Model Reduction and Feedback Control of Transitional Channel Flow

This dissertation examines the use of reduced-order models for design of linear feedback controllers for fluid flows. The focus is on transitional channel flow, a canonical shear flow case with a simple geometry yet complex dynamics. Reduced-order models of the linearized Navier-Stokes equations, which describe the evolution of perturbations in transitional channel flow, are computed using two methods for snapshot-based balanced truncation, Balanced Proper Orthogonal Decomposition (BPOD) and Eigensystem Realization Algorithm (ERA). The performance of these models in feedback control is evaluated in both linearized and nonlinear Direct Numerical Simulations (DNS) of channel flow. The first part of the dissertation describes the application of BPOD to very large systems, and the detailed evaluation of the resulting reduced-order models. Exact balanced truncation, a standard method from control theory, is not computationally tractable for very large systems, such as those typically encountered in fluid flow simulations. The BPOD method, introduced by Rowley (2005), provides a close approximation. We first show that the approximation is indeed close by applying the method to a 1-D linear perturbation to channel flow at a single spatial wavenumber pair, for which exact balanced truncation is tractable. Next, as the first application of BPOD to a very high-dimensional linear system, we show that reduced-order BPOD models of a localized 3-D perturbation capture the dynamics very well. Moreover, the BPOD models significantly outperform standard Proper Orthogonal Decomposition (POD) models, as illustrated by a striking example where models using the POD modes that capture most of the perturbation energy fail to capture the perturbation dynamics. Next, reduced-order models of a complete control system for linearized channel flow are obtained using ERA, a computationally efficient method that results in the same reduced-order models as BPOD. Linear Quadratic Gaussian (LQG) compensators, which include a reduced-order estimator based on a small number of velocity measurements, are designed for these models and used for feedback control of the energy growth of a localized perturbation near the channel wall. The performance of both a localized body-force near the channel wall and wall blowing/suction as actuation mechanisms is first studied in linearized DNS. It is found that the linear compensators are successful in reducing the growth of the perturbation energy, and that the body force actuation results in a larger decrease of the perturbation energy growth than actuation using wall blowing/suction. We then proceed to show that these compensators are also able to prevent transition to turbulence for nonlinear simulations in some cases, despite performance limitations imposed by the spatial separation of the perturbation and the actuator. Finally, since it is found that a fundamentally nonlinear mechanism of transition is not captured by the linear models, it is of interest to study nonlinear models for flow control. As a first step towards investigating nonlinear balanced truncation models of channel flow, a method for empirical nonlinear balanced truncation proposed by Lall et al. (2002) is tested on a nonlinear 1-D model problem, the Complex Ginzburg-Landau (CGL) equation. The performance of the resulting models is compared to the performance of nonlinear models obtained by projection of the full equation onto modes computed via balanced truncation of the linear part of the CGL equation. It is found that the models obtained by the latter approach are not only able to capture the dynamics of the nonlinear CGL equation, but that they also...