NDLI: Prediction of the moments in advection-diffusion lattice Boltzmann method. II. Attenuation of the boundary layers via double- $ Λ $ bounce-back flux scheme

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Year: 2017 Volume: 95

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Prediction of the moments in advection-diffusion lattice Boltzmann method. II. Attenuation of the boundary layers via double- $ Λ $ bounce-back flux scheme

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Prediction of the moments in advection-diffusion lattice Boltzmann method. II. Attenuation of the boundary layers via double- $ Λ $ bounce-back flux scheme

Content Provider	World Health Organization (WHO)-Global Index Medicus
Author	Ginzburg, Irina
Description	Country affiliation: France Author Affiliation: Ginzburg I ( Irstea, Antony Regional Centre, HBAN, 1 rue Pierre-Gilles de Gennes CS 10030, 92761 Antony cedex, France.)
Abstract	Impact of the unphysical tangential advective-diffusion constraint of the bounce-back (BB) reflection on the impermeable solid surface is examined for the first four moments of concentration. Despite the number of recent improvements for the Neumann condition in the lattice Boltzmann method–advection-diffusion equation, the BB rule remains the only known local mass-conserving no-flux condition suitable for staircase porous geometry. We examine the closure relation of the BB rule in straight channel and cylindrical capillary analytically, and show that it excites the Knudsen-type boundary layers in the nonequilibrium solution for full-weight equilibrium stencil. Although the d2Q5 and d3Q7 coordinate schemes are sufficient for the modeling of isotropic diffusion, the full-weight stencils are appealing for their advanced stability, isotropy, anisotropy and anti-numerical-diffusion ability. The boundary layers are not covered by the Chapman-Enskog expansion around the expected equilibrium, but they accommodate the Chapman-Enskog expansion in the bulk with the closure relation of the bounce-back rule. We show that the induced boundary layers introduce first-order errors in two primary transport properties, namely, mean velocity (first moment) and molecular diffusion coefficient (second moment). As a side effect, the Taylor-dispersion coefficient (second moment), skewness (third moment), and kurtosis (fourth moment) deviate from their physical values and predictions of the fourth-order Chapman-Enskog analysis, even though the kurtosis error in pure diffusion does not depend on grid resolution. In two- and three-dimensional grid-aligned channels and open-tubular conduits, the errors of velocity and diffusion are proportional to the diagonal weight values of the corresponding equilibrium terms. The d2Q5 and d3Q7 schemes do not suffer from this deficiency in grid-aligned geometries but they cannot avoid it if the boundaries are not parallel to the coordinate lines. In order to vanish or attenuate the disparity of the modeled transport coefficients with the equilibrium weights without any modification of the BB rule, we propose to use the two-relaxation-times collision operator with free-tunable product of two eigenfunctions $ Λ $ . Two different values $ Λ _{ v } $ and $ Λ _{ b } $ are assigned for bulk and boundary nodes, respectively. The rationale behind this is that $ Λ _{ v } $ is adjustable for stability, accuracy, or other purposes, while the corresponding $ Λ _{ b } ( Λ _{ v } ) $ controls the primary accommodation effects. Two distinguished but similar functional relations $ Λ _{ b } ( Λ _{ v } ) $ are constructed analytically: they preserve advection velocity in parabolic profile, exactly in the two-dimensional channel and very accurately in a three-dimensional cylindrical capillary. For any velocity-weight stencil, the (local) double- $ Λ $ BB scheme produces quasi-identical solutions with the (nonlocal) specular-forward reflection for first four moments in a channel. In a capillary, this strategy allows for the accurate modeling of the Taylor-dispersion and non-Gaussian effects. As illustrative example, it is shown that in the flow around a circular obstacle, the double- $ Λ $ scheme may also vanish the dependency of mean velocity on the velocity weight; the required value for $ Λ _{ b } ( Λ _{ v } ) $ can be identified in a few bisection iterations in given geometry. A positive solution for $ Λ _{ b } ( Λ _{ v } ) $ may not exist in pure diffusion, but a sufficiently small value of $ Λ _{ b } $ significantly reduces the disparity in diffusion coefficient with the mass weight in ducts and in the presence of rectangular obstacles. Although $ Λ _{ b } $ also controls the effective position of straight or curved boundaries, the double- $ Λ $ scheme deals with the lower-order effects. Its idea and construction may help understanding and amelioration of the anomalous, zero- and first-order behavior of the macroscopic solution in the presence of the bulk and boundary or interface discontinuities, commonly found in multiphase flow and heterogeneous transport.
File Format	HTM / HTML
ISSN	24700045
e-ISSN	24700053
Journal	Physical Review E
Issue Number	1-1
Volume Number	95
Language	English
Publisher	American Physical Society
Publisher Date	2017-01-01
Publisher Place	United States
Access Restriction	Open
Subject Keyword	Statistical and Nonlinear Physics many-body systems Condensed Matter Physics
Content Type	Text
Resource Type	Article
Subject	Statistics and Probability Statistical and Nonlinear Physics Condensed Matter Physics

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