The coupled dynamics of internal waves and hairpin vortices in stratified plane Poiseuille flow

Abstract

A simulation of stably stratified plane Poiseuille flow at a moderate Reynolds number ( $\textit {Re}_\tau = 550$ ) and Richardson number ( $\textit {Ri}_\tau = 480$ ) is presented. For the first time, the dynamics in the channel core are shown to be described as a series of internal waves that approximately obey a linear wave dispersion relationship. For a given streamwise wavenumber $k_x$ there are two internal wave solutions, a dominant low frequency mode and a weaker-amplitude high-frequency mode, respectively corresponding to ‘backward’ and ‘forward’ propagating internal waves relative to the mean flow. Analysis of linearised equations shows that the dominant low-frequency mode appears to arise due to a particularly sensitive response of the mean flow profiles to incoherent forcing. Instantaneous visualisations reveal that hairpin vortices dominate the outer region of the channel flow, neighbouring the buoyancy dominated channel core. These hairpins are fundamentally different from those observed in canonical unstratified boundary layer flows, as they arise via quasi-linear local processes far from the wall, governed by background shear. Outer region ejection events are common and can be induced by high amplitude waves. Ejected hairpins are transported into the channel core, in turn ‘ringing’ the prevailing strong buoyancy gradient and thus generating high-amplitude internal waves, high dissipation and wave breaking, induced by spanwise vortex stretching and baroclinic vorticity generation. Such spontaneous and sustained generation of quasi-linear internal waves by wall-bounded sheared turbulence may provide novel idealised solutions for, and insight into, large-scale turbulent mixing in a wide range of environmental and industrial flows.