# Large-Eddy Simulation of Droplet Deformation and Fragmentation Under Shock Wave Impact
## 4 Results and Discussion
### Droplet Breakup Dynamics  
The flow patterns for the two test cases under study are analyzed in Figure 6, showing the velocity magnitude corresponding to the (x, y) plane at various time instants. For the first configuration at \(We=205\), the breakup regime falls into the transition zone between RTI- and shear stripping-based mechanisms. This region is characterized by the simultaneous development of multiple breakup mechanisms, leading to what is referred to as a mixed or multimode regime [44]. At low \(We\) numbers, where the droplet experiences minimal deformation, the surrounding air flow resembles that corresponding to a solid sphere at Reynolds numbers between \(10^{3}\) and \(10^{4}\), characterized by a vortex separation near the midsection, as seen at \(t^{*}=0.03\). Seemingly, at later time instants (\(t^{*}=0.06\)), the wake flow is characterized by two vortical zones, which appear enlarged at \(t^{*}=0.14\), confirming previous results by [PERSON] and [PERSON] [45]. These characteristic structures remain discernible until approximately \(t^{*}=0.33\), when a toroidal vortex and reverse flow develop, accompanied by the emergence of two annular waves on the droplet's surface. Then, the wake flow loses its axial symmetry and breaks down into smaller, non-stationary structures (\(t^{*}=0.54\)). These structures form a large stagnant region with a low average velocity, similar to the velocity in the flow deceleration zone in front of the droplet's windward surface. Figure 5: Normalized CM displacement (**left**) and velocity (**right**) against time for \(M_{s}=1.47\), compared to DNS [41]. This flow results in the surface flattening and causes the air to spread radially across the rear surface. This process leads to the formation of a liquid disk whose edge facilitates the transfer of droplet fragmentation products into the aerodynamic wake. Overall, this type of droplet breakup actually corresponds to the so-called \"sheet stripping\" mechanism. The above breakup dynamics are characteristic across the range of _We_ numbers investigated. The analysis of the air velocity field reveals flow characteristics that align with experimental observations. Specifically, following [45], the external flow around the droplet is inherently unsteady, yet it comprises some stable gas-dynamic structures, each influencing the droplet in a unique manner. Initially, a prominent wave known as the generatrix bend forms. This wave emerges when the droplet is still spherical, at the separation point, where the velocity reaches its maximum. Beyond this point, a recirculation zone with a counter-current flow along the droplet's surface becomes evident. Furthermore, in the  
Figure 6: Droplet surface (lateral view) and velocity contours at the \((x,y)\) plane for \(M_{s}=1.12\)**(left)** and \(M_{s}=1.47\)**(right)** solutions as a function of time. The different rows correspond to \(t^{*}=0.03\), \(0.06\), \(0.14\), \(0.33\), and \(0.54\) (from **(top)** to **(bottom)**). wake, a secondary stable structure forms as a recirculation zone with an intense reverse flow along the longitudinal axis. This flow causes surface flattening, leading the air to spread radially across the rear surface. In this region, a second break in the generatrix occurs, which is linked to the radial spreading of liquid water. This process leads to the formation of a liquid disk whose edge facilitates the transfer of droplet fragmentation products into the aerodynamic wake. Additionally, the air flow separates near the second wave, radiating outward across the rear surface. Part of this flow re-enters the initial separation zone along the droplet's surface, while the remainder transitions into a third stable structure, represented by an isolated toroidal vortex that partially separates the previous two. Overall, this type of droplet breakup corresponds to the so-called \"sheet stripping\" mechanism.