A stable discretization of the lattice Boltzmann equation for simulation of incompressible two-phase flows at high density ratio

A stable discretization of the lattice Boltzmann equation (LBE) for non-ideal gases is presented for simulation of incompressible two-phase flows having high density and viscosity ratios. The stiffness of the discretized forcing terms in LBE for non-ideal gases is known to trigger severe numerical instability and restrict practical application of the LBE method. Use of a proper pressure updating scheme is also crucial to the stability of the LBE method because of non-negligible pressure variation across the phase interface. To deal with these issues, we propose a stable discretization scheme, which assumes the low Mach number approximation, and utilizes the stress and potential forms of the surface tension force, the incompressible transformation, and the consistent discretization of the intermolecular forcing terms. The proposed stable discretization scheme is applied to simulate 1-D advection equation with a source term, a stationary droplet, droplet oscillation and droplet splashing and deposition on a thin film at a density ratio of 1000 with varying Reynolds numbers. The numerical solutions of stationary and oscillatory droplets agree well with analytic solutions including the Laplace's law. The time history of the spread factor of the liquid sheet emitted after the droplet impact also follows the known spreading power law.