# Several Cases for the Validation of Turbulence Models Implementation
## 4 Results
### Turbulent Flat Plate  
The first case investigated was the turbulent flow over a flat plate. The turbulent flat plate case allowed one to validate the turbulence model against Coles' Law of the Wake [21]. The velocity profile and distance from the wall were nondimensionalized to highlight the different regions of the turbulent boundary layer such as the log-region sublayer. Both the sequential and parallel versions of UNS3D were compared against the results from NASA's FUN3D and CFL3D [20]. These results include dimensionless turbulent viscosity \(\mu_{T}/\mu_{\infty}\), \(\kappa\), and \(\omega\) contours. The computational domain and boundary conditions are illustrated in Figure 1. Two levels of mesh refinement were used for this case: where the parallel code used the finest grid and the sequential code used the third finest grid [20]. Figure 2 shows the coarse mesh. Table 3 specifies the freestream boundary conditions for the various turbulence models and codes. The freestream turbulence boundary conditions used in NASA's FUN3D SST and CFL3D SST were above the recommended range [4], which is  
\[\frac{U_{\infty}}{L}<\omega_{\infty}<10\frac{U_{\infty}}{L} 10^{-5}\frac{U_{\infty}^{2}}{Re_{L}}<\kappa_{\infty}<0.1\frac{U_{ \infty}^{2}}{Re_{L}}.\]  
The values used by NASA FUN3D correspond to  
\[\omega_{\infty}=125\frac{U_{\infty}}{L} \kappa_{\infty}=1.125\frac{U_{\infty}^{2}}{Re_{L}}.\]  
Therefore, FUN3D used turbulent freestream conditions that were approximately two orders of magnitude higher than the recommended range. UNS3D-PAR SST used values of \(\kappa\) and \(\omega\) that were in the recommended range:  
\[\omega_{\infty}=5\frac{U_{\infty}}{L} \kappa_{\infty}=0.05\frac{U_{\infty}^{2}}{Re_{L}}.\]  
We also generated UNS3D-PAR SST results using the turbulence boundary conditions used by the NASA codes. In this case, UNS3D-PAR SST produced \(\kappa\) and \(\omega\) contours that were approximately 10 percent smaller in magnitude than the NASA results. The results included in this paper for UNS3D-PAR SST used the turbulent boundary conditions in the recommended range, as stated in Table 3. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Code** & **Mesh** & \(\frac{\theta_{\infty}}{\
u_{L}}\) & \(\omega_{\infty}\frac{U_{\infty}}{\epsilon_{\infty}^{2}}\) & \(\kappa_{\infty}\frac{1}{\epsilon_{\infty}^{2}}\) \\ \hline UNS3D-SEQ SA & \(2\times 137\times 97\) & 3.0 & - & - \\ UNS3D-PAR SA & \(2\times 545\times 385\) & 3.0 & - & - \\ FUN3D SA & \(2\times 545\times 385\) & 3.0 & - & - \\ UNS3D-SEQ SST & \(2\times 137\times 97\) & - & \(1\times 10^{-6}\) & \(9\times 10^{-9}\) \\ UNS3D-PAR SST & \(2\times 545\times 385\) & - & \(4\times 10^{-8}\) & \(4\times 10^{-10}\) \\ FUN3D SST & \(2\times 545\times 385\) & - & \(1\times 10^{-6}\) & \(9\times 10^{-9}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Turbulent Boundary Conditions for Flat Plate. Figure 3 shows a comparison of the dimensionless turbulent viscosity predicted by the UNS3D-SEQ, UNS3D-PAR, and FUN3D codes using the Spalart-Allmaras turbulence model. A good agreement was obtained among the solutions of the three codes. A similar conclusion emerged by comparing the dimensionless turbulent viscosity predicted using the Shear Stress Transport turbulence model, shown in Figure 4. The large values of the turbulent viscosity extended farther away from the flat plate in the Spalart-Allmaras model compared to the Shear Stress Transport model, as observed by contrasting Figures 3 and 4. Figure 1: Flat plate boundary conditions [20]. Figure 2: Flat plate coarse mesh. Figure 5 shows the contours of the kinetic energy of the turbulent fluctuations, \(\kappa\), produced by the Shear Stress Transport turbulence model. Similarly to the turbulent viscosity results, a good agreement is obtained among the solutions of the three codes. Figure 6 shows that the main differences between the \(\omega\) contours generated by UNS3D-PAR and FUN3D using the Shear Stress Transport model were located in the region near the inlet of the domain.