## 4 Numerical Results
### 2D Droplet Impinging on a Thin Liquid Film
The resulting mesh cardinalities for A, B, and C are 1054, 4216, and 16864, respectively. Moreover, in each setup, a different parameter \(\alpha\) is set in order to obtain an interface thickness proportional to the mesh element dimension \(h\). Accordingly, \(\alpha=25\), 50, and 100 are fixed for setups A, B, and C, respectively. In Figure 7, for each setup, we depict the vertical velocity component and the smoothed initial density fields together with the computational mesh. Due to the high-density ratio \(r_{\rho}\), the hyperbolic tangent transformation (13) is applied for the change of density variable. Space discretization is performed with a DG polynomial approximation of degree four (\(k=4\)), while time integration with the ESDIRKS(8) scheme and the time adaptation algorithm. According to the boundary conditions setup, no mass can escape from the computational domain and the mass of fluids contained therein should be conserved. However, as pointed out in Section 3.3, and clearly shown for the density waves test case, due to the change of density variable, a mass discrepancy is introduced by space and time integration errors. Table 2 reports (for each setup) the relative mass conservation error computed at the simulation end time \(t_{f}=0.70\) together with the average time step size resulting from the adaptive temporal strategy. Time is made dimensionless with the droplet diameter and droplet initial vertical velocity. Results show that despite the current approach does not conserve the mass up to machine precision, the mass error is still small, being lower than 3 \(\times 10^{-8}\) for all the setups. Figure 8 reports the time evolution of the droplet impingement obtained with the three different setups. In particular, each column shows the results of setups A (left), B (middle), and C (right), while each row represents a different time of the flow problem: 0.45, 0.50, 0.55, and 0.70 in sequence. At time 0.45 (first row), the early droplet-film interaction, when both the bottom of the droplet and the top of the film start to deform, is appreciable. At time 0.50 (second row), the generation of the lamella is observable at the edge of the liquid-liquid contact line. Moreover, a thin layer of air is trapped between the droplet and the film. Then, the lamella develops and moves to the left pushed by the droplet penetration into the thin layer, as visible at time 0.55 (third row). The final time of the simulation 0.70 (last row) shows a fully developed lamella, which overlooks the liquid film for a global length comparable to the droplet radius, and the entrapped layer of air stretched between the two liquid regions. Roughly speaking, all three computational setups allow the reproduction of the flow behavior. However, a finer grid and, thus, thinner liquid-gas interface, provides better resolution. For example, notice that C provides reduced thickness of both the air layer and the water lamella compared to A and B setups. The thin layer of air is nonphysical as the addition of surface tension, here neglected, would lead to the formation of small air bubbles trapped in the water. Figure 7: Impinging droplet—vertical velocity component and density initial solutions and computational grid for the setup A (**left**), B (**center**), and C (**right**). \begin{table}
\begin{tabular}{c c c} \hline
**Setup** & \(\overline{\Delta t}\) & \(\boldsymbol{err_{\mathcal{M}}}\) \\ \hline A & 1.55 \(\times 10^{-3}\) & 1.39 \(\times 10^{-9}\) \\ B & 1.03 \(\times 10^{-3}\) & 1.04 \(\times 10^{-9}\) \\ C & 7.89 \(\times 10^{-4}\) & 2.43 \(\times 10^{-8}\) \\ \hline \end{tabular}
\end{table} 
Table 2: Impinging droplet—average time step width and relative mass conservation error computed at the simulation end time \(t_{f}=0.70\) for each setup.