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Convergence Study of Moment Approximations for Boundary Value Problems of the Boltzmann-BGK Equation

Part of: Equilibrium statistical mechanics Rarefied gas flows, Boltzmann equation Hyperbolic equations and systems

Published online by Cambridge University Press:  14 September 2015

Manuel Torrilhon
Manuel Torrilhon*
Affiliation:
Center for Computational Engineering Science, RWTH Aachen University, Germany
*
*Corresponding author. Email address: [email protected] (M. Torrilhon)
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Abstract

The accuracy of moment equations as approximations of kinetic gas theory is studied for four different boundary value problems. The kinetic setting is given by the BGK equation linearized around a globally constant Maxwellian using one space dimension and a three-dimensional velocity space. The boundary value problems include Couette and Poiseuille flow as well as heat conduction between walls and heat conduction based on a locally varying heating source. The polynomial expansion of the distribution function allows for different moment theories of which two popular families are investigated in detail. Furthermore, optimal approximations for a given number of variables are studied empirically. The paper focuses on approximations with relatively low number of variables which allows to draw conclusions in particular about specific moment theories like the regularized 13-moment equations.

Keywords

Kinetic gas theoryHermite expansionchannel flow

MSC classification

Secondary: 82B40: Kinetic theory of gases 76P05: Rarefied gas flows, Boltzmann equation 35L65: Conservation laws
Type
Research Article
Information
Communications in Computational Physics , Volume 18 , Issue 3 , September 2015 , pp. 529 - 557
Copyright
Copyright © Global-Science Press 2015 

