Steady Rayleigh--Bénard convection between no-slip boundaries

Abstract

The central open question about Rayleigh--Bénard convection -- buoyancy-driven flow in a fluid layer heated from below and cooled from above -- is how vertical heat flux depends on the imposed temperature gradient in the strongly nonlinear regime where the flows are typically turbulent. The quantitative challenge is to determine how the Nusselt number $Nu$ depends on the Rayleigh number $Ra$ in the $Ra\to \infty$ limit for fluids of fixed finite Prandtl number $Pr$ in fixed spatial domains. Laboratory experiments, numerical simulations, and analysis of Rayleigh's mathematical model have yet to rule out either of the proposed `classical' $Nu\sim R{a}^{1/3}$ or `ultimate' $Nu\sim R{a}^{1/2}$ asymptotic scaling theories. Among the many solutions of the equations of motion at high $Ra$ are steady convection rolls that are dynamically unstable but share features of the turbulent attractor. We have computed these steady solutions for $Ra$ up to $10^{14}$ with $Pr=1$ and various horizontal periods. By choosing the horizontal period of these rolls at each $Ra$ to maximize $Nu$, we find that steady convection rolls achieve classical asymptotic scaling. Moreover, they transport more heat than turbulent convection in experiments or simulations at comparable parameters. If heat transport in turbulent convection continues to be dominated by heat transport in steady rolls as $Ra\to \infty$, it cannot achieve the ultimate scaling.

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Cited by in corpus (6)

• Analytical bounds on the heat transport in internally heated convection
• On high Taylor number Taylor vortices
• Rigorous scaling laws for internally heated convection at infinite Prandtl number
• Time averages and periodic attractors at high Rayleigh number for Lorenz-like models
• Rotating Rayleigh-Benard convection: Attractors, bifurcations and heat transport via a Galerkin hierarchy
• Enhancing wall-to-wall heat transport with unsteady flow perturbations