On the nonlinear interaction of Görtler vortices and Tollmien-Schlichting waves in curved channel flows at finite Reynolds numbers

Abstract

The flow in a two-dimensional curved channel driven by an azimuthal pressure gradient can become linearly unstable owing to axisymmetric perturbations and/or non-axisymmetric perturbations depending on the curvature of the channel and the Reynolds number. For a particular small value of curvature, the critical Reynolds number for both these perturbations becomes identical. In the neighbourhood of this curvature value and critical Reynolds number, non-linear interactions occur between these perturbations. The Stuart-Watson approach is used to derive two coupled Landau equations for the amplitudes of these perturbations. The stability of the various possible states of these perturbations is shown through bifurcation diagrams. Emphasis is given to those cases that have relevance to external flows.