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Nonlinear evolution of subsonic and supersonic disturbances on a compressible free shear layer

Published online by Cambridge University Press:  26 April 2006

S. J. Leib
S. J. Leib
Affiliation:
Sverdrup Technology, Inc., Lewis Research Center Group, Cleveland, OH 44135, USA
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Abstract

We consider the effects of a nonlinear–non-equilibrium–viscous critical layer on the spatial evolution of subsonic and supersonic instability modes on a compressible free shear layer. It is shown that the instability wave amplitude is governed by an integro-differential equation with cubic-type nonlinearity. Numerical and asymptotic solutions to this equation show that the amplitude either ends in a singularity at a finite downstream distance or reaches an equilibrium value, depending on the Prandtl number, viscosity law, viscous parameter and a real parameter which is determined by the linear in viscid stability theory. A necessary condition for the existence of the equilibrium solution is derived, and whether or not this condition is met is determined numerically for a wide range of physical parameters including both subsonic and supersonic disturbances. It is found that no equilibrium solution exists for the subsonic modes unless the temperature ratio of the low-to high-speed streams exceeds a critical value, while equilibrium solutions for the most rapidly growing supersonic mode exist over most of the parameter range examined.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 224 , March 1991 , pp. 551 - 578
Copyright
© 1991 Cambridge University Press

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References

Dunn, D. W. & Lin, C. C., 1955 On the stability of the laminar boundary in a compressible fluid. J. Aero. Sci. 22, 455–477.Google Scholar
Goldstein, M. E. & Hultgren, L. S., 1988 Nonlinear spatial evolution of an externally excited instability wave in a free shear layer. J. Fluid Mech. 197, 295–330.Google Scholar
Goldstein, M. E. & Leib, S. J., 1988 Nonlinear roll-up of externally excited free shear layers. J. Fluid Mech. 191, 481–515.Google Scholar
Goldstein, M. E. & Leib, S. J., 1989 Nonlinear evolution of oblique waves on compressible shear layers. J. Fluid Mech. 207, 73–96 (referred to herein as I).Google Scholar
Gropengeisser, H.: 1969 Study of the stability of boundary layers and compressible fluids. NASA Translations TT-F-2.Google Scholar
Hickernell, F. J.: 1984 Time-dependent critical layers in shear flows on the beta-plane. J. Fluid Mech. 142, 431–449.Google Scholar
Jackson, T. L. & Grosch, C. E., 1989a Inviscid spatial stability of a compressible mixing layer. J. Fluid Mech. 208, 609–637.Google Scholar
Jackson, T. L. & Grosch, C. E., 1989b Inviscid spatial stability of a compressible mixing layer. Part. III. Effect of thermodynamics. NASA CR-181855.Google Scholar
Jackson, T. L. & Grosch, C. E., 1990 Absolute/convective instabilities and the convective Mach number in a compressible mixing layer. Phys. Fluids A 2, 949–964.Google Scholar
Lees, L. & Lin, C. C., 1946 Investigation of the stability of the laminar boundary layer in a compressible fluid. NAGA TN-1115.Google Scholar
Lock, R. C.: 1951 The velocity distribution in the laminar boundary layer between parallel streams. Q. J. Mech Appl. Maths 4, 42–63.Google Scholar
Pavithran, S. & Redekopp, L. G., 1989 The absolute—convective transition in subsonic mixing layers. Phys. Fluids A 1, 1736–1739.Google Scholar
Redekopp, L. G.: 1977 On the theory of solitary Rossby waves. J. Fluid Mech. 82, 725–745.Google Scholar
Schlichting, H.: 1979 Boundary Layer Theory. McGraw-Hill.