# Numerical and Experimental Analysis of the Oil Flow in a Planetary Gearbox
## 2 Materials and Methods
### Mathematical Description and Governing Equations  
The CFD code used to perform the simulations is based on OpenFOAM(r) v7 [19], a Finite Volume Method (FVM) [27] software. The complexity of the two-phase oil-air splash lubrication in the planetary gearbox was modelled with the Volume of Fluid (VOF) model [28]. The sum of the two fluids volume ratios (\(\alpha_{oil}\) and \(\alpha_{air}\)) in each control volume is equal to 1:  
\[\alpha_{oil}+\alpha_{air}=1 \tag{1}\]  
When \(\alpha_{oil}=1\), the control unit is full of oil; when \(\alpha_{oil}=0\), the control unit is filled with air; when \(0<\alpha_{oil}<1\), an oil-air interface is contained in the control unit. The solution of the continuity equation of the volume fraction allows to trace the oil-air interface.  
\[\frac{\partial\alpha}{\partial t}+\
abla(\;\alpha\mathbf{u})=0 \tag{2}\]  
The Multidimensional Universal Limiter with Explicit Solution (MULES) [29] algorithm was used to achieve a better boundedness of the volume fraction field. This is achieved by adding a dummy velocity field (\(\mathbf{u}_{c}\)) acting perpendicular to the interface to the conservation equation of the \(\alpha\) field:  
\[\frac{\partial\alpha}{\partial t}+\
abla(\alpha\mathbf{u})+\
abla(\;\mathbf{u}_{c} \alpha(1-\alpha))=0 \tag{3}\]  
The generic property \(\phi\) (as viscosity and density) of the equivalent fluid is calculated as:  
\[\phi_{eq}=\alpha_{oil}\;\phi_{oil}+(1-\alpha_{oil})\;\phi_{air} \tag{4}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Unit** & **Value** \\ \hline Kinematic viscosity (40 \({}^{\circ}\)C) & mm\({}^{2}\)/s & 95 \\ Kinematic viscosity (100 \({}^{\circ}\)C) & mm\({}^{2}\)/s & 10.7 \\ Density (40 \({}^{\circ}\)C) & kg/m\({}^{3}\) & 864 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Lubricant properties FVA3A.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Simulation** & **Lubricant** & **Planet Carrier** & **Planet Gears** & **Tangential Speed at** \\
**Name** & **Filling Level** & **Speed n\({}_{\text{t}}\) in rpm** & **Speed n\({}_{\text{p,abs}}\) in rpm** & **Pitch Circle v\({}_{\text{t}}\) in m/s** \\ \hline SIM-id1v1 & & 81 & 132 & 0.8 \\ SIM-id1v2 & 3-m\({}_{\text{n}}\) & 162 & 263 & 1.6 \\ SIM-id1v3 & & 324 & 526 & 3.2 \\ \hline SIM-id2v1 & & 81 & 132 & 0.8 \\ SIM-id2v2 & centerline & 162 & 263 & 1.6 \\ SIM-id2v3 & & 324 & 526 & 3.2 \\ \hline \hline \end{tabular} \({}^{\text{*}}\)SIM\({}^{\text{*}}\) stands for simulation, \"id\" for immersion depth, and \"v\" for velocity. As already mentioned, two immersion depths and three rotational velocities were analyzed. Therefore, \"id\" has values of 1 and 2, while \"v\" has values of 1, 2, and 3.  
\end{table}
Table 3: Operating conditions.  
The fluid-dynamic problem requires solving of the continuity and the momentum equations for the oil-air mixture:  
\[\frac{\partial\rho}{\partial t}+\
abla(\rho\mathbf{u})=0 \tag{5}\]  
\[\frac{\partial(\rho\mathbf{u})}{\partial t}+\
abla(\rho\mathbf{uu})=-\
abla p+\
abla \left[\mu\Big{(}\
abla\mathbf{u}+\
abla\mathbf{u}^{T}\Big{)}\right]+\rho g+F \tag{6}\]  
where \(\rho\) is the density, \(\mathbf{u}\) is the fluid velocity, \(\mu\) is the viscosity, \(\mathbf{g}\) is the gravity acceleration, and \(F\) represents the vector of external forces. The weighted average properties are used so that the two conservation equations are associated with the continuity equation of the volume fraction.