Twenty-Third Symposium on Naval Hydrodynamics (2001)

Below is the uncorrected machine-read text of this chapter, intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text of each book. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line SHIP STERN FLOW CALCULATIONS ON OVERLAPPING COMPOSITE GRIDS 910 Ship Stern Flow Calculations on Overlapping Composite Grids B.Regnström,1 L.Broberg,1 L.Larsson1,2 (1FLOWTECH International AB, 2Chalmers University of Technology, Sweden) ABSTRACT A method for predicting the viscous flow around ship sterns is presented. Its main advantage is the flexible high- quality grid on which the governing equations and the boundary conditions are discretized. A set of overlapping grids on the hull surface are created either by hyperbolic marching from one of the boundaries or by cutting the surface by horizontal and vertical planes. Body-fitted volume grids are then grown hyperbolically out from the surface. At the outermost edge of the computational domain a background Cartesian grid is chosen and a sequence of finer and finer Cartesian grids is automatically generated to create a sufficiently smooth transition between the coarse edge grid and the fine body-fitted grid. The algorithm guarantees that there is sufficient overlap between all grids. The Reynolds-Averaged Navier-Stokes equations are solved on the overlapping grid using finite difference discetization. The equations are partially transformed and all variables are co-located. Pressure and velocities are coupled via a SIMPLE algorithm and Rhie Chow interpolation is used to avoid checkerboard oscillations. Computed results are compared with measured data for three different hulls. INTRODUCTION The state of the art of Computational Fluid Dynamics applied to ship design was reviewed at the previous Symposium on Naval Hydrodynamics by Larsson et al (1998). While the obtainable CFD accuracy is sufficient for many purposes, especially when optimizing the hull shape, quantitative predictions of many hydrodynamic quantities must still be regarded with caution. Several reasons for the lack of absolute accuracy were listed and discussed in the review. Examples are inadequate gridding, dissipative numerical techniques and too approximate turbulence modeling. Free surface representation was also mentioned as an area were further developments are needed. During the past five years efforts have been made in the research group at Chalmers/FLOWTECH to improve the state of the art in all four areas. The work on grid generation has been reported in Petersson (1997a, b, 1998) and Li (1998a, b), while numerical developments have been presented by Carlsson (1997, 2000) and Carlsson and Petersson (1999). Turbulence modeling was reported by Svennberg (1997, 2000) and Svennberg et al (1998) and free surface developments by Kang (1996, 1997), Vogt (1997, 1998), Vogt and Kang (1997) and Vogt and Larsson (1999). In the present paper an improved gridding technique is presented together with the newly developed solver, called CHAPMAN. Free surface developments and advanced turbulence modeling are not addressed. The grid generation is presented in the next section, which is followed by a section on the Navier-Stokes solver. Thereafter the validation of the method for three different hulls is presented and finally some comments and recommendations for future work are given. GRID GENERATION Most ship flow calculations presented until now have exploited the fact that a ship hull is a smooth boundary, unlike typical boundaries of, for instance, internal flows. In fact in the vast majority of calculations to date single block structured grids have been used (see Larsson et al (1998)). Certain advantages can be envisaged, however, in using more advanced gridding techniques even for this relatively simple geometry. Thus, more complex regions, such as the sternpost and the possible stern bulb, might be better resolved. A high resolution in these regions is important for the prediction of the flow in the propeller plane just behind. Further, the singularity line(s) in the grid in front of and behind the hull can be avoided. Another advantage is that the flow near the free surface can be computed in a thin grid with sufficient resolution. If appendages are added, the geometry and possibly also the topology of the domain becomes more complex. The appendages themselves are however the authoritative version for attribution. 

