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

Accepted version
Peer-reviewed

Type

Article

Change log

Authors

Langham, Jake 
Eaves, Tom S 
Kerswell, Rich R 

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→0 and Pr, it is found that such solutions exist at arbitrarily high bulk stratification but for different physical reasons. In the Pr→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 RibO(Re−2Pr−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. In this scenario, the density stratification and the flow essentially decouple, thereby mitigating the effect of increasing Rib. An asymptotic analysis is presented in the passive scalar regime RibO(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 105 in this regime reveals that interior stably stratified density layers can form away from the boundaries, separating well-mixed regions.

Description

Keywords

physics.flu-dyn, physics.flu-dyn

Journal Title

Journal of Fluid Mechanics

Conference Name

Journal ISSN

0022-1120
1469-7645

Volume Title

Publisher

Cambridge University Press

Rights

All rights reserved
Sponsorship
Engineering and Physical Sciences Research Council (EP/K034529/1)
EPSRC