SIMULATION OF FLOW AROUND KCS-HULL

Sven Enger (CD-adapco, Germany)Milovan Perić (CD-adapco, Germany)

Robinson Perić (University of Erlangen-Nürnberg, Germany)

1.SUMMARY

The paper describes results of computations of resistance, trim and sinkage for the KRISO Container Ship (KCS) at different Froude numbers in calm water. Unstructured trimmed grids with local refinement in zones of interest have been used to obtain maximum accuracy at low cell count. The solution method is of finite-volume type and uses mostly approximations of second order. Free surface is modeled using Volume-of-Fluid approach and a high-resolution interface-capturing scheme. It is demonstrated that acceptable accuracy can be achieved with grids containing around half a million cells for half of the geometry.

2. INTRODUCTION

Industrial use of Computational Fluid Dynamics (CFD) requires high level of automation, in order to provide sufficient accuracy at low effort (both in terms of manpower and computing time). Therefore the design of computational grid requires some effort, so that grid refinement and modifications to geometry can easily be done. All computations reported here were performed using the same grid topology described in the next section. All grids were created and all computations and post-processing were performed using STAR-CCM+ code.

The solution method is of finite-volume type and uses control volumes of arbitrary polyhedral shape. The conservation equations in integral form for mass and momentum, together with an equation for volume fraction of liquid and two or more equations describing turbulence quantities, are solved using a segregated iterative solution method based on SIMPLE-algorithm. Details on discretization and solution methods can be found in literature and will not be given here (see Ferziger and Perić, 2003; Demirdžić and Muzaferija, 1995; Weiss et al, 1999). All surface and volume integrals are approximated

using midpoint rule; interpolation and gradient approximations are based on linear shape functions. Since time accuracy in the cases studied here is not of importance, first-order Euler implicit scheme is used for time integration.

The free-surface effects are modeled using the so called “Volume-of-Fluid” approach: the solution domain is assumed to be filled by a single effective fluid whose properties vary locally according to volume fraction of liquid. This equation contains only rate-of-change and convective term and its role is to track the deformation of the initially flat free surface. The convective terms are discretized using the HRIC-scheme (Muzaferija and Perić, 1999). It resolves the free surface typically with one cell when the interface is expected to be sharp.

In most computations, the standard k-ε turbulence model with wall functions was used to describe the effects of turbulence on the mean flow (Launder and Spalding, 1974). Some test calculations for the hull in fixed position and one Froude number were performed using two other two-equation turbulence models of edy-viscosity type and the full Reynolds-stress model, which requires the solution of 7 additional equations (6 for Reynolds stress tensor components and one for the dissipation rate). This was done to analyze the effects of turbulence models on the solution.

In addition, grid effects were analyzed by performing some simulations on a sequence of systematically refined grids. Grid spacing was reduced by a factor of 1.5 in the entire solution domain, except in the prism layer along walls, where the refinement was only in the directions tangential to wall. This was done in order to keep y+

values at near-wall cell centers around 50, which was found in the past to produce best solutions (Azcueta, 2001). However, one simulation was perform with varying mesh spacing in prism layer in wall-normal direction to assess the influence of y+ on the predicted resistance.

3. GRID GENERATION

In this study trimmed hexahedral grids with local refinements and prism layers along walls were used. The grid generation process is driven by specifying base mesh size, relative to which all spacings (prism layer thickness, cell size in various regions etc.) are defined. Finer meshes of the same topology are then automatically created by just reducing the base size. In order to avoid using fine grid where it is not necessary (in front of hull and at larger distance above, below, on each side and behind the hull), local volumes of different shape were created and assigned particular cell size, resulting in mesh structure shown in Fig. 1 for the coarsest mesh.

Only half of the geometry was considered due to symmetry conditions. The solution domain extended from -18 m to 18 m in flow direction, from -18 m to 9 m in vertical direction and from 0 to 18 m in lateral direction, respectively. The hull length was about 7.7 m (Lpp = 7.2786, scale factor 31.6) and the coordinate origin was at aft perpendicular and still water surface (which was 0.341772 m above keel). In addition to the symmetry plane of the hull, the lateral boundary parallel to it was also treated as symmetry plane. At the downstream boundary, hydrostatic pressure correspon-ding to the undisturbed water surface was prescribed. Upstream, top and bottom boundaries were treated as inlets with prescribed velocity and volume fraction.

