Abstract: Thermal stratification plays a profound role in a wide range of engineering and environmental contexts, with applications including reservoir management, climate prediction, thermal comfort and energy storage. In buildings, in particular, thermal stratification has major ramifications for the transport of heat and contaminant/pathogen transport in rooms and therefore affects energy usage, health and thermal comfort. While it may be possible to discern certain features of a stratification profile for some canonical flows, heating/cooling that is heterogeneous (such as a combinations of point and distributed sources) and uncertainty makes this a difficult task in the context of buildings. More generally, the underlying transport and mixing mechanisms responsible for the production, maintenance or destruction of thermal stratifications, even for canonical flows, remain poorly understood.

In this talk we will address the challenge of predicting and diagnosing the physics behind thermal stratifications by embedding the problem within a probabilistic framework. In doing so, we derive a Fokker-Planck-like equation for the (linear) evolution of the joint probability density associated with several key quantities, from which information about a flow’s energetics and thermal stratification are readily obtained as functionals. This approach will be first applied to the Lorenz system, a reduced order model of the full Boussinesq equations, which nevertheless retain some essential features of the higher dimensional system. Subsequently we will sketch out the application of this framework to the full Boussinesq equations and then conclude by providing some preliminary results.