Abstract
We examine how known unstable equilibria of the NavierStokes 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 Ri_{b} ≲ 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 → ∞. In this scenario, the density stratification and the flow essentially decouple, thereby mitigating the effect of increasing Ri_{b}. An asymptotic analysis is presented in the passive scalar regime Ri_{b }≲ 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 wellmixed regions.
Original language  English 

Journal  Journal of Fluid Mechanics 
Early online date  6 Nov 2019 
DOIs  
Publication status  Published  10 Jan 2020 
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HPC (High Performance Computing) Facility
Sadaf R Alam (Manager), Steven A Chapman (Manager), Polly E Eccleston (Other), Simon H Atack (Other) & D A G Williams (Manager)
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