The coarse, medium and fine mesh had 544138, 1220966 and 2997355 control volumes, respectively.

Fig. 1: The structure of the coarse mesh around KCS hull with rudder, showing local refinement regions near hull, near free surface, in the wake and in the wave zone.

Fig. 2: The structure of the fine mesh around KCS hull with rudder, showing local refinement regions.

There were 6 prism layers along walls (except for non-wetted walls) and the next-to-wall cells were 0.9 mm thick (prism layer was 20 mm thick, cell expansion ratio was 1.5). Figure 2 shows the fine mesh segments.

4. KCS, FIXED POSITION

This section contains results for the case of a hull fixed in its floating position at zero speed (even keel). The Froude number is 0.26. Computations were performed in a time-marching mode, starting with a flat water surface. The time step was 0.04 s and 4 iterations were performed at each time level. The standard k-ε turbulence model with wall functions was used. Figure 3 shows how forces on hull converge towards steady-state solution on the fine mesh. While shear force very quickly settles to a nearly constant value, pressure force oscillates around the steady-state value with a diminishing amplitude.

Table 1 shows predicted resistance at each grid level. The measured value of CT is 3.557 ∙10-3 (Van et al, 1998) and the numbers in parentheses show percentage difference to the experimentally obtained value (the + sign indicates that the value is over-predicted). The predicted total resistance is within 0.5% of measured value on all grids. The variation from one grid to another is not monotonic: the largest difference relative to measured value is obtained on medium grid. While the pressure drag reduces monotonically with grid refinement, the friction drag is largest on medium grid. It is not unusual that such variation is observed, especially when unstructured, locally refined grids are used.

Table 1: Predicted resistance coefficients on different grids

Grid CT (∙103) CF (∙103) CP (∙103)

Coarse 3.568 (+0.31%) 2,873 0,695

Medium 3.574 (+0.48%) 2,911 0,663

Fine 3.561 (+0.11%) 2,909 0,652

Figure 4 shows predicted wave pattern, wall shear stress on the hull and the near-hull y+ distribution. Only one half of the geometry was computed, but a mirror image of waves on the other side is also shown. The y+ values vary between 40 and 60 over the largest part of the wetted hull surface, as desired when wall functions are used. The strong reduction of wall shear stress in the stern region is due to the diffusor effect in the absence of propeller.

All three grids in these calculations had the same number of prism layers at walls: mesh refinement was done by reducing the cell size in all directions outside prism layer by a factor of 1.5, and within prism layer only in the two tangential directions, but not in the wall-normal direction. The idea here was to keep the same y+ values at near-wall cells. In order to verify the effect of variable mesh spacing inside prism layer, another coarse grid was generated with 4 prism layers instead of 6. The expansion factor was the same, but now the near-wall cell was 2.46 mm thick (compared to 0.9 mm when 6 prism layers were created). This resulted in y+ values between 100 and 120 over most of the hull surface. The total resistance computed on this mesh (which had 501342 cells, compared to 544138 cells with 6 prism layers) was 3.614∙10-3, which is 1.6% more than the measured value (compared with +0.31% more when 6 prism layers were used). Although this result is also acceptable for a relatively coarse mesh, the increase in accuracy resulting from adding two more prism layers is worth while.

We also looked at the effects of the choice of turbulence model on predicted resistance. This analysis was done using two grids provided by

Fig. 3: Convergence of friction and pressure drag during computation.

Fig. 4: Wave pattern (upper), wall shear stress (middle) and near-wall y+ (lower), computed on the finest mesh.

Germanischer Lloyd; they were of a similar type as those presented earlier and one had 745434 cells while the finer one had 2680461 cells. The results are summarized in Table 2.

