## 1 Introduction
[16] employed a combination of dual-pulse laser holography and an experimental apparatus with shock wave-driven droplets, and they acquired high-resolution experimental image data at the micron-nanosecond level. Considering that the deformation and breakup of liquid droplets are associated with fluid dynamic interface instability, further research has been initiated. Commonly encountered fluid dynamics interface instabilities include Richtmeyer-Meshkov (R-M), Rayleigh-Taylor (R-T), and Kelvin-Helmholtz (K-H) instabilities. [PERSON] [17] suggested that at high Mach numbers, the rapid growth of high-wavenumber disturbances during the breakup of Newtonian liquid droplets results from weak viscous dissipation and surface tension. The Rayleigh-Meshkov instability can be ignored due to the significant acoustic impedance mismatch at the interface, with the primary mechanisms affecting droplet breakup being the Rayleigh-Taylor and Kelvin-Helmholtz instabilities. [PERSON] and [PERSON] [18] attributed the growth and breakup of liquid sheets sandwiched between shearing airflows during droplet fragmentation to the development of Rayleigh-Taylor instability waves on the sheet. Additionally, [PERSON] [19; 20] conducted an experimental verification of Rayleigh-Taylor piercing during the droplet breakup process and quantified critical physical parameters related to shear-induced entrainment. Their findings highlight the predominant role of instability wave penetration in droplet breakup, particularly in systems with a high gas-liquid density ratio and high surface tension. Their research primarily concentrates on investigating the influence of viscosity and density ratios on the development of instability. However, a quantitative analysis of instability development at the liquid droplet interface under other parameters is conspicuously absent. Advances in computational capabilities and technology have transformed computational fluid dynamics (CFD) into a valuable complement to experimental and analytical methods. Compared to experiments, CFD numerical simulations excel at capturing the intricate details of flow fields and gas-liquid interfaces. However, the high cost of three-dimensional numerical calculations has led to the predominance of simulations conducted in two dimensions [21; 22] or in a two-dimensional axisymmetric setting [23]. [PERSON] et al. [24] performed numerical simulations analyzing liquid droplet characteristic breakup and edge tip generation. They also investigated the SIE mechanism, elucidating the shear, peel, and fracture processes of liquid droplets and the flow field evolution. [PERSON] [25] employed the volume-of-fluid (VOF) method and turbulence models for numerical simulations, revealing that the Weber number enhanced liquid droplet deformation and breakup. [PERSON] et al. [26] employed the VOF multiphase flow model and the k-\(\varepsilon\) turbulence model; they analyzed, using two-dimensional numerical simulations, the impact of different Weber numbers on the deformation and evolution processes of sub-millimeter liquid droplets. [PERSON] et al. [14] adapted dynamic grid technology and studied the structure of the flow near and in the wake of a drop, the features of the flow around a drop, the type of the shape evolution, and the character of the mass entrainment, based on the use of the volume-of-fluid (VOF) method to resolve the phase interface. [PERSON] et al. [27] creatively applied a hybrid approach using VOF and Lagrangian computational fluid dynamics methods to investigate liquid droplet fragmentation. Their research accurately predicted the deformation and fragmentation progression of liquid droplets, as well as the subsequent formation of liquid sprays, replicating essential features of droplet deformation and fragmentation under shear-induced entrainment conditions. Nevertheless, there is currently limited research on three-dimensional numerical simulations of the aerodynamic breakup of sub-millimeter-sized droplets.