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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 fr...

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Bibliographic Details
Published in:Journal of fluid mechanics 2023-04, Vol.961, Article A1
Main Authors: Christopher, T.W., Le Bars, M., Llewellyn Smith, Stefan G.
Format: Article
Language:English
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Summary: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.
ISSN:0022-1120
1469-7645
DOI:10.1017/jfm.2023.138