Analytic solutions of simple flows and analysis of nonslip boundary conditions for the lattice Boltzmann BGK model |
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Authors: | Xiaoyi He Qisu Zou Li-Shi Luo Micah Dembo |
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Institution: | (1) Center for Nonlinear Studies (CNLS), MS-B258, Los Alamos National Laboratory, 87545 Los Alamos, New Mexico;(2) Theoretical Biology and Biophysics Group (T-10), MS-K710, Theoretical Division, Los Alamos National Laboratory, 87545 Los Alamos, New Mexico;(3) Complex Systems Group (T-13), MS-B213, Theoretical Division, Los Alamos National Laboratory, 87545 Los Alamos, New Mexico;(4) Computational Science Methods Group (XCM), MS-F645, X Division, Los Alamos National Laboratory, 87545 Los Alamos, New Mexico;(5) Department of Mathematics, Kansas State University, 66506 Manhattan, Kansas;(6) ICASE, NASA Langley Research Center, Mail Stop 403, 23681 Hampton, Virginia |
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Abstract: | In this paper we analytically solve the velocity of the lattice Boltzmann BGK equation (LBGK) for several simple flows. The analysis provides a framework to theoretically analyze various boundary conditions. In particular, the analysis is used to derive the slip velocities generated by various schemes for the nonslip boundary condition. We find that the slip velocity is zero as long as fe=0 at boundaries, no matter what combination of distributions is chosen. The schemes proposed by Nobleet al. and by Inamuroet al. yield the correct zeroslip velocity, while some other schemes, such as the bounce-back scheme and the equilibrium distribution scheme, would inevitably generate a nonzero slip velocity. The bounce-back scheme with the wall located halfway between a flow node and a bounce-back node is also studied for the simple flows considered and is shown to produce results of second-order accuracy. The momentum exchange at boundaries seems to be highly related to the slip velocity at boundaries. To be specific, the slip velocity is zero only when the momentum dissipated by boundaries is equal to the stress provided by fluids. |
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Keywords: | Lattice Boltzmann BGK equations nonslip boundary conditions analytic solutions of simple flows |
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