On the stratification and induced flow in an emptying–filling box driven by a plane vertically distributed source of buoyancy

Abstract

A theoretical model is presented for the steady multi-layered flow induced by a plane vertically distributed buoyancy source producing a turbulent wall plume in a ventilated box. While aspects of the stratification and rate of fluid exchange between box and exterior have been studied previously, the streamline pattern and velocity field have not been considered until now, despite having potentially important practical implications for achieving comfort in naturally ventilated buildings and for the indoor spread of airborne contagions. The boundary condition at the wall for each layer is established by deducing the turbulent entrainment rate. Using conformal mapping techniques and Poisson's integral theorem, closed-form solutions for the streamfunction of the induced flow in each layer are established. While the flow near the ceiling was overlooked in the classic model for the multi-layered stratification, after considering the possible flow scenarios, the stratification is re-evaluated herein by incorporating an entraining ceiling current. With a markedly thinner top layer, the refined stratification matches well with the available experimental observations, the restrictions we place on the applicability of the model overcoming the previous over-prediction in the number of interfaces. The magnitude of the dimensionless flow velocity, independent of the wall buoyancy flux and physical scale of the box, decreases significantly with the number of layers. Three types of layer, each with a distinct induced flow pattern, are distinguished and their implications for room airflow considered. Notably, the flow in the base layer represents a continual and smooth flushing of air between the inlet opening and the wall plume, whereas an intermediate layer is almost entirely comprised of near-stagnant air.