Scaling arguments for the fluxes in turbulent miscible fountains

Abstract

AbstractFor established axisymmetric turbulent miscible Boussinesq fountains in quiescent uniform environments, expressions are developed for the fluxes of volume, momentum and buoyancy at the outflow from the fountain: the outflow referring to the counterflow at the horizontal plane of the source. The fluxes are expressed in terms of the fountain source conditions and two dimensionless functions of the source Froude number, ${\rm Fr}_{0}$: a radial function (relating a horizontal scale of the outflow to the source radius) and a volume flux function (relating the outflow and source volume fluxes). The forms taken by these two functions at low ${\rm Fr}_{0}$ and high ${\rm Fr}_{0}$ are deduced, thereby providing the outflow fluxes and outflow Froude number solely in terms of the source conditions. For high ${\rm Fr}_{0}$, the outflow Froude number, ${\rm Fr}_{out}$, is shown to be invariant, indicating (by analogy with plumes for which the ‘far-field’ Froude number is invariant with source Froude number) that the outflow may be regarded as ‘far-field’ since the fluxes within the fountain have adjusted to attain a balance which is independent of the source conditions. Based on ${\rm Fr}_{out}$, the fluxes in the plume that forms beyond the fountain outflow are deduced. Finally, from the results of previously published studies, we show that the scalings deduced for fountains are valid for $0.0025 \lesssim {\rm Fr}_{0} \lesssim 1.0 $ for low ${\rm Fr}_{0}$ and $ {\rm Fr}_{0} \gtrsim 3.0 $ for high ${\rm Fr}_{0}$.