Time-dependent convection in a fluid-saturated porous cube heated from below

Abstract

A numerical scheme based on the pseudospectral method has been implemented in order to study three-dimensional convection in a fluid-saturated cube of porous material. With increasing Rayleigh number R, convection first evolves from a symmetric steady state (S) to a partially non-symmetric steady state (S’, physical symmetry in the vertical direction is preserved). The transition Rayleigh number is about 550. At a Rayleigh number of 575 the flow becomes oscillatory P(1) with a single frequency that increases with R. At a value of R between 650 and 680 the oscillation becomes quasi-periodic with at least two fundamental frequencies. It returns to a simply periodic state in a narrow range about R = 725. A further increase of R transforms the simply periodic state again to a quasi-periodic state. The sequence of three-dimensional time-dependent states resembles previously studied two-dimensional cases in that evolution from more complex states to less complex states occurs with increasing R. The partial symmetry breaking prior to the onset of time dependence is unique to the three-dimensional flows, but a dependence of the S → S’ transition on the step size in R suggests the possibility that S → S’ might not occur prior to S → P(1) for sufficiently small steps in R. The quasi-periodic flows sometimes exhibit intermittency, causing difficulty in exactly defining their spectral characteristics.