Determination of a Hopf bifurcation of natural convection in a symmetric heated square cavity

In the present study, numerical investigation is carried out to analyze natural convection in symmetrically heated square cavity filled with air. Left, right and bottom walls are heated partially, whereas remaining portions of these walls are maintained at a lower constant temperature with an isolated top wall. Governing equations are solved by a finite volume method and the analysis is conducted for different Rayleigh number Ra values ranged between 10(3) and 9x10(6). The objective of the present study is to determine the critical Rayleigh number in which a transition from a stationary to an oscillatory flow takes place. The effect of the heating sources placed symmetrically at the two opposite sidewalls in addition to the heating source at the bottom wall was investigated. Results are presented in terms of streamlines, isotherms, and flow variables including the velocity and, temperature profiles and, Fourier frequency spectrum of the temperature. Obtained results shown that for Rayleigh numbers smaller than Ra-cr = 3 x 10(6), the flow inside the cavity remains stationary with perfectly symmetric patterns. Whereas, beyond this critical value, the system bifurcates in which the flow symmetry is broken and a first Hopf time dependent periodic flow patterns take place.