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References

[1]Au, J.D., Torrilhon, M., and Weiss, W., The shock tube study in extended thermodynamics, Phys. Fluids 13(7), (2001), 2423–2432.Google Scholar
[2]Cai, Z. and Li, R., Numerical regularized moment method of arbitrary order for Boltzmann-BGK equation, SIAM J. Sci. Comput. 32(5), (2010), 2875–2907.Google Scholar
[3]Chapman, S. and Cowling, T.G., The Mathematical Theory of Non-Uniform Gases, Cambridge University Press (1970).Google Scholar
[4]Cercignani, C., Theory and Application of the Boltzmann Equation, Scottish Academic Press, Edinburgh (1975).Google Scholar
[5]Duclous, R., Druboca, B., and Frank, M., A deterministic partial differential equation model for dose calculation in electron radiotherapy, Phys. Medicine Biol. 55, (2010), 3843–3857.CrossRefGoogle ScholarPubMed
[6]Eu, B.-C., A modified moment method and irreversible thermodynamics, J. Chemical Phys. 73(6), (1980), 2958–2969.Google Scholar
[7]Yuan, C. and Fox, R.O., Conditional quadrature method of moments for kinetic equations, J. Comput. Phys. 230(22), (2011), 8216–8246.Google Scholar
[8]Grad, H., On the kinetic theory of rarefied gases, Comm. Pure Appl. Math. 2, (1949), 331–407.CrossRefGoogle Scholar
[9]Grad, H., Principles of the Kinetic Theory of Gases, in Handbuch der Physik XII: Thermodynamik der Gase, Flügge, S. (Ed.), Springer, Berlin (1958).CrossRefGoogle Scholar
[10]Groth, C.P.T. and McDonald, J.G., Towards realizable hyperbolic moment closures for viscous heat-conducting gas flows based on a maximum-entropy distribution, Cont. Mech. Thermodyn. 21, (2009), 476–493.Google Scholar
[11]Gu, X. and Emerson, D.R., A computational strategy for the regularized 13 moment equations with enhanced wall-boundary conditions, J. Comput. Phys. 225, (2007), 263–283.Google Scholar
[12]Gu, X.J. and Emerson, D.R., A high-order moment approach for capturing non-equilibrium phenomena in the transition regime, J. Fluid Mech. 636, (2009), 177–216.Google Scholar
[13]Gu, X.J., Emerson, D.R., and Tang, G.H., Analysis of the slip coefficient and defect velocity in the Knudsen layer of a rarefied gas using the linearized moment equations, Phys. Rev. E 81, (2009), 016313.Google Scholar
[14]Jou, D., Casas-Vazquez, J., and Lebon, G., Extended Irreversible Thermodynamics (2nd ed.), Springer, Berlin, (1996).Google Scholar
[15]Levermore, C.D., Moment closure hierarchies for kinetic theories, J. Stat. Phys. 83(5–6), (1996), 1021–1065.Google Scholar
[16]Levermore, C.D. and Morokoff, W.J., The Gaussian moment closure for gas dynamics, SIAM J. Appl. Math. 59(1), (1998), 72–96.Google Scholar
[17]McDonald, J.G. and Groth, C.P.T., Extended fluid-dynamic model for micron-scale flows based on Gaussian moment closure, 46th AIAA Aerospace Science Meeting, (2003), 691.Google Scholar
[18]McDonald, J.G. and Torrilhon, M., Affordable robust moment closures for CFD based on the maximum-entropy hierarchy, J. Comput. Phys. 251, (2013), 500–523.CrossRefGoogle Scholar
[19]Müller, I. and Ruggeri, T., Rational Extended Thermodynamics (2nd ed.), in Springer Tracts in Natural Philosophy (Vol. 37), Springer, New York (1998).CrossRefGoogle Scholar
[20]Myong, R.-S., A computational method for Eu’s generalized hydrodynamic equations of rarefied and microscale gasdynamics, J. Comput. Phys. 168(1), (2001), 47–72.Google Scholar
[21]Öttinger, H.C., Thermodynamically admissible 13 moment equations from the Boltzmann equation, Phys. Rev. Lett. 104, (2010), 120601.Google Scholar
[22]Öttinger, H.C., Reply to the comment on ‘Thermodynamically admissible 13 moment equations from the Boltzmann equation’, Phys. Rev. Lett. 105, (2010), 128902.Google Scholar
[23]Struchtrup, H., Macroscopic Transport Equations for Rarefied Gas Flows—Approximation Methods in Kinetic Theory, Interaction of Mechanics and Mathematics Series, Springer, Heidelberg (2005).Google Scholar
[24]Struchtrup, H., Linear kinetic heat transfer: Moment equations, boundary conditions, and Knudsen layers, Physica A 387, (2008), 1750–1766.CrossRefGoogle Scholar
[25]Struchtrup, H. and Torrilhon, M., Regularization of Grad’s 13 moment equations: Derivation and linear analysis, Phys. Fluids 15(9), (2003), 2668–2680.Google Scholar
[26]Struchtrup, H. and Torrilhon, M., Comment on ‘Thermodynamically admissible 13 moment equations from the Boltzmann equation’, Phys. Rev. Lett. 105, (2010), 128901.Google Scholar
[27]Struchtrup, H., Stable transport equations for rarefied gases at high orders in the Knudsen number, Phys. Fluids 16(11), (2004), 3921–3934.Google Scholar
[28]Struchtrup, H. and Torrilhon, M., Higher-order effects in rarefied channel flows, Phys. Rev. E 78, (2008), 046301, Erratum: Phys. Rev. E 78, (2008), 069903.Google Scholar
[29]Taheri, P., Torrilhon, M., and Struchtrup, H., Couette and Poiseuille flows in microchannels: Analytical solutions for regularized 13-moment equations, Phys. Fluids 21, (2009), 017102.Google Scholar
[30]Torrilhon, M., Slow rarefied flow past a sphere: Analytical solutions based on moment equations, Phys. Fluids 22, (2010), 072001.Google Scholar
[31]Torrilhon, M., Au, J.D., and Struchtrup, H., Explicit fluxes and productions for large systems of the moment method based on extended thermodynamics, Cont. Mech. Thermodyn. 15, (2003), 97–111.Google Scholar
[32]Torrilhon, M. and Struchtrup, H., Boundary conditions for regularized 13-moment-equations for micro-channel-flows, J. Comput. Phys. 227, (2008), 1982–2011.Google Scholar
[33]Waldmann, L., Transporterscheinungen in Gasen von mittlerem Druck, in Handbuch der Physik XII: Thermodynamik der Gase, Flügge, S. (Ed.), Springer, Berlin (1958).CrossRefGoogle Scholar
[34]Young, J.B., Calculation of Knudsen layers and jump conditions using the linearised G13 and R13 moment methods, Int. J. Heat Mass Transfer 54, (2011), 2902–2912.Google Scholar