**Suggested Citation:**"Numerical Calculations of the Viscous Flow over the Ship Stern by Fully Elliptic and Partially Parabolic Navier-Stokes Equations." 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:**"Numerical Calculations of the Viscous Flow over the Ship Stern by Fully Elliptic and Partially Parabolic Navier-Stokes Equations." 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:**"Numerical Calculations of the Viscous Flow over the Ship Stern by Fully Elliptic and Partially Parabolic Navier-Stokes Equations." 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:**"Numerical Calculations of the Viscous Flow over the Ship Stern by Fully Elliptic and Partially Parabolic Navier-Stokes Equations." 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:**"Numerical Calculations of the Viscous Flow over the Ship Stern by Fully Elliptic and Partially Parabolic Navier-Stokes Equations." 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:**"Numerical Calculations of the Viscous Flow over the Ship Stern by Fully Elliptic and Partially Parabolic Navier-Stokes Equations." 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:**"Numerical Calculations of the Viscous Flow over the Ship Stern by Fully Elliptic and Partially Parabolic Navier-Stokes Equations." 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:**"Numerical Calculations of the Viscous Flow over the Ship Stern by Fully Elliptic and Partially Parabolic Navier-Stokes Equations." 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:**"Numerical Calculations of the Viscous Flow over the Ship Stern by Fully Elliptic and Partially Parabolic Navier-Stokes Equations." 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:**"Numerical Calculations of the Viscous Flow over the Ship Stern by Fully Elliptic and Partially Parabolic Navier-Stokes Equations." 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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Numerical Calculations of the Viscous Flow over the Ship Stern by Fully Elliptic ancl Partially Parabolic Navier-Stokes Equations K. J. Oh and S. H. Kang Seoul National University Seoul, Korea T. Kobayashi University of Tokyo Tokyo, Japan Abstract Two computer codes have been developed to solve the Reynolds averaged Navier-Stokes equations; namely the fully elliptic method and the partially parabolic method. These are applied to simulate flows over the stern of the SSPA model as a bench mark as well as a multi-purpose ship with a barge type stern. The numerically generated body-fitted coordinate system is used to manage the complex geometry of the ship- hull. A standard form of the k~ turbulence model is adopted for modelling of the Reynolds stresses. Simulated results by both methods are nearly identical when the longitudinal flow reversal does not appear. The partially parabolic method requires only half of the memory storage and cuts CPU time by 20% in comparison with the fully elliptic method. The capability of programs developed in the present study are confirmed by sucessfully simulating pressures, skin frictions and mean velocities over sterns of the two models. The growth of the viscous layer over the stern is well-simulated and the secondary motion is also captured, which is usually observed in the experiments. Nevertheless tl~e standard form of the k~ model is not adequate for predicting the turbulent kinetic energy over the stern. Simulated nominal wake fractions show good accordance with wake measurements. However, values of the outerpart of the wake are over-estimated, while the trends of the circumferential variations are consistent with wal;e measurements. Coefficients of the viscous resistance predicted by present methods are under-estimated by 10 percent. If further developments on the turbulence model and numerics are accomplished, this method of numerical simulations of the viscous flow over the stern would be promising for the hull form design. 1. Introduction The importance of the viscous flow simulations around the ship hull has received wide acknowledgement in the light of the hull-form design. Predictions of the viscous resistance are useful in the stage of the bare hull-form design. Ship forms of good resistance and propulsion performance cannot be developed without considering the propulsion efficiency as well as the form factor. Such design and development can be effectively attained, only if numerical method can estimate form factors, nominal and effective wakes on the propeller plane, and thrust deduction factors. These design parameters can not be reasonably obtained without complex three-dimensional turbulent flow simulations over the ship stern and in the wake. Viscous flows over the ship hull have been calculated by the three-dimensional boundary layer theory. If free surface effects are excluded, experiments and calculations indicate that the first-order boundary layer equations adequately describe the flow over a large part of a ship hull. But it begins to break down gradually over the stern, which is around 10-20 percent of the ship lengtht1,2~. Experimental information pertaining to the