The flow into an expanding spherical vortex

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Abstract
A kinematic model of the flow around and into a buoyant ‘thermal’ is discussed in some detail, and compared with existing laboratory observations. The basic assumption is that the flow is instantaneously the same as it is for Hill's spherical vortex of fixed size moving through a frictionless fluid. The equations describing the motion of a particle have been modified to allow for an expansion in radius proportional to the distance travelled, and a numerical integration of these time-dependent equations has been performed in order to find particle trajectories, with respect to both the instantaneous boundary of the spherical vortex and axes at rest. It is shown that for the expanding vortex there is no quantity corresponding to the ‘drift distance’ or the total forward displacement of particles in the flow round a sphere of constant size; particles in the wake of an expanding vortex have a finite forward velocity at large times.The model gives very close agreement with the observed behaviour of particles entering a turbulent buoyant thermal, except that in the laboratory measurements the region containing the turbulent fluid resembles an oblate spheroid rather than a sphere. The thin mixing layer over the front and the addition of fluid over a broad region at the rear of a thermal, as well as the detailed particle trajectories, can all be explained as consequences of this mean velocity distribution.

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