## 2 Materials and Methods
### Numerical Setup and Validation  
Numerical calculations were performed using the commercial software PumpLink 4.6.0. The numerical scheme is based on the finite volume method (FVM), employing a second-order upwind scheme for spatial discretization and an implicit solver to enhance stability and computational efficiency. The computational domain and monitoring points in the flow field are shown in Figure 2. In the simulation, the gears and boundaries are modeled as adiabatic slip walls. The boundary conditions consist of a pressure inlet and a pressure outlet. To ensure that the cavitation phenomenon in the simulation is not influenced by boundary conditions, the cavitation number is kept constant, eliminating the impact of factors other than temperature on cavitation.  
The cavitation number is used to characterize the degree of cavitation, as shown in Equation (12).  
\[\sigma_{c}=\frac{p_{in}-p_{v}}{0.5\rho_{l}{U_{\infty}}^{2}} \tag{12}\]  
where \(P_{in}\) is the inlet pressure of the gear pump, and \(p_{v}\) is the saturated vapor pressure. \(\rho_{l}\) is the density of the fluid at that temperature. \(U_{\infty}\) is the pitch line velocity.  
The cavitation number at 333.15 K is adopted as the reference value. Due to the properties of the medium, the saturated vapor pressure of the oil changes with temperature, so achieving the same cavitation number at different temperatures requires different pressures. The working conditions of the external gear pump are detailed in Table 2.  
For the gear pump simulation, the grid was generated using PumpLink's automatic grid generation tool, which utilizes Cartesian grids with local grid refinement to accurately capture complex flow features. PumpLink includes dynamic mesh templates for gear pumps and utilizes the sliding mesh method to model the motion of the gears.  
This paper focuses on investigating the evolution of the internal flow field and cavitation phenomena within the gear pump. The internal flow dynamics and cavitation within the gear pump are intricately tied to the pump's flow rate. Consequently, the outlet volume flow is selected to assess grid independence in this section. Figure 3 illustrates the values  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Case**} & **Temperatures** & **Gear Speed** & **Absolute Inlet** & **Absolute Outlet** \\  & **(K)** & **(r/min)** & **Pressure (Pa)** & **Pressure (Pa)** \\ \hline
1 & 303.15 & 1000 & 91,019 & 1,091,019 \\
2 & 333.15 & 1000 & 101,325 & 1,101,325 \\
3 & 363.15 & 1000 & 177,283 & 1,177,283 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Working conditions of gear pump at 303.15 K, 333.15 K, and 363.15 K.  
Figure 2: Schematic of the flow field computational domain and measurement points.  
of outlet average volume flow in the gear pump with varying total mesh elements. When the grid count exceeds 400,000, further increasing the number of grids has minimal impact on the flow. Hence, the total grid count is set at 420,000. To further ensure the reliability of the grid independence study, the Grid Convergence Index (GCI) was employed. The GCI quantifies the uncertainty associated with the discretization error and provides a measure of how much the solution changes with grid refinement [36]. As shown in Table 3, the calculated GCI values for the selected grid sizes were found to be less than 3%, which is within acceptable limits. This confirms that the chosen grid resolution of 420,000 elements is sufficient to accurately capture the flow characteristics.  
The results obtained through numerical calculations are compared with the experimental findings [37]. As illustrated in Figure 4