## 4 Alleviation of Buffet Load by Tangential Slot Blowing
### Periodic Blowing  
The effect of periodic tangential blowing was also investigated by transient numerical simulations. Given that the total pressure variation does not obviously influence the control effect of steady blowing cases, the variation ratio of total pressure was set to be 0.1, and the periodic total pressure is depicted as  
\[P_{tot\_periodic}=(1+\frac{\gamma-1}{2}M_{\infty}^{2})^{\frac{\gamma}{\gamma-1 }}+0.1\times\sin{(2\pi\times f\times t)}\times(1+\frac{\gamma-1}{2}M_{\infty }^{2})^{\frac{\gamma}{\gamma-1}} \tag{4}\]  
By referring to the PSD results above, the frequency of periodic blowing was set to be 90 Hz, and \(\gamma\) was 1.4 for calorically perfect gas. From the above context, it is known that the most effective effect was achieved when the tangential blowing slot was placed at the location denoted as \"41% x/c\". Thus, the periodic blowing slot was first placed here. The unsteady lift coefficient results under periodic blowing conditions at different incidence angles were compared with those of steady blowing cases, as shown in Figure 18. By examining the results, it is found that the unsteady lift coefficients exhibited an oscillating frequency nearly the same as that of the clean airfoil at incidence angles 5.0\({}^{\circ}\), 5.5\({}^{\circ}\), and 6.0\({}^{\circ}\). However, the mean lift coefficient values under periodic blowing conditions were a little lower compared to their corresponding steady blowing cases, as presented in Table 2. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **5.0\({}^{\circ}\)** & **5.5\({}^{\circ}\)** & **6.0\({}^{\circ}\)** \\ \hline Steady blowing & 0.73 & 0.75 & 0.71 \\ Periodic blowing & 0.71 & 0.70 & 0.65 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mean lift coefficients with and without blowing. Figure 17: Q-criterion results corresponding to maximum/median/minimum lifts with and without tangential blowing. (**a**) Maximum lift without blowing; (**b**) median lift without blowing; (**c**) minimum lift without blowing; (**d**) maximum lift with blowing; (**e**) median lift with blowing; and (**f**) minimum lift with blowing. To investigate the frequency characteristics of unsteady lift coefficients, the PSD results of different periodic blowing cases were plotted, as shown in Figure 19. It can be seen from these figures that, for the \"41% x/c\" case, the peak PSD value of the lift coefficient with periodic tangential blowing was about 5 times larger than that with steady tangential blowing. Moreover, with the increase of incidence angle, frequencies corresponding to the peak PSD values also increased. Compared to the PSD results for steady blowing cases plotted in Figure 14, there was no obvious difference in frequency characteristics. Under all three blowing locations, periodic tangential slot blowing gave higher peak PSD values compared to steady tangential slot blowing at those two blowing locations. Surface pressure coefficient distributions are plotted in Figures 20-22. Within each figure, surface pressure distribution lines corresponding to maximum, median, and minimum lifts in one oscillating period are plotted together. From these figures, it is observed that these oscillation amplitudes were smaller compared to those without tangential blowing (plotted in Figure 7), while compared to the results of steady tangential blowing cases (plotted in Figure 15), the shock oscillation amplitudes were a little larger, while the surface pressure variation downstream of the shockwave was evidently smaller. Figure 19: PSDs of unsteady lift coefficient histories under different blowing cases: (**a**) 41% x/c, (**b**) 46% x/c, and (**c**) 51% x/c. Figure 18: Unsteady lift coefficient histories under the “41% x/c” case at: (**a**) 5.0\({}^{\circ}\), (**b**) 5.5\({}^{\circ}\), and (**c**) 6.0\({}^{\circ}\). Q-criterion results corresponding to minimum lift, median lift, and maximum lift, are plotted in Figures 23-25, respectively. It can be seen from these figures that the blowing slots were all downstream of the shock foot.