The flow and heat transfer characteristics of thermal convection in confined spaces exhibit strong dependence on and complexity due to the spatial geometric configuration. This study numerically investigates the nonlinear dynamics and bifurcation transitions of natural convection in a two-dimensional circular enclosure containing four isothermal circular heat sources arranged in a diamond configuration, with the enclosure wall maintained at a constant cold temperature. The lattice Boltzmann method with a double-distribution function model (D2G9 for the flow field and D2Q4 for the temperature field) is employed to solve the governing equations under the Boussinesq approximation. Numerical simulations are conducted over a range of Rayleigh numbers (Ra) from 8.5×10⁴ to 9.5×10⁵ at a fixed Prandtl number of Pr=0.706. The flow regime transitions are identified through temperature contours and streamlines, while the system state transitions and bifurcation types are determined by analyzing time series, phase space trajectories, power spectral density (PSD), and the maximum Lyapunov exponent.
The results reveal a rich and complex bifurcation cascade as Ra increases. At low Ra, the flow remains in a steady, symmetric state. A supercritical pitchfork bifurcation occurs at Ra≈9.5×10⁴, where the symmetric steady state loses stability, giving rise to a pair of stable, mirror-image asymmetric steady solutions. This symmetry breaking is manifested by the dominance of a single vortex near the top heat source. As Ra increases further to Ra≈1.1×10⁵, the system undergoes a supercritical Hopf bifurcation, transitioning from a steady to a stable periodic oscillatory state represented by a limit cycle in phase space. This single-frequency oscillation is driven by the periodic expansion and contraction of the dominant vortex near the top heat source. Between Ra≈1.75×10⁵ and 3×10⁵, the system maintains single-periodic oscillations, with vortices periodically generated near the top heat source and moving toward one side. A fundamental change in dynamics occurs at Ra≈3.15×10⁵, where a global heteroclinic bifurcation is identified. The limit cycle destabilizes, and the system transitions directly into chaos, confirmed by a broad-band PSD and a positive maximum Lyapunov exponent. Phase space reconstruction reveals chaotic trajectories intermittently visiting regions corresponding to the two unstable mirror-image states, forming a heteroclinic network. As Ra increases through the chaotic regime, the flow structure near the top heat source reorganizes from intermittent dual-vortex motion to a persistent central thermal plume, causing the collapse of the chaotic attractor's topology. Remarkably, at Ra≈7.5×10⁵, the chaotic state destabilizes via a torus bifurcation, leading to a quasi-periodic window characterized by two incommensurate frequencies in the PSD. This is followed by a frequency-locking phenomenon at Ra≈8.75×10⁵, where the system reverts to a periodic state. The locked state eventually breaks down as Ra increases, re-entering quasi-periodicity and finally transitioning back to a fully developed chaotic state through torus breakdown at Ra≈9.5×10⁵.
In conclusion, this study elucidates a complex and non-standard route to chaos for multi-source natural convection in a circular enclosure, involving successive steady symmetry breaking, Hopf bifurcation, a direct transition to chaos via heteroclinic bifurcation, and a subsequent quasi-periodic window with frequency locking. The findings highlight the profound influence of the interplay between multiple discrete heat sources and curved boundaries on the system's nonlinear dynamics.