ON THE CONVECTION IN AN ENCLOSED CONTAINER WITH UNSTABLE SIDE WALL TEMPERATURE DISTRIBUTIONS

A numerical study of natural convection in a two-dimensional container of unity aspect ratio with unstable temperature distributions on the side walls and adiabatic top and bottom walls is discussed for a Bouss inesq fluid with unity Prandti number. For sufficiently low Rayleigh n umbers and symmetric boundary conditions a unique 2 x 2 steady cellula r flow with horizontal and vertical symmetry exists. At a critical Ray leigh number a pitchfork bifurcation occurs creating a pair of asymmet ric steady solutions, Further increasing the Rayleigh number causes th e asymmetric pair of steady solutions to undergo a subcritical Hopf bi furcation resulting in a large amplitude limit cycle. Hysteresis behav ior is observed between the stable steady flows and the stable limit c ycle for a range of Rayleigh numbers. The limit cycle disappears at a minimum Rayleigh number in what appears to be a double homoclinic orbi t. Applying asymmetric temperature boundary conditions causes an unfol ding of the pitchfork bifurcation. The character of the Hopf bifurcati ons and resulting limit cycle behavior is deeply affected by the intro duction of asymmetry. As the Rayleigh number is increased a progressio n of limit cycles containing from two to 206 small amplitude oscillati ons and one large amplitude 'relaxation' oscillation per period are se parated by what may be a series of homoclinic orbits. The steady and l imit cycle solution structure has a large influence on the heat transf er rate through the container. (C) 1998 Elsevier Science Ltd. All righ ts reserved.