Discrete Velocity Boltzmann Model for Quasi-Incompressible Hydrodynamics

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 a...

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Main Author: Oleg Ilyin
Format: Article
Language:English
Published: MDPI AG 2021-04-01
Series:Mathematics
Subjects:
Online Access:https://www.mdpi.com/2227-7390/9/9/993
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author Oleg Ilyin
author_facet Oleg Ilyin
author_sort Oleg Ilyin
collection DOAJ
description 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 <i>H</i>-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.
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spelling doaj.art-8ae79aa411064560a0bdeab53734e6e12023-11-21T17:31:29ZengMDPI AGMathematics2227-73902021-04-019999310.3390/math9090993Discrete Velocity Boltzmann Model for Quasi-Incompressible HydrodynamicsOleg Ilyin0Dorodnicyn Computing Center, Federal Research Center “Computer Science and Control” of Russian Academy of Sciences, Vavilova-40, 119333 Moscow, RussiaIn 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 <i>H</i>-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.https://www.mdpi.com/2227-7390/9/9/993discrete velocity methodlattice Boltzmann methodcomputational fluid dynamics
spellingShingle Oleg Ilyin
Discrete Velocity Boltzmann Model for Quasi-Incompressible Hydrodynamics
Mathematics
discrete velocity method
lattice Boltzmann method
computational fluid dynamics
title Discrete Velocity Boltzmann Model for Quasi-Incompressible Hydrodynamics
title_full Discrete Velocity Boltzmann Model for Quasi-Incompressible Hydrodynamics
title_fullStr Discrete Velocity Boltzmann Model for Quasi-Incompressible Hydrodynamics
title_full_unstemmed Discrete Velocity Boltzmann Model for Quasi-Incompressible Hydrodynamics
title_short Discrete Velocity Boltzmann Model for Quasi-Incompressible Hydrodynamics
title_sort discrete velocity boltzmann model for quasi incompressible hydrodynamics
topic discrete velocity method
lattice Boltzmann method
computational fluid dynamics
url https://www.mdpi.com/2227-7390/9/9/993
work_keys_str_mv AT olegilyin discretevelocityboltzmannmodelforquasiincompressiblehydrodynamics