Numerical study of instability in a horizontal porous channel with bottom heating and forced horizontal flow

We study the two-dimensional convective patterns in a long horizontal porous layer heated from below, where a nonzero cross-flow is imposed. Indeed experiments show that time-periodic planar flows are found at moderate values of the flow rate. Within the framework of the Darcy law, above the absolute threshold, varied end-conditions lead to oscillatory patterns, which are more or less similar to each other in the bulk of the device but present differences near the extremities. Depending on the boundary-conditions, the numerical simulation may produce patterns which are space-periodic traveling rolls or waves of amplitude modulated within a stationary region, with envelopes in the form of fronts. Space-periodic boundary-conditions yield wavelengths linked to the total length of the device, which sets the frequency. Input boundary-conditions breaking translational invariance along the direction of the main flow yield different structures and select the temporal period. Most attention is paid to inlet-conditions imposing a linear profile of temperature (at the entrance of the device). We study the variations of the frequency vs the seeping flow rate and the filtration Rayleigh number. The length of the resulting front is also considered.