Filling box model

A filling box model describes a turbulent plume rising through a confined fluid and spreading beneath its ceiling. Fluid entrainment transfers initially unmodified ambient fluid into the plume; the discharged fluid forms a descending filling-box first front. A maintained heat source can have negligible source volume flux while maintaining nonzero buoyancy flux. A finite injected fluid flux requires an equal exhaust if the container volume is fixed.

In a filling box model, the first front separates fluid already modified by plume discharge from the initially unmodified fluid below. If the plume occupies a negligible fraction of horizontal area $A$, its volume flux across the front is $P(h)$, and the ceiling exhaust is $F$, volume conservation gives $A\dot{h}=F-P(h)$. For a closed box with a negligible-volume buoyancy source, $F=0$. For a finite injected source flux $Q$ with ceiling overflow $Q$, $F=Q$. The first front need not delimit a uniformly mixed upper layer throughout the unventilated transient.