Numerical prediction of the first bifrucation for natrual convections of air and water closed in a square cavity

Natural convection within closed cavities is of practical and theoretical interest in many industrial applications and nonlinear sciences. The existing investigations of natural convections within a closed cavity show that a series of bifurcation are observed with the increase in the Rayleigh number for each given Pr; the corresponding fluid flow patterns change from the steady motions to the time dependent ones; with further increase in Ra, the flow finally enters to a chaotic state. The first bifurcation for natural convection flow has to be predicted successfully. In the present work, a second order Euler-Taylor-Galerkin finite element method of fractional steps, developed by the authors, are used in numerical studies for the first bifurcation (or the prediction of the corresponding critical Rayleigh number) of such flow in a closed square cavity. By considering the role of mesh resolution, the first bifurcation for two kinds of fluids, water and air, are numerical investigated.