## 3 Numerical Simulation Study of Sub-Millimeter Liquid Droplets
### Results and Discussion  
Figure 12 presents the morphological evolution of the windward and leeward surfaces at eight discrete time instances from the dimensionless times 0 to 1.497. The shapes corresponded to liquid phase isosurfaces with volume fractions of \(\alpha_{I}\) = 0.99, 0.50, and 0.01 at  
Figure 11: Evolution of the dimensionless transverse width variations in the sub-millimeter droplets. (**a**) (\(Ma\) = 1.17, \(d_{0}\) = 0.9) [26]. (**b**) (\(Ma\) = 1.36, \(d_{0}\) = 0.84) [26]. Figure 10: Evolution of the dimensionless displacement of the sub-millimeter droplets on the windward surfaces for the various grid resolutions. \(Ma=2.12\) operating conditions. Under these operating conditions, the airflow attained a supersonic state with the post-shock Mach number \(M_{2}=1.03\). This supersonic environment amplified the instant deformation and aerodynamic breakup of the sub-millimeter liquid droplets. During the initial stage (\(T=0\sim 0.191\)), the sub-millimeter liquid droplets remained largely undeformed. Surface disturbances arising from the instability began to develop gradually. At this point, the droplets could be approximated as rigid bodies with no significant axial displacements along the airflow direction. When \(T=0.322\sim 0.518\), sub-millimeter liquid droplets experienced initial distortion and deformation. The windward side retained a relatively spherical shape, while the leeward side was rapidly compressed into a flat shape. Subsequently, when \(T=0.779\sim 1.170\), vortical structures continuously developed and expanded on the leeward sides of the sub-millimeter liquid droplets due to the airflow, resulting in the formation of noticeable cavities in the leeward regions. Concurrently, the sub-millimeter liquid droplets underwent the strong shearing effect of the K-H instability. Small-scale liquid sheets and sub-droplets continually detached from the equatorial regions and downstream of the leeward sides of the droplets. High-speed airflow accelerated and stretched them away from the parent droplets. This process, influenced by flow instability, resulted in the thinning and gradual development of ligaments from the thinner parts of the liquid sheets. Over time, these ligaments became finer, and their collisions further contributed to the droplets' fragmentation into smaller liquid droplets. Consequently, in the later stages of fragmentation (\(T=1.497\)), a widespread distribution of sub-droplets became apparent in the downstream regions. Through a comprehensive examination of the sub-millimeter liquid droplet characteristics, it became evident that the deformation and breakup of the sub-millimeter liquid droplets in a supersonic airflow environment aligned with the SIE breakup mechanism as proposed by [PERSON] [11]. The comparison between the numerical simulation results and the experimental visualizations at corresponding time points is presented in Figure 13. The figure illustrates a close correspondence between the deformation and breakup morphologies of the sub  
Figure 12: Isosurfaces of the liquid volume fractions at \(\alpha_{l}=0.99\) (light purple), 0.50 (light blue), and 0.01 (white), corresponding to the eight different time instants in the interval \(0\leq T\leq 1.497\), with windward surfaces labeled (**a1–a8**) and leeward surfaces labeled (**b1–b8**). millimeter liquid droplets obtained through the numerical simulations and the experimental photographs. Additionally, Figure 14 provides a parameter comparison, depicting the dimensionless transverse deformation widths of the sub-millimeter liquid droplets in the vertical airflow direction over time. The results revealed good qualitative agreement between the numerical simulations and the experimental measurements, underscoring the effectiveness of our numerical simulation method in capturing the interface evolution during the sub-millimeter liquid droplet breakup process.