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

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 10⁶ to 6.0× 10⁷ , are found on the route to chaos of the flows at $Pr=7$ . When Ra<1.0× 10³ , 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× 10⁶ , 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× 10⁶ , 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× 10⁶ ), 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× 10⁶ , the flow remains in its periodic puffing state even though it is being perturbed.