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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’ $$Nu \sim Ra^{1/3}$$ or ‘ultimate’ $$Nu \sim Ra^{1/2}$$ 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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The Nusselt numbers of horizontal convection
In the problem of horizontal convection a non-uniform buoyancy, $$b_{s}(x,y)$$ , is imposed on the top surface of a container and all other surfaces are insulating. Horizontal convection produces a net horizontal flux of buoyancy, $$\boldsymbol{J}$$ , defined by vertically and temporally averaging the interior horizontal flux of buoyancy. We show that $$\overline{\boldsymbol{J}\boldsymbol{\cdot }\unicode[STIX]{x1D735}b_{s}}=-\unicode[STIX]{x1D705}\langle |\unicode[STIX]{x1D735}b|^{2}\rangle$$ ; the overbar denotes a space–time average over the top surface, angle brackets denote a volume–time average and $$\unicode[STIX]{x1D705}$$ is the molecular diffusivity of buoyancy $$b$$ . This connection between $$\boldsymbol{J}$$ and $$\unicode[STIX]{x1D705}\langle |\unicode[STIX]{x1D735}b|^{2}\rangle$$ justifies the definition of the horizontal-convective Nusselt number, $Nu$ , as the ratio of $$\unicode[STIX]{x1D705}\langle |\unicode[STIX]{x1D735}b|^{2}\rangle$$ to the corresponding quantity produced by molecular diffusion alone. We discuss the advantages of this definition of $Nu$ over other definitions of horizontal-convective Nusselt number. We investigate transient effects and show that $$\unicode[STIX]{x1D705}\langle |\unicode[STIX]{x1D735}b|^{2}\rangle$$ equilibrates more rapidly than other global averages, such as the averaged kinetic energy and bottom buoyancy. We show that $$\unicode[STIX]{x1D705}\langle |\unicode[STIX]{x1D735}b|^{2}\rangle$$ is the volume-averaged rate of Boussinesq entropy production within the enclosure. In statistical steady state, the interior entropy production is balanced by a flux through the top surface. This leads to an equivalent ‘surface Nusselt number’, defined as the surface average of vertical buoyancy flux through the top surface times the imposed surface buoyancy $$b_{s}(x,y)$$ . In experimental situations it is easier to evaluate the surface entropy flux, rather than the volume integral of $$|\unicode[STIX]{x1D735}b|^{2}$$ demanded by $$\unicode[STIX]{x1D705}\langle |\unicode[STIX]{x1D735}b|^{2}\rangle$$ .
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- Award ID(s):
- 1657041
- PAR ID:
- 10169194
- Date Published:
- Journal Name:
- Journal of Fluid Mechanics
- Volume:
- 894
- ISSN:
- 0022-1120
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript
- Journal Article:
- https://doi.org/10.1017/jfm.2020.269
