# Lattice Bgk Model for Incompressible Navier–Stokes Equation

Powerful Essays
Topics: Fluid dynamics
Journal of Computational Physics 165, 288–306 (2000) doi:10.1006/jcph.2000.6616, available online at http://www.idealibrary.com on

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.

References: 1. Reference removed in proofs. 2. S. Chen and G. Doolen, Lattice Boltzmann method for ﬂuid ﬂows, Ann. Rev. Fluid Mech. 30, 329 (1998). 3. S. Chen, D. Martinez, and R. Mei, On boundary conditions in lattice Boltzmann methods, Phys. Fluids. 8, 2527 (1996). 4. M. Coutanceau and R. Bouard, Experimental determination of the main features of the viscous ﬂow in the wake of a circular cylinder in uniform translation. 1. Steady ﬂow, J. Fluid Mech. 79, 231 (1977). 5. W.-N. E, and J. Liu, Essential compact scheme for unsteady viscous incompressible ﬂows, J. Comput. Phys. 126, 122 (1996). 6. O. Filippova and D. H¨ nel, Grid reﬁnement for lattice-BGK models, J. Comput. Phys. 147, 219 (1998). a 7. U. Frisch, D. d’Humi´ res, B. Hasslacher, P. Lallemand, Y. Pomeau, and J.-P. Rivet, Lattice gas hydrodynamics e in two and three dimensions. Complex Syst. 1, 649 (1987). 8. U. Ghia, K. N. Ghia, and C. T. Shin, High-Re solutions for incompressible ﬂow using the Navier–Stokes equations and a multigrid method, J. Comput. Phys. 48, 387 (1982). 9. X. He and L.-S. Luo, Lattice Boltzmann model for the incompressible Navier–Stokes equation, J. Stat. Phys. 88, 927 (1997). 10. X. He and G. Doolen, Lattice Boltzmann method on curvilinear coordinates system: Flow around a circular cylinder, J. Comput. Phys. 134, 306 (1997). 11. F. J. Higuera and S. Succi, Simulating the ﬂow around a circular cylinder with a lattice Boltzmann equation, Europhys. Lett. 8, 517 (1989). 12. S. Hou and Q. Zou, Simulation of cavity ﬂow by the lattice Boltzmann method, J. Comput. Phys. 118, 329 (1995). 306 GUO, SHI, AND WANG 13. Z. Lin, H. Fang, and R. Tao, Improved lattice Boltzmann model for incompressible two-dimensional steady ﬂows, Phys. Rev. E 54, 6323 (1997). 14. D. O. Martinez, W. H. Matthaeus, S. Chen, and D. C. Montgomery, Comparison of spectral method and lattice Boltzmann simulations of two-dimensional hydrodynamics, Phys. Fluids. 6, 1285 (1994). 15. R. Mei and Q. Shyy, On the ﬁnite difference-based lattice Boltzmann method in curvilinear coordinates, J. Comput. Phys. 143, 426 (1998). 16. F. Nieuwstadt and H. B. Keller, Viscous ﬂow past circular cylinders, Comput. Fluids. 1, 59 (1973). 17. Y. Qian, D. d’Humi´ res, and P. Lallemand, Lattice BGK models for Navier–Stokes equation, Europhys. Lett. e 17, 479 (1992). 18. R. Schreiber and H. Keller, Driven cavity ﬂow by efﬁcient numerical techniques, J. Comput. Phys. 49, 310 (1983). 19. S. P. Vanka, Block-implicit multigrid solution of Navier–Stokes equations in primitive variables, J. Comput. Phys. 65, 138 (1986). 20. L. Wagner, Pressure in lattice Boltzmann simulations of ﬂow around a cylinder, Phys. Fluids. 6, 3516 (1994). 21. L. Wagner and F. Hayot, Lattice Boltzmann simulations of ﬂow past a cylindrical obstacle, J. Stat. Phys. 81, 63 (1995). 22. Q. Zou, S. Hou, S. Chen, and G. Doolen, An improved incompressible lattice Boltzmann model for timeindependent ﬂows, J. Stat. Phys. 81, 35 (1995).

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