## 3 Numerical Results and Discussion
### Droplet Splashing on a Thin Film
To examine the correctness of the results, the spread radius was measured. A previous study [46] shows that, when splashing occurs, the spread radius obeys a power law \(r/(2R)\approx\sqrt{Ut/(2R)}\). Figure 22 shows the comparison of the present impact radius with different Reynolds numbers. It can be seen that the present results conform to the prediction of the law. On this basis, we changed the initial conditions of the droplet splashing to determine the influence of the height of the thin film and the Weber number. We chose the condition of \(Re=800\), \(We=150\), and \(H=15\) as the benchmark. All other simulations were transformed from this. Figure 23 shows the evolution of the interface shapes for the droplet splashing on a thin film at \(Re=800\), \(We=150\), and \(H=5\) or \(H=25\) with the present approach. According to Figure 23a, the height of the liquid film is pretty small and, after the droplet splashes, the shape of the sputtering liquid between the droplet and the film is straighter than the others shown in Figures 20 and 23b. It can be seen that the angle of \(H=5\) between the sputtering liquid and the film is smaller than the angles of \(H=15\) and \(H=25\) at T = 0.6 and T = 1.0. As time goes by, the clearer circle droplet appears from the top of the sputtering liquid. As for Figure 23b, the liquid deformation of the droplet and the film is similar to the condition of \(H=15\) in Figure 20; however, the angle between the sputtering liquid and the film with \(H=25\) is a little larger than the one with \(H=15\). In conclusion, the angle is bigger when the height is larger with other parameters unchanged. The film prevents the liquid from splashing after impacting, and the height of the film has a great effect when the height is pretty small. It can be speculated that the resistance and the kinetic energy loss will be larger during splashing when the film is thicker. However, the influence of the resistance and the kinetic energy loss is limited in a range such that the angle will remain ultimately unchanged even if the height is larger, according to Figures 20 and 23. For the same droplet, the simulation results for the interface shape will be almost the same when the height of the film is large enough. Figure 24 shows the development of the fluid interface with \(Re=800\), \(H=15\), and \(We=10\) or \(We=50\) with the present method. According to Figure 24a, for the case of \(We=10\), the peak of the liquid is almost absent at T = 0.2, and then, at T = 0.6, the liquid forms a hump instead of a peak. The liquid forms a wave rather than a clear liquid finger, as in Figures 20 and 24b. In this condition, the fluid movement is greatly affected by the surface tension, which limits the droplet splashing on the film. As for Figure 24b with  
Figure 22: Comparison of the predicted spread radius with different Reynolds numbers. \(We=50\), the peak of the liquid appears at T = 0.2; however, at T = 0.6, the shape of the liquid finger from the location impacted is shorter and thicker compared with the condition where \(We=150\). At T = 1.0, the shape of the liquid finger is fatter and the height of the splashing liquid is lower than the condition where \(We=150\). Differently from the small droplet at the top of the splashing liquid with \(We=150\) at T = 1.0, the droplet is larger and has a tighter link with the liquid film. In general, the surface tension prevents the liquid from splashing outward, and it is difficult for the small droplet to come into being after the impact when the surface tension is relatively large. The kinetic energy loss is larger when the Weber number is larger, such that the height of the splashing droplet is lower and the small droplet cannot arise or break away from the liquid due to the surface tension. By simulating the multiphase flows of different conditions, the reliability of the present method was validated. The improved numerical method has great potential for use in practical applications involving multiphase flows with large density ratios.