Lattice Boltzmann simulation of multiphase ¯uid ¯ows through the total variation diminishing with arti®cial compression scheme Shulong Teng * , Yu Chen, Hirotada Ohashi Department of Quantum Engineering and Systems Science, Faculty of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, 113-8656 Tokyo, Japan Received 7 May 1999; accepted 17 August 1999 Abstract The total variation diminishing with arti®cial compression (TVD/AC) scheme is applied to the lattice Boltzmann multiphase model in order to introduce a new technique to solve the traditional lattice Boltzmann equation. The TVD/AC scheme gives a much higher resolution than the well-known TVD scheme to the interface in the simulation of multiphase ¯ows. Numerical simulations also show that the new simulator is helpful in stabilizing the computation in the runs of high-density ratios. Numerical results for the coexistence curve and veri®cation of the Laplace law both in two and three spatial dimensions are presented. The detailed dynamical behaviors of the interface over a wide range of density ratios such as the variation of interface thickness and pro®les of density, the balance of pressure and interfacial stress, the distribution of spurious velocities and so on are studied. Phase separation in two- and three-dimensional systems is also demonstrated numerically. Ó 2000 Elsevier Science Inc. All rights reserved. Keywords: Multiphase ¯ow; Lattice Boltzmann method; TVD/AC scheme; Interface; Numerical simulation 1. Introduction Multiphase ¯ows are dicult to study both from the physical and computational points of view due to the com- plexity of the physics especially involved in the interfacial dynamics. In the past few years, the lattice Boltzmann method (LBM) (Benzi et al., 1992; Qian et al., 1995; Chen and Doolen, 1998) has been an attractive alternative to solve ¯uid ¯ow, taking advantage of its parallel nature, simple algorithm and easy to implement complicated boundary conditions. It also provides a method to model multiphase ¯ows at the micro- scopic scale through incorporating the non-local interaction of particles and meanwhile can fully recover the Navier±Stokes equations at the macroscopic scale. International Journal of Heat and Fluid Flow 21 (2000) 112±121 www.elsevier.com/locate/ijh Notation c microscopic particle velocity f distribution function F force experienced by molecules l length L anti-diusion term M minmod function p pressure Q function helpful to eliminate the so-called entropy violation R radius Re Reynolds number S sign function t time T temperature u velocity We Weber number x Cartesian coordinate Greeks a surface tension coecient e d h discretization error j dimensionless collision frequency k constant parameter that controls the strength of the surface tension eect m kinematic viscosity q density r 1 interfacial stress tensor r m viscous stress tensor n non-dimensional number Subscripts c critical i index r reference a; b; c spatial directions in Cartesian coordinates Superscript n time step * Corresponding author. E-mail address: teng@crimson.q.t.u-tokyo.ac.jp (S. Teng). 0142-727X/00/$ - see front matter Ó 2000 Elsevier Science Inc. All rights reserved. PII: S 0 1 4 2 - 7 2 7 X ( 9 9 ) 0 0 0 6 8 - 5