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Analytical solutions for turbulent Boussinesq fountains in a linearly stratified environment

Published online by Cambridge University Press:  05 December 2011

Rabah Mehaddi ,
Olivier Vauquelin  and
Fabien Candelier
Rabah Mehaddi*
Affiliation:
Laboratoire IUSTI, UMR CNRS 6595, Aix-Marseille Université, 5 rue Enrico Fermi, 13 453 Marseille CEDEX 13, France
Olivier Vauquelin
Affiliation:
Laboratoire IUSTI, UMR CNRS 6595, Aix-Marseille Université, 5 rue Enrico Fermi, 13 453 Marseille CEDEX 13, France
Fabien Candelier
Affiliation:
Laboratoire IUSTI, UMR CNRS 6595, Aix-Marseille Université, 5 rue Enrico Fermi, 13 453 Marseille CEDEX 13, France
*
†Email address for correspondence: rabah.mehaddi@etumel.univmed.fr
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Abstract

This paper theoretically investigates the initial up-flow of a vertical turbulent fountain (round or plane) in a linearly stratified environment. Conservation equations (volume, momentum and buoyancy) are written under the Boussinesq approximation assuming an entrainment proportional to the vertical velocity of the fountain. Analytical integration leads to exact values of both density and flow rate at the maximal height reached by the fountain. This maximal height is expressed as a function of the release conditions and the stratification strength and plotted from a numerical integration in order to exhibit overall behaviour. Then, analytical expressions for the maximal height are derived from asymptotic analysis and compared to experimental correlations available for forced fountains. For weak fountains, these analytical expressions constitute a new theoretical model. Finally, modified expressions are also proposed in the singular case of an initially non-buoyant vertical release.

JFM classification

Convection: Plumes/thermals
Type
Papers
Information
Journal of Fluid Mechanics , Volume 691 , 25 January 2012 , pp. 487 - 497
Copyright
Copyright © Cambridge University Press 2011

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References

1. Abraham, G. 1967 Jets with negative buoyancy in homogeneous fluid. J. Hydraul Res. 5, 235–248.CrossRefGoogle Scholar
2. Baines, W. D., Turner, J. S. & Campbell, I. H. 1990 Turbulent fountains in an open chamber. J. Fluid Mech. 212, 557–592.CrossRefGoogle Scholar
3. Bloomfield, L. J. & Kerr, R. C. 1998 Turbulent fountains in a stratified fluid. J. Fluid Mech. 358, 335–356.CrossRefGoogle Scholar
4. Bloomfield, L. J. & Kerr, R. C. 1999 Turbulent fountains in a confined stratified environment. J. Fluid Mech. 389, 27–54.Google Scholar
5. Bloomfield, L. J. & Kerr, R. C. 2000 A theoritical model of a turbulent fountain. J. Fluid Mech. 424, 197–216.Google Scholar
6. Clanet, C. 1998 On large-amplitude pulsating fountains. J. Fluid Mech. 366, 333–350.CrossRefGoogle Scholar
7. 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
8. Hunt, G. R. & Kaye, N. B. 2005 Lazy plumes. J. Fluid Mech. 533, 329–338.CrossRefGoogle Scholar
9. Kaminski, E., Tait, S. & Carazzo, G. 2005 Turbulent entrainment in jets with arbitrary buoyancy. J. Fluid Mech. 526, 361–376.CrossRefGoogle Scholar
10. Kaye, N. B. 2008 Turbulent plumes in stratified environments: a review of recent work. Atmos.-Ocean 46 (4), 433–441.Google Scholar
11. Kaye, N. B. & Hunt, G. R. 2006 Weak fountains. J. Fluid Mech. 558, 319–328.Google Scholar
12. Lin, W. & Armfield, S. W. 2000 Direct simulation of weak axisymmetric fountains in a homogeneous fluid. J. Fluid Mech. 403, 67–88.Google Scholar
13. Lin, W. & Armfield, S. W. 2002 Weak fountains in stratified fluid. Phys. Rev. E 66, 066308.Google Scholar
14. McDougall, T. J. 1981 Negatively buoyant vertical jets. Tellus 33, 313–320.CrossRefGoogle Scholar
15. Michaux, G. & Vauquelin, O. 2008 Solutions for turbulent buoyant plumes rising from circular sources. Phys. Fluids 20, 066601.CrossRefGoogle Scholar
16. Mizushina, T., Ogino, F., Takeuchi, H. & Ikawa, H. 1982 An experimental study of vertical turbulent jet with negative buoyancy. Warme Stoffubertr. 16, 15–21.Google Scholar
17. Morton, B. R., Taylor, G. I. & Turner, J. S. 1956 Turbulent gravitational convection from maintained and instantaneous sources. Proc. R. Soc. Lond. A 234, 1–23.Google Scholar
18. Pantzlaff, L. & Lueptow, R. M. 1999 Transient positively and negatively buoyant turbulent round jets. Exp. Fluids 27, 117–125.Google Scholar
19. Papanicolaou, P. N., Papakonstantis, I. G. & Christodoulou, G. C. 2008 On the entrainment coefficient in negatively buoyant jets. J. Fluid Mech. 614, 447–470.CrossRefGoogle Scholar
20. Phillipe, P., Raufaste, P., Kurowski, C. & Petitjeans, P. 2005 Penetration of a negatively buoyant jet in a miscible liquid. Phys. Fluids 17, 053601.Google Scholar
21. Turner, J. S. 1966 Jets and plumes with negative or reversing buoyancy. J. Fluid Mech. 26, 779–792.Google Scholar
22. Van den Bremer, T. S. & Hunt, G. R. 2010 Universal solutions for Boussinesq and non-Boussinesq plumes. J. Fluid Mech. 644, 165–192.CrossRefGoogle Scholar
23. Villermaux, E. 1994 Pulsed dynamics of fountains. Nature 371, 24–25.CrossRefGoogle Scholar
24. Williamson, N., Armfield, S. W. & Lin, W. 2011 Forced turbulent fountain flow behaviour. J. Fluid Mech. 671, 535–558.Google Scholar
25. Williamson, N., Srinarayana, N., Armfield, S. W., McBain, G. D. & Lin, W. 2008 Low-Reynolds-number fountain behaviour. J. Fluid Mech. 608, 297–317.CrossRefGoogle Scholar
26. Zhang, H. & Baddour, R. E. 1998 Maximum penetration of vertical round dense jets at small and large Froude numbers. J. Hydraul. Engng ASCE 124, 550–553.CrossRefGoogle Scholar