evolution of the flow over the stern as well as in the near wake has been reviewed by Patel[2~. Much research has been done for thick boundary layers over the stern in the past, but they have failed to provide a designer with valuable information. The partially parabolic, or the semi-elliptic type, of the Navier-Stokes equations have been recently employed to simulate the complex viscous flow over the stern instead of the full elliptic Navier-Stokes(NS) equations in consideration of physical phenomena that there is usually no region of flow reversal in the direction of ship motion. These equations can be used to describe flows between the thin boundary layer upstream and the wake far downstream from the ship. The partially parabolic Navier-Stokes(PPNS) equations have been first employed to calculate flows and heat transfer in the straight square duct by Pratap and Spaldingt3~. Abdelmeguid et al.~4] was the first to have applied to ship hulls. Markatos et al.~5], Muraokat6,7] etc. have presented further researches and several paperst8,9,10] appear in the 2nd Symposium on Ship Viscous Resistance in 1985. Chen and Patel[11] have adopted the finite analytic numerical scheme and produced reasonably accurate results for flows exte.rnn1 to an axisymmetric body of revolution and three-dimensional mathematical models. A computer program STERN/PPNS has been developed based on the partially parabolic method and applied to several models to demonstrate its performance by Kang and Oh[12,13,14]. The program proved to be reasonably accurate in describing the pressure distributions on the hull and the velocity contours. When there appears flow reversal over the hull, the NS equations should be solved. A computer code STERN/NS has been developed in the present study and it's performance has been investigated by cross checking each of their respective similated results of flows over the stern. The SSPA 720 model is selected as an bench-mark model and a multi- purpose ship with a barge type stern for the present study. The possibility for the program to be used for design purposes is investigated in the present paper by estimating the viscous resistance and nominal wakes on the propeller plane of a barge type ship form. Estimated results are compared with measured data in the towing tank. Before going further, basic equations and calculation method are briefly summarized. At' ''id I' - ~~A ·~1 At/ C111 175

2. Governing Equations and Boundary Conditions 2.1 Governing Equation Geometry of the ship hull is described in the cylindrical coordinate system (x, r, D) as shown in Fig.1. Governing equations for the incompressible, steady, and turbulent flow are given by the continuity and Reynolds averaged Navier- Stokes equations. Reynolds stresses are modelled by using the eddy viscosity. In the k~ turbulence model adopted in the present study, the. eddy viscosity is given by turbulence kinetic energy k and dissipation rate ~ which are obtained from their transport equations. In the cylindrical coordinate, above governing equations can be written in the general form as follows; a 1 d 1 ~ d df - (U¢) + - (rV¢) + - (W<l))= (rat) + ax r dr r 69 ax ax + -(rail, 'a ) + -(~+ ~ ) + SO (1) r dr fir r dD dD where ¢, Ill,, S<, stand for the flow variables, diffusion coefficient and source terms for each variables. These are summarized in Table 1. 2.2 Boundary-Fitted Coordinate System The calculation domain is bounded by the hull surface S. the center-plane C, the water surface W. the upstream section A, the downstream section B. and the outer boundary ~ far from the hull surface as shown Fig.1. A boundary fitted coordinate is adopted to transform the physical domain in the cylinderical coordinate into a rectangular computational domain in Fig.2. The numerical grid generation is widely used for such a transformation. With the values of the coordinates specified on the boundaries of the physical domains, r, D), it then remains to generate the values of these coordinates in the interior of the domain(5, at, I). This can be obtained from the solutions of the following elliptic partial differential equationst11~; V2x = 0, V2r = , V2~ = 0 (2) where 2 2 2 2 V2= gll_+g22 ~ +g33 d +2 12 ~2 ~2 dt2 d53 ~2 ~2 ~ ~ ~ 656L days 65 tell d: A: upstream B: downstream S: huh ond wolf oxis it: cuter by C: center plan. W: wet. sutoce / ! / 1 r 'Was - - ,~ fl(~5 ~ t;~=_gllX55 ~~ ' / Fig.1 Physical domain and coordinate system. Fig.2 Computational domain. w,: Fig.3 Body-fitted coordinate system and velocity components. ,gij denote metric tensors, and fi are control functions for alignment of grid points in the domain. Design parameters and expermental data are usually presented at each stations of the ship hull. Therefore it is convenient if constant ~ planes are chosen to be coincided with constant x planes. If we put =x), then following equations are obtained. g11r + g22r + g33r`; + 2gl2r~ + 2g13r`; + 2g23r + fore + ford + fork = (3) r gll065 + g220~ + g330~` + 2gl20~ + 2g130~` + 2g230~; + flea + f20 + f30 Grid control function f1 is calculated from the distribution of the x planes. In the radial direction, Mid contol function f2 is determined by grid distributions on the inlet and exit plane for grid lines to be smoothly generated. Grid control function fir is prescribed with the grid distribution on the outer boundary. x~ f (5, 71, I) =+ F2 (6) rr ,' (5) where 176 f3(5 ~ A) = _ g33[ `~]5 ~ (7 2 [(525)F1+ (551)F2] F (5, A, () = (52- 51) F1(g, A, L) = g []~=61 '

