Skip to main content Accessibility help

A Simplified Lattice Boltzmann Method without Evolution of Distribution Function

  • Z. Chen (a1), C. Shu (a1), Y. Wang (a1), L. M. Yang (a2) and D. Tan (a1)...


In this paper, a simplified lattice Boltzmann method (SLBM) without evolution of the distribution function is developed for simulating incompressible viscous flows. This method is developed from the application of fractional step technique to the macroscopic Navier-Stokes (N-S) equations recovered from lattice Boltzmann equation by using Chapman-Enskog expansion analysis. In SLBM, the equilibrium distribution function is calculated from the macroscopic variables, while the non-equilibrium distribution function is simply evaluated from the difference of two equilibrium distribution functions. Therefore, SLBM tracks the evolution of the macroscopic variables rather than the distribution function. As a result, lower virtual memories are required and physical boundary conditions could be directly implemented. Through numerical test at high Reynolds number, the method shows very nice performance in numerical stability. An accuracy test for the 2D Taylor-Green flow shows that SLBM has the second-order of accuracy in space. More benchmark tests, including the Couette flow, the Poiseuille flow as well as the 2D lid-driven cavity flow, are conducted to further validate the present method; and the simulation results are in good agreement with available data in literatures.


Corresponding author

*Corresponding author. Email: (C. Shu)


