Super-ultimate transient of turbulent thermal convection between horizontal porous walls

Turbulent thermal convection between horizontal porous walls heated from below and cooled from above has been investigated numerically and experimentally. It has been found that at low Rayleigh number Ra, the Nusselt number Nu in the bulk region scales with Ra^{1/3} as in the classical state, whereas at high Ra, Nu scales with Ra^{1/2}, implying the ultimate state in which the vertical heat flux is independent of thermal conductivity, i.e., the so-called conduction anomaly. At low Ra, vertical (wall-normal) fluid motion is not excited in the near-wall region despite wall permeability, so that the classical state can be observed. At high Ra, large-scale thermal plumes appear even near the walls from convective instabilities of near-wall thermal conduction layers to significantly intensify the wall heat flux, leading to the ultimate state. In between these two distinct scaling ranges of Ra, we have found super-ultimate transient in which Nu ~ Ra. This super-ultimate scaling is considered to be a consequence of full excitation of large-scale thermal plumes comparable with those in the ultimate state at high Ra and of less energy dissipation in the flow through porous media than in the ultimate state at high Ra.