F2(g, ~, t) = Am r ]~=62 51, 52 denote values of ~ corresponding to upstream and downstream sections respectively and ~2 denote the value at the outer boundary. Symmetric Neumann boundary conditions are used at the water plane (~=oO) and the center plane (~=90°~. Dirichlet boundary condition is used at other boundaries. the fully turbulent layer, and it is assumed that the law of the wall is satisfied and the velocity vectors.in this region are collateral. The boundary values at the first grid point are obtained by assuming the local equilibrium between turbulent kinetic energy production and dissipation. They are given as follows. Kilo P In (Enp+) low (10) ., _ 2.3 Transformed Governing Equations C b2 ~ (11) P Transformation of independent variables (x, r, D) in governing equations are considered, leaving velocity 3,4 3,2 components (U. V, W) in the original (x, r, D) coordinate in Cal k Fig.3. Then governing equations are generally represented as ~ = ~ (1~) the following formt12]. Kn [(bluish + b2lV~ + b3lWl) +(bl2U~ + b22V~ + b3Wl) +(bl3U4> + b23V~ + b33W¢~] [~r<,,Jgll ~ ~ + d Or Jg22d~ ~ + d Go Jg33d~ '+ S. (8) The above equations are still the exact equations in so far as no approximations have been made beyond those inherent in the turbulence model. The equation (8) can be rendered partially parabolic by neglecting the first term which involves the second order derivative term with respect to 5. Physically this is not the same as neglecting ~xr nor does it imply that diffusion in either x or ~ direction is neglected[11~. 2.4 Boundary Conditions Boundary conditions at each boundaries of the solution domain are summarized as below. (1) Upstream A; The position is extended to the upstream as far as thin boundary layer equations and the potential flow theory are valid. Then distributions of (U. V, W. k, c) can be prescribed from boundary layer calculation. If it is placed over the mid-ship, then distributions may be assumed by using integral parameters without exerting significant influences on downstream calculation. The streamwise velocity profile in the boundary layer is specified by 1/7th law, U A t7 = (_) Ue ~ (9) and the velocity in the inviscid region is given as the free stream velocity. The turbulent kinetic energy k and the dissipation rate ~ are also given by the flat plate correlations with the boundary layer thickness and skin- friction coefficient. (2) Downstream B.; At one ship length downstream from the stern, zero gradient condition is assumed for the all variables. In partially parabolic calculation, only the zero pressure gradient condition is required. (3) Hull surface S; The wall function is adopted in the present study. The grid points next to the wall are located in The magnitude of the velocity at the first grid point np near the wall is given as V, and n is nomal distance from the wall. (4) External boundary Hi; It is placed sufficiently far from the hull surface so that uniform flows and no turbulence condition can be assumed there. dV U = UO, W = k = ~ = 0,= 0, p = pO (13) do (5) Center plane C and water plane W; Symmetric condition are imposed. dU dV dk de W= 0, == = = 0 (14) dt dt dt d: (6) Wake center line C; Following conditions are enforced. dU dk de V= W= 0, === 0 (15) din do do 3. Numerical Scheme Uniform grid spacings are taken in the calculation domain (1~=/~=~=1) and grid control functions are determined by specified values on the boundarys. The grid construction is obtained by solving equation (3), (4) by the finite difference method. The Finite Volume Method is applied for discretizing the governing equations and the hybrid scheme is employed in the evaluation of the convection terms. The finite difference equations are obtained by integrating the governing equations over individual control volumes formed by the staggered grids system[151. The scalar variables p, k, ~ are located at the grid nodes themselves, while velocity components are positioned between the scalar nodes. Such a staggered grid has benifit of having the velocities at the boundaries of the scalar cells where they are needed in integrating convective terms. Furthermore the pressure nodes are located on either side of the velocity node and it is easy to calculate the pressure gradient terms in the momentum equations. Then the final form of the discretized governing equations are obtained. amp = aN¢N + a5~1'S + aE¢E + aW<>W + aUlu + aDID + So (16) 177 The subscript P refer to the grid node to be considered and the subscript U. D correspond to the upstream and downstream grid respectively. The other neighbouring grids in the sections are given by the subscript N. S. E, W. The a' ~D' AN' AS etc. represent the convection and diffusion at each corresponding control surface[16].

