Comptes Rendus
Discrete simulation of fluid dynamics
Local boundary reflections in lattice Boltzmann schemes: Spurious boundary layers and their impact on the velocity, diffusion and dispersion
Comptes Rendus. Mécanique, Volume 343 (2015) no. 10-11, pp. 518-532.

This work demonstrates that in advection–diffusion Lattice Boltzmann schemes, the local mass-conserving boundary rules, such as bounce-back and local specular reflection, may modify the transport coefficients predicted by the Chapman–Enskog expansion when they enforce to zero not only the normal, but also the tangential boundary flux. In order to accommodate it to the bulk solution, the system develops a Knudsen-layer correction to the non-equilibrium part of the population solution. Two principal secondary effects—(i) decrease in the diffusion coefficient, and (ii) retardation of the average advection velocity, obtained in a closed analytical form, are proportional, respectively, to freely assigned diagonal weights for equilibrium mass and velocity terms. In addition, due to their transverse velocity gradients, the boundary layers affect the longitudinal diffusion coefficient similarly to Taylor dispersion, as they grow as the square of the Péclet number. These numerical artifacts can be eliminated or reduced by a proper space distribution of the free-tunable collision eigenvalue in two-relaxation-time schemes.

Published online:
DOI: 10.1016/j.crme.2015.03.004
Keywords: Normal flux boundary condition, Advection–diffusion lattice Boltzmann schemes, Boundary layer, Tangential flux, Pure diffusion, Taylor dispersion, Boundary-layer dispersion, Equilibrium weights, Two-relaxation-time LBM scheme, Recurrence equations, Exact solutions of numerical scheme

Irina Ginzburg 1; Laetitia Roux 1; Goncalo Silva 1

1 IRSTEA, Antony Regional Centre, HBAN, 1, rue Pierre-Gilles-de-Gennes, CS 10030, 92761 Antony cedex, France
     author = {Irina Ginzburg and Laetitia Roux and Goncalo Silva},
     title = {Local boundary reflections in lattice {Boltzmann} schemes: {Spurious} boundary layers and their impact on the velocity, diffusion and dispersion},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {518--532},
     publisher = {Elsevier},
     volume = {343},
     number = {10-11},
     year = {2015},
     doi = {10.1016/j.crme.2015.03.004},
     language = {en},
AU  - Irina Ginzburg
AU  - Laetitia Roux
AU  - Goncalo Silva
TI  - Local boundary reflections in lattice Boltzmann schemes: Spurious boundary layers and their impact on the velocity, diffusion and dispersion
JO  - Comptes Rendus. Mécanique
PY  - 2015
SP  - 518
EP  - 532
VL  - 343
IS  - 10-11
PB  - Elsevier
DO  - 10.1016/j.crme.2015.03.004
LA  - en
ID  - CRMECA_2015__343_10-11_518_0
ER  - 
%0 Journal Article
%A Irina Ginzburg
%A Laetitia Roux
%A Goncalo Silva
%T Local boundary reflections in lattice Boltzmann schemes: Spurious boundary layers and their impact on the velocity, diffusion and dispersion
%J Comptes Rendus. Mécanique
%D 2015
%P 518-532
%V 343
%N 10-11
%I Elsevier
%R 10.1016/j.crme.2015.03.004
%G en
%F CRMECA_2015__343_10-11_518_0
Irina Ginzburg; Laetitia Roux; Goncalo Silva. Local boundary reflections in lattice Boltzmann schemes: Spurious boundary layers and their impact on the velocity, diffusion and dispersion. Comptes Rendus. Mécanique, Volume 343 (2015) no. 10-11, pp. 518-532. doi : 10.1016/j.crme.2015.03.004.

[1] C. Cercignani The Boltzmann Equation and Its Applications, Springer, Berlin, 1988, pp. 252-260

[2] F. Verhaeghe; L.-S. Luo; B. Blainpain Lattice Boltzmann modeling of microchannel flow in slip flow regime, J. Comput. Phys., Volume 228 (2009), pp. 147-157

[3] T. Reis; P.J. Dellar Lattice Boltzmann simulations of pressure-driven flows in microchannels using Navier–Maxwell slip boundary conditions, Phys. Fluids, Volume 24 (2012), p. 112001

[4] C. Baudet; J.-P. Hulin; D. d'Humières; P. Lallemand Lattice-gas automata: a model for the simulation of dispersion phenomena, Phys. Fluids, Volume 1 (1989), pp. 507-512

[5] A. Cali; S. Succi; A. Cancelliere; R. Benzi; M. Gramignani Diffusion and hydrodynamic dispersion with the lattice Boltzmann method, Phys. Rev. A, Volume 45 (1992) no. 8

[6] E.G. Flekkoy; U. Oxaal; J. Feder; T. Jossang Hydrodynamic dispersion at stagnation points: simulations and experiments, Phys. Rev. Lett., Volume 77 (1995), pp. 4952-4962

