A central question about Rayleigh–Bénard convection is how the Nusselt number (Nu) depends on the Rayleigh number (Ra) – i.e., how heat flux depends on imposed temperature gradient – in the turbulent limit of asymptotically large Ra. Experiments and simulations have yet to rule out either 'classical' $Ra^{1/3}$ or 'ultimate' $Ra^{1/2}$ asymptotic scaling of Nu. This talk concerns the asymptotic scaling of Nu for steady convection rolls, which we have computed numerically for both no-slip and stress-free boundary conditions on the velocity. These 2D rolls bifurcate from the static state at finite Ra in a linear instability identified by Lord Rayleigh, and they continue to exist at asymptotically large Ra but are unstable. For both velocity boundary conditions, scaling that appears to be asymptotic emerges when Ra is large, and Nu achieves classical 1/3 scaling. In the stress-free case the 1/3 exponent is achieved by rolls of any fixed horizontal period, and the solutions agree very well with prior asymptotic theory. In the no-slip case the 1/3 exponent is achieved only by rolls whose horizontal periods are chosen to maximize Nu at each Ra, which requires their periods to decrease like $Ra^{-1/5}$. Remarkably, the steady rolls that achieve the 1/3 exponent transport more heat than turbulent experiments or simulations at comparable parameters. If turbulent transport continues to be dominated by steady transport asymptotically, then Rayleigh–Bénard convection cannot achieve ultimate scaling. These observations suggest two mathematical conjectures. This is joint work with Baole Wen, Charles R. Doering, and Gregory P. Chini.
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