## 5 Conclusions  
In this study, we compared point-collocation NIPC and NISP methods for the \(\gamma-\mathrm{Re}_{\theta}\) transitional model [21]. Three model coefficients, \(\mathrm{Ca}_{2}\), \(\mathrm{Ce}_{1}\), and \(\mathrm{Ce}_{2}\), were considered as multiple random inputs with the assumption of a uniform distribution with \(\pm 10\%\) deviation. The quantities of interest were the drag and lift coefficients in transitional flow around a flat plate and Aerospatiale A-Airfoil.  
The point-collocation NIPC had the advantage of having fewer samples for uncertainty quantification analysis, even though the oversampling ratio was set to 2, followed by the NISP method. It was confirmed that the point-collocation NIPC is useful when multiple random inputs are considered. Convergence with respect to the polynomial order was faster in point-collocation NIPC than NISP. In addition, the mean value of the output was obtained with a lower order than STD. This means that a higher polynomial order was necessary to obtain a converged STD. At the highest polynomial order of 5 evaluated in this study, NISP demonstrated slightly more accurate results when compared to the Monte Carlo simulation. However, the accuracy of NISP depended on whether it was an even or odd polynomial order. Through analysis of the Sobol index, the most effective model coefficient of the \(\gamma-Re_{\theta}\) transitional model was \(Ce_{1}\), which is related to \(F_{length}\) and \(F_{onset}\) in the intermittency equation, by approximately over 90%.  
In summary, when multiple random inputs were considered, point-collocation NIPC could provide reasonable stochastic values such as the mean and STD with fewer samples than NISP. If only one or two variables were regarded as random inputs, NISP seemed to be better at a high order.  
Figure 12: Variation of the skin friction coefficient of the upper surface of the A-airfoil for the point-collocation NIPC method.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Sobol Index** & \(C_{a2}\) & \(C_{e1}\) & \(C_{e2}\) \\ \hline \(\mathrm{C}_{\mathrm{D}}\) Flat plate & 0.0034 & 0.9945 & 0.0022 \\ \(\mathrm{C}_{\mathrm{D}}\) A-airfoil & 0.0287 & 0.9554 & 0.0281 \\ \(\mathrm{C}_{\mathrm{L}}\) A-airfoil & 0.00024 & 0.9999 & 0.000234 \\ \hline \end{tabular}
\end{table}
Table 11: Sobol index of three model coefficients for NISP.  
**Author Contributions:** Conceptualization, T.H.N. and K.C.; Methodology, T.H.N.; Software, T.H.N.; Validation, T.H.N. and K.C.; Formal Analysis, T.H.N.; Investigation, K.C.; Resources, K.C.; Data Curation, T.H.N.; Writing-Original Draft Preparation, T.H.N.; Writing-Review & Editing, K.C.; Visualization, T.H.N. and K.C.; Supervision, K.C.; Project Administration, K.C.; Funding Acquisition, K.C.  
**Funding:** This research was funded by the National Research Foundation of Korea Government (NRF-2016R1D1A1B03934121).  
**Conflicts of Interest:** The authors declare no conflict of interest.  
**List of Symbols**  
c chord length (m)  
C\({}_{\rm f}\) skin friction coefficient  
M Mach number  
Re Reynolds number  
U\({}_{\infty}\) velocity freestream  
\(\alpha\) angle-of-attack (deg)  
T\({}_{\rm u}\) turbulence intensity (T\({}_{\rm u}=\sqrt{\frac{2\pi G}{\rm m}}\))  
k turbulence kinetic energy (m\({}^{2}/s^{2}\))  
\(\mu\) dynamic molecular viscosity (kg.m\({}^{-1}\).s\({}^{-1}\))  
\(\mu_{t}\) dynamic eddy viscosity (kg.m\({}^{-1}\).s\({}^{-1}\))  
\(\
u=\mu/\rho\) kinematic molecular viscosity (m\({}^{2}\)/s)  
\(\rho\) density (kg/m\({}^{3}\))  
\(\varepsilon\) turbulence kinetic energy dissipation rate (m\({}^{2}/s^{3}\))  
\(y^{+}=yu_{t}/\
u\) turbulent inner layer non-dimensional coordinate  
**Abbreviations**  
CFD Computational fluid dynamics  
NIPC Point-Collocation Non-Intrusive Polynomial  
NISP Non-intrusive Spectral Projection  
MC Monte Carlo  
PC Polynomial chaos  
PDF Probability density function  
UQ Uncertainty quantification