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.