Velocity-informed upper bounds on the convective heat transport induced by internal heat sources and sinks

TL;DR

This paper establishes mathematical upper bounds on heat transport in internally heated convection, showing that turbulent solutions can reach the maximal scaling of Nu ~ sqrt(Ra), supported by numerical and experimental data.

Contribution

It provides a rigorous mathematical framework for understanding heat transport limits in internally heated convection and identifies conditions under which maximal heat transport scaling is achieved.

Findings

Maximal Nusselt number scales as sqrt(Ra) for turbulent solutions.

Numerical and experimental data support the theoretical bounds.

Turbulent convection can attain the upper-bound heat transport scaling.

Abstract

Three-dimensional convection driven by internal heat sources and sinks (CISS) leads to experimental and numerical scaling-laws compatible with a mixing-length - or `ultimate' - scaling regime $Nu \sim \sqrt{Ra}$. However, asymptotic analytic solutions and idealized 2D simulations have shown that laminar flow solutions can transport heat even more efficiently, with $Nu \sim Ra$. The turbulent nature of the flow thus has a profound impact on its transport properties. In the present contribution we give this statement a precise mathematical sense. We show that the Nusselt number maximized over all solutions is bounded from above by const.$\times Ra$, before restricting attention to 'fully turbulent branches of solutions', defined as families of solutions characterized by a finite nonzero limit of the dissipation coefficient at large driving amplitude. Maximization of $Nu$ over such branches of solutions yields the better upper-bound $Nu \lesssim \sqrt{Ra}$. We then provide 3D numerical and experimental data of CISS compatible with a finite limiting value of the dissipation coefficient at large driving amplitude. It thus seems that CISS achieves the maximal heat transport scaling over fully turbulent solutions.