Lattice Boltzmann approach for complex nonequilibrium flows

A. Montessori, P. Prestininzi, M. La Rocca, and S. Succi

Phys. Rev. E 92, 043308 – Published 22 October 2015

Abstract

We present a lattice Boltzmann realization of Grad's extended hydrodynamic approach to nonequilibrium flows. This is achieved by using higher-order isotropic lattices coupled with a higher-order regularization procedure. The method is assessed for flow across parallel plates and three-dimensional flows in porous media, showing excellent agreement of the mass flow with analytical and numerical solutions of the Boltzmann equation across the full range of Knudsen numbers, from the hydrodynamic regime to ballistic motion.

Received 11 March 2015
Corrected 20 November 2015

DOI:https://doi.org/10.1103/PhysRevE.92.043308

Corrections

20 November 2015

Erratum

Publisher's Note: Lattice Boltzmann approach for complex nonequilibrium flows [Phys. Rev. E 92, 043308 (2015)]

A. Montessori, P. Prestininzi, M. La Rocca, and S. Succi

Phys. Rev. E 92, 069901 (2015)

Authors & Affiliations

A. Montessori^1,*, P. Prestininzi¹, M. La Rocca¹, and S. Succi²

¹Department of Engineering, University of Rome, “Roma Tre” Via Vito Volterra 62, 00146 Rome, Italy
²Istituto per le Applicazioni del Calcolo, CNR Via dei Taurini 19, 00185 Rome, Italy

^*and.montessori@gmail.com

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Vol. 92, Iss. 4 — October 2015

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Images

Figure 1
Fourth-order isotropic 19-speed lattice (red arrows) and eighth-order isotropic 41-speed lattice (red and blue arrows).
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Figure 2
Normalized mass flux across two parallel plane plates as a function of the Knudsen number. Legend: Navier-Stokes solution $1 / (12 Kn)$ (NSE) (solid line), Cercignani solution in the low-Kn slip flow regime [29] (CEL) (dotted line), Cercignani solution in the high-Kn transition and ballistic regimes (CEH) (dashed-dotted line), direct simulation Monte Carlo (DSMC) (circles), regularized D3Q19 LB with bounce-back boundary conditions (R19BB) (solid line with squares), regularized D3Q41 LB with kinetic boundary conditions (R41KB) (solid line with diamonds), regularized D3Q41 LB with bounce-back boundary conditions (R41BB) (solid line with circles), and D3Q41 LB with kinetic boundary conditions (41KB) (dotted line with triangles). Results with regularized D3Q41 LB with the kinetic boundary conditions are omitted because on the scale of the figure they overlap with R19KB. Likewise, D3Q19 LB with kinetic boundary conditions is not shown because it overlaps with the 41KB. The main message here is that higher-order lattices, regularization, and kinetic boundary conditions are required, in order to retrieve the correct behavior of the flow between two plates across all the Knudsen's regimes.
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Figure 3
Permeability correction factor as a function of the Knudsen number. The Klinkenberg (KLI) [32] and the Beskok (BES) [33] solutions are reported in solid and dashed-dotted lines, respectively. 19BB (dotted line with triangles), R19BB (dashed line with diamonds), 41BB (dashed line with crosses), and R41BB (dashed-dotted line with circles). It is evident that the nonregularized models can predict accurately the correction factor only within the continuum regime, while they deviate from the analytical solutions already in the slip regime. On the other hand, the R41BB provides correction factors in good agreement with the Klinkenberg solution across the continuum, slip, and transition regimes. When entering the ballistic region, the R41BB begins to overestimate the Klinkenberg discharge while keeping on predicting values of permeability between the Beskok and the Klinkenberg solutions. Here, 19BB stands for the D3Q19 LB with bounce-back boundary conditions, while the other labels are the same as in Fig. 2.
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Figure 4
$Q_{x x z}$ field on midplanes of the porous media at $Kn = 5$ . (a) The D3Q19 exposes in full the lattice structure, which betrays its lack of isotropy towards third-order moments. As expected, D3Q19R is powerless against this structural deficiency, as shown in (b). (c) The D3Q41R appears both isotropic and noise free.
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