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The long-wave limit in the asymptotic theory of hypersonic boundary-layer stability

Published online by Cambridge University Press:  26 April 2006

S. E. Grubin  and
V. N. Trigub
S. E. Grubin
Affiliation:
TsAGI, Zhukovsky-3, 140160, Russiaand INTECO srl. Via Mola Vecchia 2A, 03100 Frosinone, Italy
V. N. Trigub
Affiliation:
TsAGI, Zhukovsky-3, 140160, Russiaand INTECO srl. Via Mola Vecchia 2A, 03100 Frosinone, Italy
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Abstract

This paper discusses the long-wave limit of the asymptotic theory of hypersonic boundary-layer stability for a gas with the Prandtl number ½ < σ < 1 and with the viscosity–temperature law being a power function. The investigation is confined to the local-parallel approximation.

In the long-wave limit the vorticity mode starts to interact with the acoustic disturbances in the boundary-layer region. The general solution of the linear problem in the boundary-layer inner region is analysed numerically and analytically. This solution is matched with the long-wave vorticity-mode solution near the transition layer. As a result, the inviscid instability problem for a hypersonic boundary layer is formulated. The analytical solution of this problem is found and analysed. The different limits of the solution are considered and the universal forms of the dependence are obtained. A similarity parameter is found which is a function of the Prandtl number and the power in the viscosity–temperature law. A significant change of the solution behaviour is noticed when this parameter passes a critical value. The asymptotic structure of the amplification rate, as a function of the wavenumber, is described and discussed.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 246 , January 1993 , pp. 381 - 395
Copyright
© 1993 Cambridge University Press

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References

Abramowitz, M. & Stegun, I. A. 1970 Handbook of Mathematical Functions. Dover.
Balsa, T. F. & Goldstein, M. 1990 On the instabilities of supersonic mixing layers: a high Mach number asymptotic theory. J. Fluid Mech. 216, 585– 611.Google Scholar
Blackaby, N. D., Cowley, S. J. & Hall, P. 1990 On the instability of hypersonic flow past a flat plate. ICASE Rep. 90– 40.Google Scholar
Blackaby, N. D., Cowley, S. J. & Hall, P. 1993 On the instability of hypersonic flow past a flat plate. J. Fluid Mech. (in press).Google Scholar
Cowley, S. J. & Hall, P. 1990 On the instability of hypersonic flow past a wedge. J. Fluid Mech. 214, 17– 42.Google Scholar
Grubin S. E. & Trigub, V. N. 1993 The asymptotic theory of a hypersonic boundary layer stability, J. Fluid Mech. 246, 361– 380 (referred to herein as Part 1).Google Scholar
Lin, C. C. 1955 The Theory of Hydrodynamic Stability. Cambridge University Press.
Mack, L. M. 1969 Boundary layer stability theory. J.P.L. Tech. Rep. 900– 277, Part 2.Google Scholar
Nayfeh, A. H. 1973 Perturbation Methods. Wiley.
Smith, F. T. & Brown, S. X. 1990 The inviscid instability of a Blasius boundary layer at large values of the Mach number. J. Fluid Mech. 219, 499– 518.Google Scholar