On the convection in an enclosed container with unstable side wall temperature distributions

Abstract

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 Boussinesq fluid with unity Prandtl number. For sufficiently low Rayleigh numbers and symmetric boundary conditions a unique 2 × 2 steady cellular flow with horizontal and vertical symmetry exists. At a critical Rayleigh number a pitchfork bifurcation occurs creating a pair of asymmetric steady solutions. Further increasing the Rayleigh number causes the asymmetric pair of steady solutions to undergo a subcritical Hopf bifurcation resulting in a large amplitude limit cycle. Hysteresis behavior is observed between the stable steady flows and the stable limit cycle for a range of Rayleigh numbers. The limit cycle disappears at a minimum Rayleigh number in what appears to be a double homoclinic orbit. Applying asymmetric temperature boundary conditions causes an unfolding of the pitchfork bifurcation. The character of the Hopf bifurcations and resulting limit cycle behavior is deeply affected by the introduction of asymmetry. As the Rayleigh number is increased a progression of limit cycles containing from two to 206 small amplitude oscillations and one large amplitude ‘relaxation’ oscillation per period are separated by what may be a series of homoclinic orbits. The steady and limit cycle solution structure has a large influence on the heat transfer rate through the container.