o.,l ~ sly On. 0.2 0 3 \x s's ~ ~ N ,\\, AIL Fig.4 Over-view of generated grids of the SSPA model. 0.2 0.11 When the partially parabolic form of equation (8) is solved, on the discretized equations are obtained by the similar way. apt = aNIN + aS¢S + ante + avow + Cu4)u + SO (17) Since the diffusion term in the ~ direction is removed from the equation (8), only the convection term Cu is included in the ~ direction. The unknown variables (U. V, W. k, e) can be obtained by solving the equation (16) or (17) under the assumed or estimated pressures. The estimated pressures are indirectly corrected for the continuity equation to be satisfied. If SIMPLE(Semi-Implicit Pressure Linked Equation) algorithm is adopted[15i, the discretized equation for the pressure-correction is obtained. This equations can be represented by the same form with the equation (16), which has fully elliptic characteristics. In the fully elliptic calculation, the flow variables are iteratively solved and the converged solutions are obtaiined. The procedures are summarized as: (1) Construct the coordinate system, and calculate the metric tensors and Jacobian. (2) Specify initial conditions at the inlet plane. (3) Solve the velocities with assumed or previously calculated pressures. (4) Solve the pressure-correction equations (5) Correct pressure distributions and velocities. (6) Calculate the k, it. (7) Return to step(3) and repeat step(3~-~7) untill the residues are reduced by 0.1% of the reference values. To solve the partially parabolic equation, the marching procedure along the ~ -direction is employed. (U. V, W. k, c) at each sections are calculated with upstream values of each variables and previously calculated pressure. Pressure-corrections are achieved on each section without any correction of the upstream and the downstream pressures during the marching procedure. Pressure of the whole domain should be stored in the partially parabolic method and several sweeps in the 5-direction are required to obtain the converged solutions. The procedure are summarized as: (1) Construct the coordinate system, and calculate the metric tensors and Jacobian. (2) Specify initial conditions at the upstream boundary. (3) Calculate velocities at the downstream with the previously calculated pressures. . cp 0.1 r v ~x ._~- , _ of, ~ ~~ZI.3 _ STREAK LINE ~ it' I.1 1.2 13 I! ~ :~ .1121.3 1 o. 01 no- . I ~ I I ~ I 0.5 0.6 0.7 0.8 o.9 1.0 1.1 1.2 1.3 x/L Fig.5 Pressure distributions over several streamlines on the SSPA model. 0.4 _ QS ~ Cf pPNS ~ NS 0.4 · EXP(17) STREAM LINE 7 o' Fat ·: ·\ · STREAK LINE S ~ ,. .~% 08 0.9 ·~ 0.4 _ STREAK LINE 3 0.34 ~ do_ on o.S 1 o · ,~ 0.5 __ · - ~ Q4 :: ~ · ·~, STREAK LINE I Ql _ 0.0 _ 0.5 O.2 ~ O.' .0 0.6 0'7 0'8 o'g 1. l X/L Fig.6 Skin friction coefficients over several streamlines on the SSPA model. 178 (4) Correct the pressure distributions and velocities. (5) Calculate k, ~ at the downstream (6) Marching to the downstream boundary. (7) Return to step(3) and repeat step(3)-(6) untill the mass residue are reduced by 0.1% of the reference value.

