Oscillatory behaviour in an emptying–filling box

Abstract

The emptying–filling box problem is examined theoretically in a general non-Boussinesq case. The steady solutions are first reviewed and expressed both as a function of a pure geometrical parameter and as a function of a dimensionless number ${\it\Theta}$ that characterizes the strength of the buoyant source relative to the box height. A linearization of the conservation equations is performed around the steady state and a second-order differential equation is derived that facilitates consideration of the existence of underdamped oscillations in the emptying–filling process. These oscillations are quantified in terms of frequency, damping ratio and phase shift, which constitutes the major outcome of this study. It is shown that oscillations can exist whatever the strength of the buoyancy source, even if their amplitude remains extremely weak for ‘small plumes’.