Abstract
Analytical solutions of the two-dimensional triangular and square lattice Boltzmann BGK models have been obtained for the plane Poiseuille flow and the plane Couette flow. The analytical solutions are written in terms of the characteristic velocity of the flow, the single relaxation time τ, and the lattice spacing. The analytic solutions are the exact representation of these two flows without any approximation. Using the analytical solution, it is shown that in Poiseuille flow the bounce-back boundary condition introduces an error of first order in the lattice spacing. The boundary condition used by Kadanoffet al. in lattice gas automata to simulate Poiseuille flow is also considered for the triangular lattice Boltzmann BGK model. An analytical solution is obtained and used to show that the boundary condition introduces an error of second order in the lattice spacing.
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References
L. P. Kadanoff, G. R. McNamara, and G. Zanetti, A Poiseuille viscometer for lattice gas automata.Complex Systems 1:791 (1987).
L. P. Kadanoff, G. R. McNamara, and G. Zanetti, From automata to fluid flow: Comparisons of simulation and theory.Phys. Rev. A 40:4527 (1989).
M. Henon, Viscosity of a lattice gas,Complex Systems 1:763 (1987).
R. Cornubert, D. d'Humières and D. Levermore, A Knudsen layer theory for lattice gases,Physica D 47(6):241 (1991).
L. S. Luo, H. Chen, S. Chen, G. D. Doolen, and Y. C. Lee, Generalized hydrodynamic transport in lattice-gas automata,Phys. Rev. A 43:7097 (1991).
I. Ginzbourg and P. M. Adler, Boundary flow condition analysis for the three-dimensional lattice Boltzmann model,J. Phys. II France 4:191 (1994).
D. R. Noble, S. Chen, J. G. Geogiadis, and R. O. Buckius, A consistent hydrodynamic boundary condition for the lattice Boltzmann method,Phys. Fluids 7:203 (1995).
S. Chen, H. Chen, D. Martinez, and W. H. Matthaeus, Lattice Boltzmann model for simulation of magnetohydrodynamic phenomena,Phys. Rev. Lett. 67:3776 (1991).
Y. Qian, D. d'Humières, and P. Lallemand, Lattice BGK models for Navier-Stokes equation,Europhys. Lett. 17(6):479 (1992).
Y. H. Qian and S. A. Orszag, Lattice BGK models for the Navier-Stokes equation: Non-linear deviation in compressible regimes,Europhys. Lett 21(3):255 (1993).
S. Hou, Q. Zou, S. Chen, G. D. Doolen, and A. C. Cogley, Simulation of cavity flow by the lattice Boltzmann method,J. Comput. Phys. 118:329 (1995).
D. P. Ziegler, Boundary conditions for lattice Boltzmann simulations,J. Stat. Phys. 71:1171 (1993).
Y. H. Qian, Private communication.
P. A. Skordos, Initial and boundary conditions for the lattice Boltzmann method,Phys. Rev. E 48:4823 (1993).
Y. Qian, Ph.D. thesis, Université Pierre et Marie Curie (January 1990).
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Zou, Q., Hou, S. & Doolen, G.D. Analytical solutions of the lattice Boltzmann BGK model. J Stat Phys 81, 319–334 (1995). https://doi.org/10.1007/BF02179981
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DOI: https://doi.org/10.1007/BF02179981