## 4 Problem Description and Results
### Transitional Flow over a Flat Plate  
The first case was the transitional flow over a flat plate and the deterministic solution of this case was validated using the reference data of [PERSON] and [PERSON] [21]. Figure 1 shows the computational mesh for the flat plate and the boundary conditions of the computational domain over the flat plate. The freestream turbulence intensity and eddy viscosity ratio values were set to \(T_{u}=3.3\%\) and \(\mu_{t}/\mu=10\) ([PERSON] 1996) [33], respectively. A mesh with 43,000 cells was adopted and the mesh was clustered to the wall for \(y^{+}\leq 1\) in Table 4. The corresponding Reynolds number based on the free stream velocity (5.4 m/s) and length of the flat plate (0.3 m) was 1.0 \(\times\) 10\({}^{6}\). Figure 2 shows the comparison of the skin friction \(\left(C_{f}\right)\) profiles predicted by the deterministic solution with the \(\gamma-R_{\theta}\) transition model and experimental data [33; 34]. The skin friction coefficient determined by the present deterministic solution could predict the transition point and developing turbulent flow well. The slight early recovery from laminar to turbulent flow and underestimation of the skin friction right after transition agreed well with previous simulation results [21]. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Coefficient**} & **CD of Lower Boundary** & **CD of Upper Boundary** & **Difference** \\  & **(\(-\)10\%)** & **(10\%)** & **Percent (\%)** \\ \hline C\({}_{a2}\) & 0.00391 & 0.00410 & 4.60 \\ C\({}_{e1}\) & 0.00373 & 0.00421 & 11.4 \\ C\({}_{e2}\) & 0.00397 & 0.00406 & 2.22 \\ C\({}_{a1}\) & 0.00395 & 0.00402 & 1.74 \\ C\({}_{a}\) & 0.00401 & 0.00405 & 0.99 \\ \(\delta_{\gamma}\) & 0.00399 & 0.00401 & 0.49 \\ \(\sigma_{\theta t}\) & 0.00389 & 0.00392 & 0.76 \\ \(C_{\theta t}\) & 0.00402 & 0.00398 & 1.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The sensitivity analysis of each model coefficient. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Coefficient**} & **Original Value** & **Lower Boundary** & **Upper Boundary** \\  & **(\(-\)10\%)** & **(10\%)** \\ \hline C\({}_{a2}\) & 0.06 & 0.054 & 0.066 \\ C\({}_{e1}\) & 1.0 & 0.9 & 1.1 \\ C\({}_{e2}\) & 50.0 & 45.0 & 55.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The uniform distributions of the three model coefficients \(\pm\)10% deviation. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**\(n\) = 3**} & **Monte Carlo** & **Point-Collocation NIPC** & **Spectral Project** \\  & **MC** & **NISP** \\ \hline Order & - & \(2\frac{(n+p)!}{n!p!}\) & \(\left(p+1\right)^{n}\) \\ \hline
2 & & 20 & 27 \\
3 & 500 & 40 & 64 \\
4 & & 70 & 125 \\
5 & & 112 & 216 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The total numbers of selected points for Monte Carlo, NIPC, and NISP for three chosen points (\(n\) is the dimension and \(p\) is the order of stochastic computation). In this study, the drag coefficient was considered as the quantity of interest of UQ. Before discussing the stochastic results of UQ, the predicted drag coefficient obtained by the deterministic solution with the originally fixed model coefficients was 0.00401. Table 5 shows the drag coefficients predicted by the uncertainty quantification technique with the assumptions of uniform distribution of three model coefficients, \(\mathrm{c_{a2}}\), \(\mathrm{c_{e1}}\), and \(\mathrm{c_{e2}}\), and three methods: two methods are non-intrusive polynomial chaos methods (point-collocation NIPC and non-intrusive spectral [projection]) and a Monte Carlo method with 500 simulations. The mean value and standard deviation (STD) of the drag coefficient of all three methods are presented with respect to the polynomial order.