Hostname: page-component-6bb9c88b65-6vlrh Total loading time: 0 Render date: 2025-07-23T05:05:53.588Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear evolution of interacting oblique waves on two-dimensional shear layers

Published online by Cambridge University Press:  26 April 2006

M. E. Goldstein  and
S.-W. Choi
M. E. Goldstein
Affiliation:
National Aeronautics and Space Administration, Lewis Research Center, Cleveland, OH 44135, USA
S.-W. Choi
Affiliation:
National Aeronautics and Space Administration, Lewis Research Center, Cleveland, OH 44135, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the effects of critical-layer nonlinearity on spatially growing oblique instability waves on nominally two-dimensional shear layers between parallel streams. The analysis shows that three-dimensional effects cause nonlinearity to occur at much smaller amplitudes than it does in two-dimensional flows. The nonlinear instability wave amplitude is determined by an integro-differential equation with cubic-type nonlinearity. The numerical solutions to this equation are worked out and discussed in some detail. We show that they always end in a singularity at a finite downstream distance.

Information

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 207 , October 1989 , pp. 97 - 120
Copyright
© 1989 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Benney, D. J. & Maslowe, S. A. 1975 The evolution in space and time of nonlinear waves in parallel shear flows. Stud. Appl. Maths. 54, 181–205.Google Scholar
Browand, F. K. & Ho, C. M. 1983 The mixing layer: an example of quasi two-dimensional turbulence. J. Méc. Theor. Appl. 2, 99–120.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.Google Scholar
Gropengeisser, H. 1969 Study of the stability of boundary layers and compressible fluids. Deutsche Luft- und Raunfahnt, Rep. DLR-FB-69–25. (NASA translations TTF-12, 786.)
Hickernell, F. J. 1984 Time-dependent critical layers in shear flows on the beta-plane. J. Fluid Mech. 142, 431–449.Google Scholar
Ho, C. M. & Huerre, P. 1984 Perturbed free shear layers. Ann. Rev. Fluid Mech. 16, 365–424.Google Scholar
Huerre, P. 1977 Nonlinear instability of free shear layers. In Laminar-Turbulent Transition, AGARD CP, pp. 224–229.
Huerre, P. 1980 The nonlinear stability of a free shear layer in the viscous critical layer regime. Phil. Trans. R. Soc. Land. A 293, 643–672.Google Scholar
Huerre, P. 1987 On the Landau constant in mixing layers. Proc. R. Soc. Lond. A 409, 369–381.Google Scholar
Huerre, P. & Scott, J. F. 1980 Effects of critical layer structure on the nonlinear evolution of waves in free shear layers. Proc. R. Soc. Lond. A 371, 500–524.Google Scholar
Jackson, T. L. & Grosch, C. E. 1988 Spatial stability of a compressible mixing layer. NASA CR-181671.
Maslowe, S. 1986 Critical layers in shear flows. Ann. Rev. Fluid. Mech. 18, 405–432.Google Scholar
Miura, A. & Sato, T. 1978 Theory of vortex nutation and amplitude oscillation in an inviscid shear instability. J. Fluid Mech. 86, 33–47.Google Scholar
Robinson, J. L. 1974 The inviscid nonlinear instability of parallel shear flows. J. Fluid Mech. 63, 723–752.Google Scholar
Stewartson, K. 1981 Marginally stable inviscid flows with critical layer. IMA J. Appl. Maths 27, 133–175.Google Scholar