Skip to main content Accessibility help

Lattice Boltzmann method with self-consistent thermo-hydrodynamic equilibria

  • M. SBRAGAGLIA (a1), R. BENZI (a1), L. BIFERALE (a1), H. CHEN (a2), X. SHAN (a2) and S. SUCCI (a3)...


Lattice kinetic equations incorporating the effects of external/internal force fields via a shift of the local fields in the local equilibria are placed within the framework of continuum kinetic theory. The mathematical treatment reveals that in order to be consistent with the correct thermo-hydrodynamical description, temperature must also be shifted, besides momentum. New perspectives for the formulation of thermo-hydrodynamic lattice kinetic models of non-ideal fluids are then envisaged. It is also shown that on the lattice, the definition of the macroscopic temperature requires the inclusion of new terms directly related to discrete effects. The theoretical treatment is tested against a controlled case with a non-ideal equation of state.


Corresponding author

Email address for correspondence:


Hide All
Ahlers, G., Grossmann, S. & Lohse, D. 2009 Heat transfer and large-scale dynamics in turbulent Rayleigh–Bénard convection. Rev. Mod. Phys. 81, 503537.
Ansumali, S & Karlin, I. 2002 Kinetic boundary conditions in the lattice Boltzmann method. Phys. Rev. E 66, 026311.
Bathnagar, P.-L., Gross, E. & Krook, M. 1954 A model for collision processes in gases. Phys. Rev. 94, 511525.
Benzi, R., Succi, S. & Vergassola, M. 1992 The lattice Boltzmann equation: theory and applications. Phys. Rep. 222, 145197.
Boghosian, B. M. 2008 Exact hydrodynamics of the lattice BGK equation. arXiv:0810.2344v1.
Brennen, C. 2005 Fundamentals of Multiphase Flow. Cambridge University Press.
Buick, J. M. & Greated, C. A. 2000 Gravity in a lattice Boltzmann model. Phys. Rev. E 61, 53075320
Chen, S. & Doolen, G. 1998 Lattice Boltzmann method for fluid flows. Annu. Rev. Fluid Mech. 30, 329364.
Ginzbourg, I & Adler, P. M. 1994 Boundary flow condition analysis for the 3-dimensional lattice Boltzmann model. J. Phys. II 2, 191214.
Gonnella, G., Lamura, A. & Sofonea, V. 2007 Lattice Boltzmann simulation of thermal nonideal fluids. Phys. Rev. E 76, 036703.
Gradshtein, I. S. & Ryzhik, I. M. 2000 Tables of Integrals and Series, 6th edn. Academic.
Guo, Z., Zheng, C. & Shi, B. 2002 Discrete lattice effects on the forcing term in the lattice Boltzmann method, Phys. Rev. E 65, 046308.
He, X. & Doolen, G. 2001 Thermodynamic foundations of kinetic theory and lattice Boltzmann models for multiphase Flows. J. Stat. Phys. 107, 309328.
He, X. & Luo, L. S. 1997 Theory of the lattice Boltzmann method: from the Boltzmann equation to the lattice Boltzmann equation. Phys. Rev. E 56 68116817.
Ladd, 1994 Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 2. Numerical results. J. Fluid Mech. 271 311339.
Li, Q. & Wagner, A. J. 2007 Symmetric free-energy-based multicomponent lattice Boltzmann method. Phys. Rev. E 76, 036701.
Martys, N. S., Shan, X. & Chen, H. 1998 Evaluation of the external force term in the discrete Boltzmann equation. Phys. Rev. E 58, 6865.
Nie, X., Shan, X. & Chen, H. 2008 Thermal lattice Boltzmann model for gases with internal degrees of freedom. Phys. Rev. E 77, 035701(R).
Novikov, E. A. 1964 Functionals and the method of random forces in turbulence theory. Zh. Exp. Teor. Fiz. 47, 19191926.
Qi, D. W. 2006 Direct simulations of flexible cylindrical fibres suspensions in finite Reynolds numbers flows. J. Chem. Phys. 125, 114901.
Rowlinson, J. R. & Widom, B. 1982 Molecular Theory of Capillarity. Clarendon.
Shan, X. & Chen, H. 1993 Lattice Boltzmann model for simulating flows with multiple phases and components. Phys. Rev. E 47, 1815.
Shan, X. & He, X. 1998 Discretization of the velocity space in the solution of the Boltzmann equation. Phys. Rev. Lett. 80, 6568.
Shan, X., Yuan, F. & Chen, H. 2006 Kinetic theory representation of hydrodynamics: a way beyond the Navier—Stokes equation. J. Fluid Mech. 550, 413441.
Snider, R. F. 1995 Conversion between kinetic energy and potential energy in the classical non-local Boltzmann equation. J. Stat. Phys. 80, 10851117.
Swift, M. R., Osborn, W. R. & Yeomans, J. M. 1995 Lattice Boltzmann simulation of nonideal fluids. Phys. Rev. Lett. 75, 830833.
Wolf-Gladrow, D. 2000 Lattice-Gas Cellular Automata and Lattice Boltzmann Models. Springer.
Succi, S., Karlin, I. & Chen, H. 2002 Colloquium: role of the H theorem in lattice Boltzmann hydrodynamic simulations. Rev. Mod. Phys. 74, 1203.
MathJax is a JavaScript display engine for mathematics. For more information see

Lattice Boltzmann method with self-consistent thermo-hydrodynamic equilibria

  • M. SBRAGAGLIA (a1), R. BENZI (a1), L. BIFERALE (a1), H. CHEN (a2), X. SHAN (a2) and S. SUCCI (a3)...


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed