Simulation of laminar axisymmetric buoyant plumes: A case study of the anatomy of similarity solutions

This article is concerned with the numerical simulation of the laminar buoyant plume created by a heated sphere situated in an otherwise quiescent environment. The limit of vanishing sphere diameter corresponds to the point source of heat, for which extensive results have been obtained in the literature. The dual objectives of this work are the attainment of an exact characterization of the buoyant plume, and the diagnosis of an existing similarity solution for the limiting problem of the point source. These objectives were fulfilled by means of a numerically exact solution of the equations representing mass, momentum, and energy conservation. The solutions were carried out over a range of the radius-based Grashof number extending from 50 to 5106 for a Prandtl number of 0.7, which corresponds to air. The velocities in the plume were found to increase with elevation above the sphere but approach a fully developed state characterized by congruent (similar) velocity profiles at sufficiently high elevations. The temperature profiles, while attaining a self-similar shape at high elevations, actually decay with increasing elevation. The approach to both the fully developed state for the velocity and the self-similar state for the temperature occurs more rapidly at low Grashof numbers. In general, it was found that the existing similarity solutions become viable only at very large elevations above the source of buoyancy.