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Numerical Regularized Moment Method For High Mach Number Flow

Published online by Cambridge University Press:  20 August 2015

Zhenning Cai ,
Ruo Li  and
Yanli Wang
Zhenning Cai*
Affiliation:
School of Mathematical Sciences, Peking University, Beijing 100871, P.R. China
Ruo Li
Affiliation:
CAPT, LMAM & School of Mathematical Sciences, Peking University, Beijing 100871, P.R. China
Yanli Wang
Affiliation:
School of Mathematical Sciences, Peking University, Beijing 100871, P.R. China
*
Corresponding author.Email:caizn@pku.edu.cn
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Abstract

This paper is a continuation of our earlier work [SIAM J. Sci. Comput., 32(2010), pp. 2875-2907] in which a numerical moment method with arbitrary order of moments was presented. However, the computation may break down during the calculation of the structure of a shock wave with Mach number M0 ≥ 3. In this paper, we concentrate on the regularization of the moment systems. First, we apply the Maxwell iteration to the infinite moment system and determine the magnitude of each moment with respect to the Knudsen number. After that, we obtain the approximation of high order moments and close the moment systems by dropping some high-order terms. Linearization is then performed to obtain a very simple regularization term, thus it is very convenient for numerical implementation. To validate the new regularization, the shock structures of low order systems are computed with different shock Mach numbers.

Keywords

65M0865M12Boltzmann-BGK equationMaxwellian iterationregularized moment equations
Type
Research Article
Information
Communications in Computational Physics , Volume 11 , Issue 5 , May 2012 , pp. 1415 - 1438
Copyright
Copyright © Global Science Press Limited 2012

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References

[1]Abramowitz, M. and Stegun, I. A.Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York, 1964.Google Scholar
[2]Alsmeyer, H.Density profiles in argon and nitrogen shock waves measured by the absorption of an electron beam. J. Fluid. Mech., 74(3): 497–513, 1976.Google Scholar
[3]Bird, G. A.Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Oxford: Clarendon Press, 1994.CrossRefGoogle Scholar
[4]Cai, Z. and Li, R.Numerical regularized moment method of arbitrary order for Boltzmann-BGK equation. SIAM J. Sci. Comput., 32(5): 2875–2907, 2010.Google Scholar
[5]Cai, Z., Li, R., and Wang, Y.An efficient NRxx method for Boltzmann-BGK equation. J. Sci. Comput., 2011. DOI: 10.1007/s10915-011-9475-5.Google Scholar
[6]Grad, H.On the kinetic theory of rarefied gases. Comm. Pure Appl. Math., 2(4): 331–407, 1949.Google Scholar
[7]Holway, L. H.New statistical models for kinetic theory: Methods of construction. Phys. Fluids, 9(1): 1658–1673, 1966.CrossRefGoogle Scholar
[8]Ikenberry, E. and Truesdell, C.On the pressures and the flux of energy in a gas according to Maxwell’s kinetic theory I. J. Rat. Mech. Anal., 5(1): 1–54, 1956.Google Scholar
[9]Mieussens, L.Discrete velocity model and implicit scheme for the BGK equation of rarefied gas dynamics. Math. Models Methods Appl. Sci., 10(8): 1121–1149, 2000.Google Scholar
[10]Muüller, I., Reitebuch, D., and Weiss, W.Extended thermodynamics - consistent in order of magnitude. Continuum Mech. Thermodyn., 15(2): 113–146, 2002.CrossRefGoogle Scholar
[11]Muüller, I. and Ruggeri, T.Rational Extended Thermodynamics, 2nd ed., volume 37 of Springer tracts in natural philosophy. Springer-Verlag, New York, 1998.CrossRefGoogle Scholar
[12]Shakhov, E. M.Generalization of the Krook kinetic relaxation equation. Fluid Dyn., 3(5): 95–96, 1968.Google Scholar
[13]Struchtrup, H.Stable transport equations for rarefied gases at high orders in the Knudsen number. Phys. Fluids, 16(11): 3921–3934, 2004.CrossRefGoogle Scholar
[14]Struchtrup, H.Derivation of 13 moment equations for rarefied gas flow to second order accuracy for arbitrary interaction potentials. Multiscale Model. Simul., 3(1): 221–243, 2005.CrossRefGoogle Scholar
[15]Struchtrup, H.Macroscopic Transport Equations for Rarefied Gas Flows: Approximation Methods in Kinetic Theory. Springer, 2005.Google Scholar
[16]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
[17]Torrilhon, M.Two dimensional bulk microflow simulations based on regularized Grad’s 13-moment equations. SIAM Multiscale Model. Simul., 5(3): 695–728, 2006.Google Scholar
[18]Torrilhon, M.Hyperbolic moment equations in kinetic gas theory based on multi-variate Pearson-IV-distributions. Commun. Comput. Phys., 7(4): 639–673, 2010.CrossRefGoogle Scholar
[19]Weiss, W.Continuous shock structure in extended thermodynamics. Phys. Rev. E, 52(6): R5760–R5763, 1995.CrossRefGoogle ScholarPubMed