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

Whittaker, Robert J. ORCID: https://orcid.org/0000-0001-9598-5938 and Lister, John R. (2008) The self-similar rise of a buoyant thermal in very viscous flow. Journal of Fluid Mechanics, 606. pp. 295-324. ISSN 0022-1120

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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/(??) – a Rayleigh number – based on the (conserved) total buoyancy B of the thermal, and the kinematic viscosity ? and thermal diffusivity ? 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 (?t)1/2, while the wake length and rise height both increase like (RalnRa)1/2(?t)1/2, consistent with the numerical results. Previous experiments suggest that there is a significant transient regime.

Item Type:	Article
Faculty \ School:	Faculty of Science > School of Mathematics
UEA Research Groups:	Faculty of Science > Research Groups > Fluid and Solid Mechanics
Depositing User:	Robert Whittaker
Date Deposited:	10 Mar 2011 12:48
Last Modified:	26 Jan 2023 11:30
URI:	https://ueaeprints.uea.ac.uk/id/eprint/20896
DOI:	10.1017/S0022112008001730

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