Truncation errors and the rotational invariance of three-dimensional lattice models in the lattice Boltzmann method

Abstract The application of the lattice Boltzmann method (LBM) in three-dimensional isothermal hydrodynamic problems often adopts one of the following models: D3Q15, D3Q19, or D3Q27. Although all of them retrieve consistent Navier–Stokes dynamics in the continuum limit, they are expected to behave differently at discrete level. The present work addresses this issue by performing a LBM truncation error analysis. As a conclusion, it is theoretically demonstrated that differences among the aforementioned cubic lattices lie in the structure of their non-linear truncation errors. While reduced lattice schemes, such as D3Q15 and D3Q19, introduce spurious angular dependencies through non-linear truncation errors, the complete three-dimensional cubic lattice D3Q27 is absent from such features. This result justifies the superiority of the D3Q27 lattice scheme to cope with the rotational invariance principle in three-dimensional isothermal hydrodynamic problems, particularly when convection is not negligible. Such a theoretical conclusion also finds support in numerical tests presented in this work: a Poiseuille duct flow and a weakly-rotating duct flow.

[1]  L. Luo,et al.  Theory of the lattice Boltzmann method: From the Boltzmann equation to the lattice Boltzmann equation , 1997 .

[2]  J. Buick,et al.  Gravity in a lattice Boltzmann model , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[3]  Dominique d'Humières,et al.  Viscosity independent numerical errors for Lattice Boltzmann models: From recurrence equations to "magic" collision numbers , 2009, Comput. Math. Appl..

[4]  Pierre Lallemand,et al.  Lattice Gas Hydrodynamics in Two and Three Dimensions , 1987, Complex Syst..

[5]  D. d'Humières,et al.  Thirteen-velocity three-dimensional lattice Boltzmann model. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[6]  Qisu Zou,et al.  A improved incompressible lattice Boltzmann model for time-independent flows , 1995 .

[7]  Andrew Perrin,et al.  An explicit finite-difference scheme for simulation of moving particles , 2006, J. Comput. Phys..

[8]  Jens Harting,et al.  Implementation of on-site velocity boundary conditions for D3Q19 lattice Boltzmann simulations , 2008, 0811.4593.

[9]  L. Luo,et al.  A priori derivation of the lattice Boltzmann equation , 1997 .

[10]  Viriato Semiao,et al.  A study on the inclusion of body forces in the lattice Boltzmann BGK equation to recover steady-state hydrodynamics , 2011 .

[11]  P. Asinari,et al.  Factorization symmetry in the lattice Boltzmann method , 2009, 0911.5529.

[12]  A. Chorin.  A Numerical Method for Solving Incompressible Viscous Flow Problems , 1997 .

[13]  I. Ginzburg.  Consistent lattice Boltzmann schemes for the Brinkman model of porous flow and infinite Chapman-Enskog expansion. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[14]  Peter M. A. Sloot,et al.  Implementation Aspects of 3D Lattice-BGK: Boundaries, Accuracy, and a New Fast Relaxation Method , 1999 .

[15]  P. Lallemand,et al.  Theory of the lattice boltzmann method: dispersion, dissipation, isotropy, galilean invariance, and stability , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[16]  C. K. Chong,et al.  Rotational invariance in the three-dimensional lattice Boltzmann method is dependent on the choice of lattice , 2011, J. Comput. Phys..

[17]  X. He,et al.  Discretization of the Velocity Space in the Solution of the Boltzmann Equation , 1997, comp-gas/9712001.

[18]  Kai Zhang,et al.  Extended hybrid pressure and velocity boundary conditions for D3Q27 lattice Boltzmann model , 2012 .

[19]  D. d'Humières,et al.  Multiple–relaxation–time lattice Boltzmann models in three dimensions , 2002, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[20]  Q. Zou,et al.  On pressure and velocity boundary conditions for the lattice Boltzmann BGK model , 1995, comp-gas/9611001.

[21]  Qisu Zou,et al.  N ov 1 99 6 On pressure and velocity flow boundary conditions and bounceback for the lattice Boltzmann BGK model , 2008 .

[22]  S. N. Barua,et al.  Secondary flow in a rotating straight pipe , 1954, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[23]  Gábor Házi,et al.  Direct numerical and large eddy simulation of longitudinal flow along triangular array of rods using the lattice Boltzmann method , 2006, Math. Comput. Simul..

[24]  Richard Haberman,et al.  Applied Partial Differential Equations with Fourier Series and Boundary Value Problems , 2012 .

[25]  Shiyi Chen,et al.  LATTICE BOLTZMANN METHOD FOR FLUID FLOWS , 2001 .

[26]  X. Shan,et al.  Galilean invariance of lattice Boltzmann models , 2008, 0801.2924.

