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From The Boltzmann To The Lattice-Boltzmann Equation: Beyond Bgk Collision Models

Author

Listed:
  • PAULO CESAR PHILIPPI

    (Mechanical Engineering Department, Federal University of Santa Catarina, 88040-900 Florianópolis, SC, Brazil)

  • LUIZ ADOLFO HEGELE

    (Mechanical Engineering Department, Federal University of Santa Catarina, 88040-900 Florianópolis, SC, Brazil)

  • RODRIGO SURMAS

    (Mechanical Engineering Department, Federal University of Santa Catarina, 88040-900 Florianópolis, SC, Brazil)

  • DIOGO NARDELLI SIEBERT

    (Mechanical Engineering Department, Federal University of Santa Catarina, 88040-900 Florianópolis, SC, Brazil)

  • LUÍS ORLANDO EMERICH DOS SANTOS

    (Mechanical Engineering Department, Federal University of Santa Catarina, 88040-900 Florianópolis, SC, Brazil)

Abstract

In this work, we present a derivation for the lattice-Boltzmann equation directly from the linearized Boltzmann equation, combining the following main features: multiple relaxation times and thermodynamic consistency in the description of non isothermal compressible flows. The method presented here is based on the discretization of increasingly order kinetic models of the Boltzmann equation. Following a Gross-Jackson procedure, the linearized collision term is developed in Hermite polynomial tensors and the resulting infinite series is diagonalized after a chosen integerN, establishing the order of approximation of the collision term. The velocity space is discretized, in accordance with a quadrature method based on prescribed abscissas (Philippiet al.,Phys. Rev E73, 056702, 2006). The problem of describing the energy transfer is discussed, in relation with the order of approximation of a two relaxation-times lattice Boltzmann model. The velocity-step, temperature-step and the shock tube problems are investigated, adopting lattices with 37, 53 and 81 velocities.

Suggested Citation

  • Paulo Cesar Philippi & Luiz Adolfo Hegele & Rodrigo Surmas & Diogo Nardelli Siebert & Luís Orlando Emerich Dos Santos, 2007. "From The Boltzmann To The Lattice-Boltzmann Equation: Beyond Bgk Collision Models," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 18(04), pages 556-565.
    • Handle: RePEc:wsi:ijmpcx:v:18:y:2007:i:04:n:s0129183107010796
    DOI: 10.1142/S0129183107010796
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