A lattice Boltzmann front-tracking method for interface dynamics with surface tension in two dimensions

We propose to combine the lattice Boltzmann equation (LBE) and the front-tracking (FT) method to simulate interfacial dynamics with surface tension in two dimensions (2D). In the proposed LBE-FT method, the flow is modeled by the LBE on a fixed Cartesian mesh, whereas interfaces are explicitly tracked by a set of markers that are advected by the flow. The interface curvature is computed from adjacent markers and is then used to determine surface tension according to Laplace’s law. The local capillary forces evaluated at the markers are distributed to nearby Eulerian grid points according to a “smearing” function to approximate the Dirac delta function due to Peskin. To validate the proposed LBE-FT method, we simulate (1) a circular bubble in a flow either quiescent or moving with a constant velocity and (2) the decaying capillary waves at the interface of two fluids of equal viscosities and densities. For the circular bubble, the spurious current measured in the LBE-FT is weaker than that observed in the standard volume of fluid method and some existing LBE models. For the capillary waves, the numerical results of the period T (or frequency ω), the attenuation rate γ, the surface tension σ and the root-mean-square error of the wave amplitude agree well with the normal-mode and Prosperetti’s solutions. The proposed LBE-FT method is shown to have a second-order rate of convergence. We also show that the LBE-FT method is superior to the existing lattice BGK diffusive interface method in terms of accuracy of interface representation, numerical stability and computational efficiency.