Commun. Comput. Phys.,
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Volume 3.


Two-Relaxation-Time Lattice Boltzmann Scheme: About Parametrization, Velocity, Pressure and Mixed Boundary Conditions

Irina Ginzburg 1*, Frederik Verhaeghe 2, Dominique d'Humieres 3

1 Cemagref, Antony Regional Centre, Parc de Tourvoie - BP 44, 92163 Antony Cedex, France.
2 Department of Metallurgy and Materials Engineering, Katholieke Universiteit Leuven, Kasteelpark Arenberg 44, 3001, Leuven, Belgium.
3 Laboratoire de Physique Statistique, Ecole Normale Superieure, associated to CNRS and P. and M. Curie and D. Diderot Universities, 24 Rue Lhomond, 75231 Paris Cedex 05, France.

Received 14 Mar 2007; Accepted (in revised version) 22 July 2007
Available online 9 October 2007

Abstract

We develop a two-relaxation-time ($\TRT$) Lattice Boltzmann model for hydrodynamic equations with variable source terms based on equivalent equilibrium functions. A special parametrization of the free relaxation parameter is derived. It controls, in addition to the non-dimensional hydrodynamic numbers, any $\TRT$ macroscopic steady solution and governs the spatial discretization of transient flows. In this framework, the multi-reflection approach~\cite{MultiRef02,IrDiffPart2} is generalized and extended for Dirichlet velocity, pressure and mixed (pressure/tangential velocity) boundary conditions. We propose second and third-order accurate boundary schemes and adapt them for corners. The boundary schemes are analyzed for exactness of the parametrization, uniqueness of their steady solutions, support of staggered invariants and for the effective accuracy in case of time dependent boundary conditions and transient flow. When the boundary scheme obeys the parametrization properly, the derived permeability values become independent of the selected viscosity for any porous structure and can be computed efficiently. The linear interpolations~\cite{BFL2001b,Yu2003} are improved with respect to this property.


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PACS: 47.10.ad, 47.56+r, 02.60-x
Key words: Lattice Boltzmann equation, Dirichlet boundary conditions, Chapman-Enskog expansion, multiple-relaxation-time model, BGK model, TRT model, Navier-Stokes equation, Stokes equation, recurrence equations, staggered invariants.

*Corresponding author.
Email: irina.ginzburg@cemagref.fr (I. Ginzburg), frederik.verhaeghe@mtm.kuleuven.be (F. Verhaeghe), dominiq@lps.ens.fr (D. d'Humieres)
 

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