# An AMR-Based Liquid Film Simulation with Surfactant Transport Using PLIC-HF Method
## 1 Introduction
Some numerical methods, including the volume-of-fluid (VOF) method [12], level set method [13], phase field method [14], and coupled level set and VOF (CLSVOF) method [15], can describe large amounts of interface topological deformation. Considering the good conservation of mass achieved by the phase field method and the high accuracy of the complex geometric calculation in the level set method, the previous work by [PERSON] and [PERSON] proposed a coupled phase field and level set method that combines the advantages of the two methods [16]. The phase field method describes the interface between different phases as a diffusive interface with a certain thickness. In liquid film simulations, such a diffusive interface could bring uncertainty to the simulation and may not describe the true physics. Based on the VOF method, a piecewise linear interface calculation (PLIC) method was adopted in the present research. The PLIC method has become a widely used technique in computational fluid dynamics and has been applied to a range of problems involving the simulation of free-surface flows and complex fluid-fluid interactions [17; 18; 19]. We believe that the sharp interface described by the PLIC method is appropriate for use in the present simulation of liquid films. The curvature was usually calculated by level set in previous work [16; 20] due to the high accuracy of this method. However, the level set was not used as the main variable to capture the interface. The calculation of the volume fraction to height function is straightforward [21; 22; 23]. Thus, in the present research, the interface curvature was calculated by the Height Function (HF) method. It has been reported that the spurious current [24] can be greatly decreased by the present PLIC-HF method [19]. Third, the method used to solve fluid dynamics problems needs to be considered. Derived from the mesoscopic kinetic theory of gases, the lattice Boltzmann method has developed into an effective CFD method in the past two decades and can be used for research and applications [25; 26]. In contrast to the Navier-Stokes equation, which is based on macroscopic conservation law, The LBM is derived from the dynamics of fluid particles and includes collision and propagation. It is modeled by a statistical distribution function to avoid the direct tracking of each particle, as occurs in molecular dynamics methods. The most notable feature of the LBM is its explicit time advancement and local spatial dependence, which enable it to fully exploit the potential of parallel computers. Inspired by the standard LBM designed for isothermal weakly compressible fluids, weakly compressible Navier-Stokes equations can also be used to simulate incompressible flows. By introducing an isothermal state equation and ignoring density fluctuations at low Mach numbers, an independent time evolution equation of pressure can be obtained [27; 28]. This hyperbolic equation of pressure can be explicitly integrated in time through a local spatial stencil. Despite the limitation of the time step size by the speed of sound, this method greatly benefits scalable calculations. Weakly compressible Navier-Stokes equations have been successfully applied to turbulent flows [29] and two-phase flows [16]. The calculation of pressure Poisson equation is necessary for incompressible Navier-Stokes solvers. This method brings instability to the computation, and convergence in large-density-ratio two-phase flow simulations is compromised. Thus, a weakly compressible scheme [16] and evolving pressure projection method [30] were used in the current simulation. The purpose of the present research was to understand the influence of surfactant transport on the behavior of liquid film. The PLIC-HF method was used as interface capturing method and implemented on an AMR grid with extended stencil computation. The weakly compressible scheme and evolving pressure projection method was used as a fluid solver. A Langmuir model [31; 32; 33; 34] was introduced to calculate surfactant transport and the Marangoni effect.