A Viscous Multiblock Flow Solver for Free-Surface Calculations on Complex Geometries

G.Cowles, L.Martinelli (Princeton University, USA)

1 Introduction

The last two decades have brought about many major advances in the ability to predict the steady flow about an advancing ship. Solvers based on several formulations have been developed and validated using experimental data from well known ship forms. Experience has shown that in order to correctly predict the drag of even simple hulls at realistic Froude numbers, the surface cannot be linearized. That is, the dynamic and kinematic conditions must be applied to the fully deformed boundary giving a fully nonlinear update of the free surface. Also, while solution of the inviscid equations can efficiently predict the effects of altering hullform coefficients, in order to accurately predict the total drag, the interaction between the free surface and the viscous layer must be accounted for. This can be done by solving the full Reynolds averaged Navier Stokes equations (RANS) in the entire flowfield in conjunction with a proper model for the eddy viscosity. Lastly, in order to be useful within the time constraints of a typical design schedule, the run time of the solver must be on the order of several hours.

Several of the higher order domain methods include all the necessary aspects described above. However, the quality of the mesh has a large effect on both the convergence and accuracy of these methods. Realistic hullforms can be considerably more complicated than the Wigley hull. Fully appended sailboats and submarines as well as naval combatants with transom sterns give rise to topological constraints which can make it difficult, if not impossible for the grid generator to produce a high quality single block mesh with spacing suitable for Navier Stokes calculations. One solution to this problem is to eliminate the constraints of mapping a single Cartesian domain to the hull by using an unstructured mesh. However the successive generation of high quality, three dimensional unstructured meshes necessary to follow the deformation of the free surface is very expensive and complicated. Also, even with an efficient implementation, the flow solver is only about half as computationally efficient when compared with a structured solver with equal resolution. An alternative route to the improvement of grid quality is to use a multiblock implementation. It can provide exceptional topological flexibility which facilitates a more efficient placement of points as well as smooth the skewness of cells and gradients in spacing.

We decided that in order to begin making routine calculations on realistic ship geometries with run times suitable for use in a design process, a parallel, multiblock implementation was necessary. The strategy was to maintain as many elements of the efficient single processor cell-vertex based solver that was already in place. This vertex formulation [9] had benefitted from several developments. Excellent convergence rates were obtained by using multigrid acceleration, local time stepping, and residual smoothing in the bulk flow. Accuracy stemmed from the utilization of limiters in the free surface dissipation as well as the incorporation of a fully nonlinear free surface boundary condition. As an intermediate step towards our goal, a single block, parallel version of the solver was developed. This required a change from the cell-vertex stencil to a cell-center. This was because in the method of domain decomposition, a cell-vertex scheme requires processors to share flow field values at subdomain faces which can lead to difficulties with implementation at the interface. Details of the method as well as some results can be found in [17]

Once that was in place, a parallel, multiblock version was implemented in which viscous terms as well as a turbulence model are included. The multiblock strategy has only minimal effect on the efficiency of the code. Thus, similar to the single block, parallel version, inviscid flow solutions are achieved in minutes on eight processors. Due to the increased number of grid points, reduced time step, and added work in calculating the viscous terms, solutions of the Navier Stokes equations require a factor of 20 increase in CPU time over



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Twenty-Second Symposium on Naval Hydrodynamics A Viscous Multiblock Flow Solver for Free-Surface Calculations on Complex Geometries G.Cowles, L.Martinelli (Princeton University, USA) 1 Introduction The last two decades have brought about many major advances in the ability to predict the steady flow about an advancing ship. Solvers based on several formulations have been developed and validated using experimental data from well known ship forms. Experience has shown that in order to correctly predict the drag of even simple hulls at realistic Froude numbers, the surface cannot be linearized. That is, the dynamic and kinematic conditions must be applied to the fully deformed boundary giving a fully nonlinear update of the free surface. Also, while solution of the inviscid equations can efficiently predict the effects of altering hullform coefficients, in order to accurately predict the total drag, the interaction between the free surface and the viscous layer must be accounted for. This can be done by solving the full Reynolds averaged Navier Stokes equations (RANS) in the entire flowfield in conjunction with a proper model for the eddy viscosity. Lastly, in order to be useful within the time constraints of a typical design schedule, the run time of the solver must be on the order of several hours. Several of the higher order domain methods include all the necessary aspects described above. However, the quality of the mesh has a large effect on both the convergence and accuracy of these methods. Realistic hullforms can be considerably more complicated than the Wigley hull. Fully appended sailboats and submarines as well as naval combatants with transom sterns give rise to topological constraints which can make it difficult, if not impossible for the grid generator to produce a high quality single block mesh with spacing suitable for Navier Stokes calculations. One solution to this problem is to eliminate the constraints of mapping a single Cartesian domain to the hull by using an unstructured mesh. However the successive generation of high quality, three dimensional unstructured meshes necessary to follow the deformation of the free surface is very expensive and complicated. Also, even with an efficient implementation, the flow solver is only about half as computationally efficient when compared with a structured solver with equal resolution. An alternative route to the improvement of grid quality is to use a multiblock implementation. It can provide exceptional topological flexibility which facilitates a more efficient placement of points as well as smooth the skewness of cells and gradients in spacing. We decided that in order to begin making routine calculations on realistic ship geometries with run times suitable for use in a design process, a parallel, multiblock implementation was necessary. The strategy was to maintain as many elements of the efficient single processor cell-vertex based solver that was already in place. This vertex formulation [9] had benefitted from several developments. Excellent convergence rates were obtained by using multigrid acceleration, local time stepping, and residual smoothing in the bulk flow. Accuracy stemmed from the utilization of limiters in the free surface dissipation as well as the incorporation of a fully nonlinear free surface boundary condition. As an intermediate step towards our goal, a single block, parallel version of the solver was developed. This required a change from the cell-vertex stencil to a cell-center. This was because in the method of domain decomposition, a cell-vertex scheme requires processors to share flow field values at subdomain faces which can lead to difficulties with implementation at the interface. Details of the method as well as some results can be found in [17] Once that was in place, a parallel, multiblock version was implemented in which viscous terms as well as a turbulence model are included. The multiblock strategy has only minimal effect on the efficiency of the code. Thus, similar to the single block, parallel version, inviscid flow solutions are achieved in minutes on eight processors. Due to the increased number of grid points, reduced time step, and added work in calculating the viscous terms, solutions of the Navier Stokes equations require a factor of 20 increase in CPU time over

