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Instabilities and transient growth of the stratified Taylor–Couette flow in a Rayleigh-unstable regime

Published online by Cambridge University Press:  31 May 2017

Junho Park Open the ORCID record for Junho Park [Opens in a new window] ,
Paul Billant  and
Jong-Jin Baik
Junho Park*
Affiliation:
School of Earth and Environmental Sciences, Seoul National University, Seoul 151-742, Republic of Korea
Paul Billant
Affiliation:
LadHyX, CNRS, Ecole Polytechnique, F-91128 Palaiseau CEDEX, France
Jong-Jin Baik
Affiliation:
School of Earth and Environmental Sciences, Seoul National University, Seoul 151-742, Republic of Korea
*
†Email address for correspondence: [email protected]
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Abstract

The stability of the Taylor–Couette flow is analysed when there is a stable density stratification along the axial direction and when the flow is centrifugally unstable, i.e. in the Rayleigh-unstable regime. It is shown that not only the centrifugal instability but also the strato-rotational instability can occur. These two instabilities can be explained and well described by means of a Wentzel–Kramers–Brillouin–Jeffreys asymptotic analysis for large axial wavenumbers in inviscid and non-diffusive limits. In the presence of viscosity and diffusion, numerical results reveal that the strato-rotational instability becomes dominant over the centrifugal instability at the onset of instability when the axial density stratification is sufficiently strong. Linear transient energy growth is next investigated for counter-rotating cylinders in the stable regime of the Froude number–Reynolds number parameter space. We show that there exist two types of transient growth mechanism analogous to the lift up and the Orr mechanisms in homogeneous fluids but with the additional effect of density perturbations. The dominant mechanism depends on the stratification: when the stratification is strong, non-axisymmetric three-dimensional perturbations achieve the optimal energy growth through the Orr mechanism while for moderate stratification, axisymmetric perturbations lead to the optimal transient growth by a lift-up mechanism involving internal waves.

JFM classification

Instability: Instability Geophysical and Geological Flows: Stratified flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 822 , 10 July 2017 , pp. 80 - 108
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© 2017 Cambridge University Press 

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