Multiple solutions in natural convection in an air filled square enclosure: Fractal dimension of attractors

Abstract

In this study, we investigated numerically the transient natural convection in a square cavity with two horizontal adiabatic sides and vertical walls composed of two regions of same size maintained at different temperatures. The flow has been assumed to be laminar and bi-dimensional. The governing equations written in dimensionless form and expressed in terms of stream function and vorticity, have been solved using the Alternating Direction Implicit (ADI) method and the GAUSS elimination method. Calculations were performed for air (Pr = 0.71), with a Rayleigh number varying from 2.5 × 105 to 3.7 × 105. We analysed the effect of the Rayleigh number on the route to the chaos of the system. The first transition has been, found from steady-state to oscillatory flow and the second is a subharmonic bifurcation as the Rayleigh number is increased further. For sufficiently small Rayleigh numbers, present results show that the flow is characterized by four cells with horizontal and vertical symmetric axes. The attractor bifurcates from a stable fixed point to a limit cycle for a Rayleigh number varying from 2.5×105 2.51 ×105. A limit cycle settles from Ra = 3×105 and persists until Ra = 5×105. At a Rayleigh number of 2.5×105 the temporal evolution of the Nusselt number Nu(t) was stationary. As the Rayleigh number increases, the flow becomes unstable and bifurcates to a time periodic solution at a critical Rayleigh number between 2.5×105 and 2.51 ×105. After the first HOPF bifurcation at Ra = 2.51×105, the oscillatory flow undergoes several bifurcations and ultimately evolves into a chaotic flow. © 2008 Asian Network for Scientific Information.