Hostname: page-component-848d4c4894-pjpqr Total loading time: 0 Render date: 2024-06-24T04:27:59.242Z Has data issue: false hasContentIssue false
  • English
  • Français

Transition Flow with an Incompressible Lattice Boltzmann Method

Part of: Optics, electromagnetic theory - General

Published online by Cambridge University Press:  11 July 2017

J. R. Murdock ,
J. C. Ickes  and
S. L. Yang
J. R. Murdock*
Affiliation:
Department of Mechanical Engineering-Engineering Mechanics, Michigan Technological University, 1400 Townsend Drive, Houghton, MI 49931, USA
J. C. Ickes*
Affiliation:
Department of Mechanical Engineering-Engineering Mechanics, Michigan Technological University, 1400 Townsend Drive, Houghton, MI 49931, USA
S. L. Yang*
Affiliation:
Department of Mechanical Engineering-Engineering Mechanics, Michigan Technological University, 1400 Townsend Drive, Houghton, MI 49931, USA
*
*Corresponding author. Email:jrmurdoc@mtu.edu (J. R. Murdock), ickesjc@gmail.com (J. C. Ickes), slyang@mtu.edu (S. L. Yang)
*Corresponding author. Email:jrmurdoc@mtu.edu (J. R. Murdock), ickesjc@gmail.com (J. C. Ickes), slyang@mtu.edu (S. L. Yang)
*Corresponding author. Email:jrmurdoc@mtu.edu (J. R. Murdock), ickesjc@gmail.com (J. C. Ickes), slyang@mtu.edu (S. L. Yang)
Get access

Abstract

Direct numerical simulations of the transition process from steady laminar to chaotic flow are considered in this study with the relatively new incompressible lattice Boltzmann equation. Numerically, a multiple relaxation time fully incompressible lattice Boltzmann equation is implemented in a 2D driven cavity. Spatial discretization is 2nd-order accurate, and the Kolmogorov length scale estimation based on Reynolds number (Re) dictates grid resolution. Initial simulations show the method to be accurate for steady laminar flows, while higher Re simulations reveal periodic flow behavior consistent with an initial Hopf bifurcation at Re 7,988. Non-repeating flow behavior is observed in the phase space trajectories above Re 13,063, and is evidence of the transition to a chaotic flow regime. Finally, flows at Reynolds numbers above the chaotic transition point are simulated and found with statistical properties in good agreement with literature.

Keywords

Multiple relaxation timelattice Boltzmanntransitionhigh Reynolds number flowincompressible flowlid driven cavity

MSC classification

Secondary: 78A48: Composite media; random media
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 9 , Issue 5 , October 2017 , pp. 1271 - 1288
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] Bhatnagar, P. L., Gross, E. P. and Krook, M., A model for collision processes in gasses. I. Small amplitude processes in charged and neutral one-component systems, Phys. Rev., 94 (1954), pp. 511–525.Google Scholar
[2] Bouzidi, M., D'Humieres, D., Lallemand, P. and Luo, L. S., Lattice Boltzmann equation on a two-dimensional rectangular grid, J. Comput. Phys., 172 (2001), pp. 704–717.CrossRefGoogle Scholar
[3] Bruneau, C. H. and Saad, M., The 2D lid-driven cavity problem revisited, Comput. Fluids, 35 (2006), pp. 326–348.Google Scholar
[4] Burggraf, O. R., Analytical and numerical studies of the structure of steady separated flows, J. Fluid Mech., 24 (1966), pp. 113–151.Google Scholar
[5] Cazemier, W., Verstappen, R. W. and Veldman, A. E., Proper orthogonal decomposition and low-dimensional models for driven cavity flows, Phys. Fluids, 10 (1998), pp. 1685–1699.CrossRefGoogle Scholar
[6] Ghia, U., Ghia, K. N. and Shin, C. T., High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method, J. Comput. Phys., 48 (1982), pp. 387–411.Google Scholar
[7] Guo, Z., Shi, B. and Wang, N., Lattice BGK model for incompressible Navier-Stokes equation, J. Comput. Phys., 165 (2000), pp. 288–306.Google Scholar
[8] He, X., Doolen, G. D. and Clark, T., Comparison of the lattice Boltzmann method and the artificial compressibility method for Navier-Stokes equations, J. Comput. Phys., 179 (2002), pp. 439–451.Google Scholar
[9] Hou, S., Zou, Q., Chen, S., Doolen, G. and Cogley, A. C., Simulation of cavity flow by the lattice Boltzmann method, J. Comput. Phys., 118 (1995), pp. 329–347.Google Scholar
[10] 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.CrossRefGoogle ScholarPubMed
[11] Lammers, P., Beronov, K. N. and Volkert, R. et al., Lattice BGK direct numerical simulation of fully developed turbulence in incompressible plane channel flow, Comput. Fluids, 35 (2006), pp. 1137–1153.Google Scholar
[12] Li, J. and Wang, Z., An alternative scheme to calculate the strain rate tensor for the LES applications in LBM, Math. Problems Eng., (2010), 724578.Google Scholar
[13] Marchi, C. H., Suero, R. and Araki, L. K., The lid-driven cavity flow: Numerical solution with a 1024×1024 grid, J. Brazilian Society Mech. Sci. Eng., 31 (2009), pp. 186–198.Google Scholar
[14] Marie, S., Ricot, D. and Sagaut, P., Comparison between lattice Boltzmann method and Navier-Stokes high order schemes for computational aeroacoustics, J. Comput. Phys., 228 (2008), pp. 1056–1070.Google Scholar
[15] Martinez, D. O., Matthaeus, W. H., Chen, S. and Montgomery, D. C., Comparison of spectral method and lattice Boltzmann simulations of two-dimensional hydrodynamics, Phys. Fluids, 6 (2006), pp. 1285–1298.CrossRefGoogle Scholar
[16] McNamara, G. R. and Zanetti, G., Use of the Boltzmann equation to simulate lattice-gas automata, Phys. Rev. Lett., 61 (1988), 2332.Google Scholar
[17] Murdock, J. R. and Yang, S. L., Alternative and explicit derivation of the lattice Boltzmann equation for the unsteady incompressible Navier-Stokes equation, Int. J. Comput. Eng. Res., 6 (2016), pp. 47–59.Google Scholar
[18] Peng, Y. F., Shiau, Y. H. and Hwang, R. R., Transition in a 2-D lid driven cavity flow, Comput. Fluids, 32 (2003), pp. 337–352.Google Scholar
[19] Poliashenko, M. and Aidun, C. K., A direct method for computation of simple bifurcations, J. Comput. Phys., 121 (2006), pp. 246–260.Google Scholar
[20] Shi, B., He, N. and Wang, N., A unified thermal lattice BGK model for Boussinesq equations, Prog. Comput. Fluid Dyn., 5 (2005).CrossRefGoogle Scholar
[21] Wang, R. Z. and Fang, H. P., Test of the possible application of the half-way bounce-back boundary condition for lattice Boltzmann methods in complex geometry, Commun. Theor. Phys., 35 (2000), pp. 593–596.Google Scholar
[22] Zhang, C., Cheng, Y., Huang, S. and Wu, J., Improving the stability of themultiple-relaxation-time lattice Boltzmann method by a viscosity counteracting approach, Adv. Appl. Math. Mech., 8 (2016), pp. 37–51.CrossRefGoogle Scholar