On the effect of crossflow on nonlinear Görtler vortices in curved channel flows

It is known that the two-dimensional flow through a curved channel of narrow width is unstable to nonlinear vortices for sufficiently high Taylor-Görtler numbers and that the instability arises because of centrifugal effects. In many practical applications the basic flow of interest is three-dimensional and here we look for nonlinear, small-wavelength vortices in a three-dimensional and here we look for nonlinear, small-wavelength vortices in a three-dimensional flow through a curved channel and show that the crossflow needed to dramatically change the results for the case of a two-dimensional basic flow is extremely tiny. We derive the governing equation for constant-sized, streamwise-independent nonlinear vortices and by using a combination of asymptotics, numerical methods and phase-plane techniques we describe the solutions of this equation. Significantly, these solutions suggest that above a certain critical value for the crossflow the vortices cannot persist and that for lesser crossflows two different structures for the vortices are possible. The relevance of the results found here to the important case of external boundary-layer flows (where the effect of boundary-layer growth plays a crucial role) is indicated and suggestions for extensions of this work are made.