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On Singular Closures for the 5-Moment System in Kinetic Gas Theory

Published online by Cambridge University Press:  22 January 2015

Roman Pascal Schaerer  and
Manuel Torrilhon
Roman Pascal Schaerer*
Affiliation:
Center for Computational Engineering Science, RWTH Aachen University, Schinkelstr. 2, 52062 Aachen, Germany
Manuel Torrilhon
Affiliation:
Center for Computational Engineering Science, RWTH Aachen University, Schinkelstr. 2, 52062 Aachen, Germany
*
*Email addresses: [email protected] (R. P. Schaerer), [email protected] (M. Torrilhon)
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Abstract

Moment equations provide a flexible framework for the approximation of the Boltzmann equation in kinetic gas theory. While moments up to second order are sufficient for the description of equilibrium processes, the inclusion of higher order moments, such as the heat flux vector, extends the validity of the Euler equations to non-equilibrium gas flows in a natural way.

Unfortunately, the classical closure theory proposed by Grad leads to moment equations, which suffer not only from a restricted hyperbolicity region but are also affected by non-physical sub-shocks in the continuous shock-structure problem if the shock velocity exceeds a critical value. Amore recently suggested closure theory based on the maximum entropy principle yields symmetric hyperbolic moment equations. However, if moments higher than second order are included, the computational demand of this closure can be overwhelming. Additionally, it was shown for the 5-moment system that the closing flux becomes singular on a subset of moments including the equilibrium state.

Motivated by recent promising results of closed-form, singular closures based on the maximum entropy approach, we study regularized singular closures that become singular on a subset of moments when the regularizing terms are removed. In order to study some implications of singular closures, we use a recently proposed explicit closure for the 5-moment equations. We show that this closure theory results in a hyperbolic system that can mitigate the problem of sub-shocks independent of the shock wave velocity and handle strongly non-equilibrium gas flows.

Keywords

35Q2035Q35Non-equilibrium gas dynamicshyperbolic moment equationskinetic theorymaximum entropy closures
Type
Research Article
Information
Communications in Computational Physics , Volume 17 , Issue 2 , February 2015 , pp. 371 - 400
Copyright
Copyright © Global Science Press Limited 2015 

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References

[1]Ambrozie, C.-G.. Multivariate truncated moments problems and maximum entropy. Anal. Math. Phys., 3(2):145–161,2013.Google Scholar
[2]Au, J. and Weiss, W.. Shock wave structure calculations with extended thermodynamics of many moments. In Ciancio, V., Donato, A., Oliveri, F., and Rionero, S., editors, Waves and Stability in Continuous Media, Proceedings of the 10th Conference on WASCOM 99, 2001.Google Scholar
[3]Au, J.D.. Lösung nichtlinearer Probleme in der erweiterten Thermodynamik. PhD thesis, Technische Universität Berlin, 2001.Google Scholar
[4]Bhatnagar, P.L., Gross, E.P., and Krook, M.. A model for collision processes in gases. i. small amplitude processes in charged and neutral one-component systems. Phys. Rev., 94(3):511–525, 1954.Google Scholar
[5]Boillat, G. and Ruggeri, T.. Moment equations in the kinetic theory of gases and wave velocities. Continuum Mech. Thermodyn., 9(4):205–212,1997.Google Scholar
[6]Boillat, G. and Ruggeri, T.. On the shock structure problem for hyperbolic system of balance laws and convex entropy. Continuum Mech. Thermodyn., 10(5):285–292,1998.Google Scholar
[7]Curto, R.E. and Fialkow, L.A.. Recursiveness, positivity, and truncated moment problems. Houston J. Math., 17(4):603–635,1991.Google Scholar
[8]Dreyer, W., Junk, M., and Kunik, M.. On the approximation of the Fokker-Planck equation by moment systems. Nonlinearity, 14(4), 2001.Google Scholar
[9]Grad, H.. On the kinetic theory of rarefied gases. Comm. Pure Appl. Math., 2(4):331–407,1949.Google Scholar
[10]Grad, H.. The profile of a steady plane shock wave. Comm. Pure Appl. Math., 5(3):257–300,1952.Google Scholar
[11]Harten, A., Lax, P.D., and van Leer, B.. On upstream differencing and godunov-type schemes for hyperbolic conservation laws. SIAM Rev., 25(1):35–61,1983.Google Scholar
[12]Hauck, C.D., Levermore, C.D., and Tits, A.L.. Convex duality and entropy-based moment closures: Characterizing degenerate densities. SIAM J. Control Opt., 47(4):1977–2015, Jul 2008.CrossRefGoogle Scholar
[13]Junk, M.. Domain of definition of Levermore’s five-moment system. J. Stat. Phys., 93(5’6):1143–1167,1998.Google Scholar
[14]Junk, M.. Maximum entropy for reduced moment problems. Math. Models Methods Appl. Sci., 10(7):1001–1025,2000.Google Scholar
[15]Junk, M.. Maximum entropy moment problems and extended Euler equations. In Abdallah, N.B., Gamba, I.M., Ringhofer, C., Arnold, A., Glassey, R.T., Degond, P., and Levermore, C.D., editors, Transport in Transition Regimes, volume 135 of The IMA Volumes in Mathematics and its Applications, pages 189–198. Springer New York, 2004.Google Scholar
[16]Junk, M. and Unterreiter, A.. Maximum entropy moment systems and Galilean invariance. Continuum Mech. Thermodyn., 14(6):563–576,2002.Google Scholar
[17]LeVeque, R.J.. Numerical Methods for Conservation Laws. Lectures in Mathematics. ETH Zürich. Birkhauser, 1992.Google Scholar
[18]Levermore, C.D.. Moment closure hierarchies for kinetic theories. J. Stat. Phys., 83(5–6):1021–1065,1996.Google Scholar
[19]McDonald, J.G. and Groth, C.P.T.. Towards realizable hyperbolic moment closures for viscous heat-conducting gas flows based on a maximum-entropy distribution. Continuum Mech. Thermodyn., 25(5):573–603, 2013.Google Scholar
[20]McDonald, J.G. and Torrilhon, M.. Affordable robust moment closures for CFD based on the maximum-entropy hierarchy. J. Comput. Phys., 251:500–523,2013.Google Scholar
[21]Müller, I. and Ruggeri, T.. Rational Extended Thermodynamics, volume 37. Springer New York, 1998.Google Scholar
[22]Ruggeri, T.. Breakdown of shock-wave-structure solutions. Phys. Rev. E, 47(6):4135–4140,Jun 1993.CrossRefGoogle ScholarPubMed
[23]Schärer, R.P. and Torrilhon, M.. A new closure mitigating the sub-shock phenomenon in the continuous shock-structure problem. PAMM, 13(1):499–500, Dec 2013.Google Scholar
[24]Struchtrup, H. and Torrilhon, M.. Regularization of Grad’s 13 moment equations: Derivation and linear analysis. Phys. Fluids, 15(9):2668–2680,2003.Google Scholar
[25]Torrilhon, M.. Hyperbolic moment equations in kinetic gas theory based on multi-variate pearson-iv-distributions. Commun. Comput. Phys., 7(4):639–673,2010.Google Scholar
[26]Torrilhon, M. and Struchtrup, H.. Regularized 13-moment-equations: Shock structure calculations and comparison to Burnett models. J. Fluid Mech., 513:171–198,2004.CrossRefGoogle Scholar
[27]Weiss, W.. Continuous shock structure in extended thermodynamics. Phys. Rev. E, 52(6):5760–5763, Dec 1995.Google Scholar