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Taylor–Couette–Poiseuille flow with a weakly permeable inner cylinder: absolute instabilities and selection of global modes

Published online by Cambridge University Press:  26 June 2018

Nils Tilton Open the ORCID record for Nils Tilton [Opens in a new window]  and
Denis Martinand Open the ORCID record for Denis Martinand [Opens in a new window]
Nils Tilton
Affiliation:
Department of Mechanical Engineering, Colorado School of Mines, Golden, CO 80401, USA
Denis Martinand*
Affiliation:
Aix-Marseille Université, CNRS, Centrale Marseille, M2P2, 13013 Marseille, France
*
†Email address for correspondence: denis.martinand@univ-amu.fr
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Abstract

Variations in the local stability of the flow in a Taylor–Couette cell can be imposed by adding an axial Poiseuille flow and a radial flow associated with one or both of the cylinders being permeable. At a given rotation rate of the inner cylinder, this results in adjacent regions of the flow that can be simultaneously stable, convectively unstable, and absolutely unstable, making this system fit for studying global modes of instability. To this end, building on the existing stability analysis in absolute modes developing over axially invariant base flows, we consider the case of axially varying base flows in systems for which the outer cylinder is impermeable, and the inner cylinder is a weakly permeable membrane through which the radial flow is governed by Darcy’s law. The frameworks of linear and nonlinear global modes are used to describe the instabilities and assess the results of direct numerical simulations using a dedicated pseudospectral method. Three different axially evolving set-ups are considered. In the first, fluid injection occurs along the full inner cylinder. In the second, fluid extraction occurs along the full inner cylinder. Besides its fundamental interest, this set-up is relevant to filtration devices. In the third, fluid flux through the inner cylinder evolves from extraction to injection as cross-flow reversal occurs. In agreement with the global mode analyses, the numerical simulations develop centrifugal instabilities above the predicted critical rotation rates and downstream of the predicted axial locations. The global mode analyses do not fully explain, however, that the instabilities observed in the numerical simulations take the form of axial stacks of wavepackets characterized by jumps of the temporal frequency.

JFM classification

Instability: Absolute/convective instability Instability: Nonlinear instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 849 , 25 August 2018 , pp. 741 - 776
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Copyright
© 2018 Cambridge University Press 

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