# Computational Evaluation of Shock Wave Interaction with a Liquid Droplet
## 1 Introduction  
The aerodynamic breakup (aerobreakup) of liquid droplets into smaller fragments induced by the interaction with a passing shock wave (SW) is of crucial importance for many industrial gas dynamics applications [1, 2]. These include, for example, supersonic combustion air-breathing jet engines (scramjets) and raindrop distortion and demise in the flow field around supersonic acirrafts. In the former case, mixing and combustion rates of liquid fuel droplets are greatly improved as a result of the droplet fragmentation induced by detonation waves [3]. In the latter case, erosion caused by the impingement of rain droplets at high relative speeds on aircraft surfaces, is greatly alleviated through proper aerodynamic design [4]. The shock/droplet interaction plays a very important role as it represents the initial stage of the aerobreakup induced by the high-speed gas stream. Several studies have been conducted to investigate this process for correctly interpreting the underlying physical mechanisms. In particular, a number of experimental studies employing shock tube devices have been carried out, wherein a traveling planar SW is reproduced, with uniform gaseous flow conditions being established [5, 6]. In these experiments, sophisticated visualization techniques are employed, where the shock front passes over liquid droplets causing their deformation, drift and breakup, due to sudden acceleration. Theoretically, the physics of aerobreakup is controlled by the Ohnesorge number (Oh), which compares the effects of liquid viscosity and surface tension, along with the Weber number (We), which compares disruptive aerodynamic and restorative surface tension effects. At very low Ohnesorge numbers (Oh \(\ll 1\)), the influence of the liquid viscosity can be neglected so that the process is practically governed by the Weber number [7]. Previously based on a large number of different classifications of droplet crushing regimes, the interpretation and identification of the aerobreakup mechanism have been recently reviewed and greatly simplified [8]. Following this new classification, only two principal modes of fragmentation exist, according to the underlying physical mechanism of the governing interfacial instabilities. At low Weber numbers (\(10<\text{We}<10^{2}\)), the Rayleigh-Taylor instability has to be considered the main driving mechanism for the aerobreakup, which is referred to as Rayleigh-Taylor piercing (RTP) regime. Differently, at high Weber numbers (\(\text{We}>10^{3}\)), the breakup process is governed by the Kelvin-Helmholtz instability of the stretched liquid sheets that are formed at the periphery of the deforming droplet. This terminal regime, which is referred to as shear-induced entrainment (SIE), involves the disruption of the liquid surface layer with subsequent liquid sheet stripping and thinning, along with the spraying of stripped liquid [9]. From a numerical point of view, since the droplet aerobreakup under the SIE regime is characterized by a broad range of spatial and temporal scales, the accurate capture of the droplet deformation and fragmentation is particularly demanding in terms of computational power. Nevertheless, computational fluid dynamics (CFD) simulations have been effectively conducted to predict the early stages of the breakup process induced by the impact of a traveling SW. Due to the high computational cost of fully three-dimensional (3D) simulations, several numerical studies have dealt with two-dimensional shock/liquid column interactions [10; 11; 12; 13; 14]. In order to make them affordable, in the context of preliminary analyses, 3D computations have been most often conducted by neglecting the effects of molecular viscosity, while resolving the Euler equations [15; 16]. Typically, compressible multiphase solvers based on a five-equation model [17] are used to computationally study shock/droplet interactions, where the interface between liquid and gas is modeled using volume fractions.