Brinkman–Bénard convection in a box with temperature modulation

A bounded porous box saturated with Newtonian fluid and subjected to a sinusoidal temperature gradient has various practical applications, such as solar energy storage, groundwater remediation, food processing, and chemical reactors. We address the generalization of the classical Rayleigh–Bénard convection problem in a horizontal fluid layer in an infinitely large domain heated from below to a finite three-dimensional box. We also look into a more intricate form of the modulated Rayleigh–Bénard problem in which the temperature at the bottom boundary varies sinusoidally. The Rayleigh number quantifies the non-sinusoidal part of the temperature gradient, while the amplitude and frequency of modulation describe the sinusoidal one. The critical Rayleigh number is determined using linear and nonlinear stability analyses; for the latter, the energy method is used. There is a possibility of subcritical instabilities, as evidenced by the energy stability estimates being lower than the linear ones. Furthermore, eigenvalues are obtained as a function of aspect ratios, modulation amplitude, and frequency for varying Darcy numbers. Modulation amplitude more significantly triggers a change in flow patterns at the onset of convection compared to the effect of other parameters. Considering water-saturated porous media made up of different materials, we report the critical temperature difference between lower and upper surfaces required for the onset of convection. In addition, a comparison between such a temperature difference obtained from linear theory and the energy method is also provided in the same manner. It is observed that subharmonic instability occurs for all considered porous media packed densely or sparsely.