Hide All
[1] Chen, S., Chen, H., Martnez, D. and Matthaeus, W., Lattice Boltzmann model for simulation of magnetohydrodynamics, Phys. Rev. Lett., 67 (1991), 3776.
[2] Qian, Y., D’Humiéres, D. and Lallemand, P., Lattice BGK models for Navier-Stokes equation, EPL (Europhysics Letters), 17 (1992), 479.
[3] D’Humiéres, D., Multiplerelaxationtime lattice Boltzmann models in three dimensions, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 360 (2002), pp. 437451.
[4] Mei, R., Luo, L.-S. and Shyy, W., An accurate curved boundary treatment in the lattice Boltzmann method, J. Comput. Phys., 155 (1999), pp. 307330.
[5] Guo, Z., Zheng, C. and Shi, B., An extrapolation method for boundary conditions in lattice Boltzmann method, Phys. Fluids, 14 (2002), pp. 20072010.
[6] Lallemand, P. and Luo, L.-S., Theory of the lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance, and stability, Phys. Rev. E, 61 (2000), 6546.
[7] He, X. and Doolen, G., Lattice Boltzmann method on curvilinear coordinates system: flow around a circular cylinder, J. Comput. Phys., 134 (1997), pp. 306315.
[8] Chen, H., Chen, S. and Matthaeus, W. H., Recovery of the Navier-Stokes equations using a lattice-gas Boltzmann method, Phys. Rev. A, 45 (1992), R5339.
[9] Chen, S., Martinez, D. and Mei, R., On boundary conditions in lattice Boltzmann methods, Phys. Fluids, 8 (1996), pp. 25272536.
[10] Guo, Z., Shi, B. and Wang, N., Lattice BGK model for incompressible NavierStokes equation, J. Comput. Phys., 165 (2000), pp. 288306.
[11] Shu, C., Niu, X. and Chew, Y., Taylor series expansion and least squares-based lattice Boltzmann method: three-dimensional formulation and its applications, Int. J. Modern Phys. C, 14 (2003), pp. 925944.
[12] Feng, Z.-G. and Michaelides, E. E., The immersed boundary-lattice Boltzmann method for solving fluidparticles interaction problems, J. Comput. Phys., 195 (2004), pp. 602628.
[13] Zhang, Y.-H., Gu, X.-J., Barber, R. W. and Emerson, D. R., Capturing Knudsen layer phenomena using a lattice Boltzmann model, Phys. Rev. E, 74 (2006), 046704.
[14] Shan, X., Yuan, X.-F. and Chen, H., Kinetic theory representation of hydrodynamics: a way beyond the NavierStokes equation, J. Fluid Mech., 550 (2006), pp. 413441.
[15] Lim, C., Shu, C., Niu, X. and Chew, Y., Application of lattice Boltzmann method to simulate microchannel flows, Phys. Fluids, 14 (2002), pp. 22992308.
[16] He, X., Chen, S. and Doolen, G. D., A novel thermal model for the lattice Boltzmann method in incompressible limit, J. Comput. Phys., 146 (1998), pp. 282300.
[17] Peng, Y., Shu, C. and Chew, Y., Simplified thermal lattice Boltzmann model for incompressible thermal flows, Phys. Rev. E, 68 (2003), 026701.
[18] Tang, G., Tao, W. and He, Y., Lattice Boltzmann method for gaseous microflows using kinetic theory boundary conditions, Phys. Fluids, 17 (2005), 058101.
[19] He, X., Chen, S. and Zhang, R., A lattice Boltzmann scheme for incompressible multiphase flow and its application in simulation of RayleighTaylor instability, J. Comput. Phys., 152 (1999), pp. 642663.
[20] Shan, X. and Chen, H., Lattice Boltzmann model for simulating flows with multiple phases and components, Phys. Rev. E, 47 (1993), 1815.
[21] Inamuro, T., Ogata, T., Tajima, S. and Konishi, N., A lattice Boltzmann method for incompressible two-phase flows with large density differences, J. Comput. Phys., 198 (2004), pp. 628644.
[22] Shan, X., Simulation of Rayleigh-Bénard convection using a lattice Boltzmann method, Phys. Rev. E, 55 (1997), 2780.
[23] Versteeg, H. K. and Malalasekera, W., An Introduction to Computational Fluid Dynamics: the Finite Volume Method, Pearson Education, 2007.
[24] Liszka, T. and Orkisz, J., The finite difference method at arbitrary irregular grids and its application in applied mechanics, Comput. Struct., 11 (1980), pp. 8395.
[25] Lee, C. B., New features of CS solitons and the formation of vortices, Phys. Lett. A, 247 (1998), pp. 397402.
[26] Lee, C., Possible universal transitional scenario in a flat plate boundary layer: Measurement and visualization, Phys. Rev. E, 62 (2000), 3659.
[27] Lee, C. and Li, R., Dominant structure for turbulent production in a transitional boundary layer, J. Turbulence, (2007), N55.
[28] Aidun, C. K. and Clausen, J. R., Lattice-Boltzmann method for complex flows, Ann. Rev. Fluid Mech., 42 (2010), pp. 439472.
[29] Chen, S. and Doolen, G. D., Lattice Boltzmann method for fluid flows, Ann. Rev. Fluid Mech., 30 (1998), pp. 329364.
[30] He, X. and Luo, L.-S., Theory of the lattice Boltzmann method: From the Boltzmann equation to the lattice Boltzmann equation, Phys. Rev. E, 56 (1997), 6811.
[31] McNamara, G. R. and Zanetti, G., Use of the Boltzmann equation to simulate lattice-gas automata, Phys. Rev. Lett., 61 (1988), 2332.
[32] Shu, C., Wang, Y., Teo, C. and Wu, J., Development of lattice Boltzmann flux solver for simulation of incompressible flows, Adv. Appl. Math. Mech., 6 (2014), pp. 436460.
[33] Wang, Y., Shu, C. and Teo, C., Development of LBGK and incompressible LBGK-based lattice Boltzmann flux solvers for simulation of incompressible flows, Int. J. Numer. Methods Fluids, 75 (2014), pp. 344364.
[34] Wang, Y., Shu, C. and Teo, C., Thermal lattice Boltzmann flux solver and its application for simulation of incompressible thermal flows, Comput. Fluids, 94 (2014) pp. 98111.
[35] Wang, Y., Shu, C., Huang, H. and Teo, C., Multiphase lattice Boltzmann flux solver for incompressible multiphase flows with large density ratio, J. Comput. Phys., 280 (2015), pp. 404423.
[36] Wang, Y., Yang, L. and Shu, C., From lattice Boltzmann method to lattice Boltzmann flux solver, Entropy, 17 (2015), pp. 77137735.
[37] White, F. M., Fluid Mechanics, McGraw-Hill, New York, 2003.
[38] Benzi, R., Succi, S. and Vergassola, M., The lattice Boltzmann equation: theory and applications, Phys. Reports, 222 (1992), pp. 145197.
[39] Frisch, U., D’Humiéres, D., Hasslacher, B., Lallemand, P., Pomeau, Y. and Rivet, J.-P., Lattice gas hydrodynamics in two and three dimensions, Complex Systems, 1 (1987), pp. 649707.
[40] Inamuro, T., Yoshino, M. and Ogino, F., Accuracy of the lattice Boltzmann method for small Knudsen number with finite Reynolds number, Phys. Fluids, 9 (1997), pp. 35353542.
[41] Guo, Z. and Shu, C., Lattice Boltzmann Method and Its Applications in Engineering, World Scientific, 2013.
[42] Kim, J. and Moin, P., Application of a fractional-step method to incompressible Navier-Stokes equations, J. Comput. Phys., 59 (1985), pp. 308323.
[43] Anderson, J. D. and Wendt, J., Computational Fluid Dynamics, Springer, 1995.
[44] Sterling, J. D., Chen, S., Stability analysis of lattice Boltzmann methods, J. Comput. Phys., 123 (1996), pp. 196206.
[45] Niu, X., Shu, C., Chew, Y. and Wang, T., Investigation of stability and hydrodynamics of different lattice Boltzmann models, J. Stat. Phys., 117 (2004), pp. 665680.
[46] Ghia, U., Ghia, K. N. and Shin, C., High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method, J. Comput. Phys., 48 (1982), 387411.
[47] Mei, R., Luo, L.-S., Lallemand, P. and D’Humiéres, D., Consistent initial conditions for lattice Boltzmann simulations, Comput. Fluids, 35 (2006), pp. 855862.
[48] Mei, R. and Shyy, W., On the finite difference-based lattice Boltzmann method in curvilinear coordinates, J. Comput. Phys., 143 (1998), pp. 426448.
[49] Guo, Z., Zheng, C. and Shi, B., Discrete lattice effects on the forcing term in the lattice Boltzmann method, Phys. Rev. E, 65 (2002), 046308.


MSC classification


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