b) N = §^ttJ(-Ai+il2 + A6+il2)HI] 0 = §AtwI The above set of equations form a system with block pentadiagonal coefficient matrices, and can be effi- ciently solved. The coefficient matices K, L, M, N. and O for q- or (-sweep are obtained by replacing (i,A, A6) with (j,B, Be) or (k'C, C<) in the above equation. As a consequence of the approximate factorization, each line of block pentadiagonal system in any of the three sweeps is decoupled completely from others, and therefore highly vectorized coding is possible. The use of the difl~erencing forms shown in eq.~.12) results in the global conservation of momentum and mass in the discretized form. As shown in Fig. 2-1, where the area covered by a discretized governing equa- tion at a grid point is shown hatched, all the numerical flux terms are evaluated at halfway between the grid points, and therefore cancel out when they are summed up, thus the global conservation holds. The added nu- merical dissipation terms, having constant coefficients, cancel out when they are summed up, and therefore do not affect the conservation property. However, there is exception in this conservation property. In the zone be- tween the boundary point and the halfway point next to it, the conservation property does not hold unless the boundary condition is made consistent with the conser- ation property of the governing equation. In the present computation, this consistency is not ratified, and ther- fore the conservation property holds not all the way down to the boundary point but to the halfway point next to the boundary. The same is true to the nu- merical dissipation terms, since they cannot find their counterparts to cancel out near the boundary. 2.3 Boundary Conditions Fig. 2-2 (a),~b),tc) shows the grid topology and the location of the boundaries. The x-axis is in the stream- wise direction, with x=0.0 at F.P. and x=1.0 at A.P.. The z-axis is in upward direction, and the y-axis is in the lateral direction, forming a right-handed system. The grid system is in the negative y and negative z quarter. The grid topology is H-O. The (-,~-, and (- axes are approximately in streamwise, girthwise, and normal- to-surface directions respectively. The star marks in Figs. (a) and (c) are the point of mapping singular- ity. At Upstream boundary, the uniform flow condition is used. That is, q=~1,0,0,0)T and Aq=0. At Down- stream boundary, zero extrapolation in (-direction is used. They are q`=0 and i\q`=0. At (x-z) and (x-y) symmetry plane boundaries, zero-derivative condition is used, instead of an exact symmetry condtion. That is, q77=0 and f\q77=0. At Inner boundary, the solid ``all boundary condition is used. They are u=v=w=0 and an approximate boundary ondition for pressure, Lick is p<=0 and i\p<=0. Finally, at Outer boundary, the uniform flow condition is used. 73

0.4 Turbulence Model The turbulence model used is the Baldwin-Lomax (BL) zero-eq~ation algebraic model, whose original form is the Cebeci-Smith (CS) model. The kinematic eddy viscosity ~` is evaluated in the inner and outer lay- ers separately. In the inner region, the kinematic eddy viscosity Eli has the same form with the CS model. with I'd = 12 ~~ (2. 15) I = knrL1exp(<n+ /A+ )] (2.16) n+ = nRe ~/~ (2.17) where cat is the vorticity and n is the normal distance from the wall. In the present computation, as shown in Fig. 2-3, the normal distance n is determined by projecting a vector connecting the point in concern with the root point on the same ¢-direction line, onto the normal vector defined at the root point. In the outer region, the necessity of finding the edge of the boundary-layer, which existed in the CS model, is removed in the BL model. The eddy viscosity RIO has the following form. lJ`o = KCCp Fkiebnmam Fma2 ~ 2. 1 S) or Jto = KCcpFElebCwkUdiff F. (2.19) mar: with K=0.0168, Ccp =1.6, and Cask =0.25. The smaller value of the above two equations is taken. The quanti- ties Fma~ and nma.r are determined from the function F = n ~~ t1exp(<n+/A+~] (~.20) The quantity Fma2 is the maximum value of F that occurs in a velocity profile, and nmna is the value of n at that point. Uaif f is the difference btweeen maximum and minimum velocity in the profile. Ujif f = USA Umin ~ 2.21 ) where Umin is taken to be zero except in wakes. The intermittence factor FKieb is given by 1 ~ 5.5(C~`ebn/nma~6 (2.22) with C~;-'eb= 0. 3. 3. Grid Generation The grid generation method used is the geometrical method. Since the method is described in detail in t11i, descriptions are limited here on the modifications made thereafter. 