Low-dimensional models of a temporally evolving free shear layer

We develop low-dimensional models for the evolution of a free shear layer in a periodic domain. The goal is to obtain models simple enough to be analyzed using standard tools from dynamical systems theory, yet including enough of the physics to model nonlinear saturation and energy transfer between modes (e.g., pairing). Recently, experiments have suggested that high-frequency forcing of shear layers over open cavities may provide a mechanism for suppression of tones in cavities, and a long-term goal of this work is to study the dynamics of forced shear layers, to better understand these effects. In the present paper, 2D direct numerical simulations of a spatially periodic, temporally developing shear layer are performed. Low-dimensional models for these dynamics are obtained using a modified version of proper orthogonal decomposition/Galerkin projection, in which the basis functions can scale in space as the shear layer spreads. Equations are obtained for the rate of change of the shear layer thickness. When scaling is included in the shear flow dominated by k = 1 only, the first POD mode of wave number k = 1 captures 93% of the energy, which is impossible to obtain by regular POD analysis without scaling. For the flow dominated by both k = 1 and k = 2, when scaling is included, the first POD mode of wave number k = 1 and k = 2 together capture 95% of the total energy. Projection of incompressible Navier-Stokes equations to the first two POD modes of k = 1 gives a simple 2-mode model. If the projection is onto the first two POD modes of both k = 1 and k = 2, a more complex 4-mode model can be built to describe more complex flows. The 2-mode model can describe certain single-frequency features of the system, such as vortex roll-up, nonlinear saturation, and viscous damping. The 4-mode model can describe interactions between two frequencies (vortex merging) as well. The relation between the phase difference of the first (symmetric) and second (asymmetric) POD modes of the same wave number and the shear layer spreading rate can be clearly observed in both direct numerical simultions and model computations.