## 5 Results
### Functions of Plasma Actuator at Cruise Condition
Figure 13: Aerodynamic Coefficients of NACA0015 with and without DBD plasma actuator compared with Ishii airfoil. NACA0015 (\(\alpha=6^{\circ}\)), Ishii airfoil (\(\alpha=4^{\circ}\)), \(Re=6.3\times 10^{4}\), \(F^{+}=6.0\), \(Dc=0.05\), \(BR=10\%\). The pressure distributions for an Ishii airfoil plotted in Figure 15 shows a pressure plateau region in the forward part of the airfoil, which corresponds to a laminar separation bubble. Typically, reattachment occurs via a laminar-to-turbulent transition. However, the flow stays laminar even after the reattachment in this case. Figure 16 shows the power spectrum density of the chordwise velocity fluctuations. The power spectra are measured at several points in the chordwise direction along the line of the maximum turbulent-kinetic-energy (TKE) in the chordwise direction. The horizontal axis represents the Strouhal number (\(St=fc/U_{\infty}\)) and the vertical axis represents the power spectrum density. In this analysis, data are ensemble and spanwise averaged, and the power spectra are non-dimensionalized as the power spectrum density, \(PSD^{*}=PSD/u_{\infty}^{2}(u_{\infty}/c)\). The spectrum for the DBD plasma actuator off case has a peak around \(St=10\) at \(x/c=0.4\), which may correspond to the most unstable mode of the linear stability. From \(x/c=0.4\) to \(1.0\), a \(-5/3\) power decay can be clearly observed, and the flow is considered to be turbulent. This result is consistent with the pressure distributions in Figures 14 and 15. On the other hand, \(-5/3\) power decay is not observed for the Ishii airfoil almost up to the trailing edge. It seems that flow stays laminar even after the reattachment. The NACA0015 airfoil, with the DBD plasma actuator on, the location of the power decay moves toward the trailing edge and occurs somewhere between \(x/c=0.8\) and \(x/c=1.0\) (Figure 16b). The DBD plasma actuator seems to establish similar flow structures as that for the Ishii airfoil in Figure 16c. It is interesting to note that the DBD plasma actuator in this case delays the transition. At high angles of attack, the DBD plasma actuator promotes a laminar to turbulent transition to avoid flow separations. Even though the parameter settings are very similar, the DBD plasma actuator works very differently than it does at high angles of attack. Figure 17a-c shows instantaneous snap shots of the flow field. As explained for Figure 16, a two-dimensional vortex structure remains up to the rear part of the airfoil surface and the transition seems to be delayed when the DBD plasma actuator is turned on (Figure 17b). This flow structure is very similar to that for the third group plotted in Figure 8. The transition point moves upstream at a certain instance. Then, the DBD plasma actuator pushes it back toward the trailing edge by sweeping out the disturbances and the two-dimensional vortex structure induced by the actuator becomes dominant over the upper surface of the airfoil. Next, transition point moves back to forward again. This phenomenon repeats periodically, as shown in the video animation in the Supplementary Material. Figure 15: Chordwise pressure coefficient distributions over the airfoil surface: NACA0015 (\(\alpha=6^{\circ}\)), Ishii airfoil (\(\alpha=4^{\circ}\)), \(Re=6.3\times 10^{4}\), \(F^{+}=6.0\), \(Dc=0.05\), \(BR=10\%\). We still have not identified some of the flow features and we need more studies to understand in what case the DBD plasma actuator promotes the transition and in what case it delays the transition. However, this small electric device is very effective for many types of flow control applications.