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Twenty-Second Symposium on Naval Hydrodynamics the inviscid case. Thus, converged NS runs require about three hours on eight processors. The establishment of an efficient baseline multiblock flow solver is extremely important because it enables both sea-keeping analysis and optimization to become practical tools in the design process. 2 Mathematical Models For a viscous incompressible fluid moving under the influence of gravity, the differential form of the continuity equation and the Reynolds Averaged Navier-Stokes equations (RANS) in a Cartesian coordinate system can be cast, using tensor notation, in the form, Here, is the mean velocity components in the xi direction, the mean pressure, and the gravity force acting in the i-th direction, and is the Reynolds stress which requires an additional model for closure. For implementation in a computer code, it is more convenient to use a dimensionless form of the equation which is obtained by dividing all lengths by the ship (body) length L and all velocity by the free stream velocity U∞. Moreover, one can define a new variable Ψ as the sum of the mean static pressure P minus the hydrostatic component −x3Fr−2. Thus the dimensionless form of the RANS becomes: where is the Froude number and the Reynolds number Re is defined by where ν is the kinematic viscosity, and is a dimensionless form of the Reynolds stress. Figure 1 shows the reference frame and ship location used in this work. A right-handed coordinate system Oxyz, with the origin fixed at the intersection of the bow and the mean tree surface is established. The z (x3) direction is positive upwards, y (x2) is positive towards the starboard side and x (x1) is positive in the aft direction. The free stream velocity vector is parallel to the x axis and points in the same direction. The ship hull pierces the uniform flow and is held fixed in place, ie. the ship is not allowed to sink (translate in z direction) or trim (rotate in x−z plane). It is well known that the closure of the Reynolds averaged system of equation requires a model for the Reynolds stress. There are several alternatives of increasing complexity. Generally speaking, when the flow remains attached to the body, Figure 1: Reference Frame and Ship Location a simple turbulence model based on the Boussinesq hypothesis and the mixing length concept yields predictions which are in good agreement with experimental evidence. For this reason a Baldwin and Lomax turbulence model has been initially implemented and tested [5]. On the other hand, more sophisticated models based on the solution of additional differential equations for the component of the Reynolds stress may be required. Notice that when the Reynolds stress vanishes, the form of the equation is identical to that of the Navier Stokes equations. Also, the inviscid form of the Euler equations is recovered in the limit of high Reynolds numbers. Thus, a hierarchy of mathematical model can be easily implemented on a single computer code, allowing study of the controlling mechanisms of the flow. For example, it has been shown in reference [9] that realistic prediction of the wave pattern about an advancing ship can be obtained by using the Euler equations as the mathematical model of the bulk flow, provided that a non-linear evolution of the free surface is accounted for. This is not surprising, since the typical Reynolds number of an advancing vessel is of the order of 108. Free Surface Boundary Conditions When surface tension as well as tangential stresses are neglected, the boundary condition on the free surface consists of two equations. The first, the dynamic condition, states that the pressure acting on the free surface is equal to the stresses normal to the free surface. It was found that the inclusion of the viscous stresses had little to no effect on the solution since they are of the order . Thus the dynamic condition that pressure is a constant is a good approximation. The kinematic condition states that the free surface is a material surface: once a fluid particle is on the free surface, it forever remains on the surface. The dynamic and kinematic boundary conditions may be expressed as p=constant

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Twenty-Second Symposium on Naval Hydrodynamics (1) where z=β (x, y, t) is the free surface location. Hull and Farfield Boundary Conditions The remaining boundaries consist of the ship hull, the meridian, or symmetry plane, and the far field of the computational domain. In the viscous formulation, a no-slip condition is enforced on the ship hull. For the inviscid case, flow tangency is preserved. On the symmetry plane (that portion of the (x, z) plane excluding the ship hull) derivatives in the y direction as well as the υ component of velocity are set to zero. The upstream plane has u=Uo, υ=0, w=0 and ψ=0 (p=−zFr−2). Similar conditions hold on the outer boundary plane which is assumed far enough away from the hull such that no disturbances are felt. A radiation condition should be imposed on the outflow domain to allow the wave disturbance to pass out of the computational domain. Although fairly sophisticated formulations may be devised to represent the radiation condition, simple extrapolations proved to be sufficient in this work. For calculations in the limit of zero Froude number (double-hull model) the (z=0) plane is also treated as a symmetry plane. 3 Numerical Solution The formulation of the numerical solution procedure is based on a finite volume method (FVM) for the bulk flow variables (u, υ, w and ψ), coupled to a finite difference method for the free surface evolution variables (β and ψ). Bulk Flow Solution The finite volume solution for the bulk flow follows the same procedures that are well documented in references [1, 2]. The governing set of differential flow equations are expressed in the standard form for artificial compressibility [6] as outlined by Rizzi and Eriksson [7]. In particular, letting w be the vector of dependent variables: (2) Here f, g, h and fv, gv, hv represent, respectively, the inviscid and viscous fluxes. Following the general procedures used in the finite volume formulation, the governing differential equations are integrated over an arbitrary volume V. Application of the divergence theorem on the convective and viscous flux term integrals yields (3) where Sx, Sy and Sz are the projections of the area ∂V in the x, y and z directions, respectively. In the present approach the computational domain is divided into hexahedral cells. Two discretization schemes are considered in the present work. They differ primarily in that in the first, the flow variables are stored at the grid points (cell-vertex) while in the second they are stored in the interior of the cell (cell-center). While the details of the computation of the fluxes are different for the two approaches, both cell-center and cell-vertex schemes yield the following system of ordinary differential equations [4] where Cijk and Vijk are the discretized evaluations of the convective and viscous flux surface integrals appearing in equation 3 and Vijk is the volume of the computational cell. In practice, the discretization scheme reduces to a second order accurate, nondissipative central difference approximation to the bulk flow equations on sufficiently smooth grids. A central difference scheme permits odd-even decoupling at adjacent nodes which may lead to oscillatory solutions. To prevent this “unphysical” phenomena from occurring, a dissipation term is added to the system of equations such that the system now becomes (4) For the present problem a fourth derivative background dissipation term is added. The dissipative term is constructed in such a manner that the conservation form of the system of equations is preserved. The dissipation term is third order in truncation terms so as not to detract from the second order accuracy of the flux discretization. Discretization of the Viscous Terms The discretization of the viscous terms of the Navier Stokes equations requires an approximation to the velocity derivatives in order to calculate the stress tensor. In order to evaluate the derivatives one may apply the Gauss formula to a control volume V with the boundary S. where nj is the outward normal. For a hexahedral cell this gives (5) where ûi is an estimate of the average of ui over the face. This technique requires motivates the introduction of dual meshes for the evaluation of the velocity derivatives and the flux balance as sketched for for the two dimensional case in figure 2.

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Twenty-Second Symposium on Naval Hydrodynamics Figure 2: Viscous discretization for cell-centered algorithm. Multigrid time-stepping Equation 4 is integrated in time to steady state using an explicit multistage scheme. For each bulk flow time step, the grid, and thus Vijk, is independent of time. Hence equation 4 can be written as (6) where the residual is defined as and the cell volume Vijk is absorbed into the residual for clarity. The full approximation multigrid scheme of this work uses a sequence of independently generated coarser meshes by eliminating alternate points in each coordinate direction. In order to give a precise description of the multigrid scheme, subscripts may be used to indicate the grid. Several transfer operations need to be defined. First the solution vector on grid k must be initialized as where wk−1 is the current value on grid k−1, and Tk,k−1 is a transfer operator. Next it is necessary to transfer a residual forcing function such that the solution grid k is driven by the residuals calculated on grid k−1. This can be accomplished by setting where Qk,k−1 is another transfer operator. Then Rk (wk) is replaced by Rk (wk)+Pk in the time- stepping scheme. Thus, the multistage scheme is reformulated as The result wk(m) then provides the initial data for grid k+1. Finally, the accumulated correction on grid k has to be transferred back to grid k−1 with the aid of an interpolation operator Ik−1,k. Clearly the definition of Tk,k−1, Qk,k−1, Ik−1,k depends on whether a cell-vertex or a cell-center formulation is selected. A detailed account can be found in reference [13]. With properly optimized coefficients, multistage time-stepping schemes can be very efficient drivers of the multigrid process. In this work we use a five stage scheme with three evaluation of dissipation [8] to drive a W-cycle of the type illustrated in Figure 3. Figure 3: Multigrid W-cycle for managing the grid calculation. E, evaluate the change in the flow for one step; T, transfer the data without updating the solution. In a three-dimensional case the number of cells is reduced by a factor of eight on each coarser grid. On examination of the figure, it can therefore be seen that the work measured in units corresponding to a step on the fine grid is of the order of and consequently the very large effective time step of the complete cycle costs only slightly more than a single time step in the fine grid. Free Surface Solution Both a kinematic and dynamic boundary condition must be imposed at the free surface which require the adaption of the

