Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-27T21:12:19.373Z Has data issue: false hasContentIssue false
  • English
  • Français

A Comparative Study of LBE and DUGKS Methods for Nearly Incompressible Flows

Published online by Cambridge University Press:  24 March 2015

Peng Wang ,
Lianhua Zhu ,
Zhaoli Guo  and
Kun Xu
Peng Wang
Affiliation:
State Key Laboratory of Coal Combustion, Huazhong University of Science and Technology, Wuhan 430074, P.R. China
Lianhua Zhu
Affiliation:
State Key Laboratory of Coal Combustion, Huazhong University of Science and Technology, Wuhan 430074, P.R. China
Zhaoli Guo*
Affiliation:
State Key Laboratory of Coal Combustion, Huazhong University of Science and Technology, Wuhan 430074, P.R. China
Kun Xu
Affiliation:
Mathematics Department, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
*
*Corresponding author. Email addresses:sklccwangpeng@hust.edu.cn (P. Wang), lhzhu@hust.edu.cn (L. Zhu), zlguo@hust.edu.cn (Z. Guo), makxu@ust.hk (K. Xu)
Get access

Abstract

The lattice Boltzmann equation (LBE) methods (both LBGK and MRT) and the discrete unified gas-kinetic scheme (DUGKS) are both derived from the Boltzmann equation, but with different consideration in their algorithm construction. With the same numerical discretization in the particle velocity space, the distinctive modeling of these methods in the update of gas distribution function may introduce differences in the computational results. In order to quantitatively evaluate the performance of these methods in terms of accuracy, stability, and efficiency, in this paper we test LBGK, MRT, and DUGKS in two-dimensional cavity flow and the flow over a square cylinder, respectively. The results for both cases are validated against benchmark solutions. The numerical comparison shows that, with sufficient mesh resolution, the LBE and DUGKS methods yield qualitatively similar results in both test cases. With identical mesh resolutions in both physical and particle velocity space, the LBE methods are more efficient than the DUGKS due to the additional particle collision modeling in DUGKS. But, the DUGKS is more robust and accurate than the LBE methods in most test conditions. Particularly, for the unsteady flow over a square cylinder at Reynolds number 300, with the same mesh resolution it is surprisingly observed that the DUGKS can capture the physical multi-frequency vortex shedding phenomena while the LBGK and MRT fail to get that. Furthermore, the DUGKS is a finite volume method and its computational efficiency can be much improved when a non-uniform mesh in the physical space is adopted. The comparison in this paper clearly demonstrates the progressive improvement of the lattice Boltzmann methods from LBGK, to MRT, up to the current DUGKS, along with the inclusion of more reliable physical process in their algorithm development. Besides presenting the Navier-Stokes solution, the DUGKS can capture the rarefied flow phenomena as well with the increasing of Knudsen number.

Keywords

44.05.+e47.11.-j47.56.+rLattice Boltzmann equationdiscrete unified gas-kinetic schemenumerical performance
Type
Research Article
Information
Communications in Computational Physics , Volume 17 , Issue 3 , March 2015 , pp. 657 - 681
Copyright
Copyright © Global-Science Press 2015 

