## 4 Numerical Setup  
### Volume of Fluid (VOF) Multiphase Model and Governing Equations  
To track the interface between two immiscible fluids (oil and air), the VOF multiphase model presented by [PERSON] [19] was applied herein. Several scholars [10; 11; 12; 13; 16; 17; 18] have used the VOF model to study the coexistence of oil and air phases.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Measured Point** & **Temperature (\({}^{\circ}\)C)** \\ \hline
1 & 99 \\
2 & 99 \\
3 & 88 \\
4 & 88 \\
5 & 88 \\
6 & 88 \\
7 & 93 \\
8 & 90 \\
9 & 88 \\
10 & 88 \\
11 & 87 \\
12 & 80 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Experimental temperatures of the measured points.  
Figure 4: Arrangement of measuring points. 1 and 2—measuring points on the end cover near the input bearing; 3, 4, 5, and 6—measuring points on the end cover near the output bearing; 7, 8, 9, 10, 11, and 12—measuring points on the wall of the casing.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Experimental Parameter** & **Value** \\ \hline Input rotational speed (r/min) & 5000 \\ Input torque (Nm) & 286.5 \\ Oil immersion depth (mm) & 17 \\ Oil density (kg/m\({}^{3}\)) & 875.15 (at 60 \({}^{\circ}\)C) \\ Oil dynamic viscosity (kg/m\(\cdot\)s) & 0.0251 (at 60 \({}^{\circ}\)C) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Experimental parameters.  
This is accomplished by solving the continuity equation of the volume fraction of the oil phase:  
\[\frac{\partial}{\partial t}(\alpha_{\text{oil}}\rho_{\text{oil}})+\
abla\cdot( \alpha_{\text{oil}}\rho_{\text{oil}}\mathbf{u})=0 \tag{1}\]  
where \(\alpha_{\text{oil}}\) is the volume fraction of oil, \(\rho_{\text{oil}}\) is the oil density, and \(\mathbf{u}\) is the velocity vector.  
In the VOF model, the properties of the oil-air fluid, such as density, dynamic viscosity, and thermal conductivity, were calculated by averaging the volume fraction.  
\[\varphi=\alpha_{\text{oil}}\varphi_{\text{oil}}+\alpha_{\text{air}}\varphi_{ \text{air}} \tag{2}\]  
\[\alpha_{\text{oil}}+\alpha_{\text{air}}=1 \tag{3}\]  
where \(\varphi\) is the property of the oil-air fluid, \(\varphi_{\text{oil}}\) is the property of oil, \(\varphi_{\text{air}}\) is the property of air, and \(\alpha_{\text{air}}\) is the volume fraction of air.  
The continuity, momentum, fluid energy, and solid energy equations are [20]:  
\[\
abla\cdot(\rho_{\text{f}}\mathbf{u})=0 \tag{4}\]  
\[\mathbf{u}\
abla\cdot(\rho_{\text{f}}\mathbf{u})=\mu\
abla^{2}\cdot(\mathbf{u})-\
abla \cdot p+\mathbf{F} \tag{5}\]  
\[\
abla\cdot[\mathbf{u}(\rho_{\text{f}}E)]=\
abla\cdot(k_{\text{f}}\
abla T) \tag{6}\]  
\[\
abla\cdot(k_{\text{s}}\
abla T)=0 \tag{7}\]  
where \(\rho_{\text{f}}\) is the fluid density, \(\mu\) is the fluid dynamic viscosity, \(k_{\text{f}}\) is the thermal conductivity of the fluid, \(\mathbf{u}\) is the velocity vector, \(p\) is the pressure, \(\mathbf{F}\) is the external force, \(T\) is the temperature, \(E\) is the energy, and \(k_{\text{s}}\) is the thermal conductivity of the solid.  
The movements of the spiral bevel gears and double row conical roller bearings are extremely complicated, resulting in a complex two-phase flow and forming a turbulent phenomenon in the intermediate gearbox. The \(k\)-\(\epsilon\) renormalization group (RNG) turbulence model can better replicate the high strain rate and large deformation flow and is suitable for rotating flow [21]. Many scholars [10; 11; 15; 16; 17; 18; 22] have adopted the RNG \(k\)-\(\epsilon\) model to study the fluid flow characteristics in a gearbox or in a bearing cavity, and the simulation results are credible. Consequently, the RNG \(k\)-\(\epsilon\) model developed by [PERSON] and [PERSON] [23] was employed herein to simulate the turbulence. For near-wall treatment, standard wall functions were used.