## 3 Results and Analysis
### Time-Dependent Results  
The variation in the lift coefficient over time is depicted in Figure 9. Due to the formation of shock wave self-excited oscillations, the repetitive motion of the shock wave on the upper surface caused periodic fluctuation in the lift coefficient over time. To investigate the amplitude and patterns of the shock wave motion on the upper surface, eight representative flow field instances within one cycle were analyzed based on the fluctuation pattern of the lift coefficient. The specific positions at different moments in time are provided in Table 2. Due to the local fluctuations in the lift coefficient, it was challenging to precisely select a starting point for the cyclic motion of the shock wave, where the shock wave was positioned either at the limitation of the upstream or downstream. Therefore, an instantaneous cycle was defined from the recorded initial moment when the shock wave was at a certain downstream position until the moment when the shock wave returned to the same position. The time duration of this cycle is denoted as T.  
Figure 8: Root-mean-square distribution of pressure fluctuations on the upper surface of the airfoil; [PERSON] 2014 [11], [PERSON] 2012 [12]. Figure 9: Temporal evolution of the lift coefficient. Density gradient contours are presented in Figure 10 at selected instants in time. At the initial moment A, the shock wave was positioned at an approximately 30% chordwise location. At moment B, the shock wave moved downstream to around a 40% chordwise position, reaching the furthest downstream position at moment C, at an approximately 50% chordwise location. At moment D, there was no significant difference in the shock wave position compared to moment C, but the tilt angle of the shock wave increased, and small-scale high-frequency fluctuations in the lift coefficient began to appear. By moment E, the shock wave has started moving upstream and reaches x/c = 0.37. At moment F, the shock wave was positioned at x/c = 0.30, until moment G when the shock wave reached the furthest upstream position (x/c = 0.23). Subsequently, the shock wave began to move downstream again, reaching the initial position at moment H, marking the completion of one cycle. Figure 11 displays the iso-surfaces of the Q criterion in one cycle. The shock wave induced separation in the downstream boundary layer, and the shear layer exhibited strong two-dimensional characteristics. As it progressed downstream, the shear layer gradually breaks into small-scale three-dimensional turbulence structures, and some large-scale vortex shedding can be observed in the downstream wake region. Comparing the instantaneous flow at different moments revealed that the separation region induced by the shock wave undergoes periodic variations as the shock wave moves. During the downstream movement of the shock wave (from A to C), the separation region behind the shock wave gradually thinned, and the size decreased, resulting in an increase in lift. Conversely, during the upward movement of the shock wave (from E to G), the separated boundary layer thickened, and the separation region gradually enlarged, leading to a decrease in the lift coefficient. It is noteworthy that the lift coefficient did not reach its maximum peak when the shock wave reached the furthest downstream position. Combining Figures 10 and 11, it could be observed that the vortex structures of different scales at the trailing edge generated strong fluctuations. The distribution of the average friction coefficient on the wing surface at different instants is illustrated in Figure 12. The surface friction coefficient can reflect the flow separation status after the shock wave. During the shock wave downward motion, as the shock wave traveled from the upstream (t/T = 0) to the most downstream position (t/T = 40%), the separation region on the upper wing surface gradually approached the shock wave location until the entire surface was separated.