Commun. Comput. Phys.,
Study of Simple Hydrodynamic Solutions with the Two-Relaxation-Times Lattice Boltzmann Scheme
Irina Ginzburg 1*, Frederik Verhaeghe 2, Dominique d'Humieres 31 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 March 2007; Accepted (in revised version) 22 July 2007
Available online 30 October 2007
For simple hydrodynamic solutions, where the pressure and the velocity are polynomial functions of the coordinates, exact microscopic solutions are constructed for the two-relaxation-time ($\TRT$) Lattice Boltzmann model with variable forcing and supported by exact boundary schemes. We show how simple numerical and analytical solutions can be interrelated for Dirichlet velocity, pressure and mixed (pressure/tangential velocity) multi-reflection ($\MR$) type schemes. Special care is taken to adapt them for corners, to examine the uniqueness of the obtained steady solutions and staggered invariants, to validate their exact parametrization by the non-dimensional hydrodynamic and a ``kinetic'' (collision) number. We also present an inlet/outlet ``constant mass flux'' condition. We show, both analytically and numerically, that the kinetic boundary schemes may result in the appearance of Knudsen layers which are beyond the methodology of the Chapman-Enskog analysis. Time dependent Dirichlet boundary conditions are investigated for pulsatile flow driven by an oscillating pressure drop or forcing. Analytical approximations are constructed in order to extend the pulsatile solution for compressible regimes.
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PACS: 47.10.ad, 47.56+r, 02.60-x
Key words: Lattice Boltzmann equation, Dirichlet boundary conditions, pressure boundary conditions, two-relaxation-time model, Knudsen layers, exact hydrodynamic solutions, staggered invariants, pulsatile flow.
Email: email@example.com (I. Ginzburg), firstname.lastname@example.org (F. Verhaeghe), email@example.com (D. d'Humieres)