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Twenty-Second Symposium on Naval Hydrodynamics grid to conform to the computed surface wave Since flow values are stored at cell centers, they must be extrapolated up one half cell for solution of the kinematic condition. By introducing a curvilinear coordinate system that transforms the curved free surface β (x, y) into computational coordinates β (ξ, η), equation 1 can be case in a form more amenable to integration. This results in the following kinematic condition: (7) Where U and V are the contravariant velocity components give by: (8) (9) Figure 4 displays the computational stencil for the free surface update. It differs slightly from the bulk flow stencil, owing to velocity values held at the centers of the surface faces, while the free surface heights are held at surface vertices. The integration is a 3 stage Runge Kutta algorithm with dissipation evaluations on all stages. Figure 4: Free Surface Stencil In our original method, equation 7 was augmented by high order diffusion [1, 2]. Such a scheme can be obtained by introducing anti-diffusive terms in a standard first order formula. In particular, it is well known that for a one-dimensional scalar equation model, central difference approximation of the derivative may be corrected by adding a third order dissipative flux: (10) where at the cell interface. This is equivalent to the scheme which we have used until now to discretize the free surface, and which has proven to be effective for simple hulls. However, on more complex configurations of interest, such as combatant vessels and yachts, the physical wave at the bow tends to break. This phenomenon cannot be fully accounted for in the present mathematical model. In order to avoid the overturning of the wave and continue the calculations lower order dissipation must be introduced locally and in a controlled manner. This can be accomplished by borrowing from the theory of non-oscillatory schemes constructed using the Local Extremum Diminishing (LED) principle [10, 11]. Since the breaking of a wave is generally characterized by a change in sign of the velocity across the crest, it appears that limiting the antidiffusion purely from the upstream side may be more suitable to stabilize the calculations and avoid the overturning of the waves [12]. By adding the anti-diffusive correction purely from the upstream side one may derive a family of UpStream Limited Positive (USLIP) schemes: Where L (p, q) is a limited average of p and q with the following properties: P1. L (p, q)=L (q, p) P2. L (αp, αq)=αL (p, q) P3. L (p, p)=p P4. L (p, q)=0 if p and q have opposite signs. It is well known that schemes which strictly satisfy the LED principle fall back to first order accuracy at extrema even when they realize higher order accuracy elsewhere. This difficulty can be circumvented by relaxing the LED requirement. Therefore the concept of essentially local extremum diminishing (ELED) schemes is introduced as an alternative approach. These are schemes for which, in the limit as the mesh width Δx → 0, maxima are non-increasing and minima are non-decreasing. In order to prevent the limiter from being active at smooth extrema it is convenient to set where D (p, q) is a factor designed to reduce the arithmetic average, and become zero if u and v have opposite signs. Thus, for an ELED scheme we take (11) where (12) Properties P1–P3 are natural properties of an average, whereas P4 is needed for the construction of an LED scheme.

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Twenty-Second Symposium on Naval Hydrodynamics and ∈>0, r is a positive power, and s is a positive integer. Then D (p, q)=0 if p and q have opposite signs. Also if s=1, L (p, q) reduces to minmod, while if s=2, L (p, q) is equivalent to Van Leer’s limiter. By increasing s one can generate a sequence of limited averages which approach a limit defined by the arithmetic mean truncated to zero when p and q have opposite signs. These smooth limiters are known to have a benign effect on the convergence to a steady state of compressible flows. In comparison with experiment, it was found that the waterline profile calculated with ELED was more accurate when compared with LED or fourth differencing [17] [12]. Integration and Coupling with The Bulk Flow The free surface kinematic equation may be expressed as where Qij (β) consists of the collection of velocity and spatial gradient terms which result from the discretization of equation 7. Once the free surface update is accomplished the pressure is adjusted on the free surface such that The free surface and the bulk flow solutions are coupled by first computing the bulk flow at each time step, and then using the bulk flow velocities to calculate the movement of the free surface. After the free surface is updated, its new values are used as a boundary condition for the pressure on the bulk flow for the next time step. The entire iterative process, in which both the bulk flow and the free surface are updated at each time step, is repeated until some measure of convergence is attained: usually steady state wave profile and wave resistance coefficient. This method of updating the free surface works well for the Euler equations since tangency along the hull can be easily enforced. However, for the Navier-Stokes equations the no-slip boundary condition is inconsistent with the free surface boundary condition at the hull/waterline intersection. To circumvent this difficulty the computed elevation for the second row of grid points away from the hull is extrapolated to the hull. Since the minimum spacing normal to the hull is small, the error due to this should be correspondingly small, comparable with other discretization errors. 4 Parallel Multiblock Implementation Topological constraints deriving from complex configurations necessitate the integration of a multiblock algorithm into the baseline code. An intermediate step was first taken in the form of a parallel, single block version of the code. Details of this method can be found in [17]. A multiblock strategy has been developed which requires minimum deviation from the parallel single block code. A halo is constructed around each block such that the flow solution within it is transparent to the block boundaries. Since both the convective and the dissipative fluxes are calculated at the cell faces (boundaries of the control volumes), all six neighboring cells are necessary, thus requiring the existence of a single level halo around each block. The dissipative fluxes are composed of third order differences of the flow quantities. Thus, at the boundary faces of each cell in the domain, the presence of the twelve neighboring cells (two adjacent to each face) is required. Figure 5 shows the neighboring cells which are required for the calculation of convective and dissipative fluxes. For each block, some of these cells will lie directly next to an interblock boundary. If the neighboring block is contained in a different processor, it will be necessary to pass the values across processors. With halos properly updated, the blocks Figure 5: Convective and Dissipative Discretization Stencils. are simply looped through the single block integrator. Thus, the highly optimized baseline code remains unchanged. The only difference in the integration lies in the implementation of the residual averaging. In the single block case, flow information from the processor’s entire subdomain is used for smoothing. In the multiblock case, only the information from the particular block being processed is utilized for this calculation. The converged solution remains unaffected and thus far, there has not been any significant effect in the rate of convergence. Since the sizes of the blocks can be quite small, sometimes further partitioning severely limits the number of multigrid levels that can be used in the flows. For this reason, it was decided to allocate complete blocks to each processor. The underlying assumption is that there always will be more blocks than processors available. If this is the case, every

