## 2 Numerical Modeling
### Plasma Modeling  
Figure 3 shows a schematic of a single dielectric barrier discharge plasma actuator. Two electrodes are separated by a dielectric barrier material. One electrode is exposed to air, i.e., on the aerodynamic surface, and one is grounded within the dielectric material, inside the blade internal volume. When an alternate current voltage is applied between the electrodes, a plasma discharge is initiated. As a consequence, air is locally ionized in the area close to the electrodes and the flow momentum can be manipulated. The essential effect of plasma actuators is to add momentum into the flow field and in turn to modify the local velocity profile (see, for instance, Ref. [34]). In the present work, a source term is in fact added to the momentum equations for modeling the plasma. Specifically, a uniform force per unit volume is applied over a limited region always bounded on one side by the airfoil wall. Notice that this approach is consistent with that of Refs. [35; 36; 37]. The plasma is here meant to act on a completely attached flow at Re \(\sim\)10\({}^{5}\), with \(x\) component opposite to the main freestream direction--the \(y\) component is set in order to keep the plasma always tangent to the airfoil. Therefore, the body force has to be larger compared to classical low Re applications. Consistently with the assessments of Ref. [14] and with the experiments of Ref. [38], a force per unit span of 300 mN/m is used in this work. Notice that this approach is employed also in Refs. [35; 36; 37; 39]. According to [38], this body force is achieved with input voltages between 10 and 20 kV. Notice that, notwithstanding the voltage required to effectively operate plasma actuators is of the order of kV, the power requirements are usually limited to \(\sim\)10 W, the currents being employed of few mA [40; 41]. The plasma region is modeled as a quadrangle 0.1 mm thick and 10 mm wide (see Figure 4). These sizes are the same used in Refs. [12; 35] and representative of the experimental models of Refs. [14; 42; 43]. It is worth remarking that the present work is an exploratory investigation to assess whether plasma actuators could be suitable for load control in turbomachinery. The detailed design is planned for works to come and lies beyond the purposes of this manuscript. However, preliminary considerations  
Figure 3: Schematic of a plasma actuator with indicative sizes according to [13]. on the practical implementation of plasma actuators on actual blades have been done. The actuators per se are very thin, as shown in the sketch of Figure 3. The overall thickness of an actuator is expected to be \(\sim\)10\({}^{-4}\) m see Ref. [13]. Therefore, actuators could be installed also on quite thin trailing edges, by realizing appropriate hollows on the blade surface, both on the pressure and on the suction side. Optimization studies need to be performed to assess the optimal geometrical orientation of actuators. The first experimental studies envisaged in the near future will feature plasma actuators parallel to the chord and covering the whole span of the targeted linear cascade (see [22]). Successive studies are thought to be carried out on a rotating annulus, in order to investigate whether the transverse flow arising from the rotation requires orienting the actuators with a certain angle relative to the chord direction. There are different possible approaches to fit the cables into the blades. The selected configuration depends on the geometrical arrangement of the actuators along the blade span. In works to come, the optimal geometrical configuration for the actuators to control the blade loads will be assessed. If the actuators will cover the entire span, the cables will be connected at the level of the blade root, where there will be less problems in terms of space. Dedicated hollows will be realized to locate the cables.