## 2 Numerical Simulation Method
### Grid Scheme and Boundary Conditions
#### 2.2.2 Boundary Conditions, Governing Equations and Turbulence Model  
The inlet boundary condition of the vertical axial-flow pump is the total pressure, which is set on the inlet surface of the elbow inlet conduit; the total pressure is set at 1 atm. The numerical simulation results of the flow field in the axial-flow pump are not sensitive to the selection of the inlet turbulent kinetic energy parameters, so the medium turbulence intensity Tu = 5% is used on the inlet surface of the elbow inlet conduit; the outlet boundary condition is mass flow, which is set on the outlet surface of the outlet extension of the straight outlet conduit. In the near wall region, the retractable wall function is used, and each solid wall is set as a non-slip boundary condition without considering the influence of wall roughness; the convergence accuracy is set as 1.0 \(\times\) 10\({}^{-5}\). Based on the Reynolds time average equation Navier-Stokes (RANS) and the RNG \(k\)-\(\epsilon\) turbulence model, the vertical axial-flow pump was numerically solved. The RNG \(k\)-\(\epsilon\) turbulence model modified the turbulent viscosity and considered the rotation and rotation in the average flow; furthermore, the turbulence model can better deal with flow with a high strain rate and a large curvature of the streamline. The RNG \(k\)-\(\epsilon\) turbulence model was verified in [28; 29; 30] and can well simulate the internal flow field of the pump and the pump device. The fluid in the pump device can be approximately considered as an incompressible three-dimensional viscous turbulence, and the fluid motion follows the laws of mass conservation, momentum conservation and energy conservation; the fluid is liquid water at room temperature, and the heat exchange between the fluids is not considered. The continuity equation can be simplified as [28]  
\[\frac{\partial u_{i}}{\partial x_{i}} = 0\]  
with the momentum equation [28]  
\[\frac{\partial(\rho u_{i})}{\partial t}+\frac{\partial}{\partial x_{j}}(\rho u _{i}u_{j})=-\frac{\partial p}{\partial x_{i}}+\frac{\partial}{\partial xj} \left(\mu\left(\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{ \partial x_{i}}\right)\right)+F_{i}\]  
where \(\rho\) is the density of fluid, \(u_{i}\) and \(u_{j}\) are the components of the fluid velocity along \(i\) and \(j\) directions, respectively, \(x_{i}\) is the coordinate component, \(t\) is time, \(p\) is the pressure on the fluid micro element body, \(\mu\) is the kinematic viscosity coefficient of the fluid and \(F_{i}\) is the volume force component along direction \(i\). The RNG \(k\)-\(\epsilon\) model takes into account the influence of a high strain rate or large curvature flow surface so as to improve the accuracy of the model in the case of a swirling  
Figure 6: Comparison between numerical simulation and model test of axial-flow pump device. flow and large curvature. The RNG \(k\)-\(\epsilon\) model is used to simulate the axial-flow pump, and the governing equations of the RNG \(k\)-\(\epsilon\) model are as follows:  
\[ \begin{array}{c}\frac{\partial(\rho k)}{\partial t}+\frac{\partial(\rho ku_{i} )}{\partial(x_{i})}=\frac{\partial}{\partial x_{j}}\Big{[}\alpha_{k}\mu_{eff} \frac{\partial k}{\partial x_{j}}\Big{]}+\mu_{t}\Big{(}\frac{\partial u_{i}}{ \partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\Big{)}\frac{\partial u_{ i}}{\partial x_{j}}-\rho\epsilon\\ \frac{\partial(\rho v)}{\partial t}+\frac{\partial(\rho u_{i})}{\partial(x_{i} )}=\frac{\partial}{\partial x_{j}}\Big{[}\alpha_{k}\mu_{eff}\frac{\partial x }{\partial x_{j}}\Big{]}+\frac{C_{k}}{k}\mu_{t}\Big{(}\frac{\partial u_{i}}{ \partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\Big{)}\frac{\partial u_{ j}}{\partial x_{j}}-C_{2}\rho\frac{\epsilon^{2}}{K}-R\end{array}