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Twenty-Second Symposium on Naval Hydrodynamics processor in the domain would be responsible for the computations inside one or more blocks. In the case in which there are more processors than blocks available, the blocks can be adequately partitioned during a pre-processing step in order to at least have as many blocks as processors. This approach has the advantage that the number of multigrid levels that can be used in the parallel implementation of the code is always the same as in the serial version. Moreover, the number of processors in the calculation can now be any integer number, since no restrictions are imposed by the partitioning in all coordinate directions used by the single block program. The only drawback of this approach is the loss of the exact load balancing that one has in the single block implementation. All blocks in the calculation can have different sizes, and consequently, it is very likely that different processors will be assigned a different total number of cells in the calculation. This, in turn, will imply that some of the processors will be waiting until the processor with the largest number of cells has completed its work and parallel performance will suffer. The approach that we have followed to solve the load balancing problem is to assign to each processor, in a pre-processing step, a certain number of blocks such that the total number of cells is as close as possible to the exact share for perfect load balancing. One must note that load balancing based on the total number of cells in each processor is only an approximation to the optimal solution of the problem. Other variables such as the number of blocks, the size of each block, and the size of the buffers to be communicated play an important role in proper load balancing, and are the subject of current study. Figure 20 shows speedups obtained on an 18 block mesh of Series 60 hull. One can see the good scalability of the method. The viscous speedups are closer to ideal due to increased granularity from the added work in calculating the viscous fluxes. 5 Results Figures 6 and 7 show overhead wave contours and waterlines for inviscid solutions of the Series 60 at a Froude number of .316. They serve to validate the multiblock code by comparison with both experiment and the previously developed single block parallel code. In order to validate the viscous solver, calculations were performed on the Series 60 hullform for comparison with the experimental data from Iowa [19]. Figures 8 and 9 show overhead wave contours and waterlines for both viscous and inviscid solutions of the Series 60 at Froude=.16. Figures 10 and 11 show wave elevation contours and waterlines for the Series 60 hull at Froude=.316. The Reynolds numbers for these calculations were 2 million and 4 million respectively, corresponding to the Iowa data. The grid spacing near the hull on the NS mesh is such that y+values in the first cell normal to hull reside in the range .75<y+<1.5. Figures 12 and 13 show overhead wave contours and waterlines for viscous and inviscid solutions of the model 5415 geometry at a Froude number of .2756 (20 Knots full scale). The Reynolds number was 12 million, corresponding to the test data taken at David Taylor Model Basin. Since the hullform has a transom stern, attempts to use a single structured block result in very skewed cells at the intersection of the and the symmetry plane aft of the boat. With a multiblock implementation however, a second block can be fitted to the transom and extended to the outflow plane. Figures 18 and 19 show details of the transom block as well as pressure contours on the sonar bulb. As a preliminary investigation of the application of our method to underwater vehicles, several calculations on a 6:1 ellipsoid were performed. A 1.25 million point mesh consisting of an inner O-H-O layer wrapped in an H-H-H shell was generated. The use of multiple layers facilitates adjustment of the spacing in the boundary layer without affecting the stretching in the far field. Figures 14 and 15 show pressure contours on the ellipsoid moving at a length-based Froude number of .62 and a depth/L of .18. The Reynolds number for this calculation was 4 million. Initial analysis of an IACC racing yacht has also begun. Figure 16 displays the pressure contours on a double model IACC hull. The mesh topology is similar to the submarine. A O-H-O layer encompasses the hull while a H-H-H completes the far field. This topology was chosen because it will facilitate the addition of keel and rudder to the bare hull. Figure 17 shows an inviscid single block solution on another IACC hull. One can see the highly skewed elements which result from using a single block on a fully appended hullform. Complex configurations such as this sailboat were a primary motivation for the implementation of multiblock. Conclusion A cell center, viscous, parallel multiblock version of the code has been developed and implemented. Inviscid flow solutions on the order of half an hour for a grid size of one million mesh points have been achieved on 16 processors of an IBM SP2. Viscous solutions on a mesh of same dimensions demand a factor of 5 increase in CPU work. Future efforts will concentrate on reducing the discrepancy in calculation time between the inviscid and viscous solutions. Progress towards improved dissipation and smoothing in the bulk flow to accelerate convergence has already begun. Additional speedup will be acquired by improving the pseudotime interaction between the bulk flow and the free surface; specifically at the intersection of the waterline and hull. Finally, a more realistic and robust turbulence model needs to be implemented. Even without these forthcoming improvements, however, the current version still provides turn-around times of 2 hours on 16 processors, rendering the Navier-Stokes solver suitable for utilization in design.

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Twenty-Second Symposium on Naval Hydrodynamics REFERENCES [1] Farmer, J.R., Martinelli, L., and Jameson, A., “A Fast Multigrid Method for Solving the Nonlinear Ship Wave Problem with a Free Surface,” Proceedings, Sixth International Conference on Numerical Ship Hydrodynamics, pp. 155–172, 1993. [2] Farmer, J.R., Martinelli, L., and Jameson, A., “A Fast Multigrid Method for Solving Incompressible Hydrodynamic Problems with Free Surfaces,” AIAA Journal, v. 32, no. 6, pp. 1175–1182, 1994. [3] Jameson, A., “Optimum Aerodynamic Design Using CFD and Control Theory,” Proceedings, 12th Computational Fluid Dynamics Conference, San Diego, California, 1995 [4] Jameson, A., “A Vertex Based Multigrid Algorithm For Three Dimensional Compressible Flow Calculations,” ASME Symposium on Numerical Methods for Compressible Flows, Annaheim, December 1986. [5] Baldwin, B.S., and Lomax, H., “Thin Layer Approximation and Algebraic Model for Separated Turbulent Flows,” AIAA Paper 78–257, AIAA 16th Aerospace Sciences Meeting, Reno, NV, January 1978. [6] Chorin, A., “A Numerical Method for Solving Incompressible Viscous Flow Problems,” Journal of Computational Physics, v. 2, pp. 12–26, 1967. [7] Rizzi, A., and Eriksson,L., “Computation of Inviscid Incompressible Flow with Rotation,” Journal of Fluid Mechanics, v. 153, pp. 275–312, 1985. [8] Martinelli, L., “Calculations of Viscous Flows with a Multigrid Method,” Ph.D. Thesis, MAE 1754-T, Princeton University, 1987. [9] Farmer, J., “A Finite Volume Multigrid Solution to the Three Dimensional Nonlinear Ship Wave Problem,” Ph.D. Thesis, MAE 1949-T, Princeton University, January 1993. [10] A.Jameson, “Analysis and design of numerical schemes for gas dynamics 1, artificial diffusion, upwind biasing, limiters and their effect on multigrid convergence,” Int. J. of Comp. Fluid Dyn., To Appear. [11] A.Jameson, “Analysis and design of numerical schemes for gas dynamics 2, artificial diffusion and discrete shock structure,” Int. J. of Comp. Fluid Dyn., To Appear. [12] J.Farmer, L.Martinelli, A.Jameson, and G.Cowles, “Fully-nonlinear CFD techniques for ship performance analysis and design,” AIAA paper 95–1690, AIAA 12th Computational Fluid Dynamics Conference, San Diego, CA, June 1995. [13] A.Jameson, “Multigrid algorithms for compressible flow calculations,” In Second European Conference on Multigrid Methods , Cologne, October 1985. Princeton University Report MAE 1743. [14] L.Martinelli and A.Jameson, “Validation of a multigrid method for the Reynolds averaged equations,” AIAA paper 88–0414, 1988. [15] L.Martinelli, A.Jameson, and E.Malfa, “Numerical simulation of three-dimensional vortex flows over delta wing configurations,” In M.Napolitano and F.Sabbetta, editors, Proc. 13th International Conference on Numerical Methods in Fluid Dynamics, pages 534–538, Rome, Italy, July 1992. Springer Verlag, 1993. [16] F.Liu and A.Jameson, “Multigrid Navier-Stokes calculations for three-dimensional cascades,” AIAA paper 92–0190, AIAA 30th Aerospace Sciences Meeting, Reno, Nevada, January 1992. [17] G.Cowles, and L.Martinelli “Fully Nonlinear Hydrodynamic Calculations for Ship Design on Parallel Computing Platforms,” Proc. 21st International Symposium on Naval Hydrodynamics, Trondheim, Norway, June 1996 [18] G.Cowles, and L.Martinelli “A Cell-Centered Parallel Multiblock Method for Viscous Incompressible Flows with a Free Surface,” Proc. 13th AIAA Computational Fluid Dynamics Conference, Snowmass, Colorado, July 1997 [19] Y.Toda, F.Stern, and J.Longo “Mean-Flow Measurements in the Boundary Layer and Wake and Wave Field of a Series 60 Cb=.6 Ship Model for Froude Numbers .16 and .316 IIHR Report No. 352, Iowa Institute of Hydraulic Research, August 1991

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Twenty-Second Symposium on Naval Hydrodynamics Figure 6: Overhead Wave Profiles: Series 60 (Fr=.316) Figure 7: Comparison of Waterline Profiles: Inviscid Solution

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Twenty-Second Symposium on Naval Hydrodynamics Figure 8: Overhead Wave Profiles: Series 60 (Fr=.16) Figure 9: Comparison of Viscous and Inviscid Waterlines

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Twenty-Second Symposium on Naval Hydrodynamics Figure 10: Overhead Wave Profiles: Series 60 (Fr=.316) Figure 11: Comparison of Viscous and Inviscid Waterlines