cp 02r o. of o.ll PPNS NS EXP(17) ~ " . ,, o too 005 o:lo o Is 2G Fig.7 Girth-wise variation of pressure coefficient at x/L=0.9 of the SSPA model. 0.004 n non . it- -. .~ . . PPNS - - - - NS · EXP(17) o.o~ . 1 O. 00 0.05 0.10 0.15 2G Fig.8 Girth-wise variation of skin-friction coefficients at x/L=0.9 of the SSPA model. 4. Calculations and Discussion . . A computer code, STERN/PPNS based on PPNS equations has been developed by the method described above and is applied to flows over several mathematical models. Calculated results by STERN/PPNS have been discussed by Kang et al.~12,13,14~. Simulated flow fields and pressure distributions are generally in good agreement with tested data. But calculations have a marked trend to over-estimate turbulent kinetic energies near the stern. Such a trend has also been pointed out by Chen and Patel[11~. Another code STERN/NS based on NS equations has been developed in the present study. General performances of the code have been checked by simulating flows over several mathematical models by Oh[163. Performance characteristics of two programs have been intensively investigated and compared in the present study by simulating flows over the SSPA 720 model and a multi purpose barge type ship. The first one was a container ship model, which was tested in the wind-tunnel by Larssont17] and used as one of standard models of II:C-SSPA Workshopt23. The latter model has a barge type stern, which was chosen to investigate the possibility of the numerical simulation of viscous flows to be used for design purposes. 4.1 SSPA 720 model Part of the numerically generated grid system is presented in Fig.4. Numbers of meshes in the (5, a, (~-directions are (58, 25, 14) respectively. They cover the calulation domain of O.S<x/L <2.0, r/L <0.8. Circumferential grids are distributed to coincide with external stream lines along which measurements have been taken. Meshes in the radial direction are progressively closely spaced near the hull. Calculation is performed at Reynolds number 5.0x 106, where the model was tested in the wind tunnel. Converged solutions are obtained after 90 sweeps by STERN/PPNS and 190 iterations by STERN/NS. PPNS procedure save CPU time by 20 percent as well as considerable memory storage in comparison with the full elliptic procedure. 179 Fig.9 Simulated viscous layer over the stern of the SSPA model by STERN/NS. n.n5 al ResilIt he rh, NC b) Result by the PPNS . . . _ 0.05 n.nn n no y/L Fig.10 Comparison of calculated velocity contours and cross velocity vectors at x IL =0.95 of the SSPA model. Estimated pressure coefficients along each streamlines are shown in Fig.5. Both calculations are in very good agreement with blockage corrected data. The present programs can take into account the blockage effect by considering the existence of the outer wall. Even though STERN/NS shows slightly lower estimations than STERN/PPNS, differences are nigligible in this case of flow without separation. However, some attenuations appear in the estimated pressures just over the stern by using fully elliptic calculations. Such attenuations might be originated from neglected longitudinal diffusion terms in the partially parabolic method. The capability of the both programs is confirmed to simulate the interaction between the thick boundary layer and the external flow by sucessful prediction of hull pressures. On the other hand, predicted skin friction coefficients are qualitatively in good accordance with measurements as shown in Fig.6. Considerable uncertainty should be taken into account as far as skin friction measurements are concerned. It is also true that enough number of grids could not be allocated near the surface. It is seen that coefficients of skin friction reduces over the stern, where the shear layer grows thick according to the strong adverse pressure gradient and the stream line convergence except along the keel line. Girthwise

°°~1 °~'1 °'°~1 o.o 1 on 0.0_, cam_ . PPNS ~ ~ NS i · EXP(17) . STREAK _._1 LINE 7 ~ 0.05 0.00 ~ 0.5' 1. 0 004R ;1 0.03 ]- 0.02 J STREAK f 0.01 LINE ~ ~ . r °°° ~ OS I ~ 0.04 0.03 0.02 STREAM 0.01 LINE 3 ~ _.__ 0.00 ~ 0 5 0 0.04 0.03 0.02 t STREAK ·/ 0.01 LINE 2 ~ n 0 05 0-00 ~ 0.5 1 0 L o.O~ . ~ 0.03 . 0.02 O.C 0.0( 01 STREAM ~ . ~ o.o 0.5 Lo q UO Fig.11 Profiles of resultant velocity at x /L =0.95 of the SSPA model. distribution of pressure coefficients and skin friction coefficients calculated in the present study are compared with measured values in Fig.7 and Fig.8. Secondary flow is directed away from the keel and the water plane according to the girth-wise pressure gradient, and the shear layer rapidly grows thick at the mid-girth. Skin friction coefficients at the keel show their largest values and decreases along the girth to the water plane. The feature of shear layer formation over the stem by 5-l~:RN/NS program is shown in Fig.9 . The contours of the axial velocity component and the pattern of the transverse motion at x/L =0.95 are shown in Fig.10. The boundary layer remains thin along the keel line according to the divergence of streamlines. The thickness of the viscous layer over the mid-girth is almost as large as the draft of the model. The axial velocity contour is well-simulated in comparison with the measured contour. The bilge vortex, which is a general feature of the stern-flow, is also observed in the simulation. Distributions of the total velocity at several points where each streamlines intersect with the x/L = 0.95 section are compared with measured data in Fig.11. Here it should be noted that measurements are obtained nollllal to the hull and calculations are computed on transverse sections. There are good agreements between calculations and measurements, although some error might be involved due to such differences in the location. Turbulent kinetic energy distributions are compared in Fig.12, where typical characters of turbulence in the stern flow appear. Turbulence kinetic energy shows considerable reduction in the magnitude near the hull over streamlines 3, 5, and 7, which is quite a unusual 180 ~ O!eir _ __; 0.04 0.0_ PPNS ~ ~ NS 0.0 ~ STREP 0.0 ~ 0.0 ~ LINE 7 . 