Period bubbling bifurcation and transition to chaotic state of convective flow on a top-open cylinder

Abstract

Period bubbling bifurcation and transition to chaotic state of a convective flow on the top-open cylinder beneath heating are investigated using direct numerical simulation owing to its fundamental significance and extensive application. Numerical results show that a primary convective flow on the cylinder is axisymmetric and dominated by the effect of baroclinicity for small Rayleigh numbers. There exists a complex route of the transition to chaotic state involving a succession of bifurcations when the Rayleigh number is increased; that is, the pitchfork bifurcation occurs between Ra = 2.7 × 103 and Ra = 2.8 × 103 for which an axisymmetric convective flow is broken, and the first Hopf bifurcation occurs between Ra = 3.3 × 105 and Ra = 3.4 × 105. An interesting phenomenon is that the convective flow may change from periodic back to steady state with the increase in the Rayleigh number for 7.0 × 105 ≤ Ra ≤ 1.5 × 106, termed as a period bubbling bifurcation. The whole route of the transition to chaotic state can be described by a steady, a periodic, a steady again, a period doubling, a quasiperiodic, and a chaotic state as the Rayleigh number is increased. Further, the convective flow in the transition is discussed by employing the topologic index, the spectral analysis, the attractor, and the fractal dimension. Additionally, heat transfer is also quantified.