The evolution of laminar thermals

We consider the life cycle of an axisymmetric laminar thermal starting from the initial condition of a Gaussian buoyant blob. We find that, as time progresses, the thermal transitions through a number of distinct stages, undergoing several morphological changes before ending up as a vortex ring. Whilst each stage is interesting in its own right, one objective of this study is to set out a consistent mathematical framework under which the entire life cycle can be studied. This allows examination of the transition between the different stages, as well as shedding light on some unsolved questions from previous works. We find that the early stages of formation are key in determining the properties of the final buoyant vortex ring and that, since they occur on a time scale where viscosity has little effect, the final properties of the ring display an independence above a critical Reynolds number. We also find that rings consistently contain the same proportion of the initial heat and have a consistent vorticity flux. By considering the effect of Prandtl number, we show that thermal diffusion can have a significant impact on development, smoothing out the temperature field and inhibiting the generation of vorticity. Finally, by considering the wake left behind as well as the vortex ring that is generated, we observe that the wake can itself roll up to form a second mushroom cap and subsequently a secondary vortex ring that follows the first.