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TURBULENCE MODEL NUMERICAL ANALYSIS

3D Hemisphere Cylinder Validation

SA-neg Model Results

Results are shown for the 3D hemisphere cylinder at M=0.6, Re=0.35 million based on unit length, reference temperature of 540 R, and angle of attack of zero. Three different CFD codes (FUN3D, USM3D, and CFL3D) have been employed in an effort to try to discern the grid-converged result for this case. Results here are for the SA-neg variant of the SA model. However, SA-neg is passive to the original (SA) model in well-resolved flowfields, and hence is expected to yield essentially identical results as SA for the cases here. FUN3D and USM3D both used clipping method "a" with cutoff 0.00001 for $\hat S$ , while CFL3D used clipping method "c" (see Note 1 on the Spalart-Allmaras equation page). For all codes, the farfield value of the Spalart turbulence variable is $\tilde \nu_{farfield} \geq 3 \nu_{\infty}$ . In all codes the Prandtl number Pr is taken to be constant at 0.72, and turbulent Prandtl number Pr_t is taken to be constant at 0.9. The dynamic viscosity is computed using Sutherland's Law (See White, F. M., "Viscous Fluid Flow," McGraw Hill, New York, 1974, p. 28). In Sutherland's Law, the local value of dynamic viscosity is determined by plugging the local value of temperature (T) into the following formula:

$\mu = \mu_0 \left( \frac{T}{T_0} \right)^{3/2} \left( \frac{T_0 + S}{T+S} \right)$

where $\mu_0 = 1.716 \times 10^{-5} kg/(ms)$ , $T_0 = 491.6 R$ , and $S = 198.6 R$ . The same formula can be found online (with temperature constants given in degrees K and some small conversion differences). Note that in terms of the reference quantities for this particular case, Sutherland's Law can equivalently be written:

$\left( \frac{\mu}{\mu_{ref}} \right) = \left( \frac{T}{T_{ref}} \right)^{3/2} \left( \frac{T_{ref} + S}{T+S} \right)$

where $\mu_{ref}$ is the reference dynamic viscosity that corresponds to the freestream in this case, and freestream $T_{ref}$ is 540R. This latter form may be more convenient for nondimensional codes. (Specific details regarding an implementation of Sutherland's Law in nondimensional codes can be found in handwritten notes describing Sutherland's Law in CFL3D and FUN3D.)

Note that FUN3D and USM3D both ran this case with first order spatial accuracy on the turbulence advection term, whereas CFL3D used second order.

It is also important to note that unlike most other test cases on the Turbulence Modeling Resource website, for this particular test case the structured and unstructured grids were created independently of one another. Thus they are not related, other than the fact that the body shape and exit plane are the same. In particular, the outer boundary extents are different: for the structured grid, it is located approximately 20 unit lengths from the body; whereas for the unstructured grid, it is located approximately 10 unit lengths from the body. The grid differences may have some impact on the results.

Grid Convergence Behavior of Drag and Maximum Eddy Viscosity

Plots are shown to illustrate grid convergence for the drag coefficients and maximum eddy viscosity in the field (the contributions to the drag coefficient due to the viscosity and pressure are presented separately). Recall that all models were run using SA-neg. The codes appear to be consistent for the drag values, but do not appear to be approaching the same level for maximum eddy viscosity. The reason for this inconsistency is not known. These plots draw attention to the fact that the structured grids (used by CFL3D) are considerably coarser than the unstructured grids (used by FUN3D and USM3D).

For computing forces, the reference area of this case is taken to be 10 in² (for the full 360 degrees), or 5 in² (for a half-plane computation of 180 degrees).
SA - drag coefficient vs h SA - pressure drag coefficient vs h
SA - viscous drag coefficient vs h SA - max eddy viscosity vs h

Results that generated the above plots can be found in the following data files: fun3d_drag_mut_convergence_sa.dat, usm3d_drag_mut_convergence_sa.dat, cfl3d_drag_mut_convergence_sa.dat.

