A phenomenological model for fountain-top entrainment

Abstract

In theoretical treatments of turbulent fountains, the entrainment of ambient fluid into the top of the fountain, hereinafter fountain-top entrainment $Q_{top}$

(

$\text{m}^{3}~\text{s}^{-1}$

), has been neglected until now. This neglect, which modifies the energetic balance in a fountain, compromises the predictive ability of existing models. Our aim is to quantify

$Q_{top}$

by shedding light on the physical processes that are responsible for fountain-top entrainment. First, estimates for

$Q_{top}$

are obtained by applying, in turn, an entrainment closure in the vein of Morton et al. ( Proc. R. Soc. Lond. , vol. 234, 1956, pp. 1–23) and then of Shrinivas & Hunt ( J. Fluid Mech. , vol. 757, 2014, pp. 573–598) to the time-averaged fountain top. Unravelling the assumptions that underlie these approaches, we argue that neither capture the dynamical behaviour of the flow observed at the fountain top; the top being characterised by quasi-periodic fluctuations, during which large-scale eddies reverse and engulf parcels of ambient fluid into the fountain. Therefore, shifting our mindset to a periodical framework, we develop a new phenomenological model in which we emphasise the role of the fluctuations in entraining external fluid. Our model suggests that

$Q_{top}$

is similar in magnitude to the volume flux supplied to the fountain top by the upflow (

$Q_{u}$

), i.e. 

$Q_{top}\sim Q_{u}$

, in agreement with experimental evidence. We conclude by providing guidance on how to implement fountain-top entrainment in existing models of turbulent fountains.