Linear and nonlinear stability of Rayleigh–Bénard convection with zero-mean modulated heat flux

Abstract

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.