0.04 0.0 000 0.004 0.0 08 0.03 \ 1 0.o4t 0.02 ~ STREAM o oc . ; ~ 0.000 0.0 4 0.008 nor 0.02 0.0- ~ 1 o.od4 o.o.j °°;1 n 0.05 on L 0.04 . 0.Of 0.03 . 0.02 , , it. STREAK 01 000 0.004 01 08 -o 1 O.C 0.00 0.000 0.004 0.008 k STREAM LINE 3 _ Fig.12 Profiles of turbulent }kinetic energy at x/L=0.95 of the SSPA model. Fig.13 Body plan of 37K PROBOCON. feature in the thin boundary layer. It is explained that such reduction is due to strong flow convergence without enough generation in the turbulence kinetic energy over the stern. The k~ model in the present study fails to capture such a phenomena taking place over the stern. An algebraic stress model may well be a furture choice for sucessful simulations. 4.2 Barge Type Ship An object of the present study is to investigate the potentiality of programs to be used for design purposes. The selected model, 37K PROBOCON, was originally designed by KSEC(Korea Shipbuilding and Engineering Co.) and developed by SSPA through several series tests in the towing tank. The body plan is shown in Fig.13. Considerable reductions in the viscous resistance as well as increases in the propulsion efficiency have been reported in comparison with conventional stern shapes. Components of the resistance coefficient and measured nominal wake distributions in the towing tank are availablet18~. Furthermore pressure

o. l a. 3 Y/L ztL 0.2 0.3 o s 'I ~ \ ~ ~ \ \ \ \ ~ \ \ - \\ \ ~ \ \ Fig.14 Over-view of generated grids of 37K PROBOCON. ALL distributions on the eorresponsing double body have been measured in the wind-tunnelt193. Numerically generated grids are shown in Fig.14. Numbers of mesh points in the (5, a, t)-direetions are (54, 25, 32) respectively. Calculation is performed in the domain of 0.5<x/L <2.0, r/L <0.963, and at Reynolds number 8.5x106, where the model was tested in the towing tank. Converged solutions are obtained after 90 sweeps with Sl ~;RN/PPNS and 200 iterations with STERN/NS. Pressure distributions at several stations over the stern by the present calculations, by the inviseid calculations with Hess and Smith program, and by the wind-tunnel experiment are compared in Fig.15. The experiment has been carried out at Re~=2.0xlO6 and the measured data includes a small amount of blockage effects. The present methods generally show good performance of the pressure estimation on the hull. The fully elliptic procedure gives somewhat lower predictions of pressure. The potential flow theory does not properly simulate pressures, especially near ~ = 30° - 40° where the boundary layers have been grown thick according to the streamline convergence. Overall feature of axial velocity contours and velocity vectors on transverse sections to the downstream are shown in Fig.16. As observed in the experiment/19], streamlines are uniformly distributed over the stern, and consequently the gradual girth-wise variation of the boundary layer thickness is obtained. Bilge vortex is usually generated over the stern due to the girth-wise pressure distribution. Such vortex is simulated as an open type separation over the afterbody as shown in Fig.17. The region of votex extends and the strengh at the eve decreases downstream as shown in Fig.18. Although the physical propeller plane is not divided into fine enough grids to expect reasonable results, estimations of the nominal wakes are tried. It is not found how to allocate grids effectively not only for the whole flow domain but also for the local region near the propeller by adjusting grid control functions. Wake fractions have been measured at four radial positions (OAR, 0.64R, 0.88R, 1.12R) in the towing tank. R denotes the radius of the propeller. Predicted results in Fig.19 are generally acceptable at 0.64R and 0.88R, except between the stern bulb and the ship bottom, where only a few mesh points can afford to be allocated. However, wake fractions are over-estimated at the outer part of the propeller and under-estimated near the hub. The trends are qualitatively consistent with measurements. As far as grids generations are concerned, it should be stressed that enough meshes are to be allocated locally over the propeller plane, as well as globally over each ship sections. It is also noted that free surface effects are included in wake measurements. 181 PPNS ~ NS POT ~ FYPI14 _~ ~' Cp o At n -a: n I, ~ -0.! : ~ ,,;;'',- 0.90 1 . . ... _.4 . 5 . . . X/L'0.8 1. O. 30. 60. 90. ~ (deg.) Fig.15 G~rth-wise variations of pressures at several stations of 37K PROBOCON. Fig.16 Simulated viscous layer over the stern of 37K PROBOCON by STERN/NS. Finally the capability of codes to estimate the viscous resistance is investigated. It may be obtained by integrating the pressures and the skin frictions over the hull surface of the double model while free surface effects are neglected. But considerable numerical errors are expected during the integration. A wake survey method to estimate the viscous resistance component in the towing test[20] is adopted. Pgr [(HoH(Uo(Uo2 - 2g(HoH)~2~23 w 2g - 1npuO2sW where Ho=po+ pUo2/2, H=p+ pU2i2 and S is the wetted surface area of the hull. The integration position of the wake w is half the ship-length downstrean from the stern. Estimated coefficients of the viscous resistance by the partially parabolic method and by the fully elliptic method are 3.57x10-3 and 3.52x10 3 respectively. On the other hand, the measured value was 3.9x 10-3 and the form factor was 0.23. All the simulation errors, i.e. presures and velocities, are summed up to show 10% under-estimation. It should be noted that wave effects on the viscous resistance can not be talcen into account at all during the resistance test. If we take into account such uncertainties, the present predictions give us acceptable values. If changes in the viscous resistance with any modification of the hull form is reasonably estimated, the computer codes will be very useful as far as the bare hull design is concerned.

