Concerning the interaction of non-stationary crossflow vortices in a three-dimensional boundary layer

Recently there has been much work devoted to considering some of the many and varied interaction mechanisms which may be operative in three-dimensional boundary-layer flows. Here we are concerned with resonant triads of crossflow vortices. In contrast to much of the previous work we examine the effects of interactions upon resonant triads where each member of the triad has the property of being linearly neutrally stable; then the importance of the interplay between modes can be relatively easily assessed. We concentrate on investigating modes within the boundary-layer flow above a rotating disc; this choice is motivated by the similarity between this disc flow and many important practical flows and, secondly, our selected flow is an exact solution of the Navier-Stokes equations which makes its theoretical analysis especially attractive. First we demonstrate that the desired triads of linearly neutrally stable modes can exist within the chosen boundary-layer flow and then subsequently obtain evolution equations to describe the development of the amplitudes of these modes once the interaction mechanism is accounted for. It is found that the coefficients of the interaction terms within the evolution equations are, in general, given by quite intricate expressions although some elementary numerical work shows that the evaluation of these coefficients is practicable. The basis of our work lends itself to generalization to more complicated boundary layers, and effects of detuning or non-parallelism could be provided for within the asymptotic framework.