Extreme model reduction of shear layers

The aim of this research is to develop nonlinear low-dimensional models (LDMs) to describe vortex dynamics in shear layers. A modified Proper Orthogonal Decomposition (POD)/Galerkin projection method is developed to obtain models at extremely low dimension for shear layers. The idea is to dynamically scale the shear layer along y direction to factor out the shear layer growth and capture the dynamics by only a couple of modes. The models are developed for two flows, incompressible spatially developing and weakly compressible temporally developing shear layers, respectively. To capture basic dynamics, the low-dimensional models require only two POD modes for each wavenumber/frequency. Thus, a two-mode model is capable of representing single-wavenumber/frequency dynamics such as vortex roll-up, and a four-mode model is capable of representing the nonlinear dynamics involving a fundamental wavenumber/frequency and its subharmonic, such as vortex pairing/merging. Most of the energy is captured by the first mode of each wavenumber/frequency, the second POD mode, however, plays a critical role and needs to be included. In the thesis, we first apply the approach on temporally developing weakly compressible shear layers. In compressible flows, the thermodynamic variables are dynamically important, and must be considered. We choose isentropic Navier-Stokes equations for simplicity, and choose a proper inner product to present both kinetic energy and thermal energy. Two cases of convective Mach numbers are studied for low compressibility and moderate compressibility. Moreover, we study the sensitivity of the compressible four-mode model to several flow parameters: Mach number, the strength of initial perturbations of the fundamental and its subharmonic, and Reynolds number. Secondly we apply the approach on spatially developing incompressible shear layers with periodicity in time. We consider a streamwise parabolic form of the Navier-Stokes equations. When we add arbitrary excitation at different harmonics to the model, we observe the promoting or delaying/eliminating of vortex merging events as a result of mode competition. To study coherent structures in shear layers, we solve the Direct Lyapunov Exponents (DLEs) to identify the Lagrangian coherent structures (LCS). The negative-time LCS provide structures similar to the ones shown by flow visualization in experiments. The positive-time LCS are also important in describing the dynamics. Both negative and positive LCS are plotted together to give a complete picture of dynamics in shear layers.