# Large-Eddy Simulation of Droplet Deformation and Fragmentation Under Shock Wave Impact
## 4 Results and Discussion
### Verification and Validation  
To validate the present LES model, the simulation results are compared with experimental data by [PERSON] et al. [26], experimental images provided by [PERSON] et al. [14], and numerical results by [PERSON] and [PERSON] [41]. Figure 3 shows some side views of the deforming droplet in the meridian plane, plotted against non-dimensional time. An analysis of snapshots within the time frame \(0.074\leq t^{*}\leq 0.49\) indicates that the numerical results for the configuration at \(M_{s}=1.47\) (as detailed in Table 1), correspond well to the reference experiment examining the breakup of a water droplet at \(We=780\) for a post-shock flow Mach number of 0.32 [14]. Generally, the qualitative agreement with experimental photographs is evident, both in terms of droplet shape and deformation dynamics. However, at the last time point, a slight difference can be observed, where the numerical solutions show a cupcake-like shape, while the experimental shape looks like an inverted bowl, which is likely due to the different Weber number. Beyond the previous qualitative analysis, a key quantitative indicator of droplet deformation is represented by the time increasing cross-sectional extent of the droplet, which also affects the droplet dynamics. Figure 4 illustrates the relative growth of the droplet's midsection, represented as \(D_{cro}/D_{0}\), which is the ratio of the current transverse droplet dimension to its initial size, for \(M_{s}=1.12\) (see Table 1). The figure indicates a strong correlation between the average deformation rate obtained from an experiment [26] and the current LES data. In fact, despite minor discrepancies, the change in the droplet's cross-stream deformation generally aligns acceptably well with the experimental findings in terms of both trend and magnitude. Though the reference data seem slightly underpredicted, the proposed VOF-LES method is demonstrated to effectively capture the droplet  
Figure 2: Compressible flow structures for \(M_{s}=1.47\) at different time instants. (**a**) \(t^{*}=0.005\); (**b**) \(t^{*}=0.03\); (**c**) \(t^{*}=0.06\); (**d**) \(t^{*}=0.33\). deformation and initial breakup characteristics, even considering that the experimental uncertainty is not known. Basically, the primary objective of numerical simulations focusing on the early stages of shock-droplet interactions is to accurately predict the motion and deformation of the wa  
Figure 4: Temporal evolution of dimensionless cross-stream diameter \(D_{\textit{con}}/D_{0}\) from an experiment [26] and numerical calculation, for \(M_{s}=1.12\). Figure 3: Droplet morphology (lateral view) for \(M_{s}=1.12\) and \(1.47\) compared to corresponding experimental images [14] (from (**left**) to (**right**)) against time. The different rows correspond to \(t^{*}=0.074\), \(0.13\), \(0.32\), and \(0.49\) (from (**top**) to (**bottom**)), and the airflow is from right to left). ter body, which is critical for understanding the subsequent breakup phase. The droplet's kinematic characteristics are usually assessed by analyzing the streamwise position and velocity of its center-of-mass (CM). These time-dependent parameters are defined, according to the adopted VOF formulation, as follows:  
\[x_{CM}(t)=\frac{\int_{\Omega}F\rho_{l}x\ d\Omega}{\int_{\Omega}F\rho_{l}d\Omega}, \tag{18}\]  
\[u_{CM}(t)=\frac{\int_{\Omega}F\rho_{l}u\ d\Omega}{\int_{\Omega}F\rho_{l}\ d \Omega}. \tag{19}\]  
The above integrals, while spanning the entire computational domain \(\Omega\), exclusively account for contributions from space regions where the liquid phase is present, as determined by a non-zero volume fraction \(F\). The displacement and velocity parameters are normalized here as \(\Delta x^{*}=\frac{x_{CM}(t)-x_{CM}(0)}{D_{0}}\) and \(u^{*}=\frac{u_{CM}(t)}{u_{g}}\), respectively. Figure 5 shows the time histories of these two parameters at \(M_{s}=1.47\). The present LES results demonstrate a good correlation with the reference DNS data [41].