## 2 Numerical Model
### Mathematical Model
#### 2.3.1 Two-Fluid Model  
In two-fluid model [18], the air and oil phase are both treated as interconnected continuous media. As the volume occupied by one phase cannot be occupied by the other, the phase volume fraction is introduced. The volume fraction is a continuous function of space and time, and the sum of the volume fractions of each phase is 1. The volume fraction equation is as follows,  
\[\alpha_{q}=\frac{V_{q}}{V},\;0\leq\alpha_{q}\leq 1 \tag{1}\]  
\[\sum_{q=1}^{n}\alpha_{q}=1 \tag{2}\]  
where \(V\) is the total volume of air oil two-phase, \(V_{q}\) is the volume of the \(q\) phase, \(\alpha_{q}\) is the volume fraction of the \(q\) phase, and \(n\) has two phases: gas is labeled as phase 1 and oil droplet is labeled as phase 2. It should be noted that a two-fluid model can be considered a valid approximation only up to Stokes number \(\sim\)0.1-0.2. A similar set of continuity equations and momentum equations can be derived for each phase. Figure 3: Verification of grid independence, the number of meshes is 487,585, 609,719, and 1,530,258, respectively, when air inflow is 12 g/s, the air phase velocities at four positions (\(-\)0.11,0,0), (\(-\)0.09,0,0), (0.09,0,0), and (0.11,0,0) in z = 0 plane. The continuity equation is  
\[\frac{\partial(\alpha_{q}\rho_{q})}{\partial t}+\
abla\cdot(\alpha_{q}\rho_{q} \mathbf{U}_{q})=0 \tag{3}\]  
where \(\rho_{q}\) is the density of the \(q\) phase; \(\mathbf{U}_{q}\) is the velocity vector of the \(q\) phase. The momentum equation is  
\[ \begin{array}{l}\frac{\partial(\alpha_{q}\rho_{q}\mathbf{U}_{q})}{\partial t} +\
abla\cdot(\alpha_{q}\rho_{q}(\mathbf{U}_{q}\otimes\mathbf{U}_{q}))+\
abla \cdot(\alpha_{q}\mathbf{\tau}_{q})+\
abla\cdot(\alpha_{q}\mathbf{R}_{q})\\ =-\alpha_{q}\
abla p+\alpha_{q}\rho_{q}\mathbf{g}\pm\mathbf{M}_{q}\end{array}  \tag{4}\]  
where \(\mathbf{\tau}_{q}\) is the viscous-stress tensors of the \(q\) phase, \(p\) is the pressure share by gas phase and the liquid phase, \(\mathbf{R}_{q}\) is the Reynolds stress tensors, \(\mathbf{g}\) is the gravitational acceleration vector, and \(\mathbf{M}_{q}\) is the interfacial force term between the gas phase and liquid phase. When q is oil droplet, \(\mathbf{M}_{q}\) is positive; when q is air, \(\mathbf{M}_{q}\) is negative. The term \(\mathbf{M}_{q}\) is usually composed by three forces: drag, virtual mass, and lift. In present work, only the drag force is taken into account, whereas the others are neglected. The drag force is calculated as follows,  
\[\mathbf{M}_{q}=\alpha_{1}\alpha_{2}(\frac{3}{4}C_{D}\frac{\rho_{l}}{d_{32}}| \mathbf{U}_{l})\mathbf{U}_{r} \tag{5}\]  
where \(\mathbf{U}_{r}\)= \(\mathbf{U}_{g}-\mathbf{U}_{l}\) is the slip velocity and \(\mathbf{U}_{g}\), \(\mathbf{U}_{l}\) are the average vector of the air and oil, respectively: \(C_{D}\) is the drag coefficient; \(d_{32}\) is the mean Sauter diameter, calculated as the ratio between the moments of order three and two of the droplet size distribution. In the two-fluid model, the turbulent Reynolds stress should be closed. The Reynolds stress arises from the operation of Reynolds averaging on the momentum equation. In this paper, the \(k\)-\(\varepsilon\) turbulence model [19; 20] is selected to close the turbulence term in two fluid model.