## 2 Numerical Methodology
### Validation of the Numerical Approach  
To better simulate the motion of shock waves over the airfoil and the influence of the trailing-edge wake on the shock waves, the grid for the shock wave motion region on the upper surface and the downstream wake region behind the trailing edge was moderately refined (greater than or equal to c). Before the computations commenced, three sets of grids with different densities were established, and specific parameters can be found in Table 1. The grid cross-section is illustrated in Figure 1.  
The three sets of grids were utilized to simulate the flow field at a freestream Mach number of 0.73 and an AOA of 2.5 degrees. According to the experimental results, no transonic shock buffet was observed in the flow field in this condition, and the shock wave position remained fixed. Steady-state computations were performed for the three grid sets by the SST model. The surface pressure distributions obtained from different grids are compared in Figure 2. It is evident that the pressure distributions from the three sets of grids are quite similar, showing a good agreement with experimental data. The calculated shock wave positions also align with the experimental observations. The streamline patterns shown in Figure 3 indicate that the shock wave induces a small localized separation region within the boundary layer, which is consistent with experimental observations. For subsequent calculations in this section, the second grid (Grid-B) was consistently employed.  
The surface pressure distributions obtained from different turbulence models are compared in Figure 4, including the S-A model, the SST model, and the k-\(\omega\) model. The shock wave occurred at x/c \(\approx\) 0.50 in the experiment. Comparing the performance of the three models, the SST model predicted the shock wave position being in good agreement with the experiment; The S-A model was slightly downstream, while the Wilcox k-\(\omega\) model deviated significantly from the experiment, placing the shock wave at x/c \(\approx\) 0.57, noticeably further downstream. Since transition strips were applied at 7% chordwise positions on both the upper and lower surfaces of the airfoil in the experiment, we further compared the impact of adding a fixed transition (FT) into the computational results. Compared to full-turbulence simulation, the shock wave positions predicted with the S-A model and  
\begin{table}
\begin{tabular}{c c c c c} \hline  & **Streamwise** & **Chordwise** & **Spanwise** & **Total Nodes** \\  & **Nodes** & **Nodes** & **Nodes** & **Total Nodes** \\ \hline
**Grid-A (Coarse)** & 172 & 89 & 32 & \(5.05\times 10^{5}\) \\
**Grid-B (Medium)** & 220 & 137 & 40 & \(1.20\times 10^{6}\) \\
**Grid-C (fine)** & 265 & 192 & 40 & \(2.04\times 10^{6}\) \\ \hline \end{tabular}
\end{table}
Table 1: The distribution of grid points.  
Figure 1: Topology and grid.  
SST model are slightly advanced when a fixed transition is added. To better simulate the experimental conditions in unsteady simulation, a fixed transition was added at the corresponding positions in subsequent calculations in this section.  
The computational domain for the ZDES of OAT15A is shown in Figure 5, where DES was applied to the upper surface and wake region prone to trans  
Figure 4: Comparison of pressure distributions with different turbulence models (\(\alpha=2.5^{\circ}\)).  
Figure 3: Streamline plots (Grid-B, \(\alpha=2.5^{\circ}\)).  
Figure 2: Surface pressure distributions obtained from grids with different densities.  
the rest of the domain utilized RANS. The time step for the unsteady simulation was set to \(1\times 10^{-6}\) s, with 15 sub-iterations performed within each time step, which depends on a Courant-Friedrichs-Lewy number based on acoustic velocity and spanwise grid spacing of less than one.