Low-dimensional models of a temporally evolving free shear layer
M. Wei and C. W. Rowley
Journal of Fluid Mechanics 618 : 113–134, January 2009.
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). 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. A model with 2 complex modes can describe certain single-frequency features of the system, such as vortex roll-up, nonlinear saturation, and viscous damping. A model with 4 complex modes 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 simulations and model computations.
Full text: pdf