## 4 Computational Approach
### Modeling of the Body Force Induced by the Plasma Actuator  
Several body force models have been proposed for the plasma actuator in the past, such as those in Refs. [21; 22; 23; 24]. Among them, we adopted the Suzen and Huang model [23; 24]. As described above, the body force terms introduced for the plasma actuator are evaluated by the computation of the electric charge density \(q_{c}\) and the distribution of the electric field vector \(E_{i}\). The Suzen and Huang model provides these variables as the solutions of the following two equations. \[\frac{\partial E_{k}}{\partial x_{k}}=\frac{\partial}{\partial x_{k}}(\varepsilon _{r}\frac{\partial\phi}{\partial x_{k}})=0 \tag{4}\]\[\frac{\partial}{\partial x_{k}}(\varepsilon_{r}\frac{\partial q_{c}}{\partial x_{ k}})=\frac{q_{c}}{\lambda_{d}^{2}} \tag{5}\]  
where \(\varepsilon_{r}\) is the relative permittivity of the medium, \(\phi\) is the scalar potential and \(\lambda_{d}\) is the Debye length. \(\varepsilon_{r}\) is set to be 2.7, considering the material for the polyimide dielectric layers used in the experiment. \(\lambda_{d}\) is set to be 0.001 as is given by the Suzen and Huang model. These equations are solved with the proper boundary conditions. More details about the Suzen and Huang model can be found in Refs. [23; 24] and the formulation in this study can be found in Ref. [25]. The unsteady feature of the body force in Equations (2) and (3) is modeled as follows:  
\[S(x,y,z,t)=S_{suzen}(x,y,z)\sin^{2}(2\pi F_{base}t) \tag{6}\]  
where \(F_{base}\) is the non-dimensional base frequency of the input AC voltage and defined as \(F_{base}=f_{base}c/u_{\infty}\). Here, \(f_{base}\) is the base frequency. \(S_{suzen}(x,y,z)\) is the spatial body force distributions obtained from the Suzen and Huang model above with the solution of Equations (4) and (5). Unsteady change of the body force based on the base frequency is modeled by the sinusoidal term in Equation (6). The body force generated by the plasma actuator is assumed to be a push-push type in one cycle and the square of the sine function is simply applied as shown in Equation (6) [26; 27]. The magnitude of the body force has a parameter \(D_{c}\) for the scaling of electrical forces to inertial forces in the freestream. Here, \(D_{c}\) is defined as follows:  
\[D_{c}=\frac{q_{c,ref}E_{ref}c}{\rho_{\infty}u_{\infty}^{2}}=\frac{q_{c,ref} \phi_{ref}}{\rho_{\infty}u_{\infty}^{2}} \tag{7}\]  
The subscript _ref_ denotes the reference value. In this equation, \(q_{ref}\) and \(E_{ref}\) are set as the maximum values of \(q_{c}\) and \(E\) obtained by the Suzen and Huang model in each simulation. Therefore, change of the \(D_{c}\) value may be considered as a change of the input voltage. As shown in Equation (7), \(D_{c}\) represent the ratio of the electromagnetic force to the freestream dynamic pressure. As the maximum value of \(q_{c}\) could not be determined, we needed to examine the dependency of the maximum induced velocity on \(D_{c}\) relative to the model scale and the freestream velocity scale used in the simulations of the present study. [PERSON] et al. conducted simulations of the induced velocity fields in a quiescent field under the model scale and the freestream velocity (as a result, Reynolds number of 6.3 \(\times\) 10\({}^{4}\)) used here [28]. Good agreement with the experiment was obtained for the distribution of the induced velocity profiles when the maximum induced velocity is adjusted to be the same value as that in the experiment. [PERSON] et al. showed that plasma-induced body forces under quiescent air conditions should remain unchanged for the external flows up to 100 m/s or so [29]. From the studies by [PERSON] et al. [28] and [PERSON] et al. [29], we concluded that adjusting the maximum induced velocity under the quiescent air would result in the agreement of the velocity profiles and would lead to the proper value of \(D_{c}\) that properly scales the electrical forces to the inertial forces in the freestream.