# Control of Flow around an Oscillating Plate for Lift Enhancement by Plasma Actuators
## 4 Results and Discussion  
### Lift Fluctuation in Baseline Flow  
In this section, the relationship between the lift fluctuations and the flow field are discussed for the baseline flow. The lift force exerted on the plate periodically fluctuates along the oscillation motion. Figure 12 demonstrates the time fluctuations of the lift coefficient in an oscillation period in the baseline case, where the dashed line indicates the displacement of the flat plate without the dimension of amplitude. These fluctuations are related to the vortices and pressure field around the plate. Figure 13a depicts the iso-surfaces of the second invariant of the velocity gradient tensor (vortices) with the contours of the pressure, subtracting ambient pressure, \(p_{0}\). As shown in the figures, the vortical structures are two-dimensional in the span-wise direction, where the pressure distributions are also uniform in the span-wise direction. In an actual MAV, the vortical structures around the flapping wing possibly become three-dimensional due to the influence of the tip of the wing or three-dimensional flapping. In the present simulation, the flapping was two-dimensional up-and-down movement and the tip of the wing was not considered. Along with the low-Reynolds-number conditions, this possibly leads to the occurrence of the two-dimensional vortices. Also, it is noted that two-dimensional vortices were also confirmed in the above-mentioned computational results with finer spanwise grid resolution width with the factor of 0.67 compared with the original resolution. Also, Figure 13b illustrates the contours of the vorticity around the oscillating plate. Figure 11: Measured and predicted profiles of mean streamwise velocity \(U_{\mathrm{h}}\) at the downstream position from the leading edge with the distance of 4.5\(b\) from the leading edge are shown in the left figure, where \(y/b=0.5\) is corresponding to position of plate wall. The right figure shows the profiles of turbulent intensity. These figures show both the predicted profiles agree with those measured for the cases with and without control by the plasma actuator [31]. Figure 12 expresses that the lift coefficient decreases with time as \(t/T=0.0\)-0.18 in the up-stroke motion. As shown in Figure 13a, the large separation vortices with a low-pressure region generated in the down-stroke motion (\(t/T=0.94\)) become weaker on the top surface. As a result, the pressure on the top surface increases in the up-stroke motion, which subsequently decreases lift. The lift coefficient increases and decreases again between \(t/T=0.18\)-0.35 as shown in Figure 12. The low lift coefficients at \(t/T=0.18\) and 0.35 are due to the vortex shedding from the trailing edge on the bottom surface (vortex 1 and vortex 2 shown in Figure 13). This phenomenon decreases the pressure on the bottom surface and consequently the lift. At \(t/T=0.25\), the flow is largely separated on the top surface as displayed in Figure 10 and the vortex shed from the bottom surface at \(t/T=0.18\)  
Figure 12: The time fluctuations of lift coefficient in the baseline flow. The dashed line represents the displacement of oscillating plate without the dimension of amplitude. Figure 13: The flow field around the oscillating plate in the baseline flow. (**a**) The iso-surfaces of second invariant (\(q/(U_{0}/b)^{2}=0.1\)) of velocity gradient tensor and contours of pressure coefficient (\(p\)-\(p_{0}\))/(\(0.5p_{0}U_{0}{}^{2}\)), where \(p_{0}\) is the ambient pressure; (**b**) The contours of vorticity. of the lift obtained in other time, leads to a much-improved time-averaged lift. This is a future problem to be solved. As displayed in Figure 14b, the deeper decrease appears at the above-mentioned time of \(t/T=0.22\) in the control on both surfaces compared with the control only on the top surface.