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line SHIP STERN FLOW CALCULATIONS ON OVERLAPPING COMPOSITE GRIDS 911 mostly streamlined. There are thus a limited number of flow regions, each of which having smooth boundaries. A completely unstructured grid in this case would waste the regularity offered by the boundaries. It should be better to take advantage of the smoothness in each region and to create separate structured grids in each such domain. The question is how to join the different grids. Butt-joined grids are usually unnecessarily restrictive. A more flexible solution is to let the grids overlap arbitrarily and to handle the overlapping parts in a separate procedure. Component grids with suitable resolution of the boundary layer can the be fitted to each part of the hull and appendages, while the outer flow is computed in a Cartesian or cylindrical grid. Overlapping grids have been in use for about ten years and the most well known code for generating such grids is CHIMERA, originally developed at NASA. In the USA this gridding technique is often referred to as the CHIMERA technique. A special series of symposia is devoted to this grid type and the 5th Symposium on Overset Composite Grid & Solution Technology will be held at the University of California at Davis, USA in September 2000. Applications in hydrodynamics have been presented by Weems et al (1994), Lin et al (1998) and Masuko (1998). It should be pointed out that other advanced gridding techniques are also rapidly being introduced in hydrodynamic calculations. Most popular is the standard multi-block technique, see e.g. Beddhu et al (1998) and Wilson et al (1998), but unstructured grids have also been used, e.g. Hino (1998) and Löhner and Onate (1998). The present grid generator The grid generation process starts by importing a CAD surface description file. Often these files have to be improved, for instance by closing the propeller shaft opening at the stern. A separate module creates an ellipsoid that is fitted to the bossing and faired to the hull. Having fixed the surface, surface grids are generated on the hull (and the appendages, if any). This can be done in one of two ways. The simplest possibility is to cut the surface by horizontal or vertical planes to obtain one set of lines. The other set is obtained thereafter by connecting points at a given percentage of the total arc length along each one of these lines, measured from one of the patch boundaries. If grids of this type get too skewed hyperbolic marching is used. Starting from a patch side, grids are grown inwards in a stepwise manner. One step consists of moving all the points on one line to a new one. This is done for each point by taking a step at right angles to the original line and in a direction tangential to the surface. The point found will generally be away from the surface, so it is moved along the normal down to the surface. Each step length is determined from the cell area defined by the start and end positions of two successive points. The grid may be forced to follow given lines at the side boundaries. Having completed the grid generation on the hull, body-fitted volume grids are grown hyperbolically outwards from the surface. The procedure is a three-dimensional generalization of the two-dimensional one just described. Points are first moved at right angles to the hull surface and thereafter at right angles to the surface defined by the grid points from the previous step. The size of the step, and the capability of following side boundaries are as described above. The description so far concerned the curvilinear body-fitted grids. These are embedded into one or more background grids, which are normally Cartesian (cylindrical grids have also been used) and extend to the boundaries of the computational domain. When all component grids have been generated the overlap algorithm starts. Each grid is given a unique priority, the initial background grid always being the lowest. The overlapping grid is then constructed in the following steps: 1. Cut surface holes, i.e. mark all points outside of the computational domain and lying on grid faces that are part of the physical boundary as dead. 2. Cut volume holes, i.e. mark the remaining points that are outside of the computational region as dead. 3. Set up exact interpolation points. 4. Classify (initially) all remaining points, see below. 5. Check the consistency of the interpolation points. 6. If there are any bad interpolation points, refine the background grid and go to 1. 7. Trim unneeded interpolation points. Except for step 6, this is the same algorithm as Peterson (1997a, b, 1998). After step 1 and 2 all points that are outside of the computational region are marked as dead. The exact interpolation points in step 3 occur where a grid has been split in order to accommodate a change of boundary condition. The grids share the same mapping function and overlap by one grid cell and consequently two layers of grid points coincide. The importance of this step is that the interpolation stencil for these points has zero width so there is no risk for it to incorporate any dead points. In the initial classification the following is done for each unclassified point in each grid, starting with the highest priority: • Check if it interpolates from a higher priority grid. • If not, check if it can be a discretization point. • If not, check if it interpolates from a lower priority grid. • If not, mark point as dead. the authoritative version for attribution. 

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line SHIP STERN FLOW CALCULATIONS ON OVERLAPPING COMPOSITE GRIDS 912 After step 4 each grid point is classified as an interpolation point, discretization point or dead point. Points in non- background grids and not close to physical boundaries are always allowed to interpolate to the background grid and refinements thereof. If the background grid is still too coarse to yield an interpolation stencil free of dead points, the interpolation point is marked as bad. In step 5 all interpolation and discretization stencils are inspected and each point that uses a stencil that contains a dead point is put on the list of bad points. All points in the stencils of bad interpolation points due to too coarse background, see above, are also put on this list. If the list is not empty after all grid points are classified, the bad points in the finest background grids are used to calculate a set of boxes that encloses the bad points, Bell et. al. (1994). For each such box a refined Cartesian background grid is inserted in the overlapping grid, and the algorithm restarts from the beginning. SOLVER Governing equations In a Cartesian coordinate system the Reynolds-averaged Navier-Stokes equations for incompressible flow may be written (1) and the continuity equation reads (2) Ui are the mean velocity components and xi the space coordinates. P is the pressure, ν the kinematic viscosity, τij the Reynolds stress tensor, and t is the time. Cartesian coordinates may be used in the background grids, while the body fitted grids require curvilinear coordinates. Transforming only the independent variables the above equations become (3) and (4) In the calculations of the present paper the two-layer k-ε model of Chen and Patel (1988) is used, i.e. transport equations for k and ε are solved in the major part of the computational region, while a prescribed length scale is used together with the k-equation in a thin layer close to the wall. The transport equations may be written (5) and (6) These can be transformed to curvilinear coordinates like when (1) is transformed to (3). The production term is (7) with τij calculated using the Boussinesq assumption as follows: (8) (9) This means that the last term in (1), can be handled by adding the turbulent viscosity νt to the kinematic viscosity. (10) The constants have their traditional values: (11) the authoritative version for attribution. 

