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Centrifugal/elliptic instabilities in slowly varying channel flows

Published online by Cambridge University Press:  28 September 2020

Jitesh S. B. Gajjar Open the ORCID record for Jitesh S. B. Gajjar [Opens in a new window]  and
Philip Hall Open the ORCID record for Philip Hall [Opens in a new window]
Jitesh S. B. Gajjar
Affiliation:
Department of Mathematics, University of Manchester, ManchesterM13 9PL, UK
Philip Hall*
Affiliation:
School of Mathematics, Monash University, Clayton, Victoria 3800, Australia
*
†Email address for correspondence: phil.hall@monash.edu
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Abstract

The instability of the flow in a two-dimensional meandering channel of slowly varying depth is considered. The flow is characterised by $\delta$ the typical slope of the channel walls and the modified Reynolds number $R_m$ which is the usual Reynolds number multiplied by $\delta$. The modified Reynolds number is shown to be the appropriate parameter controlling the instability of the flow to streamwise vortices periodic in the spanwise direction. In particular, channels periodic in the streamwise direction are considered and it is found that the most unstable mode can correspond to either a subharmonic or synchronous disturbance. The instability problem at finite $R_m$ is discussed first and then the inviscid and large wavenumber regimes are discussed in detail. The instability is shown to be a hybrid form of centrifugal instability having properties of both Görtler vortices and a parametric resonance usually referred to as an elliptic instability. The limiting case of small wall modulation amplitudes is investigated and the results suggest that at small amplitudes the subharmonic mode is always dominant.

JFM classification

Instability: Absolute/convective instability Instability: Parametric instability Low-Reynolds-number flows: Lubrication theory
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 903 , 25 November 2020 , A27
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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