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Twenty-Second Symposium on Naval Hydrodynamics with 7 models. The Fu et al. (1987) pressure/rate-of-strain model (labeled FLT1) as it was modified by Shih and Lumley (1993) (labeled FLT2) worked best among the tested models for the 2D flow normal stresses. The Shih-Lumley/Choi-Lumley (SLCL) (1984) PRS model worked best for the shearing stress transport equation for the 2D flow. The FLT2 PRS model for the v’2 transport equation worked best for the 3D flow stations that were tested which were away from the chaotic bimodal vortices that were mentioned above. For the same stations, the PRS terms in the shear stress transport equations were best described by the FLT1 and SLCL models. There is still a need for improved PRS models for 3D flows, especially near the wall in the -vw shearing stress transport equation. Ölçmen and Simpson (1997a) used several turbulent diffusion models to compare with the experimentally measured profiles. All models that were tested failed to capture the magnitude of the triple products, apparently due to the scaling factor q2/ε. Note that since the experimentally measured turbulent diffusion terms were used in the transport equation budgets, Ölçmen and Simpson’s conclusions about the PRS models do not depend upon a turbulent diffusion model. Concluding Comments and Future Work Because of the non-equilibrium nature of 3D turbulent boundary layers, it is necessary to use Reynolds-averaged transport equations which can mimic the lags between the mean flow and the shearing stress structure. Work discussed and referenced here show that v’ is closely related to the shear stress magnitude by S in a variety of non-equilibrium 3D experiments over a range of Reynolds numbers. The parameters are nearly constant across the outer boundary layer in these flows. These parameters alone are not enough for closure and relationships for the pressure diffusion, pressure/rate-of-strain (PRS), turbulent diffusion, and dissipation are needed. The work quoted here shows that several uncertainties exist in the modeling of these terms, especially in the near-wall region. Better models for the turbulent diffusion are needed. The relationships among pressure and velocity fluctuations remains an important modeling issue. Data are needed for the pressure diffusion and PRS in order to verify existing models and to develop better models. Since the direct measurement of the pressure fluctuation within these flows is presently unfeasible with a fine-resolution sensor, one must use the Poisson volumetric integral of the turbulent velocity fluctuation correlation contributions to the pressure fluctuation (Chou, 1945). To this end, recently considerable vi-vj and vi-vj2 LDV spatial correlation wind tunnel data have been obtained for a 2-D TBL and at Station 5 of the wing/body junction wind tunnel flow (Ölçmen et al. 1998). These data are being used in the Poisson pressure-fluctuation volume integral in order to better understand the effects of 3-D skewing on the pressure-fluctuation structure. Acknowledgment Portions of the referenced work at VPI&SU have been supported by: Office of Naval Research (current Grant N00014–94–1–0092; Dr. L.P.Purtell, Program Manager). The authors gratefully acknowledge this support. References 1. Ailinger, K.G. and Simpson, R.L. (1990) Measurements of Surface Shear Stresses under a Three-Dimensional Turbulent Boundary Layer Using Oil-Film Laser Interferometry, VPI-AOE-173; DTIC Report ADA2294940XSP. 2. Barberis, D. and Molton, P. (1993) Experimental Study of 3-D Separation on a Large Scale Model, AIAA 24th Fluid Dynamics Conference, Orlando, FL USA, AIAA-93–3007. 3. Baskaran, V., Pontikis, Y.G., and Bradshaw, P. (1990) An Experimental Investigation of a Three-dimensional Turbulent Boundary Layer on ‘Infinite’ Swept Curved Wings,” J. Fluid Mech., Vol. 211, p. 95–122. 4. Chesnakas, C.J., Simpson, R.L., and Madden, M.M. (1994) Three-dimensional Velocity Measurements on a 6:1 Prolate Spheroid at 10° Angle of Attack, VPI-AOE-202REV; DTIC report ADA2764850XSP. 5. Chesnakas, C.J. and Simpson, R.L. (1992) An Investigation of the Three-dimensional Turbulent Flow in the Cross-flow Separation Region of a 6:1 Prolate Spheroid, Sixth International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, 20–23 July, 1992; Exp. in Fluids, Vol. 17, pp. 68–74, 1994. 6. Chesnakas, C.J. and Simpson, R.L. (1997) Detailed Investigation of the Three-Dimensional Separation About a 6:1 Prolate Spheroid, AIAA Journal,

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Twenty-Second Symposium on Naval Hydrodynamics Vol. 35, no. 6, pp. 990–999. 7. Choi, K.S. and Lumley, J.L. (1984) Turbulence and Chaotic Phenomena in Fluids, Proceedings IUTAM Symposium (Kyoto, Japan), edited by T.Tatsumi, North-Holland, Amsterdam, p. 267. 8. Chou, P.Y. (1945) On Velocity Correlations and the Solutions to the Equations of Turbulent Motions, Quarterly Applied Math., vol. 3, pp. 38–54. 9. Ciochetto, D.S. and Simpson, R.L. (1995) An Investigation of 3-D Turbulent Shear Flow Experiments and New Modeling Parameters, Turbulent Shear Flows 10, pp. 7–25 thru 7–30, Penn State, August 14–16, 1995. 10. Ciochetto, D.S. and Simpson, R.L. (1997) Analysis of Three-Dimensional Turbulent Shear Flow Experiments with Respect to Algebraic Modeling Parameters, Report VPI-AOE-248, submitted to DTIC. 11. Devenport, W.J. and Simpson, R.L. (1990) A Time-dependent and Time-averaged Turbulence Structure Near the Nose of a Wing-body Junction”, J. Fluid Mech., 210, pp. 23–55. 12. Devenport, W.J. and Simpson, R.L. (1990) The Flow Past a Wing-body Junction—an Experimental Evaluation of Turbulence Models, 18th Symposium on Naval Hydrodynamics, August 20–22, Ann Arbor, Michigan; also AIAA J., 30, pp. 873–881, 1992. 13. Devenport, W.J. and Simpson, R.L. (1990) An Experimental Investigation of the Flow Past an Idealized Wing-body Junction,” VPI&SU Report VPI-AOE-172; DTIC report ADA2296028XSP. 14. Driver, D.M. and Johnston, J.P. (1990) Experimental Study of a Three-dimensional Shear-driven Turbulent Boundary Layer with Streamwise Adverse Pressure Gradient,” NASA TM 102211; also Report MD-57, Thermo. Div., Dept. Mechanical Engineering, Stanford University. 15. Dussauge, J., Fernholz, H., Finley, P., Smith, R., Smits, A., and Spina, E. (1996) Turbulent Boundary Layers in Subsonic and Supersonic Flow, AGARD-AG-335. 16. Fernholz, H.H., Krause, E., Nockemann, M., and Schober, M. (1995) Comparative measurements in the canonical boundary layer at Reθ≤6×104 on the wall of the German-Dutch windtunnel, Phys. Fluids, Vol. 7, no. 6, pp.1275–1281. 17. Fleming, J.L. and Simpson, R.L. (1997) Experimental Investigation of the Near Wall Flow Structure of a Low Reynolds Number 3-D Turbulent Boundary Layer,” Report VPI-AOE-247, submitted to DTIC. 18. Fleming, J., Simpson, R.L., and Devenport, W.J. (1991) An Experimental Study of a Turbulent Wing-body Junction and Wake Flow, VPI-AOE-179; DTIC Report ADA2433886XSP; AIAA-92–0434; Exp. Fluids, 14, pp. 366–378, 1993. 19. Fu, S., Launder, B.E., and Tselepidakis, D.P. (1987) Accommodating the Effects of High Strain Rates in Modeling the Pressure-strain Correlation, UMIST Mechanical Engineering Dept. Rept. TFD/87/5. 20. Ha, S. and Simpson, R.L. (1993) An Experimental Investigation of a Three-dimensional Turbulent Boundary Layer Using Multiple-sensor Probes, Ninth Symposium on Turbulent Shear Flows, Kyoto, Japan, 16–18 Aug. 1993, p. 2–3–(1–6). 21. Hallbäck, M., Groth, J., and Johansson, A.V. (1990) An Algebraic Model for Nonisotropic Turbulent Dissipation Rate in Reynolds Stress Closure, Phys. Fluids A, vol. 2, no. 10, pp. 1859–1866. 22. Johnston, J.P. and Flack, K.A. (1996) Review—Advances in Three-Dimensional Turbulent Boundary Layers with Emphasis on the Wall-Layer Regions, J. Fluids Engineering, Vol. 118, no. 2, pp. 219–232. 23. Kreplin, H.P. and Stäger, R. (1993) Measurements of the Reynolds Stress Tensor in the Three Dimensional Boundary Layer of an Inclined Body of Revolution, Ninth Symposium on Turbulent Shear Flows, Kyoto, Japan, 16–18 Aug. 1993, p. 2–4–(1–6). 24. Lewis, D.J., Simpson, R.L., and Diller, T. (1993) Time-Resolved Surface Heat Flux Measurements in the Wing/body Junction Vortex, AIAA-93–0291, AIAA 31st Aerospace Sciences Meeting, Reno, NV, Jan. 11–14; also AIAA J. Thermo. and Heat Trans., 8, pp. 656–663, 1994. 25. Lewis, D.J. and Simpson, R.L. (1996) An Experimental Investigation of Heat Transfer in Three-dimensional and Separating Turbulent Boundary Layers, VPI-AOE-229; submitted to DTIC. 26. Lewis, D.J. and Simpson, R.L. (1998) Turbulence Structure of Heat Transfer Through a Pressure-driven Three-dimensional Turbulent Boundary

