Steady Rayleigh-Benard convection between stress-free boundaries

Steady two-dimensional Rayleigh-Benard convection between stress-free isothermal boundaries is studied via numerical computations. We explore properties of steady convective rolls with aspect ratios pi/5 <= G <= 4 pi, where Gamma is the width-to-height ratio for a pair of counter-rotating rolls, over eight orders of magnitude in the Rayleigh number, 10(3) <= Ra <= 10(11), and four orders of magnitude in the Prandtl number, 10(-2) <= Pr <= 10(2). At large Ra where steady rolls are dynamically unstable, the computed rolls display Ra -> infinity asymptotic scaling. In this regime, the Nusselt number Nu that measures heat transport scales as Ra-1/3 uniformly in Pr. The prefactor of this scaling depends on Gamma and is largest at Gamma approximate to 1.9. The Reynolds number Re for large-Ra rolls scales as (Pr-1Ra2/3) with a prefactor that is largest at Gamma approximate to 4.5. All of these large-Ra features agree quantitatively with the semi-analytical asymptotic solutions constructed by Chini & Cox (Phys. Fluids, vol. 21, 2009, 083603). Convergence of Nu and Re to their asymptotic scalings occurs more slowly when Pr is larger and when Gamma is smaller.