# Large-Eddy Simulation of Droplet Deformation and Fragmentation Under Shock Wave Impact
## 1 Introduction  
Droplet aerobreakup, also known as secondary atomization, refers to the disintegration of liquid droplets induced by high-speed gas flows. This phenomenon poses a complex challenge in multiphase flow investigations owing to its highly transient characteristics spanning both spatial and temporal dimensions. Atomization driven by aerodynamic forces is critically important in various engineering and industrial applications, including liquid jet atomization, agricultural spraying, energy systems, and nuclear fusion processes [1, 2, 3, 4]. Also, the breakup of liquid droplets in high-pressure airflows is particularly important during compression strokes in internal combustion engines and compressors in gas turbines [5]. Additionally, aerodynamic breakup triggered by passing shock waves is a fundamental aerospace engineering concern, with advanced applications ranging from fuel injection in rocket engines to erosion damage in supersonic flights [6]. Understanding droplet deformation and breakup is especially important in applications where increased airflow pressure adds complexity to droplet behavior. Thus, gaining a deeperunderstanding of droplet deformation and breakup is crucial under both atmospheric and higher-pressure conditions. In the last decades, extensive experimental and numerical research has been carried out to investigate droplet aerobreakup. Various classifications of droplet breakup regimes are documented in the literature, defined by critical dimensionless parameters. The latter include the Weber number (\(We\)), which characterizes the predominance of aerodynamic (inertial) forces over surface tension (restorative) forces in free-stream conditions, and the Ohnesorge number (\(Oh\)), which accounts for the influence of liquid viscosity on breakup dynamics. These two parameters are defined as follows:  
\[We=\frac{\rho_{g}u_{g}^{2}D_{0}}{\sigma},\] \[Oh=\frac{\mu_{l}}{\sqrt{\rho_{l}\sigma D_{0}}},\]  
depending on factors such as initial droplet diameter (\(D_{0}\)), surface tension coefficient (\(\sigma\)), post-shock gas density (\(\rho_{g}\)) and velocity (\(u_{g}\)), liquid density (\(\rho_{l}\)), and dynamic viscosity (\(\mu_{l}\)). For instance, [PERSON] et al. [7] documented the breakup of droplets through the bag and stamen process at elevated Weber numbers for both viscoelastic and Newtonian fluids. Importantly, as \(Oh\) increases, a delay in the deformation and breakup process is observed, with the viscosity resulting in having minimal influence for \(Oh<0.1\)[8]. Several comprehensive reviews, such as those by [PERSON] [9] and [PERSON] [10], have individuated various modes of droplet fragmentation, along with their associated criteria for regime transitions. Traditional works have identified five aerobreakup modes, characterized by distinct ranges at an increasing Weber number, namely, bag breakup, bag-stamen/plume breakup, multibag breakup, sheet thinning/stripping breakup, and catastrophic breakup [11; 12; 13]. However, some breakup investigations conducted for air-jet flows, at moderately low \(We\) values, have revealed multimode structural formations due to Rayleigh-Taylor instability (RTI). Furthermore, the transition from other modes to shear breakup regime, distinguished by equatorial liquid sheet formation, was demonstrated to be influenced by viscosity and the density ratios between the two phases [8]. Therefore, also based on these studies, various breakup modes have been reclassified taking into account the hydrodynamic instabilities responsible for the physical phenomenon [10]. Specifically, for low Weber numbers (\(We<10^{2}\)), the previously identified bag, bag-stamen, and multibag breakup modes have been consolidated into the single Rayleigh-Taylor piercing (RTP) mode, attributed to RTI. Moreover, at higher Weber numbers (\(We>10^{3}\)), the previously identified sheet thinning/stripping and catastrophic modes have been combined into the shear-induced entrainment (SIE) regime, associated with Kelvin-Helmholtz (KH) instability.