## 2 Computational Methodology
### DBD Plasma Actuator Model  
The phenomenological Shyy model is considered a weakly ionized, non-thermal plasma. This concept simulates the \"bulk\" motion of fluid flow due to the induced plasma force and is based on the model developed by [PERSON] et al. [21] and [PERSON] et al. [30]. An illustration of the model is shown in Figure 2.  
This phenomenological model assumes a linearized electric field [11; 18; 35; 36]. Here, the body force generated from the Lorentzian collisions between ions is modeled as a source term in the momentum equation. At a high frequency, electrons travel with high velocity and produce a short duration plasma in which fluid species dissociate and recombine. Due to the greater mass of ions compared to electrons, the plasma body force is induced by momentum transfer of the ions. This is a key concept in the discharge process of the plasma actuator. The electric field lines, shown in Figure 2, roughly display the formation and strength of the electric field due to an asymmetric electric configuration and cause unidirectional force generation. Additionally, the electric field is strongest in the region near the electrodes and decreases in strength further away from the electrodes. Outside of this region, we assumed there was no plasma formation; hence, no force was generated. The force generated by the plasma can be formulated in a qualitative expression. [PERSON] assumed that the electric field shown was uniform, linear, and parallel. This linearized electric field distribution is shown in Figure 3.  
The electric field was assumed to decrease in strength as it moved further away from the source. The electric field, \(E\)m varies linearly and is expressed as:  
\[|\bar{E}|=E_{0}-k_{1}x-k_{2}y \tag{6}\]  
Figure 2: Electric field distributions in the actuator arrangement. where \(E_{0}\) is the maximum electric field and \(k_{1}\) and \(k_{2}\) are constants representing the gradient of the electric field intensity along the \(x\) and \(y\) directions, respectively. Equation (6) shows that the electric field strength, \(E\), varies with position and decreases with distance from the electrodes. The maximum electric field strength is expressed as:  
\[E_{0}=\frac{U}{d} \tag{7}\]  
\(d\) is the distance between the electrodes. The constants \(k_{1}\) and \(k_{2}\) are calculated based on the plasma boundary conditions by introducing the breakdown field strength, \(E_{b}\), at locations \(A\) and \(B\) from Figure 3. The equations for the electric field components are shown in (8) and (9), and the magnitude of the body force is formulated based on the definition of the electrostatic force components in the Lorentz force, shown in Equations (10) and (11). \[E_{x}=\frac{Ek_{2}}{\sqrt{k_{1}^{2}+k_{2}^{2}}} \tag{8}\]  
\[E_{y}=\frac{Ek_{1}}{\sqrt{k_{1}^{2}+k_{2}^{2}}} \tag{9}\]  
\[f_{x}=\delta E_{x}\rho_{c}e_{c} \tag{10}\]  
\[f_{y}=\delta E_{y}\rho_{c}e_{c} \tag{11}\]  
\(\rho_{c}\) is the charge density, and \(e_{c}\) is the electron charge; both are constants. Thus, the force components are only a function of electric field strength. As a result, there is a linear increase of the induced body force magnitude with electric field strength. The function \(\delta\) is used to describe the effectiveness of the plasma-induced body force. Outside the electric field, \(\delta\) is zero and is unity inside the electric field. Figure 3.0  
Figure 3.1  
In our case, \(E_{cr}\) is also represented by the breakdown electric field strength, \(E_{b}\). The discharge time, \(\Delta t\), of the plasma actuator is very small and is a fraction of the time of the duty cycle. The force is produced for a short amount of time, \(\Delta t\), of the first half cycle. The force during the second half cycle is negligible due to little or no plasma formation. Moreover, the force is considered as a constant because of the high-voltage frequency of the discharge.