Entrainment and dilution in a fountain top

Abstract

Experiments are carried out on a turbulent fountain to investigate the entrainment of ambient fluid and the dilution of scalar concentration in the fountain top (also referred to as the ‘cap’). The source Froude number ( $Fr_o$ ), defined based on the density difference between the source and the ambient fluids, is varied between 10 and 30, while the Reynolds number ( $Re_o$ ) is set to a minimum of 3000 to ensure a fully turbulent flow at the source. High-fidelity velocity and concentration measurements are obtained using particle image velocimetry and planar laser induced fluorescence techniques, respectively. The mean concentration field indicates that the cap is approximately hemispherical and its base is characterised by the local Froude number $Fr_z \approx 1.5$ . It is observed that the ratio of the entrained ( $Q_{top}$ ) volume flux in the fountain top and the volume flux supplied ( $Q_{in}$ ) at the base of the cap varies between 1.5 and 3.5 at different $Fr_o$ , exceeding the values of $Q_{top}/Q_{in} (= 0.5\text {--}0.8)$ for a fountain developing across a density interface (Lin & Linden, J. Fluid Mech., vol. 542, 2005, pp. 25–52). The present experimental results for $Q_{top}/Q_{in}$ are in excellent agreement with published numerical simulations of turbulent fountains at similar $Fr_o$ . Lastly, a robust metric to estimate the dilution of scalar in the fountain top has been proposed. The results clearly indicate that dilution is not equal to the entrainment ratio; however, the self-similarity of non-dimensional dilution profiles at different $Fr_o$ suggests that the phenomenological model for entrainment in the fountain top put forth by Debugne & Hunt (J. Fluid Mech., vol. 796, 2016, pp. 195–210), can also be effectively used to model the dilution of scalar concentration in the cap. Further, it is found that the variation of the dilution ratio is closely linked to the local Reynolds number ( $Re_z$ ) at the base of the cap. This is verified using an analytical model that describes the dependency of $Re_z$ on the local and source parameters – $Fr_z$ , $Re_o$ and $Fr_o$ .