## 2 Materials and Methods
### Turbulence Model and Cavitation Model  
Assuming that the hydraulic oil is an incompressible and continuous medium, the hydraulic oil in the gear pump is subjected to the high-speed rotary motion of the gears, and the flow characteristics are complex, so the oil flow is regarded as turbulent while the k-\(\varepsilon\) turbulence model is chosen here to describe the turbulent state. k-\(\varepsilon\) model is a two-equation model, which describes the turbulence effect by introducing the turbulent kinetic energy k and turbulent dissipation rate \(\varepsilon\); it is one of the most widely used turbulence models because of its robustness, economy, and reasonable prediction of large-scale turbulence it is also often used in the calculation of the cavitation flow [26], whose equation is:  
\[\left\{ \begin{array}{l}\frac{\partial}{\partial t}(\rho k)+\frac{\partial}{ \partial x_{j}}(\rho u_{j}k)=\frac{\partial}{\partial x_{j}}\Big{[}(\mu+\frac {\mu_{t}}{\varepsilon_{k}})\frac{\partial k}{\partial x_{j}}\Big{]}+A\\ A=G_{k}+G_{b}-\rho\varepsilon-Y_{m}\\ \frac{\partial}{\partial t}(\rho\varepsilon)+\frac{\partial}{\partial x_{j}}( \rho u_{j}\varepsilon)=\frac{\partial}{\partial x_{j}}\Big{[}(\mu+\frac{\mu_ {t}}{\varepsilon_{k}})\frac{\partial k}{\partial x_{j}}\Big{]}+B\\ B=C_{k}\varepsilon\big{(}G_{k}+C_{3k}G_{b}\big{)}-C_{2}\rho\xi^{2}\end{array} \right. \tag{1}\]  
where \(G_{k}\) is the turbulent kinetic energy generated by the mean velocity gradient, \(G_{b}\) is the turbulent kinetic energy generated by the buoyancy force, and \(Y_{m}\) is the effect of compressibility on the expansion of the turbulent pulsation. \(\mu_{t}\) is the turbulent viscosity, which can be calculated as follows:  
\[\mu_{t}=\rho C_{\mu}\frac{k^{2}}{\varepsilon} \tag{2}\]  
\(k\) for the turbulent kinetic energy, \(\varepsilon\) for the flow energy dissipation rate, and the five empirical constants \(C_{1x}\), \(C_{2x}\), \(C_{\mu}\), \(\sigma_{k}\), and \(\sigma_{\varepsilon}\) are 1.44, 1.92, 0.9, 1.0, and 1.3. Cavitation is a phase change process; the pressure change of the flow field leads to the mutual transformation of gas and liquid phases; how to describe the mass transfer between the two phases becomes a key issue in the numerical simulation of cavitation [27]. The gas phase in the cavitation regions is assumed to behave as an ideal gas, which simplifies the calculations and is a reasonable approximation to the conditions studied. In the present paper, the cavitation model proposed by [PERSON] et al. [28; 29] is adopted, which takes into account the effects of phase change, pressure variations, vacuole movement, and the change in density of the medium with gas content. Its basic expression is as follows:  
\[ \begin{array}{l}\frac{\partial}{\partial t}\int_{\Omega(t)}\rho f_{v}d \Omega+\int_{\sigma}\rho\Big{(}\bigg{(}\overrightarrow{v}-\overrightarrow{v} _{\sigma}\bigg{)}\cdot\overrightarrow{n}\Big{)}f_{v}d\sigma\\ =\int_{\sigma}\bigg{(}D_{f}+\frac{u_{l}}{\sigma_{f}}\bigg{)}\cdot\Big{(} \
abla f_{v}\cdot\overrightarrow{n}\Big{)}d\sigma+\int_{\Omega}(R_{e}-R_{c})d \Omega\end{array}  \tag{3}\]  
where \(\rho\) represents the density of the gas-bearing fluid, \(f_{v}\) is the mass fraction of the vapor, \(v\) is the fluid velocity, \(\sigma_{\sigma}\) is the fluid surface velocity, \(D_{f}\) is the vapor diffusion coefficient, \(u_{t}\) is the turbulent viscosity, \(\sigma_{f}\) is the turbulent Schmidt number, \(R_{c}\) is the vapor generation rate, and \(R_{c}\) is the gas phase condensation rate.