Onset of low-frequency shear-driven instability in differentially heated cavities

Abstract

The low-frequency instability in a square differentially heated cavity at different Prandtl numbers (0.4 ≤ Pr ≤ 1.4) is investigated using two-dimensional numerical simulations. For 0.4 ≤ Pr ≤ 0.45 and 0.71 ≤ Pr ≤ 1.4, a low-frequency shear-driven instability in the detached flow structure bifurcates the steady flow into a periodic state. However, for 0.5 ≤ Pr ≤ 0.65, this instability is observed in the unsteady regime with the flow bifurcating due to a different high-frequency boundary-layer instability. The frequency of the shear-driven instability is always less than the buoyancy frequency and excites internal waves in the stably stratified core, which re-orient themselves to form different standing internal wave modes. The standing internal wave modes govern the stability of the shear-driven instability as the instability, and internal wave modes are phase-locked in the limit cycle regime governed by this instability. The low-order internal wave modes increase the stability of the shear-driven instability and delay its onset to higher Rayleigh numbers. Feedback between the instability, stratification and the viscosity of the fluid causes a global absolute instability despite the instability being localised to the detached flow structure.