## 3 Numerical Model  
### Numerical Method  
The commercial computational fluid dynamics code ANSYS CFX 18.0 is used to study the pump's internal flow characteristics. Considering the high curvature and complex flow in the pump, this paper chooses the Reynolds-averaged Navier-Stokes (RANS) equations of the shear stress transport (SST) \(k\)-\(\omega\) turbulence model to simulate the flow in the pump. The turbulent viscosity equations of the SST model, k equations, \(\omega\) equations, and weighting functions are provided as follows:  
\[v_{t}=\frac{a_{1}k}{\max(a_{1}\omega,SF_{2})} \tag{7}\]  
\[\frac{\partial k}{\partial t}+u_{j}\frac{\partial k}{\partial x_{j}}=\frac{ \partial}{\partial x_{j}}\Bigg{[}(v+\sigma_{k}v_{t})\frac{\partial k}{ \partial x_{j}}\Bigg{]}+\rho p_{k}-\beta^{*}k\omega \tag{8}\]  
\[\frac{\partial\omega}{\partial t}+u_{j}\frac{\partial\omega}{\partial x_{j}}= \frac{\partial\omega}{\partial x_{j}}\Bigg{[}(v+\sigma_{\omega}v_{t})\frac{ \partial\omega}{\partial x_{j}}\Bigg{]}+\gamma\frac{\omega}{k}\rho p_{k}- \beta\omega^{2}+2(1-F_{1})\sigma_{\omega}2\frac{1}{\omega}\frac{\partial k}{ \partial x_{j}}\frac{\partial\omega}{\partial x_{j}} \tag{9}\]  
\[F_{1}=\tanh\Bigl{(}arg_{1}^{4}\Bigr{)} \tag{10}\]  
Figure 5: Cavitation performance curve. \[F_{2}=\tanh\Bigl{(}arg_{2}^{2}\Bigr{)} \tag{11}\]  
\[arg_{1}=\min\left[\max\left(\frac{\sqrt{k}}{\beta^{\prime}\omega y^{\prime}},\frac {500v}{y^{2}\omega}\right),\frac{4 pk}{CD_{k\omega}\sigma_{\omega 2}y^{2}}\right] \tag{12}\]  
\[arg_{2}=\max\left(\frac{2\sqrt{k}}{\beta^{\prime}\omega y^{\prime}},\frac{500v} {y^{2}\omega}\right) \tag{13}\]  
\[CD_{k\omega}=\max\left(2\rho\frac{1}{\sigma_{\omega 2}\omega}\frac{\partial k}{ \partial x_{j}}\frac{\partial\omega}{\partial x_{j}},\;1.0\times 10^{-10}\right) \tag{14}\]  
where \(S\) and \(y\) denote the value of vortices and the first layer grid from the wall. \[\psi=F_{1}\psi_{1}+(1-F_{1})\psi_{2} \tag{15}\]  
\[\frac{\partial k}{\partial t}+u_{j}\frac{\partial k}{\partial x_{j}}=\frac{ \partial}{\partial x_{j}}\Bigg{[}(v+\sigma_{k}v_{t})\frac{\partial k}{ \partial x_{j}}\Bigg{]}+\rho p_{k}-\beta^{*}k\omega \tag{8}\]0  
where the model constants are \(a_{1}=0.31\), \(\sigma_{k1}=0.85\), \(\sigma_{\omega 1}=0.5\), \(\beta_{1}=0.075\), \(\beta^{*}=0.09\), and \(\kappa=0.41\). \[\frac{\partial k}{\partial t}+u_{j}\frac{\partial k}{\partial x_{j}}=\frac{ \partial}{\partial x_{j}}\Bigg{[}(v+\sigma_{k}v_{t})\frac{\partial k}{ \partial x_{j}}\Bigg{]}+\rho p_{k}-\beta^{*}k\omega \tag{8}\]1  
where the model constants are \(a_{2}=0.31\), \(\sigma_{k2}=1.0\), \(\sigma_{\omega 2}=0.856\), \(\beta_{2}=0.0828\), \(\beta^{*}=0.09\), and \(\kappa=0.41\). The flow was assumed to be at a steady state and incompressible and isothermal. The wall boundary conditions are adiabatic. A no-slip wall with a smooth surface is utilized for all physical barriers in this model. The frozen rotor is applied at the interface between the rotating and stationary components. R134a refrigerant is the working fluid. The inlet and outlet boundary conditions are mass flow and pressure boundaries, respectively. The root mean square residual error is employed for calculations, and the convergence criterion is set to a value below \(10^{-5}\) to ensure accuracy. In this study, the Sauer-Schnerr cavitation model is used to conduct numerical simulation research on the cavitation flow process of the working fluid pump. The Sauer-Schnerr model is a cavitation model based on the mass transport equation. This model approximates the liquid cavitation and bubble collapse during the cavitation process as the mass equation's pressure-driven mass change term, as shown in Equation (7). \[ \begin{cases}\dot{m^{+}}=\frac{\rho_{v}\rho_{l}}{\rho_{m}}\frac{3\alpha(1- \alpha)}{R_{b}}\sqrt{\frac{2}{3}\frac{(p-p_{sat})}{\rho_{l}}}&(p>p_{sat})\\ \dot{m^{-}}=\frac{\rho_{v}\rho_{l}}{\rho_{m}}\frac{3\alpha(1-\alpha)}{R_{b}} \sqrt{\frac{2}{3}\frac{(p_{sat}-p)}{\rho_{l}}}&(p<p_{sat})\end{cases}