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line SHIP STERN FLOW CALCULATIONS ON OVERLAPPING COMPOSITE GRIDS 913 The k equation (5) and the production term (7) are maintained in the near-wall layer, but the ε equation (6) is reduced to: (12) and the calculation of the eddy viscosity (10) is changed to: (13) where, with y the normal distance to the wall: (14) The inner layer extends from the surface to Ry=250. Numerical method In order to make the interpolation equations as simple as possible and to keep the number of interpolation points small some care has to be exercised when discretizing the equations. First the discretization stencil has to be as small as possible, otherwise the overlap regions will be wide. Since the equations are second order the smallest stencil possible is 3 nodes wide. Second, collocated storage is required since staggered storage gives four interpolation points per cell. Collocated storage also facilitates the use of Cartesian components so that base vectors will not have to be transformed when interpolating between the grids. Thus, some methods like higher order upwind differences and staggered grids cannot be used, even though they are numerically attractive. The challenge in the present work is to find a stable, efficient and accurate scheme with collocated node-centred storage with a three node wide stencil. Another requested property of the method is the capability to simulate thin boundary layers. At model scale the Reynolds number for a ship hull is typically 107 and at full scale 109. To resolve the corresponding boundary layers, grids with cell aspect ratios up to 106, Regnström (1994), Eca and Hoekstra (1996), close to the wall are needed. If a time- stepping scheme is used it must not have too severe limitations on the time step size for such grids and this excludes purely explicit schemes. For implicit and semi-implicit methods the high aspect ratio cells will give a poorly conditioned coefficient matrix. If a traditional stretched grid is used an iterative method capable of handling this must be used. The last requirement on the algorithm is that it must be robust. Even if this means lower accuracy the results can be used as a basis for improvements. The discretization of the governing equations and the boundary conditions will now be described. A shift operator is defined by: (15) Normally the index i will refer to one of the coordinate directions. Using the operator the second order central and first order forward and backward finite differences are defined by: (16) (17) (18) Next the terms in the partially transformed equations are expressed by replacing the partial derivatives with finite differences. This is done using the operators defined in Table 1: the authoritative version for attribution. 

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line SHIP STERN FLOW CALCULATIONS ON OVERLAPPING COMPOSITE GRIDS 914 Table 1: Definition of operators Operator Symbol Cartesian form Curvilinear form Convection (1st order upwind) K1u K Convection (2nd order central) 2c V Diffusion Gradient G Divergence (face centred) Df Divergence D Laplacian L where the contravariant velocity components are vi=ui/h(i) in Cartesian and in curvilinear coordinates. The SIMPLE scheme is used with the Rhie-Chow (1983) method to suppress chequer board oscillations that will otherwise occur when using collocated storage. First comes a predictor step that is implicit in u: (19) Superscript with one or more marks an approximation to the variable at time level n+1. The operator K is a hybrid of the first and second order convection operators: (20) The corrector step is explicit in u and implicit in p: (21) The difference between these is (22) Taking the discrete divergence D of this equation results in a sparse Laplacian DG that will give solutions with chequer board oscillations. Its stencil is also wider than the three nodes postulated earlier for minimizing the overlap region. It also requires that boundary conditions are specified in two layers. Replacing the sparse with the dense Laplacian will not affect the order of the scheme, but it will make the predictor and corrector steps incompatible, so that no true steady state where both equations are simultaneously satisfied can be reached. The remedy is Rhie-Chow interpolation where a staggered storage scheme is approximated. Denoting quantities at control volume faces by subscript f the predictor equation is expressed as (23) Approximate the convective and diffusive terms by interpolating from the nodes (overbarred terms): (24) the authoritative version for attribution. 

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line SHIP STERN FLOW CALCULATIONS ON OVERLAPPING COMPOSITE GRIDS 915 Note that the pressure values are located on the grid nodes in this formula. Another approximate expression is obtained by interpolating the whole of the predictor equation: (25) Taking the difference between these gives the Rhie-Chow formula for the face-centred velocity: (26) Requiring the (face-centred) divergence of u** to be zero gives the pressure equation: (27) where (28) The last term in (27) means that no attempt at removing divergence because of the difference between the sparse and dense Laplacian will be made if a steady state is approached. After the velocity components and the pressure have been updated the k and ε equations are solved implicitly like in the predictor step. u is used as the convective velocity. (29) (30) In the inner layer the explicit (12) is used for ε. After solving (29) and (30) the eddy viscosity is calculated according to (10) and (13). On each face of a component grid that coincides with the boundary of the fluid domain one of the four boundary conditions given in Table 2 are applied. A boundary face is identified by the coordinate direction not tangential to the face, denoted by B, and the direction of the normal +1 if it points into the grid and −1 otherwise. The index B is excluded from the summation convention. For the continuous problem explicit boundary conditions are only given for the velocity. The pressure boundary conditions are derived by applying the continuity equation on the boundary, except for the outflow where the pressure is constant. When applying the continuity equation on the boundary it must be remembered that a staggered storage scheme is mimicked. The control volume adjacent to the boundary is half of an interior volume, see the figure below, The divergence in a boundary point is expressed as (31) The boundary expression for the face-centred Laplacian is derived like before by applying the divergence operator, now (31), on the normal flow components of the pressure gradient. On all the boundary types where the Laplacian will be applied, the normal flow component of u is constant so that the velocity correction is zero. By this the normal component of the gradient of the pressure correction on these boundaries is zero and can be excluded from the expression for the Laplacian. Application of the slip and outflow conditions for the velocities are deferred until the velocity correction. For the predictor step the Dirichlet boundary condition is used instead. the authoritative version for attribution. 

