Route to chaos in porous-medium thermal convection

Abstract

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 f₁ varies as $R^{\frac{7}{8}} $ and the average Nusselt number $\overline{Nu}$ is proportional to $R^{\frac{2}{3}}$; in P⁽²⁾, f₁ 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 f₁ 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.