## 4 Problem Description and Results
### Transitional Flow over Aerospatiale-A Airfoil
The percent errors of the mean and STD values were 0.029% and 0.09% for point-collocation NIPC and 0.03% and 0.97% for NISP when the two methods with fifth-order were compared to the Monte Carlo method. When the convergence of the drag coefficient with respect to the polynomial order was checked, the overall pattern was similar to the previous case for the drag coefficient of the flat plate flow. The mean value of the drag coefficient already converged within 0.5% (\(\Delta\)C\({}_{\text{D}}\)\(\sim\) 0.0001) at the second order, whereas the STD converged after the second polynomial order as shown in Figure 7. As the polynomial order increased, the mean and STD approached the values obtained by the Monte Carlo method (long dashed line). The distributions of the probability density functions of the drag coefficients are plotted with variation of the order in Figure 8. In this case, the distributions of the PDF revealed different patterns depending the adopted method and the order of the polynomial. In the case of point-collocation NIPC, the distribution of the case with second order was very different form other order cases and had similar patterns starting from the third order. NISP had a similar trend as point-collocation and the results were similar starting from the fourth-order case. Interestingly, there were two peaks in the distribution of the PDF in high-order cases for the two methods. The transitional flow simulation of Aerospatiale A-Airfoil was more sensitive to used grid, convergence criteria and the adopted scheme than the flat plate simulation. This means that the quantities of interest, such as drag and lift coefficients, were sensitive to the computational conditions (grid, convergence criteria and so on) and sampling \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Order** & **Mean\_NIPC** & **STD\_NIPC** & **Mean\_NISP** & **STD\_NISP** & **Mean\_MC** & **STD\_MC** \\ \hline
2 & 0.020561 & 0.0001252 & 0.020598 & 0.0000895 & \\
3 & 0.020570 & 0.000111 & 0.020508 & 0.0001090 & \\
4 & 0.020551 & 0.000113 & 0.020587 & 0.0001060 & 0.02057489 & 0.0001119 \\
5 & 0.020569 & 0.000112 & 0.020581 & 0.0001130 & \\ \hline \hline \end{tabular}
\end{table} 
Table 8: The mean and standard deviation of the drag coefficient (C\({}_{\text{D}}\)) values for A-airfoil. Figure 7: The mean and standard deviations of the drag coefficient of the three methods for Orders 2–5: (**a**) mean value; and (**b**) standard deviation. points that were selected at the given order of polynomial. We presume that the sampling points were insufficient to resolve the exact probabilistic distribution in the lower Orders 2 and 3. Table 9 shows the mean and STD of the lift coefficient obtained by the three methods. The mean values of all methods were predicted to be approximately 1.399, whereas the deterministic solution was 1.390. Similar to the convergence of the drag coefficient, the mean value of the lift coefficient converged early from the second order but the STD showed a large difference when the order was 2. At the fifth order, there was a 0.85% difference in point-collocation NIPC and 0.46% in NISP as shown in Figure 9. Even though the convergence rate of point-collocation NIPC was faster than that of NISP, the accuracy at the highest order (fifth order) was higher in NISP than point-collocation NIPC. The difference between predicted and the deterministic solution value resulted because the distribution of PDF was biased toward the positive difference value, as confirmed in Figure 10.  
In the lift coefficient, similar patterns of the PDF were obtained depending on the method and the order. Even though the peak of PDF was located at a lower value, the flat distributions of the lift coefficients ranged to the value of 1.43 or 1.44 in every case. This broad distribution towards a higher lift coefficient made the predicted mean lift coefficient move to a higher value than the deterministic solution.