Non-stationary cross-flow vortices in three-dimensional boundary-layer flows

Abstract

The upper-branch neutral stability of three-dimensional disturbances imposed on a three-dimensional boundary-layer profile is considered and in particular we investigate non-stationary cross-flow vortices. The wave speed is taken to be small initially and with a disturbance structure analogous to that occurring in two-dimensional boundary-layer stability the linear and nonlinear eigenrelations are derived for profiles with more than one critical layer. For the flow due to a rotating disc we show that linear viscous neutral modes exist for all wave angles between 10.6° and 39.6°. As the extremes of this range are approached the flow structure evolves to another, either the viscous mode of Hall ( Proc. R. Soc. Lond . A 406, 93(1986)) or the non-stationary inviscid modes considered by Stuart in Gregory et al . ( Phil. Trans. R. Soc. Lond . A 248, 155 (1955)). In the former case, corresponding to wave angles of 39.6°, the waves become almost stationary and in the latter case, with wave angles of 10.6° the waves are travelling much faster with a disturbance structure based on the Rayleigh scalings. The analysis is extended to include O (1) wavespeeds and we show that as the wavespeed of the cross-flow vortex approaches the tree-stream value, the corresponding disturbance amplitude increases, the growth here being slower than that for two-dimensional boundary layers.