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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, Zhukowsky-3, 140160, Russiaand INTECO srl, Via Mola Vecchia 2A, 03100 Frosinone, Italy
V. N. Trigub
Affiliation:
TsAGI, Zhukowsky-3, 140160, Russiaand INTECO srl, Via Mola Vecchia 2A, 03100 Frosinone, Italy
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Abstract

In this paper the linear stability of the hypersonic boundary layer is considered in the local-parallel approximation. It is assumed that the Prandtl-number ½ < σ < 1 and the viscosity-temperature law is a power function: μ/μ∞ = (T/T∞)ω. The asymptotic theory in the limit M∞ → ∞ is developed.

Smith & Brown found for the Blasius base flow and Balsa & Goldstein for the mixing layer that, in this limit, the disturbances of the vorticity mode are located in the thin region between the boundary layer and the external flow. The gas model with σ = 1, ω = 1 was exploited in these studies. Here it is demonstrated that the vorticity mode also exists for gas with ½ < σ 1, ω < 1, but its structure and characteristics are considerably different. The nomenclature is discussed, i.e. what an acoustic mode and a vorticity mode are. The numerical solution of the inviscid instability problem for the vorticity mode is obtained for helium and compared with the solution of the complete Rayleigh equation at finite Mach numbers.

The limit M∞ → ∞ in the local-parallel approximation for the Blasius base flow is considered so as to understand the viscous structure of the vorticity mode. The viscous stability problem for the vorticity mode is formulated under these assumptions. The problem contains only a single similarity parameter which is a function of the Mach and Reynolds numbers, the temperature factor and wave inclination angle. This problem is numerically solved for helium. The universal upper branch of the neutral curve is obtained as a result. The asymptotic results are compared with the numerical solutions of the complete problem.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 246 , January 1993 , pp. 361 - 380
Copyright
© 1993 Cambridge University Press

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References

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. 247, 369– 416.Google Scholar
Brown, S. N., Khorrami, A. F., Neish, A. & Smith, F. T. 1991 On hypersonic boundary-layer interaction and transition. Phil. Trans. R. Soc. Lond. A 335, 139– 152.Google Scholar
Bush, W. B. 1966 Hypersonic strong-interaction similarity solutions for flow past a flat plate. J. Fluid Mech. 25, 51– 64.Google Scholar
Bush, W. B. & Cross, A. K. 1967 Hypersonic weak interaction similarity solutions for flow past a flat plate. J. Fluid Mech. 29, 349– 359.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
Freeman, N. C. & Lam, S. H. 1959 On the Mach number independence principle for a hypersonic boundary layer. Princeton University Rep. 471.Google Scholar
Gapanov, S. A. & Maslov, A. A. 1981 Development of the Perturbations in Compressible Flows. Novosibirsk: Nauka.
Goldstein, M. E. & Wundrow, D. W. 1990 Spatial evolution of nonlinear acoustic mode-instabilities on hypersonic boundary layers. J. Fluid Mech. 219, 585– 607.Google Scholar
Grubin, S. E., Simakin, I. N. & Trigub, V. N. 1992 Investigation of stability characteristics for the hypersonic boundary layer on a flat plate. Zh. Prikl. Mek. Tek. Fiz (to appear).Google Scholar
Grubin, S. E. & Trigub, V. N. 1993 The long-wave limit in the asymptotic theory of a hypersonic boundary layer. J. Fluid Mech. 246, 381– 395.Google Scholar
Hayes, W. D. & Probstein, R. F. 1959 Hypersonic Flow Theory, Academic.
Lees, L. 1947 The stability of the laminar boundary layer in a compressible fluid. NACA Tech. Rep. 876.Google Scholar
Lees, L. & Lin, C. C. 1946 Investigation of the stability of the laminar boundary layer in a compressive fluid. NACA Tech. Note 1115.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
Mack, L. M. 1987 Review of linear compressible stability theory. In Stability of Time Dependent and Spatially Varying Flows (ed. D. L. Dwoyer & M. Y. Hussaini). Springer.
Seddougui, S. O., Bowles, R. I. & Smith, F. T. 1991 Surface-cooling effects on compressible boundary layer instability. Eur. J. Mech. B: Fluids 10, 117– 145.Google Scholar
Smith, F. T. 1989 On the first-mode instability in subsonic, supersonic or hypersonic boundary layers. J. Fluid Mech. 198, 127– 153.Google Scholar
Smith, F. T. & Brown, S. N. 1990 The inviscid instability of a Blasius boundary layer at large values of the Mach number. J. Fluid Mech. 219, 499– 518.Google Scholar
Zhigulev, V. N. & Tumin, A. M. 1987 Origin of the Turbulence. Novosibirsk: Nauka.