## 3 Computational Approach
### Plasma Actuator Modeling  
The body force term for PA was modeled with \(D_{\mathrm{c}}S_{i}\) and \(D_{\mathrm{c}}u_{\mathrm{s}}S_{k}\) in the Navier-Stokes equations, as in Equations (3) and (4). The Suzen-Huang model [34] is utilized to obtain \(S_{i}\). The non-dimensional body force vector \(S_{i}\) is represented as follows:  
\[S_{i}=\rho_{\mathrm{c}}\bigg{(}-\frac{\partial\phi}{\partial x_{i}}\bigg{)}f( t)^{2}, \tag{6}\]  
where \(f(t)\) is the waveform function of the input voltage, \(\rho_{\mathrm{c}}\) is the net charge density, and \(\phi\) is the electric potential. The following equations are solved to obtain \(\rho_{\mathrm{c}}\) and \(\phi\):  
\[\frac{\partial}{\partial x_{j}}\bigg{(}\epsilon_{\mathrm{r}} \frac{\partial\phi}{\partial x_{j}}\bigg{)} = 0, \tag{7}\] \[\frac{\partial}{\partial x_{j}}\bigg{(}\epsilon_{\mathrm{r}} \frac{\partial\rho_{\mathrm{c}}}{\partial x_{j}}\bigg{)} = \frac{\rho_{\mathrm{c}}}{\lambda_{\mathrm{d}}^{2}}, \tag{8}\]  
where \(\epsilon_{\mathrm{r}}\) is the relative permittivity of the medium, and \(\lambda_{\mathrm{d}}\) is the Debye length. The following plasma actuator was considered in the present study: The exposed electrode was 2 mm wide, and the insulated electrode was 8.75 mm wide. The electrodes were 0.1 mm thick and separated by a 0.1 mm thick dielectric. The streamwise spacing of electrodes was 0.5 mm. The dielectric was Kapton, and \(\epsilon_{\mathrm{r}}\) was 2.7. In the air, \(\epsilon_{\mathrm{r}}\) was 1.0. For \(\lambda_{\mathrm{d}}\), the same 1 mm as in the previous study was used [34]. These length parameters were non-dimensionalized by the reference length \(c=0.1\) m. Equations (7) and (8) were solved by the successive over-relaxation (SOR) method using a 1201 \(\times\) 801 two-dimensional mesh. Boundary conditions for Equation (7) are given as follows:  
\[ \begin{array}{ll}\text{on outer boundaries,}&\partial\phi/\partial n_{i}= 0,\\ \text{on exposed electrode,}&\phi=1.0,\\ \text{on insulated electrode,}&\phi=0,\end{array} \]  
where \(n_{i}\) is the unit normal vector. The boundary conditions for Equation (8) are given as:  
\[ \begin{array}{ll}\text{on outer boundaries and in dielectric,}&\rho_{\mathrm{c}}= 0,\\ \text{on the surface of dielectric above the insulated electrode,}&\rho_{\mathrm{c}}=G(x^{\prime}),\\ \text{on the other surface of PA,}&\partial\rho_{\mathrm{c}}/\partial n_{i}=0.\end{array} \]  
\(G(x^{\prime})\) is given by a half Gaussian distribution as follows:  
\[G(x^{\prime})=\exp\bigg{(}-\frac{x^{\prime 2}}{2\delta^{2}}\bigg{)}, \tag{9}\]  
where \(x^{\prime}\) is the chordwise length measured from the edge of the insulated electrode, and \(\delta\) was 0.3\(l_{\mathrm{e}}\) in the present study. \(l_{\mathrm{e}}\) is the insulated electrode length. In the present study, the input voltage is a standard alternating current. Therefore, the waveform \(f(t)\) is the sinusoidal wave:  
\[f(t)=\sin(2\pi F_{\mathrm{base}}t). \tag{10}\]  
Figure 3 shows the distribution of the body force magnitude in the \(x\) direction (\(S_{x}\)) when \(f(t)=1\). The body force vectors are also shown in an enlarged view near PA. The length of the model region is 0.15\(c\) in the chordwise direction and 0.05\(c\) in the wall-normal direction. The body force in the spanwise direction was not implemented so that  
no disturbance in the spanwise direction could be included in the plasma actuation of the present simulations. The magnitude of the body force was determined by the non-dimensional input voltage parameter, \(D_{\rm c}\), which is defined as follows:  
\[D_{\rm c}=\frac{\rho_{\rm c,max}\phi_{\rm max}}{\rho_{\infty}U_{\rm c}^{2}}, \tag{11}\]  
where \(\rho_{\rm c,max}\) and \(\phi_{\rm max}\) are the maximum values of \(\rho_{\rm c}\) and \(\phi\). In the present study, \(D_{\rm c}=0.04\) was used. The maximum induced velocity produced by continuous actuation with \(D_{\rm c}=0.04\) in quiescent air reaches approximately \(3.4\) m/s if the reference velocity is \(U_{\infty}=10\) m/s [32].