Instability of the buoyancy layer on an evenly heated vertical wall

Abstract

The stability of the buoyancy layer on a uniformly heated vertical wall in a stratified fluid is investigated using both semi-analytical and direct numerical methods. As in the related problem in which the excess temperature of the wall is specified, the basic laminar flow is steady and one-dimensional. Here flows varying in time and with height are considered, the behaviour being determined by the fluid's Prandtl number and a Reynolds number proportional to the ratio of two temperature gradients: the horizontal one imposed at the wall and the vertical one existing in the far field. For low Reynolds numbers, the flow is stable with variation only in the wall-normal direction. For Reynolds numbers greater than a critical value, depending on the Prandtl number, the flow is unstableand supports two-dimensional travelling waves. The critical Reynolds number and other properties have been obtained via linearized stability analysis and are shown to accuratelypredict the behaviour of the full nonlinear solution obtained numerically for Prandtl number 7. The stability analysis employs a novel Laguerre collocation scheme while the direct numerical simulations use a second-order finite volume method.