## 4 Results of Traveling Wave Mode Simulations  
Traveling wave mode simulations are carried out on the clean and on the plasma equipped blade cascade. For these simulations, an alternate PS/SS actuation is imposed over the pitching oscillating cycle of the blade. The frequency of oscillation for the simulations presented hereinafter is f = 19.17 Hz, yielding a reduced frequency based on the semi-chord of \(\mathbf{k}=2\pi\mathrm{fc}/2\mathrm{U}_{\infty}=0.4597\)--being the blade  
Figure 10: Lift, drag and mid-chord moment coefficient on the clean blade and with PS/SS actuation; Re \(\sim\)350,000; \(\alpha\in[0\,12]\) degrees. chord and U\({}_{\infty}\) = 19.65 m/s the freestream velocity. The mean angle of attack is set to two degrees and the oscillation amplitude is set to one degree. The motion law of the \(n\)th blade of the cascade reads: \(\alpha=2+\sin\left(2\pi\mathrm{f}t+\mathrm{n}\times\mathrm{IBPA}\frac{\pi}{180}\right)\). Notice that these specific simulation parameters have been chosen because a significant rise in the vibratory loads--alongside a drop in the aeroelastic stability--was detected numerically at IBPA = \(-\)51.43 deg. [22] and experimentally at IBPA \(\sim\)\(-\)51 deg. [26], on the same cascade in these conditions. The blade displacement is achieved via a mesh adaptation algorithm implemented within the flow solver. One hundred time steps per oscillation cycle are employed for the numerical computations, yielding a time step size of \(\sim\)5.2 \(\times\)10\({}^{-4}\) s. A total of three periods are simulated, in order to ensure the convergence of the numerical solution. The time histories of the unsteady loads for the plasma equipped cascade are illustrated in the following part of this section. Several IBPs between \(-\)180 deg. and 180 deg. are considered, including \(\pm\)51.43 degrees. In order to maximize the actuation effectiveness, the triggering of PS/SS actuation during the oscillation period has to be set properly. To this aim, the phase of unsteady airloads relative to the blade pitching cycle for the clean cascade is evaluated for different IBPAs, at k = 0.4597. This allows to tune the triggering of PS/SS actuation in order to: (i) maximize the airloads when a minimum of the oscillating loads is observed; and (ii) minimize the airloads when a maximum in the oscillating airloads is encountered. Indeed--due to a combination of vortex shedding phenomena and inertial effects caused by the flow acceleration--the unsteady airloads are out of phase relative to the airfoil harmonic motion. The lead/lag phasing of loads is determined by the reduced frequency and by the thickness of the airfoil. A detailed discussion of these phenomena for an isolated airfoil is provided in Ref. [50]. Additional out of phasing effects may be caused by the mutual interaction of the blades. Figure 11a,b show the phase of the mid-chord moment and lift coefficient, computed at IBPAs between \(-\)180 and 180 degrees. Notice that the same out of phasing has been computed for the different blades of the cascade, at each of the interblade phase angles. Consistently with Ref. [50] and with Ref. [51], chapter 8, the computed phase is different for the two force coefficients. Only the phase of the moment coefficient--which has been proved to yield the most detrimental vibratory loads on the targeted cascade [26]--is used here to tune the actuation triggering. In fact, the main target of the conceived actuation is--at least for the present work--to manipulate the pitching moment. Indeed, the blade bending stiffness is much larger than the torsional counterpart, for the cascade under consideration [22; 26]. Moreover, the experimental results of [PERSON] [24] showed that the torsional mode is the most critical for the cascade under consideration. Therefore, acting on the pitching moment, can have the further beneficial effect of increasing the aeroelastic stability of the cascade.