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)→QP2→P(2)→NP, where P(1) and P(2) are simply periodic regimes and QP2 is a quasi-periodic state with two basic frequencies. The transitions (from onset of convection to chaos) occur at Rayleigh numbers of 4π2, 380–400, 500–520, 560–570, and 850–1000. In the first simply periodic regime the fundamental frequency f 1 varies as $R^{\frac{7}{8}} $ and the average Nusselt number $\overline{Nu}$ is proportional to $R^{\frac{2}{3}}$; in P(2), f 1 varies as $R^{\frac{3}{2}}$ and $\overline{Nu}\propto R^{\frac{11}{10}}$. Convection in QP2 exhibits hysteresis, i.e. if the QP2 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 $R^{\frac{7}{8}}(R^{\frac{3}{2}})$. 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.