Rise Heights of Lazy Fountains

The initial rise height $z_m$ of turbulent Boussinesq fountains is determined analytically for small and large source Froude numbers $Fr_0$. Solutions were obtained after recasting the plume conservations equations of Morton, Taylor \& Turner (1956) in terms of the inverse square of a local Froude number and a local dimensionless fountain width. For large $Fr_0$, the fountain is `forced' and the well-established linear increase of the rise height with $Fr_0$ is obtained, i.e. $z_m/r_0 \sim Fr_0$; $r_0$ denoting the source radius. However, for small $Fr_0$ the fountain is `lazy' and the dependence $z_m/r_0 \sim Fr_0^2$ more sensitive. Additionally, the rise height for lazy fountains is predicted to be independent of the entrainment coefficient $\alpha$. Comparison of our solutions with existing experimental and numerical results of fountain rise height, as well as with our own experimental results, show good agreement and support the derived scalings. Experimental results suggest that the entrainment coefficient for highly-forced fountains is $\alpha_f\approx0.058$, i.e. closer to that of a jet than of a plume. \\ Morton, B. R., Taylor, G. I. \& Turner, J. S. (1956), ‘Turbulent gravitational convection from maintained and instantaneous sources’, \it{Proc. Roy. Soc. A} \textbf{234}, 1–23.