74 3.1 Body Surface Grid In the grid generation method, first tile surface grids at inner and outer boundaries are generated. The inner boundary usually corresponds to a body surface. The body surface geometry is represented using para- metric bi-cubic splines. Then the grid points on the sur- face can be shifted along the surface according to four geometrical requirements, i.e., orthogonality, smooth- ing, clustering, and minimum spacing. Fig. 3-1 shows a spline surface on a Series 60 (Cb=0.6) hull, which has been determined from given offset points. Fig. 3-9 shows a surface grid on the same hull. Clustering has been imposed toward bow and stern edges, as well as toward bottom and waterline edges. Orthogonality has been imposed at all the four edges. The same method has been applied to the surface grid on a propeller blade. Fig. 3-3 (a), (b) show surface grids on a propeller blade before and after mo~lif~ca- tion. Clustering is made toward leading, baling, and tip edges, with various degree of clustering. Here all the points on the surface are allowed to move except for th\ose at the four corners, and therefore highly fle~c- ible grid generation is possible. 3.2 Total Grid After the surface grid is generated, intermediate grid points, which are the points between the inner and outer boundaries, are generated, by simply connecting the corresponding points on the inner and outer bound- aries with a straight line, and disributing points on it. Then the initial grid is modified iteratively under the same geometrical requirements as those used in the sur- face grid modification. The grid points on the bound- aries are kept fixed. The most important change since ref.~11] is in the combination of the grid point modifications. In the orig- inal method, grid points were shifted separately dice to each modification requirement, whereas in the present method the grid pains are shifted only once in each iteration as a result of the combined shifts. The com- bination is made with weights which change locally ac- cording to the local grid geometry. The parameter vised to represent local grid geometry is the aspect ratio of a grid cell. The value of the parameter varies greatly depending on the location, since the grid is highly cl~s- tered near the solid wall surface. The minimum spacing requirement has been modified. In the old version, if a spacing between two neighboring points in a certain di- rection is smaller than a given value, that spacing is made larger to the given value in that direction by re- distributing points along that direction with the total length kept unchanged. This requirement was found to be too "active" in some cases, especially near the solid wall boundary where the grid is highly clustered in the normal direction. There the re-di.ctribution in the along-the-wall direction sometimes results in kinks in the other direction. Therefore in the present version

the requirement has been changed to a "passive" one, where, if a grid spacing is found to be smaller than a given value, the modification which makes the spacing even smaller is made zero. After the grid is modified according to the foure requirements, the final grid for high Reynolds num- ber flow computation is obtained by re-clustering. Re-clustering is the process in which the grid is re- distributed, in a single sweep, in the normal-to-wall direction according to a given minimumm spacing ad- jacent to the solid wall and a new given number of grid points in that direction, making the information inher- ent in the original grid point distribution reflected in the new distribution. This process makes possible to use smaller number of grid points and much larger min- imum spacing near the solid wall in the iterative mod- ification stage, thus making the grid generation much faster and easier. No change has been made to the re- clustering process. Fig. 3-4 (a),~b),tc) shows the gener- ated grid around a Series 60 (Cb=0.6) hull. The i=38 section shown in (b) is approximately at midship, and the i=60 section is at the stern edge (see Table 4-1~. 