Numerical Simulation of Thermal Lattice Boltzmann Model with a Modified In-Ternal Energy Non-Equilibrium First-Order Extrapolation Boundary Condition

수정된 내부 에너지 비평형 1차 외삽 경계조건을 적용한 열 유동 격자 볼츠만 모델에 관한 수치적 연구

  • Published : 2007.07.01


In this paper, we adapt a modified internal energy non-equilibrium first-order extrapolation thermal boundary condition to the thermal lattice Boltzmann model (TLBM). This model is the double populations approach to simulate hydrodynamic and thermal fields. The bounce-back boundary condition which is a traditional boundary condition of lattice Boltzmann method has only a first order in numerical accuracy at the boundary and numerical instability. A non-equilibrium first-order extrapolation boundary condition has been verified to be of better numerical stability than the bounce-back boundary condition and this boundary condition is proved to be of second-order accuracy for the flat boundaries. The two-dimensional natural convection flow in a square cavity with Pr=0.71 and various Rayleigh numbers are simulated. The results are found to be in good agreement with those of previous studies.



  1. S. Chen and G. D. Doolen, 1998, 'Lattice Boltzmann method for fluid flows', Ann. Rev. Fluid Mech., Vol. 30, pp. 329-364
  2. D. O. Martınez,W. H. Matthaeus, S. Chen and D. C. Montgomery, 1994, 'Comparison of spectral method and lattice Boltzmann simulations of two-dimensional hydrodynamics', Phys. Fluids, Vol. 6, pp. 1285-1298
  3. S. Hou, Q. Zou, S. Chen, G. D. Doolen and A. Cogley, 1995, 'Simulation of cavity flow by the lattice Boltzmann method', J. Comput. Phys., Vol. 118, pp. 329-347
  4. X. He, L. S. Luo and M. Dembo, 1996, 'Some progress in lattice Boltzmann method. I. nonuniform mesh grids', J. Comput. Phys., Vol. 129, pp. 357-363
  5. X. He and G. D. Doolen, 1997, 'Lattice Boltzmann method on curvilinear coordinates system: Flow around a circular cylinder', J. Comput. Phys., Vol. 134, pp. 306-315
  6. G. McNamara and B. Alder, 1993, 'Analysis of the lattice Boltzmann treatment of hydrodynamics', Physica A, Vol. 194, pp. 218-228
  7. F. J. Alexander, S. Chen and J. D. Sterling, 1993, 'Lattice Boltzmann thermohydrodynamics', Phys. Rev. E, Vol. 47, R2249
  8. Y. Chen, H. Ohashi and M. Akiyama, 1994, 'Thermal lattice Bhatnagar–Gross–Krook model without nonlinear deviations in macrodynamic equations', Phys. Rev. E, Vol. 50, pp. 2776-2783
  9. A. Bartoloni, C. Battista and S. Cabasino, 1993, 'LBE simulation of Rayleigh–Benard convection on the APE100 parallel processor', Int. J. Mod. Phys. C, Vol. 4, pp. 993-1006
  10. X. Shan, 1997, 'Simulation of Rayleigh–Benard convection using a lattice Boltzmann method', Phys. Rev. E, Vol. 55, pp.2780-2788
  11. X. He, S. Chen and G. D. Doolen, 1998, 'A novel thermal nodel for the lattice Boltzmann method in incompressible limit', J. Comput. Phys.,Vol. 146, pp. 282-300
  12. R. Cornubert, D. d'Humières and D. Levermore, 2002, 'A Knudsen layer theory for lattice gases', Physica D, Vol. 47, pp.241-259
  13. D. P. Ziegler, 1992, 'Boundary conditions for lattice Boltzmann simulations', J. Stat. Phys., Vol.71, pp.1171-1177
  14. X. He, Q. Zou, L.S. Luo and M. Dembo, 1997, 'Analytic solutions of simple flows and analysis of nonslip boundary conditions for the lattice Boltzmann BGK model', J. Stat. Phys., Vol. 87, No. 1-2, pp.115-136
  15. T. Inamuro, M. Yoshino and F. Ogino, 1995, 'A non-slip boundary condition for lattice Boltzmann simulations', Phys. Fluids, Vol.7, No.12, pp. 2928-2930
  16. Q.S. Zou and X.Y. He, 1997, 'On pressure and velocity boundary conditions for the lattice Boltzmann BGK model', Phys. Fluids, Vol.9, No.6, pp.1591-1598
  17. S. Chen, D. Martinez and R. Mei, 1996, 'On boundary conditions in lattice Boltzmann methods', Phys. Fluids, Vol.8, No.9, pp.2527-2536
  18. Z.L. Guo, C.G. Zheng and B.C. Shi, 2002, 'Nonequilibrium extrapolation method for velocity and pressure boundary conditions in the lattice Boltzmann method', Chinese Phys., Vol.11, pp.366-374
  19. A. D'Orazio, S. Succi and C. Arrighetti, 2003, 'Lattice Boltzmann simulation of open flows with heat transfer', Phys. Fluids, Vol.15, No.9, pp.2778-2781
  20. G.H. Tang, W.Q. Tao and Y.L. He, 2005, 'Thermal boundary condition for the thermal lattice Boltzmann equation', Phys Rev E, Vol.72, 016703
  21. Z.L. Guo, B. Shi and C. Zheng, 2002, 'A couple lattice BGK model for the Boussinesq equations', Int. J. Numer. Meth. Fludis, Vol.39, pp.325-342
  22. P.G. Drazin, and W.H. Reid, 1981, 'Hydrodynamic Stability', Cambridge University Press
  23. J.R. Lee and M.Y. Ha, 2005, 'A numerical study of natural convection in a horizontal enclosure with a conducting body', Int. J. Heat Mass Transfer, Vol. 48, pp.3308-3318
  24. A. D'Orazio, M. Corcione and G.P. Celata, 2004, 'Application to natural convection enclosed flows of a lattice Boltzmann BGK coupled with a general purpose thermal boundary condition', Int. J. Thermal Science, Vol. 43, pp. 575-586
  25. De Vahl Davis, 1983, 'Natural Convection of Air in a Square Cavity A Benchmark Numerical Solution', Int. J. Numer. Meth. Fluids, Vol.3, pp.249-264
  26. W. Brown, 1973, 'Heat flux transition at low Rayleigh number', J. Fluid Mech., Vol.60, pp.539-559