## 4 Numerical Method and Verification  
### Governing Equations and Turbulence Models  
The fluid movement follows the law of mass conservation, momentum conservation and energy conservation. The flow inside the pump can be considered as incompressible three-dimensional viscous turbulence, ignoring the heat transfer and following the basic laws of the conservation of mass and momentum.  
Continuity equation [16]:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial(\rho u_{i})}{\partial x_{i}}=0 \tag{9}\]  
In this paper, we used the method of classical uncertainty assessment. The continuity equation is expressed as:  
\[\frac{\partial\rho}{\partial t}=\frac{\partial u_{i}}{\partial x_{i}}=0 \tag{10}\]  
Momentum equation [17]:  
\[\frac{\partial(\rho u_{i})}{\partial t}+\frac{\partial(\rho u_{j}u_{i})}{ \partial x_{j}}=-\frac{\partial p}{\partial x_{i}}+\frac{\partial}{\partial x _{j}}\left[\mu_{eff}\left(\frac{\partial u_{i}}{\partial x_{j}}+\frac{ \partial u_{j}}{\partial x_{i}}\right)\right]+F_{i} \tag{11}\]  
where \(\rho\) is the fluid density, \(u_{i}\) and \(u_{j}\) are the components of the fluid velocity along the \(i\) and \(j\) directions, respectively, and \(x_{i}\) is the coordinate component, \(t\) is the time, \(P\) is the pressure on the fluid microelement body, \(\mu\) is the kinematic viscosity coefficient of the fluid, \(\mu_{eff}\) is the effective viscosity coefficient and \(F_{i}\) is the volume force component in the \(i\) direction.  
The RNG \(k\)-\(\epsilon\) turbulence model was proposed by [PERSON] and [PERSON] [18] in 1986. It is derived from the instantaneous Navier-Stokes equation by using the mathematical method of group recombination. In the RNG \(k\)-\(\epsilon\) turbulence model, the small-scale motion is systematically removed from the governing equation by reflecting the small-scale effect in the large-scale motion and the modified viscosity term. The RNG \(k\)-\(\epsilon\) turbulence model can better simulate the flow with a high strain rate and large curvature of the streamline by modifying the turbulent viscosity and considering the rotating and swirling flows in the average flow. At the same time, it can effectively predict the three-dimensional flow characteristics of the axial flow pump device, It has been adopted by many related researches [19; 20]. Therefore, RNG was adopted in this paper. The \(k\)-\(\epsilon\) turbulence model was used to calculate the three-dimensional steady flow of the vertical axial flow pump with a siphon outlet conduit.  
The \(k\) equation and \(\epsilon\) equation:  
\[\frac{\partial(\rho k)}{\partial t}+\frac{\partial(\rho ku_{i})}{\partial(x_ {i})}=\frac{\partial}{\partial\big{(}x_{j}\big{)}}\left[\alpha_{k}\mu_{eff} \frac{\partial_{k}}{\partial x_{j}}\right]+\mu_{t}\left(\frac{\partial u_{i} }{\partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\right)\frac{\partial u _{i}}{\partial x_{j}}-\rho\epsilon \tag{12}\]\[\frac{\partial(\rho\varepsilon)}{\partial t}+\frac{\partial(\rho\varepsilon u_{i})}{ \partial(x_{i})}=\frac{\partial}{\partial(x_{j})}\left[\alpha_{\varepsilon}\mu_ {eff}\frac{\partial_{\varepsilon}}{\partial x_{j}}\right]+\frac{C_{1\varepsilon }}{k}\mu_{l}\left(\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}} {\partial x_{i}}\right)\frac{\partial u_{i}}{\partial x_{j}}-C_{2\varepsilon }\rho\frac{\varepsilon^{2}}{k}-R \tag{13}\]  
where coefficient \(C_{\mu}=0.0845\), \(\alpha_{k}\) and \(\alpha_{\varepsilon}\) are the effective Prandtl numbers for the high Reynolds number flow, \(\alpha_{k}=\alpha_{\varepsilon}=1.39\), coefficient \(C_{1\varepsilon}=1.39\), \(C_{2\varepsilon}=1.68\), \(\eta_{0}=4.377\) and \(\beta=0.012\).