## 4 Results
### Droplet Deformation  
As a consequence of the exposure to the post-shock high-speed flow, after the reaction time has elapsed, during which the solution practically corresponds to the flow past a rigid sphere, the deformation of the droplet begins to be observed. At the early stages of the process, following the small-scale instabilities originating on the liquid surface, the water body is flattened in the streamwise direction. That is mainly due to the high pressure difference between the front and back droplet sides. The droplet evolution is illustrated, in Figures 1 and 2, by drawing the isosurfaces of the volume fraction of water at \(\alpha=0.5\), to identify the droplet boundary, along with the instantaneous pressure contours at the midplane \(y=0\). Both front and back views are reported, for eight typical instants in the time interval \(0.044\leq t^{*}\leq 0.83\), with the pressure field being normalized by the post-shock freestream value. To correctly illustrate the droplet drift and deformation, the size and position of the zoomed images are the same for each snapshot of these two figures. Initially, as the droplet shape changes very slightly, both reflected and transmitted wavefronts are apparent in Figure 1. Then, the formation of ligaments at the droplet surface is observed, which are further stretched and, eventually, disintegrated by the aerodynamic shearing force of the surrounding airflow, as shown in Figure 2. The water body is continuously eroded, with sub-droplets being stripped from the parent droplet, as well as from the water ligaments. In the later stages, the liquid droplet shatters into smaller fragments distributed widely in the flow field. These pictures confirm the qualitative features of the so-called shear stripping breakup mechanism, which is typical at high Weber numbers, under the SIE regime [2]. To further analyze the CFD solution, in Figure 3, the vortical structures are visualized by plotting the contours of the second component of the normalized vorticity vector field in the midplane \(y=0\), as well as on the droplet boundary. The equatorial recirculation region that is formed around the droplet is apparent, corresponding to the muffin-like shape of the deforming water body. As observed in [15], this leads to the formation of liquid sheets that are simultaneously drawn out from the hemispherical lip and the planar back of the droplet. When making a comparison with URANS [19], the present SAS-URANS method allows for a better representation of unsteady turbulent flow features, with the capture of some fine-scale structures. In Figure 4, the morphology of the deforming droplet is demonstrated by showing the side view of the water body traced by nondimensional time, making a direct comparison with the experimental images provided by [PERSON] and co-workers [8]. The simulation results partially agree with the observations made in the experiment investigating the water droplet breakup at \(\text{We}=7.8\times 10^{2}\) with a post-shock flow Mach number of 0.32. In the CFD analysis, owing to the higher Mach number, the surface instability at the equator is more pronounced than that one originating at the rear part of the body. The comparison with experimental findings is initially quite good while, at later times, the URANS solution inherently cannot represent the fine scale structure of the two-phase flow. Nevertheless, these results suggest that the SAS formulation is capable of accurately reproducing the transition process by resolving, at least partially, the small-scale flow structures at the gas-liquid interface. In practice, the proposed multiscale URANS approach is found to predict the early stage breakup mechanisms with an accuracy comparable to more expensive methods. Figure 1: Normalized pressure contours depicted in the midplane \(y=0\), as well as on the droplet boundary, at four typical time instants with \(0.044\leq t^{*}\leq 0.25\). Both front (**top**) and back (**bottom**) views are shown.