Numerical computation of strongly nonlinear large-wavenumber steady states in Rayleigh–Bénard convection*

Recent investigations confirm that steady convective flows share many structural features with turbulent Rayleigh–Bénard convection (RBC) and organize the 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 large-wavenumber regime. In this study, we numerically explore the large-wavenumber steady convection roll solutions that bifurcate supercritically from the motionless conductive state for 2D RBC between stress-free boundaries. Our new computations confirm the existence of a local heat-flux-maximizing solution in the large-wavenumber regime. To elucidate the asymptotic properties of this solution, we perform computations over eight orders of magnitude in the Rayleigh number, 10⁸ ≤ Ra ≤ 10^16.5, and two orders of magnitude in the Prandtl number, 10^-1 ≤ Pr ≤ 10^3/2. The numerical results indicate that as Ra → ∞, the local heat-flux-maximizing aspect ratio Γ_loc* ~ Ra^-1/4, the Nusselt number Nu(Γ_loc*) ~ Ra^3/10, and the Péclet number Re(Γ_loc*)Pr ~ Ra^2/5, where Re is the Reynolds number. Moreover, we demonstrate that the interior flow is accurately described by an analytical heat-exchanger solution, and discuss the connection to the large-wavenumber asymptotic solution given by Blennerhassett & Bassom (IMA J. Appl. Math., 1994). With a fixed aspect ratio 0.06 ≤ Γ ≤ π/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 (Phys. Fluids, 2009), with scalings Nu ~ Ra^1/3 and RePr ~ Ra^2/3. Finally, we construct a phase diagram to demarcate distinct regimes of steady solutions in the large-Rayleigh-number-wavenumber plane.