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Exact coherent structures in stably stratified plane Couette flow

Published online by Cambridge University Press:  08 August 2017

D. Olvera Open the ORCID record for D. Olvera [Opens in a new window]  and
R. R. Kerswell
D. Olvera*
Affiliation:
School of Mathematics, University of Bristol, BristolBS8 1TW, UK
R. R. Kerswell
Affiliation:
School of Mathematics, University of Bristol, BristolBS8 1TW, UK
*
†Email address for correspondence: do12542@bristol.ac.uk
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Abstract

The existence of exact coherent structures in stably stratified plane Couette flow (gravity perpendicular to the plates) is investigated over Reynolds–Richardson number ($Re$–$Ri_{b}$) space for a fluid of unit Prandtl number $(Pr=1)$ using a combination of numerical and asymptotic techniques. Two states are repeatedly discovered using edge tracking – EQ7 and EQ7-1 in the nomenclature of Gibson & Brand (J. Fluid Mech., vol. 745, 2014, pp. 25–61) – and found to connect with two-dimensional convective roll solutions when tracked to negative $Ri_{b}$ (the Rayleigh–Bénard problem with shear). Both these states and Nagata’s (J. Fluid Mech., vol. 217, 1990, pp. 519–527) original exact solution feel the presence of stable stratification when $Ri_{b}=O(Re^{-2})$ or equivalently when the Rayleigh number $Ra:=-Ri_{b}Re^{2}Pr=O(1)$. This is confirmed via a stratified extension of the vortex wave interaction theory of Hall & Sherwin (J. Fluid Mech., vol. 661, 2010, pp. 178–205). If the stratification is increased further, EQ7 is found to progressively spanwise and cross-stream localise until a second regime is entered at $Ri_{b}=O(Re^{-2/3})$. This corresponds to a stratified version of the boundary region equations regime of Deguchi, Hall & Walton (J. Fluid Mech., vol. 721, 2013, pp. 58–85). Increasing the stratification further appears to lead to a third, ultimate regime where $Ri_{b}=O(1)$ in which the flow fully localises in all three directions at the minimal Kolmogorov scale which then corresponds to the Osmidov scale. Implications for the laminar–turbulent boundary in the ($Re$–$Ri_{b}$) plane are briefly discussed.

JFM classification

Nonlinear Dynamical Systems: Bifurcation Nonlinear Dynamical Systems: Nonlinear Dynamical Systems Turbulent Flows: Stratified turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 826 , 10 September 2017 , pp. 583 - 614
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© 2017 Cambridge University Press 

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