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line SHIP STERN FLOW CALCULATIONS ON OVERLAPPING COMPOSITE GRIDS 916 Table 2: Boundary conditions Boundary Predictor eq. Pressure eq. Corrector eq. u*=un Noslip ∆tLp'=Du* u**=u* Inflow Slip Outflow p'=0 Here nB is the boundary surface normal unit vector. The Neumann boundary condition on slip boundaries for the predictor and corrector steps really only applies to the components tangential to the surface, the normal component is zero. Implementing this directly would couple the component equations, so instead the normal component of velocity is removed in a separate step. VALIDATION During the spring of 1999 the CHAPMAN code was validated through a large number of calculations within the EU- project CALYPSO (Tuxen et al 1998). The computed cases included the following: • Flat plate • Ellipsoid • HSVA tanker • Ryuko Maru tanker • Modern container ship, two variants • Modern ferry with flat stern, two variants • Modern ferry with tunnel stern, two variants For the plate, the ellipsoid and the Ryuko Maru tanker the calculations were carried out both at model and full scale Reynolds numbers. The best experimental data available are from the old tankers, so rather detailed comparison with computed results has been made for these two cases. Some of these comparisons will be presented here. The reason for incorporating both tankers is that the HSVA hull is outstanding when it comes to boundary layer and wake measurements at model scale, while the Ryuko Maru data include results from three Reynolds numbers, corresponding to model and full scale, as well as an intermediate scale. More scarce experimental data have been available for the modern ships, but some comparison between calculations and measurements has been possible in all cases. Unfortunately, the modern ships are confidential but permission has been granted to show a few examples from the container ship. The geometry of the modern Korean hulls used in the Gothenburg 2000 CFD workshop was not available by the time these calculations were carried out. the authoritative version for attribution. Figure 1. Overlapping grid around the aft half of the HSVA tanker. Top: overview, bottom: close-up near the stern 

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line SHIP STERN FLOW CALCULATIONS ON OVERLAPPING COMPOSITE GRIDS 917 HSVA Tanker The HSVA tanker is probably the most widely used test case for ship hydrodynamics CFD. It was one of the cases in all three workshops on ship viscous flows held in the eighties and the nineties (Larsson 1981, Larsson et al 1991, Kodama et al 1994). Reference is made to the workshops for the hull geometry. Only the aft half of the hull was computed and the grid layout is shown in Figure 1. Three body fitted grids are used to represent the hull and its immediate neighborhood. These grids have 30 points in the direction out from the hull and they are stretched in this direction using a hyperbolic tangent function. The distance from the closest point to the surface is approximately 6x10−6 L, which corresponds to y+ around 1. The automatically refined background grid is Cartesian with 5 components at 4 levels of grid density. In the Figure there is also a block surrounding the stern where the density has been increased only in the two transverse directions. Calculations were carried out both with and without this block. The results below were obtained with the block included and the total number of grid points was 289 703. Note that this is for only the aft half of the hull (one side). Figure 2. Comparison between results from the two blend ratios. Top: isobars, bottom: isowakes in the propeller plane Boundary conditions for the calculations are given in Table 2. The quantity un was obtained from a flat plate boundary layer solution for the forebody back to the inflow station amidship. This velocity profile (with an undisturbed velocity outside the boundary layer) was also used to initialize the solution in the computational domain. The same technique was used for all hulls in the validation studies, although the intention is to incorporate the new solver in the SHIP-FLOW zonal system (Larsson et al (1990)), thereby increasing the accuracy of the inflow conditions. The computation was run with a hybrid first order upwind—second order central discretization of the convective terms. For the turbulence equations the blend was 50/50, while for the momentum equations two blends were tested: 20/80 and 5/95, where the first number corresponds to the first order percentage. In total 180 iterations were made, 90 with a time step of 0.1 and 90 with a step of 0.01. In Figure 2 comparisons are made between results from the two blend ratios. It is seen that the differences are very small, both in the pressure distribution and in the isowakes at the propeller plane. It is unlikely that a full second order discretization would the authoritative version for attribution. 

