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Extension of classical stability theory to viscous planar wall-bounded shear flows

Published online by Cambridge University Press:  02 September 2019

Harry Lee Open the ORCID record for Harry Lee [Opens in a new window]  and
Shixiao Wang Open the ORCID record for Shixiao Wang [Opens in a new window]
Harry Lee
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA
Shixiao Wang*
Affiliation:
Department of Mathematics, University of Auckland, Auckland 1142, New Zealand
*
†Email address for correspondence: wang@math.auckland.ac.nz
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Abstract

A viscous extension of Arnold’s inviscid theory for planar parallel non-inflectional shear flows is developed and a viscous Arnold’s identity is obtained. Special forms of the viscous Arnold’s identity have been revealed that are closely related to the perturbation’s enstrophy identity derived by Synge (Proceedings of the Fifth International Congress for Applied Mechanics, 1938, pp. 326–332, John Wiley) (see also Fraternale et al., Phys. Rev. E, vol. 97, 2018, 063102). Firstly, an alternative derivation of the perturbation’s enstrophy identity for strictly parallel shear flows is acquired based on the viscous Arnold’s identity. The alternative derivation induces a weight function. Thereby, a novel weighted perturbation’s enstrophy identity is established, which extends the previously known enstrophy identity to include general streamwise translation-invariant shear flows. Finally, the validity of the enstrophy identity for parallel shear flows is rigorously examined and established under global nonlinear dynamics imposed with two classes of wall boundary conditions. As an application of the enstrophy identity, we quantitatively investigate the mechanism of linear instability/stability within the normal modal framework. The investigation reveals a subtle interaction between a critical layer and its adjacent boundary layer, which determines the stability nature of the disturbance. As an implementation of the relaxed wall boundary conditions imposed for the enstrophy identity, a control scheme is proposed that transitions the wall settings from the no-slip condition to the free-slip condition, through which a flow is stabilized quickly in an early stage of the transition.

JFM classification

Flow Control: Instability control Boundary Layers: Boundary layer stability Vortex Flows: Vortex dynamics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 877 , 25 October 2019 , pp. 1134 - 1162
Copyright
© 2019 Cambridge University Press 

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