Drag Force for Flow over a Sphere
Daniel Prosser 11/4/10
The purpose of this assignment is to model fluid flow over a sphere and to illustrate how forces, pressures, and flow behavior change as Reynolds number increases. Turbulence models are employed to show the differences between laminar and turbulent flow behavior and to attempt to determine the Reynolds number at which the boundary layer on the sphere becomes turbulent, and to quantify the effect of a turbulent boundary layer on the total drag force on the sphere. The sphere is modeled inside a large box such that the walls of the box are distant from the sphere by at least 10 times the sphere diameter at all points. Figure 1 shows the geometry and boundary conditions, while Table 1 shows inlet velocities of air for the Reynolds numbers tested.
Figure 1: Geometry and boundary conditions
Table 1: Inlet Velocities and Reynolds Numbers
V (m/s) Re 0.292147 1000 1.460735 5000 2.921469 10000 5.842939 20000 7.303673 25000 8.764408 30000 10.22514 35000 11.68588 40000 29.21469 100000 87.64408 300000
2.0 Mesh
A simple, unstructured tetrahedral mesh was created in Gambit using sizing functions. Grid
independence is ensured by adapting the grid in Fluent at a Reynolds number of 1000, using total drag on the sphere as the comparison criterion for the different grid sizes. Figure 2 depicts the five different grid sizes tested.
Figure 2: Cross-sectional views of different grid sizes
Table 2 lists the mesh number, number of cells, and total drag force at a Reynolds number of 1000 for the grid independence test. Figure 3 plots the drag results. Mesh 2 was selected because it gives a good result with a relatively low number of cells. The very large grids showed drag force divergence, as can be seen in Figure 3.
Table 2: Mesh sizes and total drag
Mesh Cells Drag (N) 1 187571 0.002127 2 326885 0.002018 3 731408 0.001961 4 856981 0.001806 5 874257 0.001637
Figure 3: Grid independence study
The maximum skew for the final mesh (mesh 2) is 0.92 and the maximum aspect ratio is 14.1.
3.0 Numerical Procedures
Two different turbulence models are compared: the k-epsilon model and the k-omega model. All tests are performed using the default turbulence settings: The k-epsilon model uses a Cmu of 0.09, C1- epsilon of 1.44, C2-epsilon of 1.92, TKE Prandtl number of 1, and TDR Prandtl number of 1.3. The k- omega model uses Alpha_inf of 0.52, Beta*_inf of 0.09, Beta_i of 0.072, Zeta* of 1.5, Mt0 of 0.25, and Prandtl numbers set to 2. Both methods use the SIMPLE pressure-velocity coupling, Green-Gauss Node- Based gradient discretization, PRESTO! pressure discretization, 2nd Order Upwind momentum, and 1st Order Upwind turbulent kinetic energy and dissipation rate. All residuals are set to 1e-6; however, neither model was able to reach this level for all residuals. The velocities generally did reach that level of convergence or better, but continuity and the turbulence terms were generally an order of magnitude higher for both turbulence models. Figure 4 shows residual plots at a Reynolds number of 300,000 for each solver. The left-side plot is the k-epsilon model and the right-side plot is the k-omega model. Note that at this high Reynolds number the k-epsilon model gave somewhat better residual convergence, but
turbulence models.
Figure 4: Side-by-side residual histories for the two turbulence models
4.0 Results
The goal of this study is to determine whether either of the two solvers does a better job of predicting drag force on the sphere in turbulent flow. Figure 5 shows the compared drag coefficients from the two turbulence models as a function of Reynolds number.
