## 3 Results and Discussion
### The High Weber Number Conditions
concluded that R-T instability does not occur at the front end of the windward side (\(-30\)-\(30^{\circ}\) region) but did not explicitly state whether R-T instability exists at the windward edge of the droplet. Regarding the difference between \(We=184\) and \(We=696\) conditions in the windward side of the droplet breakup step, it has been analyzed that the interfacial tension can inhibit the development of K-H instability at \(We=184\) due to a narrower gas-liquid velocity difference. In addition, the droplet first experience compressional deformation at \(We=184\), and the change of droplet shape led to the reduction of gas flow shear velocity, thus limiting the development of K-H instability. Moreover, at \(We=696\), the increased gas flow velocity causes the droplet to fold on the windward side. Reducing the droplet diameter under the same \(We\) condition enhances the difficulty of droplet breakup and lengthens the deformation time so that the droplet can experience more deformation behaviors. Figure 9 shows the evolutionary morphology and time scale of small and large droplets (experimental photographs taken by [PERSON] et al. [11]) during deformation under the same \(We\). It can be found that when the droplet diameter decreases, the droplet takes longer to undergo deformation before it breaks up. As the droplet diameter decreases, its curvature increases, making it more difficult to undergo deformation under external forces, leading to the above changes. The droplet move in the \(x\)-axis direction under the action of high-speed gas flow after the shock wave impacts the droplet, and the windward displacement can be a good measure of the droplet deflection. The windward displacement of the droplet is nondimensionalized by Equation (5):  
\[X=\frac{S}{d_{0}} \tag{5}\]  
Here \(S\) is the windward displacement of the droplet, and \(d_{0}\) is the initial droplet diameter. Figure 10 illustrates the windward displacement of a hundred-micron droplet at low \(We\) and high \(We\). As \(We\) increases, the droplet's windward displacement growth rate gradually decreases. Still, the droplet's windward displacement is not simply uniformly accelerated, as depicted in Figure 10a. Under the \(We=184\) condition, the droplet undergoes bag and stamen breakup at \(\tau=2.1245\). Prior to this event, the windward displacement of the droplet is significantly slower than that of the bag breakup at \(We=15\), and the gap between them expanded with the deformation of the droplet. After the breakup, the  
Figure 9: The evolutionary morphology and time scale of small and large droplets (experimental photographs taken by [PERSON] et al.). difference in their windward displacement decreases, which indicates that the bag and stamen breakup has faster windward acceleration after the breakup than before. This is because the droplet edge moves much faster than the stamen under the action of R-T instability, and the droplet mass hardly decreases while the droplet area in the vertical gas flow direction increases during the droplet film formation and the gradual development of the bag structure. The stamen needs to maintain the liquid film from breakup under interfacial tension, so the gap between the displacement of the windward of _We_ = 19 and _We_ = 15 increases with the deformation of the droplet. After \(\tau\) = 2.1245, the droplets are left with only the stamers, which move at a faster acceleration under the action of gas flow. The gas velocity behind the wave of _We_ = 184 and _We_ = 696 are 184.96 m/s and 289.32 m/s, respectively, which are substantial differences. However, we can find from Figure 10b that differences in windward displacement between sheet-thinning breakup and catastrophic breakup are minimal, especially in the deformation step and the early breakup step.