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

TL;DR

This study investigates the complex transition to chaos in natural convection on a heated horizontal surface, identifying key bifurcations and flow states through numerical simulations and stability analysis across a wide range of Rayleigh numbers.

Contribution

It provides a detailed analysis of bifurcation sequences and flow states in natural convection, including the first observation of a rotating state at high Rayleigh numbers.

Findings

Flow transitions from conduction to chaotic states with increasing Rayleigh number.

Identification of a unique rotating flow state at high Rayleigh numbers.

Flow stability varies near bifurcation points, with some states being conditionally unstable.

Abstract

The transition route and bifurcations of the buoyant flow developing on a heated circular horizontal surface are elaborated using direct numerical simulations and direct stability analysis. A series of bifurcations, as a function of Rayleigh numbers (Ra) ranging from $10^1$ to $6\times10^7$, are found on the route to the chaos of the flow at $Pr=7$. When $Ra<1.0\times10^3$, the buoyant flow above the heated horizontal surface is dominated by conduction, because of which distinct thermal boundary layer and plume are not present. At $Ra=1.1\times10^6$, 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 $Ra=1.9\times10^6$, 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 doubling) and finally leading to chaos, are diagnosed using spectral analysis and two-dimensional Fourier Transform (2DFT). 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 bifurcation points (e.g., $Ra=2.0\times10^6$), 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 $Ra=1.5\times10^6$, the flow remains its periodic puffing state even though it is being perturbed.