## 3 Numerical Simulation Study of Sub-Millimeter Liquid Droplets
### Computational Analysis
For unsteady flow environments, it was essential to control the entire computational time by standardizing the gradient values. In the scaling method, the \"Scale by Global Maximum\" option was applied. Finally, for the phase fraction gradient adaptation, specific refinement  
Figure 8: Definitions of the droplet’s windward displacement, width of the vertical airflow-induced deformation, and width of the deformation along the direction of the airflow. and coarsening values were provided, with a refinement value of 0.06, a coarsening value of 0.05, and a maximum refinement level that was set to 4. Grid refinement was automatically conducted during the calculations based on the liquid volume fraction gradient to achieve an accurate solution domain. Furthermore, under these thermodynamic conditions, the environmental pressure in the flow field exceeded the saturated vapor pressure of the water. Consequently, during the sub-millimeter liquid droplet deformation and breakup, no phase change occurred, and it could be assumed that the gas-liquid two-phase was immiscible. In this research, the multiphase flow model selected was the VOF (volume-of-fluid) model due to its excellent mass conservation properties and effective interface capturing capabilities. It can accurately describe mass transfer in the fluid and interface diffusion [14; 32; 35]. This method is based on the idea of treating liquid and gas as a single two-component medium. Within the computational domain, the spatial distribution of the liquid phase was characterized by a step function, denoted as F (x, y, z, t), with values that ranged between 0 and 1. If the entire computational grid was occupied by the liquid phase, F (x, y, z, t) was equal to one, if it was entirely occupied by the gas phase, F (x, y, z, t) was equal to zero. In cases where both the liquid and gas phases coexisted within the computational grid, 0 < F (x, y, z, t) < 1. As the value of F (x, y, z, t) changed over time, solving the transport equation for the volume fraction of the liquid phase within the grid allowed for the determination of the distribution of the two-phase system. This, in turn, enabled the tracking of the evolution of the liquid phase interface. Our simulation calculations assumed that the liquid phase was incompressible and treated the gas phase as an ideal gas governed by the gas state equation. Gravity effects were not taken into consideration. The governing equations included the continuity equation, momentum equation, and energy equation, as follows:  
\[\frac{\partial}{\partial t}(\alpha_{g}\rho_{g})+\frac{\partial}{\partial x_{j} }(\alpha_{g}\rho_{g}u_{g,j})=0 \tag{6}\]  
\[\frac{\partial}{\partial t}\alpha_{l}+\frac{\partial}{\partial x_{j}}\alpha_{ l}u_{l,j}=0. \tag{7}\]  
In the above continuity equation, the subscripts \(g\) and \(l\) correspond to the gas phase and liquid phase, respectively, and \(u\) represents velocity, \(x_{j}\) is a Cartesian coordinate component, and \(\alpha_{l}\) and \(\alpha_{g}\) denote the volume fractions of the liquid phase and gas phase, respectively, which were needed to satisfy Equation (8). \[\left\{ \begin{array}{l}\alpha_{g}+\alpha_{l}=1\\ \rho=\alpha_{g}\rho_{g}+\alpha_{l}\rho_{l}\end{array} ,\right. \tag{8}\]  
Figure 9: Computational domain for the shock-driven sub-millimeter droplet breakup.