Multi-scale steady solutions representing classical and ultimate scaling in thermal convection

Rayleigh–Bénard convection is one of the most canonical flows widely observed in nature and engineering applications. The effect of buoyancy on a flow is characterised by the Rayleigh number Ra, and the flow becomes turbulent eventually as Ra increases. One of the primary interests in convective turbulence is the scaling law of the Nusselt number Nu (dimensionless vertical heat flux) with Ra. A one-third power law for Nu with Ra, referred to as the 'classical' scaling, has been reported in many experiments and numerical simulations. On the other hand, a one-half power law, referred to as the 'ultimate' scaling, has not been observed yet in conventional Rayleigh–Bénard convection (buoyancy-driven convection between horizontal impermeable walls with a constant temperature difference). In this talk, I will first discuss a multi-scale steady solution in the conventional Rayleigh–Bénard convection. It is a three-dimensional steady solution to the Boussinesq equations, found using a homotopy from the wall-to-wall optimal transport solution (Motoki et al. 2018 J. Fluid Mech., 851, R4). The exact coherent thermal convection exhibits the classical scaling and reproduces structural and statistical properties of convective turbulence. Next, I will draw attention to thermal convection between permeable walls. The permeable wall is a simple model mimicking a Darcy-type porous wall (Jiménez et al. 2001 J. Fluid Mech. 442, 89-117). The wall permeability leads to the ultimate scaling, meaning that a wall heat flux being independent of thermal conductivity, although the heat transfer on the wall is dominated by thermal conduction. Finally, I will discuss the physical mechanisms of classical and ultimate scaling.