Three-dimensional secondary instability of a spatially developing von Kármán vortex street in a far wake

Abstract

This paper presents studies of a three-dimensional secondary instability of a spatially developing von Kármán vortex street. It develops owing to the nonlinear interaction between a two-dimensional mean far-wake flow and its most unstable disturbances. This forms a nonlinear primary wake flow. Sections of this flow are selected to perform a temporal secondary stability study under the assumption of parallel flow. The eigenvalue characteristics of the secondary instability are compared with the results from the use of a linear primary flow comprising unmodified mean wake flow coexisting with a linear primary fundamental disturbance with an empirical amplitude as a parameter, resulting in a simpler Floquet analysis. The maximum amplification rates occur at about the same spanwise wavenumber for both the nonlinear and linear primary flows, in qualitative agreement. But the amplification rate versus the spanwise wavenumber spectrum are both qualitatively and quantitatively different, the nonlinear primary flow results in a lower magnitude of the amplification rates. Some interpretations of controlled experiments are made, and it is concluded that the two- and three-dimensional disturbances so obtained appeared to be from the primary instability, where the amplification mechanisms come from the unmodified mean flow. A general discussion of the nonlinear interaction between the primary two-dimensional flow and the three-dimensional secondary instability is given, which may well form the basis for further nonlinear studies.