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The self-similar rise of a buoyant thermal in very viscous flow

Published online by Cambridge University Press:  10 July 2008

ROBERT J. WHITTAKER  and
JOHN R. LISTER
ROBERT J. WHITTAKER
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK
JOHN R. LISTER
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK
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Abstract

An exact similarity solution is obtained for the rise of a buoyant thermal in Stokes flow, in which both the rise height and the diffusive growth scale like t1/2 as time t increases. The dimensionless problem depends on a single parameter Ra = B/(νκ) – a Rayleigh number – based on the (conserved) total buoyancy B of the thermal, and the kinematic viscosity ν and thermal diffusivity κ of the fluid. Numerical solutions are found for a range of Ra. For small Ra there are only slight deformations to a spherically symmetric Gaussian temperature distribution. For large Ra, the temperature distribution becomes elongated vertically, with a long wake containing most of the buoyancy left behind the head. Passive tracers, however, are advected into a toroidal structure in the head. A simple asymptotic model for the large-Ra behaviour is obtained using slender-body theory. The width of the thermal is found to increase like (κt)1/2, while the wake length and rise height both increase like (RalnRa)1/2(κt)1/2, consistent with the numerical results. Previous experiments suggest that there is a significant transient regime.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 606 , 10 July 2008 , pp. 295 - 324
Copyright
Copyright © Cambridge University Press 2008

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