The onset of penetrative convection in a three-layered porous-air-porous system with internal heat source

The study is devoted to a linear stability problem for a system of two horizontal porous sublayers separated by an air interlayer under the influence of the gravity field. The porous sublayers are capable of generating heat with a constant volumetric strength of heat source proportional to the solid volume fraction. Equal temperatures are maintained at the external boundaries of the three-layered system. The onset value for internal convection is numerically determined in the limiting case of a thin air sublayer. The critical wave number is also found, upon reaching which the equilibrium stability is lost in the system, and convective patterns begin to arise. The influence on the convective stability of the ratio of the permeabilities of the layers, the relative thickness of the lower porous sublayer, and the parameter of the air interlayer is analyzed. This parameter is the ratio of the air sublayer thickness raised to the third power to the Darcy number. New results are obtained that take into account the localization of the convective flow. It is shown that the permeability ratio can both destabilize and stabilize the motionless state. An increase in the interlayer parameter unambiguously reduces the convection onset value. A map for the local and large-scale convective regimes has been constructed in the coordinates of permeability ratio versus the air interlayer parameter. A demarcation line of the transition from one regime to another on the map corresponds to the parameters at which the critical internal Rayleigh-Darcy numbers for the two minima of the bimodal neutral curve have equal values. One of the minima belongs to the region of small wave numbers (a long-wave minimum) and defines the onset value for large-scale convection, and the other one is in the region of large wave numbers (a short-wave minimum) and indicates the onset value for local convection. It is shown that with an increase in the air interlayer parameter, the ratio of permeability required for a transition from large-scale to local convective regimes decreases.