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

philosophical transactions - royal society. mathematical, physical and engineering sciences/philosophical transactions - royal society. mathematical, physical and engineering sciences

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