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 [Formula: see text]. However, asymptotic analytic solutions and idealized two-dimensional simulations have shown that laminar flow solutions can transport heat even more efficiently, with [Formula: see text]. 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 [Formula: see text], before restricting attention to 'fully turbulent branches of solutions', defined as families of solutions characterized by a finite non-zero limit of the dissipation coefficient at large driving amplitude. Maximization of [Formula: see text] over such branches of solutions yields the better upper-bound [Formula: see text]. We then provide three-dimensional 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. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 1)'.
Keywords: thermal convection; turbulence; upper bounds.