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Transient growth in linearly stable Taylor–Couette flows

Published online by Cambridge University Press:  21 February 2014

Simon Maretzke ,
Björn Hof  and
Marc Avila
Simon Maretzke*
Affiliation:
Faculty of Physics, University of Göttingen, 37073 Göttingen, Germany
Björn Hof
Affiliation:
Max Planck Institute for Dynamics and Self-Organization (MPIDS), 37077 Göttingen, Germany
Marc Avila
Affiliation:
Institute of Fluid Mechanics, Friedrich-Alexander-Universität Erlangen-Nürnberg, 91058 Erlangen, Germany
*
†Email address for correspondence: simon.maretzke@googlemail.com
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Abstract

Non-normal transient growth of disturbances is considered as an essential prerequisite for subcritical transition in shear flows, i.e. transition to turbulence despite linear stability of the laminar flow. In this work we present numerical and analytical computations of linear transient growth covering all linearly stable regimes of Taylor–Couette flow. Our numerical experiments reveal comparable energy amplifications in the different regimes. For high shear Reynolds numbers $\mathit{Re}$, the optimal transient energy growth always follows a $\mathit{Re}^{2/3}$ scaling, which allows for large amplifications even in regimes where the presence of turbulence remains debated. In co-rotating Rayleigh-stable flows, the optimal perturbations become increasingly columnar in their structure, as the optimal axial wavenumber goes to zero. In this limit of axially invariant perturbations, we show that linear stability and transient growth are independent of the cylinder rotation ratio and we derive a universal $\mathit{Re}^{2/3}$ scaling of optimal energy growth using Wentzel–Kramers–Brillouin theory. Based on this, a semi-empirical formula for the estimation of linear transient growth valid in all regimes is obtained.

JFM classification

Instability: Instability Instability: Transition to turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 742 , 10 March 2014 , pp. 254 - 290
Copyright
© 2014 Cambridge University Press 

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