Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-27T20:04:48.188Z Has data issue: false hasContentIssue false
  • English
  • Français

A Simplified Lattice Boltzmann Method without Evolution of Distribution Function

Part of: Rarefied gas flows, Boltzmann equation Basic methods in fluid mechanics

Published online by Cambridge University Press:  11 October 2016

Z. Chen ,
C. Shu ,
Y. Wang ,
L. M. Yang  and
D. Tan
Z. Chen
Affiliation:
Department of Mechanical Engineering, National University of Singapore 10 Kent Ridge Crescent, Singapore 119260
C. Shu*
Affiliation:
Department of Mechanical Engineering, National University of Singapore 10 Kent Ridge Crescent, Singapore 119260
Y. Wang
Affiliation:
Department of Mechanical Engineering, National University of Singapore 10 Kent Ridge Crescent, Singapore 119260
L. M. Yang
Affiliation:
Department of Aerodynamics, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Yudao Street, Nanjing, Jiangsu 210016, China
D. Tan
Affiliation:
Department of Mechanical Engineering, National University of Singapore 10 Kent Ridge Crescent, Singapore 119260
*
*Corresponding author. Email:[email protected] (C. Shu)
Get access

Abstract

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.

Keywords

Chapman-Enskog expansion analysislattice Boltzmann equationNavier-Stokes equationsmemory coststability

MSC classification

Secondary: 76M28: Particle methods and lattice-gas methods 76P99: None of the above, but in this section
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 9 , Issue 1 , February 2017 , pp. 1 - 22
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Chen, S., Chen, H., Martnez, D. and Matthaeus, W., Lattice Boltzmann model for simulation of magnetohydrodynamics, Phys. Rev. Lett., 67 (1991), 3776.Google Scholar
[2] Qian, Y., D’Humiéres, D. and Lallemand, P., Lattice BGK models for Navier-Stokes equation, EPL (Europhysics Letters), 17 (1992), 479.Google Scholar
[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. 437–451.Google Scholar
[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. 307–330.Google Scholar
[5] Guo, Z., Zheng, C. and Shi, B., An extrapolation method for boundary conditions in lattice Boltzmann method, Phys. Fluids, 14 (2002), pp. 2007–2010.Google Scholar
[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.Google Scholar
[7] He, X. and Doolen, G., Lattice Boltzmann method on curvilinear coordinates system: flow around a circular cylinder, J. Comput. Phys., 134 (1997), pp. 306–315.Google Scholar
[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.Google Scholar
[9] Chen, S., Martinez, D. and Mei, R., On boundary conditions in lattice Boltzmann methods, Phys. Fluids, 8 (1996), pp. 2527–2536.Google Scholar
[10] Guo, Z., Shi, B. and Wang, N., Lattice BGK model for incompressible NavierStokes equation, J. Comput. Phys., 165 (2000), pp. 288–306.Google Scholar
[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. 925–944.Google Scholar
[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. 602–628.Google Scholar
[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.Google Scholar
[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. 413–441.CrossRefGoogle Scholar
[15] Lim, C., Shu, C., Niu, X. and Chew, Y., Application of lattice Boltzmann method to simulate microchannel flows, Phys. Fluids, 14 (2002), pp. 2299–2308.Google Scholar
[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. 282–300.Google Scholar
[17] Peng, Y., Shu, C. and Chew, Y., Simplified thermal lattice Boltzmann model for incompressible thermal flows, Phys. Rev. E, 68 (2003), 026701.Google Scholar
[18] Tang, G., Tao, W. and He, Y., Lattice Boltzmann method for gaseous microflows using kinetic theory boundary conditions, Phys. Fluids, 17 (2005), 058101.CrossRefGoogle Scholar
[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. 642–663.Google Scholar
[20] Shan, X. and Chen, H., Lattice Boltzmann model for simulating flows with multiple phases and components, Phys. Rev. E, 47 (1993), 1815.Google Scholar
[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. 628–644.Google Scholar
[22] Shan, X., Simulation of Rayleigh-Bénard convection using a lattice Boltzmann method, Phys. Rev. E, 55 (1997), 2780.Google Scholar
[23] Versteeg, H. K. and Malalasekera, W., An Introduction to Computational Fluid Dynamics: the Finite Volume Method, Pearson Education, 2007.Google Scholar
[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. 83–95.Google Scholar
[25] Lee, C. B., New features of CS solitons and the formation of vortices, Phys. Lett. A, 247 (1998), pp. 397–402.Google Scholar
[26] Lee, C., Possible universal transitional scenario in a flat plate boundary layer: Measurement and visualization, Phys. Rev. E, 62 (2000), 3659.Google Scholar
[27] Lee, C. and Li, R., Dominant structure for turbulent production in a transitional boundary layer, J. Turbulence, (2007), N55.Google Scholar
[28] Aidun, C. K. and Clausen, J. R., Lattice-Boltzmann method for complex flows, Ann. Rev. Fluid Mech., 42 (2010), pp. 439–472.Google Scholar
[29] Chen, S. and Doolen, G. D., Lattice Boltzmann method for fluid flows, Ann. Rev. Fluid Mech., 30 (1998), pp. 329–364.CrossRefGoogle Scholar
[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.Google Scholar
[31] McNamara, G. R. and Zanetti, G., Use of the Boltzmann equation to simulate lattice-gas automata, Phys. Rev. Lett., 61 (1988), 2332.Google Scholar
[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. 436–460.Google Scholar
[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. 344–364.Google Scholar
[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. 98–111.Google Scholar
[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. 404–423.Google Scholar
[36] Wang, Y., Yang, L. and Shu, C., From lattice Boltzmann method to lattice Boltzmann flux solver, Entropy, 17 (2015), pp. 7713–7735.Google Scholar
[37] White, F. M., Fluid Mechanics, McGraw-Hill, New York, 2003.Google Scholar
[38] Benzi, R., Succi, S. and Vergassola, M., The lattice Boltzmann equation: theory and applications, Phys. Reports, 222 (1992), pp. 145–197.Google Scholar
[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. 649–707.Google Scholar
[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. 3535–3542.Google Scholar
[41] Guo, Z. and Shu, C., Lattice Boltzmann Method and Its Applications in Engineering, World Scientific, 2013.Google Scholar
[42] Kim, J. and Moin, P., Application of a fractional-step method to incompressible Navier-Stokes equations, J. Comput. Phys., 59 (1985), pp. 308–323.Google Scholar
[43] Anderson, J. D. and Wendt, J., Computational Fluid Dynamics, Springer, 1995.Google Scholar
[44] Sterling, J. D., Chen, S., Stability analysis of lattice Boltzmann methods, J. Comput. Phys., 123 (1996), pp. 196–206.Google Scholar
[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. 665–680.Google Scholar
[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), 387–411.Google Scholar
[47] Mei, R., Luo, L.-S., Lallemand, P. and D’Humiéres, D., Consistent initial conditions for lattice Boltzmann simulations, Comput. Fluids, 35 (2006), pp. 855–862.Google Scholar
[48] Mei, R. and Shyy, W., On the finite difference-based lattice Boltzmann method in curvilinear coordinates, J. Comput. Phys., 143 (1998), pp. 426–448.CrossRefGoogle Scholar
[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.CrossRefGoogle ScholarPubMed