Plume dynamics across scales: from laminar flow, instability to turbulent flow

Buoyant plumes generated from localised sources arise throughout the natural world and across a wide variety of scales. The Prandtl number (Pr) in particular varies from ~10^{-4} in planetary applications to ~10^{20} in geological flows. We use direct numerical simulation based on a spectral element method combined with scaling/asymptotic analysis to investigate the full life cycle of evolving plumes in both unstratified and stratified environments, spanning initially laminar to turbulent flow. We thereby explore the general effects of viscosity and thermal diffusivity on the dynamics of plumes. Our numerical simulations reveal characteristics of plumes without reference to simplified models, thereby allowing us to scrutinise these simplified descriptions of plume dynamics. Further, they yield insight into the conditions necessary for direct numerical solutions (typically limited to Re < O(10⁴)) to effectively simulate flow in large-scale phenomena such as volcanic eruptions (where Re ~ O(10¹²)).

By considering intrinsic scales derived directly from the Navier-Stokes equations, we find that all plumes can be fully classified in terms of a spectrum of Pr alone. Through a systematic analysis of solutions across this spectrum, we determine the transition length on which laminar to turbulent flow occurs, separating the plume into three regions - the laminar, the turbulent and an intervening transition zone - and explore the relative size and growth rate of these regions.

In the stratified case, an additional dimensionless number is introduced, Re, representing the relative importance of viscosity. Remarkably, the rise height of a plume varies non-monotonically with this parameter, reaching a global maximum for Re ≈ 700. As Re is increased beyond this maximal value, the rise height decreases slowly towards an asymptote of 4.3 as Re → ∞, recovering the limit in which the plume becomes independent of both viscosity and thermal diffusivity.