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Asymptotic solutions in non-equilibrium nozzle flow

Published online by Cambridge University Press:  28 March 2006

P. A. Blythe
P. A. Blythe
Affiliation:
National Physical Laboratory, Teddington, Middlesex
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Abstract

Analytical solutions for the quasi-one-dimensional flow of a gas not in thermodynamic equilibrium are presented for two distinct types of rate equation, namely, the linear rate equation which governs vibrational relaxation, and the nonlinear rate equation which governs dissociation. The solutions are derived for the case when, to a first approximation, the rate equation is uncoupled from the remaining flow equations.

There are, in general, three distinct regions in non-equilibrium nozzle flow. First, a so-called near equilibrium region where a perturbation solution is expected to hold. This region is followed by a narrow transition-layer in which there is a rapid departure from equilibrium. Finally, downstream of this layer, the energy in the lagging mode tends asymptotically to some constant ‘frozenout’ value. The solutions applicable to each of these three regions are derived for both rate equations, the boundary conditions for the transition-layer solution and the asymptotic solution are obtained from appropriate matching procedures.

In particular the structures of the asymptotic solutions are discussed. Several approximate methods for determining the asymptotic frozen level of theenergy in the lagging mode have been proposed in the literature. For the present case, when there is only a small amount of energy in the lagging mode, it is shown that none of these approximate methods is mathematically correct.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 20 , Issue 2 , October 1964 , pp. 243 - 272
Copyright
© 1964 Cambridge University Press

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References

Bloom, M. H. & Ting, L. 1960 On near-equlibrium and near-frozen behaviour of one-dimensional flow. AEDC-TN-60–156, PIBAL-R-525.
Blythe, P. A. 1963 Non-equilibrium flow through a nozzle. J. Fluid Mech. 17, 126.Google Scholar
Bray, K. N. C. 1959 Atomic recombination in a hypersonic wind tunnel nozzle. J. Fluid Mech. 6, 1.Google Scholar
Bray, K. N. C. & Wilson, J. A. 1961 A preliminary study of ionic recombination of argon in wind tunnel nozzles, Part II. A.R.C. 23 290-Hyp. 145a.
Eschenroeder, A. Q. 1962 Ionization non-equilibrium in expanding flows. A.R.S. Journal, 32, 196.Google Scholar
Freeman, N. C. 1958 Non-equilibrium flow of an ideal dissociating gas. J. Fluid Mech. 4, 407.Google Scholar
Freeman, N. C. 1959 Non-equilibrium theory of an ideal dissociating gas through a conical nozzle. A.R.C. C.P. 438.Google Scholar
Hall, J. G. & Russo, A. L. 1959 Studies of chemical non-equilibrium in hypersonic nozzle flows. Cornell Aero. Lab. Rep. AD-1118-A-6, AFOSR TN 59–1090.Google Scholar
Heims, S. P. 1958 Effects of oxygen recombination on one-dimensional flow at high Mach numbers. N.A.C.A. TN 4144.Google Scholar
Herzfeld, K. F. & Litovitz, T. A. 1959 Absorption and Dispersion of Ultrasonic Waves. New York: Academic Press.
Jeffreys, H. & Jeffreys, B. S. 1946 Methods of Mathematical Physics. Cambridge University Press.
Johannesen, N. H. 1961 Analysis of vibrational relaxation regions by means of the Rayleigh line method. J. Fluid Mech. 10, 25.Google Scholar
Lighthill, M. J. 1957 Dynamics of a dissociating gas. Part I. Equilibrium flow. J. Fluid Mech. 2, 1.Google Scholar
Rosner, D. E. 1962 Estimation of electrical conductivity at rocket nozzle exit sections. A.R.S. Journal, 32, 1602.Google Scholar
Shapiro, A. H. 1953 The Dynamics and Thermodynamics of Compressible Fluid Flow. Ronald Press.
Shuler, K. E. 1959 Relaxation processes in multistate systems. Phys. Fluids, 2, 442.Google Scholar
Smith, F. T. 1958 On the analysis of recombination reactions in an expanding gas stream. Seventh Symposium on Combustion. London: Butterworth's Scientific Publications.
Spence, D. A. 1961 Unsteady shock propagation in a relaxing gas. Proc. Roy. Soc. A, 264, 221.Google Scholar
Stollery, J. L. & Park, C. 1963 Computer solutions to the problem of vibrational relaxation in hypersonic nozzle flows. Imperial College Report, no. 115.Google Scholar
Stollery, J. L. & Smith, J. E. 1962 A note on the variation of vibrational temperature along a nozzle. J. Fluid Mech. 13, 225.Google Scholar
Tsuchiya, S. 1962 Relaxation of chemical equilibrium in gases flowing through de Laval nozzle. Aero. Res. Inst. Tokyo Report, no. 371.Google Scholar
Watson, G. N. 1952 Theory of Bessel functions. Cambridge University Press.
Wood, G. P. 1956 Calculations of the rate of thermal dissociation of air behind normal shock waves at Mach numbers of 10, 12, and 14. N.A.C.A. TN 3634.Google Scholar