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A resonance mechanism in plane Couette flow

Published online by Cambridge University Press:  19 April 2006

L. HÅKan Gustavsson  and
Lennart S. Hultgren
L. HÅKan Gustavsson
Affiliation:
Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA 02139
Lennart S. Hultgren
Affiliation:
Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA 02139 Present address: Department of Mechanics, Mechanical and Aerospace Engineering, Illinois Institute of Technology, Chicago, Illinois 60616.
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Abstract

The temporal evolution of small three-dimensional disturbances on viscous flows between parallel walls is studied. The initial-value problem is formally solved by using Fourier–Laplace transform techniques. The streamwise velocity component is obtained as the solution of a forced problem. As a consequence of the three-dimensionality, a resonant response is possible, leading to algebraic growth for small times. It occurs when the eigenvalues of the Orr–Sommerfeld equation coincide with the eigenvalues of the homogeneous operator for the streamwise velocity component. The resonance has been investigated numerically for plane Couette flow. The phase speed of the resonant waves equals the average mean velocity. The wavenumber combination that leads to the largest amplitude corresponds to structures highly elongated in the streamwise direction. The maximum amplitude, and the time to reach this maximum, scale with the Reynolds number. The aspect ratio of the most rapidly growing wave increases with the Reynolds number, with its spanwise wavelength approaching a constant value of about 3 channel heights.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 98 , Issue 1 , 15 May 1980 , pp. 149 - 159
Copyright
© 1980 Cambridge University Press

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References

Cantwell, B., Coles, D. & Dimotakis, P. 1978 Structure and entrainment in the plane of symmetry of a turbulent spot. J. Fluid Mech. 87, 641.Google Scholar
Collatz, L. 1960 The numerical treatment of differential equations. 3rd edn. Springer.
Davey, A. 1973 On the stability of plane Couette flow to infinitesimal disturbances. J. Fluid Mech. 57, 369.Google Scholar
Davey, A. & Reid, W. H. 1977 On the stability of stratified viscous plane Couette flow. Part 1. Constant buoyancy frequency. J. Fluid Mech. 80, 509.Google Scholar
Elder, J. W. 1960 An experimental investigation of turbulent spots and breakdown to turbulence. J. Fluid Mech. 9, 235.Google Scholar
Gallagher, A. P. 1974 On the behaviour of small disturbances in plane Couette flow. Part 3. The phenomenon of mode-pairing. J. Fluid Mech. 65, 29.Google Scholar
Gallagher, A. P. & Mercer, A. McD. 1962 On the behaviour of small disturbances in plane Couette flow. J. Fluid Mech. 13, 91.Google Scholar
Gallagher, A. P. & Mercer, A. McD. 1964 On the behaviour of small disturbances in plane Couette flow. Part 2. The higher eigenvalues. J. Fluid Mech. 18, 350.Google Scholar
Gaster, M. 1975 A theoretical model of a wave packet in the boundary layer on a flat plate. Proc. Roy. Soc. A 347, 271.Google Scholar
Gustavsson, L. H. 1979 Initial-value problem for boundary layer flows. Phys. Fluids 22, 1602.Google Scholar
Joseph, D. D. 1969 Eigenvalue bounds for the Orr-Sommerfeld equation. Part 2. J. Fluid Mech. 36, 721.Google Scholar
Komoda, H. 1967 Nonlinear development of disturbance in a laminar boundary layer. Phys. Fluids Suppl. 10, S87.Google Scholar
Lin, C. C. 1955 The Theory of Hydrodynamic Stability. Cambridge University Press.
Nishioka, N., Iida, S. & Ichikawa, Y. 1975 An experimental investigation of the stability of plane Poiseuille flow. J. Fluid Mech. 72, 731.Google Scholar
Ross, J. A., Barnes, F. H., Burns, J. G. & Ross, M. A. S. 1970 The flat plate boundary layer. Part 3. Comparison of theory with experiment. J. Fluid Mech. 43, 819.Google Scholar
Wygnanski, I., Haritonidis, J. H. & Kaplan, R. E. 1979 On a Tollmien-Schlichting wave packet produced by a turbulent spot. J. Fluid Mech. 92, 505.Google Scholar