High velocity flow simulation using lattice Boltzmann method with no-free-parameter dissipation scheme

  • J. H. B. Erdembilegt
  • Wei-bing Feng (封卫兵)
  • Wu Zhang (张 武)
Information Technology
First Online:
Received:
Revised:

Abstract

A finite difference lattice Boltzmann method of second-order accuracy in time is developed based on non-oscillatory scheme with no-free-parameter dissipation (NND) difference scheme in this paper. The NND lattice Boltzmann method is used to simulate high-speed flows by constructing a new equilibrium distribution function of the lattice Boltzmann method. Compared with a variation of lattice Boltzmann method developed by Qu, et al., the present method can capture shock waves and handle oscillations of high velocity flows accurately in larger time steps and in shorter computing time. Numerical results indicate the correctness and capability of simulating shock wave interactions of the NND lattice Boltzmann method.

Keywords

lattice Boltzmann method (LBM) no-free-parameter dissipation (NND) scheme high velocity flow 
This is a preview of subscription content, log in to check access.

References

  1. [1]
    Hardy J, de Phazzis O, Pomeau Y. Molecular dynamics of a classical lattice gas: Transport properties and time correlation functions [J]. Physical Review A, 1976, 13(5): 1949–1961CrossRefGoogle Scholar
  2. [2]
    Frisch U, Hasslacher B, Pomeau Y. Lattice-gas automata for the Navier-Stokes equation [J]. Physical Review Letters, 1986, 56(14): 1505–1508.CrossRefGoogle Scholar
  3. [3]
    Qian Y H, d’Hummieries D, Lallemand P. Lattice BGK models for Navier-Stokes equation [J]. Europhysics Letters, 1992, 17(6): 479–484.MATHCrossRefGoogle Scholar
  4. [4]
    Bhatnagar P L, Gross E P, Krook M. A model for collision processes in gases, I: Small amplitude processes in charged and neutral one component systems [J]. Physical Review, 1954, 94(3): 511–525.MATHCrossRefGoogle Scholar
  5. [5]
    Shakov E M. Generalization of the krook kinetic relaxation equation [J]. Fluid Dynamics, 1968, 3(5): 142–145.Google Scholar
  6. [6]
    Wolf-Gladrow D A. Lattice-gas cellular automata and lattice Boltzmann models [M]. Berlin: Springer-Verlag, 2000.Google Scholar
  7. [7]
    Sukop M, Thorne D T. Lattice Boltzmann modeling: An introduction for geoscientists and engineers [M]. Berlin: Springer-Verlag, 2005.Google Scholar
  8. [8]
    Succi S. The lattice Boltzmann equation for fluid dynamics and beyond [M]. Oxford: Oxford University Press, 2001.Google Scholar
  9. [9]
    He X, Luo L S. A priori derivation of the lattice Boltzmann equation [J]. Physical Review E, 1997, 55(6): 6333–6336.CrossRefGoogle Scholar
  10. [10]
    Shan X W, Yuan X F, Chen H D. Kinetic theory representation of hydrodynamics: A way beyond Navier-Stokes equation [J]. Journal of Fluid Mechanics, 2006, 550: 413–441.MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    Qu K, Shu C, Chew Y T. Alternative method to construct equilibrium distribution functions in lattice Boltzmann method simulation of inviscid compressible flows at high Mach number [J]. Physical Review E, 2007, 75(3): 6706–6718.CrossRefMathSciNetGoogle Scholar
  12. [12]
    Alexander F J, Chen H, Chen S, Doolen G D. Lattice Boltzmann model for compressible fluids [J]. Physical Review A, 1992, 46(4): 1967–1970.CrossRefGoogle Scholar
  13. [13]
    Chen Y, Ohashi H, Akiyama M. Thermal BGK model without non-linear deviations in macrodynamic equations [J]. Physical Review E, 1994, 50(4): 2776–2783.CrossRefGoogle Scholar
  14. [14]
    Yan G W, Chen Y S, Hu S H. Simple lattice Boltzmann model for simulating flows with shock wave [J]. Physical Review E, 1999, 59(1): 454–459.CrossRefGoogle Scholar
  15. [15]
    Shi W, Shyy W, Mei R. Finite-difference-based lattice Boltzmann method for inviscid compressible flows [J]. Numerical Heat Transfer, Part B, 2001, 40(1): 1–21.Google Scholar
  16. [16]
    Kataoka T, Tsutahara M. Lattice Boltzmann model for the compressible Euler equations [J]. Physical Review E, 2004, 69(5): 1–14.CrossRefMathSciNetGoogle Scholar
  17. [17]
    Kataoka T, Tsutahara M. Lattice Boltzmann model for the compressible Navier Stokes equations with flexible specific heat ratio [J]. Physical Review, 2004, 69(3): 1–4.MathSciNetGoogle Scholar
  18. [18]
    Xu K. A gas kinetic BGK scheme for the Navier-Stokes equations and its connections with artificial dissipation and godunov method [J]. Journal of Computational Physics, 2001, 171(1): 289–335.MATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    Sun C H. Lattice-Boltzmann models for high speed flows [J]. Physical Review E, 1998, 58(6): 7283–7287.CrossRefGoogle Scholar
  20. [20]
    Zhang Han-xin. The exploration of the spatial oscillations in finite difference solutions for navier stokes shocks [J]. Acta Aerodynamica Sinica, 1988, 6(1): 12–19 (in Chinese).Google Scholar
  21. [21]
    Zhang Han-xin. NND scheme [J]. Acta Aerodynamica Sinica, 1984, 2(1): 143–165 (in Chinese).Google Scholar
  22. [22]
    Wu W Y, Cai Q D. A new NND difference scheme of second order in time and space [J]. Applied Mathematics and Mechanics, 2000, 21(6): 617–630.MATHCrossRefMathSciNetGoogle Scholar
  23. [23]
    Chen S Y, Doolen G D. Lattice Boltzmann method for fluid flows [J]. Annual Review of Fluid Mechanics, 1998, 30(1): 329–364.CrossRefMathSciNetGoogle Scholar
  24. [24]
    Anderson J D. Modern compressible flow: with historical perspective [M]. New York: McGraw-Hill Science, 2002.Google Scholar
  25. [25]
    Einfeldt B, Munz C D, Roe P L, Sjogreen B. On godunov type methods near low densities [J]. Journal of Computational Physics, 1991, 92(2): 273–295.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Shanghai University and Springer Berlin Heidelberg 2009

Authors and Affiliations

  • J. H. B. Erdembilegt
    • 1
  • Wei-bing Feng (封卫兵)
    • 1
  • Wu Zhang (张 武)
    • 1
  1. 1.School of Computer Engineering and ScienceShanghai UniversityShanghaiP. R. China

Personalised recommendations