On the meaning of mixing efficiency for buoyancy-driven mixing in stratified turbulent flows

Abstract

The concept of a mixing efficiency is widely used to relate the amount of irreversible diabatic mixing in a stratified flow to the amount of energy available to support mixing. This common measure of mixing in a flow is based on the change in the background potential energy, which is the minimum gravitational potential energy of the fluid that can be achieved by an adiabatic rearrangement of the instantaneous density field. However, this paper highlights examples of mixing that is primarily 'buoyancy-driven' (i.e. energy is released to the flow predominantly from a source of available potential energy) to demonstrate that the mixing efficiency depends not only on the specific characteristics of the turbulence in the region of the flow that is mixing, but also on the density profile in regions remote from where mixing physically occurs. We show that this behaviour is due to the irreversible and direct conversion of available potential energy into background potential energy in those remote regions (a mechanism not previously described). This process (here termed 'relabelling') occurs without requiring either a local flow or local mixing, or any other process that affects the internal energy of that fluid. Relabelling is caused by initially available potential energy, associated with identifiable parcels of fluid, becoming dynamically inaccessible to the flow due to mixing elsewhere. These results have wider relevance to characterising mixing in stratified turbulent flows, including those involving an external supply of kinetic energy.

Stratified shear instability in a field of pre-existing turbulence

“…The sorted profile b (z) may be calculated from the full three-dimensional buoyancy field either by sorting or by computing its probability density function, as we do here (Winters et al 1995;Tseng & Ferziger 2001;Davies Wykes et al 2015). The available potential energy P a is then the energy that may be used to mix the flow and arises due to horizontal buoyancy gradients or from statically-unstable regions of the flow (Lorenz 1955).…”

Harmonic to subharmonic transition of the Faraday instability in miscible fluids

“…To disentangle reversible exchanges from irreversible mixing, it is proposed to use the concepts of sorted density fields and background and available potential energies [9,10,20,21]. Recall that within the Boussinesq approximation, the density variations are linked to concentration through ρ ∼ 2A t ρ 0 c. The total potential energy being E p = ρ gz dV , we refer hereafter unambiguously to the normalized potential energy per unit surface as…”

“…The second definition can be found in Refs. [10,21,23] and is based on the background potential energy, so that it should be more relevant in terms of quantifying the irreversible mixing. The instantaneous mixing efficiency reads…”

Energetics of mixing for the filling box and the emptying-filling box

“…the source of energy in the system is initially entirely E a ), high mixing efficiencies have been observed, with values of the cumulative mixing efficiency greater than 75% measured in experiments of Rayleigh-Taylor instability [9], and tending towards 100% in experiments of horizontal convection [10]. In some cases of buoyancy-driven stratified mixing, the mixing efficiency depends on the density profile in regions remote from where the mixing takes place [11]. Nevertheless, the mixing efficiency is often discussed in the literature as if it were a constant value or some property of the turbulence itself, therefore it seems useful to examine and understand cases where this is not true, to extend our intuition about such cases.…”