Rayleigh-Bénard Convection with Multiple Solutions in Trapezoidal Closed Cavities

Rayleigh-Bénard convection in symmetric trapezoidal closed cavities with cavity angle ϕ = 70° − 110°, filled with air, is studied using numerical simulations where inclined side walls are adiabatic. In contrast to rectangular cavities, where no flow exists below a threshold value, there is a weak convection even at a low Rayleigh number (Ra) due to the fact that there is a component of thermal gradient in the horizontal direction in these cavities. Interestingly, these cavities show sudden and significant jumps in the convection, similar to square cavities (Rac = 2585.02 for ϕ = 90°), as Ra increases beyond a critical value (Rac). It is noted here that these Rac represent symmetry-breaking pitchfork bifurcations. These bifurcations are seen in both acute (Rac = 8000 for ϕ = 70°) and obtuse (Rac = 2300 for ϕ = 110°) angle trapezoidal cavities. Moreover, it is observed that multiple steady-state solutions (MSSS) exist as Ra is further increased. A forward and backward continuation approach for numerical simulations is used to track the co-existence of MSSS. These steady states have co-existing one-roll and two-roll convective patterns beyond another threshold value of Ra. Here, two types of critical Ra have been identified for different cavity angles; one shows the sudden jump in the convection, and the other is the one beyond which MSSS co-exist. Furthermore, a co-dimension two bifurcation analysis is carried out with Ra and ϕ as two parameters. The bifurcation analysis divides the parameter space into different regions based on the multiplicity of the solutions.