Finite-amplitude crossflow vortices, secondary instability and transition in the rotating-disk boundary layer

Abstract

In the three-dimensional boundary layer produced by a rotating disk, the experimentally well-documented sharp transition from laminar to turbulent flow is shown to coincide with secondary absolute instability of the naturally selected primary nonlinear crossflow vortices. Fully saturated primary finite-amplitude waves and the associated nonlinear dispersion relation are first numerically computed using a local parallel flow approximation. Exploiting the slow radial development of the basic flow, the naturally selected primary self-sustained flow structure is then derived by asymptotic analysis. In this state, outward-spiralling nonlinear vortices are initiated at the critical radius where primary absolute instability first occurs. A subsequent secondary stability analysis reveals that as soon as the primary nonlinear waves come into existence they are absolutely unstable with respect to secondary perturbations. Secondary disturbances growing in time at fixed radial locations continuously perturb the primary vortices, thus triggering the direct route to turbulence prevailing in this configuration.