# Computational Evaluation of Shock Wave Interaction with a Cylindrical Water Column
## 1 Introduction  
The aerodynamic breakup process of liquid droplets induced by the interaction with plane shock waves is of crucial importance for many industrial gas dynamics applications. The latter include, just to mention a couple of them, supersonic combustion airbreathing jet engines (scramjets) and rain impact damage to supersonic aircraft [1, 2]. The shock wave interaction with a water droplet actually represents the initial stage of the breakup process induced by the high-speed gas stream. Since this process plays an important role in the droplet breakup, a number of studies have been conducted to investigate the shock/droplet interaction for correctly interpreting the mechanism of aerobreakup. In particular, this process has been the subject of several experimental studies employing shock tube devices, wherein a traveling planar shock wave is reproduced, with uniform gaseous flow conditions being established, e.g., [3, 4]. In these experiments, the shock front passes over the droplet and causes its deformation and successive breakup. The physical mechanism of aerobreakup is controlled by the Ohnesorge number (Oh), which compares liquid viscosity to surface tension effects, as well as the Weber number (We), which compares disruptive aerodynamic forces to restorative surface tension ones. However, at low Ohnesorge numbers (Oh \(<0.1\)), the influence of liquid viscosity can be neglected, and the process is governed essentially by the Weber number [5]. The interpretation and identification of breakup mechanisms have been recently reviewed, based on experimental studies using highly resolved images of liquid droplets suddenly exposed to supersonic gas streams [6]. According to this reclassification, at low Weber numbers (\(10<\text{We}<10^{2}\)), Rayleigh-Taylor instability has to be considered the main driving mechanism for the aerobreakup, which is referred to as Rayleigh-Taylor piercing (RTP). Differently, at high Weber numbers (\(\text{We}>10^{3}\)), significant shear-induced motions exist, and the breakup process is governed by the instability of the stretched liquid sheets thatare formed at the periphery of the deforming droplet. This regime is referred to as shear-induced entrainment (SIE) [7]. Another non-dimensional parameter that is sometimes considered when examining this process is the ratio between liquid and gas densities. As for other challenging engineering applications, computational fluid dynamics (CFD) can be effectively employed to predict the early stages of the aerobreakup of a water droplet due to the impact of a traveling shock wave. In the context of preliminary analyses, the shock-droplet interaction can be evaluated by considering a water cylinder with a circular cross-section in the high-speed flow behind a planar shock front. The cylindrical geometry of the water body can be efficiently modeled using a two-dimensional computational domain that provides shorter turnaround times compared to three-dimensional simulations. Several numerical studies exist in the literature dealing with two-dimensional shock-column interactions, e.g., [8; 9; 10], where the sheet-thinning process was simulated and the column deformation and drift were measured. Usually, such works focus on the early stages of aerobreakup and neglect the effects of both surface tension and molecular viscosity. The two-dimensional approach is also justified based on experimental investigations of liquid columns that allow for easier visualization of the wave structures [11; 12]. Indeed, in the early stages of the interaction process, the deformation of two-dimensional liquid columns has been found to be very similar to that of three-dimensional spherical droplets [13]. Compressible multi-phase solvers based on a five-equation model [14] are often developed to computationally study shock-water column interactions, where the interface is modeled using volume fractions.