## 4 Results and Discussion
### Unsteady Flow Feature
The PSD shows the value at the position where the TKE is the largest (indicated by the black cross symbol in Figure 17) at each cord length position. The TKE distribution differs at each angle of attack, reflecting the characteristics of the instantaneous flow fields. At \(\alpha=6\), the TKE increases from \(x/c\simeq 0.3\) toward downstream due to the passing of the two-dimensional vortex structures. A turbulent transition occurs at \(0.7\lesssim x/c\lesssim 0.8\), and the region where TKE is particularly high (red region) locally spreads. In the PSD of Figure (a)a, at \(x/c=0.3\), the frequency \(St=6\) and its harmonics are dominant due to the passing of the two-dimensional vortex structures induced by the burst drive of PA. At \(x/c=0.7\), the energy in the high-frequency range begins to increase due to the turbulent transition, and at \(x/c=0.9\), a PSD decay of the \(-5/3\) slope can be confirmed. The position, where the \(-5/3\) slope reveals, is downstream of that in the PA-OFF case (Figure (a)a). These PSD characteristics indicate that transporting the two-dimensional vortex structures suppresses the turbulent transition. At \(\alpha=8\), the TKE increases from \(x/c\simeq 0.2\) by the passing of the two-dimensional vortex structures, similar to that at \(\alpha=6\). At \(x/c\simeq 0.8\), the two-dimensional vortex structures begin to move  
Figure 16: Instantaneous flows with the isosurface of the second invariant of the velocity gradient tensor colored with the chordwise velocity \((u/U_{\infty})\) in the Burst cases. Figure 15: Power spectrum densities (PSDs) of the chordwise velocity fluctuations at the point of the maximum TKE directly above each code length at \(\alpha=6\). away from the airfoil surface; thus, the high TKE region is also seen away from the airfoil surface. The two-dimensional vortex structures are relatively stable compared to those at the other angles of attack, and the turbulent transition is suppressed. Therefore, even at \(x/c=0.9\), the PSD decay of the \(-5/3\) slope is not seen (Figure 18b). At \(\alpha=10\), TKE increases sharply at \(0.1\lesssim x/c\lesssim 0.3\) due to rapid turbulent transition. The PSD also has high energy in the high-frequency range at \(x/c=0.3\), and the slope of the energy decay is close to the \(-5/3\) slope. In Section 4.3, the possibility that the trailing edge separation affects the peak value of \(C_{\text{p}}\) at \(\alpha=8\) in the burst case is discussed. At \(\alpha=8\), as shown in Figures 16b and 18b, the two-dimensional vortex structures induced by PA keep their structure and move away from the airfoil surface, near the trailing edge, while the flows in the burst cases at \(\alpha=6\) and \(8\) occur as turbulent transitions and maintain attached near the trailing edge. Therefore, the promotion of turbulent transition at \(\alpha=8\) could suppress the trailing edge separation, the negative \(C_{\text{p}}\) peak value could become lower, and \(C_{\text{L}}\) could be improved. In the previous study [27], the control effects of \(F^{+}=1\) and \(10\) were discussed by performing LESs at the angle of attack of \(\alpha=6\). In the case of \(F^{+}=1\), a turbulent transition occurred relatively upstream after a large-scale two-dimensional vortex structure shedding, while in the case of \(F^{+}=10\), two-dimensional vortex structures keep their structures flowing up to the trailing edge of the airfoil. This result suggests the possibility of promoting a turbulence transition by using frequencies lower than the burst frequency, which induces stable two-dimensional structures, in a flow with a relatively small pressure gradient at low angles of attack. In other words, for the Bust case at \(\alpha=8\) of the present study, using frequencies lower than \(F^{+}=6\) could promote the turbulent transition and reduce the low-velocity region at the trailing edge. The characteristics of the controlled flow at the pre-stall condition differ depending on the angle of attack and the PA drive condition.