Multiple states, topology and bifurcations of natural convection in a cubical cavity

A numerical investigation has been conducted to explore the steady nonlinear low Prandtl number flow/thermal transition in a differentially heated cubic cavity. For small values of Rayleigh number (Ra), it is observed that initially there was only one symmetric steady-state solution. When the Ra was amplified, the system bifurcates from one fixed-point solution to the two stationary solutions, namely, Mode I and Mode II pitchfork bifurcations. This is due to the symmetric nature existing along the vertical and diagonal planes. The flow structure in the present nonlinear system consists of a pair of asymmetric counter-rotating helical cells in a double helix structure, foliated with invariant helically symmetric surfaces containing the fibre-like fluid particle orbits. Also the evolution of different symmetry-breaking orientations on the transverse and diagonal planes of the cavity was noticed. In the Mode I orientation a symmetric vortex coreline was observed. However, in the Mode II orientation a pair of anti-symmetric vortex corelines was observed. Detailed topological study was made based on the rule of Hunt and the structural stability criteria. Also the simulated results were corroborated with numerical evidence. The existence of the critical Ra values was ascertained with the aid of the predicted L₂-error norms, thermal/flow iso-contours and streamlines. The route of Mode I orientation was made of the alternate symmetric and asymmetric flows as Ra was augmented.