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Numerical and theoretical study of the shock stand-off distance in non-equilibrium flows

Published online by Cambridge University Press:  30 June 2008

N. BELOUAGGADIA ,
H. OLIVIER  and
R. BRUN
N. BELOUAGGADIA
Affiliation:
Université de Provence, Marseille, France
H. OLIVIER
Affiliation:
RWTH Aachen University, Aachen, Germany
R. BRUN
Affiliation:
Université de Provence, Marseille, France
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Abstract

A theoretical model based on a quasi-one-dimensional formulation is developed which allows determination of the shock stand-off distance at the stagnation point of blunt bodies in hypersonic non-equilibrium flows. Despite the simple ideal dissociating gas model implemented in the theoretical approach, it gives insight into the main physics governing the shock stand-off problem. More detailed and precise data are obtained by a numerical simulation where vibrational and chemical relaxation processes as well as their interactions are taken into account. The physical modelling of these processes is based on a kinetic approach and on a generalized Chapman–Enskog method of solving the Boltzmann equation. Explicit formulae for rate constants and vibrational energy consumption are derived and incorporated into the general conservation equations. Good agreement between theoretical, numerical and experimental results is achieved which ensures a reliable and mutual validation of the different methods.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 607 , 25 July 2008 , pp. 167 - 197
Copyright
Copyright © Cambridge University Press 2008

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References

REFERENCES

Belouaggadia, N. & Brun, R. 1998 Chemical rate constants in non-equilibrium flows. J. Therm. Heat Transfer 12, 482–488.CrossRefGoogle Scholar
Belouaggadia, N. & Brun, R. 2006 Statistical model for vibration-chemical reaction interaction: Extension to gas mixtures. J. Therm. Heat Transfer 20, 148–150.CrossRefGoogle Scholar
Brun, R., Villa, M. P. & Meolans, J. G. 1984 Generalised transport terms in vibrationally relaxing flows. In Rarefied Gas Dynamics (ed. Oguchi, M.). pp. 593–599. University of Tokyo Press.Google Scholar
Brun, R. 1988 Transport properties in reactive gas flows. AIAA Paper 88-2655.CrossRefGoogle Scholar
Brun, R. 1991 Transport phenomena in relaxing gas mixtures: Models and applications. In Rarefied Gas Dynamics (ed. Beylich, A. E.). pp. 379–390. VCH, Weinheim.Google Scholar
Freeman, N. C. 1958 Non-equilibrium flow of an ideal dissociating gas. J. Fluid Mech. 4, 407–425.CrossRefGoogle Scholar
Furudate, M., Nonaka, S. & Sawada, K. 1999 Behavior of two-temperature model in intermediate hypersonic regime. J. Therm. Heat Transfer 13, 424–430.CrossRefGoogle Scholar
Garr, L. J. & Marrone, P. V. 1963 Inviscid, non-equilibrium flow behind bow and normal shock waves, part II. Cornell Aeron. Lab. Rep. QM-1626-A-12(II).Google Scholar
Hall, J. G., Eschenroeder, A. Q. & Marrone, P. V. 1962 Blunt-nose inviscid airflows with coupled non-equilibrium processes. J. Aero Space Sci. 29, 1038–1051.CrossRefGoogle Scholar
Hashimoto, T. 2003 Analytical and experimental study of hypersonic nozzle flows in free piston shock tunnel. PhD Thesis AOTD1606, Tohoku University, Japan.Google Scholar
Hayes, W. D. & Probstein, R. F. 1959 Hypersonic Inviscid Flow. Academic.Google Scholar
Hornung, H. G. 1972 Non-equilibrium dissociating nitrogen flow over spheres and circular cylinders. J. Fluid Mech. 53, 149–176.CrossRefGoogle Scholar
Kogan, M. N., Galkin, V. S. & Makashev, N. K. 1979 Generalised Chapman-Enskog method: Derivation of the non-equilibrium gasdynamics equations. In Rarefied Gas Dynamics (ed. Campargue, R.). pp. 693–734. CEA Paris.Google Scholar
Lick, W. 1960 Inviscid flow of a reacting mixture of gases around a blunt body. J. Fluid Mech. 7, 128–144.CrossRefGoogle Scholar
Lin, S. C. & Shen, S. F. 1951 An analytical determination of the flow behind a symmetrical curved shock in a uniform stream. NACA Tech. Note 2506.Google Scholar
Lobb, R. K. 1964 Experimental measurement of shock detachment distance on spheres fired in air at hypervelocities. In The High Temperature Aspects of Hypersonic Flow (ed. Nelson, W. C.). pp. 519–527. Pergamon.Google Scholar
MacCormack, R. W. & Baldwin, B. S. 1975 A numerical method for solving the Navier-Stokes equations with application to shock-boundary interactions. AIAA Paper 75-1.Google Scholar
MacCormack, R. W. & Candler, G. 1989 The solution of the Navier-Stokes equations by Gauss-Seidel line relaxation. Computers Fluids 17, 135–155.CrossRefGoogle Scholar
Mazoue, F., Chikhaoui, A. & Brun, R. 1994 Non-equilibrium species concentration behind a normal shock wave in CO2. In Aerothermochemistry of Spacecraft and Associated Hypersonic Flows. Jouve, Paris.Google Scholar
Millikan, R. C. & White, D. R. 1969 Systematics of vibrational relaxation. J. Chem. Phys. 39, 3209–3213.CrossRefGoogle Scholar
Nonaka, S. 2000 Experimental and numerical study on hypersonic flows in ballistic range. Dept. of Aeronautics and Space Engineering, Tohoku University, Sendai, Japan.Google Scholar
Olivier, H. 2000 A theoretical model for the shock stand-off distance in frozen and equilibrium flow. J. Fluid Mech. 413, 345–353.CrossRefGoogle Scholar
Park, C. 1985 On convergence of computation of chemically reacting flows. AIAA Paper 85-0247.CrossRefGoogle Scholar
Park, C. 1989a A review of reaction rates in high temperature air. AIAA Paper 89-1740.CrossRefGoogle Scholar
Park, C. 1989b Assessment of two temperature kinetic model for ionizing air. J. Therm. Heat Transfer 3, 233–244.CrossRefGoogle Scholar
Park, C. 1990 Review of finite-rate chemistry models for air dissociation and ionisation. In Molecular Physics and Hypersonic Flows (ed. Capitelli, M.). pp. 581–596. NATO-ASI Series, Kluwer.Google Scholar
Pascal, S. & Brun, R. 1993 Tranport properties in non-equilibrium gas mixtures. Phys. Rev. 44, 3251–3267.Google Scholar
Stupochenko, Y. V., Losev, S. A. & Osipov, A. I. 1967 Relaxation in Shock Waves. Springer.CrossRefGoogle Scholar
Treanor, C. E. & Marrone, P. V. 1962 Effect of dissociation on the rate of vibrational relaxation. Phys. Fluids 5, 1022–1027.CrossRefGoogle Scholar
Van Dyke, M. D. 1958 The supersonic blunt body problem - review and extension. J. Aero Space Sci. 25, 485–496.CrossRefGoogle Scholar
Wen, C. Y. & Hornung, H. G. 1995 Non-equilibrium dissociating flow over spheres. J. Fluid Mech. 299, 389–405.CrossRefGoogle Scholar
Zlotnick, M. & Newman, D. J. 1957 Theoretical calculation of the flow on blunt-nosed axisymmetric bodies in a hypersonic stream. Avco Mfg. Corp. Lawrence, MA, Tech. Rep. 2-57-29.Google Scholar