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Critical transitions on a route to chaos of natural convection on a heated horizontal circular surface
Published online by Cambridge University Press: 04 June 2024
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.
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Jiang et al. supplementary movie 1
The x-z plane temperature contour plot for Ra = 1.1×106 at equilibrium state.
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Jiang et al. supplementary movie 2
The temperature iso-surface at T = 0.1 and for Ra = 1.1×106 at equilibrium state.
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Jiang et al. supplementary movie 3
The x-z plane temperature contour plot for Ra = 2×106 at equilibrium state.
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Jiang et al. supplementary movie 4
The temperature iso-surface at T = 0.1 and for Ra = 2×106 at equilibrium state.
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Jiang et al. supplementary movie 5
The x-z plane temperature contour plot for Ra = 2.5×106 at equilibrium state.
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Jiang et al. supplementary movie 6
The temperature iso-surface at T = 0.1 and for Ra = 2.5×106 at equilibrium state.
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