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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: [email protected]
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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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