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Forced fountains

Published online by Cambridge University Press:  03 August 2016

Gary R. Hunt  and
Antoine L. R. Debugne Open the ORCID record for Antoine L. R. Debugne [Opens in a new window]
Gary R. Hunt*
Affiliation:
Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK
Antoine L. R. Debugne
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

We present a three-region model for the time-averaged behaviour of established turbulent axisymmetric fountains at high source Froude numbers $(Fr_{0})$ in which we uniquely account for entrainment of ambient fluid both laterally and at the fountain top. High-$Fr_{0}$ ‘forced’ fountains, as originally investigated experimentally by Turner (J. Fluid Mech., vol. 26 (4), 1966, pp. 779–792), are characterised by an upflow, a counterflow and a fountain top where the flow reverses direction. Through the inclusion of the flow-reversal region and by accounting for fountain-top entrainment, which is neglected in all existing models, close agreement is achieved between our solutions and existing experimental data. Moreover, our predictions of the fluxes within the fountain are in accord with scaling arguments deduced in recent studies. Our model reveals five key ratios that characterise the fountain asymptote to constant values in the high-$Fr_{0}$ limit. These are the ratios of the (1) initial and mean rise heights, (2) vertical extents of the fountain top and upflow regions, (3) fluxes of volume entrained into the fountain top and entrained laterally into the counterflow, (4) forces of inertia and buoyancy acting on the counterflow at the level of the source and (5) average times taken for fluid to rise through the upflow and fall through the counterflow. Attributing the invariance of these ratios to the global self-preserving behaviour of the fountain, we propose a threshold source Froude number for which a continuous negatively buoyant release may be regarded as giving rise to a ‘forced’ fountain.

JFM classification

Convection: Convection Wakes/Jets: Jets Convection: Plumes/thermals
Type
Papers
Information
Journal of Fluid Mechanics , Volume 802 , 10 September 2016 , pp. 437 - 463
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Copyright
© 2016 Cambridge University Press 

