# Large-Eddy Simulation of Droplet Deformation and Fragmentation Under Shock Wave Impact
## 2 Hybrid Eulerian-Lagrangian Formulation
### LES Approach  
As far as fluid turbulence is concerned, in this work, the LES approach is followed, where part of the turbulent scales are directly resolved, while modeling the effect of the residual part. Specifically, the spatially filtered Navier-Stokes equations are numerically solved [24], which allows us to accurately describe the space-time evolution of the mixture flow with affordable computational complexity. In contrast, the Reynolds-averaged Navier-Stokes (RANS) models provide only averaged flow characteristics and do not adequately capture the development of instabilities in droplet-shock wave interactions, e.g., [25]. Note that classical numerical estimates for this particular two-phase flow problem indicate that DNS calculations would require computational grids with hundreds of millions of cells for typical configurations, while LES represents a viable numerical technique for aerobreakup studies, e.g., [26; 27].  
The system of compressible LES governing equations can be expressed as follows:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial(\rho\overline{u}_{i})}{\partial x _{i}}=0, \tag{11}\]  
\[\frac{\partial(\rho\overline{u}_{i})}{\partial t}+\frac{\partial}{\partial x_ {j}}(\rho\overline{u}_{i}\overline{u}_{j})=-\frac{\partial\overline{p}}{ \partial x_{i}}+\frac{\partial\overline{v}_{ij}}{\partial x_{j}}+\frac{ \partial\overline{v}_{ij}}{\partial x_{j}}+F_{i}^{S}, \tag{12}\]  
where \(\overline{\sigma}_{ij}\) is the filtered viscous stress tensor, which is defined in terms of resolved velocity gradients by  
\[\overline{\sigma}_{ij}=\mu\left(\frac{\partial\overline{u}_{i}}{\partial x_{j }}+\frac{\partial\overline{u}_{j}}{\partial x_{i}}\right)-\frac{2}{3}\mu \frac{\partial\overline{u}_{k}}{\partial x_{k}}\delta_{ij}. \tag{13}\]  
The tensor components \(\tau_{ij}\), which are referred to as subgrid scale (SGS) stresses, represent the effect of unresolved turbulent fluctuations upon the dynamics of resolved turbulent eddies. Analogous to RANS models, these unknown quantities can be determined using the Boussinesq approximation:  
\[\tau_{ij}-\frac{1}{3}\tau_{kk}\delta_{ij}=-2\mu_{l}\overline{S}_{ij}, \tag{14}\]where \(\overline{S}_{ij}\) is the resolved strain-rate tensor, namely,  
\[\overline{S}_{ij}=\frac{1}{2}\Bigg{(}\frac{\partial\overline{u}_{i}}{\partial x_ {j}}+\frac{\partial\overline{u}_{j}}{\partial x_{i}}\Bigg{)}. \tag{15}\]  
Following the classical Smagorinsky model [28], the SGS viscosity parameter is provided as \(\mu_{t}=\rho L_{S}^{2}|\overline{S}|\), where \(L_{S}\) denotes the mixing length of the (unresolved) subgrid scales, namely:  
\[L_{S}=\min(\kappa d,C_{S}V_{\rm cell}^{1/3}), \tag{16}\]  
where \(\kappa\) is the von Karman constant, \(d\) the distance to the nearest wall, \(V_{\rm cell}\) the volume of the computational cell, and \(C_{S}\) the Smagorinsky coefficient. In this study, employing the dynamic formulation of this model, the latter parameter is not constant but evaluated based on the localized instantaneous flow conditions, improving the accuracy of the LES solution [29, 30]. The dynamic model equations, which are not reported here for brevity, can be found, for instance, in [31], where an assessment of various SGS models, in a similar DPM-LES framework, was performed for a different fluid engineering application, namely, water-reactive fire suppression.