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

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.