## 4 Results and Discussion
### Unsteady Flow Feature  
In this section, we discuss the relationship between the instantaneous flow field for each case and the plots up to Section 4.3. Figure 13 shows the instantaneous flow fields of the PA-OFF and Continuous cases at \(\alpha=6\). The isosurface is the second invariant of the velocity gradient tensor (\(Q=6250\)), colored by the chordwise velocity (\(u/U_{\infty}\)). Figures 14 and 15 show the turbulent-kinetic-energy (TKE) distributions and the power spectral densities (PSDs) of the chordwise velocity fluctuations of the PA-OFF and Continuous cases, respectively. The PSD shows the value at the position where the TKE is the largest (indicated by the black cross symbol in Figure 14) at each cord length position. The gray dashed line indicates the slope of [PERSON]'s \(-5/3\) power law. As shown in Figure 13, the instantaneous flow fields of the PA-OFF and Continuous cases are similar. In  
Figure 12: Displacement thickness (\(\delta^{*}\)) on the airfoil’s upper surface. Figure 15, the spectrum shows a \(-5/3\) power slope at \(x/c=0.5\) and downstream there in both PA-OFF and Continuous cases, indicating the flow transitions to turbulent flow. Although, at first glance, the instantaneous fields of the PA-OFF and Continuous cases are very similar, due to the momentum addition by PA, the laminar flow separation in the Continuous case is delayed more than that in the PA-OFF case, as shown in Figure 10, and the separated shear layer of the Continuous case is closer to the airfoil surface. As a result, as shown in Figure 12a, the displacement thickness of the Continuous case is thinner than that of PA-OFF in most regions. The position of the turbulent transition in the Continuous case is delayed, and a region where TKE is high is seen downstream compared to that in the PA-OFF case. Although only the case at \(\alpha=6\) was discussed here, the flow characteristics of the PA-OFF and Continuous cases are the same for other angles of attack. Figure 16 shows the instantaneous flow fields of the burst case at each angle of attack. The isosurface is the second invariant of the velocity gradient tensor (\(Q=6250\)), colored by the chordwise velocity (\(u/U_{\infty}\)). The flow fields in the Burst cases differ from those in the PA-OFF and Continuous cases, where the large separated region is formed under the separated shear layer. At \(\alpha=6\) and \(8\) in Figure 16, two-dimensional vortex structures are induced by the burst drive of PA near the leading edge and move downstream. These two-dimensional vortex structures maintain their spanwise shape up to near the trailing edge. Such stable two-dimensional vortex structures are not observed in the controlled flows at post-stall angles. These two-dimensional vortex structures merge several times in the process of moving. Where large two-dimensional vortex structures merge, local variations of \(C_{\text{f}}\) and \(C_{\text{p}}\) are seen. Specifically, in the region of \(0.4\lesssim x/c\lesssim 0.5\), a small plateau region is seen in Figure 9a,b, and a local decrease in \(C_{\text{f}}\) is seen in Figure 10a,b. Figure 14: Turbulent kinetic energy (TKE) distributions at \(\alpha=6\). Figure 13: Instantaneous flows with the isosurface of the second invariant of the velocity gradient tensor colored with the chordwise velocity (\(u/U_{\infty}\)) at \(\alpha=6\). The difference between the flow fields at \(\alpha=6\) and \(\alpha=8\) is remarkable near the trailing edge. As shown in Figure (b)b, the flow at \(\alpha=8\) separates near the trailing edge while maintaining the two-dimensional vortex structures. The flow at \(\alpha=10\) is different from those at \(\alpha=6\) and \(8\), the two-dimensional vortex structures induced by PA rapidly collapse into the two-dimensional vortex structures (Figure (c)c). This flow resembles the controlled flows investigated in the previous study [8] at the post-stall angles. Figures 17 and 18 show the TKE distributions and the PSDs of the chordwise velocity fluctuations for the Burst case at each angle of attack.