3-DIMENSIONAL TRANSITION OF NATURAL-CONVECTION FLOWS

The stability with respect to two- and three-dimensional perturbations of natural-convection flow of air in a square enclosure with differen tially heated vertical walls and periodic boundary conditions in the l ateral direction has been investigated. The horizontal walls are eithe r conducting or adiabatic. The solution is numerically approximated by Chebyshev-Fourier expansions. In contrast to the assumption made in e arlier studies, three-dimensional perturbations turn out to be less st able than two-dimensional perturbations, giving a lower critical Rayle igh number in the three-dimensional case for the onset of transition t o turbulence. Both the line-symmetric and line-skew-symmetric three-di mensional perturbations are found to be unstable. The most unstable wa velengths in the lateral direction typically are of the same size as t he enclosure. In the nonlinear solution new symmetry breaking occurs, giving either a steady or an oscillating final state. The three-dimens ional structures in the nonlinear saturated solution consist of counte r-rotating longitudinal convection rolls along the horizontal walls. T he energy balance shows that the three-dimensional instabilities have a combined thermal and hydrodynamic nature. Besides the stability calc ulations, two- and three-dimensional direct numerical simulations of t he weakly turbulent flow were performed for the square conducting encl osure at the Rayleigh number 10(8). In the two-dimensional case, the t ime-dependent temperature shows different dominant frequencies in the horizontal boundary layers, vertical boundary layers and core region, respectively. In the three-dimensional case almost the same frequencie s are found, except for the horizontal boundary layers. The strong thr ee-dimensional mixing leaves no, or only very weak, three-dimensional structures in the time-averaged nonlinear solution. Three-dimensional effects increase the maximum of the time- and depth-averaged wall-heat transfer by 15%.