## 2 Numerical Simulation Methodology
### Numerical Methods
#### 2.3.2 Governing Equation  
For convenience, the main equations used in the mathematical model are described as follows. Continuity equation:  
\[\frac{\partial(\rho u)}{\partial x}+\frac{\partial(\rho v)}{\partial y}+\frac{ \partial(\rho w)}{\partial z}=0 \tag{2}\]  
Momentum equation:  
\[\
abla\cdot(\rho uu)=-\frac{\partial p}{\partial x}+\frac{\partial\tau_{xx}}{ \partial x}+\frac{\partial\tau_{yx}}{\partial y}+\frac{\partial\tau_{zx}}{ \partial z}+F_{x} \tag{3}\]  
\[\
abla\cdot(\rho vu)=-\frac{\partial p}{\partial y}+\frac{\partial\tau_{xy}}{ \partial x}+\frac{\partial\tau_{yy}}{\partial y}+\frac{\partial\tau_{zy}}{ \partial z}+F_{y} \tag{4}\]  
\[\
abla\cdot(\rho wu)=-\frac{\partial p}{\partial z}+\frac{\partial\tau_{xz}}{ \partial x}+\frac{\partial\tau_{yz}}{\partial y}+\frac{\partial\tau_{zz}}{ \partial z}+F_{z} \tag{5}\]  
Energy equation:  
\[\
abla\cdot\left(\rho_{f}uT\right)=\
abla\cdot\left(\frac{k}{c_{p}}grad(T) \right)+S_{T} \tag{6}\]  
Concentration equation:  
\[\
abla\cdot\left(\rho_{f}uc_{s}\right)=\
abla\cdot\left(D_{s}grad\left(\rho_{ f}c_{s}\right)\right)+S_{s} \tag{7}\]  
Turbulent energy equation:  
\[\
abla\cdot\left(\rho_{f}uk\right)=\
abla\cdot\left(a_{k}u_{eff}gradk\right) +G_{k}+\rho_{f}\varepsilon \tag{8}\]  
Dissipation rate equation:  
\[\
abla\cdot\left(\rho_{f}u\varepsilon\right)=\
abla\cdot\left(a_{\varepsilon} u_{eff}grad\varepsilon\right)+\frac{\varepsilon}{k}\left({C_{1e}}^{*}G_{k}-C_{2 \varepsilon}\rho_{f}\varepsilon\right) \tag{9}\]  
\[\mu_{eff}=\mu+\mu_{t} \tag{10}\]  
\[\mu_{t}=\rho_{f}C_{\mu}\frac{k^{2}}{\varepsilon} \tag{11}\]  
\[\
abla\cdot(\rho uu)=-\frac{\partial p}{\partial x}+\frac{\partial\tau_{xx}}{ \partial x}+\frac{\partial\tau_{yx}}{\partial y}+\frac{\partial\tau_{zx}}{ \partial z}+F_{x} \tag{3}\]0  
\[\
abla\cdot(\rho uu)=-\frac{\partial p}{\partial x}+\frac{\partial\tau_{xx}}{ \partial x}+\frac{\partial\tau_{yx}}{\partial y}+\frac{\partial\tau_{zx}}{ \partial z}+F_{x} \tag{3}\]1  
\[\
abla\cdot(\rho uu)=-\frac{\partial p}{\partial x}+\frac{\partial\tau_{xx}}{ \partial x}+\frac{\partial\tau_{yx}}{\partial y}+\frac{\partial\tau_{zx}}{ \partial z}+F_{x} \tag{3}\]2  
where \(u\) is the fluid velocity vector and \(u\), \(v\), \(w\) is the component of the velocity vector in the \(x\), \(y\), \(z\) direction [m/s]; \(\mu\) is the dynamic viscosity of the gas [Pa\(\cdot\)s]; \(P\) is the pressure [Pa]; \(\rho_{f}\) is the mass density of fluid [kg/m\({}^{3}\)]; \(F\) is the body force [N]; \(c_{s}\) is the component volume fraction; \(S_{T}\) is the heat source term; \(S_{s}\) is the mass source term; \(D_{s}\) is the mass diffusion coefficient of the component; \(C_{\mu}=0.085\); \(a_{k}=a_{\varepsilon}=1.39\); \(C_{1e}=1.42\); \(C_{2e}=1.68\); \(\eta_{0}=4.38\); \(\beta=0.012\). In the ESP, the electrostatic field caused by the corona discharge process was solved with [PERSON]'s equation and current continuity equation, which are expressed as follows. \[\
abla\cdot(\rho uu)=-\frac{\partial p}{\partial x}+\frac{\partial\tau_{xx}}{ \partial x}+\frac{\partial\tau_{yx}}{\partial y}+\frac{\partial\tau_{zx}}{ \partial z}+F_{x} \tag{3}\]3  
\[\
abla\cdot(\rho uu)=-\frac{\partial p}{\partial x}+\frac{\partial\tau_{xx}}{ \partial x}+\frac{\partial\tau_{yx}}{\partial y}+\frac{\partial\tau_{zx}}{ \partial z}+F_{x} \tag{3}\]4  
where \(\varphi\) is the electric potential [V]; \(\varepsilon_{0}\) is the permittivity of free space, 8.85 \(\times\) 10\({}^{-12}\) [C\({}^{2}\)/ (N \(\cdot\) m\({}^{2}\))]; \(\rho_{ion}\) is the ion charge density [C/m\({}^{3}\)], \(\mathbf{E}\) is the electric field strength [V/m], \(\mathbf{u}\) is the fluid velocity [m/s], \(k_{ion}\) is the mobility of ions, and \(D_{i}\) is the ion diffusion coefficient. The corona onset electric strength on the electrode surface can be calculated according to [PERSON]'s law using the equation.