x/L 1.0 ~ ~ PPNS , · ~ 0.8 ~ ~ EXP (I -, I.0~0.6 ~~ y'L o. Os , ,, ',,f',, '!,_ o. to y/L Fig.17 Simulation of bilge vortex over the stern of 37K PROBOCON by STERN/NS. Fig.18 Variation of longitudinal component of vorticity In the wake of 37K PROBOCON. ,18,' Uo~U l-0 - 0.6 mu: - Uo ·N 0.8 O. ~ %\ O. ~ · ~ o 1 o. L o. \ 120. 150. 180. ·: 120. 150. 180. ·\ 1 ·t 30. 60. 90. 120. 150. 180. ~ (a.) Fig.19 Variation of nominal wake of 37K PROBOCON (x/L=0.975). 5. Conclusion (1) Two computer program have been developed in the present study. STERN/PPNS simulates flows over the stern by the partially parabolic method, and the SI~ERN/NS by the fully elliptic method. Simulated results are shown to be nearly identical. This indicates that the effects of stream-wise dffusion terms are negligible when the flow reversal does not appear over the stern. They also cross check each others' numerical scheme. The partially parabolic method requires only half of the memory storage and reduces CPU time by 20% in comparison with the fully elliptic method. (2) The capability of programs developed in the present study is confirmed by sucessfully simulating pressures, skin frictions and mean velocities over the stern of the both models. The growth of the viscous layer over the stern is well-simulated and the secondary motion is also captured, which is usually observed in the experiments. (3) There appears to be some deficiency of the k~ model enough to simulate the turbulence fields over the stern. The standard form of the model usually over-predicts the turbulent kinetic energy. It is also investigated that the model cannot properly account for the reduction of the turbulent kinetic energy near the wall when the viscous layer becomes thick over the stern. (4) The streamlines over the stern of the barge type ship form show uniform distributions, consequently the gradual girth-wise variations of the boundary layer thickness and the pressure distribution are noted. 182

Table 1. +, At, and 5~', for the governing equation. rO W vat ''ote; G = v,{2[(_)2+( dV )2+( 1 dW + V 32] ax Or r do r DU dV 2 1 dU dW 2 1 dV dW W 2 +(+ ) +( + ) +( + - ) } dr ax r do ax r dD dr r vat = v + v,, v, = C~k2/e C '=0~09, CD =1.0, C'=1.44, C2=1.92, (rk =1.0, crc=1.3 54 o 1 dp p fix ~ dU 1 d dV 1 ~ a4W + - v' )+ ~rv' )+ ~V' ) ax ax r or ax r 80 ax 1 ap p fir d dU 1 d dV 1 d dW w2 + - v, )+ ~rv, )+ ~v, )+ ax Br r Br Br r R.q Ear r 1 ~ 2ve aW V V r2 ~(V,W) 2 do vie 2 v, 2 _ 1 UP pr 88 d v, dU 1 d dV 1 d v, dW + - )+ ~v,)+ (- ) ax r dB r dr do r do r dD 1 d 2 d v, W VW ~v,W)+~(V)ve 2 - 2 v, dW vie dV v dV + + + r dr r2 89 r2 do GCDE (C IGC26) (5) Simulated nominal wake fractions show good accordance with wake measurements. But values of the outerpart of the wake are over-predicted, while the trends of the circumferential variation are consistent with wake measurements. Setting aside the question of the validity of the turbulence model, it should be studied further how to allocate enough meshes locally over the propeller plane, as well as globally over each ship sections. (6) The capability of the codes to estimate the viscous resistance is investigated. A coefficients of the viscous resistance predicted by the present method are shown to be under-estimated by 10 percent. If further developments on the turbulence model and numerics are accomplished, the numerical simulation of the viscous flow over the stern will be promising for the hull form design. References 1. L.Larsson, SSPA-IITC Workshop on Boundary Layers-Proceedings, SSPA-Publication No. 90., (1981). 