## 3 Numerical Results and Discussion
### Droplet Splashing on a Thin Film  
A droplet splashing on a liquid film is significant in studying multiphase flows, so we studied it to test the reliability of the proposed method for solving multiphase flows with large density ratios and high Reynolds numbers. The schematic diagram of the study is illustrated in Figure 19, where a droplet moves down with the velocity \(U\) to a film without considering gravity. The heavier droplet is in an ambient lighter vapor field. The radius of the droplet is \(R\) and the height of the film is \(H\). The density and the dynamic viscosity of the liquid droplet and vapor are given respectively as \((\rho_{H},\mu_{H})\) and \((\rho_{L},\mu_{L})\). This problem can be governed by two non-dimensional parameters: the Reynolds number \(Re=\frac{2 RU}{\mu_{H}}\) and the Weber number \(We=\frac{2\rho_{H}U^{2R}}{\sigma}\), where \(\sigma\) is the surface tension. The computational zone only occupies half of the flow field due to the symmetric flow. To mimic this process, a mesh size of \(401\times 251\) was adopted. The natural boundary condition was adopted on the top wall and the right wall, while the no-slip boundary condition was adopted on the  
Figure 16: Interface positions of the bubble and spike for the Rayleigh–Taylor instability of Re = 3000 compared with references [10; 45]. bottom wall and the symmetric boundary condition was adopted on the left wall. The flow parameters were set as follows: \(R=50\), \(H=15\), \(U=0.02\), \(\rho_{H}=1\), \(\rho_{L}=0.001\), \(v_{L}=0.05\), \(\bar{\xi}=4\), and \(We=150\), while \(Re\) was defined as \(100\), \(400\), \(800\), and \(2000\). As for the coefficient \(k\) in Equation (11), \(k\) was defined as \(k=k_{1}\Big{(}\frac{\Delta x}{L}\Big{)}^{2}\) and, in this section, \(k_{1}=0.4\) and \(L=2R\). Figure 17: Development of the interface for the Rayleigh–Taylor instability of \(\mathrm{Re}=3000\) with the present approach (\(k_{1}=10\)). Figure 18: Interface shapes for Rayleigh–Taylor instability of \(\mathrm{Re}=3000\) (\(\mathrm{T}=5\)): (**a**) original method, (**b**) present method with \(k_{1}=1\), (**c**) present method with \(k_{1}=10\). Figure 20 shows the development of the interface from T = 0-1 at \(Re=800\) and \(We=150\) with the present method. It can be clearly observed that a peak appears from the location between the droplet and the film, and then a small droplet appears on the top of the peak due to the surface tension. A clear liquid finger can be found after T = 0.6. In this condition, the finger is strong enough such that the breakup of the splashing liquid cannot occur. Figure 19: Illustration of the droplet splashing on a thin film. Figure 20: Development of the interface at Re = 800 and We = 150 with the present method. Figure 21 shows a comparison of the interface shape at different Reynolds numbers with the present approach and the original DI approach. The contour lines of the order parameter are divided into 0.05, 0.5, and 0.95 to observe the interface zone of the multiphase flow. We chose the results for \(Re=100\), 800, and 2000 to examine whether the present method has a great influence on the interface zone compared with the original DI method. In the condition where \(Re=100\), the interface thickness obtained with the present approach is smaller than the thickness from the original approach. With the Reynolds number increasing, the difference in the interface thickness is larger. The interface thickness of \(Re=800\) and 2000 with the present method is much thinner, especially with the prominence of the liquid, which reflects the fact that the present method shows good improvement in the interface zone. It is proved that the present method has wonderful applicability in maintaining interfacial shapes for simulating the flows of high Reynolds numbers. Figure 21: Comparison of the interface shapes of different Reynolds number with two methods; the three contour lines from up to down: 0.05, 0.5, 0.95. (**a**) \(\text{Re}=100\), (**b**) \(\text{Re}=800\), (**c**) \(\text{Re}=2000\).