# Control of Flow around an Oscillating Plate for Lift Enhancement by Plasma Actuators
## 2 Method
### Computational Methodologies  
#### 2.4.1 Governing Equations and Finite Difference Scheme  
In this simulation, the three-dimensional Navier-Stokes equations along with the mass and energy conservation laws were solved using a six-order-accurate compact finite difference scheme (fourth-order-accuracy at the boundary) for both the inviscid and viscous flux vectors [32]. The time integration was explicitly performed with a third-order-accurate Runge-Kutta method. To reproduce a moving object, such as the oscillating plate, in a static rectangular computational grid, the volume penalization (VP) method [33; 34] was utilized.  
The external penalization term, \(\mathbf{V}\) and the plasma actuator term, \(\mathbf{D}\), which is described in detail in Section 2.4.2, were added to the right-hand side of the governing equations of the three-dimensional Navier-Stokes equations as follows:  
\[\mathbf{Q}_{t}+\left(\mathbf{F}_{x}-\mathbf{G}_{x}\right)_{x}+\left(\mathbf{F}_{y}-\mathbf{G}_{y} \right)_{y}+\left(\mathbf{F}_{z}-\mathbf{G}_{z}\right)_{z}=\mathbf{V}+\mathbf{D}, \tag{1}\]  
where \(\mathbf{Q}\) is the vector of the conservative variables, \(\mathbf{F}\) are the inviscid flux vectors, \(\mathbf{G}\) are the viscous flux vectors, \(\mathbf{V}\) is the penalization term as follows:  
Figure 5: The experimental setup with link mechanism for oscillation of the plate. (**a**) Photograph; (**b**) Schematic sketch.  
Figure 6: Visualized flow field in experiment and predicted streamline for a fixed plate with \(\alpha=4^{\circ}\).  
\[\mathbf{Q}=\left(\begin{array}{c}\rho\\ \rho u\\ \rho v\\ \rho w\\ e\end{array}\right),\mathbf{F}_{x}=\left(\begin{array}{c}\rho u\\ \rho uu+p\\ \rho vu\\ \rho wu\\ (e+p)u\end{array}\right),\mathbf{F}_{y}=\left(\begin{array}{c}\rho v\\ \rho uv\\ \rho vv+p\\ \rho wv\\ (e+p)v\end{array}\right),\mathbf{F}_{z}=\left(\begin{array}{c}\rho w\\ \rho w\\ \rho vw\\ \rho ww+p\\ (e+p)w\end{array}\right), \tag{2}\]  
\[\mathbf{G}_{x}=\left(\begin{array}{c}0\\ \tau_{xx}\\ \tau_{yx}\\ \tau_{zx}\\ G_{x5}\end{array}\right),\mathbf{G}_{y}=\left(\begin{array}{c}0\\ \tau_{xy}\\ \tau_{yy}\\ \tau_{zy}\\ G_{y5}\end{array}\right),\mathbf{G}_{z}=\left(\begin{array}{c}0\\ \tau_{xz}\\ \tau_{yz}\\ \tau_{zz}\\ G_{z5}\end{array}\right),G_{i5}=u_{j}\tau_{ij} \tag{3}\]  
\[\mathbf{V}=-\left(\frac{1}{\phi}-1\right)\chi\left(\begin{array}{c}\partial\rho u _{i}/\partial x_{i}\\ 0\\ 0\\ 0\\ 0\end{array}\right), \tag{4}\]  
\[\varphi=0.25, \tag{5}\]  
\[\chi=\left\{\begin{array}{ll}\min(1,|d/\Delta y|)&(\text{inside object})\\ 0&(\text{outside object})\end{array}\right., \tag{6}\]  
where \(\rho\) is the density of the fluid, \(u\), \(v\) and \(w\) are the velocities in each direction, \(p\) is the pressure, \(e\) is the total energy per unit volume, \(\tau_{ij}\) is the viscous stress tensor.  
The variable \(\chi\) was the mask function, where \(\chi\) was set to be zero outside the object and \(1\) inside the object, excepting the region near the surface of the oscillating plate. To prevent the generation of artificial pressure waves, the mask function was smoothly changed between the inside and outside of the object by the ratio of the distance from the surface of the object, \(d\) and the grid resolution, \(\Delta y\), in the object near the surface (\(d/\Delta y<1\)). In the objects, the velocity was set to zero and the constant ambient temperature (293 K) was given. The coefficient, \(\varphi\), which is defined as the porosity of the porous medium in the original VP method [34], was here determined as 0.25 considering the computational stability. It has been confirmed that the flow field around an oscillating plate was correctly predicted by the present methodologies [35].