Three-dimensional stability of natural convection flows in inclined square enclosures

The three-dimensional stability of two-dimensional natural convection flows in a heated, square enclosure inclined to the horizontal is investigated numerically. First, the computational procedure is validated by comparison of base flow solutions to results reported in literature across a range of inclinations. A bi-global linear stability analysis is then conducted to investigate the stability of these two-dimensional base flows to infinitesimal three-dimensional perturbations, and the effect that buoyancy forces (defined by a buoyancy number R_N) and enclosure inclination θ have on these stability characteristics. The flow is first observed to become three-dimensionally unstable at buoyancy number R_N = 213.8 when θ is 180^◦; this increases to R_N = 2.54 × 10⁴ at inclination θ = 58^◦. It is found that the two-dimensional base flow is more unstable to three-dimensional perturbations with the critical R_N corresponding to three-dimensional instability being significantly lower than its two-dimensional counterpart across all considered inclinations except 83^◦ ≤ θ ≤ 88^◦, where the most unstable mode is a two-dimensional oscillatory mode that develops in the boundary layers along the conducting walls. Eight different leading three-dimensional instability modes are identified, with inclinations 58^◦ ≤ θ < 88^◦ transitioning through an oscillatory mode, and inclinations 88^◦ ≤ θ ≤ 180^◦ transitioning through a stationary mode. The characteristics of the primary instability modes corresponding to inclinations 88^◦ ≤ θ ≤ 179^◦ indicate the presence of a Taylor-Görtler instability.