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 → 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 well-mixed regions.

Structured input-output analysis of stably stratified plane Couette flow

“…This observation that the product P rRi b determines flow behavior in the low Prandtl number limit is widely observed in stratified shear flows; see e.g., (Lignieres 1999;Garaud et al 2015Garaud et al , 2017Garaud 2021). Conversely, in the high Prandtl number regime, exact coherent structures in stratified PCF (Langham et al 2020) show that a nearly uniform density region forms near the channel center, and the influence of bulk Richardson number on the averaged properties of the flow are significantly reduced. Moreover, Taylor & Zhou (2017) proposed a multi-parameter criterion for the formation of a 'staircase' in the density distribution (i.e.…”

“…The Prandtl number (P r = ν/κ, where ν is the kinematic viscosity and κ is the diffusivity of the density field) plays a perhaps unsurprisingly important role in determining flow structures. For example, for sufficiently small P r, flows with the same value of the product P rRi b develop the same averaged vertical density profile (Langham et al 2020). This observation that the product P rRi b determines flow behavior in the low Prandtl number limit is widely observed in stratified shear flows; see e.g., (Lignieres 1999;Garaud et al 2015Garaud et al , 2017Garaud 2021).…”

“…The Prandtl number is known to play an important role in the types of flow structures characterizing stratified PCF (Zhou et al 2017b,a;Taylor & Zhou 2017;Langham et al 2020). The Prandtl number also varies over a wide range in different applications.…”

Shear-induced modulation on thermal convection over rough plates

“…A similar result was also found in the case of sheared stably stratified turbulence (Zonta & Soldati 2018). The effects of Pr on wall-sheared RB convection (Blass et al 2021) and on sheared stably stratified flow (Langham, Eaves & Kerswell 2020) were further studied. Moreover, Scagliarini et al (2015) studied the BL characteristics of sheared RB turbulence.…”

Free-stream coherent structures in parallel compressible boundary-layer flows at subsonic and moderate supersonic Mach numbers

“…Faisst & Eckhardt (2003), Waleffe (2001, 2003), Wedin & Kerswell (2004) and Wang, Gibson & Waleffe (2007). The study of exact coherent structures in two-parameter space has previously only been conducted in the context of stably stratified flows (Eaves & Caulfield 2015; Deguchi 2017; Lucas & Caulfield 2017; Lucas, Caulfield & Kerswell 2017; Olvera & Kerswell 2017), where it is shown that the Prandtl number plays a key role in the structure of the states found (Langham, Eaves & Kerswell 2020).…”