Low-dimensional model of spatial shear layers

Abstract

The aim of this work is to develop nonlinear low-dimensional models to describe vortex dynamics in spatially developing shear layers with periodicity in time. By allowing a free variable g(x) to dynamically describe downstream thickness spreading, we are able to obtain basis functions in a scaled reference frame and construct effective models with only a few modes in the new space. To apply this modified version of proper orthogonal decomposition (POD)/Galerkin projection, we first scale the flow along y dynamically to match a template function as it is developing downstream. In the scaled space, the first POD mode can capture more than 80% energy for each frequency. However, to construct a Galerkin model, the second POD mode plays a critical role and needs to be included. Finally, a reconstruction equation for the scaling variable g is derived to relate the scaled space to physical space, where downstream spreading of shear thickness occurs. Using only two POD modes at each frequency, our models capture the basic dynamics of shear layers, such as vortex roll-up (from a one-frequency model) and vortex-merging (from a two-frequency model). When arbitrary excitation at different harmonics is added to the model, we can clearly observe the promoting or delaying/eliminating vortex merging events as a result of mode competition, which is commonly demonstrated in experiments and numerical simulations of shear layers.