# Large-Eddy Simulation of Droplet Deformation and Fragmentation Under Shock Wave Impact
## 1 Introduction
In fact, the above-mentioned hydrodynamic instabilities are crucial for determining droplet morphology during the aerodynamic atomization process. RTI typically drives breakup modes involving bag formation, with various types of bag breakups observed. For instance, recent studies by [PERSON] and co-workers [10; 14] have utilized finite-thickness RTI analysis to identify the conditions leading to initial criticality for the bag formation process. In contrast, shear-induced droplet breakup is driven by KH instability on the liquid droplet surface [14; 15]. Several studies have explored the complex process of liquid droplet breakup using the shock tube technique. For instance, [PERSON] et al. [16] combined experimental and numerical methods to investigate the dynamics of reflected, transmitted, and diffracted waves during the interaction of an incident shock wave with a cylindrical water column [17]. Their findings indicated that, at higher shock Mach numbers, the focusing of expansion waves within the droplet could also lead to cavitation. Moreover, a number of numerical simulations have demonstrated the formation of a recirculating flow in the equatorial region, together with an upstream jet flow in the droplet's wake, which promotes droplet deformation and breakup [18; 19; 20]. The rapid aerodynamic fragmentation of liquid droplets, which is particularly pronounced at high Weber numbers, poses significant challenges for numerical simulations due to its complex multiscale nature and short characteristic time scales. Achieving high spatio-temporal resolutions would be essential to observe the transformation of millimeter-sized primary droplets into micron-sized secondary droplets within these brief time intervals. While increasing computational resources can enhance numerical simulations, comprehensive studies that effectively align with experimental data across a range of parameters, and fully capture the evolving dynamics of this phenomenon, are still lacking. Indeed, the limitations and challenges associated with current numerical methodologies for simulating droplet aerobreakup highlight a significant gap in the literature regarding a detailed analysis of dispersed mist development under the SIE regime. This gap primarily arises from the prevalence of multiscale structures and the substantial computational costs required for thorough analyses. Despite extensive research, the exploration of these fundamental mechanisms remains limited, and comprehensive three-dimensional numerical investigations under high-pressure conditions are relatively rare. Direct numerical simulation (DNS), with a full resolution of the phase interface, would be the only universal tool. However, due to the associated immense computational cost, DNS can be applied only to a very limited class of simple model problems, while being unaffordable for applications of engineering interest. Moreover, some limitations of the commonly employed volume of fluid (VOF) method are well-documented, particularly in resolving fine-scale features and accurately capturing the interface dynamics of sub-droplets. These limitations include interface smearing, resolution dependency, and a restricted particle tracking capability. In this complex scenario, the main aim of this study was to test a hybrid Eulerian-Lagrangian approach, combining VOF with the discrete phase model (DPM), e.g., [21]. This method exploits the VOF technique's strength in capturing initial breakup dynamics, together with the DPM's ability to accurately track individual sub-droplets. The proposed computational model makes use of the large-eddy simulation (LES) approach, where the large turbulent structures are resolved, while modeling the effect of unresolved small turbulent eddies. The LES-based hybrid VOF-DPM method allows us to explore the multiscale characteristics of droplet aerobreakup at a feasible computational cost.