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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MDPI AG
2021-04-01
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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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id | doaj.art-8ae79aa411064560a0bdeab53734e6e1 |
institution | Directory Open Access Journal |
issn | 2227-7390 |
language | English |
last_indexed | 2024-03-10T11:53:11Z |
publishDate | 2021-04-01 |
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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 |