Flow simulation by the lattice Boltzmann method with multiple-relaxation times

Authors

  • A.M. Zakharov Lomonosov Moscow State University
  • D.S. Senin Lomonosov Moscow State University
  • E.A. Grachev Lomonosov Moscow State University

Keywords:

computational fluid dynamics, lattice Boltzmann method, multiple-relaxation time, two-dimensional flows of Newtonian fluid, turbulent flows, Poiseuille flow, second Stokes problem

Abstract

The lattice Boltzmann method with a multirelaxation collision integral is considered to simulate two-dimensional flows of an incompressible viscous Newtonian fluid. On the basis of this method, a software package that allows one to simulate two-dimensional turbulent flows in a medium with a given shape of obstacles is developed. This package is verified using the following three test problems: a steady Poiseuille flow in a plane channel, a flow in a square cavity with a moving top wall, and the second Stokes problem. The numerical results are in good agreement with the theoretical results and with the results of previous studies. For the second Stokes problem, the dependence of the penetration depth of oscillations on their period and the viscosity coefficient is studied.

Author Biographies

A.M. Zakharov

D.S. Senin

E.A. Grachev

References

  1. Calore E., Schifano S.F., Tripiccione R. A portable OpenCL lattice Boltzmann code for multi- and many-core processor architectures // Procedia Computer Science. 2014. 29. 40-49.
  2. Januszewski M., Kostur M. Sailfish: a flexible multi-GPU implementation of the lattice Boltzmann method // Computer Physics Communications. 2014. 185, N 9. 2350-2368.
  3. Habich J., Feichtinger C., Köstler H., Hager G., Wellein G. Performance engineering for the lattice Boltzmann method on GPGPUs: architectural requirements and performance results // Computers &; Fluids. 2013. 80. 276-282.
  4. Crimi G., Mantovani F., Pivanti M., Schifano S.F., Tripiccione R. Early experience on porting and running a lattice Boltzmann code on the Xeon-phi co-processor // Procedia Computer Science. 2013. 18. 551-560.
  5. Delbosc N., Summers J.L., Khan A.I., Kapur N., Noakes C.J. Optimized implementation of the lattice Boltzmann method on a graphics processing unit towards real-time fluid simulation // Computers &; Mathematics with Applications. 2014. 67, N 2. 462-475.
  6. McNamara G.R., Zanetti G. Use of the Boltzmann equation to simulate lattice-gas automata // Phys. Rev. Lett. 1988. 61, N 20. 2332-2335.
  7. Bhatnagar P.L., Gross E.P., Krook M. A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems // Phys. Rev. 1954. 94, N 3. 511-525.
  8. Qian Y.H., d’Humiéres D., Lalemand P. Lattice BGK models for Navier-Stokes equation // Europhys. Lett. 1992. 17, N 6. 479-484.
  9. Chen H., Chen S., Matthaeus W.H. Recovery of the Navier-Stokes equations using a lattice-gas Boltzmann method // Phys. Rev. A. 1992. 45, N 8. R5339-R5342.
  10. Behrend O., Harris R., Warren P.B. Hydrodynamic behavior of lattice Boltzmann and lattice Bhatnagar-Gross-Krook models // Phys. Rev. E. 1994. 50, N 6. 4586-4595.
  11. d’Humiéres D. Generalized lattice Boltzmann equations // Rarefied Gas Dynamics: Theory and Simulations. Reston: Amer. Inst. Aeronaut. Astronaut., 1994. 450-458.
  12. d’Humiéres D., Ginzburg I., Krafczyk M., Lallemand P., Luo L.-S. Multiple-relaxation-time lattice Boltzmann models in three dimensions // Phil. Trans. R. Soc. Lond. A. 2002. 360. 437-451.
  13. Rettinger C. Fluid flow simulations using the lattice Boltzmann method with multiple relaxation times. Erlangen: Friedrich-Alexander-Universität Erlangen-Nürnberg, 2013.
  14. Razzaghian M., Pourtousi M., Darus N. Simulation of flow in lid driven cavity by MRT and SRT // Proc. Int. Conf. on Mechanical and Robotics Engineering (ICMRE’2012), Phuket, Thailand, May 16-27, 2012. Johannesburg: Planetary Scientific Research Center, 94-97.
  15. Zhen-Hua C., Bao-Chang S., Lin Z. Simulating high Reynolds number flow in two-dimensional lid-driven cavity by multi-relaxation-time lattice Boltzmann method. Chinese Phys. 2006. 15, N. 8. 1855-1863.
