Unifying laminar and turbulent dynamics of plumes

Buoyant plumes are widespread physical phenomena, present across a large range of scales, from the plume generated by a candle to a mantle plume beneath the crust of the Earth, and spanning laminar to fully turbulent flow. In its most basic form, a plume is the vertical transport of heat emanating from a buoyancy source. Despite the ubiquity of plumes and their role in heat and mass transport, many aspects of their dynamics are still not fully understood.

In the literature, plumes are often separated into laminar and turbulent regimes and are yet to be combined into a cohesive theory of plume evolution and structure. In this thesis, I perform a theoretical and numerical analysis of pure plumes generated by a localised point source of heat in order to unify these separate theories in a description of a hybrid laminar-turbulent plume. The governing equations for a plume are shown to be dependent on only one dimensionless parameter, the Prandtl number, which is the ratio of kinematic viscosity to thermal diffusivity. Using direct numerical simulations (DNS) combined with scaling analysis, I establish some universal properties of pure plumes. Following the laminar regime, the height at which instability occurs, hereafter referred to as height to instability, for the case Pr = 1, is found. The turbulent regime is described using similarity scalings and a virtual origin (differing from the height to instability) is found, implying that the transition region exists over a nonzero spatial range.

Expanding on this work, further DNS are undertaken to derive a unified laminar-turbulent theory of a plume in an unstratified environment emanating from a point source over a range of Prandtl numbers. The height to instability is found to increase sublinearly with Prandtl number and a formula for interpolating the height to instability within the range Pr = 0.1 to 2.0 is fitted to the results.

My investigation into developing a unified laminar-turbulent theory of plume rise from a point source is further expanded by introducing a linear stratification to the background. This introduces another length scale to the problem and the relationship between height to instability and maximum rise height is investigated. I performed DNS of plumes across a wide spectrum of Reynolds numbers and developed a simplified theory of low-Re plumes, resulting in excellent agreement with the predictions of the DNS. Interestingly, the results demonstrate a non-monotonic relationship between rise height and Reynolds number, with a global maximum at Reynolds number is approximately 1500.

Finally, the unstratified results are applied to the transport (e.g. of pathogens) in the built environment by local sources of heat. A hybrid laminar-turbulent theory of particle transport in rooms is developed using the results of height to instability. Remarkably, a strongly non-monotonic relationship between transport rates of particles and the initial buoyancy flux, with two turning points, is discovered.