On the generation of mean flows by the interaction of Gortler vortices and Tollmien-Schlichting waves in curved channel flows

There are many fluid flows where the onset of transition can be caused by different instability mechanisms which compete in the nonlinear regime. Here the interaction of a centrifugal instability mechanism with the viscous mechanism which causes Tollmien-Schlichting waves is discussed. The interaction between these modes can be strong enough to drive the mean state; here the interaction is investigated in the context of curved channel flows so as to avoid difficulties associated with boundary layer growth. In the first instance the mean state driven by a vortex of short wavelength in the absence of a Tollmien-Schlichting wave is considered. When Tollmien-Schlichting waves are present then the nonlinear differential equation to determine the mean state is modified. The vortex and Tollmien-Schlichting wave structure in the nonlinear regime has viscous wall layers and internal shear layers; the thickness of the internal layers is found to be a function of the Tollmien-Schlichting wave amplitude.