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References

Bloomfield, L. J. & Kerr, R. C. 2000 A theoretical model of a turbulent fountain. J. Fluid Mech. 424, 197–216.Google Scholar
Burridge, H. C. & Hunt, G. R. 2012 The rise heights of low- and high-Froude-number turbulent axisymmetric fountains. J. Fluid Mech. 691, 392–416.Google Scholar
Burridge, H. C. & Hunt, G. R. 2013 The rhythm of fountains: the length and time scales of rise height fluctuations at low and high Froude numbers. J. Fluid Mech. 728, 91–119.Google Scholar
Burridge, H. C. & Hunt, G. R. 2014 Scaling arguments for the fluxes in turbulent miscible fountains. J. Fluid Mech. 744, 273–285.Google Scholar
Burridge, H. C. & Hunt, G. R. 2016 Entrainment by turbulent fountains. J. Fluid Mech. 790, 407–418.Google Scholar
Burridge, H. C., Mistry, A. & Hunt, G. R. 2015 The effect of source Reynolds number on the rise height of a fountain. Phys. Fluids 27, 1–17.Google Scholar
Campbell, I. H. & Turner, J. S. 1989 Fountains in magma chambers. J. Petrol. 30, 885–923.Google Scholar
Carazzo, G., Kaminski, E. & Tait, S. 2010 The rise and fall of turbulent fountains: a new model for improved quantitative predictions. J. Fluid Mech. 657, 265–284.Google Scholar
Cotel, A. J., Gjestvang, J. A., Ramkhelawan, N. N. & Breidenthal, R. E. 1997 Laboratory experiments of a jet impinging on a stratified interface. Exp. Fluids 23, 155–160.Google Scholar
Cresswell, R. W. & Szczepura, R. T. 1993 Experimental investigation into a turbulent jet with negative buoyancy. Phys. Fluids 5 (11), 2865–2878.Google Scholar
Debugne, A. L. R. & Hunt, G. R. 2016 A phenomological model for fountain-top entrainment. J. Fluid Mech. 796, 195–210.Google Scholar
Devenish, B. J., Rooney, G. G. & Thomson, D. J. 2010 Large-eddy simulation of a buoyant plume in uniform and stably stratified environments. J. Fluid Mech. 652, 75–103.Google Scholar
Ezzamel, A., Salizzoni, P. & Hunt, G. R. 2015 Dynamical variability of axisymmetric buoyant plumes. J. Fluid Mech. 765, 576–611.Google Scholar
Fischer, H. B., List, E. J., Koh, R. C. Y., Imberger, J. & Brooks, N. H. 1979 Mixing in Inland and Coastal Waters. Academic.Google Scholar
Hunt, G. R. & Burridge, H. C. 2015 Fountains in industry and nature. Annu. Rev. Fluid Mech. 47, 195–220.Google Scholar
Hunt, G. R. & Kaye, N. B. 2005 Lazy plumes. J. Fluid Mech. 533, 329–338.Google Scholar
Kaye, N. B. & Hunt, G. R. 2006 Weak fountains. J. Fluid Mech. 558, 319–328.Google Scholar
Koh, R. C. Y. & Brooks, N. H. 1975 Fluid mechanics of waste-water disposal in the ocean. Annu. Rev. Fluid Mech. 7, 187–211.Google Scholar
Kotsovinos, N. E. & List, E. J. 1977 Plane turbulent buoyant jets. Part 1. Integral properties. J. Fluid Mech. 81, 25–44.Google Scholar
Lin, Y. J. P. & Linden, P. F. 2005 The entrainment due to a turbulent fountain at a density interface. J. Fluid Mech. 542, 25–52.Google Scholar
Mcdougall, T. J. 1981 Negative buoyant vertical jets. Tellus 33, 313–320.Google Scholar
Mehaddi, R., Vauquelin, O. & Candelier, F. 2012 Analytical solutions for turbulent Boussinesq fountains in a linearly stratified environment. J. Fluid Mech. 691, 487–497.Google Scholar
Mehaddi, R., Vaux, S., Candelier, F. & Vauquelin, O. 2015 On the modelling of steady turbulent fountains. Environ. Fluid Mech. 15, 1115–1134.Google Scholar
Mizushina, T., Ogino, F., Takeuchi, H. & Ikawa, H. 1982 An experimental study of vertical turbulent jet with negative buoyancy. Wärme-Stoffübertrag. 16, 15–21.Google Scholar
Morton, B. R. 1959 Forced plumes. J. Fluid Mech. 5, 151–163.Google Scholar
Morton, B. R. 1962 Coaxial turbulent jets. Intl J. Heat Mass Transfer 5, 955–965.Google Scholar
Morton, B. R., Taylor, G. & Turner, J. S. 1956 Turbulent gravitational convection from maintained and instantaneous sources. Proc. R. Soc. Lond. A 234, 1–23.Google Scholar
Pantzlaff, L. & Lueptow, R. M. 1999 Transient positively and negatively buoyant turbulent round jets. Exp. Fluids 27, 117–125.Google Scholar
van Reeuwijk, M. & Craske, J. 2015 Energy-consistent entrainment relations for jets and plumes. J. Fluid Mech. 782, 333–355.Google Scholar
Shrinivas, A. B. & Hunt, G. R. 2014 Unconfined turbulent entrainment across density interfaces. J. Fluid Mech. 757, 573–598.Google Scholar
Shy, S. S. 1995 Mixing dynamics of jet interaction with a sharp density interface. Exp. Therm. Fluid Sci. 10 (94), 355–369.Google Scholar
Turner, J. S. 1966 Jets and plumes with negative or reversing buoyancy. J. Fluid Mech. 26 (4), 779–792.Google Scholar
Williamson, N., Armfield, S. W. & Lin, W. 2011 Forced turbulent fountain flow behaviour. J. Fluid Mech. 671, 535–558.Google Scholar
Williamson, N., Srinarayana, N., Armfield, S. W., Mcbain, G. D. & Lin, W. 2008 Low-Reynolds-number fountain behaviour. J. Fluid Mech. 608, 297–317.Google Scholar