## 3 Numerical Simulation Study of Sub-Millimeter Liquid Droplets
### Computational Analysis  
A shock wave itself exerts limited influence on the breakup of sub-millimeter liquid droplets. Instead, the primary factor is the significant relative velocity generated in the flow field surrounding a droplet after a shock wave's passage, initiating a sequence of deformations and fragmentations [33]. In practice, a droplet experiences acceleration under the influence of air that is thousands of times greater than gravitational acceleration. Consequently, gravity's effect is usually neglected in the description of droplet evolution and quantitative parameter analysis [34]. When dimensionally analyzing droplet evolution, we can assess interactions between various parameters. The dimensionless windward displacement, denoted by \(X\), is defined as the ratio of the distance moved along the  
Figure 7: Relationship between the K–H instability wave growth rates and the wavenumbers of sub-millimeter droplets with different initial diameters. (**a**) We = 19. (**b**) We = 196. (**c**) We = 713.  
airflow from the beginning of the windward to the initial droplet diameter (\(S/d_{0}\)). The dimensionless transverse deformation width, denoted as \(d_{c}\), signifies the ratio of the width of the transverse deformation along the vertical airflow direction during the droplet's evolution to the initial diameter (\((D-d_{0})/d_{0}\)). Furthermore, the dimensionless axial deformation width, denoted as \(w\), is defined as the ratio of the axial deformation width in the direction of the airflow propagation to the initial diameter (\((d_{l}-d_{0})/d_{0}\)), as illustrated in Figure 8. To eliminate the influence of airflow-related parameters, the following formulas were used to nondimensionalize the fragmentation time:  
\[T=u_{g}\frac{t}{d_{o}\sqrt{\rho_{l}/\rho_{g}}} \tag{5}\]  
where \(t\) is the dimensional time, \(d_{o}\) is the initial diameter of the droplet, and \(\rho_{l}\) and \(\rho_{g}\) represent the densities of the liquid phase and the gas phase, respectively. The moment when the shock wave contacts the droplet is defined as time zero. In the simplified representation shown in Figure 9, the computational domain model depicts a left-to-right propagating shock wave. A sub-millimeter liquid droplet is symmetrically positioned on the central axis. The shock tube's cross-sectional area is \(60\) mm \(\times\)\(60\) mm, and the experimental section has a length of \(500\) mm. This study exclusively examined the physical attributes of the flow field around sub-millimeter liquid droplets, eliminating the need for a numerical simulation of the entire shock tube structure. The simplified calculation spatial domain size was chosen such that \(-20<x/d_{0}<30\), \(-10<y/d_{0}<10\), and \(-10<z/d_{0}<10\). This approach not only mitigated the influence of reflected waves but also conserved computational resources, to some extent. Given the high flow field symmetry, a one-quarter model was selected for the numerical simulation. \"Symmetry\" boundary conditions were imposed on the two central planes containing sub-millimeter liquid droplets. At the left boundary of the computational domain, pressure inlet boundary condition was applied and at the right boundary, a pressure outlet was assigned. Other boundary conditions, which included the top and back boundaries of the computational domain, were set to symmetry. The flow field used structured grids, and the sub-millimeter liquid droplet section employed a grid gradient adaptation to improve the computational efficiency. Dynamic grid adaptation based on phase fraction gradient was implemented inANSYS Fluent V20.2. This involved adjusting the mesh dynamically by refining regions with high gradient values and coarsening regions with low gradient values. To achieve this, the refinement thresholds were set to refine the grids in areas with large gradients, while coarsening thresholds were set to revert the mesh in areas with small gradients.