Linear stability analyses of natural convection flows in a differentially heated square cavity with conducting horizontal walls

Abstract

The stability of two-dimensional (2D) natural convection flows with respect to both two- and three-dimensional perturbations is investigated numerically. Several methods (Arnoldi’s method, preconditioned Newton’s iteration and preconditioned continuation method) are put together for this purpose and applied to natural convection in a differentially heated square cavity with conducting horizontal walls for a large range of Prandtl numbers. These methods are first validated by comparison with results reported in the literature for several Prandtl numbers for both two-dimensional and three-dimensional perturbations. They are then used to extend the stability of the 2D base flows to two-dimensional perturbations for other Prandtl numbers, and to investigate in detail the stability of these two-dimensional base solutions with respect to three-dimensional perturbations for a large range of Prandtl number. For Prandtl number equal to 1 the base solutions are more unstable to three-dimensional perturbations and the most unstable three-dimensional mode, which is oscillatory, is connected to the third most unstable two-dimensional mode; for larger Prandtl numbers (3, 7, and 20), the 2D base flows are found more stable to three-dimensional perturbations than to the 2D ones and the most unstable modes are thus two dimensional; for smaller Prandtl numbers (0.71, 0.1, and 0.015) the base solutions are again more unstable to three-dimensional perturbations. These three-dimensional modes are however no longer connected to the most unstable two-dimensional oscillatory modes and correspond to stationary perturbations. Furthermore for Pr=0.015 and 0.1 the critical Rayleigh number is approximately 40 times smaller than that found for two-dimensional perturbations, indicating a dramatic change in the nature of the instability which seems to be linked to Taylor–Görtler-type instability.

Radiation effects on bifurcation and dual solutions in transient natural convection in a horizontal annulus

“…He observed dual steady solutions at Ra > Ra c ≈ 3800 for wide gap annuli (R = 2) by the vorticity-streamfunction method. Similar results were provided later by Mizushima et al, 20,21 xin et al 22 and Mercader et al 23 Petrone et al 24 performed a stability analysis of numerical steady-state solutions, and provided a detail of the bifurcation diagram near the imperfect bifurcation for different radius ratio R = 1.2, 1.4 and 2 at Pr = 0.7. Angeli et al 25 provided a critical review of buoyancy-induced flow transitions in horizontal annuli.…”

Quasi-periodicity and chaos in a differentially heated cavity

“…In the case of a laterally heated cavity, the convective response does not have to overcome a finite threshold, since it occurs for an arbitrarily small Rayleigh number. The successive transitions from the primary convective steady state to the oscillating and chaotic motions studied recently [1,2,3,4, among others], show a strong dependence on geometrical conditions, such as the aspect ratio, on boundary conditions and on material parameters.…”

Bifurcations and chaos in single-roll natural convection with low Prandtl number

“…Despite its apparent simplicity, the problem has already revealed a rich variety of complex behaviors and a rather strong sensitivity to small variations of parameters, such as the aspect ratio ͑length over height͒ or the Prandtl number, and to the boundary conditions ͑both thermal and mechanical͒. [1][2][3][4][5][6][7][8][9][10][11] From the applied point of view, the phenomenon of natural convection is ubiquitous. As opposed to the Rayleigh-Bénard convection, the quiescent state is not a solution; consequently, fluid motion is present without the need to overcome a threshold value of any parameter.…”