The effect of a shear flow on convection in a layer heated non-uniformly from below

Abstract

In an earlier paper (Walton 1982) we showed that, when a fluid layer is heated non-uniformly from below in such a way that the vertical temperature difference maintained across the layer is a slowly varying monotonic function of a horizontal coordinate x, then convection occurs for x > xc, where xc is the point where the local Rayleigh number is equal to the critical value for a uniformly heated layer. Furthermore, the amplitude of the convection increases smoothly from exponentially small values for x [Lt ] xc and asymptotes to a value given by Stuart–Watson theory for x [Gt ] xc.At the present time no solutions of this kind are available for a class of problems in which the onset of instability is heavily influenced by a shear flow (e.g. Görtler vortices in a boundary layer on a curved wall, convection in a heated Blasius boundary layer). In a first step to bridge the gap between these problems and in order to elucidate the difficulties associated with the presence of a shear flow, we investigate the effect of a (weak) shear flow on our earlier convection problem. We show that the onset of convection is delayed and that it appears more suddenly, but still smoothly. The role of horizontal diffusion is shown to be of paramount importance in enabling a solution of this kind to be found, and the implications of these results for instabilities in higher-speed shear flows are discussed.