The results are similar for the two different solvers, although the k-omega (red) gives consistently higher drag coefficients for all but the lowest Reynolds number. Most notably, there is no noticeable sudden drop in drag coefficient for either model, which would indicate a transition from a laminar to turbulent boundary layer. There are a couple potential reasons for this. The first idea is that the mesh is not fine enough in the boundary layer and wake region to capture all effects of turbulence. Grid adaption was performed at the lowest Reynolds
Figure 5: CD comparison for the two turbulence models
number (as specified in the homework handout). It may be a better idea to adapt the grid at a higher Reynolds number in order to capture the effects of turbulence. The other possibility is that the trip to a turbulent boundary layer does not occur until a higher Reynolds number. In order to determine
whether the mesh fineness had an effect on turbulence results, the case with Reynolds number of 40,000 was run again, with the grid adapted to increase the refinement in the boundary layer and wake region. Figure 6 shows the adapted grid for this case, which may be compared to the previous grid used (the second picture in Figure 2). Figure 7 shows the CD result for the k-epsilon and k-omega turbulence models, with the new data point shown for the newly adapted grid at Reynolds number 40,000.
Figure 6: Adapted grid for turbulence Figure 7: CD for k-w with adapted grid at Re = 40,000
Note in Figure 6 the improved mesh refinement in the boundary layer and wake regions for the newly adapted grid, compared to that of Figure 2. Figure 7 shows that, with the newly adapted grid at Re of 40,000, the drag coefficient has decreased noticeably for the k-omega model. This suggests that the mesh refinement is indeed important in capturing the effects of turbulence. The k-epsilon model had
poor convergence with this grid.
Figure 8 shows the pressure coefficient distribution on the top surface of the sphere for different Reynolds numbers. The
magnitude of Cp decreases as Reynolds number increases. At the lowest Reynolds number, the gage pressure is significantly less than zero at the back of the sphere. This is most likely due to high degrees of flow separation on the back half of the sphere, which produces a low pressure wake and high drag. This result is consistent with the very high drag coefficient calculated for this case (see Figure 5).
Figure 8: Cp on top surface of sphere for different Re
Additionally, the pressure coefficient on the front half of the sphere decreases significantly as Reynolds number increases, which also helps to explain why the drag coefficient decreases as Reynolds number increases. It is important to note, however, that though the drag and pressure coefficients decrease in magnitude for increasing Reynolds number, the actual drag force and pressure increase due to increased flow velocity. However, the non-dimensional coefficients do a better job of illustrating the flow
phenomena than the total pressures and forces. Figures 9 and 10 compare velocity vectors and pressure contours, respectively, for laminar and turbulent cases. Figure 9 shows that for the turbulent case, the wake actually narrows behind the sphere, while for the laminar case it remains fairly wide, which results in high pressure drag behind the sphere. Figure 10 confirms that the low pressure
contours behind the sphere are significant for laminar flow but are much smaller for the turbulent case.
Figure 9: Velocity vectors for Re = 1,000 and 40,000 (newly adapted grid)
Figure 10: Pressure contours for Re = 1,000 and 40,000 (newly adapted grid)
5.0 Conclusion
The major conclusion to be drawn from this study is that the mesh refinement in the boundary layer and wake region is important if turbulence is expected. The k-omega model showed a significant reduction in drag coefficient when the grid was adapted at a higher Reynolds number. For this reason, it is recommended that future offerings of this assignment instruct the grid adaption to be performed at a higher Reynolds number. The k-omega model was better able to account for turbulence with the newly adapted grid at Reynolds number of 40,000; convergence was poor for the k-epsilon model for this same case. However, the results of both need to be validated against experimental data, especially to see if the Reynolds number at which the boundary layer becomes turbulent is accurate. Further study into this subject would probably be better served by using a boundary layer mesh in Gambit as opposed to using a simple sizing function with grid adaption.
The results show that as Reynolds number increases, the drag coefficient decreases, likely because the boundary layer stays attached longer while the width of the wake is reduced. When the boundary is turbulent, this effect is even more pronounced. The velocity vectors and pressure contour plots for laminar and turbulent flow clearly illustrate these characteristics. The pressure contours especially show a very strong region of low pressure in the wake for the laminar case, but for the turbulent case this region is much less pronounced. This large region of low pressure in the wake for the laminar case contributes to the high drag coefficient, as does a similarly large region of high pressure region in front.
