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.

Linear stability and sensitivity of the flow past a fixed oblate spheroidal bubble

“…(14) below. Note that in previous studies 16,17,23 , the surface (or 'wall') forcing was obtained only through a disturbance applied to the normal velocity (resulting in blowing or suction), whereas the present formulation allows us to study also the effect of a disturbance of the surface shear stress. Here, blowing or suction is of course irrelevant since we are dealing with a bubble and not with a rigid body.…”

“…16 for a review). Applying this approach to the flow past a bubble may shed some light on the similarities and differences resulting from the presence of a shear-free condition at the bubble surface, as opposed to the no-slip condition involved in all available studies, most of which focused on two-dimensional circular cylinders [17][18][19][20] and axisymmetric rigid bodies such as spheres, disks 21,22 or bullet-shaped bodies 23 .…”

On long-term boundedness of Galerkin models

“…The methodology is well established for linear systems (see e.g. Kim & Bewley 2007;Sipp et al 2010) and generalised for nonlinear systems (see e.g. Aamo & Krstić 2002;Khalil 2002).…”

Some observations regarding steady laminar flows past bluff bodies

“…However, such 'passive' flow control approaches will be unable to control subsequent instability modes (with the initial laminar ones analysed in [44]). Flow past a two-dimensional cylinder and over backwards steps have served as test problems for several active flow control strategies, both for the finite-Re case [45,46] and for Euler (point vortex) flows [47].…”