# Computational Evaluation of Shock Wave Interaction with a Liquid Droplet
## 2 Computational Model
### Case Study  
The virtual shock tube device is built for the generation of well-defined air flow conditions to which the spherical water body is exposed. According to the 3D approach, the geometry of the shock tube is represented by the cylindrical domain sketched in Figure 1. The reference cylindrical coordinates system \((r,\theta,z)\) is chosen with the \(z\)-axis aligned with the streamwise direction, while the origin corresponds to the leading-edge of the water droplet in its initial position. Note that the normal SW, and thus the post-shock air flow, moves in the positive \(z\)-axis direction. Furthermore, based on previous numerical simulations conducted for similar flow configurations [14], the spatial domain size is chosen such that \(-21<z/d_{0}<26\) and \(r/d_{0}<10\), where \(d_{0}\) stands for the initial diameter of the spherical droplet, which represents the natural reference length of the problem. The shock tube is divided into two parts by means of a virtual cross diaphragm, which is initially located at \(z/d_{0}=-2\) (upstream of the droplet). In the following, the gas conditions at the right and left sides of the tube are indicated by the subscripts 1 and 4, respectively. The two tube sections are initialized with the same temperature (\(T_{4}/T_{1}=1\)) and very different pressure and density levels (\(p_{4}/p_{1}=\rho_{4}/\rho_{1}\gg 1\)), with the ideal gas air being at rest on either side (\(V_{4}=V_{1}=0\)). Starting from these initial conditions, a normal SW develops and travels towards the driven section on the right, while a set of expansion waves propagate towards the driver section on the left. Given the pressure level \(p_{1}\) on the right side of the tube, as well as the shock Mach number (\(\text{Ma}_{\text{s}}\)) and, thus, the compression ratio across the SW, that is, \(\beta=p_{2}/p_{1}\), the thermodynamic parameters to be initially imposed on the left side are analytically determined from theoretical considerations, e.g., [24]. Note that the contact surface (where \(p_{3}=p_{2}\)) between the two regions at different density and temperature travels at a lower velocity in the same direction as the shock. In this study, the shock tube flow is simulated with a SW strength corresponding to \(\text{Ma}_{\text{s}}=1.47\), which stands for the case mostly studied in the literature, starting from the pioneering experimental works of [PERSON] and [PERSON] [22; 23]. The actual thermodynamic parameters that are imposed as initial conditions are summarized in Table 1. The air conditions in the driven section correspond to atmospheric pre-shock conditions, while the ambient conditions imposed to the air surrounding the water droplet, which are indicated by the subscript 2, correspond to the high-speed stream behind the shock. These conditions, together with the droplet parameters, are provided in Table 2, where \(V_{\text{s}}\) represents the constant velocity of the traveling SW front. Note that, given the present post-shock thermodynamic conditions, the ambient pressure results in being greater than the saturation pressure of water. This way, the liquid body does not undergo phase change and the two different fluids can be considered immiscible. It is worth stressing that the above condition could be no more valid for increasing Mach number. Figure 1: Sketch of spatial domain and physical model with the traveling shock front. The compressible aerodynamics of the problem under investigation is governed by the flow's Mach and Reynolds numbers at free-stream conditions, defined as  
\[\text{Ma}=\frac{V_{2}}{\sqrt{\gamma p_{2}/\rho_{2}}}\, \tag{1}\]  
and  
\[\text{Re}=\frac{\rho_{2}V_{2}d_{0}}{\mu_{2}}\, \tag{2}\]  
respectively.