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]van Leer, B., Computational fluid dynamics: science or toolbox, AIAA Report No. 2001–2520, 2001.Google Scholar
[2]Karttunen, M., Vattulainen, I., and Lukkarinen, A. (Eds.), Novel Methods in Soft Matter Simulations, vol. 640 of Lecture Notes in Physics, Springer, Berlin, 2004.Google Scholar
[3]Rothman, D.H. and Zaleski, S., Lattice-gas cellular automata: simple models of complex hydrodynamics, Cambridge University Press, 2004.Google Scholar
[4]Succi, S., The lattice Boltzmann equation: for fluid dynamics and beyond, Oxford university press, 2001.Google Scholar
[5]Xu, K., A gas-kinetic BGK scheme for the NavierCStokes equations and its connection with artificial dissipation and Godunov method, J. Comput. Phys., 171 (1) (2001), 289–335.Google Scholar
[6]Xu, K. and Huang, J.C., A unified gas-kinetic scheme for continuum and rarefied flows, J. Comput. Phys., 229 (2010), 7747–7764.Google Scholar
[7]Guo, Z.L., Xu, K., and Wang, R.J., Discrete unified gas kinetic scheme for all Knudsen number flows: Low-speed isothermal case, Phys. Rev. E, 88 (2013), 033305.Google Scholar
[8]Gingold, R.A. and Monaghan, J.J., Smoothed particle hydrodynamics-theory and application to non-spherical stars, Mon. Roy. Astron. Soc., 181 (1977), 375–389.CrossRefGoogle Scholar
[9]Yu, D., Mei, R., Luo, L.-S., and Shyy, W., Viscous flow computations with the method of lattice Boltzmann equation, Prog. Aerospace Sci., 39 (2003), 329–367.Google Scholar
[10]Xu, K. and Lui, S.H., Rayleigh-Benard simulation using the gas-kinetic BhatnagarCGrossCK-rook scheme in the incompressible limit, Phys. Rev. E, 60 (1) (1999), 464–470.Google Scholar
[11]Shan, X. and Chen, H., Lattice Boltzmann model for simulating flows with multiple phases and components, Phys. Rev. E, 47 (1993), 1815–1819.CrossRefGoogle ScholarPubMed
[12]Luo, L.-S., Unified theory of the lattice Boltzmann models for nonideal gases, Phys. Rev. Lett., 81 (1998), 1618–1621.CrossRefGoogle Scholar
[13]Luo, L.-S., Theory of lattice Boltzmann method: lattice Boltzmann models for nonideal gases, Phys. Rev. E, 62 (2000), 4982–4996.Google Scholar
[14]Guo, Z.L. and Zhao, T.S., Discrete velocity and lattice Boltzmann models for binary mixtures of nonideal fluids, Phys. Rev. E, 68 (2003), 035302.Google Scholar
[15]Luo, L.-S. and Girimaji, S.S., Lattice Boltzmann model for binary mixtures, Phys. Rev. E, 66 (2002), 035301(R).Google Scholar
[16]Luo, L.-S. and Girimaji, S.S., Theory of the lattice Boltzmann method: two-fluid model for binary mixtures, Phys. Rev. E, 67 (2003), 036302.Google Scholar
[17]Xu, K., Super-Burnett solutions for Poiseuille flow, Phys. Fluid, 15 (2003), 2077–2080.Google Scholar
[18]Xu, K. and Li, Z.H., Microchannel flow in the slip regime: gas-kinetic BGK-Burnett solutions, J. Fluid Mech., 513 (2004), 87–110.Google Scholar
[19]Qian, Y., d’Humires, D., and Lallemand, P., Lattice BGK models for Navier-Stokes equation, Europhys. Lett., 17 (1992), 479.CrossRefGoogle Scholar
[20]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-R5342.Google Scholar
[21]Hou, S., Zou, Q., Chen, S., and Doolen, G., Cogley, A.C., Simulation of cavity flow by the lattice Boltzmann method, J. Comput. Phys., 118 (1995), 329–347.CrossRefGoogle Scholar
[22]Luo, L.-S., Liao, W., Chen, X., Peng, Y., and Zhang, W., Numerics of the lattice Boltzmann method: Effects of collision models on the lattice Boltzmann simulations, Phys. Rev. E, 83 (2011), 056710.Google Scholar
[23]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
[24]Guo, Z.L., and Shu, C., Lattice Boltzmann method and its application in engineering, World Scientific press, 2013.Google Scholar
[25]Cao, N., Chen, S. Y., and Martinez, D., Physical symmetry and lattice symmetry in the lattice Boltzmann method, Phys. Rev. E, 55 (1997), 21–24.Google Scholar
[26]Guo, Z.L., and Zhao, T. S., Explicit finite-difference lattice Boltzmann method for curvilinear coordinates, Phys. Rev. E, 67 (2003), 066709Google Scholar
[27]Peng, G., Xi, H., Duncan, C. and Chou, S. H., Finite volume scheme for the lattice Boltzmann method on unstructured meshes, Phys. Rev. E, 59 (1999), 4675.Google Scholar
[28]Rossil, N., Ubertinil, S., Bellal, G., and Succi, S., Unstructured lattice Boltzmann method in three dimensions. Int. J. Numer. Meth. Fluids, 49 (2005), 619–633.Google Scholar
[29]Ubertini, S. and Succi, S., Recent advances of lattice Boltzmann techniques on unstructured grids, Prog. Comput. Fluid Dyn., 5(1/2) (2005), 85–96.CrossRefGoogle Scholar
[30]Ubertini, S. and Succi, S., A generalised lattice Boltzmann equation on unstructured grids, Commun. Comput. Phys., 3 (2008), 342Google Scholar
[31]Lee, T., and Lin, C.-L., A characteristic Galerkin method for discrete Boltzmann equation, J. Comput. Phys., 171 (2001), 336–356.Google Scholar
[32]Bhatnagar, P.L., Gross, E.P., and Krook, M., A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems, Phys. Rev., 94 (1954), 511.CrossRefGoogle Scholar
[33]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
[34]Botella, O. and Peyret, R., Benchmark spectral results on the lid-driven cavity flow, Comput. Fluid, 27 (1998), 421–433.CrossRefGoogle Scholar
[35]Breuer, M., Bernsdorf, J., Zeiser, T., and Durst, F., Accurate computations of the laminar flow past a square cylinder based on two different methods: lattice-Boltzmann and finite-volume, Int. J. Heat Fluid Flow, 21 (2000), 186–196.Google Scholar
[36]Guo, Z.L., Liu, H.W., Luo, L.S., and Xu, K., A comparative study of the LBE and GKS methods for 2D near incompressible laminar flows, J. Comput. Phys., 227 (2008), 4955–4976.Google Scholar
[37]Ladd, A. and Verberg, R., Lattice-Boltzmann simulations of particle-fluid suspensions, J. Stat. Phys., 104 (2001), 1191–1251.Google Scholar
[38]Turki, S., Abbassi, H., and Nasrallah, S.B., Effect of the blockage ratio on the flow in a channel with a built-in square cylinder, Comput. Mech., 33 (2003), 22–29.Google Scholar