The discharge plume parameter and its implications for an emptying–filling box

Abstract

The natural ventilation flow driven by an internal buoyant plume in a box involving an upper opening (vent) located at the ceiling (for the outflow) and a large lower opening at the floor (for the inflow) is examined theoretically in a general non-Boussinesq case. Analytical solutions of this emptying–filling box problem allow the characteristics of the flow at the vent to be determined. From these characteristics, a non-dimensional parameter $\unicode[STIX]{x1D6E4}_{d}$ (called the discharge plume parameter) is expressed. This parameter characterizes the initial balance of volume, buoyancy and momentum fluxes in the plume-like flow that forms above the vent. We then note that the value of $\unicode[STIX]{x1D6E4}_{d}$ allows the buoyant fluid layer depth in the box to be estimated, which is a new and interesting result for natural ventilation problems. Following previous experimental results, the decrease of the vent discharge coefficient $C_{d}$ when $\unicode[STIX]{x1D6E4}_{d}$ increases is discussed and a theoretical model based on plume necking is proposed. The emptying–filling box model is then extended for a variable $C_{d}$ (depending on $\unicode[STIX]{x1D6E4}_{d}$). Even though the discharge coefficient may be markedly reduced at high values of $\unicode[STIX]{x1D6E4}_{d}$, our results show that this only affects transients and the steady state of an emptying–filling box for relatively thin buoyant fluid layers.