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Discrete Rotational Symmetry, Moment Isotropy, and Higher Order Lattice Boltzmann Models

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Abstract

Conventional lattice Boltzmann models only satisfy moment isotropy up to fourth order. In order to accurately describe important physical effects beyond the isothermal Navier-Stokes fluid regime, higher-order isotropy is required. In this paper, we present some basic results on moment isotropy and its relationship to the rotational symmetry of a generating discrete vector set. The analysis provides a geometric understanding for popular lattice Boltzmann models, while offering a systematic procedure to construct higher-order models.

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References

  1. Agarwal, R., Yun, K.-Y., Balakrishnan, R.: Beyond Navier-Stokes: Burnett equations for flows in the continuum-transition regime. Phys. Fluids 13, 3061–3085 (2001)

    Article  Google Scholar 

  2. Benzi, R., Succi, S., Vergassola, M.: The lattice Boltzmann equation: Theory and applications. Phys. Rep. 222, 145–197 (1992)

    Article  Google Scholar 

  3. Bhatnagar, P.L., Gross, E., Krook, M.: A model for collision process in gases I. Small amplitude process in charged and neutral one-component systems. Phys. Rev. 94(3), 511–525 (1954)

    Article  MATH  Google Scholar 

  4. Bishop, R., Goldberg, S.: Tensor Analysis on Manifolds. Dover, New York (1968)

    MATH  Google Scholar 

  5. Cercignani, C.: Theory and Application of the Boltzmann Equation. Elsevier, New York (1975)

    MATH  Google Scholar 

  6. Chapman, S., Cowling, T.: The Mathematical Theory of Non-Uniform Gases, 3rd edn. Cambridge University Press, Cambridge (1990)

    Google Scholar 

  7. Chen, H.: H-theorem and generalized semi-detailed balance conditions for lattice gas systems. J. Stat. Phys. 81, 347–359 (1995)

    Article  MATH  Google Scholar 

  8. Chen, H.: Volumetric formulation of the lattice Boltzmann method for fluid dynamics: Basic concept. Phys. Rev. E 58(3), 3955–3963 (1998)

    Article  Google Scholar 

  9. Chen, H., Teixeira, C.: H-theorem and origins of instability in thermal lattice Boltzmann models. Comput. Phys. Commun. 129, 21–31 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  10. Chen, H., Chen, S., Matthaeus, W.H.: Recovery of Navier-Stokes equations using a lattice-gas Boltzmann method. Phys. Rev. A 45, R5339–42 (1992)

    Article  Google Scholar 

  11. Chen, H., Orszag, S., Staroselsky, I., Succi, S.: Expanded analogy between Boltzmann kinetic theory of fluids and turbulence. J. Fluid Mech. 519, 301–314 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  12. Chen, H., Teixeira, C., Molvig, K.: Digital physics approach to computational fluid dynamics: Some basic theoretical features. Int. J. Mod. Phys. C 8, 675–684 (1997)

    Article  Google Scholar 

  13. Chen, S., Doolen, G.: Lattice Boltzmann method for fluid flows. Ann. Rev. Fluid Mech. 30, 329–364 (1998)

    Article  MathSciNet  Google Scholar 

  14. Chen, Y., Ohashi, H., Akiyama, M.: Thermal lattice Bhatnagar-Gross-Krook model without nonlinear deviation in macrodynamic equations. Phys. Rev. E 50, 2776–2783 (1994)

    Article  Google Scholar 

  15. Coxeter, H.S.M.: Regular Polytopes, 3rd edn. Dover, New York (1973)

    Google Scholar 

  16. Frisch, U., Hasslacher, B., Pomeau, Y.: Lattice-gas automata for the Navier-Stokes equation. Phys. Rev. Lett. 56, 1505 (1986)

    Article  Google Scholar 

  17. Gad-el-Hak, M.: The fluid mechanics of microdevices—The Freeman scholar lecture. J. Fluid Eng. 121, 5–33 (1999)

    Google Scholar 

  18. Higuera, F., Succi, S., Benzi, R.: Lattice gas dynamics with enhanced collisions. Europhys. Lett. 9, 345–349 (1989)

    Article  Google Scholar 

  19. Qian, Y., d’Humieres, D., Lallemand, P.: Lattice BGK models for Navier-Stokes equation. Europhys. Lett. 17, 479–484 (1992)

    Article  MATH  Google Scholar 

  20. Schwarzenberger, R.L.E.: N-Dimensional Crystallography. Pitman, London (1980)

    MATH  Google Scholar 

  21. Shan, X., He, X.: Discretization of the velocity space in solution of the Boltzmann equation. Phys. Rev. Lett. 80, 65 (1998)

    Article  Google Scholar 

  22. Shan, X., Yuan, X., Chen, H.: Kinetic theory representation of hydrodynamics: A way beyond the Navier-Stokes equation. J. Fluid Mech. 550, 413–441 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  23. Succi, S., Karlin, I., Chen, H.: Role of the H theorem in lattice Boltzmann hydrodynamic simulations. Rev. Mod. Phys. 74, 1203–1220 (2002)

    Article  Google Scholar 

  24. Teixeira, C.: Continuum limit of lattice gas fluid dynamics. Ph.D. Thesis, M.I.T. (1992)

  25. Teixeira, C., Chen, H., Freed, D.: Multispeed thermal lattice Boltzmann method stabilization via equilibrium under-relaxation. Comput. Phys. Commun. 129, 207–226 (2000)

    Article  MATH  Google Scholar 

  26. Vainshtein, B.K.: Modern Crystallography. Springer, Berlin (1981)

    Google Scholar 

  27. Watari, M., Tsutahara, M.: Possibility of constructing a multispeed Bhatnagar-Gross-Krook thermal model of the lattice Boltzmann method. Phys. Rev. E 70, 016703–9 (2004)

    Article  Google Scholar 

  28. Wolfram, S.: Cellular automaton fluid 1: Basic theory. J. Stat. Phys. 45, 471–526 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  29. Xu, K.: A gas-kinetic BGK scheme for the Navier-Stokes equations and its connection with artificial dissipation and Godunov method. J. Comput. Phys. 171(48), 289–335 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  30. Xu, K., Martinelli, L., Jameson, A.: Gas-kinetic finite volume methods, flux vector splitting and artificial dissipation. J. Comput. Phys. 120(48), 48–65 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  31. Zhang, R., Chen, H., Qian, Y., Chen, S.: Effective volumetric lattice Boltzmann scheme. Phys. Rev. E 63, 056705(1)–056705(6) (2001)

    Google Scholar 

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Authors and Affiliations

  1. Exa Corporation, 3 Burlington Woods Drive, Burlington, MA, 01821, USA

    Hudong Chen, Isaac Goldhirsch & Steven A. Orszag

  2. School of Mechanical Engineering, Tel-Aviv University, Ramat-Aviv, Israel

    Isaac Goldhirsch

  3. Department of Mathematics, Yale University, New Haven, CT, 06510, USA

    Steven A. Orszag

Authors
  1. Hudong Chen
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  2. Isaac Goldhirsch
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  3. Steven A. Orszag
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Correspondence to Steven A. Orszag.

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Chen, H., Goldhirsch, I. & Orszag, S.A. Discrete Rotational Symmetry, Moment Isotropy, and Higher Order Lattice Boltzmann Models. J Sci Comput 34, 87–112 (2008). https://doi.org/10.1007/s10915-007-9159-3

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  • DOI: https://doi.org/10.1007/s10915-007-9159-3

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