Table 2: Predicted resistance coefficients on two grids using different turbulence models

Grid k-ε standard

k-ε two-layer

k-ω SST RSM

Coarse 3.610(+1.49%)

3.591(+0.96%)

3.425(-3.71%)

3.638(+2.28%)

Fine 3.588(+0.87%)

3.560(+0.08%)

3.415(-4.86%)

3.623(+1.86%)

The structure of the meshes used here was less optimized, and hence the effect of mesh refinement is more visible. For all models but k-ω, the values computed on the finer mesh are closer to experimentally obtained value of 3.557 ∙10-3. The Reynolds-stress model (RSM) predicts slightly higher resistance than both versions of k-ε model, while the k-ω SST model under-predicts the measured value by almost 5%. The fact that for this model the deviation compared to experiment increases with grid refinement is due to the fact that modeling and discretization errors vary locally in both sign and magnitude; the finding that the k-ε model produces the best result is most probably due to the partial cancellation of the two error types. Also, the model which predicts resistance closest to the measured value may not necessarily give the best velocity distribution in the propeller plane (RSM is expected to do the best job in this respect).

5. KCS, HEAVE & PITCH

In the second set of simulations, the hull with rudder was free to heave and pitch. The same grids were again used, but the mesh was now moving with the hull as a rigid body. All simulations started with a flat water surface and hull's bottom parallel to free surface. The mass of the half hull with rudder was set to 823.0451 kg, the center of gravity was at x = 3.5315 m (relative to coordinate origin in CAD-file, which was at aft perpendicular) and 0.1113924 m below free surface. The moment of inertia for rotation around y axis, needed for prediction of pitching motion, was set to 2725.117 kg m2. Simula-tions were performed for four Froude numbers: 0.1949, 0.2274, 0.2599 and 0.2816. For the two smaller Froude numbers, computations were performed only on the coarsest grid, while for the two highest Froude numbers, all three grids were used. The time step was set to 0.04 s for the Froude number 0.2599, and for other Froude numbers it was scaled according to velocity variation. For body

motion, 4 to 5 iterations per time steps were performed, while for the flow computation, 8 to 10 iterations were performed. The time step was the same for all grids, since temporal accuracy is not of interest – only the final steady-state solution is sought.

Figures 5 and 6 show convergence of ship motion and forces acting on it during simulation on the finest mesh for the two largest Froude numbers. Both the amplitude and frequency of oscillations are

Fig. 5: Convergence of pitch and heave motion (upper) and friction and pressure forces (lower) for Fn = 0.2599 and the finest mesh.

Fig. 6: Convergence of pitch and heave motion (upper) and friction and pressure forces (lower) for Fn = 0.2816 and the finest mesh.

higher for lower Froude number. This behavior has been observed in other applications of the solution methods to similar problems. No attempts were made here to vary the parameters in the simulation to minimize the computing effort needed to obtain the steady-state solution.

Figure 7 shows comparison of predicted and measured total resistance, sinkage and trim. The predicted drag agrees well with experimental data: the largest discrepancy is 1.66% on the coarsest grid for Fn = 0.2816. For other Froude numbers, already on the coarsest grid the error is of the order of 1% or

less (0.4%, 1.1% and 0.6% in the order of increasing Froude number). For Fn = 0.2816, the error is reducing with grid refinement to 0.2% on the finest mesh. However, for Fn = 0.2599, the error first increases to 1.3% and then drops to 1% on the finest mesh. It is also interesting that the predicted friction resistance slightly reduces with increasing Froude number (from 3.035 to 2.985, 2.071 and 2.961), while the pressure drag increases (from 0.455 to 0.520, 0.701 and 1.549, in the order of increasing Froude number). The contribution of pressure drag to total resistance increases sharply from 19.1% at Fn = 0.2599 to 34.3% at Fn = 0.2816.

Fig. 7: Comparison of predicted and measured total resistance (upper), sinkage (middle) and trim (lower)

Fig. 8: Computed wave pattern around KCS-hull for Froude numbers 0.1949, 0.2274, 0.2599 and 0.2816 (from top to bottom; first two computed on the coarsest, last two on the finest grid).