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line SHIP STERN FLOW CALCULATIONS ON OVERLAPPING COMPOSITE GRIDS 918 change the results noticeably. Comparisons with measurements are shown in Figure 3 to Figure 5. The pressure distribution along the waterline and the keel is presented in Figure 4, and it is seen that the correspondence with the data is quite good. In this figure and the following x is a dimensionless coordinate along the hull, with the origin at the FP and a value of 1.0 at the AP. Limiting streamlines are shown in Figure 3. Note that the thick lines correspond to the block boundaries and that the streamlines pass the boundaries without any distortion. The topology of the calculated lines is the same as in the oil flow picture from the measurements, but the regions of upward flow from the bilge and downward flow from the stern seem too large. Most likely, this is an effect of the too simple turbulence model. Figure 3. Limiting streamlines. Top: calculations, bottom oil flow picture Figure 4. Pressure distribution along the waterline and keel the authoritative version for attribution. 

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line SHIP STERN FLOW CALCULATIONS ON OVERLAPPING COMPOSITE GRIDS 919 Isowakes are shown at station x=0.976 (the propeller plane) in Figure 5. The outermost contours are relatively well predicted, but the innermost ones are too smooth, as can be expected with the k-ε turbulence model Figure 5. Isowakes at x=0.976 (propeller plane). Top: calculations, bottom: measurements The cross-flow in the propeller plane is shown in Figure 6. Thanks to the Cartesian coordinate system used in both experiments and calculations it is fairly easy to compare the vector plots. There is, however, a difference in the reference length for the vectors. In the experimental data there are two clearly distinguishable vortices on top of each other and these vortices are seen also in the computed results, although they are not as clearly separated. Figure 6. Cross-flow at x=0.976 (propeller plane). Top: calculations, bottom: measurements Ryuko Maru The Ryuko Maru is one of the test cases recommended by the 22nd ITTC Resistance Committee, 1999. Measured data have been reported by Namimatsu and Muraoka (1974) at three different Reynolds numbers, corresponding to model the authoritative version for attribution. scale, full scale and one intermediate scale. Velocity contours are given at all three scales at a station 8 m (full scale) in front of the propeller. The reader is referred to the references 

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line SHIP STERN FLOW CALCULATIONS ON OVERLAPPING COMPOSITE GRIDS 920 for a body plan. Figure 7. Overlapping grid around the aft half of the Ryuko Maru tanker. Top: overview, middle: volume grids at the stern, bottom: surface grids at the stern The hull surface is covered with three overlapping surface grids. The foremost grid covers most of the aft hull and is generated by cutting the surface with planes of constant x. The grid on the overhang is generated from constant z cuts. Finally the trailing edge and lower tip of the skeg is covered with a grid that is generated with a hyperbolic method, starting at the trailing edge and marching forward, while following the bottom line. The surface covered by the foremost grid cuts the boundary of the computational region at fairly large angles, so a boundary-fitted volume grid with faces coinciding with the physical boundaries except at the rear edge is suitable. The overhang is more difficult since the intersection angle with the symmetry (y=0) plane varies from large at the water plane, to small down on the skeg. A grid of the same type as the first is nevertheless generated, with faces fitted to the symmetry and water planes. This makes the cells highly skewed in the region where the overhang blends into the skeg, and here the tangential resolution is increased by refining the volume grid twice. The surface grid at the skeg trailing edge has two neighbouring sides in the same plane, so if any or both of the volume grid sides growing out from these lines were made to follow the symmetry plane, the volume grid would be singular. Instead the surface grid is extended out onto the symmetry plane, and a volume grid is generated from this extended surface. Since it is only possible to assign a single boundary condition to each grid face, the volume is split after generation in three parts, one fitted to the original patch on the hull and two fitted to the symmetry plane. The two latter grids form an L around the skeg tip. All boundary fitted grids are generated with a hyperbolic method. The grids are the same for all Reynolds numbers, except for the distance to the first grid point and the number of grid points out from the surface. The number of points in this direction was chosen so that the grids became approximately equally thick. Number of grid points 47 82 82 1.6 10−6 5 10−9 5 10−9 ds The computational region is then filled with a coarse Cartesian background grid which is automatically and iteratively refined until a valid overlapping grid is obtained. To obtain converged results at the highest Reynolds number a pure first order discretization was used. This was not necessary at the lower Reynolds numbers, but for consistency the same discretization was used throughout. In order to investigate the effects of the low order a calculation was carried out with a 50/50 blend at the model scale. A comparison the authoritative version for attribution. with the pure first order results is made in Figure 8. The differences are visible, but very small. It was therefore conjectured that the grid was fine enough to get reliable results also with the first order method. 