2. 13. V.C.Patel, "Some Aspects of Thick Three-Dimensional Boundary Layers," Proc. 14th ONR Symp., (1982). 3. V.S.Pratap, D.B.Spalding, 'Fluid Flow and Heat Transfer in Three-Dimensional Duct Flows," Int. Journal of Heat and Mass Transfer, Vol.19, (1979). 4. A.M.Abdelmeguid, N.C.Markatos, K.Muraoka, D.B.Spalding, "A Comparison Between the Parabolic and Partially Parabolic Solution Procedures for Th-ree- Dimensional Turbulant Flows around Ship's Hull," Appl. Math. Modelling, Vol.3, (1979). N.C.Markatos, M.R.Malin, D.G.Tatcheel, "Computer Analysis of Three-Dimensional Turbulant Flows around Ship's Hulls," Proc. Inst. Engrs., London, Vol.194, (1980). 6. K.Muraoka, "Calculation of Thick Boundary Layer and Wake of Ships by a Partially Parabolic Method," Proc. 13th ONR Symp., Tokyo, (1980). 7. K.Muraoka, "Calculation of Viscous Flow around Ships with Parabolic and Partially Parabolic Solution Procedures," Trans. West Japen Soc. Naomi Arah., Vol.63, (1982). 8. C.E.J~nson, L.Larsson, "Ship Flow Calculations Using the PHOENICS Computer Code," Proc. 2nd Int. Symp. on Ship Viscous Resistence, SSPA, (1985~. 9. H.C.Raven, M.Hoekstra, "A Parabolized Navier-Stokes Solution Method for Ship Stern Flow Calculations," 2nd Int. Symp. on Ship Viscous Resistence, SSPA, (1985). 10. G.D.Tzabiras, "On the Calculation of the 3-D Reynolds Stress Tensor by Two Algorithm," 2nd Int. Symp. on Ship Viscous Resistence, SSPA, (1985). 11. H.C.Chen, V.C.Patel, "Calculation of Stern Flows by a Time Marching Solution of the Partially-Parabolic Equations," Proc. 16th ONR Symp, (1986~. S.H.Kang, K.J.Oh, S.B.Lee, "Study on the Stern Design by Using Viscous Flow Simulations," RIIS Rept.87-092, Coellege of Eng., Seoul N. University, (1987). S.H.Kang and K.J.Oh, "Numencal Calculations of Three-Dimensional Viscous Stern-Flows by Semi- Elliptic Equations,"26(1),(1989). 14. S.H.Kang, K.J.Oh,"Numencal Calculation of Three- Dimensional Using Viscous Flow over a Barge Type Stern by Semi-Elliptic Equations," Seminar on Ship Hyrodynamics, Seoul,(1988). 15. S.V.Pantankar, Numerical Heat Transfer and Fluid Flow, McGrow-Hill, (1980). 16. K.J.Oh, "Numerical Study on the Viscous Flows over the Ship Stem," Ph.D. Thesis, Seoul N. University, (1989). 17. L.Larsson, "Boundary Layers of Ships. Part 3: An Experimental Investigation of the Turbulent Boundary Layer on a Ship Model," SSPA Rept.46, Goteborg, Sweden, (1974). 18. SSPA, 37K PROBOCON Wake Measurements, Rept.3202-12, (1987~. 19. S.H.Kang, J.Y.Yoo, B.Y.Shon, S.B.Lee, and S.J.Bail;, "Experimental Study on Viscous Flows around Ship Sterns by Using the Hot-Wire Anemometer in the Wind-Tunnel,"JSNAK, 11(5), (1987). 20. S.H.Kang and B.S.Hyun, "A Simple Estimation of the Viscous Resistance of Ships by Wake Survey," JSNAK, Vol.19, No.2, (1982). 183