## 2 Numerical Modeling
### Grid Sensitivity Study and Comparison with Experimental Data  
The reliability of the Computational Fluid Dynamic (CFD) simulations is checked first by assessing the sensitivity of the numerical solution to the spacial refinement. Namely, the steady state airloads computed with the reference grid--550,949 elements--are compared to those obtained with a finer and coarser mesh. The finer grid is composed of 1,138,921 elements, whereas the coarser grid features 274,946 elements. The clean cascade, as well as the configurations with PS/SS plasma,  
Figure 4: Sketch of the blade section with the modeling of plasma actuators. are considered. A low and a moderate angle of attack, i.e., \(\alpha=2\) deg. and \(\alpha=8\) deg., are taken into account. Tables 1 and 2 show the percent difference between the airloads computed with the reference grid and with the finer or coarser spacial resolutions. The percent difference is depicted for both the non-actuated and the actuated configurations. Both for the actuated and the non-actuated case, the reference grid features small discrepancies with respect to the finer mesh. This occurs at \(\alpha=2\) deg. and at \(\alpha=8\) degrees. Namely, the percent differences in loads are always smaller than 1% beside two cases, in which the drag values differ by 1.2% and 3.1% for reference and finer mesh. This is regarded as acceptable, as an accurate quantification of drag is not the main focus of the paper. The assumption of fully turbulent flow, together with the employment of RANS computations adopted here, do not allow for predicting properly the drag coefficient. On the other hand, the coarser grid provides larger differences not only in terms of drag, but also in terms of lift and moment coefficients, already at \(\alpha=2\) deg. As a consequence, the reference grid is regarded as the best trade-off between accuracy of the solution and computational burden. Figure 5 shows the pressure distribution achieved experimentally on the reference test rig of the Chair of Aero Engines at TU Berlin and the counterpart obtained with CFD simulations. Specifically, the pressure coefficient \(\mathrm{C_{P}}\) is displayed. A steady state configuration is considered, with angle of attack of 2 degrees, and freestream velocity of 34.36 m/s, yielding a chord/freestream velocity based Reynolds number of \(\sim\)350,000. A very good agreement is found between the numerical computations and the experiments, including the leading edge and the trailing edge area of the blade. Figure 6 shows the results of time-resolved traveling wave mode simulations performed on the clean cascade and compared to the experimental counterpart, as well as to the measured data of [PERSON] [23] and [PERSON] [24] on the same geometry. The comparison is performed for the blades oscillating at frequency of 5.25 Hz, corresponding to a reduced frequency \(\mathrm{k}=2\pi\mathrm{fc}/U_{\infty}=0.1440\)--with the same inlet velocity used for the comparisons at constant angle of attack. The IBPA is equal to 0 degrees. The time step employed for the simulations is \(1.9048\times 10^{-3}\) s, corresponding to 100 time steps per period. Three oscillation periods are simulated, in order to allow the flow to reach a fully \begin{table}
\begin{tabular}{l l c c c} \hline \hline  & **Grids–Airloads** & \(\Delta\mathbf{C_{1}\ \%}\) & \(\Delta\mathbf{C_{d}\ \%}\) & \(\Delta\mathbf{C_{m_{\alpha}2}\ \%}\) \\ \hline \multirow{2}{*}{**CLEAN BLADE**} & reference/finer & 0.1 & 0.4 & 0.1 \\  & reference/coarser & 0.2 & 0.6 & 0.2 \\ \hline \multirow{2}{*}{**BLADE WITH PS PLASMA**} & reference/finer & 0.7 & 1.2 & 0.2 \\  & reference/coarser & 9.4 & 41.2 & 3.8 \\ \hline \multirow{2}{*}{**BLADE WITH SS PLASMA**} & reference/finer & 0.4 & 0.2 & 0.3 \\  & reference/coarser & 7.5 & 4.3 & 6.4 \\ \hline \hline \end{tabular}
\end{table}