Advances in the Design of High-Performance Flow Control

Abstract

This thesis tackles challenges in feedback control design for fluid flows, from multiple angles and approaches. It covers three major facets—stability theory, control, and reduced-order modeling—and it investigates three major challenges of these facets: nonlinearity, high dimensionality, and non-normality.

The dissertation begins with a discussion of global stability via linearized Navier–Stokes eigendecompositions, including numerical algorithms for this analysis. This section then investigates the global stability of a pipe flow through a T-shaped bifurcation at mid-hundred Reynolds numbers, which exhibits vortex breakdown. The recirculation and sensitivity regions closely coincide, which we explain using an inviscid short-wavelength perturbation theory. We also discuss the stability and receptivity properties of this flow.

The second part discusses feedback control design for fluid flows, including optimal actuator and sensor placement. It presents an algorithm that computes the gradient of a control measure with respect to such placements, allowing an efficient gradient-based optimization. The implementation on the linearized Ginzburg–Landau and the Orr–Sommerfeld/Squire models of fluid flow reveals that common methods for placement, such as global mode analysis, are suboptimal. We discuss heuristics, including sensitivity, that may predict optimal placements.

The third part covers reduced-order flow modeling. It examines previously unknown properties of dynamic mode decomposition (DMD)—a data-based modeling technique—including the uniqueness of the numerical algorithm and the boundary conditions of DMD-based models. We also propose an “optimized” DMD that produces less spurious decompositions, and gives the user control over the number of output modes. We show examples from the two-dimensional laminar flow over a cylinder. This part also investigates the stability and performance of high dimensional (e.g., fluid) systems in closed-loop with reduced-order controllers, since such control design is typically necessary for computational tractability. Theorems based on the normalized coprime factorization and ν-gap metric provide sufficient conditions for stability and performance. These conditions can also determine model reduction orders for which the stability or performance of reduced-order control is guaranteed. We demonstrate this on the control of the linearized Ginzburg–Landau system.