Similarity solutions of thermal vortex rings: Vorticity–dynamics-based derivation

Abstract

This study derives the classical similarity solutions for the thermal vortex ring, originally deduced by Scorer [J. Fluid Mech. 2, 583–594 (1957)] based on dimensional analysis from basic principles of vorticity dynamics. Derivations are performed by two methods: (1) by completing the impulse-dynamics analysis initiated by Turner [Proc. R. Soc. London, Ser. A 239, 61–75 (1957)] and (2) by a method of separation of variables. The former analysis elucidates various conditions (e.g., absence of buoyancy along the vortex-ring vertical axis), whereas the latter approach leads to the similarity solution only if one assumes that both the potential vorticity and the buoyancy evolve in a similar manner by sharing the same dependency on the vortex-ring size. It suggests a potentially richer behavior of a thermal vortex ring than predicted by similarity solutions.