# Computational Evaluation of Shock Wave Interaction with a Liquid Droplet
## 2 Computational Model
### Case Study
Moreover, as discussed in the previous section, the droplet aerobreakup induced by the SW impact is controlled by the Ohnesorge number for the liquid droplet,  
\[\text{Oh}=\frac{\mu_{l0}}{\sqrt{\rho_{l0}\sigma d_{0}}}\, \tag{3}\]  
which is based on the water properties, and the Weber number,  
\[\text{We}=\frac{\rho_{2}V_{2}^{2}d_{0}}{\sigma}\, \tag{4}\]  
which is based on the post-shock air conditions. The constant surface tension \(\sigma\) at the air-water interface appearing in both definitions (3) and (4) is assumed constant. In addition, the droplet-phase-to-ambient-phase density ratio, \(\varepsilon=\rho_{l0}/\rho_{2}\), and droplet-phase-to-ambient-phase viscosity ratio, \(N=\mu_{l0}/\mu_{2}\), are usually taken into account. Basically, the size of the liquid droplet plays a crucial role in determining both the air flow regime and the droplet breakup characteristics. The current value for the initial droplet diameter, that is, \(d_{0}=4.8\) mm, corresponding to the classical reference experiments [22; 23], has been widely used in numerical investigations of the aerobreakup in the SIE regime [10; 15]. Based on this reference length and the other physical parameters that are involved in their definitions, the above-mentioned characteristic dimensionless groups take the values provided in Table 3. \begin{table}
\begin{tabular}{c c c} \hline
**Parameter** & **Symbol** & **Value** \\ \hline Driven section pressure & \(p_{1}\) & 101.3 kPa \\ Driven section density & \(\rho_{1}\) & 1.204 kg/m\({}^{3}\) \\ Shock compression ratio & \(p_{2}/p_{1}\) & 2.35 \\ Driver section pressure & \(p_{4}\) & 0.641 MPa \\ Driver section density & \(\rho_{4}\) & 7.61 kg/m\({}^{3}\) \\ Temperature & \(T_{1}=T_{4}\) & 293.15 K \\ \hline \end{tabular}
\end{table} 
Table 1: Shock tube flow parameters. \begin{table}
\begin{tabular}{c c c} \hline
**Parameter** & **Symbol** & **Value** \\ \hline Air temperature & \(T_{2}\) & 381 K \\ Air density & \(\rho_{2}\) & 2.18 kg/m\({}^{3}\) \\ Air viscosity & \(\mu_{2}\) & 2.23 \(\times\) 10\({}^{-5}\) Pa/s \\ Air velocity & \(V_{2}\) & 225.9 m/s \\ Shock front velocity & \(V_{\text{S}}\) & 504.5 m/s \\ Droplet diameter & \(d_{0}\) & 4.80 \(\times\) 10\({}^{-3}\) m \\ Water density & \(\rho_{l0}\) & 998 kg/m\({}^{3}\) \\ Water viscosity & \(\mu_{l0}\) & 1.003 \(\times\) 10\({}^{-3}\) Pa/s \\ Surface tension & \(\sigma\) & 7.286 \(\times\) 10\({}^{-2}\) N/m \\ \hline \end{tabular}
\end{table} 
Table 2: Post-shock air flow and water body parameters. Owing to the very low Ohnesorge and the relevant Weber numbers, the current aerobreakup configuration does belong to the SIE regime, where the gas is expected to go around the liquid mass producing a surface layer peeling-and-ejection action [25]. Capillary effects could be neglected with respect to the inertial ones in the early stages of the aerodynamic breakup, where shear stripping is the dominant mechanism. Furthermore, based on the relatively high Reynolds number, viscous effects could be considered negligible. In fact, differently from this work, several numerical investigations of the droplet aerobreakup have been conducted by solving the compressible Euler equations in absence of surface tension [15].