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

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 t(1/2) as time t increases. The dimensionless problem depends on a single parameter Ra = B/(v kappa) - a Rayleigh number - based on the (conserved) total buoyancy B of the thermal, and the kinematic viscosity v and thermal diffusivity kappa 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 (kappa t)(1/2), while the wake length and rise height both increase like (Ra ln Ra)(1/2) (kappa t)(1/2), consistent with the numerical results. Previous experiments suggest that there is a significant transient regime.