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.

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