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Analytic solutions of simple flows and analysis of nonslip boundary conditions for the lattice Boltzmann BGK model

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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 Σαfαeα=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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Authors and Affiliations

  1. Center for Nonlinear Studies (CNLS), MS-B258, Los Alamos National Laboratory, 87545, Los Alamos, New Mexico

    Xiaoyi He & Qisu Zou

  2. Theoretical Biology and Biophysics Group (T-10), MS-K710, Theoretical Division, Los Alamos National Laboratory, 87545, Los Alamos, New Mexico

    Xiaoyi He & Micah Dembo

  3. Complex Systems Group (T-13), MS-B213, Theoretical Division, Los Alamos National Laboratory, 87545, Los Alamos, New Mexico

    Qisu Zou & Li-Shi Luo

  4. Computational Science Methods Group (XCM), MS-F645, X Division, Los Alamos National Laboratory, 87545, Los Alamos, New Mexico

    Li-Shi Luo

  5. Department of Mathematics, Kansas State University, 66506, Manhattan, Kansas

    Qisu Zou

  6. ICASE, NASA Langley Research Center, Mail Stop 403, 23681, Hampton, Virginia

    Li-Shi Luo

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  1. Xiaoyi He
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  2. Qisu Zou
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He, X., Zou, Q., Luo, LS. et al. Analytic solutions of simple flows and analysis of nonslip boundary conditions for the lattice Boltzmann BGK model. J Stat Phys 87, 115–136 (1997). https://doi.org/10.1007/BF02181482

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  • DOI: https://doi.org/10.1007/BF02181482

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