## 3 CFD Model of the Internal Flow Field in the Gearbox
### Physical Model and Boundary Setting  
This paper focuses on investigating the transient variation dynamics of the flow field within the gearbox under oil injection lubrication conditions, thus necessitating transient CFD simulations. The Fluent software is capable of studying oil injection lubrication for high-speed helical gears, with the modeling of the helical gearbox employing dynamic mesh overlapping grid technology. The preprocessed mesh model is imported into Fluent for transient solution [35]. This paper employs a pressure-based solver within Ansys Fluent for our transient simulations of the flow field in the gearbox. The pressure-velocity coupling was achieved using the SIMPLE algorithm. The realizable k-epsilon turbulence model is utilized to simulate the fluid within the gearbox for turbulence modeling, as it provides a good balance between accuracy and computational cost for complex industrial flows. The finite volume method was used to discretize the governing equations. Additionally, a first-order implicit transient formulation was used for time discretization [36].  
Concurrently, the VOF two-phase flow model is employed to simulate the flow field within the gearbox, wherein the primary phase is air with a density of 1.225 kg/m\({}^{3}\) and a dynamic viscosity of 1.7894 \(\times\) 10\({}^{-5}\) kg/(m\(\cdot\)s), while the secondary phase is lubricating oil with a density of 850 kg/m\({}^{3}\) and a dynamic viscosity of 0.04 kg/(m\(\cdot\)s). The oil injection port is set as a velocity inlet, with an inlet velocity of 15 m/s, and the oil outlet is set as a pressure outlet, with an outlet pressure of 0 Pa. The driving gear rotates clockwise at a speed of 8000 rpm, and the driving gear rotates counterclockwise at a speed of 5405 rpm. The gearbox wall and gear surfaces are set as no-slip solid walls, with the rotation of the gears achieved through dynamic meshing technology. To improve the resolution of the boundary layer, we generated prism layers near the solid wall. The number of prism layers  
Figure 4: Internal structure of gearbox: (1) driving wheel, (2) fuel injector, (3) gear box, (4) driven wheel, and (5) oil outlet.  
Figure 3: Three-dimensional model of involute variable tooth thickness gear: (1) driving wheel and (2) driven wheel.  
is 3, and the growth rate of the prism layers is set to 1.1 to ensure a smooth transition of the grid size. The initial value of non-dimensional y+ was set to 45, and the first layer height of the boundary layer was calculated as 0.91 mm; the total thickness of the prism layers is 3.0121 mm.  
In order to reduce the impact of time steps on the results, this paper selected five-time steps, 1000 steps, 5000 steps, 10,000 steps, 15,000 steps, and 20,000 steps, for time independence verification. The time step size was set to \(1\times 10^{-5}\) s, and the above four time steps were used as independent variables to simulate and calculate the oil volume fraction at a monitoring point above the detection gear meshing. The specific verification data are shown in Figure 5. From the data in the figure, it can be seen that when the time step is set to 1000 and 5000 steps, the change in oil volume fraction is not very stable. At 15,000 and 20,000 steps, the oil volume fraction at the monitoring point also slightly increases, but the increase is significantly reduced. Considering the calculation time and efficiency, 10,000 steps were ultimately chosen as the simulation calculation time step, and the CFD transient calculation time is 0.1 s.