# Review on the Recent Numerical Studies of Liquid Atomization
## 5 Hybrid of Interface Capturing and Particle Tracking
### Application Cases  
[PERSON] et al. [63] proposed a criterion to judge if a connect liquid domain or so-called liquid inclusion should be transferred into Lagrangian particles. Taking 3D simulation for example, \(N_{1}\) was noted as the number of cells in a liquid inclusion. If \(N_{1}\) is less than 8, the inclusion is automatically treated as a Lagrangian phase and if \(N_{1}\) is larger than 216, the inclusion is still treated by the Eulerian CLSVOF method until a possible stabilization or fragmentation stemming from Eulerian calculation occurs. For liquid inclusion \(8<N_{1}<216\), a sphericity criterion is used. \(r_{\text{p}}\) is taken as the radius of the sphere of equivalent volume to the liquid inclusion, \(V_{1}\) is the inclusion volume,and \(\delta_{\rm V}\) are defined as the difference of characteristic length and volume, respectively, as shown below:  
\[\left\{ \begin{aligned} \delta_{l}=r_{p}-L\\ \delta_{v}=V_{cell}\sum{x_{i}}\end{aligned} \right. \tag{33}\]  
where \(L\) is the smallest distance between the sphere center and inclusion interface, \(i\) is the cell index for each grid of the inclusion, \(V_{\rm cell}\) is the volume of each grid, and \(\chi\) is 1 if the cell is located outside the sphere and 0 if inside the sphere. Upon the above definition, one liquid inclusion can be judged as Lagrangian particle if:  
\[\frac{\delta_{l}}{r_{p}}<C_{a}\frac{\delta_{V}}{V_{l}}<C_{b} \tag{34}\]  
where \(C_{\rm a}\) and \(C_{\rm b}\) are coefficients being set as 0.5 and 1, respectively. [PERSON] et al. [64] performed a comparison study between VOF and fine grid volume tracking (FGVT) methods with their in-house MFSim code. They focused on the primary atomization of liquid jet in crossflow. LES turbulence model was adopted in the Eulerian region and a simple transition criterion to Lagrangian region was based on two prerequisites show below:  
\[V_{drop}<V_{criterion}\ \&\ r_{max}<2\sqrt[3]{\frac{3}{4\pi}V_{drop}} \tag{35}\]  
The transition occurs if the continuous liquid ligament has a \(V_{\rm drop}\) smaller than \(V_{\rm criterion}\), and its maximum radius \(r_{\rm max}\) must be less than twice that of the sphere of equivalent volume. [PERSON] et al. [65] analyzed the atomizing powder preparation of 24 CrNiMoY alloy steel for additive manufacturing. For the primary atomization very close to the nozzle exit, LES was cooperated into VOF to resolve the interface, and for the secondary atomization DPM was used together with SST \(k\)-\(\omega\) model. The secondary breakup was also accounted for by the competition between breaking mechanisms of KH and TAB breakup models. Two grid systems with different grid refinement levels were employed for the VOF and DPM, respectively. The transition from Eulerian phase to Lagrangian was through sampling the  
Figure 4: Two different ways of transiting the liquid Eulerian phase to Lagrangian phase. (**a**) transiting in the whole region. (**b**) transiting after a distance. particles near the end of primary atomization, saving to an extra file, and importing that particle information to another simulation. The coupling between VOF and DPM was further improved to a twice incorporation manner by [PERSON] et al. [66] in their powder preparation of AlSi10 Mg. The VOF in the Fluent was used to take into account the liquid shear and primary atomization, after which the particles from three main injections were sampled and imported to a subsequent DPM simulation. In the DPM, the TAB model was used regarding the possible secondary breakup. After the breakup finishes, each droplet with the information from the DPM was once again imported into a VOF simulation, which only covers a smaller area with the same gas flow field extracted from the full-field simulation. The solidifying powders including hollow, satellite, and irregular powders, which are all unwished defects in the powder production, were analyzed, and some measures to avoid these defects were proposed.