Stationary instability of the convective flow between differentially heated vertical planes

Abstract

An asymptotic theory describes the stationary instability of convective flow between differentially heated vertical planes at large Prandtl numbers. The theory is concerned with the structure for A [Gt ] 1 where A is a Rayleigh number based on the horizontal temperature difference and the distance between the planes. As such it is relevant to the instability of flow in a vertical slot of aspect ratio h [Gt ] 1 where the convective regime corresponds to order-one values of a non-dimensional parameter γ which partly depends on the vertical temperature gradient generated in the slot and can be approximated by γ4 = A/8h. Instability is shown to set in at a critical value of γ that compares well with experimental observation. The lower branch of the neutral curve conforms to a boundary-layer type approximation while the upper branch has a critical-layer structure midway between the planes which becomes fully developed as the first reversal of the vertical velocity of the base flow is encountered near the centreline.