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Discrete Velocity Boltzmann Model for Quasi-Incompressible Hydrodynamics

Author

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  • Oleg Ilyin

    (Dorodnicyn Computing Center, Federal Research Center “Computer Science and Control” of Russian Academy of Sciences, Vavilova-40, 119333 Moscow, Russia)

Abstract

In this paper, we consider the development of the two-dimensional discrete velocity Boltzmann model on a nine-velocity lattice. Compared to the conventional lattice Boltzmann approach for the present model, the collision rules for the interacting particles are formulated explicitly. The collisions are tailored in such a way that mass, momentum and energy are conserved and the H -theorem is fulfilled. By applying the Chapman–Enskog expansion, we show that the model recovers quasi-incompressible hydrodynamic equations for small Mach number limit and we derive the closed expression for the viscosity, depending on the collision cross-sections. In addition, the numerical implementation of the model with the on-lattice streaming and local collision step is proposed. As test problems, the shear wave decay and Taylor–Green vortex are considered, and a comparison of the numerical simulations with the analytical solutions is presented.

Suggested Citation

  • Oleg Ilyin, 2021. "Discrete Velocity Boltzmann Model for Quasi-Incompressible Hydrodynamics," Mathematics, MDPI, vol. 9(9), pages 1-14, April.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:9:p:993-:d:545041
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    Cited by:

    1. Mikhail Posypkin & Andrey Gorshenin & Vladimir Titarev, 2022. "Preface to the Special Issue on “Control, Optimization, and Mathematical Modeling of Complex Systems”," Mathematics, MDPI, vol. 10(13), pages 1-8, June.

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