## 4 Results
### Sub-Droplets Tracking  
A very important but less investigated aspect of droplet aerobreakup under the SIE regime concerns the dynamics of the dense mist of sub-droplets that is forming as a result of the droplet disintegration. In this study, due to the adoption of the hybrid VOF/DPM methodology, we can effectively simulate all stages of the shock-induced breakup, including the birth and the dynamics of small liquid particles that are consequently originated. Differently from what was recently performed in the Eulerian/Lagrangian model proposed in [42], here, the number of Lagrangian particles is not predetermined, but is increasing with time depending on the verification of the two transition criteria discussed in Section 2.2. The VOF-to-DPM method employs a dynamic mesh adaptation strategy, where the mesh resolution is automatically increased in the transition region between the two different phases, according to the volume fraction gradient field. This is exemplified in Figure 8, where the plane views of the FV mesh at four typical time instants are shown, in the midplane \(y=0\), along with the corresponding contours of the volume fraction gradient magnitude, which is normalized by the reciprocal of the minimum mesh size. The coupling of adaptive meshing with the VOF method allows for capturing the gas-liquid interfaces with increased resolution, improving the numerical predictions for given computational resources. The simulation of the discrete phase, through the solution of the momentum balance Equation (4) for each particle, allows determining the spatial-temporal evolution of the sub-droplets in the mean turbulent flow. To illustrate the results, in Figure 9, the instantaneous positions of the DPM particles are reported at three typical time instants, while being colored by either the nondimensional diameter \(d_{p}/d_{0}\) (top) or the nondimensional speed \(u_{p}/u_{\text{g}}\) (bottom). Apparently, the sub-droplets evolve following the gas stream flow in the complex wake of the residual water body. By inspection of these pictures, there is a lack  
Figure 8: Plane views in the midplane \(y=0\) of adaptive numerical mesh (**top**) and corresponding contours of normalized volume fraction gradient magnitude (**bottom**) at four typical time instants. of correspondence between size and speed, as the space-time evolution of DPM particles results in being quite complicated. The statistical distribution of sub-droplets is examined in Figure 10, where the histograms report the percentage of DPM particles in terms of either normalized diameter or speed, at two different time instants. Seemingly, the particle-size and particle-speed distributions appear poly-disperse and time-dependent. This is not surprising since the aerobreakup under the SIE regime is a continuous and multi-stage phenomenon [8]. Indeed, the small-scale sub-droplets are continuously formed and entrained during the whole process, while the fragmentation occurs not only from the parent body but also from sheets and ligaments, as well as from smaller child droplets. Moreover, finer particles achieve higher speeds faster. When looking at the particles arising from the shear-stripping of liquid ligaments, the size distribution should resemble a Gamma function distribution [43]. This is approximately observed on the left side of Figure 10, where the theoretical behavior seems to be somewhat approached with increasing time. Converting the smallest liquid lumps resulting from the dissolution of the parent body into point masses that can be tracked using the DPM model, the hybrid modeling approach provides insight into the sub-droplets dynamics, allowing to bypass the limits of experimental studies, where the dense mist investigation is extremely difficult. This way, the proposed CFD model is very promising to be used in the fluids engineering research. For instance, these results could be of particular importance when predicting the rain erosion on supersonic aircraft [44].