Critical transitions on a route to chaos of natural convection on a heated horizontal circular surface

Abstract

The transition route and bifurcations of the buoyant flows developing on a heated horizontal circular surface are elaborated using direct numerical simulations and direct stability analysis. A series of bifurcations, as a function of Rayleigh numbers ( $Ra$ ) ranging from $10^6$ to $6.0\times 10^7$ , are found on the route to chaos of the flows at $Pr=7$ . When $Ra<1.0\times 10^3$ , the buoyant flows above the heated horizontal surface are dominated by conduction, because of which the distinct thermal boundary layer and plume are not present. At $Ra=1.1\times 10^6$ , a Hopf bifurcation occurs, resulting in the flow transition from a steady state to a periodic puffing state. As $Ra$ increases further, the flow enters a periodic rotating state at $Ra=1.9\times 10^6$ , which is a unique state that was rarely discussed in the literature. These critical transitions, leaving from a steady state and subsequently entering a series of periodic states (puffing, rotating, flapping and period-doubling) and finally leading to chaos, are diagnosed using two-dimensional Fourier transforms. Moreover, direct stability analysis is conducted by introducing random numerical perturbations into the boundary condition of the surface heating. We find that when the state of a flow is in the vicinity of critical values (e.g. $Ra=2.0\times 10^6$ ), the flow is conditionally unstable to perturbations, and it can bifurcate from the rotating state to the flapping state in advance. However, for relatively stable flow states, such as at $Ra=1.5\times 10^6$ , the flow remains in its periodic puffing state even though it is being perturbed.