The linear instability of the stratified plane Couette flow

Abstract

We present the stability analysis of a plane Couette flow which is stably stratified in the vertical direction orthogonally to the horizontal shear. Interest in such a flow comes from geophysical and astrophysical applications where background shear and vertical stable stratification commonly coexist. We perform the linear stability analysis of the flow in a domain which is periodic in the stream-wise and vertical directions and confined in the cross-stream direction. The stability diagram is constructed as a function of the Reynolds number Re and the Froude number F r, which compares the importance of shear and stratification. We find that the flow becomes unstable when shear and stratification are of the same order (i.e. F r ∼ 1) and above a moderate value of the Reynolds number Re 700. The instability results from a resonance mechanism already known in the context of channel flows, for instance the unstratified plane Couette flow in the shallow water approximation. The result is confirmed by fully non linear direct numerical simulations and to the best of our knowledge, constitutes the first evidence of linear instability in a vertically stratified plane Couette flow. We also report the study of a laboratory flow generated by a transparent belt entrained by two vertical cylinders and immersed in a tank filled with salty water linearly stratified in density. We observe the emergence of a robust spatio-temporal pattern close to the threshold values of F r and Re indicated by linear analysis, and explore the accessible part of the stability diagram. With the support of numerical simulations we conclude that the observed pattern is a signature of the same instability predicted by the linear theory, although slightly modified due to streamwise confinement.Key words: Authors should not enter keywords on the manuscript, as these must be chosen by the author during the online submission process and will then be added during the typesetting process (see http://journals.cambridge.org/data/relatedlink/jfmkeywords.pdf for the full list)

Linear instability of tilted parallel shear flow in a strongly stratified and viscous medium

“…Given the buoyancy Reynolds number argument, this implies that the primary instability, which originates from horizontal inflectional base flow, should be two dimensional as long as Fr is below O(1). This also explains why the low-Froude-number instability mode observed in the experiment of Meunier (2012) worth mentioning the recent work by Facchini et al (2018), where a new type of threedimensional linear instability was reported in horizontal Couette flow. However, in this case, Fr ∼ O(10 −1 − 1) and Re > O(10 3 ).…”

“…relatively high buoyancy Reynolds number), it is important to note that the Squire-like theorem, which we demonstrated previously, does not precisely apply, as will be discussed in §3.3. Indeed, the recent work by Facchini et al (2018) has shown that a three-dimensional instability can arise in horizontal plane Couette flow where inflectional instability mechanism is ruled out by its base flow. While we have not observed such a three-dimensional instability as the most unstable primary instability in the present study, we do not rule out such possibility in other flow configurations where buoyancy Reynolds number is not small.…”

Layer formation and relaminarisation in plane Couette flow with spanwise stratification

“…the possibility of turbulence being sustained. This linear instability is the analogous instability (for flows with unit Prandtl number) to the instability identified by Facchini et al (2018) at infinite Schmidt number. Figure 2 shows some typical flow field snapshots at Re = 5000 and F −2 h = 0.1, which is near to the largest possible stratification possible at this Reynolds number for which subcritically-triggered turbulence can be sustained.…”

“…The flow is forced naturally by moving boundaries in a plane Couette flow (pCf) system with spanwise stratification (hereafter referred to as HSPC for horizontal stratified plane Couette). This paper constitutes, along with the recent paper by Facchini et al (2018), the first exploration of the effect of spanwise stratification on plane Couette flow dynamics. Mainly considering flows at high Prandtl number, Facchini et al (2018) have shown that for the spanwise stratified version of plane Couette flow, a new linear instability appears when the geometry permits resonances between internal gravity waves.…”

“…This paper constitutes, along with the recent paper by Facchini et al (2018), the first exploration of the effect of spanwise stratification on plane Couette flow dynamics. Mainly considering flows at high Prandtl number, Facchini et al (2018) have shown that for the spanwise stratified version of plane Couette flow, a new linear instability appears when the geometry permits resonances between internal gravity waves. However, the instability only arises in geometries which allow the required critical relationship between vertical and streamwise wavelengths, given by the appropriately Doppler-shifted dispersion relationship.…”