Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-23T06:31:54.643Z Has data issue: false hasContentIssue false
  • English
  • Français

On Initial Conditions for the Lattice Boltzmann Method

Published online by Cambridge University Press:  30 July 2015

Juntao Huang ,
Hao Wu  and
Wen-An Yong
Juntao Huang
Affiliation:
Zhou Pei-Yuan Center for Applied Mathematics, Tsinghua University, Beijing 100084, China
Hao Wu*
Affiliation:
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
Wen-An Yong
Affiliation:
Zhou Pei-Yuan Center for Applied Mathematics, Tsinghua University, Beijing 100084, China
*
*Corresponding author. Email addresses: huangjt13@mails.tsinghua.edu.cn (J. Huang), hwu@tsinghua.edu.cn (H.Wu), wayong@tsinghua.edu.cn (W.-A. Yong)
Get access

Abstract

In this paper, we propose two initialization techniques for the lattice Boltzmann method. The first one is based on the theory of asymptotic analysis developed in [M. Junk and W.-A. Yong, Asymptotic Anal., 35(2003)]. By selecting consistent macroscopic quantities, this initialization leads to the second-order convergence for both velocity and pressure. Another one is an improvement of the consistent initial conditions proposed in [R. W. Mei, L.-S. Luo, P. Lallemand and D. d’Humières, Comput. Fluids, 35(2006)]. The improvement involves a modification of the collision term and a reconstruction step. Numerical examples confirm the accuracy and efficiency of our techniques.

