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Unsteady turbulent plume models

Published online by Cambridge University Press:  12 March 2012

M. M. Scase  and
R. E. Hewitt
M. M. Scase*
Affiliation:
Faculty of Engineering, University of Nottingham, Nottingham NG7 2RD, UK
R. E. Hewitt
Affiliation:
School of Mathematics, The University of Manchester, Oxford Road, Manchester M13 9PL, UK
*
†Email address for correspondence: matthew.scase@nottingham.ac.uk
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Abstract

Four existing integral models of unsteady turbulent plumes are revisited. We demonstrate that none of these published models is ideal for general descriptions of unsteady behaviour and put forward a modified model. We show that the most recent (top-hat) plume model (Scase et al. J. Fluid Mech., vol. 563, 2006, p. 443), and the earlier (Gaussian) plume models (Delichatsios J. Fluid Mech., vol. 93, 1979, p. 241; Yu Trans. ASME, vol. 112, 1990, p.186), are all ill-posed. This ill-posedness arises from the downstream growth of short-scale waves, which have an unbounded downstream growth rate. We show that both the top-hat and the Gaussian (Yu) models can be regularized, rendering them well-posed, by the inclusion of a velocity diffusion term. The effect of including this diffusive mechanism is to include a vertical structure in the model that can be interpreted as representing the vertical extent of an eddy. The effects of this additional mechanism are small for steady applications, and cases where the plume forcing can be considered to follow a power law (both of which have been studied extensively). However, the inclusion of diffusion is shown to be crucial to the general initial-value problem for unsteady models.

JFM classification

Convection: Plumes/thermals
Type
Papers
Information
Journal of Fluid Mechanics , Volume 697 , 25 April 2012 , pp. 455 - 480
Copyright
Copyright © Cambridge University Press 2012

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