# Computational Evaluation of Shock Wave Interaction with a Cylindrical Water Column
## 2 Computational Model
### Case Study  
According to the two-dimensional approach, the simplified geometry of the shock tube device is represented by the rectangular domain sketched in Figure 1, where the planar shock, and thus the post-shock air flow, moves from left to right. The reference coordinates system \((x,y)\) was chosen with the \(x\)-axis aligned with the streamwise direction, while the origin corresponded to the leading-edge of the water cylinder in its initial position. Based on previous numerical simulations conducted for the same flow configuration, the spatial domain size was chosen such that \(-8<x/d_{0}<16\) and \(|y|/d_{0}<10\), where \(d_{0}\) stands for the initial diameter of the cylindrical water column, which represents the natural reference length. The shock tube was divided into two parts by means of a virtual cross diaphragm, which was initially located at \(x/d_{0}=-2\) (upstream of the column). 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 were 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 planar shock wave develops and travels towards the right side (driven section), while a set of expansion waves propagate towards the left one (driver section). The virtual diaphragm, which corresponds to the contact surface between the two different density regions, travels in the same direction of the shock, though at a lower velocity. Given and knowing the pressure level \(p_{1}\) on the right side, as well as the compression ratio across the shock \(\beta=p_{2}/p_{1}\), corresponding to the prescribed Mach number, the thermodynamic parameters to be imposed on the left side can be analytically determined. In fact, combining the moving shock wave equations with the expansion region relations, it is possible to derive an analytical expression for the ratio \(p_{4}/p_{1}\), that is:  
\[\frac{p_{4}}{p_{1}}=\beta\Bigg{(}1-\frac{(\gamma-1)(\beta-1)}{\sqrt{4\gamma^{2 }+2\gamma(\gamma+1)(\beta-1)}}\Bigg{)}^{-\frac{2\gamma}{\gamma-1}}\,, \tag{1}\]  
where \(\gamma=1.4\) was assumed for the specific heat ratio. Therefore, based on the above estimation, the initial conditions in the driver section were uniquely determined, e.g., [19]. In this study, the shock tube flow was simulated at a Mach number of 1.73, which corresponds to one of the cases recently examined in [20], where the water cylinder was immersed in supersonic air crossflow. The thermodynamic parameters that were imposed are summarized in Table 1, while \(V_{\rm S}=593.7\) m/s and \(V_{\rm 2}=329.3\) m/s represent the constant velocities of the traveling shock wave and post-shock air flow, which were determined from theoretical considerations. The high-speed airstream behind the shock represents the ambient conditions to which the water column is exposed, which are indicated by the subscript 2. For clarity, the water column and post-shock air flow parameters of interest are tabulated in Table 2.  
Figure 1: Sketch of the physical model with traveling shock wave. The compressible aerodynamics of the interaction process is governed by the flow Mach and Reynolds numbers, which are:  
\[\text{Ma}=\frac{V_{2}}{\sqrt{\gamma p_{2}/\rho_{2}}}\, \tag{2}\]  
and:  
\[\text{Re}=\frac{\rho_{2}V_{2}d_{0}}{\mu_{2}}\, \tag{3}\]  
respectively. Moreover, as discussed in the previous section, the aerobreakup of the water column induced by the impact of the shock wave is controlled by the Ohnesorge and Weber numbers. Based on the liquid water properties and the surface tension \(\sigma\), which was assumed constant, these dimensionless parameters are defined as:  
\[\text{Oh}=\frac{\mu_{l}}{\sqrt{\rho_{l}\sigma}d_{0}}\, \tag{4}\]  
and:  
\[\text{We}=\frac{\rho_{2}V_{2}^{2}d_{0}}{\sigma}\, \tag{5}\]  
respectively.