# Review on the Recent Numerical Studies of Liquid Atomization
## 6 Multi-Fluid and Multi-Scale Eulerian Method
### Theoretical Bases  
The transitions from continuous fluids to dispersed fluids are bidirectional, and there can be momentum transfer between the couples of continuous liquid and continuous gas, continuous liquid and dispersed gas, and continuous gas and dispersed liquid. The control equations for the fluids, the mass and momentum transfer among fluids, as well as the breakup and coalescence of the different scales of dispersed phases are all in the Eulerian framework. It relies on various semi-empirical correlations to obtain the mass transitions and interface scale transformation. A high-resolution turbulence model was usually unnecessary, and thus this method is computationally inexpensive. The momentum interactions exerted by the different interface topologies between continuous liquid and continuous gas can be dealt with by an algebraic interfacial area density (AIAD) model [69]. Regardless of the specific interface geometry, the volumetric drag force density in between is expressed as:  
\[\left|M_{cl}\right|=\left|M_{cg}\right|=0.5C_{D}A\rho\left|\mathbf{U}\right|^{2} \tag{36}\]  
where \(C_{\text{D}}\) is the dimensionless drag coefficient, \(A\) is the interfacial area density, \(\mathbf{U}\) is the relative velocity and \(\rho\) is the volume fraction-weight averaged density of the two phases. \(M_{\text{cl}}\) and \(M_{\text{cg}}\) are the interfacial forces exerted to each phase by another phase, of course through the interface. The AIAD mixes the contributions of three interface topologies, namely dispersed droplets, dispersed bubbles, and continuous surface into interfacial area density A and drag coefficient \(C_{\text{D}}\) by three blending functions [70], as shown below:  
\[ \begin{cases}f_{D}=\left[1+e^{a_{\text{D}}\left(\alpha_{cl}-\alpha_{\text{D}, \text{limit}}\right)}\right]^{-1}\\ f_{B}=\left[1+e^{a_{\text{B}}\left(\alpha_{cg}-\alpha_{B,\text{limit}} \right)}\right]^{-1}\\ f_{\text{FS}}=1-f_{D}-f_{B}\end{cases}  \tag{37}\]  
where D, B, and FS represent droplet, bubble, and continuous free surface, respectively, and \(\alpha_{\text{cl}}\) and \(\alpha_{\text{cg}}\) are the volume fraction of continuous liquid and gas. Note that since there are more than two phases, the summation of \(\alpha_{\text{cl}}\) and \(\alpha_{\text{cg}}\) is less than unit. The other constants usually have the following value: \(a_{\text{D}}=a_{\text{B}}=70\) and \(\alpha_{\text{D,limit}}=\alpha_{\text{B,limit}}=0.3\). By adopting proper models for the interfacial area densities and drag force coefficients for each of the three interface topologies, the \(C_{\text{D}}\) and \(A\) in Equation (36) can be obtained by their summation weighted by the blending functions in Equation (37). The momentum interactions between a continuous phase and a dispersed phase (continuous liquid with bubbles, continuous gas with droplets) can still be taken care of by the particle model. Comparing the relative magnitudes, drag force, lift force, and turbulent dispersion force are usually considered to be important, and their formulations are no difference from those in particle tracking method. The disintegration of droplets from the continuous liquid or the primary breakup has to be accounted for by sub-grid models since the coarsen grid system is not precise enough to produce exact interface structure. According to the theory of [PERSON] et al. [71], a droplet with size of the eddy can be disintegrated out of continuous surface if the addition of turbulent energy of an eddy and the aerodynamic energy exerted by the flowing gas can overcome the surface tension energy, which can be written as  
\[CR=\Big{(}\rho_{cl}v_{fl}^{2}+C_{a}\rho_{cg}u_{g}^{2}\Big{)}\lambda^{3}-C_{s} \sigma\lambda^{2}>0 \tag{38}\]  
where \(C_{a}\) and \(C_{s}\) are empirical constants. \(v_{fl}\) is the velocity fluctuations vertical to the surface of the continuous liquid.