# Several Cases for the Validation of Turbulence Models Implementation
## 4 Results
### Two-Dimensional Airfoil Near-Wake  
This test case is based off Model A airfoil, a convential airfoil that was experimentally investigated by [PERSON] [22]. Model A airfoil is a 10%-thick conventional airfoil with a 61 cm chord. The angle of attack was 0 deg, the wind tunnel speed was 30 m/s, and the freestream turbulence level was 0.02%. The boundaries of the computational domain, shown in Figure 10, were 20 chords away from the airfoil. A detail of the mesh near the airfoil is shown in Figure 11. The input flow parameters used in the numerical simulation are listed in Table 4.  
Figure 8: Skin friction coefficients for turbulent flat plate: (**a**) Spalart–Allmaras, and (**b**) Shear Stress Transport. Figure 9: Velocity profiles on turbulent flat plate: (**a**) Spalart–Allmaras, and (**b**) Shear Stress Transport. Figure 11: Detail of airfoil mesh. Figure 10: Airfoil boundary conditions [20]. As with the turbulent flat plate case, the freestream values of \(\kappa\) and \(\omega\) for UNS3D-PAR SST were different from the values used for the FUN3D simulation. Figure 12 shows the contour plots of the dimensionless turbulent viscosity \(\mu_{T}/\mu_{\infty}\) at the airfoil's trailing edge predicted using the Spalart-Allmaras model. There is good agreement between the results generated by the UNS3D-PAR and FUN3D codes. The turbulent viscosity predicted by the UNS3D-SEQ code, which used the coarse grid, was similar to the two codes; however, the lack of grid refinement at the trailing edge leads to a somewhat thicker wake. Figures 13 and 14 show the contour plots of the turbulent kinetic energy, \(\kappa\), and the specific dissipation rate, \(\omega\), predicted using the Shear Stress Transport model. The agreement among the results generated by the three codes is good. While there are few differences on the turbulent kinetic energy contour plots, the specific dissipation rates are almost identical. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(\mathbf{p}_{\text{ref}}\) & \(\mathbf{T}_{\text{ref}}\) & \(\mathbf{M}_{\text{inlet}}\) & \(\mathbf{\text{Re}}\) & \(\mathbf{\mu}_{\text{ref}}\) & \(\frac{\hat{\varrho}}{\
u}\) & \(\omega_{\infty}\frac{\
u_{L}}{c_{\infty}^{2}}\) & \(\mathbf{\kappa}_{\infty}\frac{1}{c_{\infty}^{2}}\) \\ \([\text{Pa}]\) & \([\text{K}]\) & [-] & [-] & [Pa s] & [-] & [-] & [-] \\ \hline
62,000 & 300 & 0.088 & 1,200,000 & 1.838 \(\times\) 10\({}^{-5}\) & 3 & 4 \(\times\) 10\({}^{-8}\) & 4 \(\times\) 10\({}^{-10}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Input parameters for Nakayama airfoil. Figure 12: \(\mu_{t}/\mu_{\infty}\) contours on Nakayama’s airfoil predicted with Spalart–Allmaras model: (**a**) UNS3D-SEQ, (**b**) UNS3D-PAR, and (**c**) FUN3D. Figure 13: \(\kappa\) contours on Nakayama’s airfoil predicted with Shear Stress Transport model: (**a**) UNS3D-SEQ, (**b**) UNS3D-PAR, and (**c**) FUN3D. Figure 15 compares the measured velocity profiles against the predicted ones at seven locations in the airfoil's wake. These locations, measured from the airfoil leading edge, are: 1.01, 1.05, 1.20, 1.40, 1.80, 2.19, and 3.0 \(x\)/chord. The numerical predictions were done using the UNS3D-PAR code with the Spalart-Allmaras and the Shear Stress Transport models. The velocity profiles predicted using the Shear Stress Transport model matched the experimental data better than the Spalart-Allmaras model, although the Spalart-Allmaras model predicted the velocity profiles quite well. Figure 16 compares the measured turbulent shear stresses \(-\overline{u^{\prime}v^{\prime}}\) against the predicted ones at the same seven locations in the wake. Both the Spalart-Allmaras and Shear Stress Transport models were used in the UNS3D-PAR code. The two turbulence models predicted the general shape of the shear stresses, although there were some differences between the predicted results and the experimental data. Figure 14: \(\omega\) contours on [PERSON]’s airfoil predicted with Shear Stress Transport model: (**a**) UNS3D-SEQ, (**b**) UNS3D-PAR, and (**c**) FUN3D.