Hostname: page-component-7c8c6479df-fqc5m Total loading time: 0 Render date: 2024-03-28T21:31:30.468Z Has data issue: false hasContentIssue false

A phenomenological model for fountain-top entrainment

Published online by Cambridge University Press:  28 April 2016

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

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.

Type
Papers
Copyright
© 2016 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bloomfield, L. J. & Kerr, R. C. 2000 A theoretical model of a turbulent fountain. J. Fluid Mech. 424, 197216.CrossRefGoogle Scholar
Burridge, H. C. & Hunt, G. R. 2012 The rise heights of low- and high-Froude-number turbulent axisymmetric fountains. J. Fluid Mech. 691, 392416.CrossRefGoogle 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, 91119.CrossRefGoogle 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, 265284.CrossRefGoogle Scholar
Cetegen, B. M. 1998 A phenomenological model of near-field fire entrainment. Fire Safety J. 31, 299312.CrossRefGoogle 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, 155160.CrossRefGoogle Scholar
Cresswell, R. W. & Szczepura, R. T. 1993 Experimental investigation into a turbulent jet with negative buoyancy. Phys. Fluids 5, 28642878.CrossRefGoogle 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, 75103.CrossRefGoogle Scholar
Friedman, P. D., Vadakoot, V. D., Meyer, W. J. & Carey, S. 2007 Instability threshold of a negatively buoyant fountain. Exp. Fluids 42, 751759.CrossRefGoogle Scholar
Hunt, G. R. & Burridge, H. C. 2015 Fountains in industry and nature. Annu. Rev. Fluid Mech. 47, 195220.CrossRefGoogle Scholar
Hunt, G. R. & Debugne, A. L. R. Forced fountains. J. Fluid Mech. (in press).Google Scholar
Kaye, N. B. & Hunt, G. R. 2006 Weak fountains. J. Fluid Mech. 558, 319328.CrossRefGoogle Scholar
Koh, R. C. Y. & Brooks, N. H. 1975 Fluid mechanics of waste-water disposal in the ocean. Annu. Rev. Fluid Mech. 7, 187211.CrossRefGoogle Scholar
Lin, Y. J. P. & Linden, P. F. 2005 The entrainment due to a turbulent fountain at a density interface. J. Fluid Mech. 542, 2552.CrossRefGoogle Scholar
McDougall, T. J. 1981 Negative buoyant vertical jets. Tellus 33, 313320.CrossRefGoogle Scholar
Mehaddi, R., Vaux, S., Candelier, F. & Vauquelin, O. 2015 On the modelling of steady turbulent fountains. Environ. Fluid Mech. 15, 11151134.CrossRefGoogle Scholar
Mizushina, T., Ogino, F., Takeuchi, H. & Ikawa, H. 1982 An experimental study of vertical turbulent jet with negative buoyancy. Wärme-Stoffübertrag. 16, 1521.CrossRefGoogle Scholar
Morton, B. R., Taylor, G. & Turner, J. S. 1956 Turbulent gravitational convection from maintained and instantaneous sources. Proc. R. Soc. Lond. 234, 123.Google Scholar
Pantzlaff, L. & Lueptow, R. M. 1999 Transient positively and negatively buoyant turbulent round jets. Exp. Fluids 27, 117125.CrossRefGoogle Scholar
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
Shrinivas, A. B. & Hunt, G. R. 2014 Unconfined turbulent entrainment across density interfaces. J. Fluid Mech. 757, 573598.CrossRefGoogle Scholar
Shy, S. S. 1995 Mixing dynamics of jet interaction with a sharp density interface. Exp. Therm. Fluid Sci. 10, 355369.CrossRefGoogle Scholar
Sreenivas, K. R. & Prasad, A. K. 2000 Vortex-dynamics model for entrainment in jets and plumes. Phys. Fluids 12, 21012107.CrossRefGoogle Scholar
Turner, J. S. 1966 Jets and plumes with negative or reversing buoyancy. J. Fluid Mech. 26, 779792.CrossRefGoogle Scholar
Turner, J. S. 1986 Turbulent entrainment: the development of the entrainment assumption, and its application to geophysical flows. J. Fluid Mech. 173, 431471.CrossRefGoogle Scholar
Van Dyke, M. 1982 An Album of Fluid Motion. Parabolic Press.CrossRefGoogle Scholar
Williamson, N., Armfield, S. W. & Lin, W. 2011 Forced turbulent fountain flow behaviour. J. Fluid Mech. 671, 535558.CrossRefGoogle Scholar