Route to chaos in porous-medium thermal convection

A pseudo-spectral numerical scheme is used to study two-dimensional, single-cell, time-dependent convection in a square cross-section of fluid saturated porous material heated from below. With increasing Rayleigh number R convection evolves from steady S to chaotic NP through the sequence of bifurcations S→P (1) →QP 2 →P (2) →NP, where P (1) and P (2) are simply periodic regimes and QP 2 is a quasi-periodic state with two basic frequencies. The transitions (from onset of convection to chaos) occur at Rayleigh numbers of 4π 2 , 380–400, 500–520, 560–570, and 850–1000. In the first simply periodic regime the fundamental frequency f 1 varies as R7/8 and the average Nusselt number Nū is proportional to R2/3 ; in P (2) , f 1 varies as R3/2 and Nū∝ R11/10 . Convection in QP 2 exhibits hysteresis, i.e. if the QP 2 state is reached from P (1) (P (2) ) by increasing (decreasing) R then the frequency with the largest spectral power is the one consistent with the extrapolation of f 1 according to R7/8(R3/2) . The chaotic states are characterized by spectral peaks with at least 3 fundamental frequencies superimposed on a broadband background noise. The time dependence of these states arises from the random generation of tongue-like disturbances within the horizontal thermal boundary layers. Transition to the chaotic regime is accompanied by the growth of spectral components that destroy the centre-symmetry of convection in the other states. Over-truncation can lead to spurious transitions and bifurcation sequences; in general it produces overly complex flows.