## 2 Numerical Calculation Method
### Euler Gas-Liquid Two-Phase Model
#### 2.1.2 Control Equation  
(1) Volume fraction equation  
The definition of LWC is led in by treating fluid as a continuum \(\alpha_{q}\) (LWC of phase \(q\)), it is the share of space by each phase, and the volume \(V\) of phase \(q\) is obtained from the following formulae [28]:  
\[V=\int\limits_{V}\alpha_{q}dV. \tag{1}\]  
and  
\[\sum\limits_{q=1}^{n}\alpha_{q}=1. \tag{2}\]  
The actual density of phase \(q\) is:  
\[\rho=\alpha_{q}\rho_{q}. \tag{3}\]  
where \(\rho_{q}\) is the density of phase \(q\).  
(2) Conservation equation  
When the Euler-Euler approach is adopted to solve the water-drop motion, the continuity equation and momentum conservation equation of the water drop are as follows [35]:  
\[\
abla\cdot(\rho_{l}\mathbf{\alpha}\mathbf{u})=0. \tag{4}\]\[\
abla(\rho_{l}\alpha\mathbf{u})=\rho_{l}\alpha F+\rho_{l}\alpha(\mathbf{u}_{a}-\mathbf{u})\cdot K. \tag{5}\]  
where \(\rho_{l}\) is the water-drop density, \(\alpha\) is the volume fraction of the water drop, \(\mathbf{u}\) is the velocity vector of the water-drop, \(\mathbf{u}_{a}\) is the velocity vector of the air, \(F\) is the force exerted on the water drop except resistance, and \(K\) is the transmission coefficient obtained from the following formula [35]:  
\[K=\frac{18\mu_{a}f}{\rho_{l}d_{p}{}^{2}}. \tag{6}\]  
where \(\mu_{a}\) is the air viscosity, \(d_{p}\) is diameter, and \(f\) is the resistance function. According to the Schiller-Neumann model, \(f\) can be obtained from the following formula [35]:  
\[f=C_{d}Re/24. \tag{7}\]  
where \(C_{d}\) is the resistance index and \(Re\) is the relative Reynolds number of the water-drop diameter, which is obtained from the following formula [35]:  
\[C_{d}=\begin{cases}24(1+0.15 Re^{0.687})/Re&Re\leq 1000\\ 0.44&Re>1000\end{cases}. \tag{8}\]  
\[Re=\frac{\rho_{a}|\mathbf{u}_{a}-\mathbf{u}|d_{p}}{\mu_{a}}. \tag{9}\]  
where \(\rho_{a}\) is the air density.  
When the Eulerian multiphase flow model is adopted, the influence of turbulent dispersion force on the momentum transfer of two-phase turbulence can be considered. In this paper, the diffusion in the VOF model is used. At this time, the water-drop mass conservation equation can be rewritten as [28]:  
\[\
abla\cdot(\rho_{l}\alpha\mathbf{u})=\
abla\left[\gamma_{q}\
abla\alpha_{q} \right]. \tag{10}\]  
where \(\gamma_{q}\) is the diffusion coefficient of phase \(q\). \(\
abla[\gamma_{q}\
abla\alpha_{q}]\) is the turbulent diffusion and the following conditions shall be met [28]:  
\[\sum_{q=1}^{n}\
abla\left[\gamma_{q}\
abla\alpha_{q}\right]=0. \tag{11}\]  
The diffusion coefficient of the second phase can be determined by the turbulent viscosity \(\mu_{t,q}\) estimate [28]:  
\[\gamma_{q}=\frac{\mu_{t,q}}{\sigma_{q}},q=2\cdot\cdot\cdot n. \tag{12}\]  
where \(\sigma_{q}\) = 0.75.  
\[\mu_{t,q}=\frac{\rho_{a}k}{\omega}\frac{1}{\max\left[\frac{1}{\alpha^{*}}, \frac{SF_{2}}{\alpha_{1}\omega}\right]}. \tag{13}\]  
The transport equation governing the turbulent kinetic energy, denoted as \(k\), and the specific dissipation rate, denoted as \(\omega\), as well as the control parameters and solution strategies can be obtained from the authoritative source referenced as [28].  
The diffusion phase \(D_{q}\) (\(q\) = 1) of the main phase meets the following formula [28]:  
\[D_{q}=-\sum_{p=2}^{n}D_{p}. \tag{14}\]