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

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 Ri_b ≤ O(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 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 10⁵ in this regime reveals that interior stably stratified density layers can form away from the boundaries, separating well-mixed regions.