Numerical Algorithms

, Volume 45, Issue 1–4, pp 389–408 | Cite as

Stable simulation of fluid flow with high-Reynolds number using Ehrenfests’ steps

  • R. Brownlee
  • A. N. Gorban
  • J. Levesley
Original Paper


The Navier–Stokes equations arise naturally as a result of Ehrenfests’ coarse-graining in phase space after a period of free-flight dynamics. This point of view allows for a very flexible approach to the simulation of fluid flow for high-Reynolds number. We construct regularisers for lattice Boltzmann computational models. These regularisers are based on Ehrenfests’ coarse-graining idea and could be applied to schemes with either entropic or non-entropic quasiequilibria. We give a numerical scheme which gives good results for the standard test cases of the shock tube and the flow past a square cylinder.


Navier–Stokes equations Ehrenfests’ steps Numerical stabilisation 


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  1. 1.
    Ansumali, S., Karlin, I.V.: Kinetic boundary conditions in the lattice Boltzmann method. Phys. Rev., E 66(2), 026311.1–026311.6 (2002)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Ansumali, S.S, Karlin, I.V., Ottinger, H.C.: Minimal entropic kinetic models for hydrodynamics. Europhys. Lett. 63(6), 798–804 (2003)CrossRefGoogle Scholar
  3. 3.
    Ansumali, S., Chikatamarla, S.S., Frouzakis, C.E., Boulouchos, K.: Entropic lattice Boltzmann simulation of the flow past square-cylinder. Int. J. Mod. Phys. C 15, 435–445 (2004)zbMATHCrossRefGoogle Scholar
  4. 4.
    Baskar, G., Babu, V.: Simulation of the unsteady flow around rectangular cylinders using the ISLB method. In: 34th AIAA Fluid Dynamics Conference and Exhibit, pp. 2004–2651. AIAA, Washington, DC (2004)Google Scholar
  5. 5.
    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. Phys. Rev. 94(3), 511–525 (1954)zbMATHCrossRefGoogle Scholar
  6. 6.
    Brownlee, R.A., Gorban, A.N., Levesley, J.: Stabilisation of the lattice-Boltzmann method using the Ehrenfests’ coarse-graining. Phys. Rev., E 74, 037703 (2006)CrossRefGoogle Scholar
  7. 7.
    Brownlee, R.A., Gorban, A.N., Levesley, J.: Stability and stabilization of the lattice Boltzmann method. Phys. Rev., E 75(3), 036711 (2007)CrossRefGoogle Scholar
  8. 8.
    Chen, H., Chen, S., Matthaeus, W.: Recovery of the Navier–Stokes equation using a lattice–gas Boltzmann method. Phys. Rev., A 45, R5339–R5342 (1992)CrossRefGoogle Scholar
  9. 9.
    Davis, R.W., Moore, E.F.: A numerical study of vortex shedding from rectangles. J. Fluid Mech. 116, 475–506 (1982)zbMATHCrossRefGoogle Scholar
  10. 10.
    Dellar, P.J.: Compound waves in a thermohydrodynamic lattice BGK scheme using non-perturbative equilibria. Europhys. Lett. 57, 690–696 (2002)CrossRefGoogle Scholar
  11. 11.
    Ehrenfest, P., Ehrenfest, T.: The conceptual foundations of the statistical approach in mechanics. Dover, New York (1990)Google Scholar
  12. 12.
    Gorban, A.N.: Basic types of coarse-graining. In: Gorban, A.N., Kazantzis, N., Kevrekidis, I.G., Öttinger, H.-C., Theodoropoulos, C. (eds.) Model Reduction and Coarse-graining Approaches for Multiscale Phenomena, pp. 117–176. Springer, Berlin (cond-mat/0602024) (2006)CrossRefGoogle Scholar
  13. 13.
    Gorban, A., Kaganovich, B., Filippov, S., Keiko, A., Shamansky, V., Shirkalin, I.: Thermodynamic Equilibria and Extrema: Analysis of Attainability Regions and Partial Equilibrium. Springer, Berlin (2006)Google Scholar
  14. 14.
    Gorban, A.N., Karlin, I.V.: Invariant manifolds for physical and chemical kinetics. In: Lecture Notes in Physics, vol. 660. Springer, Berlin (2005)Google Scholar
  15. 15.
    Gorban, A.N., Karlin, I.V., Öttinger, H.C., Tatarinova, L.L.: Ehrenfest’s argument extended to a formalism of nonequilibrium thermodynamics. Phys. Rev., E 62, 066124 (2001)CrossRefGoogle Scholar
  16. 16.
    Higuera, F., Succi, S., Benzi, R.: Lattice gas – dynamics with enhanced collisions. Europhys. Lett. 9, 345–349 (1989)CrossRefGoogle Scholar
  17. 17.
    Karlin, I.V., Ansumali, S., Frouzakis, C.E., Chikatamarla, S.S.: Elements of the lattice Boltzmann method I: linear advection equation. Commun. Comput. Phys. 1, 616–655 (2006)Google Scholar
  18. 18.
    Karlin, I.V., Chikatamarla, S.S., Ansumali, S.: Elements of the lattice Boltzmann method II: Kinetics and hydrodynamics in one dimension. Commun. Comput. Phys. 2, 196–238 (2007)MathSciNetGoogle Scholar
  19. 19.
    Karlin, I.V., Ferrante, A., Öttinger, H.C.: Perfect entropy functions of the lattice Boltzmann method. Europhys. Lett. 47, 182–188 (1999)CrossRefGoogle Scholar
  20. 20.
    Karlin, I.V., Gorban, A.N., Succi, S., Boffi, V.: Maximum entropy principle for lattice kinetic equations. Phys. Rev. Lett. 81, 6–9 (1998)CrossRefGoogle Scholar
  21. 21.
    Kullback, S.: Information Theory and Statistics. Wiley, New York (1959)zbMATHGoogle Scholar
  22. 22.
    Okajima, A.: Strouhal numbers of rectangular cylinders. J. Fluid Mech. 123, 379–398 (1982)CrossRefGoogle Scholar
  23. 23.
    Shan, X., He, X.: Discretization of the velocity space in the solution of the Boltzmann equation. Phys. Rev. Lett. 80, 65–68 (1998)CrossRefGoogle Scholar
  24. 24.
    Succi, S.: The Lattice Boltzmann Equation for Fluid Dynamics and Beyond. OUP, New York (2001)zbMATHGoogle Scholar
  25. 25.
    Qian, Y.H., D. d’Humieres, Lallemand, P.: Lattice BGK models for Navier–Stokes equation. Europhys. Lett. 17, 479–484 (1992)zbMATHCrossRefGoogle Scholar
  26. 26.
    Vickery, B.J.: Fluctuating lift and drag on a long cylinder of square cross-section in a smooth and in a turbulent stream. J. Fluid Mech. 25, 481–494 (1966)CrossRefGoogle Scholar
  27. 27.
    Yablonskii, G.S., Bykov, V.I., Gorban, A.N., Elokhin, V.I.: Kinetic models of catalytic reactions. In: Compton, R.G. (ed.) Series Comprehensive Chemical Kinetics, vol. 32. Elsevier, Amsterdam (1991)Google Scholar
  28. 28.
    Zeldovich, Y.B.: Proof of the uniqueness of the solution of the equations of the law of mass action. In: Ostriker, J.P. (ed.) Selected Works of Yakov Borisovich Zeldovich, vol. 1, pp. 144–148. Princeton University Press, Princeton, NJ (1996)Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of LeicesterLeicesterUK

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