Transition to chaos of convective flows in a convectively heated cubic cavity

The present study is aimed at the transition route of convective flows in a cubic cavity with convectively heated sidewalls. Three dimensional numerical simulation is carried out for a wide range of Rayleigh number (Ra) from 10° to 5 × 108 with air as the working fluid in the cavity (Pr = 0.71) but water (Pr = 7.74) as the working fluid in the water bathes for convective heating. The present study shows a complex transition route to chaos concerning a succession of bifurcations. The pitchfork bifurcation first occurs between Ra = 3.3 × 104 and Ra = 3.4 × 104 based on the topologic index in a steady flow state. Further, the second, third, and more pitchfork bifurcations in the steady flow state occur with the increase in the Rayleigh number. Numerical results show that convective flows are steady for Ra ≤ 3.9 × 107 but unsteady for Ra = 4 × 107; that is, Hopf bifurcation occurs between Ra = 3.9 × 107 and Ra = 4 × 107. If the Rayleigh number is further increased, convective flows undergo the transition from periodic to quasi-periodic flow state between Ra = 1.2 × 108 and Ra = 1.3 × 108. For higher Rayleigh numbers of, e.g., Ra = 4 × 108, convective flows become chaotic. In order to investigate typical convective flows in the transition to chaos, topologic point, power spectral density, maximum Lyapunov exponents, fractal dimension, and attractor are applied. In addition, heat transfer is quantified under different dominances and different scaling laws are obtained.