We investigate the onset of transient natural convection in fluid layers subject to volumetric radiative heating and surface cooling. Linear stability analysis reveals a non-monotonic evolution of stability in deep layers, where the flow undergoes successive stages of initial destabilisation, intermediate suppression and eventual restabilisation. This complex temporal behaviour necessitates the definition of dual critical Rayleigh numbers: a lower bound marking the onset of initial instability and an upper bound required for sustained convection. To efficiently predict these thresholds, we develop a local Rayleigh number model that depends solely on the instantaneous conductive temperature profile. When the Rayleigh number exceeds the lower bound, the critical time tc for flow onset is determined through transient linear stability analysis, and scaling laws are derived to characterise the dependence of tc on the Rayleigh number Ra, Prandtl number Pr, cooling parameter and layer depth H. Two distinct instability triggering mechanisms are identified: a top-triggered regime, where tc Ra Pr / (2+Pr) ^-1/2, and a bottom-triggered regime, where tc Ra Pr (-H) / (2+Pr) ^-1/2. All theoretical predictions are rigorously validated against direct numerical simulations, providing a unified predictive framework for convective onset in systems governed by coupled effects of radiation absorption and surface cooling.
Yang et al. (Wed,) studied this question.