The trim and sinkage differ more from experimental data. The discrepancies in sinkage are 7.7%, 0.8%, 7.0% and 4.1% in the order of increasing Froude numbers, respectively (the last two values cor-respond to the finest mesh). Grid refinement did not result in improvement: the error increased from 5.3% to 7% for Fn = 0.2599, and from 2.5% to 4.1% for Fn = 0.2816. The errors in trim angle are 1%, 3.3%, 0.7% and 8.6% (from the lowest to the highest Froude number). At Fn = 0.2599, the discrepancy reduced from 4.7% to 0.7% with grid refinement, but for Fn = 0.2816, the discrepancy to experiment increased slightly from 7.5% to 8.6%.

Figure 8 shows predicted wave patterns for the four Froude numbers. The highest water elevation is always just behind ship stern, ranging from 45 mm at Fn = 0.1949 to 95.1 mm at Fn = 0.2816. The minimum water level is always at hull shoulder, ranging between -26 mm at Fn = 0.1949 to -64 mm at Fn = 0.2816. Water elevation is over-predicted on coarse grids. Figure 9 shows a comparison of wave pattern at Fn = 0.2816 computed on the coarse and the fine grid, respectively. The maximum predicted water elevation on the coarse grid was 98.6 mm, which dropped to 95.1 mm on the finest mesh (3.7% change). The lowest predicted point on water using coarse mesh was at -68.1 mm, which changed to -63.8 mm on the finest mesh (6.7% change). The change in resistance, however, was just below 1.5%.

Finally, Fig. 10 shows the resolution of free surface in the symmetry plane and along hull in the stern region on coarse grid. Almost everywhere the interface falls into one cell, which is the maximum possible resolution by interface-capturing schemes.

6. CONCLUSIONS

The presented results of simulations show that one can obtain a reliable prediction of resistance of container vessels using relatively coarse grids with about half a million cells (for half of the geometry) when the grid is well designed and locally refined in critical zones. The standard k-ε turbulence model with wall functions is adequate for this purpose and for optimum results, the prism layers near wall should be arranged so that y+ values around 50 are obtained. While the k-ε and Reynolds-stress model over-predict resistance by up to 2%, the k-ω SST model under-predicts the measured value by almost 5%.

REFERENCES

[1] Ferziger, J.H. and Perić, M. (2003): “Computa-tional Methods for Fluid Dynamics”, 3rd Ed., Springer Verlag, Berlin, Heidelberg.

[2] Demirdžić, I., Muzaferija, S. (1995): Numerical method for coupled fluid flow, heat transfer and stress analysis using unstructured moving meshes with cells of arbitrary topology, Comput. Methods Appl. Mech. Engrg., 125, 235-255.

[3] Weiss, J., Maruszewski, J.P., Smith, W.A. (1999): Implicit solution of preconditioned Navier-Stokes equations using algebraic multigrid. AIAA J., 37, 29-36.

[4] Muzaferija, S., Perić, M. (1999): Computation of free surface flows using interface-tracking and interface-capturing methods, In O. Mahrenholtz, M. Markiewicz (eds.), Nonlinear Water Wave Inter-action, Chap. 2, 59-100, WIT Press, Southampton.

[5] Launder, B.E., and Spalding, D.B. (1974): The numerical computation of turbulent flows, Comput. Meth. Appl. Mech. Eng., 3, 269–289.

[6] Azcueta, R. (2001): Computation of turbulent free-surface flows around ships and floating bodies, PhD Thesis, Technical University of Hamburg-Harburg, Germany.

[7] Van, S.H., Kim, W.J., Yim, G.T., Kim, D.H., and Lee, C.J. (1998): Experimental Investigation of the Flow Characteristics Around Practical Hull Forms, Proc. 3rd Osaka Colloquium on Advanced CFD Applications to Ship Flow and Hull Form Design, Osaka, Japan.

Fig. 9: Wave pattern around KCS-hull for Froude number 0.2816 computed on the coarse (upper) and fine (lower) grid.

Fig. 10: Resolution of free surface by HRIC scheme.