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On the instability of vortex–wave interaction states

Published online by Cambridge University Press:  08 August 2016

Kengo Deguchi  and
Philip Hall
Kengo Deguchi*
Affiliation:
Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK
Philip Hall*
Affiliation:
School of Mathematics, Monash University, Melbourne, VIC 3800, Australia
*
†Email addresses for correspondence: k.deguchi@imperial.ac.uk, phil.hall@monash.edu
†Email addresses for correspondence: k.deguchi@imperial.ac.uk, phil.hall@monash.edu
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Abstract

In recent years it has been established that vortex–wave interaction theory forms an asymptotic framework to describe high Reynolds number coherent structures in shear flows. Comparisons between the asymptotic approach and finite Reynolds number computations of equilibrium states from the full Navier–Stokes equations have suggested that the asymptotic approach is extremely accurate even at quite low Reynolds numbers. However, unlike the situation with an approach based on solving the full Navier–Stokes equations numerically, the vortex–wave interaction approach has not yet been developed to study the instability of the structures it describes. In this work, a comprehensive study of the different instabilities of vortex–wave interaction states is given and it is shown that there are three different time scales on which instabilities can develop. The most dangerous type is a rapidly growing Rayleigh instability of the streak part of the flow. The least dangerous type is a slow mode operating on the diffusion time scale of the roll–streak part of the flow. The third mode of instability, which we will refer to as the edge mode of instability, occurs on a time scale midway between those of the other two modes. The existence of the latter mode explains why some exact coherent structures can act as edge states between the laminar and turbulent attractors. These stability results are compared to results from Navier–Stokes calculations.

JFM classification

Instability: Instability Turbulent Flows: Shear layer turbulence Turbulent Flows: Turbulent Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 802 , 10 September 2016 , pp. 634 - 666
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Copyright
© 2016 Cambridge University Press 

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