The three-dimensional evolution of a plane mixing layer: the Kelvin–Helmholtz rollup

The Kelvin–Helmholtz rollup of three-dimensional temporally evolving plane mixing layers with an initial Reynolds number of 500 based on vorticity thickness and half the velocity difference have been simulated numerically. All simulations were begun from a few low-wavenumber disturbances, usually derived from linear stability theory, in addition to the mean velocity profile. A standard set of ‘clean’ structures develops in the majority of the simulations. The spanwise vorticity rolls up into a corrugated spanwise roller with vortex stretching creating strong spanwise vorticity in a cup-shaped region at the bends of the roller. Predominantly streamwise rib vortices develop in the braid region between the rollers. For sufficiently strong initial three-dimensional disturbances these ribs ‘collapse’ into compact axi-symmetric vortices. The rib vortex lines connect to neighbouring ribs and are kinked in the direction opposite to that of the roller vortex lines. Because of this, these two sets of vortex lines remain distinct. For certain initial conditions, persistent ribs do not develop. In such cases, the development of significant three-dimensionality is delayed. In addition, simulations of infinitesimal three-dimensional disturbances evolving in a two-dimensional mixing layer were performed. Many features of the fully nonlinear flows are remarkably well predicted by the linear computations. Such computations can thus be used to predict the degree of three-dimensionality in the mixing layer even after the onset of nonlinearity. Several nonlinear effects can also be identified by comparing linear and nonlinear computations. These include the collapse of rib vortices, the formation of cups of spanwise vorticity, and the appearance of spanwise vorticity with sign opposite that of the mean vorticity. These nonlinear effects have been identified as precursors of the transition to turbulence (Moser & Rogers 1991).