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Stably stratified exact coherent structures in shear flow: the effect of Prandtl number

Published online by Cambridge University Press:  06 November 2019

Jake Langham Open the ORCID record for Jake Langham [Opens in a new window] ,
Tom S. Eaves Open the ORCID record for Tom S. Eaves [Opens in a new window]  and
Rich R. Kerswell
Jake Langham*
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW, UK
Tom S. Eaves
Affiliation:
Department of Mathematics, University of British Columbia, 1984 Mathematics Rd., Vancouver BC V6T 1Z2, Canada
Rich R. Kerswell
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW, UK Centre for Mathematical Sciences, University of Cambridge, Cambridge CB3 0WA, UK
*
†Email address for correspondence: j.langham@bristol.ac.uk
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Abstract

We examine how known unstable equilibria of the Navier–Stokes equations in plane Couette flow adapt to the presence of an imposed stable density difference between the two boundaries for varying values of the Prandtl number $Pr$, the ratio of viscosity to density diffusivity and fixed moderate Reynolds number, $Re=400$. In the two asymptotic limits $Pr\rightarrow 0$ and $Pr\rightarrow \infty$, it is found that such solutions exist at arbitrarily high bulk stratification but for different physical reasons. In the $Pr\rightarrow 0$ limit, density variations away from a constant stable density gradient become vanishingly small as diffusion of density dominates over advection, allowing equilibria to exist for bulk Richardson number $\mathit{Ri}_{b}\lesssim O(Re^{-2}Pr^{-1})$. Alternatively, at high Prandtl numbers, density becomes homogenised in the interior by the dominant advection which creates strongly stable stratified boundary layers that recede into the wall as $Pr\rightarrow \infty$. In this scenario, the density stratification and the flow essentially decouple, thereby mitigating the effect of increasing $\mathit{Ri}_{b}$. An asymptotic analysis is presented in the passive scalar regime $\mathit{Ri}_{b}\lesssim O(Re^{-2})$, which reveals $O(Pr^{-1/3})$-thick stratified boundary layers with $O(Pr^{-2/9})$-wide eruptions, giving rise to density fingers of $O(Pr^{-1/9})$ length and $O(Pr^{-4/9})$ width that invade an otherwise homogeneous interior. Finally, increasing $Re$ to $10^{5}$ in this regime reveals that interior stably stratified density layers can form away from the boundaries, separating well-mixed regions.

JFM classification

Turbulent Flows: Stratified turbulence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 882 , 10 January 2020 , A10
Copyright
© 2019 Cambridge University Press 

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