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Fractal sets of neutral curves for stably stratified plane Couette flow

Published online by Cambridge University Press:  13 June 2019

Jonathan J. Healey Open the ORCID record for Jonathan J. Healey [Opens in a new window]
Jonathan J. Healey*
Affiliation:
Department of Mathematics, Keele University, Keele ST5 5BG, UK
*
†Email address for correspondence: [email protected]
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Abstract

The linear stability of plane Couette flow is investigated when the plates are horizontal, and the fluid is stably stratified with a cubic basic density profile. The disturbances are treated as inviscid and diffusion of the density field is neglected. Previous studies have shown that this density profile can develop multiple neutral curves, despite the stable stratification, and the fact that plane Couette flow of homogeneous fluid is stable. It is shown that when the neutral curves are plotted with wave angle on one axis, and location of the density inflexion point on the other axis, they produce a self-similar fractal pattern. The repetition on smaller and smaller scales occurs in the limit when the waves are highly oblique, i.e. longitudinal vortices almost aligned with the flow; the corresponding limit for two-dimensional waves is that of strong buoyancy/weak flow. The fractal set of neutral curves also represents a fractal of bifurcation points at which nonlinear solutions can be continued from the trivial state, and these may be helpful for understanding turbulent states. This may be the first example of a fractal generated by a linear ordinary differential equation.

JFM classification

Geophysical and Geological Flows: Internal waves Geophysical and Geological Flows: Stratified flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 872 , 10 August 2019 , pp. 697 - 728
Copyright
© 2019 Cambridge University Press 

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