Convection at an isothermal wall in an enclosure and establishment of stratification

Abstract

In this experimental–theoretical investigation, we consider a turbulent plume generated by an isothermal wall in a closed cavity and the formation of heat stratification in the interior. The buoyancy of the plume near the wall and the temperature stratification are measured across a vertical plane with the temperature laser induced fluorescence method, which is shown to be accurate and efficient (precision of $0.2\,^{\circ }$C) for experimental studies on convection. The simultaneous measurement of the velocity field with particle image velocimetry allows for the calculation of the flow characteristics such as the Richardson number and Reynolds stress. This enables us to give a refined description of the wall plume, as well as the circulation and evolution of the stratification in the interior. The wall plume is found to have an inner layer close to the heated boundary with a laminar transport of hardly mixed fluid which causes a relatively warm top layer and an outer layer with a transition from laminar to turbulent at a considerable height. The measured entrainment coefficient is found to be dramatically influenced by the increase in stratification of the ambient fluid. To model the flow, the entrainment model of Morton, Taylor & Turner (Proc. R. Soc. Lond. A, vol. 234 (1196), 1956, pp. 1–23) has first been adapted to the case of an isothermal wall. Differences due to their boundary condition of a constant buoyancy flux, modelled with salt by Cooper & Hunt (J. Fluid Mech., vol. 646, 2010, pp. 39–58), turn out to be small. Next, to include the laminar–turbulent transition of the boundary layer, a hybrid model is constructed which is based on the similarity solutions reported by Worster & Leitch (J. Fluid Mech., vol. 156, 1985, pp. 301–319) for the laminar part and the entrainment model for the turbulent part. Finally, the observed variation of the global entrainment coefficient, which is due to the increased presence of an upper stratified layer with a relatively low entrainment coefficient, is incorporated into both models. All models show reasonable agreement with experimental measurements for the volume, momentum and buoyancy fluxes as well as for the evolution of the stratification in the interior. In particular, the introduction of the variable entrainment coefficient improves all models significantly.