The nonlinear development of three-dimensional disturbances at hyperbolic stagnation points: A model of the braid region in mixing layers

Abstract

The properties of steady, two-dimensional flows with spatially uniform strain rates ε and rotation rates γ where ε2⩾γ2, and hence open, hyperbolic, streamlines are investigated. By comparison with a high resolution numerical simulation of a free shear layer, such a quadratic flow is an idealized local model of the “braid” region which develops between neighboring saturated Kelvin–Helmholtz billows in an unstable free shear layer. A class of exact three-dimensional nonlinear solutions for spatially periodic perturbations is derived. These solutions satisfy the condition that the amplitude of the time-varying wave number of the perturbation remains bounded in time, and hence that pressure plays an asymptotically small role in their dynamics. In the limit of long time, the energy of such perturbations in an inviscid flow grows exponentially, with growth rate 2ε2−γ2, and the perturbation pressure plays no significant role in the dynamic evolution. This asymptotic growth rate is not the maximal growth rate accessible to general perturbations, which may grow transiently at rate 2ε, independently of γ. However, almost all initial conditions lead to, at most, transient growth and hence finite asymptotic perturbation energy in an inviscid flow as time increases, due to the finite amplitude effects of pressure perturbations. Perturbations which do undergo significant transient growth take the form of streamwise-aligned perturbation vorticity which varies periodically in the spanwise direction. By comparison of this local model with a numerically simulated mixing layer, appropriately initialized “hyperbolic instabilities” appear to have significantly larger transient growth rates than an “elliptical instability” of the primary billow core. These hyperbolic instabilities appear to be a simple model for the spanwise periodic perturbations which are known to lead to the nucleation of secondary rib vortices in the braid region between adjacent billow cores.