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

High-wavenumber steady solutions of two-dimensional Rayleigh--B\'enard convection between stress-free boundaries