Wavenumber selection for small-wavelength Goertler vortices in curved channel flows

The problem of wavenumber selection for fully nonlinear, small-wavelength Goertler vortices in a curved channel flow is considered. These types of Goertler vortices were first considered by Hall & Lakin (1988) for an external boundary layer flow. They proved particularly amenable to asymptotic description, it was possible to consider vortices large enough so that the mean flow correction driven by them is as large as the basic state, and this prompted the authors to consider them in a curved channel flow as an initial application of the phase-equation approach to Goertler vortices. This involves the assumption that the phase variable of these Goertler vortices varies on slow spanwise and time scales, then an analysis of both inside and outside the core region, to which vortex activity is restricted, leads to a system of partial differential equations which can be solved numerically for the wavenumber. The authors consider in particular the effect on the wavenumber of the outer channel wall varying on the same slow spanwise scale as the phase variable.