## 2 Theoretical Framework for Numerical Analysis of Oil Injection Lubrication
### Numerical Theory of Flow  
The VOF model represents the density of mixed fluids and tracks the position of the free surface by introducing the fluid volume fraction function and its governing equations. If \(\alpha\) denotes the volume fraction of the liquid phase, a volume fraction within a grid cell of 1 indicates that the cell is entirely occupied by liquid; conversely, a volume fraction of 0 signifies that the grid cell is filled with gas. An intermediate state, where \(0<\alpha<1\), indicates the presence of a gas-liquid interface within the grid cell. In the simulation of oil spray lubrication within high-speed gearboxes, employing the VOF model can resolve issues related to the reconstruction of the gas-liquid interface and the temporal advancement of the interface.  
Assuming that the volume fraction of the second phase is tracked, the density property equation for the fluid is expressed as follows:  
\[\rho=\alpha_{1}\rho_{1}+\alpha_{2}\rho_{2}=(1-\alpha_{2})\rho_{1}+\alpha_{2} \rho_{2} \tag{8}\]  
where \(\rho\) represents the density of the mixed fluid; \(\rho_{1}\) and \(\rho_{2}\) are the densities of the first and second phases, respectively; and \(\alpha_{1}\) and \(\alpha_{2}\) are the volume fractions of the first and second phases, respectively.  
The continuity equation is as follows:  
\[\frac{\partial}{\partial t}(\rho)+\frac{\partial}{\partial x_{i}}(\rho u_{i})=0 \tag{9}\]  
The Reynolds averaged Navier-Stokes equation for mixed fluids is as follows:  
\[\frac{\partial}{\partial t}\rho u_{j}+\frac{\partial}{\partial x_{i}}\rho u_{ i}u_{j}=\frac{\partial}{\partial x_{i}}(\mu+\mu_{i})(\frac{\partial u_{j}}{ \partial x_{i}}+\frac{\partial u_{i}}{\partial x_{j}})-\frac{\partial p}{ \partial x_{j}}+\rho g_{j} \tag{10}\]  
In the formula, \(u_{i}\) and \(u_{j}\) are the velocities of the mixed fluid; \(t\) is time; \(g\) is the acceleration due to gravity; \(\mu_{i}\) is the turbulent viscosity coefficient; \(\mu\) is the viscosity coefficient of the mixed fluid; and \(R\) is the net phase transition rate. Moreover, when \(R\) is very small, this term is ignored in the solution.  
The turbulent kinetic energy equation of a mixed fluid is as follows:  
\[\rho\frac{\partial k}{\partial t}=\frac{\partial}{\partial x_{i}}[(\mu+\frac{\mu_ {i}}{\sigma_{k}})\frac{\partial k}{\partial x_{i}}]-\rho\varepsilon-\rho\overline {u^{\prime}_{i}u^{\prime}}\frac{\partial u_{i}}{\partial x_{i}} \tag{11}\]  
The energy dissipation rate equation for the turbulent flow of mixed fluids is as follows:  
\[\rho\frac{\partial\varepsilon}{\partial t}=\frac{\partial}{\partial x_{i}}[( \mu+\frac{\mu_{i}}{\sigma_{\varepsilon}})\frac{\partial\varepsilon}{\partial x _{i}}]+C_{1\varepsilon}\frac{\varepsilon}{k}G_{K}-C_{2\varepsilon}\rho\frac{ \varepsilon^{2}}{k} \tag{12}\]  
In the formula, \(k\) is turbulent kinetic energy; \(\varepsilon\) is the turbulence energy dissipation rate; \(\sigma_{k}\), \(\sigma_{\varepsilon}\), \(C_{1\varepsilon}\), and \(C_{2\varepsilon}\) are constants; and \(G_{K}\)= \(-\rho\overline{u^{\prime}_{i}u^{\prime}}\frac{\partial u_{i}}{\partial x_{i}}\).