A Numerical investigation of Two Critical Conditions for the First Steady Bifurcation of Natural Convection Flow of Air Enclosed within Rectangular Cavities

Abstract

Abstract For a given fluid, the aspect ratio and the corresponding Rayleigh number are the two critical conditions for bifurcations of natural convection flow enclosed in rectangular cavities. As none given references has directly shown the critical conditions of the natural convection flow of air encolsed in rectangular cavities, the second-order Euler-Taylor-Galerkin (ETG) finite element method developed by the authors was used for computing the critical aspect ratios and the critical Ra for the first steady bifurcation of natural convection flow of air within rectangular cavities (vertical, horizontal, and square cavities). The bisection method was used for finding the critical ratio of W/H or H/W accordingly. As the results indicate, the critical W/H for horizontal rectangular cavities is about 2.12, and the critical H/W for vertical rectangular cavities is about 3.62. Based on the investigation of the critical aspect ratios, the first steady bifurcations for the cavities having the critical aspect ratios were then computed numerically by using different Ra. It was found that for the rectangular cavities having critical aspect ratios, the first natural air convection flow bifurcation was that of an inverse pitchfork bifurcation with two given vortices converged into one core cell while the Ra was larger than the critical value. The critical Ra for a critical vertical rectangular cavity was about 434.5 when using the bisection method, and the critical Ra for a horizontal rectangular cavity was about 55.5. For a better understanding of the variation of the first steady bifurcation modes in rectangular cavities with W/H (H/W), the natural convection flow of air within a square cavity computed to investigate the first steady bifurcation in it and to find the critical Ra. It was found for the square cavity that the first steady bifurcation was a pitchfork bifurcation with one core cell broken into two vortices while Ra>31314. The critical Ra lay between (31313, 31314). It can be deduced that the aspect ratio is one key factor for the flow structure evolution of natural convection flows enclosed within the cavities. Moreover, the bifurcation diagram and the mechanism of eddy emergence for such flow needs to be numerically computed in a systematic manner.