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Twenty-Second Symposium on Naval Hydrodynamics Layer, AIAA J. of Thermophysics and Heat Transfer, Vol. 12, 2, pp. 248–255, April–June. 27. Littell, H.S. and Eaton, J.K. (1991) An Experimental Investigation on the Three-dimensional Boundary Layer on a Rotating Disk, Thermo. Div., Dept. Mechanical Engineering, Stanford Univ., Report No. MD-60. 28. Lumley, J.L. (1978) Computation and Modeling of Turbulent Flows, Adv. in Applied Mech., vol. 18, pp. 124–176. 29. McMahon, H., Hubbartt, J., and Kubendran, L. (1982) Mean Velocities and Reynolds Stresses in a Juncture Flow, NASA CR-3605. 30. Nockemann, M., Abstiens, R., Schober, M., Bruns, J., and Eckert, D. (1994) Messungen in einer turbulenten Wandgrenzschicht bei grossen Reynolds-Zahlen im Deutsch-Niederländischen Windkanal Messbereicht, Aero. Inst., Tech. Hochschule Aachen, Germany. 31. Ölçmen, S.M. and Simpson, R.L. (1990) An Experimental Investigation of Pressure-Driven Three-dimensional Turbulent Boundary Layer. VPI-AOE-178; DTIC Report ADA2294957XSP. 32. Ölçmen, S.M. and Simpson, R.L. (1995a) An Experimental Study of Three-dimensional Pressure-Driven Turbulent Boundary Layer, J. Fluid Mech., 290, pp. 225–262. 33. Ölçmen, S.M. and Simpson, R.L. (1995b) A 5-Velocity-Component Laser-Doppler Velocimeter for Measurements of a Three-Dimensional Turbulent Boundary Layer, paper 4.2, Seventh International Symposium on Applications of Laser Techniques to Fluid Mechanics, 11–14, July, 1994, Lisbon, Portugal; revised version Invited Paper, Measurement Science and Technology, 6, pp. 702–716, 1995. Highlighted in The Year in Review, Fluid Dynamics, pp. 20–21, Aerospace America, Vol. 33, no. 12, Dec. 1995. 34. Ölçmen, S.M., and Simpson, R.L. (1996a) An Experimental Study of a Three-Dimensional Pressure-Driven Turbulent Boundary Layer: Data Bank Contribution, J. Fluids Engineering, vol. 118, pp. 416–418, 1996. 35. Ölçmen, S.M., and Simpson, R.L. (1996b) Experimental Transport-rate Budgets in Complex Three-dimensional Turbulent Flows at a Wing/Body Junction, 27th AIAA Fluid Dynamics Conference, AIAA-96–2035, New Orleans, LA, June 17–20. 36. Ölçmen, S.M., and Simpson, R.L. (1996c) Theoretical and Experimental Pressure-strain Comparison in a Pressure-driven Three-dimensional Turbulent Boundary Layer, 1st AIAA Theoretical Fluid Mechanics Meeting, AIAA-96–2141, New Orleans, LA, June 17–20. 37. Ölçmen, S.M. and Simpson, R.L. (1996d) Experimental Evaluation of Pressure-Strain Models in Complex 3-D Turbulent Flow Near a Wing/Body Junction, VPI-AOE-228; DTIC Report ADA3071164XSP. 38. Ölçmen, S.M. and Simpson, R.L. (1996e) Higher Order Turbulence Results for a Three-Dimensional Pressure-Driven Turbulent Boundary Layer, VPI-AOE-237; submitted to DTIC. 39. Ölçmen, S.M. and Simpson, R.L. (1997a) Experimental Evaluation of Turbulent Diffusion Models in Complex 3-D Flow Near a Wing/Body Junction, AIAA-97–650, 35th AIAA Aerospace Sciences Meeting, Jan. 6–10. 40. Ölçmen, S.M. and Simpson, R.L. (1997b) Some Features of a Turbulent Wing-Body Junction Vortical Flow, AIAA-97–0651, 35th AIAA Aerospace Sciences Meeting, Jan. 6–10; VPI-AOE-238 submitted to DTIC, 1996. 41. Ölçmen, S.M. and Simpson, R.L. (1997c) Higher Order Turbulence Results for a Three-Dimensional Pressure-Driven Turbulent Boundary Layer, under revision for J. Fluid Mechanics, 1997. 42. Ölçmen, S.M., Simpson, R.L. and George, J. (1999) Experimental Study of a High Reynolds Number (Reθ=23000) Turbulent Boundary Layer Around a Wing-Body Junction, submitted for 37th AIAA Aerospace Sciences Meeting, Jan.. 43. Ölçmen, S.M., Simpson, R.L. and Goody, M. (1998) An Experimental Investigation of Two-Point Correlations in Two- and Three-Dimensional Turbulent Boundary Layers, AIAA-98–0427, 36th AIAA Aerospace Sciences Meeting and Exhibit, Jan. 12–15. 44. Pompeo, L.P. (1992) An Experimental Study of Three-dimensional Turbulent Boundary Layers, Ph.D. Diss., ΕΤΗ no. 9780, Swiss Fed. Inst. Tech., Zurich.

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Twenty-Second Symposium on Naval Hydrodynamics 45. Pompeo, L.P., Bettelini, M.S.G., and Thomann, H. (1993) Laterally Strained Turbulent Boundary Layers Near a Plane of Symmetry, J. Fluid Mech., Vol. 257, pp. 507–532. 46. Rizzi, A. and Vos, J. (1998) Towards Establishing Credibility in Computational Fluid Dynamics Simulations, AIAA Journal, Vol. 36, no. 5, pp. 668–675. 47. Schwarz, W.R. and Bradshaw, P. (1992) Three-dimensional Turbulent Boundary Layer in a 30 Degree Bend: Experiment and Modeling, Thermosciences Division Report No. MD-61, Stanford. 48. Schwarz, W.R. and Bradshaw, P. (1993) Measurements in a Pressure Driven Three-dimensional Turbulent Boundary Layer During Development and Decay, AIAA Journal, Vol. 31, No. 7, p. 1207–1214. 49. Shih, T.H. and Lumley, J.L. (1993) Critical Comparison of Second-order Closures with Direct Numerical Simulations of Homogeneous Turbulence, AIAA Journal, Vol. 31, no. 4, pp. 663–670. 50. Simpson, R.L. (1996) Aspects of Turbulent Boundary Layer Separation, Prog. Aerospace Sci., Vol. 32, pp. 457–521.

