Stability of two-dimensional (2D) natural convection flows in air-filled differentially heated cavities: 2D/3D disturbances

Abstract

Following our previous two-dimensional (2D) studies of flows in differentially heated cavities filled with air, we studied the stability of 2D natural convection flows in these cavities with respect to 3D periodic perturbations. The basis of the numerical methods is a time-stepping code using the Chebyshev spectral collocation method and the direct Uzawa method for velocity–pressure coupling. Newton's iteration, Arnoldi's method and the continuation method have been used in order to, respectively, compute the 2D steady-state base solution, estimate the leading eigenmodes of the Jacobian and perform linear stability analysis. Differentially heated air-filled cavities of aspect ratios from 1 to 7 were investigated. Neutral curves (Rayleigh number versus wave number) have been obtained. It turned out that only for aspect ratio 7, 3D stationary instability occurs at slightly higher Rayleigh numbers than the onset of 2D time-dependent flow and that for other aspect ratios 3D instability always takes place before 2D time-dependent flows. 3D unstable modes are stationary and anti-centro-symmetric. 3D nonlinear simulations revealed that the corresponding pitchfork bifurcations are supercritical and that 3D instability leads only to weak flow in the third direction. Further 3D computations are also performed at higher Rayleigh number in order to understand the effects of the weak 3D fluid motion on the onset of time-dependent flow. 3D flow structures are responsible for the onset of time-dependent flow for aspect ratios 1, 2 and 3, while for larger aspect ratios they do not alter the transition scenario, which was observed in the 2D cases and that vertical boundary layers become unstable to traveling waves.