[27]  Yassin A. Hassan,et al.  The effect of lattice models within the lattice Boltzmann method in the simulation of wall-bounded turbulent flows , 2013, J. Comput. Phys..

[28]  Viriato Semiao,et al.  First- and second-order forcing expansions in a lattice Boltzmann method reproducing isothermal hydrodynamics in artificial compressibility form , 2012, Journal of Fluid Mechanics.

[29]  Wei Shyy,et al.  Lattice Boltzmann Method for 3-D Flows with Curved Boundary , 2000 .

[30]  Nicos Martys,et al.  Breakdown of Chapman-Enskog expansion and the anisotropic effect for lattice-Boltzmann models of porous flow , 2007 .

[31]  Robert S. Bernard,et al.  Boundary conditions for the lattice Boltzmann method , 1996 .

[32]  L. Luo,et al.  Theory of the lattice Boltzmann equation: symmetry properties of discrete velocity sets. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[33]  D. d'Humières,et al.  Study of simple hydrodynamic solutions with the two-relaxation-times lattice Boltzmann scheme , 2008 .

[34]  A. Ladd.  Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 1. Theoretical foundation , 1993, Journal of Fluid Mechanics.

[35]  B. Shi,et al.  Discrete lattice effects on the forcing term in the lattice Boltzmann method. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[36]  Skordos,et al.  Initial and boundary conditions for the lattice Boltzmann method. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[37]  Cyrus K. Aidun,et al.  Lattice-Boltzmann Method for Complex Flows , 2010 .

[38]  Orestis Malaspinas,et al.  Straight velocity boundaries in the lattice Boltzmann method. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[39]  L. Luo,et al.  Lattice Boltzmann Model for the Incompressible Navier–Stokes Equation , 1997 .

[40]  G. Doolen,et al.  Comparison of the Lattice Boltzmann Method and the Artificial Compressibility Method for Navier-Stokes Equations , 2002 .

[41]  P. Adler,et al.  Boundary flow condition analysis for the three-dimensional lattice Boltzmann model , 1994 .

[42]  Jun Yang,et al.  Study of force-dependent and time-dependent transition of secondary flow in a rotating straight channel by the lattice Boltzmann method , 2009 .

[43]  X. Yuan,et al.  Kinetic theory representation of hydrodynamics: a way beyond the Navier–Stokes equation , 2006, Journal of Fluid Mechanics.

[44]  D. Patil,et al.  Higher-order Galilean-invariant lattice Boltzmann model for microflows: single-component gas. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[45]  G. H. Lee,et al.  A numerical study of the similarity of fully developed laminar flows in orthogonally rotating rectangular ducts and stationary curved rectangular ducts of arbitrary aspect ratio , 2002 .

[46]  P. Bhatnagar,et al.  A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems , 1954 .

[47]  R. Benzi,et al.  The lattice Boltzmann equation: theory and applications , 1992 .

[48]  Yan Peng,et al.  Numerics of the lattice Boltzmann method: effects of collision models on the lattice Boltzmann simulations. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[49]  Dierk Raabe,et al.  Shear stress in lattice Boltzmann simulations. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[50]  Jonathan R Clausen,et al.  Entropically damped form of artificial compressibility for explicit simulation of incompressible flow. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[51]  Sarah E. Harrison,et al.  The use of the lattice Boltzmann method in thrombosis modelling. , 2007 .

[52]  Bernie D. Shizgal,et al.  Rarefied Gas Dynamics: Theory and Simulations , 1994 .

[53]  Y. Qian,et al.  Lattice BGK Models for Navier-Stokes Equation , 1992 .

[54]  Paul S. Hammond,et al.  Error analysis and correction for Lattice Boltzmann simulated flow conductance in capillaries of different shapes and alignments , 2012, J. Comput. Phys..

[55]  D. d'Humières,et al.  Two-relaxation-time Lattice Boltzmann scheme: About parametrization, velocity, pressure and mixed boundary conditions , 2008 .

[56]  B. Shizgal,et al.  Generalized Lattice-Boltzmann Equations , 1994 .

[57]  I. Ginzburg.  Equilibrium-type and link-type lattice Boltzmann models for generic advection and anisotropic-dispersion equation , 2005 .

[58]  Irina Ginzburg,et al.  Truncation Errors, Exact and Heuristic Stability Analysis of Two-Relaxation-Times Lattice Boltzmann Schemes for Anisotropic Advection-Diffusion Equation , 2012 .

[59]  Haroon S. Kheshgi,et al.  Viscous flow through a rotating square channel , 1985 .

[60]  Hiroshi Ishigaki,et al.  Analogy between laminar flows in curved pipes and orthogonally rotating pipes , 1994, Journal of Fluid Mechanics.

[61]  P. Philippi,et al.  From the continuous to the lattice Boltzmann equation: the discretization problem and thermal models. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.