## 4 Results of Traveling Wave Mode Simulations
Additionally, because the geometry of the reference cascade is span-wise uniform and meant to represent a blade section, the lift is not an accurate picture of the blade loading at the hub, which should be quantified with three-dimensional bladings. It is also worth noting that the choice of focusing on the pitching moment only and not on the lift is consistent with other works (see e.g., [2; 23]). These works deal indeed with the unsteady aerodynamics and with the aeroelastic response of linear two-dimensional cascades. The results at constant angle of attack presented in Section 3 show that a larger body force is required on the suction side, in order to obtain percentage gains similar to the PS counterpart. As a consequence, a body force equal to 75% of the reference 300 mN/m, i.e., 225 mN/m, is applied on the pressure side. On the other hand, a force equal to 150% of 300 mN/m, i.e., 450 mN/m, is applied on the suction side--this value being feasible for modern plasma actuators [38]. Therefore, the body force on the suction side is double relative to that applied on the pressure side. The body force is modeled as constant during the actuation-on cycle, hence its time history would be a squared wave. In order to avoid undesired peaks in the pressure distribution, an exponential law modulation is applied to the body force amplitude at the very beginning and end of the operating cycle. This allows a smooth transition between the non-actuated and actuated configuration and vice versa. The triggering of the PS/SS actuation is defined on the basis of Figure 11a. The previous section showed that PS actuation increases the positive nose-down pitching moment. Therefore, PS plasma should be operated when the pitching moment features negative oscillations, relative to the mean value. On the contrary SS actuation, which provides a reduction of the positive nose-up pitching moment, has to be operated when the C\({}_{\text{m}}\) oscillations have positive amplitude. Because Figure 11a shows that the moment is out of phase relative to the motion, this lead/lag has to be accounted for, when setting the PS/SS actuation triggering. For an oscillation law in the form \(\alpha=2+\sin\left(2\pi ft+\text{n}\times\text{IBPA}\frac{\pi}{\text{ISQ}}\right)\), the phase of the PS actuation on the \(n\)th blade is \(\phi_{\text{PS}}=\phi_{\text{C}_{m}}+\pi\). Consistently, the phase of the SS actuation relative to the blade motion is \(\phi_{\text{SS}}=\phi_{\text{C}_{m}}\). Figure 12 shows the time history of lift, drag and moment coefficient on the central blade of the plasma-equipped cascade at IBPA = \(-51.43\) degrees. In this case, four oscillation periods are simulated. Figure 12a depicts the four simulated periods, whereas Figure 12b is a blowing up on the second, third and fourth simulated periods, where the reaching of a well-established period regime can be better detected. Specifically, the transient results are deemed quasi-converged when the force coefficient relative difference between two consecutive periods falls below 0.02%. This condition must be achieved for all the force coefficients considered, i.e., lift, drag and moment. It is found that the difference in lift, drag and moment coefficient falls below the fixed threshold already between the second and the third oscillation period. Notice that also [PERSON] et al. (2012), who performed numerical computations on a cascade with pitching blades, found that their numerical solution reaches a good  
Figure 11: Phase of mid-chord moment coefficient and of lift coefficient at different IBPAs; Re \(\sim\)195,000; \(\alpha=2+\sin\left(2\pi ft+4\times\text{IBPA}\frac{\pi}{\text{ISQ}}\right)\) degrees; f = 19.17 Hz, T = 1/f. periodicity after two periods of oscillation. Namely, three oscillating cycles are simulated for small amplitude oscillations and four oscillating cycles for high amplitude oscillations are simulated in [2]. Figure 13 shows the velocity magnitude field, normalized by the freestream velocity, at four different instants of the pitching cycle.