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Multiscale LBM Simulations

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Multiscale and Multiphysics Flow Simulations of Using the Boltzmann Equation

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Abstract

For flow problems of the continuum regime, the lattice Boltzmann method (LBM) is a good alternative to the traditional CFD solvers based on the N–S-like equations. It is efficient in modeling dynamic problems and very powerful for pore-scale applications, where the simulation of interface dynamics on the real irregular pore surface is challenging, if not impossible, to most of the traditional CFD solvers. We start in this chapter with the basic LBM algorithm to show its correlation with the N–S equation through the Chapman–Enskog expansion. Then, the widely used Shan–Chen model will be introduced to simulate multiphase multicomponent flow systems, having its applications detailed in the subsequent sections. We also present the extension of LBM to the Darcy-scale simulations, where the LBM works as a unified framework for simulations at different scales, i.e., both pore and Darcy scales, and the detailed results are given at the end of this chapter. In the ordinary application of LBM for computing the absolute permeability, we clarify the prevailed confusion interpreted as viscosity-dependent permeability and reveal the underlying rarefaction mechanism that has been commonly oversighted. Additionally, we also discuss the application of large eddy simulation of turbulence in the LBM framework and the same idea can be extended to model non-Newtonian fluids.

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Notes

  1. 1.

    By a slight abuse of notation, Q denotes the total number of lattice velocities introduced here and auxiliary variables used elsewhere for notation clarity.

  2. 2.

    By a slight abuse of notation, \(\omega \) denotes the power of the dependence of \(\mu \) on T introduced in Sect. 2.3, \(\omega _\alpha \) denotes the weighting factor associated with \(\mathbf {e}_\alpha \) introduced here, and \(\omega \) denotes the acentric factor introduced in Sect. 4.5.3.

  3. 3.

    The lattice Boltzmann method actually solves a BGK-type equation rather than the Boltzmann equation. Additionally, the relaxation time \(\tau \) of the original BGK equation (2.81) has physical unit (second) but \(\tau \) in LBM is dimensionless. The correlation between \(\tau \) and viscosity in LBM, i.e., Eq. (4.19), is completely different from that in the BGK equation, i.e., Eq. (2.83).

  4. 4.

    By a slight abuse of notation, n denotes the molecular number density introduced in Sect. 2.1, superscript n denotes the term index of model property introduced in Eq. (4.2) and the term indices of Taylor and Chapman–Enskog expansions introduced here.

  5. 5.

    By a slight abuse of notation, \(f^{(2)}\) denotes a two-particle distribution function introduced in Sect. 2.4 and \(f^{(2)}_\alpha \) denotes the second-order term in the expansion of \(f_\alpha \) introduced here.

  6. 6.

    As a general formula, Eq. (4.16) contains arbitrary \(\varDelta t\) and c.

  7. 7.

    By a slight abuse of notation, \(\nu \) denotes the intermolecular collision frequency introduced in Sect. 2.8 and the kinematic viscosity introduced here.

  8. 8.

    The summation over \(\alpha \) in computing force interactions should be \(\sum _{\alpha =1}^{Q-1}\), which is replaced by \(\sum _\alpha =\sum _{\alpha =0}^{Q-1}\) for notation clarity thanks to \(\mathbf {e}_0=\mathbf {0}\).

  9. 9.

    By a slight abuse of notation, \(M^\mathrm{LBM}\) denotes the mobility in LBM and M denotes the total mass of a system introduced in Sect. 2.7.

  10. 10.

    By a slight abuse of notation, \(\mu ^\mathrm{LBM}\) denotes the chemical potential in LBM and \(\mu \) denotes the dynamic viscosity elsewhere.

  11. 11.

    We denote the volumetric flow velocity of the Darcy-scale simulation by \(\mathbf {u}\), instead of \(\mathbf {U}\) that is used in Sect. 3.4.4, where \(\mathbf {u}\) is used as the pore-scale flow velocity.

  12. 12.

    Similar idea can be used to model some kinds of non-Newtonian fluids.

  13. 13.

    To recover the incompressible N–S equation from the LBM, we only need the zero- and first-order moments of Eq. (4.9).

  14. 14.

    Changing \(\varDelta t\) alone at fixed \(\tau \) and \(\varDelta x\) doesn’t affect the computed permeability.

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  1. Center for Integrative Petroleum Research, College of Petroleum Engineering and Geosciences, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia

    Jun Li

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Li, J. (2020). Multiscale LBM Simulations. In: Multiscale and Multiphysics Flow Simulations of Using the Boltzmann Equation. Springer, Cham. https://doi.org/10.1007/978-3-030-26466-6_4

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  • DOI: https://doi.org/10.1007/978-3-030-26466-6_4

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