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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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