On the Non-linear Evolution of Görtler Vortices in Non-parallel Boundary Layers

Abstract

The non-linear development of finite amplitude Görtler vortices in a non-parallel boundary layer on a curved wall is investigated using perturbation methods based on the smallness of e, the non-dimensional wavelength of the vortices. The crucial stage in the growth or decay of the vortices takes place in an interior viscous layer of thickness O(ε2) and length O(ε). In this region the downstream velocity component of the perturbation contains a mean flow correction of the same order of magnitude as the fundamental which is driving it. Moreover, these functions satisfy a pair of coupled non-linear partial differential equations which must be solved subject to some initial conditions imposed at a given downstream location. It is found that, depending on whether the boundary layer is more or less unstable downstream of this location, the initial disturbance either grows into a finite amplitude Görtler vortex or decays to zero. For the Blasius boundary layer on a concave wall it is found that Görtler vortices can only develop if the rate of increase of curvature of the wall is sufficiently large. In this case the finite amplitude solution which develops initially in an ε-neighbourhood of the position where the disturbance is introduced changes its structure further downstream. This structure is investigated at a distance O(εδ) (with 0< δ<1) downstream of the above ε-neighbourhood. In this régime the downstream fundamental velocity component has an elliptical profile over most of the flow field. However, in two thin boundary layers located symmetrically either side of the centre of the viscous layer the fundamental velocity component decays exponentially to zero. The locations of these layers are determined by an eigenvalue problem associated with the one-dimensional diffusion equation. The mean flow correction persists both sides of the boundary layer and ultimately decays exponentially to zero. This large amplitude motion is not sensitive to the imposed initial conditions and appears to be the ultimate state of any initial disturbance. However, in the initial stages of the growth of the vortex, some surprising flows are possible. For example, it is possible to set up a vortex flow similar to that observed by Wortmann (1969) which consists of a sequence of cells inclined at an angle to the vertical.