Three-dimensional bifurcations in a cubic cavity due to buoyancy-driven natural convection

Rich and complex buoyancy-driven flow field due to natural convection will be studied numerically over a wide range of Rayleigh numbers in a cubic cavity by virtue of the simulated bifurcation diagram, limit cycle, power spectrum and phase portrait. When increasing the Rayleigh number, the predicted flow is found to evolve from the conductive state to the state with the onset of convection, which is featured with the steady and symmetric laminar solution, and then to the asymmetric state (pitchfork bifurcation), which will not be discussed in this paper. As the Rayleigh number was further increased, a limit cycle branching from the fixed point of the investigated dynamical system is observed. Supercritical Hopf bifurcation is confirmed to be the birth of the orbitally stable limit cycle that separates the vortex flow into an inner unstable region (moving away from the vortex coreline) and an outer stable region (moving towards the vortex coreline). As the Rayleigh number is increased still, the investigated buoyancy-driven flow became increasingly destabilized through quasi-periodic bifurcation and then through two predicted frequency-doubling bifurcations. Thanks to the power spectrum analysis, bifurcation scenario was confirmed to have an initially single harmonic frequency, which is featured with a driving amplitude. Then an additional ultraharmonic frequency showed its presence. Prior to chaos, in the five predicted arithmetically related frequencies there exists one frequency that is incommensurate to the other two fundamental frequencies. This computational study enlightens that the investigated nonlinear system, which involves frequency-doubling bifurcations, loses its stability to a quasi-periodic bifurcation featured with the formation of a subharmonic frequency. Subsequent to the formation of three frequency-doubling bifurcations and one quasi-periodic bifurcation, an infinite number of frequencies was observed in flow conditions with the continuously increasing Rayleigh numbers. Finally, the chaotic attractor was predicted to be evolved from the strange attractor in the corresponding phase portraits.