If there is a difference in temperature between bulk fluid and a wall surface in wall-bounded turbulent flows, heat will be transferred between the fluid and the wall. Such heat transfer is dominated by thermal conduction on the wall where turbulent heat flux is null, although it highly depends on turbulence characteristics. In this talk, turbulent heat transfer in wall-bounded thermal convection and shear flow is discussed with emphasis on the so-called ultimate state in which a wall heat flux is independent of thermal diffusivity, i.e. conduction anomaly (or anomalous scalar dissipation), while energy dissipation is independent of kinematic viscosity, i.e. the Taylor dissipation law implying inertial energy dissipation or anomalous energy dissipation. The classical scaling widely observed in turbulent Rayleigh-Bénard convection is first reviewed to differentiate the ultimate state from the classical state. Feasibility of the ultimate heat transfer is then explored numerically. Wall permeability, which can be implemented on a porous wall, is introduced in Rayleigh-Bénard convection. It is found that in thermal convection between the horizontal permeable walls, the ultimate heat transfer can be achieved at high Rayleigh numbers. We discuss the reason why wall permeability can lead to the ultimate scaling in wall-bounded convective turbulence. We further pursue the ultimate heat transfer numerically in turbulent channel flow by introducing the wall permeability. The ultimate heat transfer can be accomplished even in shear flow between the parallel permeable walls at high Reynolds numbers. It is also demonstrated that the ultimate heat transfer can be achieved in realistic configurations by numerical simulation and experiment of turbulent thermal convection between horizontal porous walls. At low Rayleigh numbers, 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 Rayleigh numbers, however, large-scale thermal plumes appear even near the walls from convective instabilities of near-wall thermal conduction layers to intensify the vertical heat flux, leading to the ultimate state. In between these two distinct scaling ranges of the Rayleigh number, we have found super-ultimate behaviour represented by the higher value of the scaling exponent of the heat flux than that in the ultimate state. 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 and of less energy dissipation in the flow through porous walls than in the ultimate state at the high Rayleigh numbers.

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