High-wavenumber steady solutions of two-dimensional Rayleigh--Bénard convection between stress-free boundaries

Recent investigations show that steady solutions share many features with turbulent Rayleigh–Bénard convection (RBC) and form the state space skeleton of turbulent dynamics. Previous computations of steady roll solutions in two-dimensional (2D) RBC between no-slip boundaries reveal that for fixed Rayleigh number $Ra$ and Prandtl number $Pr$, the heat-flux-maximizing solution is always in the high-wavenumber regime. In this study, we explore the high-wavenumber steady convection roll solutions that bifurcate supercritically from the motionless conductive state for 2D RBC between stress-free boundaries. Our computations confirm the existence of a local heat-flux-maximizing solution in the high-wavenumber regime. To elucidate the asymptotic properties of this solution, we perform computations over eight orders of magnitude in the Rayleigh number, ${10}^8\le Ra\le {10}^{16.5}$, and two orders of magnitude in the Prandtl number, ${10}^{-1}\le Pr\le {10}^{3\mathrm{/}2}$. The numerical results indicate that as $Ra\to \mathrm{\infty }$, the local heat-flux-maximizing aspect ratio ${\mathrm{\Gamma }}_{loc}^{\ast }\simeq R{a}^{-1\mathrm{/}4}$, the Nusselt number $Nu({\mathrm{\Gamma }}_{loc}^{\ast })\simeq R{a}^{0.29}$, and the Reynolds number $Re({\mathrm{\Gamma }}_{loc}^{\ast })\simeq P{r}^{-1}R{a}^{2\mathrm{/}5}$, with all prefactors depending on $Pr$. Moreover, the interior flow of the local $Nu$-maximizing solution can be well described by an analytical heat-exchanger solution, and the connection to the high-wavenumber asymptotic solution given by Blennerhassett & Bassom is discussed. With a fixed aspect ratio $0.06\le \mathrm{\Gamma }\le \pi \mathrm{/}5$ at $Pr=1$, however, our computations show that as $Ra$ increases, the steady rolls converge to the semi-analytical asymptotic solutions constructed by Chini & Cox, with scalings $Nu\sim R{a}^{1\mathrm{/}3}$ and $Re\sim P{r}^{-1}R{a}^{2\mathrm{/}3}$. Finally, a phase diagram is delineated to gain a panorama of steady solutions in the high-Rayleigh-number-wavenumber plane.