  16. Leclaire S., Pellerin N., Reggio M., Trépanier J.-Y. Enhanced equilibrium distribution functions for simulating immiscible multiphase flows with variable density ratios in a class of lattice Boltzmann models // International Journal of Multiphase Flow. 2013. 57. 159-168.
  17. Favier J., Revell A., Pinelli A. A lattice Boltzmann-immersed boundary method to simulate the fluid interaction with moving and slender flexible objects // Journal of Computational Physics. 2014. 261. 145-161.
  18. Anderl D., Bogner S., Rauh C., Rüde U., Delgado A. Free surface lattice Boltzmann with enhanced bubble model // Computers &; Mathematics with Applications. 2014. 67, N 2. 331-339.
  19. Елизарова Т.Г. Квазигазодинамические уравнения и методы расчета вязких течений. Лекции по математическим моделям и численным методам в динамике газа и жидкости. М.: Научный Мир, 2007.
  20. Ландау Л.Д., Лифшиц Е.М. Гидродинамика. М.: Наука, 1980.
  21. Квасников И.А. Теория неравновесных систем. М.: Изд-во МГУ, 2003.
  22. Бикулов Д.А., Сенин Д.С., Демин Д.С., Дмитриев А.В., Грачев Н.Е. Реализация метода решеточных уравнений Больцмана для расчетов нa GPU-кластере // Вычислительные методы и программирование. 2012. 13. 13-19.
  23. Кривовичев Г.В. О расчете течений вязкой жидкости методом решеточных уравнений Больцмана // Компьютерные исследования и моделирование. 2013. 5, № 2. 165-178.
  24. Куперштох А.Л. Трехмерное моделирование двухфазных систем типа жидкость-пар методом решеточных уравнений Больцмана на GPU // Вычислительные методы и программирование. 2012. 13. 130-138.
  25. Кривовичев Г.В. Об устойчивости конечно-разностных решеточных схем Больцмана // Вычислительные методы и программирование. 2013. 14. 1-8.
  26. Кривовичев Г.В. О применении интегро-интерполяционного метода к построению одношаговых решеточных кинетических схем Больцмана // Вычислительные методы и программирование. 2012. 13. 19-27.
  27. Куперштох А.Л. Трехмерное моделирование методом LBE на гибридных GPU-кластерах распада бинарной смеси жидкого диэлектрика с растворенным газом на систему парогазовых каналов // Вычислительные методы и программирование. 2012. 13. 384-390.
  28. Grazyna K. The numerical solution of the transient heat conduction problem using the lattice Boltzmann method // Scientific Research of the Institute of Mathematics and Computer Science. 2006. N 11. 23-30.
  29. Latt J. Choice of units in lattice Boltzmann simulations. [Электронный ресурс]. (http://wiki.palabos.org/).
  30. Wolf-Gladrow D.A. Lattice-gas cellular automata and lattice Boltzmann models: an introduction. Berlin: Springer, 2000.
  31. Mattila K. Implementation techniques for the lattice Boltzmann method. Jyväskylä: Univ. of Jyväskylä, 2010.
  32. Sukop M.C., Thorne D.T. Lattice Boltzmann modeling: an introduction for geoscientists and engineers. New York: Springer, 2006.
  33. Zou Q., He X. On pressure and velocity boundary conditions for the lattice Boltzmann BGK model // Phys. of Fluids. 1997. 9, N 6. 1591-1599.
  34. Narváez A., Harting J. Evaluation of pressure boundary conditions for permeability calculations using the lattice-Boltzmann method // Advances in Applied Mathematics and Mechanics. 2010. 2, N 5. 685-700.
  35. Chen S., Martinez D., Mei R. On boundary conditions in lattice Boltzmann methods // Phys. of Fluids. 1996. 8, N 9. 2527-2536.
  36. Raabe D. Overview of the lattice Boltzmann method for nano- and microscale fluid dynamics in materials science and engineering // Modelling Simul. Mater. Sci. Eng. 2004. 12, N 6. R13-R46.
  37. Boyd J., Buick J., Green S. A second-order accurate lattice Boltzmann non-Newtonian flow model // J. Phys. A.: Mathematical and General. 2006. N 39, N 46. 14241-14247.

Published

2014-11-21

How to Cite

Захаров А.М., Сенин Д.С., Грачев Е.А. Flow Simulation by the Lattice Boltzmann Method With Multiple-Relaxation Times // Numerical methods and programming. 2014. 15. 644-657

Issue

Section

Section 1. Numerical methods and applications