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

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 $Pr\to 0$ and $Pr\to \infty$, it is found that such solutions exist at arbitrarily high bulk stratification but for different physical reasons. In the $Pr\to 0$ 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 $R{i}_b\lesssim O(R{e}^{-2}P{r}^{-1})$. 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 $Pr\to \infty$. In this scenario, the density stratification and the flow essentially decouple, thereby mitigating the effect of increasing $R{i}_b$. An asymptotic analysis is presented in the passive scalar regime $R{i}_b\lesssim O(R{e}^{-2})$, which reveals $O(P{r}^{-1/3})$-thick stratified boundary layers with $O(P{r}^{-2/9})$-wide eruptions, giving rise to density fingers of $O(P{r}^{-1/9})$ length and $O(P{r}^{-4/9})$ 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.

arXiv comment29 pages, 21 figures

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Cited by in corpus (3)

• Structured input-output analysis of stably stratified plane Couette flow
• Free-stream coherent structures in parallel compressible boundary-layer flows at subsonic Mach numbers
• Centrifugal instability of Taylor-Couette flow in stratified and diffusive fluids