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Asymptotic Analysis of Lattice Boltzmann Outflow Treatments

Published online by Cambridge University Press:  20 August 2015

Michael Junk  and
Zhaoxia Yang
Michael Junk*
Affiliation:
FB Mathematik und Statistik, Universität Konstanz, Postfach D194, 78457 Konstanz, Germany
Zhaoxia Yang*
Affiliation:
FB Mathematik und Statistik, Universität Konstanz, Postfach D194, 78457 Konstanz, Germany
*
Email:[email protected]
Corresponding author.Email:[email protected]
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Abstract

We show the methodology and advantages of asymptotic analysis when applied to lattice Boltzmann outflow treatments. On the one hand, one can analyze outflow algorithms formulated directly in terms of the lattice Boltzmann variables, like the extrapolation method, to find the induced outflow conditions in terms of the Navier-Stokes variables. On the other hand, one can check the consistency and accuracy of lattice Boltzmann outflow treatments to given hydrodynamic outflow conditions like the Neumann or average pressure condition. As example how the gained insight can be used, we propose an improvement of the well known extrapolation method.

Keywords

65N0676D0582C40Outflow conditionasymptotic analysislattice Boltzmann method
Type
Research Article
Information
Communications in Computational Physics , Volume 9 , Issue 5 , May 2011 , pp. 1117 - 1127
Copyright
Copyright © Global Science Press Limited 2011

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References

[1]Abe, T., Derivation of the lattice Boltzmann method by means of the discrete ordinate method for the Boltzmann equation, J. Comput. Phys., 131 (1997), 241–246.Google Scholar
[2]Benzi, R., Succi, S., and Vergassola, M., The lattice-Boltzmann equation: theory and applications, Phys. Rep., 222 (1992), 145–197.CrossRefGoogle Scholar
[3]Chikatamarla, S. S., Ansumali, S., and Karlin, I. V., Grad’s approximation for missing data in lattice Boltzmann simulations, Europhys. Lett., 74(2) (2006), 215–221.CrossRefGoogle Scholar
[4]Christer, B. and Johansson, V., Well-posedness in the generalized sense for boundary layer suppressing boundary conditions, J. Sci. Comput., 6-4 (1991), 391–414.Google Scholar
[5]Christer, B. and Johansson, V., Well-posedness in the generalized sense for the incompressible Navier-Stokes equation, J. Sci. Comput., 6-2 (1991), 101–127.Google Scholar
[6]Christer, B. and Johansson, V., Boundary conditions for open boundaries for the incompressible Navier-Stokes equation, J. Comput. Phys., 105 (1993), 233–251.Google Scholar
[7]Blazy, S., Nazarov, S. and Specovius-Neugebauer, M., Artificial boundary conditions of pressure type for viscous vlows in a system of pipes, J. Math. Fluid. Mech., 9 (2007), 1–33.Google Scholar
[8]d’Humiéres, D., Generalized lattice-Boltzmann equations. in rarefied gas dynamics: theory and simulations, (eds.shizgal, B. D. and Weaver, D. P.), Progr. Astronaut. Aero., 59 (1992), 450–548.Google Scholar
[9]d’Humiéres, D., Ginzbourg, I., Krafczyk, M., Lallemand, P., and Luo, L.-S., Multiple-relaxation-time lattice Boltzmann models in three dimensions, Phil. Trans. R. Soc. Lond. A., 360 (2002), 437–451.Google Scholar
[10]Frisch, U., d’Humiéres, D., Hasslacher, B., Lallemand, P., Pomeau, Y., and Rivet, J. P., Lattice gas hydrodynamics in two and three dimensions, Complex. Sys., 1 (1987), 649–707.Google Scholar
[11]He, X. and Luo, L.-S., A priori derivation of the lattice Boltzmann equation, Phys. Rev. E., 55 (1997), 6333–6336.CrossRefGoogle Scholar
[12]He, X. and Luo, L.-S., Lattice Boltzmann model for the incompressible Navier-Stokes equation, J. Stat. Phys., 88(3/4) (1997), 927–944.Google Scholar
[13]He, X. and Luo, L.-S., Theory of the lattice Boltzmann method: from the Boltzmann equation to the lattice Boltzmann equation, Phys. Rev. E., 56 (1997), 6811–6817.Google Scholar
[14]Heywood, J., Rannacher, R., and Turek, S., Artificial boundaries and flux and pressure conditions for the incompressible Navier–Stokes equations, Int. J. Numer. Math. Fluids., 22 (1996), 325–352.Google Scholar
[15]Higuera, F. and Jiménez, J., Boltzmann approach to lattice gas simulations, Europhys. Lett., 9 (1989), 663.Google Scholar
[16]Izquierdo, S., Martinez-Lera, P., and Fueyo, N., Analysis of open boundary effects in unsteady lattice Boltzmann simulations, Comput. Math. Appl., doi:10.1016/j.camwa.1009.02.014 (2009).Google Scholar
[17]Junk, M., Klar, A., and Luo, L.-S., Asymptotic analysis of the lattice Boltzmann equation, J. Comput. Phys., 210 (2005), 676–704.CrossRefGoogle Scholar
[18]Junk, M. and Yang, Z., Asymptotic analysis of lattice Boltzmann boundary conditions, J. Stat. Phys., 121 (2005), 3–35.CrossRefGoogle Scholar
[19]Junk, M. and Yang, Z., Outflow boundary conditions for the lattice Boltzmann method, Progr. Comput. Fluid. Dyn., 8 (2008), 38–48.Google Scholar
[20]Lallemand, P. and Luo, L.-S., Theory of the lattice Boltzmann method: dispersion, dissipation, isotropy, Galilean invariance, and stability, Phys. Rev. E., 61 (2000), 6546–6562.Google Scholar
[21]McNamara, G. and Zanetti, G., Use of Boltzmann equation to simulate lattice-gas automata, Phys. Rev. Lett., 61 (1988), 2332.Google Scholar
[22]Qian, Y., Lattice Gas and Lattice Kinetic Theory Applied to the Navier-Stokes Equations, PhD thesis, 1990.Google Scholar
[23]Qian, Y., d’Humiéres, D., and Lallemand, P., Lattice BGK models for Navier-Stokes equation, Europhys. Lett., 17 (1992), 479–484.CrossRefGoogle Scholar
[24]Schäfer, M. and Turek, S., Benchmark computations of laminar flow around a cylinder, in Flow simulation with high-performance computers II, Volume 52 of notes on numericalfluid dynamics, Viehweg, I. E. Hirschel, ed., 1996, 547–566.Google Scholar
[25]Verhaeghe, F., Luo, L.-S., and Blanpain, B., Lattice Boltzmann modeling of microchannel flow in slip flow regime, J. Comput. Phys., 228 (2009), 147–157.Google Scholar
[26]Yang, Z., Lattice Boltzmann outflow treatments: convective conditions and others, to appear.Google Scholar
[27]Yang, Z.., Analysis of Lattice Boltzmann Boundary Conditions, Dissertation, Uni. Konstanz, 2007.Google Scholar
[28]Yu, D., Mei, R., and Shyy, W., Improved treament of the open boundary in the method of lattice Boltzmann equation, Progr. Comput. Fluid. Dyn., 5(1/2) (2005), 1–11.CrossRefGoogle Scholar