Heat-transport-maximizing steady roll solutions in Rayleigh–Bénard convection

Steady two-dimensional convection rolls can transport heat at nearly the same rate as turbulent flows in Rayleigh–Bénard convection, despite the roll solutions being unstable at large Rayleigh numbers (Ra). This talk concerns the large-Ra asymptotic properties of steady roll solutions at various Prandtl numbers (Pr) between no-slip top and bottom boundaries. Waleffe et al. (Phys. Fluids 2015) and Sondak et al. (J. Fluid Mech. 2015) examined how the Nusselt number (Nu) of steady rolls depends on their horizontal period. At large Ra, they report two local maxima in Nu as a function of the period. Based on computations up to Ra ~ 10⁹, they find that the global maximum of Nu occurs at the larger-period local maximum when Pr is less than approximately 7, but at the smaller-period local maximum when Pr is greater than about 7. Using new numerical methods, we compute steady roll solutions up to Ra values larger than 10¹⁴ for various Pr ≥ 1. Our computations seem to reach the large-Ra asymptotic regime, where there remain two local maxima of Nu, but where the global maximum of Nu always occurs at the larger-period local maximum for all Pr. These Nu-maximizing steady rolls achieve Nu ~ c Ra^1/3—the so-called classical scaling. The large-Ra asymptotic structure of these Nu-maximizing roll solutions will be discussed.