From PDEs to Scaling Laws: Quantitative Analysis on Turbulent Convection

Abstract

In this seminar, we will explore the derivation and significance of scaling laws within the context of Rayleigh-Bénard convection, a key problem in fluid dynamics that is essential for understanding geophysical flows in the atmosphere, oceans, and various industrial applications. Scaling laws offer a framework for characterizing how crucial quantities, such as heat transport, behave in extreme parameter regimes that cannot be reached through direct experimental methods.
Our focus will be on the Nusselt number Nu, which quantifies the efficiency of heat transport compared to pure conduction, and its relationship with the Rayleigh number Ra, which measures the thermal driving force of the system. We will review recent developments on rigorous upper bounds for Nu under various boundary conditions in both flat and rough domains.
A key part of this seminar will involve presenting simplifications and refinements of the techniques first introduced by Doering and Constantin in their seminal works from the 1990s. By refining their approach, we not only simplify the proofs but also obtain improved bounds on Nu, which shed light on the fundamental limits of heat transport in convective flows.
 

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