Time-periodic convective patterns in a horizontal porous layer with through-flow

We investigate the time-periodic convective patterns which set on in an infinite porous layer saturated by a fluid, under the influence of a vertical temperature gradient, superimposed to a horizontal seeping through-flow. When the seepage velocity keeps moderate, it obeys the Darcy law, while the heat equation rules the evolution of the temperature. The transition towards convection is governed by the filtration Rayleigh number, and the above-mentioned system of partial differential equations has Galilean invariance: travelling waves, stationary with respect to a moving frame, solve this problem near the threshold. This follows directly from the study of natural convection. With the help of the center manifold theory, we show that other kinds of time-periodic two-dimensional structures exist: upstream to a fixed zone, their amplitude vanishes. Downstream, they resemble the travelling waves. Numerical simulation of the governing equations reproduces elements of both sets of convective structures, indexed by definite values of the time-frequency.