4. Computed Results 4.1 Tested Cases and Parameters Computations were made for seven cases with five different bodies. The bodies vary in the degree of full- ness, from a flat plate to a Series 60 (Cb=0.8) hull. Ta- ble 4-1 shows the cases and the parameters. There im, jm, and km are the number of grid points in ~ (stream- wise), ~ (girthwise) and ~ (normal-tc~wall) directions. The experiments with Froude number values (Fn) were made in cirulating water channels with free surface, and all the others were made using double models in wind tunnels. Followings are the parameters common to all the computational cases: ifp=16 iap=60 Xupstream end = ~0 5 Xfp = 0.0 Cap = 1.0 Xdo~vnstream end = 2 0 Outer boundary radius = 1.0 ,l? = 0.1 where ifp and lap are the numberings in i at bow and stern (or leading and trailing for a flat plate) edges. As shown in Fig. 4-1, the minimum spacing of grid points adjacent to the inner boundary (solid shall or symmetry plane) /\min is determined as /\mi77~ = /\min-14 = /\ mini = /\ mini 0.05 /\ m ~ 77 ~ = ~ v Re (x < 0.1) (0.1 < x < 1.0) (1.0 < x) (my) (4.1) Computations were made using the Stellar GS- 1000 graphic workstation with a rector processor unit and 3~) SIB main memory. The NS solver took approx- imately 40 sees (Cases .5,6,7) CPU time per timestep, 7s using a highly vectorized code. /\t varied from 0.005 NTU (Nondimensional Time Unit) for the Wigley hull case, to 0.001 NTU for the Series 60 Cb=0.8 case. It took approximately 4 to 5 NTU to reach the stea<ly state. It corresponds to a day or two in the real time. The grid generation and post-processing were made us- ing interactive graphics on the Sun-4/260 engineering workstation with 32 MB main memory. 4.1 Flat Plate Fig. 4-2 shows the flat plate grid. The grid was generated based on the Wigley grid at the (x-z) sym- metry plane, with y values made zero on the hull sur- face. The coordinates are nondimensionalized using the length of the flat plate. Transition switch in the turbu- lence model has been suppressed, i.e. the flow is as- sumed turbulent starting from the leading edge. Fig. 4-3, ~4, and 4-5 show respectively the wall shear stress law, the displacement thickness be, and the shape factor H. The solid lines are empirical curves t12] JEW = 2Cf = 2~2109joRx0.651-23 (i.l) 0 0~733R 0.861 (~.~\) H = 1.3 (~.3) where Of is the local skin friction coefficient. They are in good agreement. Figs. 4-6 (a),~b), and (c) show logarithmic plot of velocity at three streamwise locations. The empirical curves are us-= n+ in sublayer uw = 2.541n(n+) + 5.56 in log region (4.4) where uw is the friction velocity. They are in good agreement, both in the sublayer and the logarithmic regions. Fig. 4-7 shows the kinematic eddy viscosity i/` distributions. They are nondimensiona]ized using uw and the momentum thickness 52. They fall on a sin- gle curve. The straight line stemming from the origin has slope value of ~ =0.41 t133. The horizontal line has the value of 0.828, derived by comibining an empirical velocity distribution with L,t/uw5=0.065, where ~ is the boundary-layer thickness. The disagreement with the computed values seems to come from the ambiguity in determining the boundary layer edge. 4.2 Wigley Hull The results shown in this section are mostly for Case 2 (see Table 4-1), the standard case for the compll- tation of flow past a Wigley hull. Experimental results shown for comparison in this section are all taken from ref.~144. The flow is assumed laminar lip to x=0.0.5, and

turbulent thereafter, to be consistent with the experi- ment, where the studs were pla.cecl at the same location. This applies to all the cases with ship hulls. Fig. 4-S shows pressure contours on the hull sur- fa.ce, (x-z) symmetry plane, and (x-y) symmetry plane. The flow is from left to right. The contour lines are wiggle-free. In the previous computation [5], where the non-conservation form was used, there were wiggles at fore and aft ends of the hulll. Fig. 4-9 (a),tb),~c),td) show wake contours at x=constant sections, from midship (~=0.5) to down- strea.m (x=1.5~. The contours at an x=constant section is obtained by interpolating the values at grid point lo- cations using parametric tri-cubic splines, which is an extension of bi-cubic splines to three-dimensions. The computed values show good agreement with the mea- sured values. Fig. ~10 shows the kinematic eddy vis- cosity u~ contours at i=53 section (x~ 0.95~. The step- wise change of z'` in the girthwise direction occurs be- ca.use the location of FmaX in (-direction is determined at either of the grid points, which are widely spaced in the outer region. Fig. ~ , ~ is, rameters on the wake contour at x=1.0. The figure (a) shows the result with ~ = 5.0 (Case 3~. The difference from the ~ = 1.0 result is small, which implies that the value ~ = 1.0 in Case 2 is small enough such that the added numerical dissipation terms do not affect the computed result. The figure (b) shows the result where the number of grid points in ¢-direction is doubled. The difference is