## 2 Physical Model and Control Equation
### Governing Equation  
(1) Continuity equation:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial(\rho u)}{\partial x}+\frac{ \partial(\rho v)}{\partial y}+\frac{\partial(\rho w)}{\partial z}=0 \tag{4}\]  
where \(\rho\) is density in kg/m\({}^{3}\); t is time in s; and \(u\), \(v\), and \(w\) are the components of velocity vector u in the \(x\), \(y\), and \(z\) directions in m/s. Because it is an incompressible liquid, \(\partial\rho/\partial t=0\)  
\[\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w }{\partial z}=0 \tag{5}\]  
(2) Navier-Stokes equations:  
\[\left\{ \begin{array}{l}\rho\left(\frac{\partial u}{\partial t}+u\frac{ \partial u}{\partial x}+\frac{\partial u}{\partial y}+w\frac{\partial u}{ \partial z}\right)=\rho f_{x}-\frac{\partial p}{\partial x}+\mu\left(\frac{ \partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y^{2}}+\frac{ \partial^{2}u}{\partial z^{2}}\right)\\ \rho\left(\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+\frac{ \partial w}{\partial y}+w\frac{\partial w}{\partial z}\right)=\rho f_{y}- \frac{\partial p}{\partial y}+\mu\left(\frac{\partial^{2}v}{\partial x^{2}}+ \frac{\partial^{2}v}{\partial y^{2}}+\frac{\partial^{2}v}{\partial z^{2}} \right)\\ \rho\left(\frac{\partial w}{\partial t}+u\frac{\partial w}{\partial x}+\frac{ \partial w}{\partial y}+w\frac{\partial w}{\partial z}\right)=\rho f_{z}- \frac{\partial p}{\partial z}+\mu\left(\frac{\partial w}{\partial x}+\frac{ \partial^{2}w}{\partial y^{2}}+\frac{\partial^{2}w}{\partial z^{2}}\right) \end{array} \right. \tag{6}\]  
where \(\rho\) is the fluid density in kg/m\({}^{3}\); \(t\) is time in s;\(f_{x}\),\(f_{y}\), and \(f_{z}\) are the components of the volume force per unit volume of fluid in the \(x\), \(y\), and \(z\) directions in N; \(\mu\) is the dynamic viscosity in N\(\cdot\)s/m\({}^{2}\); \(u\), \(v\), and \(w\) are the velocity components of the fluid at \(x\), \(y\), and \(z\) at time t in m/s; and \(p\) is the pressure in Pa. In this paper, the simulation environment is an incompressible liquid, and there is complex turbulence. Therefore, the \(k\)-\(\epsilon\) two-equation model is used for numerical simulation. (3) The momentum equation:  
\[\left(\frac{\partial\rho u_{i}}{\partial t}+\rho u_{j}\frac{ \partial u_{j}}{\partial x_{j}}\right)=\rho f_{i}-\frac{\partial\rho}{ \partial x_{i}}+\frac{\partial}{\partial x_{j}}\left[\mu_{eff}\left(\frac{ \partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\right)\right] \tag{7}\]  
\[\mu_{eff}=\mu+\mu_{t} \tag{8}\]  
\[\mu_{t}=C_{\mu}\rho\frac{k^{2}}{\epsilon} \tag{9}\]  
(4) Turbulent kinetic energy equation:  
\[\frac{\partial(\rho k)}{\partial t}+\partial u_{j}\frac{\partial k}{\partial x _{j}}=\frac{\partial}{\partial x_{j}}\left[\left(\mu+\frac{\mu_{t}}{\sigma_{ k}}\right)\frac{\partial k}{\partial x_{j}}\right]+P_{k}-\rho\epsilon \tag{10}\]  
\[\frac{\partial(\rho\epsilon)}{\partial t}+\rho u_{j}\frac{\partial\epsilon}{ \partial x_{j}}=\frac{\partial}{\partial x_{j}}\left[\left(\mu+\frac{\mu_{t}}{ \sigma_{k}}\right)\frac{\partial\epsilon}{\partial x_{j}}\right]+\frac{ \epsilon}{k}(C_{\epsilon 1}P_{k}-C_{\epsilon 2}\rho\epsilon) \tag{11}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Scheme**} & \multicolumn{2}{c}{**Nozzle Diameters**} & \multicolumn{2}{c}{**Rear Nozzle**} & \multirow{2}{*}{**Initial Velocity (m/s)**} \\  & \multicolumn{2}{c}{**(mm)**} & & \\ \hline
1 & 2 & 25\({}^{\circ}\) & 25 \\
2 & 2 & 30\({}^{\circ}\) & 25 \\
3 & 2 & 35\({}^{\circ}\) & 25 \\
4 & 2 & 40\({}^{\circ}\) & 25 \\
5 & 3.5 & 30\({}^{\circ}\) & 25 \\
6 & 3.5 & 35\({}^{\circ}\) & 25 \\
7 & 3.5 & 40\({}^{\circ}\) & 25 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Conceptual design.