# Review on the Recent Numerical Studies of Liquid Atomization
## 3 Interface Capturing
### Mathematical Description  
As a single-fluid Eulerian method, the gas and liquid phases in VOF and LS share the same conservation equations, which are shown as follows:  
\[\frac{\partial\rho}{\partial t}+\
abla\cdot\left(\rho\overset{\cdot}{U} \right)=0 \tag{1}\] \begin{table}
\begin{tabular}{c c c} \hline
**Name** & **Abbreviation** & **Expression** \\ \hline Weber number & _We_ & \(\frac{\partial t^{2}d}{\beta_{s}}\) \\ Ohnesorge number & _Oh_ & \(\sqrt{\rho_{s}d\sigma}\) \\ Density Ratio & \(\varepsilon\) & \(\frac{\rho_{s}}{\rho_{s}}\) \\ Viscosity Ratio & \(N\) & \(\frac{\mu_{s}}{\mu_{e}}\) \\ Reynolds number & _Re_ & \(\frac{\partial t\Delta d}{\mu}\) \\ Momentum Ratio & \(q\) & \(\frac{\rho_{s}U_{s}^{2}}{\rho_{s}U_{s}^{2}}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Dimensionless groups in liquid atomization. \[\frac{\partial\left(\rho\overset{\rightharpoonup}{U}\right)}{\partial t}+\
abla \cdot\left(\rho\overset{\rightharpoonup}{U}\times\overset{\rightharpoonup}{U} \right)=-\
abla P+\
abla\cdot T+\rho\overset{\rightharpoonup}{g}+\overset{ \rightharpoonup}{F_{\sigma}} \tag{2}\]  
\[\frac{\partial\left(\rho e\right)}{\partial t}+\
abla\cdot\left(\rho\overset{ \rightharpoonup}{U}\left(e+\frac{p}{\rho}\right)\right)=-\
abla\cdot q+\
abla \cdot\left(T\cdot\overset{\rightharpoonup}{U}\right)+\kappa\sigma\
abla a\cdot \overset{\rightharpoonup}{U} \tag{3}\]  
where \(U\) is velocity, \(\alpha\) is the volume fraction of the liquid phase, \(\rho\) is density, \(P\) is pressure, \(\sigma\) is the surface tension coefficient, \(\kappa\) is the local interface curvature, \(e\) is the total energy per unit mass, \(g\) is gravity, \(T\) is the viscous stress tensor, which can be obtained from [PERSON] internal friction law, \(q\) is the heat flux vector, which can be obtained from Fourier's law, and \(F_{\sigma}\) is the surface tension force, which can be calculated using the continuous surface force (CSF) method as follows [36]:  
\[\overset{\rightharpoonup}{F_{\sigma}}=\kappa\sigma\overset{\rightharpoonup }{n} \tag{4}\]  
where \(n\) is the interfacial normal vector. The interface normal vector and curvature are calculated as:  
\[\overset{\rightharpoonup}{n}=\
abla\alpha,\text{ and }\kappa=\
abla\cdot \overset{\rightharpoonup}{n} \tag{5}\]  
As a single set of N-S equations are employed for two fluids, the properties including density, thermal conductivity and viscosity must be replaced by fraction-weighted or mass-weighted average value. In the VOF method, volume fraction of 1 and 0 represent liquid and gas phases, respectively, while intermediate values represent the interface. The tracking of the interfaces is accomplished by the solution of a continuity equation for the volume fraction of secondary (liquid) phase, as shown in Equation (6), and the volume fraction of primary (gas) phase can be taken as 1-\(\alpha\). \[\frac{\partial\alpha\epsilon\rho_{l}}{\partial t}+\
abla\cdot\left(\alpha \rho_{l}\overset{\rightharpoonup}{U_{l}}\right)=0 \tag{6}\]  
In the LS method, the interface is represented by the zero value of a level set function \(\varphi(x,\,t)\)[37], which equals the signed distance \(d\) to the nearest interface location. The evolution of \(\varphi\) can be expressed similar to the volume fraction:  
\[\frac{\partial\varphi}{\partial t}+\
abla\cdot\left(\varphi\overset{ \rightharpoonup}{U_{l}}\right)=0 \tag{7}\]  
The momentum equation is the same with Equation (2), and the surface tension force can be given by:  
\[\overset{\rightharpoonup}{F_{\sigma}}=-\kappa\sigma\delta(\varphi)\overset{ \rightharpoonup}{n} \tag{8}\]  
where \(\delta(\varphi)\), n, and \(\kappa\) can be expressed by:  
\[\delta(\varphi)=\left\{ \begin{array}{ll}0,&|\varphi|>a\\ \frac{1+\cos\left(\frac{\pi a}{a}\right)}{2a},&|\varphi|<a\end{array} \right.