again small, which implies that the number of grid points used in Case 2 is large enough. Fig. 4-12 (a),~b),tc) show the wall shear stress Id distributions at z/D= 0.2, 0.5, and 0.8 sections, where D is the depth of the hull. The empirical curves are the same as those in the flat plate case. The agreement with the measured values are generally good. The id values agree well with the flat plate values in most of the regions. Fig. ~13 (a),tb),tc) show the displacement thick- ness {~ distributions at z/D= 0.9, 0.5, and 0.8 sections. The agreement is again generally good. The b~ values deviate considerably from the empirical flat plate values shown as solid lines in the figures. Fig. 4-14 shows the shape factor H at z/D= 0.2, 0.5, and 0.8 sections. They show tendencies similar to those of flat plate results, except near the stern. Fig. 4-15 (a),tb),~c) show logarithmic plot of ve- locity at three i=consta.nt sections, i.e. i=25 (x~ 0.07), i=38 (x=0.5), and i=53 (x~ 0.95~. It can be seen that logarithmic law holds in every case. Fig. ~16 (a),tb),tc) shows the distribution of the kinematic eddy viscosity id, the locations of which correspond to those in Fig. 4-15. The tedency is similar to that of the flat plate. 4-11 (a).(b) show the effect of chancing oa- 4.3 Series :60 (Cb=0.6, 0.7, 0.8) Hulls In this section, the measured data. shown are taken 76 from [15] (Cb=0.6), [16] (Cb=0.7), and [17] (Cb=0.8). Fig. 4-17 (a),tb),~c) shows computed pressure contours for Series 60 Cb=0.6, 0.7, and 0.8 hulls. They are plotted at the same pressure values as those for the Wigley case (Fig. ~8~. There are slight pressure oscil- lations just below the points of mapping singularity in the wake. Fig. 4-18 shows comparison of the computed pres- sure distribution near the stern region on the C1.~=0.6 hull with the measurement [1S]. The agreement is good, though there is systematic deviation which increases toward midship, and the adverse pressure gradient is greater in the computed result. Fig. 4-19 (a) to. (h) show the wake contours. The agreement with the measurements are good in general, except in the wake. There the downward movement of the low speed region near the (x-z) symmetry plane is not captured in the computation. There are two possi- ble reasons for this failure. One is that the intensity of the downwash due to the pair of longitudial vortices is insufficient. The other is in the way the eddy viscosity is determined. In the wake, the normal distance n+ is- ta.ken from the (x-z) symmetry plane. However, when the flow near the stern is highly three-dimensional, the (x-z) symmetry plane is not necessarily suital:)le for this purpose, in contrast to the situation with the Wigley hull. There the longitudinal vortices are not strong, and the flow in the wake remains similar to that of a flat plate. Figs. 4-20,~21,~22 show the wall shear stress tow, the displacement thickness 5~, and the shape factor H at three z=constant locations. There at z/D=0.8, the separation occurs near the stern. Fig. 4-23 (a),~b),tc) show the logarithmic plot of velocity at three streamwise locations. It is seen that all the points adjacent to the solid wall are well within the viscous sublayer as the turbulence model used demands, and that all the velocity profiles follow the logarithmic distribution law in the inner layer. They suggest that the Baldwin-Lomax turbulence model, a simple zero- equation model, can be used for this type of flow. Fig. 4-24 (a),~b),~c) show the kinematic eddy viscosity ~~` distributions at the same three strea.mwise locations as in the logarithmic velocity plot. The clistributions are similar to those in the Wigley hull case shown in Fig. 4-16. Finally, Fig. 4-25 (a),tb) show the wake contours for the Cb=0.7 case (Case 6 in Table 4-1), and Fig. ~1- 26 shows the wake contours for the Cb=0.8 case scene 7~. In the Cb=0.7 case, the computed wakes still show reasonable agreement with the measurements in those streamwise locations. However, in the Cb=0.8 case, the agreement becomes poor. By looking at the three com- puted wake contours, i.e. the cases Cb=0.6, Cb=0.7, and Cb=0.8, a systematic trend is observed. In the stern region, the wake near the bottom becomes thin- ner than the measured one, and the wake in the mid- de;pth region becomes very thick, as the fullness of the

