Consistent two-phase Lattice Boltzmann model for gas-liquid systems

Abstract: A new lattice Boltzmann method for simulating multiphase flows is developed theoretically. The method is adjusted such that its continuum limit is the Navier-Stokes equation, with a driving force derived from the Cahn-Hilliard free energy. In contrast to previous work, however, the bulk and interface terms are decoupled, the former being incorporated into the model through the local equilibrium populations, and the latter through a forcing term. We focus on gas-liquid phase equilibria with the possibility to implement an arbitrary equation of state. The most novel aspect of our approach is a systematic Chapman-Enskog expansion up to the third order. Due to the third-order gradient in the interface forcing term, this is needed for full consistency with both hydrodynamics and thermodynamics. Our construction of a model that satisfies all conditions is based upon previous work by Chen, Goldhirsch, and Orszag (J. Sci. Comp. 34, 87 (2008)), and implies 59 and 21 velocities in three and two dimensions, respectively. Applying the conditions of positivity of weights, existence of a two-phase region in the phase diagram, and positivity of the bulk viscosity, we find substantial restrictions on the permitted equation of state, which can only be lifted by an even more refined model. Moreover, it turns out that it is necessary to solve a self-consistent equation for the hydrodynamic flow velocity, in order to enforce the identity of momentum density and mass current on the lattice. The analysis completely identifies all spurious terms in the Navier-Stokes equation, and thus shows how to systematically eliminate each of them, by constructing a suitable collision operator. (continued - see article PDF)