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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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Linear and nonlinear stability of Rayleigh–Bénard convection with zero-mean modulated heat flux
Linear and nonlinear stability analyses are performed to determine critical Rayleigh numbers ( $${Ra}_{cr}$$ ) for a Rayleigh–Bénard convection configuration with an imposed bottom boundary heat flux that varies harmonically in time with zero mean. The $${Ra}_{cr}$$ value depends on the non-dimensional frequency $$\omega$$ of the boundary heat-flux modulation. Floquet theory is used to find $${Ra}_{cr}$$ for linear stability, and the energy method is used to find $${Ra}_{cr}$$ for two different types of nonlinear stability: strong and asymptotic. The most unstable linear mode alternates between synchronous and subharmonic frequencies at low $$\omega$$ , with only the latter at large $$\omega$$ . For a given frequency, the linear stability $${Ra}_{cr}$$ is generally higher than the nonlinear stability $${Ra}_{cr}$$ , as expected. For large $$\omega$$ , $${Ra}_{cr} \omega ^{-2}$$ approaches an $O(10)$ constant for linear stability but zero for nonlinear stability. Hence the domain for subcritical instability becomes increasingly large with increasing $$\omega$$ . The same conclusion is reached for decreasing Prandtl number. Changing temperature and/or velocity boundary conditions at the modulated or non-modulated plate leads to the same conclusions. These stability results are confirmed by selected direct numerical simulations of the initial value problem.
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- Award ID(s):
- 1829919
- PAR ID:
- 10406225
- Date Published:
- Journal Name:
- Journal of Fluid Mechanics
- Volume:
- 961
- ISSN:
- 0022-1120
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/jfm.2023.138
