## 3 Numerical Simulation Study of Sub-Millimeter Liquid Droplets
### Results and Discussion
Figure 15 displays the streamline, velocity contours, and pressure contours illustrating the deformation and fragmentation processes of the sub-millimeter liquid droplets subjected to shock wave conditions with an _Ma_ value of 2.12. Small surface waves formed on the windward surfaces and propagated toward the equators from the initial moment. On the leeward surfaces of the sub-millimeter liquid droplets, airflow separation occurred, resulting in the formation of small vortexes. At this point, the airflow near the windward surfaces demonstrated high pressures and low velocities. The pressures on the windward surfaces were not uniform, and the surface waves created high-pressure stagnation points on the windward sides by impeding the airflow behind the waves. In the equatorial regions of the sub-millimeter liquid droplets, low pressures and high velocities prevailed, with the highest airflow speed observed at this position. When considering the droplets' stagnation states, it became evident that the maximum tangential velocity differences between the gas and liquid phases occurred on the equatorial planes, resulting in the onset of Kelvin-Helmholtz instability. Subsequently, due to the significantly lower pressures in the equatorial regions compared to the pressures at the front and rear stagnation points, the sub-millimeter liquid droplets underwent flattening. This flattening reduced the curvatures on the windward sides of the droplets, leading to increased variations in the airflow velocities as they traversed the droplets. This, in turn, promoted the continuous growth of vortices on the leeward sides and propelled the propagation of the surface waves towards the equatorial planes. When examining Figure 12, it is evident that the sub-millimeter liquid droplets expanded laterally in the vertical airflow direction, while the leeward regions adopted concave shapes, causing sub-droplets to continually detach from the vicinity of the equatorial planes. Figure 14: Evolution of the dimensionless transverse width variations in the sub-millimeter droplets. Figure 13: Comparison of the numerical calculation results with the experimental data for Ma = 2.12 at the three different time points. Experimental image (**left**) and numerical \(\alpha_{l}=0.5\) isosurface (**right**). The velocity contours revealed flame-like tails extending in the airflow direction on the leeward sides of the sub-millimeter liquid droplets. In the influence of the vortices, there were observable annular low-speed regions in the middles of the velocity distributions on the leeward sides, corresponding to the outer vortex regions. Finally, the flow fields and vortex structures on the leeward sides continue to develop and became more complex. The main sizes and masses of the droplets decreased continuously due to the detachment of small droplets, while the annular low-speed regions at the tails enlarged continuously due to the expansion of the vortexes. A quantitative investigation into the deformation process of the sub-millimeter liquid droplets was subsequently carried out. Figure 16 illustrates the dimensionless axial deformation widths aligned with the flow direction and the dimensionless transverse deformation widths aligned with vertical airflow over time. The results are presented for volume fractions of 0.01, 0.5, and 0.99. The axial widths of the sub-millimeter liquid droplets decreased approximately linearly with time in the airflow direction, while the evolution of the transverse widths in the vertical airflow direction underwent the following  
Figure 15: Velocity contours and streamlines (**top, images a–f**) and pressure contours (**bottom, images a–f1**) of the sub-millimeter droplets (with corresponding dimensionless times of \(0.06\leq T\leq 0.974\)). two stages: a slow flattening deformation stage and a strong shear stretching deformation stage, which aligned with the characteristics observed in the numerical simulations for the sub-millimeter liquid droplets' deformation and fragmentation, as shown in Figure 12.