Steady thermal convection representing the ultimate scaling

Nonlinear simple invariant solutions representing the ultimate scaling have been discovered to the Navier- Stokes equations for thermal convection between horizontal no-slip permeable walls with a distance H and a constant temperature difference triangle T. On the permeable walls, the vertical transpiration velocity is assumed to be proportional to the local pressure fluctuations, i.e. w = +/-beta p/rho (Jimenez et al. 2001 J. Fluid Mech., 442, 89-117. (doi:10.1017/S0022112001004888)). Two-dimensional steady solutions bifurcating from a conduction state have been obtained using a Newton- Krylov iteration up to the Rayleigh number Ra -10(8) for the Prandtl number Pr =1, the horizontal period L/H = 2 and the permeability parameter beta U = 0-3, U being the buoyancy-induced terminal velocity. The wall permeability has a significant impact on the onset and the scaling properties of the found steady "wall bounded' thermal convection. The ultimate scaling Nu similar to Ra1/2 has been observed for beta U > 0 at high Ra, where Nu is the Nusselt number. In the steady ultimate state, large-scale thermal plumes fully extend from one wall to the other, inducing the strong vertical velocity comparable with the terminal velocity U as well as intense temperature variation of O(triangle T) even in the bulk region. As a consequence, the wall-to-wall heat flux scales with U triangle T independent of thermal diffusivity, although the heat transfer on the walls is dominated by thermal conduction. This article is part of the theme issue "Mathematical problems in physical fluid dynamics (part 1)'.