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The rise and fall of turbulent fountains: a new model for improved quantitative predictions

Published online by Cambridge University Press:  10 June 2010

G. CARAZZO ,
E. KAMINSKI  and
S. TAIT
G. CARAZZO*
Affiliation:
Department of Earth and Ocean Sciences, The University of British Columbia, 6338 Stores Rd, Vancouver, BC, CanadaV6T 1Z4
E. KAMINSKI
Affiliation:
Equipe de Dynamique des Fluides Géologiques, Université Paris Diderot and Institut de Physique du Globe de Paris, 4 place Jussieu, 75252 Paris CEDEX 05, France
S. TAIT
Affiliation:
Equipe de Dynamique des Fluides Géologiques, Université Paris Diderot and Institut de Physique du Globe de Paris, 4 place Jussieu, 75252 Paris CEDEX 05, France
*
†Email address for correspondence: [email protected]
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Abstract

Turbulent fountains are of major interest for many natural phenomena and industrial applications, and can be considered as one of the canonical examples of turbulent flows. They have been the object of extensive experimental and theoretical studies that yielded scaling laws describing the behaviour of the fountains as a function of source conditions (namely their Reynolds and Froude numbers). However, although such scaling laws provide a clear understanding of the basic dynamics of the turbulent fountains, they usually rely on more or less ad hoc dimensionless proportionality constants that are scarcely tested against theoretical predictions. In this paper, we use a systematic comparison between the initial and steady-state heights of a turbulent fountain predicted by classical top-hat models and those obtained in experiments. This shows scaling agreement between predictions and observations, but systematic discrepancies regarding the proportionality constant. For the initial rise of turbulent fountains, we show that quantitative agreement between top-hat models and experiments can be achieved by taking into account two factors: (i) the reduction of entrainment by negative buoyancy (as quantified by the Froude number), and (ii) the fact that turbulence is not fully developed at the source at intermediate Reynolds number. For the steady-state rise of turbulent fountains, a new model (‘confined top-hat’) is developed to take into account the coupling between the up-flow and the down-flow in the steady-state fountain. The model introduces three parameters, calculated from integrals of experimental profiles, that highlight the dynamics of turbulent entrainment between the up-flow and the down-flow, as well as the change of buoyancy flux with height in the up-flow. The confined top-hat model for turbulent fountains achieves good agreement between theoretical predictions and experimental results. In particular, it predicts a systematic increase of the ratio between the initial and steady-state heights of turbulent fountains as a function of their source Froude number, an observation that was not handled properly in previous models.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 657 , 25 August 2010 , pp. 265 - 284
Copyright
Copyright © Cambridge University Press 2010

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