Dynamics and Control of Global Instabilities in Open-Flows: A Linearized Approach

Abstract

This review article addresses the dynamics and control of low-frequency unsteadiness, as observed in some aerodynamic applications. It presents a coherent and rigorous linearized approach, which enables both to describe the dynamics of commonly encountered open-flows and to design open-loop and closed-loop control strategies, in view of suppressing or delaying instabilities. The approach is global in the sense that both cross-stream and streamwise directions are discretized in the evolution operator. New light will therefore be shed on the streamwise properties of open-flows. In the case of oscillator flows, the unsteadiness is due to the existence of unstable global modes, i.e., unstable eigenfunctions of the linearized Navier–Stokes operator. The influence of nonlinearities on the dynamics is studied by deriving nonlinear amplitude equations, which accurately describe the dynamics of the flow in the vicinity of the bifurcation threshold. These equations also enable us to analyze the mean flow induced by the nonlinearities as well as the stability properties of this flow. The open-loop control of unsteadiness is then studied by a sensitivity analysis of the eigenvalues with respect to base-flow modifications. With this approach, we manage to a priori identify regions of the flow where a small control cylinder suppresses unsteadiness. Then, a closed-loop control approach was implemented for the case of an unstable open-cavity flow. We have combined model reduction techniques and optimal control theory to stabilize the unstable eigenvalues. Various reduced-order-models based on global modes, proper orthogonal decomposition modes, and balanced modes were tested and evaluated according to their ability to reproduce the input-output behavior between the actuator and the sensor. Finally, we consider the case of noise-amplifiers, such as boundary-layer flows and jets, which are stable when viewed in a global framework. The importance of the singular value decomposition of the global resolvent will be highlighted in order to understand the frequency selection process in such flows.