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Twenty-Second Symposium on Naval Hydrodynamics Investigation of Viscous Flow Field Around an Appended Revolution Body with Guide Vane Propeller L.-D.Zhou (China Ship Scientific Research Center, China), F.Zhao (National Laboratory of Hydrodynamics, China) Abstract A RANS solver is presented to numerically simulate the viscous wake of an appended revolution body with guide vane propeller at Reynolds number 107. The k-ε turbulence model together with wall function is used. The resulting finite difference equations are solved by SIMPLEC, ADI. The technique of rising up the bottom surface is presented to overcome radial contraction problem in Cartesian coordinate system. The three dimensional body forces are separately adopted to model the affection of the guide vane and propeller. The detailed flow characteristics, especially the counter-swirl component generated by the guide vane in the propeller inflow are numerical seized successfully. Comparing with the experimental data, computational axial velocity on the propeller disk plane comes up to engineering requirement. 1. Introduction The flow around an appended revolution body at high Reynolds number is part of a competitive benchmark problem to evaluate existing computational fluid dynamics (CFD) capabilities for incompressible viscous flow. There are always appendages equipped in the submarine’s stern. The interaction of flow around the hull/appendage creates complex juncture vortices shedding downstream. Thus, the flow into the propulsor becomes quite complex and non-uniform even in steady straight motion. Such undesired inflow may cause propulsor efficiency decreased and generate noise. The prediction of this flow is of great importance to the propeller designer. To numerical simulate this complicated inflow to the propulsor for an appended submarine configuration, the current rapidly developed computational fluid dynamics has provided an effective and feasible approach. The powerful combination of interactive computer-aided design (CAD) and grid generation packages allow one to model and generate grids quite effectively around such complicated geometry. In addition, the numerical techniques to solve the Navier-Stokes equation has been substantially improved in the areas of accuracy, robustness and computational speed. Also, the multi-zonal concept has proven to be quite capable in effectively handling complicated grid topology resulting from complex hull/appendages configurations. For a zone with appendages, using the multiblock grid structure, we could solve the 3-D Reynolds-averaged Navier-Stokes equations in a simple region (without any appendage). The treatment of the interface boundary between neighboring zones and sub-blocks is very important. Authors have developed a multiblock grid generation [1] and corresponding flow field multiblock coupled computational method [2]. It is well known that the adoption of stators locating in front of propeller can result in a reduction of rotational energy losses, so that propulsion efficiency could be increase. In addition, a non-axisymmetric upstream stator can alter the inflow to downstream propeller in such a way that unsteady forces and cavitation could be reduced. The Guide Vane, as an effective equipment of improving the inflow of the propeller, has been adopted in surface ship design. Recently, Professor Shen H.C. et al were taking the lead in Guide Vane design for submergible appended revolution body [4]. The Guide Vane is composed of 4 radially placed vanes of aerofoil section and installed at an appropriate position between the stern appendages and propeller. By adopting guide vane, the inflow state of propeller and the interactions among ship hull, appendages and propeller were regulated, so that powering performance could be improved and noise reduced. Based on the model experimental results [4], about 5~10 percent saving of DHP and 2~3 dB noise reduction were expected. Regarded as a special appendage of submergible body, the Guide Vane is simple to construct, easy to manufacture and convenient to install. It should be thought as a promising practical device for submergible vehicles. Besides experimental research, CFD is very helpful to understand the flow structure around vane and in the wake, and also to evaluate and optimize the design of guide vane. The purpose of this paper is trying to develop an effective and practicable CFD method in the guide vane design.The

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Twenty-Second Symposium on Naval Hydrodynamics purpose of this paper is trying to develop an effective and practicable CFD method in the guide vane design. The geometry of a general submergible body is composed of streamlined appendages (one sail and four fins) attached to an axisymmetric hull. The whole flow field around this appended revolution body is divided into two big zones, sail zone and fins zone. The detailed results of the sail zone was proposed in authors’ earlier papers [2]~[3]. In this paper, the fins zone with and without guide vane is specially presented. A 3-D RANS equation solver [3] is applied to predict the viscous wake of an appended submergible body. The k-ε two equations turbulence model together with wall function are used. The resulting finite difference equations discreted in non-staggered grid system are solved by SIMPLEC, ADI. Computational Reynolds number is 107. The guide vane is also a kind of thin foil body located between fins and propeller, something like a steady propeller, as shown in Fig. 1. So, the body force model is used in the present paper to simulate the action of propeller and also the guide vane. Using the panel method [5], the distribution of 3-D body force by the guide vane and propeller are both gotten. These additional source items, substituted into RANS equations at proper grid nodes, are the representation of the guide vane and propeller respectively. Fig. 1 Illustration of guide vane location A couple of plotting package is also integrated into the system for the efficient visualization of computed results and for the meaningful comparison with experimental data or other numerical results. The numerical results have been compared with experimental data in wake and on the propeller disk plane. It shows that most of the important features of the mean flow, such as block effect of appendages, preswirl of guide vane and drawing of propeller, could be successfully predicted. Instead of many qualitative contours or vectors, the most important and interesting quantitative data of the nominal mean velocity radial distributions on the propeller disk plane were provided here. The average error of comparison data between the computation and the experimental data in wind tunnel is 2.54% for guide vane case, and 4.63% without guide vane. 2. Numerical Method 2.1 Governing Equations We consider the governing equations in Cartesian coordinate (xi,t)=(x,y,z,t) for unsteady, three-dimensional, incompressible flow. The complete three-dimensional Reynolds-averaged equations in the general form are (1) which represents any one of the convective transport quantities (1,U,V,W,k,ε). The scalar diffusivity and source terms for Ui, k and ε respectively are (2a) (2b) (2c) (2d) 2.2 Body-fitted Grid System For the present application to the flow around an appended submergible body, a single integrated numbering grid system is very difficult to generate. The idea of multizone and multiblock is

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Twenty-Second Symposium on Naval Hydrodynamics adopted in grid generation and flow solver. The whole appended submergible body external flow field was divided into two big zones, the one is the front part of the submergible body, hull with sail, the other is stern part, body with fins. For the sail zone, a multiblock grid generation method [1] is used to generate body-fitted grid surrounding the sail. The computation of flow field also used multiblock coupled method [2]. The body-fitted numerical grids were generated by a system of elliptic partial differential (Poisson) equations of the form of (3) here, ∇2 is the Laplacian operator in Cartesian coordinates xi. The nonhomogeneous source function fi may be assigned appropriate values to yield desirable grids distributions. In practical applications, the inverse transformation of equation (3) is used to obtain the coordinate transformation relations xi=xi(ξi), i.e., (4) which ∇2 is the Laplacian operator in the transformed plane (ξi). Using the transformation relations, the equation (4) can be rewritten as (5) which gjk is the inverse metric tensor. The grid control function fj were determined by the specified boundary node distribution. In present study, a corrective method is used to generate nearly orthogonal grids [3]. 2.3 Rising up of bottom surface Generally, the hull of submarine is an axisymmetric body, contracting into a point at stern. This will cause grids near the hull converging into one point in Cartesian coordinate system (i.e. r=0 case). To deal with this kind problem, cylindrical coordinate system or some geometry appropriate in the stern may be adopted. The problems with r=0 are solved undoubtedly in cylindrical coordinate system, but many computer codes developed under Cartesian coordinate system could not use directly. Though the stern shape changed a little, the method [6] of trailing out a “thin hub” is often applied to make geometry appropriate. The technique of rising up the bottom surface [7] is presented here to overcome this radial contraction problem in Cartesian coordinate system. Possessing 3-d body-fitted grid generation method, the present paper use TDMBGG software [1], the key of question is how to properly give six surfaces’ grid systems of the controlling volume. Fig. 2 illustrates the six surfaces and computational zone (fins zone). The bottom surface is the hull surface, including wake. During the stern contraction, we should arrange the bottom surface crawl up to both side surfaces. Thus, the good quality grid system could always be gotten, even in r=0 case. Illustrated in Fig. 3 is the process of rising up. Fig. 2 Six surfaces of fins zone Fig. 3 Process of rising up the bottom surface

