Analytic solutions of simple flows and analysis of nonslip boundary conditions for the lattice Boltzmann BGK model


In this paper we analytically solve the velocity of the lattice Boltzmann BGK equation (LBGK) for several simple flows. The analysis provides a framework to theoretically analyze various boundary conditions. In particular, the analysis is used to derive the slip velocities generated by various schemes for the nonslip boundary condition. We find that the slip velocity is zero as long as Σαfαeα=0 at boundaries, no matter what combination of distributions is chosen. The schemes proposed by Nobleet al. and by Inamuroet al. yield the correct zeroslip velocity, while some other schemes, such as the bounce-back scheme and the equilibrium distribution scheme, would inevitably generate a nonzero slip velocity. The bounce-back scheme with the wall located halfway between a flow node and a bounce-back node is also studied for the simple flows considered and is shown to produce results of second-order accuracy. The momentum exchange at boundaries seems to be highly related to the slip velocity at boundaries. To be specific, the slip velocity is zero only when the momentum dissipated by boundaries is equal to the stress provided by fluids.

This is a preview of subscription content, access via your institution.


  1. 1.

    D. R. Noble, S. Chen, J. G. Georgiadis, and R. O. Buckius, A consistent hydrodynamics boundary condition for the lattice Boltzmann method.Phys. Fluids 7:203 (1995).

    Google Scholar 

  2. 2.

    D. R. Noble, J. G. Georgiadis, and R. O. Buckius, Direct assessment of lattice Boltzmann hydrodynamics and boundary conditions for recirculating flows,J. Stat. Phys. 81:17 (1995).

    Google Scholar 

  3. 3.

    T. Inamuro, M. Yoshino, and F. Ogino, A non-slip boundary condition for lattice Boltzmann simulations,Phys. Fluids 7:2928 (1995); Erratum,8:1124 (1996).

    Google Scholar 

  4. 4.

    G. McNamara and G. Zanetti, Use of the Boltzmann equation to simulate lattice-gas automata,Phys. Rev. Lett. 61:2332 (1988).

    Google Scholar 

  5. 5.

    F. Higuera and J. Jimenez, Boltzmann approach to lattice gas simulationsEurophys. Lett. 9:663 (1989).

    Google Scholar 

  6. 6.

    H. Chen, S. Chen, and W. H. Matthaeus, Recovery of the Navier-Stokes equations using a lattice Boltzmann method,Phys. Rev. A 45:R5339 (1991).

    Google Scholar 

  7. 7.

    Y. H. Qian, D. d'Humières, and P. Lallemand, Lattice BGK models for the Navier-Stokes equation,Europhys. Lett. 17:479 (1992).

    Google Scholar 

  8. 8.

    R. Benzi, S. Succi, and M. Vergassola, The lattice Boltzmann equation: Theory and applications,Phys. Rep. 222:145 (1992).

    Google Scholar 

  9. 9.

    X. Shan and H. Chen, Lattice Boltzmann model for simulating flows with multiple phases and components,Phys. Rev. E 47: 1815 (1993); Simulation of non-ideal gases and liquid-gas phase transitions by the lattice Boltzmann equation,Phys. Rev. E 49:2941 (1994).

    Google Scholar 

  10. 10.

    A. J. C. Ladd, Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part I. Theoretical foundation,J. Fluid Mech. 271:285 (1994); Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part II. Numerical results,J. Fluid Mech. 271:311 (1994).

    Google Scholar 

  11. 11.

    S. Chen, H. Chen, and W. H. Matthaeus, Lattice Boltzmann magnetohydrodynamics,Phys. Rev. Lett. 67:3776 (1991).

    Google Scholar 

  12. 12.

    S. P. Dawson, S. Chen, and G. Doolen, Lattice Boltzmann computations for reaction-diffusion equations,J. Chem. Phys. 98:1514 (1993).

    Google Scholar 

  13. 13.

    S. Hou, Q. Zou, S. Chen, G. D. Doolen, and A. Cogley, Simulation of cavity flow by the lattice Boltzmann method,J. Comp. Phys. 118:329 (1995).

    Google Scholar 

  14. 14.

    P. A. Skordos, Initial and boundary conditions for the lattice Boltzmann method,Phys. Rev. E 48:4823 (1993).

    Google Scholar 

  15. 15.

    R. Cornubert, D. d'Humières, and D. Levermore, A Knudsen layer theory for lattice gases.Physica D 47(6):241 (1991).

    Google Scholar 

  16. 16.

    I. Ginzbourg and P. M. Adler Boundary condition analysis for the three-dimensional lattice Boltzmann model,J. Phys. II France 4:191 (1994).

    Google Scholar 

  17. 17.

    L.-S. Luo, H. Chen, S. Chen, G. D. Doolen, and Y.-C. Lee, Generalized hydrodynamic transport in lattice-gas automata.Phys. Rev. A 43:R7097 (1991).

    Google Scholar 

  18. 18.

    Q. Zou, S. Hou and G. D. Doolen, Analytical solutions of the lattice Boltzmann BGK model,J. Stat. Phys. 81:319 (1995).

    Google Scholar 

  19. 19.

    U. Frisch, D. d'Humières, B. Hasslacher, P. Lallemand, Y. Pomeau, and J.-P. Rivet, Lattice gas hydrodynamics in two or three dimensions,Complex Systems 1:649 (1987).

    Google Scholar 

  20. 20.

    A. K. Gunstensen and D. H. Rothman, Microscopic modeling of immiscible fluids in three dimensions by a lattice-Boltzmann method. MIT Porous Flow Project, Report No.4, 20 (1991).

    Google Scholar 

  21. 21.

    L. P. Kadanoff, G. R. McNamara, and G. Zanetti, From automata to fluid flow: Comparisons of simulation and theory,Phys. Rev. A 40:4527 (1989).

    Google Scholar 

Download references

Author information



Rights and permissions

Reprints and Permissions

About this article

Cite this article

He, X., Zou, Q., Luo, LS. et al. Analytic solutions of simple flows and analysis of nonslip boundary conditions for the lattice Boltzmann BGK model. J Stat Phys 87, 115–136 (1997).

Download citation

Key Words

  • Lattice Boltzmann BGK equations
  • nonslip boundary conditions
  • analytic solutions of simple flows