Basile Gallet - Commissariat à l'Énergie Atomique (CEA), Condensed matter physics Decades of investigation of the Rayleigh-Bénard (RB) thermal convection setup indicate that the heat transport is strongly restricted by boundary layers near the hot and cold solid plates. This prevents the observation of the 'ultimate' scaling-regime of thermal convection, where bulk turbulence controls the convective heat flux independently of molecular diffusivities. In contrast to the RB setup, many geophysical and astrophysical convective flows are driven radiatively: absorption of incoming light by a body of fluid induces local internal heating. We have developed a laboratory experiment that reproduces such radiative heating: heat is input radiatively, directly inside the bulk turbulent flow and away from the boundary layers. After providing experimental and numerical evidence that this setup naturally leads to the ultimate regime of thermal convection, I will discuss the maximum theoretical Nusselt number that can be achieved by such internally heated and cooled convection. I will show that there exist steady laminar solutions that transport heat more efficiently than the ultimate regime, with a scaling behavior Nu~Ra. These solutions can be stable in 2D, but they are unstable in 3D and quickly evolve into a turbulent state. I will show that a maximization of the heat transport over turbulent flows only (i.e., over flows that satisfy the zeroth law of turbulence) leads to an upper bound on the Nusselt number that is proportional to the square root of the Nusselt number, in line with the experimental and numerical data.