## 4 Problem Description and Results
### Comparison of the Two Methods  
Figure 11 describes the distribution of the PDF of point-collocation NIPC and NISP for the fifth-order polynomial and Monte Carlo method with 500 sampling points. In the case of flat plate, the peak values of the drag coefficients, 0.0041, agreed well in the three methods, NISP, PC-NIPC and Monte Carlo. This peak was located at a higher value than that (0.00401) of the deterministic solution. In the drag coefficients of Aerospatiale A-Airfoil simulation, there were two peaks in the PDF and the left peak was lower than the right one in every method. The predicted peak values in NISP and PC-NIPC methods were approximately 0.0206 with 0.03% error with that of Monte Carlo method. The pattern of the distribution of the lift coefficient in A-Airfoil simulation was similar to that of the drag coefficient in the flat plate simulation. The peak value of NISP showed smaller difference with 0.46% than that of PC-NIPC (0.86%) when compared with the Monte Carlo result. Even though the number of samplings (500) in the present work was not sufficient to get the converged solution of MC, the overall pattern distribution and the number of the peak were consistent between MC and two adopted methods. When the number of sample points for the UQ technique was considered, even though point-collocation NIPC required fewer samples than NISP (Table 3), similar accuracy and  
Figure 10: PDF analysis of the lift coefficient (CL): (**a**) point-collocation non-intrusive polynomial; and (**b**) non-intrusive spectral projection. Figure 9: The mean and standard deviations of the lift coefficient of the three methods for Orders 2–5: (**a**) mean value; and (**b**) standard deviation. PDF as the mean and STD of the quantities of interest could be obtained. In particular, the number of random variables increased and the point-collocation NIPC was useful from a computational cost and time point of view. Sensitivity analysis of three model coefficients, \(\text{c}_{\text{a2}}\), \(\text{c}_{\text{e1}}\), and \(\text{c}_{\text{e2}}\), was carried out using the Sobol index of the two methods (Tables 10 and 11). The Sobol indices provided quantitative information regarding how much effect the corresponding random parameters had on quantities of interest. In the point-collocation NIPC and NISP methods, the most effective model coefficient was calculated as \(C_{e1}\), whose Sobol index was generally over 90% in all outputs of both the flat plate and A-airfoil cases. The model coefficient \(C_{e1}\) was included in the source term of the transport equation of the intermittency and was related to the \(F_{lgorth}\) and \(F_{onset}\) terms. The other two variables, \(C_{e2}\) and \(C_{a2}\), showed similar effects on the results of less than 5% of Sobol indices. This means that more careful calibration of the model coefficient \(C_{e1}\) was required compared to the other coefficients for better prediction of transitional flow. \begin{table}
\begin{tabular}{c c c c} \hline
**Sobol Index** & \(\mathbf{C_{a2}}\) & \(\mathbf{C_{e1}}\) & \(\mathbf{C_{e2}}\) \\ \hline \(\text{C}_{\text{D}}\) Flat plate & 0.01934 & 0.983 & 0.01955 \\ \(\text{C}_{\text{D}}\) A-airfoil & 0.32256 & 0.8963 & 0.15859 \\ \(\text{C}_{\text{L}}\) A-airfoil & 0.05347 & 0.9793 & 0.03839 \\ \hline \end{tabular}
\end{table} 
Table 10: Sobol index of three model coefficients for point-collocation NIPC. Figure 11: The distributions of the PDF at the fifth order of NIPC, NISP, and Monte Carlo (CL): (**a**) CD of flat plate; and (**b**) CD of A-airfoil, and (**c**) CL of A-airfoil. Figure 12 displays the distribution of the skin friction coefficient from point-collocation NIPC at the fifth order. It was confirmed that the change of the model coefficients had a significant effect on the variation of the skin friction coefficient at the location of the flat plate. There was a small STD before the transition point (i.e., laminar flow region) and then a large STD after the transition (i.e., developing turbulent flow region).