ship hull (Cl:) becomes greater. Clearly the tubulence remodel needs n~c~clification there (see Supplement. 5. Conclusions Flows past ship hulls were computed and compared with measurements. The NS solver used is the IAF scheme, where the pseudo-compressibility is introduced in the continuity equation, in order to make the system of equations hyperbolic. The accuracy and convergence of the computed results were tested by computing flows rising different numl:>er of grid points, or using different amount of the added numerical dissipation terms. The Baldwin-Lomax zero-equation turbulence model was used. The validity and limitation of the tur- bulence model was tested by computing flows past five different bodies. They are a flat plate, a Wigley hull, Series 60 Cb=06, 0.7, and 0.8 hulls. They vary in the de- gree of fullness, from complete flatness of the flat plate to high fullness of the Series 60 C=0.8 hull. By com- paring the computed results with measurements, the turl:'ulence model was found to be useful for fine hull forms, such as a flat plate, a Wigley hull, and a Se- ries 60 Cb=0.6 hull. However, the agreement between the computed and measured data was not satisfactory for the Series 60 Cb=0.7 or 0.8 hulls. This suggests that the turbulence model needs modification for such flows where strong adverse pressure gradient exists and three-di~nensionality becomes important. The grid generation method called the geometrical method was used to generate grid around those ship hulls mentioned above. The method was also applied to generate a surface grid on a propeller blade. The body surface was represented using parametric bi-cubic splines, and the grid points on the body surface were allowed to move along the surface, in order to meet the requirements imposed, i.e. orthogonality, cluster- ing, smoothing, and minimum spacing. Sato,T. et al."Finite-Di~erence Simulation Method for Wave and Viscous Flows al:'out a Ship", J. of SNAJ vol. 160, (Dec. 1986). 4] Masuko, A. et al."Numerical Simulation of Viscous Flow around a Series of Mathematical Ship Mod- els", J. of SNAJ vol. 162, pp.1-10 (Dec. 1987). 5] Kodama, Y."Computation of High Reynolds Num- ber Flows Past a Ship Hull Using the IAF Scheme" J. of SNAJ vol. 161, pp.25-34 (1987~. 6] Beam, R.M. and Warming, R.F."An Implicit Fac- tored Scheme for the Compressible Navier-Stokes Equations", AIAA Journal, Vol.16, No.4, (April 1978). 7] Kodama, Y."Three-Dimensional Grid Genera- tion around a Ship Hull Using the Geometrical Method", J. of SNAJ vol. 164, pp.9-16 (1988). 8] Cebeci, T. and Smith, A.M.O."A Finite-Difference Method for Calculating Compressible Laminar and Turbulent Boundary Layers", J. of Basic Engi neer- ing, Trans. of the ASME, pp.523-535 (Sept. 1970~. 9] Rodi, W."Turbulence Models and Their Applica- tion in Hydraulics", IAHR (1980~. t10; Stock, H.W. and Haase, W." Determination of Length Scales in Algebraic Turbulence Models for Navier-Stokes Methods", AIAA Journal, Vol.97, No.1, (January 1989~. [11] Kodama, Y."Three-Dimensional Grid Genera- tion around a Ship Hull Using the Geometrical Method", J. of SNAJ vol. 164, pp.9-16 (Dec. 1988~. [12] Acknowledgements [13] The author would like to thank Profs. I. Tanaka and T. Suzuki of Osaka University, Prof. T. Okuno of University of Osaka Prefecture, and Prof. V.C. Patel of University of Iowa for providing references and material on the measured data used in the present work. The au- t.hor also thanks the members of the CFD group at the Ship Research Institute for many valuable discussions. References 1] Himeno? Y."Calculation Method of the Twos Dimensional Turbulent Boundary-Layers", Pros ceedings of the Symposium on Viscous Resistance, SNAJ, pp.59-93 (1973) (In Japanese). 2] Chen, H. C. and Patel, V.C." Calculation of Trailing-Edge, Stern and Wake Flows by a Time- Marching Solution of the Partially-Parabolic Equa- tions", IIHR Report No. 285, IIHR, University of Iowa (198.5). 77 Schlichting, H."Boundary-Layer Theory", 6th Edi- tion, McGrawHill (1968). Cebeci, T. and Bradshaw, P." Momentum Trans- fer in Boundary Layers", Hemisphere Publishing, p. 168 ( 1977). [14] Sarda, O.P."Turbulent Flow Past Ship Hulls- An Experimental and Computational Study", Ph.D. Thesis, Univ. of Iowa (Aug. 1986). [15] Toda, Y. et al."Mean-Flow Measurements in the Boundary Layer and Wake of a Series 60 Cb=0.6 Model Ship With and without Propeller", IIHR Report No.326, Iowa Institute of Hydraulic Re- search, Univ. of Iowa (Nov. 1988~. [16] Okuno,T." Study on Three-dimensional Boundary- Layers on Ship Hulls", Ph.D. Thesis, Univ. of Os- aka Prefecture (In Japanese) (Nov. 1980~. t17] Fukuda, K. and Fujii, A."Turbulence Measure- ments in Three Dimensional Boundary Layer and Wake around a Ship Model", J. of SNAJ, Vol.1.50 pp.85-98 (In Japanese) (Dec. 1981~.

