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

TL;DR

This paper analyzes the mixing efficiency of plumes in filling and emptying-filling boxes, revealing that steady-state efficiencies depend on the box configuration and ambient layer depth, with a constant efficiency of 1/2 in filling boxes.

Contribution

It provides a theoretical calculation of mixing efficiencies in filling and emptying-filling boxes, highlighting the dependence on steady-state conditions and ambient layer depth.

Findings

Mixing efficiency in filling boxes at steady state is 1/2.

In emptying-filling boxes, efficiency depends on ambient layer depth, given by 1 - xi.

Efficiency varies with the configuration and state of the box.

Abstract

The mixing efficiency of a plume in a filling box and an emptying-filling box is calculated for both transient and steady states. The mixing efficiency of a plume in a filling box in an asymptotic steady state is 1/2, independent of the details of this state or how the plume is modelled. The mixing efficiency of a plume in an emptying-filling box in steady state is 1 - xi, where xi = h/H, the depth of the ambient layer h non-dimensionalised by the height of the box H. A deeper mixed layer therefore corresponds to a higher mixing efficiency.