We investigate the detailed nature of the ‘mixing transition’ through which turbulence
may develop in both homogeneous and stratified free shear layers. Our focus is
upon the fundamental role in transition, and in particular the associated ‘mixing’
(i.e. small-scale motions which lead to an irreversible increase in the total potential
energy of the flow) that is played by streamwise vortex streaks, which develop once
the primary and typically two-dimensional Kelvin–Helmholtz (KH) billow saturates
at finite amplitude.Saturated KH billows are susceptible to a family of three-dimensional secondary
instabilities. In homogeneous fluid, secondary stability analyses predict that the stream-wise vortex streaks originate through a ‘hyperbolic’ instability that is localized in the
vorticity braids that develop between billow cores. In sufficiently strongly stratified
fluid, the secondary instability mechanism is fundamentally different, and is associated
with convective destabilization of the statically unstable sublayers that are created as
the KH billows roll up.We test the validity of these theoretical predictions by performing a sequence of
three-dimensional direct numerical simulations of shear layer evolution, with the
flow Reynolds number (defined on the basis of shear layer half-depth and half
the velocity difference) Re = 750, the Prandtl number of the fluid Pr = 1, and
the minimum gradient Richardson number Ri(0) varying between 0 and 0.1. These
simulations quantitatively verify the predictions of our stability analysis, both as to
the spanwise wavelength and the spatial localization of the streamwise vortex streaks.
We track the nonlinear amplification of these secondary coherent structures, and
investigate the nature of the process which actually triggers mixing. Both in stratified
and unstratified shear layers, the subsequent nonlinear amplification of the initially
localized streamwise vortex streaks is driven by the vertical shear in the evolving
mean flow. The two-dimensional flow associated with the primary KH billow plays
an essentially catalytic role. Vortex stretching causes the streamwise vortices to extend
beyond their initially localized regions, and leads eventually to a streamwise-aligned
collision between the streamwise vortices that are initially associated with adjacent
cores.It is through this collision of neighbouring streamwise vortex streaks that a final
and violent finite-amplitude subcritical transition occurs in both stratified and
unstratified shear layers, which drives the mixing process. In a stratified flow with
appropriate initial characteristics, the irreversible small-scale mixing of the density
which is triggered by this transition leads to the development of a third layer within
the flow of relatively well-mixed fluid that is of an intermediate density, bounded by
narrow regions of strong density gradient.
On the Finite Amplitude Oscillations in a Viscous Two-Dimensional Fluid