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Twenty-Second Symposium on Naval Hydrodynamics 2.4 Body force modeling of guide vane and propeller Using the method in [5], the radial distributions of 3-D body force of guide vane and propeller are given in their located position respectively (6) In circumferential direction we share out equally onto their computational grids controlling point (7) If the volume of grid unit (i, j, k) is Vol(i, j, k), then the momentum equation could be written as (8) where the additional source term is the affection of guide vane and propeller on their own located computational grids (9) 2.5 Numerical algorithm and solution procedure The discretisation of the governing differential equation and wall function are detailed in reference [7]. Non-staggered grid method is adopted, and velocity-pressure coupling SIMPLEC approach is used. In a time step, the momentum equations are solved first, then, the equations for pressure correction, and the k-ε turbulence model equations are solved last. The iteration is used at two levels, an inner iteration to solve for the spatial coupling of each variable and an outer iteration to solve for coupling between variables. Thus each variable is taken in sequence, regarding all other variables as fixed and reforming coefficients by updated values of the variables. The detailed description was given in reference [2]~[3], [6]~[7]. 3. Results and Discussion In the presented paper, viscous wake of a submarine model with and without guide vane and propeller are numerical simulated. As mentioned above, the computational Reynolds number is 107. All the computation are carried on CONVEX computer with good convergence characteristics. The computational grids are shown in Fig. 4. There were four computational cases, without guide vane and propeller, with guide vane only, with propeller only and both with guide vane and propeller. The axial drawing of the propeller is shown in Fig. 5. The comparisons of computational velocity vectors on the cross plane after propeller disk are emphatically given in Fig. 6. The main flow characteristics, such as the preswirl affection of guide vane and the reduction of the rotative kinetic energy losses of propeller, are clearly simulated. Instead of many other qualitative contours or vectors, the present paper specially provides an more interesting quantitative comparison data. Fig. 7 shows the comparison curves of the radial distribution of axial velocities on the propeller disk plane between the computation and the experiment data in wind tunnel. The average error is 2.54% for guide vane case, and 4.63% without guide vane. Fig. 4 Computational grids of stern wake Fig. 5 Axial drawing of the propeller

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Twenty-Second Symposium on Naval Hydrodynamics 4. Conclusion A 3-D RANS solver is applied to predict the viscous wake of an appended submergible body with a guide vane propeller. The technique of rising up the bottom surface is presented to overcome radial contraction problem in Cartesian coordinate system. The 3-D body force model is successfully used to represent the guide vane and propeller actions. The qualitative and quantitative consistence between experiment and calculation is shown. This would prove the present method to be competent at simulation the submergible body wake with guide vane propeller. Although the validation of this analysis system is not fully completed at the present time, the continuous improvement of system will provide an efficient procedure to evaluate viscous flow phenomena around appended axisymmetric body, with and without guide vane and propeller, such as prediction of nominal wake distribution at the propeller disk plane. Improvements on the accuracy of each module in the system are to be continued necessarily. 5. Acknowledgments The authors greatly acknowledge to National Laboratory of Hydrodynamics for its Fund supporting. Fig. 6 Computational cross vectors of four cases Fig. 7 Comparison curves of axial velocity on the propeller disk plane

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Twenty-Second Symposium on Naval Hydrodynamics 6. References [1] ZHAO, F., ZHOU, L.D., and REN, A.L., “General Coupled Generation Method of 3-D Multiple Block Body-fitted Grid,” Journal of Hydrodynamics, Ser. A, Vol. 9, No. 4, 1994, (in Chinese) [2] ZHAO, F., and ZHOU, L.D., “Multiblock Coupled Computation of the Viscous Flow Field Around a Submarine Sail Zone,” Journal of Hydrodynamics, Ser. A, Vol. 10, No. 4, 1995, (in Chinese) [3] ZHAO, F., “Numerical Simulation of the Viscous Flow Around an Appended Submarine Model,” Proc. of Second International Conference on hydrodynamics, Vol. 1, Hongkong, 1996 [4] SHEN, H.C., et al, “Development of Guide Vane As a Device for Improving Powering Performance and Reducing noise for Body of Revolution with Appendages,” Proc. of China-Korea Marine Hydrodynamics Meeting, Shanghai, 1997 [5] DANG, J., CHEN, J.D., and TANG, D.H., “The Prediction of Time Dependent and Time Independent Propeller Hydrodynamic Performance by Panel Method,” Research Report 92237, China Ship Scientific Research Center, 1992, (in Chinese) [6] ZHAO, F., and ZHOU, L.D., “Research of Flow Field Around Submarine Fin Appendages and In Wakes,” Research Report 95374, China Ship Scientific Research Center, 1995, (in Chinese) [7] ZHOU, L.D., and ZHAO, F., “Numerical Simulation and Experiment Comparison of Submarine’s Wake (with and without guide vane and propeller),” Research Report 96108, China Ship Scientific Research Center, 1996, (in Chinese) DISCUSSION M.Abdel-Maksoud Potsdam Model Basin, Germany The operation of propeller in behind condition is one of the most important hydrodynamic problems for naval architects under the point of view of high efficiency, pressure pulses and vibration. The presented paper deals with some interesting aspects of application of body force method for analysis of interaction between body of revolution, guide vane and propeller. It will be interesting to know which boundary condition is applied in the circumferential direction at the boundaries of the calculation domain. According to the results shown in Figure 6, a symmetry boundary condition might be applied. This could be done for the case without guide vane and without propeller; see Fig. 6a. For the other cases with guide vane and propeller, a cyclic boundary condition is necessary to capture the physics of the investigated swirl flow around the symmetry axis. The application of symmetry boundary condition instead of the cyclic one leads to incorrect results specially in the circumferential direction. It is known that the quality of the numerical results can be improved when the interaction between propeller inflow and distribution of the body forces is considered during the iterations. This means that after a certain number of iterations with the viscous flow code, the calculated propeller inflow velocities should be used to update the body force distribution. It is not clear whether the authors have already considered such effects in the computations or not. I think the deviation between the numerical and experimental results may be reduced, when the cyclic boundary condition in the circumferential and updating of body force distribution during the iterations would be taken into account. AUTHORS’ REPLY Thanks for Dr. M.Abdel-Maksoud’s interesting questions. The numerical technique in the present paper was developed using our former method, dealing with no additional source items. Our tentative purpose is to set up an effective method for analysis of the interaction between appended revolution body and propeller, including guide vane. As you pointed out, boundary condition in the circumferential direction is very important for this kind of problem. In this paper, the body-force field of guide vane and propeller was averaged in the circumferential direction, so the symmetry boundary condition would be appropriate.

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Twenty-Second Symposium on Naval Hydrodynamics We will consider the circumferential distribution of the body force field in the further research. The propeller/wake interaction is one of the main concerns in our group [1], [2]. Thank you for the suggestion of the iteration between the propeller for these application cases. But only a first step was undertaken in this paper to capture the flow characteristics of the wake with guide vane propeller. Similarly, we need to consider the iterations for the further work. References 1. Liandi Zhou and Feng Zhao, “An integrated method for computing the internal and external viscous flow field around the ducted propulsor behind an axisymmetric body,” 20th ONR Symp., 1994. 2. Liandi Zhou and Qiuxin Gao, “Numerical simulation of the interaction between the ship stern flow and multi-propeller,” Proc. of ICHD’98, 1996.