# Large-Eddy Simulation of Droplet Deformation and Fragmentation Under Shock Wave Impact
## 2 Hybrid Eulerian-Lagrangian Formulation
### VOF Method  
The VOF method [22] represents the most widely utilized technique among algorithms relying on continuous volume markers. Its popularity derives from the simple implementation and high efficiency, while it has demonstrated excellent performance in calculating interfacial flows. The core concept involves treating the liquid and gas as a single two-phase medium, where a special marker function \(F(x,y,z,t)\) defines the space-time distribution of the phases within the computational domain. The volume fraction of the liquid phase within a computational cell is characterized as follows: \(F=0\) indicates a cell without liquid, \(F=1\) denotes a cell entirely filled with liquid, and \(0<F<1\) represents a cell containing the air-liquid interface. The movement of the interface is tracked by solving the following transport equation for the liquid volume fraction, as the interface moves along with the liquid:  
\[\frac{\partial F}{\partial t}+\
abla\cdot(F\mathbf{V})=0, \tag{1}\]  
where \(\mathbf{V}\) is the velocity vector of the two-phase medium. The latter is determined by solving a system of compressible flow governing equations, including the mass conservation equation  
\[\frac{\partial\rho}{\partial t}+\
abla\cdot(\rho\mathbf{V})=0, \tag{2}\]  
and the momentum balance equation  
\[\frac{\partial(\rho\mathbf{V})}{\partial t}+\
abla\cdot(\rho\mathbf{V})=- \
abla p+\
abla\cdot\mathbf{\sigma}^{v}+\mathbf{F}^{S}. \tag{3}\]  
Here, \(\rho\) is the density of the two-phase medium, \(p\) is the static pressure, \(\mathbf{\sigma}^{v}\) is the viscous stress tensor, and \(\mathbf{F}^{S}\) is the volumetric force vector field. The components of the viscous stress tensor are expressed as follows:  
\[\sigma_{ij}^{v}=\mu\left(\frac{\partial V_{i}}{\partial x_{j}}+\frac{\partial V _{j}}{\partial x_{i}}\right)-\frac{2}{3}\mu\frac{\partial V_{k}}{\partial x_{ k}}\delta_{ij}, \tag{4}\]  
where \(\mu\) represents the dynamic viscosity of the two-phase medium. The local density and molecular viscosity of the mixture fluid are determined using the volume fraction of the liquid in the cell, according to the mixture rule, namely:  
\[\rho=F\rho_{l}+(1-F)\rho_{g}, \tag{5}\]  
\[\mu=F\mu_{l}+(1-F)\mu_{g}, \tag{6}\]  
where \(\rho_{l}\) and \(\mu_{1}\) are the density and viscosity of the liquid, and \(\rho_{g}\) and \(\mu_{g}\) are the density and viscosity of the gas, respectively. When analyzing fluid flows with a gas-liquid interface, special consideration has to be given to the phenomenon of surface tension. In fact, studying flows affected by surface tension forces represents a complex and distinct task. One of the benefits of the VOF method is in the ability to facilitate the relatively straightforward inclusion of surface tension effects. The typical continuous surface force (CSF) algorithm proposed by [PERSON] et al. [23] is employed here to simulate surface tension within the current VOF framework. This algorithm introduces the additional volumetric force \(\mathbf{F}^{S}\) into the momentum equation, which is determined by the following relation:  
\[\mathbf{F}^{S}=\sigma\kappa\
abla F, \tag{7}\]where \(\sigma\) is the air-liquid surface tension coefficient, and \(\kappa\) is the curvature of the free surface, defined as the divergence of the normal unit vector, namely:  
\[\kappa=\
abla\cdot\frac{\mathbf{n}}{|\mathbf{n}|}. \tag{8}\]  
Finally, the energy conservation equation reads  
\[\frac{\partial(\rho E)}{\partial t}+\
abla\cdot\left(\rho\mathbf{V}(E+\frac{p} {\rho})\right)=\
abla\cdot(\lambda\
abla T), \tag{9}\]  
where \(E\) represents the total energy, which is the sum of the internal and kinetic energies. To account for the thermal effects, the thermal conductivity and heat capacity of the two-phase medium are defined using the mixing rule:  
\[\lambda=F\lambda_{l}+(1-F)\lambda_{g},\quad C_{p}=FC_{pl}+(1-F)C_{pg}, \tag{10}\]  
where \(\lambda_{l}\) and \(\lambda_{g}\) are the thermal conductivities of the liquid and gas phases, respectively, and \(C_{pl}\) and \(C_{pg}\) are the corresponding heat capacities.