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

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.