Linear and nonlinear stability of Rayleigh–Bénard convection with zero-mean modulated heat flux
Résumé
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 ω 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 ω, with only the latter at large ω. For a given frequency, the linear stability Ra cr is generally higher than the nonlinear stability Ra cr , as expected. For large ω, Ra cr ω −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 ω. 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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