Fig. 2-1 Area covered by the discretized governing equations. ( x-y ) Symmetry Plane ~ Fig. 3-2 Surface grid on a Series 60 (Cb=0.6) hull BY n (b) Body section (c) Wake region Fig. 0-2 Grid topology and boundaries n ~ = from body surface ~ Bay Surface Mapping Singularity Fig. ~>-3 Norma1 distance Fig. 3-1 Spline surface on a Series 60 (Cb=0.6) hull (a) Init,ia,1 grid (b) Modified grid Fig. 3-3 Surface grid on a propeller blade (a) Perspective view ~/~/~N (b) i=.3(S section (midship) (c) i=60 section (stern) Fig. 3-~1 Gricl around a Series 60 (Cb=0.6) hull 78

0.005 Iw o O. Case 1 2 4 5 6 7 n ~ \ ~ u 0 0.1 1.0 ~ x Tal~le 4-1 Computation paran~etei~ Body . f l a t p l a t e Wi g I ey Wi g l ey Wi g ley S. 60 Cb=O. 6 S. 60 Cb=O. 7 S. 60 Cb=O. 8 . _ i~ _ 8 . 8 8 8 8 . 8l _ 81 km _ _ 3l l. 3l 1. 31 5. 61 1. 3l l. 31 1. _ 31 1. . Re 4. Ox106 4. Ox106 . 4. OX106 4. OxlO6 4. Ox106 1. 7X106 . 2. 1x106 . E xper i men t Re=1. 7X106 to 18xlO6 Re=4. 5x106 Re=4. 5X106 Re=4. 5x106 Re=3. 94X106' Pn=O. 16 Re=1. 7x106, Fn=O. 21 Re=2. 1X106 Fig. 4-1 Minimum sDacine ~ ~ Fig. 4-2 Flat plate grid '1 ' ' ' ' ' ' ' ' ' 1 O Computed Empirical v v ~ > Fig. 4-3 NVall shea.r stress ~w on a flat plate ~1 o 0 1 30 1 u 0 Computed _ Empirical O ^~V 1,~. ~ O x 1 Fig. 4-4 Displacement thickness 51 on a flat plate 1 . O _ H O o _ Ref. I _ , [12] [14] 1 [14] [14 [15] [~ _ [17] 1 30L J~ ~ . oOO~OOOOOOOOOOOOO 0 O Computed ~v , ~ Empirical | 2 n+ 104 1o6 (a) x=0.07 I ~ r ~ r ~ ~ I i ' ~' ~ y~U~ O Computed Empirlcal 2 n+ 104 (~)) ~=0.5 ' ',~ 1 T _ '~ 1 _ /0 _ , . . . . 1 102 n+ 104 1o6 (c) X=0.953 0 Computed - Empirical Fig. 4-6 Log plot of velocity on a flat plate ~ooo° O 0 0 0 0 0 0 Oooomcmn" <' Computed Fig. 4-5 Shape fa.ctor H on a flat plate ~ x=O.953 O /\ O 10 15 Fig. 4-/ Kinema.t.ic eddy visco~ity ~t on a flat pla.te 1 79

- - - t~P 1 Fig. 4-8 Pressure contours of a. \\Tigley hull Computed ~ Computed u=1.0,0.9,0.8, ~ u=1.0, 0.95,0. --Measured u=0.9,0.8,0.7 | ----Measured (a) z/D = 0.2 (a.) x=0.5 -- km=31 ~ '1 (a) ~ values (b) Number of grid points. 4-11 Effect of changing parameters on wake at O Computed ~ ~< Measured (z/D=0.25) t~q~O O `3 n ~ - - tO.8rO.7, .. ~ u=0~95~0~9~0~8 ,0.7 r O. . 6 (b) x=O.9 1 1 1 1 1 1 l I 1 1 ~ 1 \ ~ Computed ~ \~ u=0.9,0.8,0.7 \\ rOe6' ~ ~ ~ \ ----Measured u=0.9,0.8,0.7,0.6 (c) x=1.0 Computed ~ \~\\ u=1.0,0.95, ~ `\~\\ 0.9,0.85,0.8 \ \\\ ---Measured u=0.95,0.9,0.85,0.8 1 (d) x=1.5 Fia. 4-9 Wake contours of a TViale~ hull Fig. 4-10 Ixinematic eddy viscosity ~` of a Wigley hull at i=53 (x~0.95) 80 Iw 0.005 Iw (c) z/D = 0.8 Fig. 4-12 Wall shear stress ~w on a \Vigley hull 0 x 0 Computed r ~ ~ _ _ · ~ _ o O x 1 (b) z/D = 0.5 J r O Computed Measured (z/D=0.80) 1

