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 ( Racr ) for a Rayleigh–Bénard convection configuration with an imposed bottom boundary heat flux that varies harmonically in time with zero mean. The Racr value depends on the non-dimensional frequency ω of the boundary heat-flux modulation. Floquet theory is used to find Racr for linear stability, and the energy method is used to find Racr 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 Racr is generally higher than the nonlinear stability Racr , as expected. For large ω , Racr 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.