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The inviscid nonlinear instability of parallel shear flows

Published online by Cambridge University Press:  29 March 2006

J. L. Robinson
J. L. Robinson
Affiliation:
Physics and Engineering Laboratory, D.S.I.R., Lower Hutt, New Zealand
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Abstract

In this paper we assume the existence of a nonlinear boundary layer centred on the critical point, and explore its effect on the development of unstable parallel shear flows. A velocity matching condition derived in a qualitative discussion suggests a growth of harmonics which differs from that predicted by previous theories; however, the prediction is in excellent agreement with experimental data. A hyperbolic-tangent velocity profile, subjected to perturbations with wavenumbers and frequencies close to marginal values, is then chosen as a mathematical model of the nonlinear development, both temporal and spatial instability growth being considered.

A singularity in the analysis which has been treated in previous theories by the introduction of viscosity is dealt with in the present work by the introduction of a growth boundary layer. The asymptotics are non-uniform and the time-dependent solution does not resemble the steady viscous solutions, even as the growth rate tends to zero. The theory suggests that the instability will develop as a series of temporally growing spiral vortices, a description differing from that of a cat's-eye pattern predicted by existing theories, but in accord with experimental and field observations.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 63 , Issue 4 , 15 May 1974 , pp. 723 - 752
Copyright
© 1974 Cambridge University Press

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