On the lattice Boltzmann method for multiphase flows with large density ratios

is a reduced-order kinetic model to reproduce the Navier-Stokes hydrodynamics (and beyond) at a macroscopic level. Since the pioneering work of McNamura and Zanetti (1988), the LB method has been extended to various complex flows involving, for example, multicomponent and interfacial phenomena , and has been particularly successful for multiphase flows in complex geometry and porous media (Succi 2001; Chen & Doolen 1998). However, in computation of multiphase flows, its application range has been limited because of numerical instability, especially for flows with large density ratio between different phases. The objective of this study is to develop a more stable and efficient LB method for multiphase flows with large density ratios. Most multiphase LB methods that have been developed so far that an interface is numerically resolved with a few grid points across it. A specific formulation of the discrete kinetic model can be derived from a continuum kinetic equation such as the Enskog equation or can be constructed a posteriori in such a way that the corresponding macroscopic equation follows a desired hydrodynamic equation for multi-phase flows. The former approach includes the model of Luo and Girimaji (2003) and of He et al. (1997), while the latter includes the Shan-Chen model (Shan & Chen 1993) and the free energy model (Swift et al. 1996). Typically, the density ratio that can be simulated by these methods is restricted to O(10). Several approaches have been developed to resolve the large density ratio problem in the LB method (Inamuro et al. 2004; Lee & Lin 2005; Zheng et al. 2006). Of these remedies, all the existing methods that can simulate density ratios up to O(1000) are based on the use of the order parameter to describe the interface dynamics. Based on the free energy LB method, Inamuro et al. (2004) employed the projection method for which the pressure Poisson equation is solved at each time step to enforce the divergence-free condition for the velocity field. Lee and Lin (2005) proposed a stable discretization scheme for the mean-field LB method of He et al. (1999). Zheng et al. (2006) proposed an LB method that reproduces the Cahn-Hillard equation for the order parameter. The nominal density is, however, used for the continuity and momentum equations and thus does not correctly describe the momentum transport in flows with large density ratios in the model of Zheng et al. (2006). In this paper, we develop a …