Modal package convection in a porous layer with boundary imperfections

Thermal convection with a continuous finite bandwidth of modes in a porous layer with horizontal walls at different mean temperatures is considered when a spatially non-uniform temperature is prescribed at the lower wall. The nonlinear problem of three-dimensional convection for values of the Rayleigh number close to the classical critical value is solved by using multiple scales and perturbation techniques. The preferred flow solutions are determined by a stability analysis. It is found that for the case of near-resonant wavelength excitation regular or non-regular solutions in the form of superposition of small-scale multi-modal solutions with large-scale multimodal (or non-modal) amplitude can become preferred, provided the wave vectors of the solutions are contained in the set of wave vectors due to the modal form of the boundary imperfections and the form of the large-scale part is the same as that due to the boundary imperfections. For the case of non-resonant wavelength excitation some three-dimensional solutions in the form of superposition of small-scale multi-modal solutions with large-scale multi-modal (or non-modal) amplitudes can be preferred, provided that the wavelength of the small-scale modulation is not too small. Large-scale flow structures are quite different from the small-scale flow structures in a number of cases and, in particular, they can exhibit kinks and can be non-modal in nature. The resulting flow patterns are affected accordingly, and they can provide quite unusual and non-regular three-dimensional preferred patterns. In particular, they are multiples of irregular rectangular patterns, and they can be non-periodic.