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line SHIP STERN FLOW CALCULATIONS ON OVERLAPPING COMPOSITE GRIDS 921 Figure 8. Comparison between first order and hybrid first and second order (50/50) results. Pressure distribution and streamlines on the afterbody Axial velocity contours at the measurement station for the three Reynolds numbers are shown in Figure 9. Unfortunately the measured region is quite small, especially at full scale. The thinning of the boundary layer with increasing Reynolds number is however apparent, and it is rather well predicted. Note that the computations were carried out without the propeller in operation during the measurements. This effect should be rather small, since the measurement station was about a propeller diameter in front of the propeller, but it cannot be completely ignored. The limiting streamlines at the three scales are shown in Figure 10. As can be expected, the separated region near the stern is gradually removed when the Reynolds number is increased. At full scale the separation seems to have disappeared completely. As the separation is removed and the boundary layer becomes thinner the pressure at the stern increases. This is clearly seen in the figures to the left and means that the viscous resistance is reduced. Figure 9. Axial velocity contours at the measurement station 8m (full scale) in front of the propeller. Top: Rn=7.4×106, middle: Rn=6.6x107, bottom: Rn=2.4×109. Port side: calculations, starboard side: measurements the authoritative version for attribution. 

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line SHIP STERN FLOW CALCULATIONS ON OVERLAPPING COMPOSITE GRIDS 922 Figure 10. Limiting streamlines and isobars. Top: Rn=7.4x106, middle: Rn=6.6x107, bottom: Rn=2.4x109. Modern container ship This hull was one of the test cases in the European project CALYPSO (Tuxen et al 1998). For confidentiality reasons the body plan cannot be shown, but some results from the stern flow predictions may still be of interest. Figure 11 shows a close-up of the aftmost part of the hull with the stern bulb. It is seen that the surface grids cover the surface very well. One important detail is the grid on the tip of the bulb. In the original CAD surface the hole for the propeller shaft was open, so the surface had to be closed by a cap. This was generated by a part of an ellipsoid, which was fitted to the edges of the hole. However, since the grid on the ellipsoid had a singularity at the tip this part was cut and replaced by a rectangular grid patch. Figure 11. Surface grids at the stern of the container ship The predicted pressure distribution and limiting the authoritative version for attribution. 

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line SHIP STERN FLOW CALCULATIONS ON OVERLAPPING COMPOSITE GRIDS 923 streamlines are shown in Figure 12. There is an interesting similarity in the topology of the streamlines with the corresponding plot for the HSVA tanker, i.e. a branching of the lines near the waterline with a downflow near the trailing edge and a separation line where the flow from the bottom meets the downflow from the side. This separation line is however much shorter and has moved to the aft end of the bulb. Much weaker bilge vortices may be expected from this stern. Figure 12. Pressure distribution and limiting streamlines Iso-wakes are presented in Figure 13. As expected both the measured and calculated contours are rather smooth, indicating that the bilge vortex is weak. A hook is noted for the innermost measured contour, but this may well be an effect of the shaft that was present in the measurements, but not in the calculations. In general, there is a rather good correspondence between the predictions and the data. For non-vortical flows this is possible even with the simple k-ε turbulence model. FUTURE WORK While the present work has demonstrated the viability of the overlapping grid method for ship hydrodynamic calculations at both model and full scale Reynolds numbers, the full potential of it has not been used. The overlapping grids used here are best suited to fully explicit methods, where the updating of the solution is done on the structured grids completely separated from the unstructured updating of the interpolation points. In contrast to this the requirement to have an implicit solver that stems both from the incompressible flow and the thin boundary layers encountered, forces us to somehow solve the fluid flow and interpolation equations simultaneously. This is presently done by assembling sparse coefficient matrices from the structured flow equations and the interpolation equations and feeding those to an iterative solver. Figure 13. Calculated and measured iso-wakes in the propeller disk. Solid line: calculations, dashed line: measurements What appears to be the most promising way out is the Non-Aligned Multigrid method, Johnson (1992). Here the smoothing takes place on the component grids, completely separated from the possibly unstructured restriction and prolongation operations that handle the inter-grid communication. The cost for this is that whole grids instead of only the boundaries of the overlapping regions have to be interpolated. Experience from the present work has however showed that the number of interpolation points is so large that the simpler data structures, arrays vs. linked lists, possible when interpolating the whole grids gives approximately the same overhead for the two approaches. ACKNOWLEDGEMENTS the authoritative version for attribution. A major part of this work was sponsored by the European Commission under the contract BE95–1721 (the CALYPSO project). The authors are indebted to Mr. Niklas Wikström, who carried out the calculations for the Ryuko Maru tanker. REFERENCES Bell, J, Berger, M, Saltzman, J, Welcome, M (1994): Three-Dimensional Mesh Refinement for Hyperbolic Conservation Laws. SIAM J. Sci. Comput., Vol. 15, No 1, pp. 127–138, January Beddhu, M., Jiang, M.Y., Whitfield, D.L., Taylor, L.K., Arabshahi, A. (1998): CFD validation of the free surface flow around DTMB Model 5415, 3rd Osaka Col 

