## 3 Results and Analysis
### Time-Dependent Results
During the shock wave upward motion, the upper wing surface was in a mostly separated state from the shock wave location to the trailing edge until the end of the upward motion. Figure 13 depicts the temporal and spatial variations in flow separation and reattachment on the upper surface of the airfoil based on the distribution of the friction coefficients. The vertical axis represents the progression of time, which is dimensionless with respect to the transonic shock buffet period T, corresponding to the global time shown in Figure 9. The horizontal axis represents the streamwise position on the upper surface of the airfoil (x/c = 0 for the leading edge, 1 for the trailing edge). The white region indicates the attached flow, while the dark region represents the separated flow. The figure clearly reflects the changing positions of the shock wave and flow separation regions over time. At the initial instant (t = 0), the shock wave was in the downward phase. Local separation appeared in the trailing-edge region, while the flow in other areas remained fully attached. As the shock wave further moved downstream, separation bubbles occurred simultaneously at the base of the shock wave and in the trailing-edge region. When reaching the most downstream position, these two separation regions converged, leading to complete separation on the upper surface after the shock wave. During the upward motion of the shock wave, the region behind the wave remained fully separated until the shock wave approached \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Transient Location** & **A** & **B** & **C** & **D** & **E** & **F** & **G** & **H** \\ \hline _t/T=_ & 0 & 11\% & 35\% & 49\% & 60\% & 73\% & 90\% & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Transient position selection. the most upstream position. At this point, the separation region divided into two parts again. As the shock wave moved downstream once more, these two regions gradually shrank, then expanded again, repeating the cycle. Although the results presented here represent only one cycle of flow separation, the flow separation patterns in other cycles were qualitatively consistent, with potential quantitative differences. Figure 10: Instantaneous density gradient contours in one cycle. Figure 11: Instantaneous turbulent structures in one cycle (**A**–**H**). The variation trend of the shock wave movement speed during the transonic shock buffet process can be observed from Figure 13. In the first 40% of the time during the shock wave ascent or descent, the trajectory slope of the shock wave position showed a relatively large change with time, indicating a fast movement speed of the shock wave. The displacement during this period accounted for approximately 60% of the entire range of shock wave motion. During the subsequent 40%, the trajectory slope of the shock wave position over time decreased, indicating a slower movement speed. When approaching the most upstream or downstream position, the speed reached its minimum. Furthermore, within one cycle, the time taken for shock wave ascent and descent in this study was in a 1:1 ratio. In contrast, the DDES predicted that the time for shock wave ascent occupied approximately one-third of the single cycle, which was related to the shorter transonic shock buffet period predicted with DDES compared to the experimental period. Figure 12: Surface friction coefficient distribution. Figure 13: The evolution of flow separation on the upper surface of the airfoil. Figure 14 illustrates the instantaneous wave field of the transonic shock buffet at different moments. During the downstream movement of the shock wave, the instability at the trailing edge is relatively weak. At this point, the pressure wave generated by the trailing edge is not enough to push the shock wave upstream. The shock wave-induced separation also caused the instability of the boundary layer to propagate downstream with the movement of the shock wave.