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A phenomenological model for fountain-top entrainment

Published online by Cambridge University Press:  28 April 2016

Antoine L. R. Debugne  and
Gary R. Hunt
Antoine L. R. Debugne
Affiliation:
Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK
Gary R. Hunt*
Affiliation:
Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK
*
†Email address for correspondence: gary.hunt@eng.cam.ac.uk
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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.

JFM classification

Wakes/Jets: Jets Convection: Plumes/thermals Turbulent Flows: Turbulent convection
Type
Papers
Information
Journal of Fluid Mechanics , Volume 796 , 10 June 2016 , pp. 195 - 210
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Copyright
© 2016 Cambridge University Press 

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