2 1 H o Fig. 4-13 Displacement thickness 5~ on a Wigley hull 1 _ O ~ 00 Q ~ ~ ~ Q ~ Q ~ 0 ~Q ~ O z/D=0.2 _ _ ~ z/D=0.5 ~ z/D=0.8 . . . . . . . . . O x Fig. 4-14 Shape factor H on a Wigley h~ll 0.005 1 ~ I I ' ' ' ' ' ' O 30 ~ . /,,, ~ C Computed O - + _ ~ _ ~ Measured (z/D=0.25) Oi u ~ O j=13 (z/D=0.22) 61 ~ O 1 ~ ~ ~ j=9 (z/D=0.51) 1 O ~ ~ ~ j=5 (z/D=0, 79) ~ 1 1o2 n+ 104 1o6 ~ 0 _ (a) i = 25 (x~ 0.07) ~ ~ O , , , , , , 1 30 ~ , , ~ ~ O ~ 1 _ ~ (a) z/D = 0.2 u+ ~ o j=14 (z/D=O.l9) °-°°51 ' ' ' ' ' ' ' ' ' ~ ~ ~ ~ j=10 (z/D=0.5 j=6 (z/D=0.7E Computed ~ O , , , Measured (z/D=0.54) ~ 1 102 n 104 1C 6 ~ (b) i = 38 (x~ 0.5) 61 1 ^~' 1 4OI . . , . . ~=~ u+~ ~ ~ 0 1 ~ O j=13 (z/D=0.22 O x 1 ~ ~ j=9 (z/D=0.51) _ (b) z/D= 0.5 ~ 0 j=5 (z/D=0.79) 1 o 0.0~ 1 1o2 n+ 104 10 _ (c) i = 53 (x~ 0.95) ~ Computed Fig. 4-15 Log plot of velocity on a Wigley hull -1 1 ~ Measured (z/D=0.80) l 1 ~ ~ 0: ~: ~ L; , 0~ O x 1 0 5 n/62 10 15 (c) z/D = 0.8 (a) i = 25 (x~ 0.07) vt ~~ ~ ~ ao uw6 0 1 0 1 ;~ ~~o~ 0 0 ~ - o 5 n/62 10 15 (b) i = 38 (x~ 0.5) 81

1 At uw62 r Sopor on on Lo 0 o n/62 1° 15 (c) i = 53 (x~ 0.95) t-16 Kinematic eddy viscosity u~ on a Wigley hull Computed ---Measured u=1.0,0.9,0.8,... (a) x = 0.5 ; (b) Cb=0.7 / . . . (C) C13=0.8 Fig. 4-17 Pressure contours on Series 60 hulls x=o .8 P~-'O O 9 P-O .05 1 0 (A ~'~\~\L'L§LL~ Computed Measured Fig. 4-18 Pressure contours at the stern of a Series 60 (Cb=0.6) hull Computed --- Measured u=1.0,0.9,0.8,... (b) x = 0.7 Computed ---Measured u=0.9,0.8,0.7,., (d) x = o.g Computed ---Measured u=0.9,0.8,0.7,. (e) x = 0.95 - Computed i:' ---Measured u=0.9,0.8,0.7,..1. (g) X = 1.05 - Computed ---Measured u=0.9,0.8,0. (I) x= 1.0 --Computed -- Measured u=0.9,0.8,0.7 (h) x = 1.1 Fig. 4-19 Wake contours on a Series 60 (Cb=O.G) hull 82

o . 408 ~w O z/D=0.2 z/D=0.5 $~8096~^ ·: o O x 1 Fig. 4-20 Wall shear stress ~w on a Series 60 (Cb=0.6) hull o.oo ~1 H + u O x=0.507 z/D=0.240 x=0.501 z/D=0.495 x=0.496 z/D=0.834 (b) i = 38 0 z/D=0.2 z/D=0.5 0 z/D=0.8 ~ ^0 /\ O x 1 Fig. 4-21 Displacement thickness 5~ on a Series 60 (Cb=0.6) hull I l , o Q ~ oo O o /\ o ,!\ o '\ o ~ _ '` ° 8 o - ~ _ ~4o~ 0 z/D=0.2 ~ z/D=0.5 O z/D=0.8 Fig. 4-22 Shape factor H on a Series 60 (Cb=O.~) hull o L: x=0.073 z/D=0.221 x=0.074 z/D=0.489 x=0.074 z/D=0.800 . . . -1 1o2 n+ 104 1`' (a) i = 05 Jo6 - ,L/~~ _ . . . . ~ _ 0 o° / ~/ O x=0.942 z/D=0.184 x=0.935 z/D=0.493 _ 0 x=0.935 z/D=0.805 1 ~ I I 1o2 n+ 104 1o6 (c) i = 53 Fig. 4-23 Log plot of velocity on a Series 60 (Cb=0.6) hull vt uw62 83 ol - /0 tP dP ~o /B ~' 0 5 n/62 1 (a) i = _5 0 x=0.073 z/D=0.221 x=0.074 z/D=0.489 ~ x=0.074 z/D=0.800 _ =0 ~ _0 15 ,1 . ---. . o 0 5 n/62 10 15 _ i / 0 x=0.507 z/D=0.240- iO 0 0 ~ O ~3 x 0.501 z/D=0.495_ ~ x=0.496 z/D=0.834 (b) i = 38

