Formation of a pitchfork bifurcation in thermal convection flow inside an isosceles triangular cavity

Abstract

Transient, laminar thermal convection of air confined to an isosceles triangular cavity heated from the base and symmetrically cooled from the upper inclined walls has been investigated numerically. The system of conservation equations, subject to the proper boundary conditions, along with the equation of state assuming the air behaves as a perfect gas are solved with the finite volume method. In the conservation equations, the second-order-accurate QUICK scheme was used for the discretization of the convective terms and the SIMPLE scheme for the pressure-velocity coupling. The maximum height-to-base aspect ratio A is fixed at 0.5, while the Grashof number extends from a low Gr=103 to a high Gr=106. The influence of Gr on the flow and temperature patterns is analyzed and discussed for two opposing scenarios, one corresponding to increasing Gr and the other corresponding to decreasing Gr. It is found that two steady-state solutions are possible, excluding their solution images through a vertical mirror plane. The symmetrical solution prevails for relatively low Grashof numbers. However, as the Gr is gradually increased, a transition occurs at a critical value of Gr. Above this critical value of Gr, an asymmetrical solution exhibiting a pitchfork bifurcation arises and eventually becomes steady. The existing ranges of these unsteady and steady solutions are reported for the two opposing scenarios. Also, issues related to the observed hysteresis phenomenon are discussed in detail.

Natural convection and heat transfer in attics subject to periodic thermal forcing

“…The main objective of their research was to validate the existence of the numerical prediction of the symmetry-breaking bifurcation of the heated air currents that arise with gradual increments in Rayleigh number. Ridoune and Campo [15] and Lei et al [16] have also investigated the numerical prediction of the symmetry-breaking bifurcation. However, water has been used as the working fluid by Lei et al [16].…”

“…However, water has been used as the working fluid by Lei et al [16]. Ridoune and Campo [15] reported that as Ra is gradually increased, the symmetric plume breaks down and fades away. Thereafter, a subcritical pitchfork bifurcation is created giving rise to an asymmetric plume occurring at a critical Rayleigh number, Ra ¼ 1.42 Â 10 5 .…”

Effect of aspect ratio on natural convection in attics subject to periodic thermal forcing

“…The observed asymmetric flow structure in the isosceles triangular enclosure was previously reported by Holtzmann et al [7] in the case with constant cooling at the inclined surfaces and simultaneous heating at the base. The formation of the asymmetric flow is due to a pitchfork bifurcation as discussed by Ridouanea and Campo [6] At t = 2.8125 P , Figure 2 In Figure 2(f) the large cell has crossed the centreline of the cavity and the smaller cell, closer to this large cell, fills up this gap and grows itself. The flow is then gradually returning to symmetry.…”

Heatline analysis of heat recovery and thermal transport in materials confined within triangular cavities

“…However, many of the recent works are targeted more towards the application aspect of the process. Ridouane et al [32] and Ridouane and Campo [33] studied extensively the natural convection in an isosceles triangle and right triangular cavity in their work. Ben-Nakhi and Mahmoud [34] studied the natural convection in roof cavities inside buildings.…”