# An AMR-Based Liquid Film Simulation with Surfactant Transport Using PLIC-HF Method
## 5 Numerical Results
### Two-Dimensional Single Rising Bubble  
To verify the accuracy of the proposed AMR-PLIC-HF method and the ability to solve two-phase flows, a two-dimensional rising bubble problem was simulated as a benchmark test. This is a typical benchmark test for two-phase flow methods [13; 49; 50; 51], because gravity and surface tension need to be taken into consideration. In the simulation, as illustrated by Figure 10, the size of the computational domain was \([0,1]\times[0,2]\). There was a circular bubble with a diameter of \(D=0.5\) whose center was located at \((0.5,0.5)\). It consisted of light fluid and was surrounded by heavy fluid. It was at rest at the beginning of the simulation. The upper and lower walls were non-slip walls, while the left and right walls were free-slip walls. Gravitational acceleration was applied towards the negative Y direction. Surface tension existed on the interface between light and heavy fluids. Two different sets of physical properties were tested as case A and case B, as shown in Table 1. Two different AMR grids were used to simulate both cases. The initial tree node was \(4\times 8\), so the initial resolution was \(16\times 32\). For maximum AMR refinement levels 4 and 5, the finest resolutions were \(256\times 512\) and \(512\times 1024\). One NVIDIA GeForce RTX 3080 GPU and one 12-core 2.1 MHz Intel Core i7-12700F CPU were used in the current simulation. Figure 9: Time evolution of the total cell number on the AMR grid. Maximum AMR refinement level 2 with a finest resolution of \(256\times 256\) is shown in blue, and maximum AMR level 3 with a finest resolution of \(512\times 512\) is shown in pink. A typical velocity was chosen, \(U=0.5\), and to make sure that the low Mach number condition was satisfied during the simulation, an initial Mach number of \(Ma=0.03\) was set. The speed of sound was constant in this simulation and was set according to Equation (3) as  
\[c_{s}=\frac{U}{Ma}=\frac{0.5}{0.03}\simeq 16.67 \tag{62}\]  
Since the computational domain was enclosed by walls and Neumann boundary conditions were applied for pressure at all the walls, the acoustic waves could easily be reflected and were hard to dampen. The acoustic sound waves caused obvious oscillations in the bubble rising velocity. In order to avoid the impact on the initial pressure field, for both simulations and AMR settings, the following pressure Poisson equation is solved by the red-black successive over-relaxation (SOR) method in the first five time steps:  
\[\
abla\cdot\left(\frac{1}{\rho}\
abla p\right)=\frac{\
abla\cdot\mathbf{u}}{\Delta t} \tag{63}\]  
The convergence criterion of the pressure Poisson equation is  
\[\left|\
abla\cdot\left(\frac{1}{\rho}\
abla p\right)-\frac{\
abla\cdot\mathbf{u}}{ \Delta t}\right|\leq 10^{-7} \tag{64}\]  
Since the pressure Poisson equation was only solved in the first five time steps, the influence on the overall computational performance was negligible. The evolving pressure projection method [30] was also introduced to the present simulations with 10 iterations in every time step. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Case** & \(\rho_{Heavy}\) & \(\rho_{light}\) & \(\rho_{heavy}\) & \(\mu_{light}\) & \(g\) & \(\sigma\) \\ \hline Case A & 1000.0 & 100.0 & 10.0 & 1.0 & 0.98 & 24.5 \\ Case B & 1000.0 & 1.0 & 10.0 & 0.1 & 0.98 & 1.96 \\ \hline \end{tabular}
\end{table} 
Table 1: Physical properties of the two-dimensional rising bubble problem. Figure 10: A schematic diagram of the two-dimensional rising bubble problem as a benchmark test. The region in light blue is occupied by heavy fluid and the region in white is occupied by light fluid. The shapes of the interface at \(t=3.0\) in case A and case B with AMR grids with different of maximum AMR refinement levels are shown in Figure 11. In case A, both AMR grids generated similar shapes. In case B, the higher resolution AMR grid gave a result with better agreement with [49].