# Computational Evaluation of Shock Wave Interaction with a Liquid Droplet
## 3 Results
### Quantitative Validation  
The quantitative validation of the proposed CFD model was performed by examining the movement and deformation of the water droplet against time, while making a comparison with reference numerical data [15; 16]. The time-dependent axial position of the center-of-mass (CM) of the water droplet is simply determined by:  
\[z_{\text{CM}}=\frac{\int_{\Omega}\alpha\rho_{l}z\ d\Omega}{\int_{\Omega}\alpha \rho_{l}\ d\Omega}\, \tag{9}\]  
where \(\Omega\) represents the overall computational domain. Analogously, the axial velocity of the droplet CM is given by:  
\[w_{\text{CM}}=\frac{\int_{\Omega}\alpha\rho_{l}w\ d\Omega}{\int_{\Omega}\alpha \rho_{l}\ d\Omega}. \tag{10}\]  
Owing to the VOF formulation, the above equations inherently exclude the contribution from the flow region that is actually occupied by the gaseous phase (\(\alpha=0\)). By inspection of Figure 6, wherein the time histories of the CM drift and velocity are given for two different FV grid resolutions, the present CFD data appear to be acceptably accurate when compared to high-fidelity numerical results [15]. In practice, the present engineering analysis provides slightly anticipated movement and deformation of the liquid droplet. However, since the above integral parameters result in being quite insensitive to small-scale interface structures, which are not resolvable by means of the current mean flow analysis, the overall movement of the droplet is satisfactorily captured. Furthermore, the positive effect of increasing the mesh resolution is apparent. Furthermore, the streamwise acceleration \(a_{\text{cm}}\) of the droplet CM was obtained by differentiating the discrete velocity data using second-order finite difference approximations, for both the present CFDs and reference numerical solutions [13]. Looking at the time evolution of this variable, which is reported suitably normalized on the left side of Figure 7, the good agreement with reference data in [15] is confirmed. Note that a similar result also holds for the aerodynamic drag coefficient of the water body. As to the droplet deformation, the time history of the decreasing axial width \(d\), normalized by the initial diameter \(d_{0}\), is shown on the right side of Figure 7. Apparently, an acceptable agreement is achievedbetween the present solution and reference numerical data in [16], where the Mach number is almost the same, with the comparison improving for increasing resolution. It is worth stressing that both reference numerical simulations were conducted without considering the effects of molecular viscosity and surface tension. In particular, the finest mesh resolution that was used in [15] corresponded to 100 cells per original droplet diameter, while no modeling procedure was introduced for the residual unresolved flow scales. Therefore, that solution can be interpreted as either under-resolved DNS or no-model LES, where the numerical scheme inherently included artificial viscosity. Differently, the present study aims to investigate the mean flow by solving the unsteady RANS equations supplied with a suitable closure model. Finally, based on the acceptable results that were obtained in terms of both qualitative analysis and quantitative validation with respect to reference numerical and experimental findings, the proposed CFD model was shown to be able to capture the salient features of this particular industrial fluid dynamics application, without being computationally cumbersome. This highlights the benefits of the present relatively simple and straightforward numerical approach. Figure 6: Time history of the normalized CM drift (**left**) and velocity (**right**) of the droplet: present solutions CFD II (black solid line) and CFD III (green dotted line), compared to reference numerical solutions (red dashed line).