## 2 Numerical Modeling  
### Computational Geometry and Flow Solver  
The compressor cascade is modeled with seven blades, as depicted in Figure 1. The blade section is an airfoil of the NACA 65 series. The chord length is 0.15 m. The pitch to chord ratio corresponds to 0.75. The cascade resembles the experimental facility for aeroelastic assessments of the Chair of Aero Engines at TU Berlin [22]. This allows for comparisons between numerical and experimental data for the clean configuration. Plasma actuators are applied on the three central blades of the cascade, in order to take into account the relative effects of plasma between the blades. Two-dimensional numerical simulations are performed, as the flow has been proved to have negligible three-dimensional effects, at least for the conditions considered here [22]. Additionally, plasma actuators are meant to cover the entire blade span when installed, so that their effects can be assumed as mostly two-dimensional. A multiblock structured grid composed of 550,949 elements is realized by means of the commercial software Ansys ICEM CFD(r)(Version 17, Ansys, Canonsburg, PA, USA).The domain extends approximately two chords both upstream the leading edge and downstream the trailing edge. In order to reproduce correctly the main features of the boundary layer, a targeted O-grid is realized around the airfoil. The O-grid features 100 elements in the direction normal to the airfoil and 410 elements on the airfoil surface. Refinements are introduced in the area where the plasma is modeled, as well as on the leading edge and on the trailing edge. The height of first cell in the boundary layer is set to keep the y\({}^{+}\) equal to one, for the Reynolds number of \(\sim\)10\({}^{5}\), based on the airfoil chord and on the freestream velocity. One block of the grid is realized specifically for the wake region. In this block, the grid is kept finer, compared to the other far field areas. This allows for capturing accurately the modifications in the airfoil circulation--and in turn on lift and moment--induced by the blade motion. With the aim to maximize the quality of the numerical solution, the elements of the boundary layer and of the wake are all kept approximately squared. The results of grid sensitivity studies are presented in the following. Figure 2 is a detail of the computational grid around one blade with a blowing up of the trailing edge region. The angle of attack is set here to \(\alpha=2\) degrees. Freestream velocities of 34.36 m/s and 19.17 m/s are imposed as a boundary condition at the inlet, consistently with the experimental tests of Refs. [22; 26]. An average static pressure condition of 101,325 Pa is imposed at the outlet. Periodic boundary conditions are applied on the top and on the bottom of the computational domain (see again Figure 1). Figure 1: Sketch of the computational geometry for the blade cascade. The finite volume solver Ansys CFX\({}^{\textregistered}\) is employed for the viscous computations on the present cascade. The advection terms are solved with a high resolution scheme, whereas a second order backward scheme is employed for the transient terms. The flow is assumed as fully turbulent, and the system of Reynolds-Averaged Navier Stokes (RANS) equations is closed with the SST k-\(\omega\) Menter turbulence model [27]. Wall functions are employed close to the solid walls. In [28], the performance of different turbulence models in describing fully turbulent flows around oscillating airfoils in pre-stall and in deep-stall conditions was investigated. The contents shown in [28] shows that the k-\(\omega\) SST model [27], the Spalart-Allmaras [29] model and the Baldwin-Barth [30] model are those that better capture the unsteady flow physics associated with airfoils in oscillating motion. The k-\(\omega\) SST model [27] is employed here because it is appropriate to describe the flow around oscillating airfoils and because it allows to get a good accuracy both close to the walls and in external flow.