1 at uwd2~ ~ 5 n/62 10 _ r ^~^ a A o,~08 V @^ O o ~ ALU . ~ -;) J I/ L,I . OU:) Cal x=0.942 z/D=0.184 ~ x=0.935 z/D=0.493 - ~ .~_^ ~ ~ ~ ~ /~_m O^C o o ~ O ~ A (c) i = 53 Fig. 4-94 Kinematic eddy viscosity zig on a Series 60 (Cb=0.6) hull , ,, \` Computed u=l.O,O.9, . . Computed ~ ~ ~~ ~ n) _ u=l.O,O.9, . . . ---Measured |~-1.O | ---Measured 1. u=0.9,0.8,... \ (a) x = 0.9 u=0.9,0.8,... (b) x= 0.95 Fig. 4-25 Wake contours on a Series 60 (Cb=0.7) hull Computed ~1 u=1.0,0.9,...1.0- - Measured u=0.9,0.8,... Fig. 4-96 Wa.ke contours on a Series 60 (Cb=0.8) hull. x= 0.95 Supplement After the present paper was submitted for the pre- sentation at the Conference, the following disagreement between the two measured results t16],(17] on the wake contours of the Series 60 (Cb=O.~) model has been found out. In Fig. A-1 the computed wake is compared u=o.9 with the ~~easure~ent obtained by Okuno t164? while in Fig. 4-26 it is compared with the the measured re- sult ol:'tainecl l:>,y Ful;uda. and Fujii [174. The co~nputecl wake in Fig. A-1 shows reasonable agreement with the Ol;uno's measurement, whereas it shows significant dis- crepa.ncy from the Fukuda's measurement in Fig. 4-26. This clearly shows that measurements must be carefully valiclat.ed before they are used in the validation of com- 5 licit. a,tio~. it Computed 1 0 u=l.O,O.9, . . . Measured[16] 1 ~4 u=0.9,0.8,... Fig. A-1 Wake contours on a Series 60 (Cb=0.8) hull. x=0.95

DISCUSSION by C.M. Lee I know that a few people in the past have shown the NS-Solver results for the DTRC body forms. I am wondering if further progresses in computational techniques for a fully appended submerged body with propeller have been made. From the paper of Dr. Fu jii this morning, the CDF people in aeronautics seem to have progressed to the stage that they can compute the f low about a fully appended airplane. Have you tried to include the sail and stern control surfaces in your computations? Author's Reolv I have tried to compute a ship-stern f low with propeller effect, using the body force method. Though I got a converged result, there was pressure oscillation, which is due to the use of central differencing. I have not yet tried to compute f lows with appendages. In order to do that, I think a multi-block ap- proach should be used. 85