Ultimate heat transfer in `wall-bounded' convective turbulence

Direct numerical simulations have been performed for turbulent thermal convection between horizontal no-slip, permeable walls with a distance $H$ and a constant temperature difference $\Delta T$ at the Rayleigh number $Ra=3\times10^{3}-10^{10}$. On the no-slip wall surfaces $z=0$, $H$ the wall-normal (vertical) transpiration velocity is assumed to be proportional to the local pressure fluctuation, i.e. $w=-\beta p'/\rho, +\beta p'/\rho$ (Jim\'enez et al., J. Fluid Mech., vol. 442, 2001, pp. 89-117), and the property of the permeable wall is given by the permeability parameter $\beta U$ normalised with the buoyancy-induced terminal velocity $U={(g\alpha\Delta TH)}^{1/2}$, where $\rho$, $g$ and $\alpha$ are mass density, acceleration due to gravity and volumetric thermal expansivity, respectively. A zero net mass flux through the wall is instantaneously ensured, and thermal convection is driven only by buoyancy without any additional energy inputs. The critical transition of heat transfer in convective turbulence has been found between the two $Ra$ regimes for fixed $\beta U=3$ and fixed Prandtl number $Pr=1$. In the subcritical regime at lower $Ra$ the Nusselt number $Nu$ scales with $Ra$ as $Nu\sim Ra^{1/3}$, as commonly observed in turbulent Rayleigh-B\'enard convection. In the supercritical regime at higher $Ra$, on the other hand, the ultimate scaling $Nu\sim Ra^{1/2}$ is achieved, meaning that the wall-to-wall heat flux scales with $U\Delta T$ independent of the thermal diffusivity, although the heat transfer on the wall is dominated by thermal conduction. The physical mechanisms of the achievement of the ultimate heat transfer are presented.