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line SHIP STERN FLOW CALCULATIONS ON OVERLAPPING COMPOSITE GRIDS 924 loquium, Osaka Carlsson, L (1997): A Second-Order Accurate Solver for the 2-D Incompressible Navier-Stokes Equations on Overlapping Grids. MSc thesis, Chalmers Univ. of Technology, Dept. Naval Arch. and Ocean Eng. Carlsson, L (2000): Validation of a Semi-Implicit Overlapping Grid Solver for the Navier-Stokes Equations. Electronic Journal of Grid Generation, Vol 1, No 1, 2000 Chen, H C and Patel, V C (1988): Near-Wall Turbulence Models for Complex Flows Including Separation. AIAA Journal, Vol 26, No 6 Eca, L and Hoekstra, M (1996): Numerical Calculations of Ship Stern Flows at Full-Scale Reynolds Numbers, 21st Symposium on Naval Hydrodynamics, Trondheim Hino, T. (1998), Navier-Stokes Computations of Ship Flows on Unstructured Grids, 22nd Symp. on Naval Hydrodynamics, Washington, August 1998 Johnson, R.A. (1992): A Multigrid Approach to Embedded-Grid Solvers, Wright Laboratory, Report WL-TR-92–7093 (or AD-A256 405) Kang, K-J (1996): Numerical Simulation of Non-Linear Waves about a Submerged Hydrofoil. 11th Workshop on Water Waves and Floating Bodies, Hamburg Kang, K-J (1997): Numerical Simulations of Non-Linear Waves Generated by Submerged Bodies, Chalmers Univ. of Technology, Dept. Naval Arch. and Ocean Eng., CHA/NAV/R-96/0043 Kodama, Y, Hino, T, Murashige, S, Takeshi, H, Uto, S, Hinatsu, M and Hirata, N (editors) (1994): Proceedings of the CFD Workshop Tokyo 1994, Ship Research Institute, Tokyo Larsson, L. (1981), SSPA-IITC Workshop on Ship Boundary Layers 1980, Proceedings, SSPA Report No. 90, Göteborg Larsson, L., Broberg, L., Kim, K.-J., Zhang, D.H. (1990): A Method for Resistance and Flow Prediction in Ship Design, SNAME Transactions Vol. 98 Larsson, L, Patel, V C & Dyne, G (editors) (1991): Ship Viscous Flow. Proceedings of the SSPA-CTH-IIHR Workshop, FLOWTECH International, Research Report No 2 Larsson, L., Regnström, B., Broberg, L., Li, D.-Q., Janson, C.-E. (1998): Failures, Fantasies and Feats in the Theoretical/Numerical Prediction of Ship Performance, Keynote lecture, 22nd Symp. on Naval Hydrodynamics, Washington DC, August 1998 Li, D-Q. (1998a): Development of Hyperbolic Overlapping Surface Grid Generation Techniques, Chalmers Univ. of Technology, Dept. Naval Arch. and Ocean Eng, Report No CHA/NAV/R-98/0058, 1998 (Presented at the 4th Symposium on Overset Composite Grid and Solution Technology, Aberdeen, USA, September 1998) Li, D-Q. (1998b): Geometric Representation and Overset Surface Grid Generation of Marine propellers, Chalmers Univ. of Technology, Dept. Naval Arch. and Ocean Eng, Report No CHA/NAV/R-98/0059, November 1998 Lin, C.W., Percival, S., Fisher, L. (1998), Viscous flow computations on an appended ship by chimera RANS scheme, 3rd Osaka Colloquium, Osaka Löhner, R., Qnate, E. (1998): Viscous Free Surface Hydrodynamics Using Unstructured Grids, 22nd Symp. on Naval Hydrodynamics, Washington, August 1998 Masuko, A. (1998): Numerical simulation of the viscous flow for complex geometries using overset method, 3rd Osaka Colloquium, Osaka Namimatsu, M. and Muraoka, K.: Wake distribution of full form ship, IHI Engineering Review, 1974 Petersson, A N (1997a): An Algorithm for Constructing Overlapping Grids. Chalmers Univ. of Technology, Dept. Naval Arch. and Ocean Eng, Report No CHA/ NAV/R-97/0049 Petersson, A N (1997b): Hole-Cutting for 3-D Overlapping Grids. Chalmers Univ. of Technology, Dept. Naval Arch. and Ocean Eng, Report No CHA/ NAV/R-97/ 0052 Petersson, A (1998): Hole-Cutting for 3-D Overlapping Grids. Accepted for publication in the SIAM J. of Scientific Computing, 1998 Regnström, B (1994): Prediction of Wing Section Lift and Drag from Numerical Solutions of the Navier-Stokes Equations. Chalmers Univ. of Technology, Dept. Naval Arch. and Ocean Eng, Ph.D. Thesis Rhie, C.M. and Chow, W.L. (1983): Numerical study of the turbulent flow pas an airfoil with trailing edge separation, AIAA Journal, 21(11) Svennberg, U S (1997): A Test of Turbulence Models for a Vortex in a Free-Stream, Chalmers University of Technology, Dept. of Naval Architecture and Ocean Engineering, Report No CHA/NAV/R-96/0047 Svennberg, U.S., Regnström, B., Larsson, L.: A Test of Turbulence Models for Vortices, 3rd Osaka Colloquium, Osaka, May 1998 Svennberg, U.S. (2000): A Test of Turbulence Models the authoritative version for attribution. 

