Buoyant flow and instability in a vertical cylindrical porous slab with permeable boundaries

TL;DR

This paper investigates the stability of buoyant flow in a vertical cylindrical porous slab with permeable boundaries, revealing how aspect ratio influences the onset of convective instability.

Contribution

It provides a numerical analysis of the linear stability and critical Rayleigh number for buoyant flow in a cylindrical porous system with permeable boundaries, highlighting the effect of aspect ratio.

Findings

Instability increases with aspect ratio.

Critical Rayleigh number decreases as aspect ratio increases.

Flow becomes more unstable as the system approaches an infinite aspect ratio.

Abstract

The basic stationary buoyant flow in a vertical annular porous passage induced by a boundary temperature difference is investigated. The vertical cylindrical boundaries are considered both isothermal and permeable to external fluid reservoirs. There exists a stationary parallel velocity field with a zero flow rate and pure conduction heat transfer. Its linear stability is analysed with normal mode perturbations of the pressure and temperature fields. The transition to convective instability is caused by the basic horizontal temperature gradient. Hence, its nature differs from that of the usual Rayleigh-B\'enard instability. The linear dynamics of the perturbed flow is formulated as an eigenvalue problem, solved numerically. Its solution provides the neutral stability curve at each fixed aspect ratio between the external radius and the internal radius. The critical Rayleigh number triggering the instability is evaluated for different aspect ratios. It is shown that the system becomes more an more unstable as the aspect ratio increases, with the critical Rayleigh number dropping to zero when the aspect ratio tends to infinity.