Disruption of SSP/VWI states by a stable stratification

TL;DR

This paper investigates how stable stratification affects the minimal initial conditions needed to trigger turbulence in plane Couette flow, revealing that stratification alters coherent structures and suppresses vertical motions, impacting turbulence transition.

Contribution

It identifies minimal seeds for turbulence in stratified plane Couette flow and analyzes how stratification modifies the self-sustaining process and coherent structures.

Findings

Stratification alters the form of coherent structures at moderate Reynolds numbers.

Vertical motion suppression by stratification disrupts wave-to-roll energy transfer.

Higher stratification levels prevent waves from reinforcing roll structures, hindering turbulence onset.

Abstract

We identify `minimal seeds' for turbulence, i.e. initial conditions of the smallest possible total perturbation energy density $E_c$ that trigger turbulence from the laminar state, in stably stratified plane Couette flow using the `direct-adjoint-looping' (DAL) method for finding nonlinear optimal perturbations that optimise the time averaged total dissipation of energy in the flow. These minimal seeds are located adjacent to the edge manifold, the manifold in state space that separates trajectories which transition to turbulence from those which eventually decay to the laminar state. The edge manifold is also the stable manifold of the system's `edge state'. The trajectories from the minimal seed initial conditions spend a large amount of time in the vicinity of some states: the edge state; another state contained within the edge manifold; or even in dynamically slowly varying regions of the edge manifold, allowing us to investigate the effects of a stable stratification on any coherent structures associated with such states. In unstratified plane Couette flow, these coherent structures are manifestations of the self-sustaining process (SSP) deduced on physical grounds by Waleffe (1997), or equivalently finite Reynolds number solutions of the vortex-wave interaction (VWI) asymptotic equations initially derived mathematically by Hall & Smith (1991). The stratified coherent states we identify at moderate $Re$ display an altered form from their unstratified counterparts for bulk Richardson numbers $Ri_B=\textit{O}(Re^{-1})$, and exhibit chaotic motion for larger $Ri_B$. We demonstrate that at high $Re$ the suppression of vertical motions by stratification strongly disrupts input from the waves to the roll velocity structures, thus preventing the waves from reinforcing the viscously decaying roll structures adequately, when $Ri_B=\textit{O}(Re^{-2})$.