Keywords

65M0676M20Lattice Boltzmann methodinitial conditionsasymptotic analysisincompressible Navier-Stokes equation
Type
Research Article
Information
Communications in Computational Physics , Volume 18 , Issue 2 , August 2015 , pp. 450 - 468
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]Banda, M.K., Yong, W.-A., and Klar, A.. A stability notion for lattice boltzmann equations. SIAM Journal on Scientific Computing, 27:2098–2111, 2006.Google Scholar
[2]Caiazzo, A.. Analysis of lattice Boltzmann initialization routines. Journal of Statistical Physics, 121:37–48, 2005.CrossRefGoogle Scholar
[3]Chen, H.D., Chen, S.Y., and Matthaeus, W.. Recovery of the Navier-Stokes equations using a lattice-gas Boltzmann method. Physical Review A, 45:5339–5342, 1992.Google Scholar
[4]Chen, S.Y. and Doolen, G.D.. Lattice Boltzmann method for fluid flows. Annual Review of Fluid Mechanics, 30:329–364, 1998.Google Scholar
[5]Guo, Z.L. and Zhao, T.S.. Lattice Boltzmann model for incompressible flows through porous media. Physical Review E, 66:036304, 2002.Google Scholar
[6]Guo, Z.L. and Zhao, T.S.. Explicit finite-difference lattice Boltzmann method for curvilinear coordinates. Physical Review E, 67:066709, 2003.Google Scholar
[7]Guo, Z.L. and Zheng, C.G.. Theory and Applications of Lattice Boltzmann Method. Science Press, Beijing, 2008. in Chinese.Google Scholar
[8]Guo, Z.L., Zheng, C.G., and Shi, B.C.. Discrete lattice effects on the forcing term in the lattice Boltzmann method. Physical Review E, 65:046308, 2002.CrossRefGoogle ScholarPubMed
[9]Guo, Z.L., Zheng, C.G., Shi, B.C., and Zhao, T.S.. Thermal lattice Boltzmann equation for low Mach number flows: Decoupling model. Physical Review E, 75:036704, 2007.Google Scholar
[10]He, X.Y., Chen, S.Y., and Doolen, G.D.. A novel thermal model for the lattice Boltzmann method in incompressible limit. Journal of Computational Physics, 146:282–300, 1998.Google Scholar
[11]He, X.Y. and Luo, L.-S.. Lattice Boltzmann model for the incompressible Navier-Stokes equation. Journal of Statistical Physics, 88:927–944, 1997.Google Scholar
[12]He, X.Y. and Luo, L.-S.. A priori derivation of the lattice Boltzmann equation. Physical Review E, 55:6333–6336, 1997.Google Scholar
[13]He, X.Y. and Luo, L.-S.. Theory of the lattice Boltzmann method: From the Boltzmann equation to the lattice Boltzmann equation. Physical Review E, 56:6811–6817, 1997.Google Scholar
[14]Huang, J.T., Liu, T., Wu, H., and Yong, W.-A.. Numerical analysis and high order initialization of MRT-LBM, 2013. submitted.Google Scholar
[15]Inamuro, T., Yoshino, M., and Ogino, F.. Accuracy of the lattice Boltzmann method for small Knudsen number with finite reynolds number. Phys. Fluids, 9:3535–3542, 1997.CrossRefGoogle Scholar
[16]Junk, M. and Yang, Z.X.. Convergence of lattice Boltzmann methods for Navier-Stokes flows in periodic and bounded domains. Numerische Mathematik, 112:65–87, 2009.CrossRefGoogle Scholar
[17]Junk, M. and Yong, W.-A.. Rigorous Navier-Stokes limit of the lattice Boltzmann equation. Asymptotic Analysis, 35:165–185, 2003.Google Scholar
[18]Junk, M. and Yong, W.-A.. Weighted ℓ 2-stability of the lattice Boltzmann method. SIAM Journal on Numerical Analysis, 47:1651–1665, 2009.CrossRefGoogle Scholar
[19]Koelman, J.M.V.A.. A simple lattice Boltzmann scheme for Navier-Stokes fluid flow. Euro-physics Letters, 15:603–607, 1991.CrossRefGoogle Scholar
[20]Lallemand, P. and Luo, L.-S.. Theory of the lattice Boltzmann method: Dispersion, dissipation, isotropy, Galiean invariance, and stability. Physical Review E, 61:6546–6562, 2000.Google Scholar
[21]Lallemand, P. and Luo, L.-S.. Theory of the lattice Boltzmann method: Acoustic and thermal properties in two and three dimensions. Physical Review E, 68:036706, 2003.CrossRefGoogle ScholarPubMed
[22]Luo, L.-S.. Unified theory of lattice Boltzmann models for nonideal gases. Physical Review Letters, 81:1618–1621, 1998.CrossRefGoogle Scholar
[23]Mei, R.W., Luo, L.-S., Lallemand, P., and d’Humières, D.. Consistent initial conditions for lattice Boltzmann simulations. Computers & Fluids, 35:855–862, 2006.Google Scholar
[24]Pan, C.X., Luo, L.-S., and Miller, C.T.. An evaluation of lattice Boltzmann schemes for porous medium flow simulation. Computers & Fluids, 35:898–909, 2006.Google Scholar
[25]Qian, Y.H.. Simulating thermohydrodynamics with lattice BGK models. Journal of Scientific Computing, 8:231–242, 1993.Google Scholar
[26]Qian, Y.H., d’Humières, D., and Lallemand, P.. Lattice BGK models for Navier-Stokes equation. Europhysics Letters, 17:479–484, 1992.Google Scholar
[27]Qian, Y.H., Succi, S., and Orszag, S.A.. Recent advances in lattice Boltzmann computing. Annual Reviews Of Computational Physics, 3:195–242, 1995.Google Scholar
[28]Shan, X.W. and Chen, H.D.. Lattice Boltzmann model for simulating flows with multiple phases and components. Physical Review E, 47:1815–1819, 1993.Google Scholar
[29]Skordos, P.A.. Initial and boundary conditions for the lattice Boltzmann method. Physical Review E, 48:4823–4842, 1993.Google Scholar
[30]Sone, Y.. Kinetic Theory and Fluid Dynamics. Birkhäuser, 2002.Google Scholar
[31]Van Leemput, P., Rheinläder, M., and Junk, M.. Smooth initialization of lattice Boltzmann schemes. Computers and Mathematics with Applications, 58:867–882, 2009.CrossRefGoogle Scholar
[32]Yong, W.-A. and Luo, L.-S.. Nonexistence of H theorems for the athermal lattice Boltzmann models with polynomial equilibria. Physical Review E, 67:051105, 2003.CrossRefGoogle ScholarPubMed
[33]Yong, W.-A. and Luo, L.-S.. Nonexistence of H theorem for some lattice Boltzmann models. Journal of Statistical Physics, 121:91–103, 2005.Google Scholar
[34]Yong, W.-A. and Luo, L.-S.. Accuracy of the lattice Boltzmann method for the stress. Physical Review E, 86:5701, 2012.Google Scholar
[35]Yu, H.D., Girimaji, S.S., and Luo, L.-S.. Lattice Boltzmann simulations of decaying homogeneous isotropic turbulence. Physical Review E, 71:016708, 2005.Google Scholar