## 4 Problem Description and Results
### Transitional Flow over Aerospatiale-A Airfoil  
In this section, the transitional flow around the Aerospatiale A-airfoil is presented under the conditions of the maximum lift angle of attack (\(\alpha=13.1^{\circ}\)) and \(\mathrm{Re}_{c}=2.07\times 10^{6}\) based on the freestream velocity and chord length as shown in Figure 5. The information of the geometry and the experimental results for the Aerospatiale-A airfoil is available from the Office National d'Etudes et Recherches AeIrospatiales F1 (ONERA F1) wind tunnel ([PERSON] 1997) [32]. The inflow conditions and grid system are summarized in Table 6. The turbulent intensity and eddy viscosity ratio were set to \(T_{u}=0.2\%\) and \(\mu_{t}/\mu=10\)[20], respectively. The length of the upstream part and downstream part were 20c and 25c and the height of the computational domain was 30c. At the upstream and side parts, the velocity inlet boundary condition was given and, at the downstream, the Neumann boundary condition Was set for the outlet. To satisfy \(y^{+}\leq 1\) for correct resolving of the turbulent boundary layer, the distance of the first mesh off the wall was \(10^{-5}\) c and the growth ratio was 1.2. Figure 4: PDF analysis of the drag coefficient: (**a**) point-collocation non-intrusive polynomial chaos; and (**b**) non-intrusive spectral projection. Figure 5: Computational mesh for Aerospatiale A-airfoil. Figure 6 shows the skin friction coefficient distributions over the upper wall of the airfoil predicted by the deterministic solver with the \(\gamma-R_{\theta}\) transition model as well as the comparison with experimental data [32]. The transition point was predicted to be between 10% and 15% of the chord length, which agreed well with other simulation results [35]. The behavior with developing turbulent flow after transition showed consistent results with experimental data. Further, the coefficients of the drag and lift forces were calculated and compared to experimental data, F1 and F2 [32] as show in Table 7. Unfortunately, there were large discrepancies between the experimental data, F1 and F2, especially for the drag coefficient. When the measurement sensitivity of this transitional flow was considered, the present simulation showed reasonable results similar to previous simulation results [35; 36]. It was confirmed that the present simulation adopts adequate methodologies and grid system. The mean value and STD of the drag and lift coefficients of the Aerospatiale A-Airfoil were calculated using the point-collocation NIPC, NISP, and Monte Carlo methods. The number of samples \begin{table}
\begin{tabular}{c c c} \hline \hline
**Inflow Condition** & \(U_{\infty}\) & 32.8 m/s \\ \(T_{u}\) & 0.2\% \\ \(\mu_{t}/\mu\) & 10.0 \\ \hline \multirow{3}{*}{**Grid System**} & \(\alpha\) & 13.1\({}^{\circ}\) \\ \cline{2-3}  & (Lx,Ly) & (45c,30c) \\ \cline{1-1}  & (Nx,Ny) & (350,120) \\ \cline{1-1}  & \(\text{Re}_{c}\) & \(2.07\times 10^{6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Information about the inflow conditions and grid system for flow over Aerospatiale A-airfoil. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Model** & \(\mathbf{C_{L}}\) & \(\mathbf{C_{D}}\) \\ \hline \(\gamma-R_{\theta}\) & 1.390 & 0.0206 \\ F1 exp & 1.562 & 0.0208 \\ F2 exp & 1.515 & 0.0308 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Lift and drag at \(\alpha=13.1^{\circ}\)  
Figure 6: Skin friction coefficient distribution for the original formulation over Aerospatiale A-airfoil. of Monte Carlo was 500, which was the same as in the previous case. Table 8 shows the drag coefficient with respect to the polynomial order. When the polynomial order is 5, the mean drag coefficient was predicted to be 0.020569 by PC-NIPC and 0.020581 by NISP. When these were compared with the deterministic solution of the original model coefficients, it was found that the drag coefficient was not as biased as in the flat plate case.