# Large-Eddy Simulation of Droplet Deformation and Fragmentation Under Shock Wave Impact
## 4 Results and Discussion
### Sub-Droplets Distribution  
Sub-droplets are formed when liquid sheets, ligaments, and lips detach from the parent droplet [9]. Practically, the initial breakup event occurs when the first ligament or sub-droplet separates from the droplet under shear breakup conditions. Due to the DPM methodology, in the framework of the employed CFD solver, the progression from initiation to the completion of the ligament fragmentation process can be tracked while counting the number of sub-droplets that are formed. Here, differently from previous research employing the unsteady RANS approach [21; 25], the LES formulation allows for a more detailed analysis of sub-droplet size distributions while investigating the impact of varying the \(We\) parameter.  
Figure 8 demonstrates the generation of sub-droplets in terms of the percentage distribution of diameters at various Mach and Weber numbers, making a comparison against experimental data by [PERSON] et al. [47]. The distribution of droplet sizes tends to shift towards smaller values as \(We\) increases, which represents a trend well documented in the literature. In fact, at higher \(We\) values, the proportion of smaller droplets increases, whereas at lower \(We\) values, larger droplets make up a greater portion of the total liquid mass. Note that sub-droplets generated by the direct stripping of KH waves result in being  
generally smaller than those formed from sheets, ligaments, and lips detaching from the edges and leeward side of the parent droplet. Initially, aerodynamic forces can effectively break apart larger sheets and ligaments into multiple larger sub-droplets. Over time, these larger fragments may further disintegrate into smaller sub-droplets. Notably, the present trend in particle size distribution aligns with the experimental data, confirming the accuracy of the LES approach. It is important to highlight that the reference experimental distribution actually corresponded to the temporal range \(0.33<t^{*}<0.67\), whereas the preset numerical data corresponded to a given time instant: \(t^{*}=0.55\). This difference represents a key factor contributing to the observed discrepancy in the diameter sizes. Nevertheless, the overall predicted log-normal distribution results in a consistency between the two different approaches, supporting the validity of the scale-resolving DPM-LES approach.  
Figure 7: Droplet surface (leeward view) and vorticity contours at \((x,y)\) and \((x,z)\) planes for \(M_{s}=1.47\) as a function of time. (**a**) \(t^{*}=0.03\). (**b**) \(t^{*}=0.06\). (**c**) \(t^{*}=0.14\). (**d**) \(t^{*}=0.33\). (**e**) \(t^{*}=0.54\).  
Furthermore, Figure 9 displays the relationship between particle velocities and associated diameters. Apparently, higher Weber and Mach numbers correspond to increased particle velocities. To examine the sub-droplets distribution, the Sauter mean diameter (SMD) can be computed. This latter parameter is defined as follows:  
\[D_{32}=\frac{\sum_{i}n_{i}d_{i}^{3}}{\sum_{i}n_{i}d_{i}^{2}}, \tag{20}\]  
where \(n_{i}\) represents the number of droplets with a (discretized) diameter \(d_{i}\). Practically, this expression calculates the mean diameter by weighting the volume and surface area of the tracked droplets. SMD is highly effective for characterizing the distribution of spray droplets, particularly in evaporation processes. The SMD values calculated throughout the current atomization process are reported in Table 2, indicating a significant reduction in SMD as the Weber number increases.  
\begin{table}
\begin{tabular}{c c} \hline
**We** & **SMD [\(\mu\)m]** \\ \hline
205 & 61.9 \\
7000 & 15.0 \\ \hline \end{tabular}
\end{table}
Table 2: Overall SMD value as a function of Weber number.  
Figure 8: Percentage distribution of diameters of daughter droplets at various Weber numbers, compared to experiment by [PERSON] et al. [47].  
Figure 9: Velocity distribution of daughter droplets as a function of their diameter.