On turbulent particle fountains

Abstract

We describe new experiments in which particle-laden turbulent fountains with source Froude numbers $20>Fr_{0}>6$ are produced when particle-laden fresh water is injected upwards into a reservoir filled with fresh water. We find that the ratio $U$ of the particle fall speed to the characteristic speed of the fountain determines whether the flow is analogous to a single-phase fountain ($U\ll 1$) or becomes a fully separated flow ($U\geqslant 1$). In the single-phase limit, a fountain with momentum flux $M$ and buoyancy flux $B$ oscillates about the mean height, $h_{m}=(1.56\pm 0.04)M^{3/4}B^{-1/2}$, as fluid periodically cascades from the maximum height, $h_{t}=h_{m}+{\rm\Delta}h$, to the base of the tank. Experimental measurements of the speed $u$ and radius $r$ of the fountain at the mean height $h_{m}$, combined with the conservation of buoyancy, suggest that $Fr(h_{m})=u(g^{\prime }r)^{-1/2}\approx 1$. Using these values, we find that the classical scaling for the frequency of the oscillations, ${\it\omega}\sim BM^{-1}$, is equivalent to the scaling $u(h_{m})/r(h_{m})$ for a fountain supplied at $z=h_{m}$ with $Fr=1$ (Burridge & Hunt, J. Fluid Mech., vol. 728, 2013, pp. 91–119). This suggests that the oscillations are controlled in the upper part of the fountain where $Fr\leqslant 1$, and that they may be understood in terms of a balance between the upward supply of a growing dense particle cloud, at the height where $Fr=1$, and the downward flow of this cloud. In contrast, in the separated flow regime, we find that particles do not reach the height at which $Fr=1$: instead, they are transported to the level at which the upward speed of the fountain fluid equals their fall speed. The particles then continuously sediment while the particle-free fountain fluid continues to rise slowly above the height of particle fallout, carried by its momentum.