Lattice BGK Model for Incompressible Navier–Stokes Equation Zhaoli Guo,∗ Baochang Shi,† and Nengchao Wang†

∗ National Laboratory of Coal Combustion, and Department of Computer Science, Huazhong University of Science and Technology, Wuhan 430074, People’s Republic of China; and †Department of Mathematics, Huazhong University of Science and Technology, Wuhan 430074, People’s Republic of China E-mail: sbchust@public.wuhan.cngb.com Received May 10, 1999; revised February 28, 2000

Most of the existing lattice Boltzmann BGK models (LBGK) can be viewed as compressible schemes to simulate incompressible ﬂuid ﬂows. The compressible effect might lead to some undesirable errors in numerical simulations. In this paper a LBGK model without compressible effect is designed for simulating incompressible ﬂows. The incompressible Navier–Stokes equations are exactly recovered from this incompressible LBGK model. Numerical simulations of the plane Poiseuille ﬂow, the unsteady 2-D shear decaying ﬂow, the driven cavity ﬂow, and the ﬂow around a circular cylinder are performed. The results agree well with the analytic solutions and the results of previous studies. c 2000 Academic Press Key Words: Lattice BGK method; Incompressible Navier–Stokes equation.

1. INTRODUCTION

The Lattice Boltzmann BGK (LBGK) method is a new numerical scheme for simulating viscous compressible ﬂows in the subsonic regime [2]. In recent years, LBGK has achieved great success in simulations of ﬂuid ﬂows and in modeling physics in ﬂuids. Through multiscaling expansion [7], the compressible Navier–Stokes equations can be recovered from the lattice Boltzmann BGK equation on the assumptions that (i) the Mach number is small, and (ii) the density varies slowly. Therefore, theoretically the LBGK model can only be used to simulate compressible ﬂows in the incompressible limit. When used for incompressible ﬂows, it must be viewed as an artiﬁcal compressible method for the incompressible Navier–Stokes equations. In such circumstances, the LBGK solutions might depart from the direct solutions of the incompressible Navier–Stokes equations [14]; at least part of the departures might be attributable to the effects of the compressibility of the LBGK model. Some efforts have been made to reduce or eliminate such errors [7, 9, 13, 22]. However, most of these existing incompressible LBGK models can be used only to simulate steady 288

0021-9991/00 $35.00 Copyright c 2000 by Academic Press All rights of reproduction in any form reserved.

LBGK FOR INCOMPRESSIBLE NAVIER–STOKES EQUATION

289

ﬂows. By neglecting terms of higher order Mach number in the equilibrium density distribution function, He and Luo [9] proposed an incompressible LBGK model in which the distribution function is of pressure representation. From the model, the incompressible Navier–Stokes equations in artiﬁcial compressible form can be derived. He’s model can be used for both incompressible steady and unsteady ﬂows. However, before the model is used, the average pressure of the ﬂow must be speciﬁed in advance. In some cases, especially in practical problems, the average pressure is not known or cannot be prescribed precisely. Furthermore, when used to simulate unsteady incompressible ﬂows, the model requires an additional condition, T L/cs (T and L are characteristic time and length, respectively), to neglect the artiﬁcal compressible effect. Considering the signiﬁcance of the incompressible Navier–Stokes equations in theory and applications, it is necessary to establish a LBGK model which can exactly model the incompressible Navier–Stokes equations in general. It is well known that the small Mach number limit is equivalent to the incompressible limit, so it is possible to set up a LBGK model which can properly model the incompressible Navier–Stokes equations only with the...