Grid Convergence Behavior of Cp

The first plot below shows grid convergence of surface Cp at three x-locations on the body: x=0.25, x=0.5, and x=1.50. When plotted at this scale, results look reasonably close between the three codes. However, when zoomed in, there are differences at x=1.5 (see second plot below). The reason for this difference is believed to primarily be the fact that the structured grid (used by CFL3D) has a farfield extent of approximately 20 unit lengths, whereas the unstructured grid (used by FUN3D and USM3D) has a smaller farfield extent of 10 unit lengths.
SA - Cp vs h at 3 x-locations SA - Cp vs h at x=1.5

Results that generated the above plots can be found in the following data files: fun3d_cp_convergence_sa.dat, usm3d_cp_convergence_sa.dat, cfl3d_cp_convergence_sa.dat.

To test the hypothesis that the farfield extent/shape has an influence on the solution, some limited results were obtained with all 3 codes on the same grids. The first plot below shows the FUN3D and USM3D Cp results on the default unstructured grids at x=1.5 (these are the same results already shown above, with FF=10). Then, the second plot shows results for all 3 codes on the default structured grids (FF=20) used by CFL3D in the results shown above. Finally, a new set of structured grids was created with FF=10, and the 3 codes produced the results shown in the third figure below. Two conclusions can be made from this assessment: (1) the codes all approach the same results when run on the same grid, and (2) the farfield boundary has an impact on the Cp at this location. The pressure coefficient computed on the structured-grid domain with the farfield extent of 20 converges to a value about 2.5% lower (larger in magnitude) than the pressure coefficient computed on the structured domain with the farfield extent of 10. There is also apparently a sensitivity to the shape of the farfield boundary when located in a relatively close proximity to the surface: the pressure coefficients computed on domains with similar locations and different shapes of the farfield boundaries differ by about 1.3% (compare the 2 results with FF=10).
SA - Cp vs h at x=1.5, unstructured grid with FF=10 SA - Cp vs h at x=1.5, structured grid with FF=20

Results that generated the above plots can be found in the following data files: fun3d_cp_convergence_sa_str_grids.dat, usm3d_cp_convergence_sa_str_grids.dat, cfl3d_cp_convergence_sa_str_grids.dat.

Cp and Cf Axially

The Cp and Cf distributions (axially, on the body) from the three codes on the finest grid are shown in the following figures. All results are very close to plotting accuracy.
SA - Cp vs x SA - Cf vs x

Results that generated the above Cp plots can be found in the following data files: fun3d_cp_axial_sa.dat, usm3d_cp_axial_sa.dat, cfl3d_cp_axial_sa.dat. Results that generated the above Cf plots can be found in the following data files: fun3d_cf_axial_sa.dat, usm3d_cf_axial_sa.dat, cfl3d_cf_axial_sa.dat.

Cp, U-Velocity, and Eddy Viscosity Along a Spanwise Grid Line

The following plots show Cp and u-velocity along a radial line emanating from the surface at x=0.5:
SA - Cp along radial line at x=0.5 SA - u/a_ref along radial line at x=0.5

Results that generated the above plots can be found in the following data files: fun3d_various_radial_sa.dat, usm3d_various_radial_sa.dat, cfl3d_various_radial_sa.dat.

The following plot shows eddy viscosity along the same radial line emanating from the surface at x=0.5:
SA - eddy viscosity along radial line at x=0.5

Results that generated the above plots can be found in the following data files: fun3d_mut_radial_sa.dat, usm3d_mut_radial_sa.dat, cfl3d_mut_radial_sa.dat.

Radial Variations on the Surface

The following plots show radial variations of Cp and Cf at the location x=0.25, using 3 grid levels. There is very little circumferential variation in the structured-grid CFL3D results, but on the unstructured grids both FUN3D and USM3D show variations, particularly near the boundaries of 60-degree sectors, where there are abrupt changes in the surface grid triangulation.
SA - Cp circumferentially at x=0.25, CFL3D vs. FUN3D, three grid levels

Results that generated the above plots can be found in the following data files: fun3d_cp_circum_sa.dat, usm3d_cp_circum_sa.dat, cfl3d_cp_circum_sa.dat.
SA - Cf circumferentially at x=0.25, CFL3D vs. FUN3D, three grid levels

Results that generated the above plots can be found in the following data files: fun3d_cf_circum_sa.dat, usm3d_cf_circum_sa.dat, cfl3d_cf_circum_sa.dat.

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Last Updated: 11/05/2021