Evolution of coherent structures in shear flows: A phase dynamics approach

Abstract

A general amplitude evolution model is proposed to describe the development of instability waves in parallel free shear flows which are invariant under both space translations and Galilean transformations. It is argued that temporal and spatial modulations of finite-amplitude states are strongly coupled with a marginally unstable large-scale horizontal velocity field. As a result, an array of spatially periodic coherent structures is shown to exhibit two distinct types of phase instabilities: an Eckhaus-like modulational instability dominated by two-dimensional disturbances and a fully three-dimensional secondary instability generated by coupling between pattern deformation and advection by the large scale field. Anisotropic phase waves can also travel on the basic array, the global velocity field acting as a restoring force. Applications of these concepts to the phase dynamics of Kelvin-Helmholtz vortices are briefly considered.