# An Adaptive Multi-Fidelity Surrogate Model for Uncertainty Propagation Analysis
## 5 Test Cases  
To evaluate the proposed method's effectiveness, we analyze the impact of SA turbulence model coefficient uncertainties on flow field predictions. The SA model (widely used in aerospace [33]) assumes fully turbulent flow with the transition term excluded, yielding nine uncertain coefficients: \(c_{b1},\sigma,c_{b2},\kappa,c_{w2},c_{w3},c_{v1},c_{t3},c_{t4}\). These coefficients exhibit epistemic uncertainty, modeled probabilistically via uniform distributions (Table 1) with ranges from the literature [34]. Note that uncertainty parameter characterization (distributions/types) requires consultation with domain experts. The present study primarily focuses on the propagation of uncertainty given a mathematical description of the uncertain parameters, rather than on the precise quantification of uncertainty for the model parameters. The analysis examines two numerical cases: wall friction coefficient distribution in low-speed NACA0012 airfoil flow and wall pressure coefficient distribution in transonic M6 wing flow. Detailed presentations follow for each configuration. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Minimum Value** & **Maximum Value** & **Standard Value** \\ \hline \(c_{b1}\) & 0.12893 & 0.137 & 0.1355 \\ \(\sigma\) & 0.6 & 1.0 & 2/3 \\ \(c_{b2}\) & 0.60983 & 0.6875 & 0.622 \\ \(\kappa\) & 0.38 & 0.42 & 0.41 \\ \(c_{w2}\) & 0.055 & 0.3525 & 0.3 \\ \(c_{w3}\) & 1.75 & 2.5 & 2.0 \\ \(c_{v1}\) & 6.9 & 7.3 & 7.1 \\ \(c_{t3}\) & 1.0 & 2.0 & 1.2 \\ \(c_{t4}\) & 0.3 & 0.7 & 0.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Intervals and standard values of SA model parameters. Low-speed flow around the NACA0012 airfoil  
The first case under investigation is wall friction coefficient distribution in low-speed NACA0012 airfoil flow under the following computational conditions:  
\[M_{\infty}=0.15,\alpha=5.0^{\circ},T_{\infty}=288.15K,\text{Re}_{\infty}=6\times 1 0^{6}\]  
Two distinct grid resolutions (Figure 2) were employed for multi-fidelity sampling: a coarse mesh comprising 3584 cells and a refined mesh of 57,344 elements. The coarse grid consists of 25% of the cell count in each direction relative to the fine grid. Figure 3 displays the wall friction coefficient distribution predicted with standard model parameters. While high- and low-fidelity results show similar trend patterns--particularly the sharp leading-edge variations on both airfoil surfaces followed by stabilization--significant magnitude discrepancies emerge. These abrupt local variations present critical modeling challenges. (2) Transonic flow around the M6 wing  
The transonic flow over the ONERA M6 wing--a standard validation case for compressible flow solvers--was analyzed under the following conditions:  
\[M_{\infty}=0.8395,\alpha=3.06^{\circ},T_{\infty}=255.56K,\text{Re}_{\infty}=1. 172\times 10^{7}\]  
Multi-fidelity simulations employed two computational meshes (illustrated in Figure 4): a coarse grid comprising 990,360 cells (13,638 surface nodes, yielding pressure  
Figure 3: The computed wall friction coefficient distribution for the NACA0012 airfoil using the standard model parameters. Figure 2: Computational grids for the NACA0012 airfoil: (**a**) coarse grid, (**b**) dense grid. coefficient vector in \(\mathbb{R}^{13,638}\)) and a refined grid containing 3,594,863 cells (29,684 surface nodes, corresponding response vector in \(\mathbb{R}^{29,684}\)). At transonic conditions, a characteristic \(\lambda\)-shaped shock structure forms on the wing's upper surface (Figure 5), presenting significant modeling challenges due to its complex pressure gradients. While both grid resolutions produce qualitatively similar pressure distributions, the fine-grid solution demonstrates improved fidelity in capturing the suction peak and shock wave position, as validated against reference experimental measurements in Figure 6.  
Figure 4: Computational grid for the M6 wing: (**a**) coarse grid, (**b**) dense grid. Figure 5: Pressure contours of the M6 wing under standard model parameters using the fine grid.