[7] G.I. Taylor Dispersion of soluble matter in solvent flowing slowly through a tube, Proc. R. Soc. Lond. A, Volume 219 (1953), pp. 186-203

[8] R. Aris On the dispersion of a solute in a fluid flowing through a tube, Proc. R. Soc. Lond., Volume 235 (1956), pp. 67-77

[9] G. Drazer; J. Koplik Tracer dispersion in two dimensional rough fractures, Phys. Rev. E, Volume 63 (2001)

[10] I. Ginzburg Generic boundary conditions for Lattice Boltzmann models and their application to advection and anisotropic–dispersion equations, Adv. Water Resour., Volume 28 (2005), pp. 1196-1216

[11] I. Ginzburg; D. d'Humières; A. Kuzmin Optimal stability of advection–diffusion lattice Boltzmann models with two relaxation times for positive/negative equilibrium, J. Stat. Phys., Volume 139 (2010) no. 6, pp. 1090-1143

[12] I. Ginzburg Truncation errors, exact and heuristic stability analysis of two-relaxation-time lattice Boltzmann schemes for anisotropic advection–diffusion equation, Commun. Comput. Phys., Volume 11 (2012) no. 5, pp. 1439-1502

[13] I. Ginzburg; F. Verhaeghe; D. d'Humières Two-relaxation-time lattice Boltzmann scheme: about parametrization, velocity, pressure and mixed boundary conditions, Commun. Comput. Phys., Volume 3 (2008), pp. 427-478

[14] I. Ginzburg; F. Verhaeghe; D. d'Humières Study of simple hydrodynamic solutions with the two-relaxation-time lattice Boltzmann scheme, Commun. Comput. Phys., Volume 3 (2008), pp. 519-581

[15] Y. Qian; D. d'Humières; P. Lallemand Lattice BGK models for Navier–Stokes equation, Europhys. Lett., Volume 17 (1992), pp. 479-484

[16] I. Ginzbourg; D. d'Humières Local second-order boundary method for lattice Boltzmann models, J. Stat. Phys., Volume 84 (1996), pp. 927-971

[17] I. Ginzburg; D. d'Humières Multi-reflection boundary conditions for lattice Boltzmann models, Phys. Rev. E, Volume 68 (2003)

[18] L. Li; R. Mei; J.F. Klausner Boundary conditions for thermal lattice Boltzmann equation method, J. Comput. Phys., Volume 237 (2013), pp. 366-395

[19] T. Gebäck; A. Heintz A lattice Boltzmann method for the advection–diffusion equation with Neumann boundary conditions, Commun. Comput. Phys., Volume 15 (2014) no. 2, pp. 487-505

[20] D. d'Humières; I. Ginzburg Viscosity independent numerical errors for lattice Boltzmann models: from recurrence equations to “magic” collision numbers, Comput. Math. Appl., Volume 58 (2009) no. 5, pp. 823-840

[21] R. Cornubert; D. d'Humières; D. Levermore A Knudsen layer theory, Physica D, Volume 47 (1991), pp. 241-259

[22] I. Ginzbourg Les problèmes des conditions aux limites dans les méthodes de gaz sur réseaux à plusieurs phases, University of Paris-6, 1994 (PhD thesis)

[23] I. Ginzbourg; P.M. Adler Boundary flow condition analysis for the three-dimensional lattice Boltzmann model, J. Phys. II, Volume 4 (1994), pp. 191-214

[24] S. Khirevich; I. Ginzburg; U. Tallarek Coarse- and fine-grid numerical behavior of MRT/TRT lattice Boltzmann schemes in regular and random sphere packings, J. Comput. Phys., Volume 281 (2015), pp. 708-742

[25] X. Yin; G. Le; J. Zhang Mass and momentum transfer across solid-fluid boundaries in the lattice-Boltzmann method, Phys. Rev. E, Volume 86 (2012)

[26] L. Roux; I. Ginzburg Truncation effect on Taylor–Aris dispersion in two-relaxation-times Lattice Boltzmann scheme: accuracy towards stability, J. Comput. Phys., Volume 299 (2015), pp. 974-1003

[27] I. Ginzburg Multiple anisotropic collisions for advection–diffusion lattice Boltzmann schemes, Adv. Water Resour., Volume 51 (2013), pp. 381-404

[28] A. Kuzmin; I. Ginzburg; A.A. Mohamad A role of the kinetic parameter on the stability of two-relaxation-time advection–diffusion lattice Boltzmann scheme, Comput. Math. Appl., Volume 61 (2011) no. 12, pp. 3417-3442

[29] I. Ginzburg; G. Silva; L. Talon Analysis and improvement of Brinkman lattice Boltzmann schemes: bulk, boundary, interface. Similarity and distinctness with finite elements in heterogeneous porous media, Phys. Rev. E, Volume 91 (2015) no. 2

Cited by Sources:

Comments - Policy