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line SHIP STERN FLOW CALCULATIONS ON OVERLAPPING COMPOSITE GRIDS 925 for Steady Flow Around Ships, Numerical Towing Tank Conf., Tjärnö, 2000 Tuxen, J., Hoekstra, M., Nowacki, H., Larsson, L., van Walree, F., Terkelsen, M.: The CALYPSO Project: Computational Fluid Dynamics in the Design Process, Symposium on the Practical Design of Ships and Offshore Structures, PRADS ‘98, Wageningen, September 1998 Vogt, M and Kang, K-J (1997): A Level Set Technique for Computing 2-D Free Surface Flows, 12th Workshop on Water Waves and Floating Bodies, Marseille Vogt, M (1997): A Comparison Between Moving Grid and a Level Set Technique for Solving 2-D Free Surface Flows. ASME Fluids Engineering Summer Meeting, Vancouver Vogt, M. (1998): A Numerical Investigation of The Level Set Method For Computing Free Surface Waves. Chalmers Univ. of Technology, Dept. Naval Arch. and Ocean Eng, Report No CHA/NAV/R-98/0054, March 1998 Vogt, M and Larsson, L (1999): Two Level Set Methods for Predicting Viscous Free Surface Flows. 7th International Conference on Numerical Ship Hydrodynamics, Nantes Weems, K., Korpus, R., Lin, W.M., Fritts, M., Chen, H.C. (1994): Near-field flow prediction for ship design, 20th Symp. on Naval Hydrodynamics, Santa Barbara Wilson, R.Paterson, E., Stern, F. (1998): Unsteady RANS CFD Method for Naval Combatants in Waves, 22nd Symp. on Naval Hydrodynamics, Washington, August 1998 the authoritative version for attribution. 

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line SHIP STERN FLOW CALCULATIONS ON OVERLAPPING COMPOSITE GRIDS 926 DISCUSSION M.Visonneau Ecole Centrale de Nantes, France The authors present an interesting method for computing the flow around ship sterns based on a set of overlapping grids on and around the hull surface. This work illustrates very clearly that the flexible Chimera-like methodology which is often employed to solve complex flows around bodies with appendages in motion, may also be used with great profit to reduce the discretisation errors by improving the mesh quality in terms of orthogonality and refinement in the regions presenting physical challenges. Incidentally, this work brings a final illustration of the prominent role played by the turbulence modelling for the prediction of ship stern flows (see fig. 2). When the configuration which is studied does not consist of mobile appendages, it is also possible to have recourse to unstructured grids to improve the grid quality as well. As long as no really efficient unstructured grid generation tools are available for high-Re turbulent flows, the use of a set of overlapping structured grids may be considered as more convenient, even if the modern finite-volume methods may treat control volumes of arbitrary shape. However, the main difficulty with the Chimera-like methods may be the loss of physical properties such as mass and momentum conservation, transportivity at the interpolation points. Can you comment on the relative merits of these two approaches and indicate if special ad-hoc procedures have been implemented in CHAPMAN to enforce some of these physical properties? AUTHOR'S REPLY Overall we believe that all viable methods will give similar results for a similar level of development effort, i.e. “there is no silver bullet”. The main difference between the methods is that spatial discretization is easier on a smooth, structured grid, especially if a high order of accuracy is desired, while conservation is easier to enforce on a grid with non-overlapping cells. For the present case, and also with appendages added, the following hints towards preferring overlapping grids: • Mostly smooth boundary, so that few component grids are required and the relative effort spent on interpolation is low. • High Re requiring high aspect-ratio cells. Such grids are easier to generate as separate structured component grids. • No shocks, so the solution can be resolved and the inaccuracy related to not satisfying some discrete conservation condition exactly can (probably) be neglected. • A Dirichlet boundary condition can be used for the pressure on the outlet boundary, so there is no solvability constraint on the pressure Poisson equation. For the case of rigid bodies moving relative to one another, it appears that it is more efficient to use a fixed grid around each body and interpolate, than to regenerate the grid for each time step. In this case most of the machinery required is already present in an overlapping grid method, while adding this capability to an unstructured grid method takes more work. No ad hoc procedures have been implemented in CHAPMAN to enforce conservation of physical properties across interpolation boundaries. the authoritative version for attribution.