Closed-loop control of unsteadiness over a rounded backward-facing step

Abstract

International audienceThe two-dimensional, incompressible flow over a rounded backward-facing step at Reynolds number Re = 600 is characterized by a detachment of the flow close to the step followed by a recirculation zone. Even though the flow is globally stable, perturbations are amplified as they are convected along the shear layer, and the presence of upstream random noise renders the flow unsteady, leading to a broadband spectrum of excited frequencies. This paper is aimed at suppressing this unsteadiness using a controller that converts a shear-stress measurement taken from a wall-mounted sensor into a control law that is supplied to an actuator. A comprehensive study of various components of closed-loop control design - covering sensor placement, choice and influence of the cost functional, accuracy of the reduced-order model, compensator stability and performance - shows that successful control of this flow requires a judicious balance between estimation speed and estimation accuracy, and between stability limits and performance requirements. The inherent amplification behaviour of the flow can be reduced by an order of magnitude if the above-mentioned constraints are observed. In particular, to achieve superior controller performance, the estimation sensor should be placed upstream near the actuator to ensure sufficient estimation speed. Also, if high-performance compensators are sought, a very accurate reduced-order model is required, especially for the dynamics between the actuator and the estimation sensor; otherwise, very minute errors even at low energies and high frequencies may render the large-scale compensated linearized simulation unstable. Finally, coupling the linear compensator to nonlinear simulations shows a gradual deterioration in control performance as the amplitude of the noise increases

Nonlinear control of unsteady finite-amplitude perturbations in the Blasius boundary-layer flow

“…For more complex flows, Fourier modes cannot be used, but the degrees of freedom of the problem need to be reduced as well. This has been accomplished by means of system identification techniques (see the ARMAX model used by Herve et al (2012) for controlling the flow over a backward-facing step) and projection techniques, such as Galerkin projection onto POD or balanced POD modes (for instance see the work by Barbagallo et al (2012) for controlling unsteadiness over a rounded backward-facing step, and Semeraro et al (2011) for a three-dimensional boundary-layer flow). The applicability of such linear strategies to the control of transition to turbulence is based on the hypothesis that linear models accurately represent the input-output dynamics of the transitional flow systems.…”

Feedback control of unstable flows: a direct modelling approach using the Eigensystem Realisation Algorithm

“…Åkervik et al 2007;Henningson & Åkervik 2008;Barbagallo et al 2011;Ehrenstein, Passaggia & Gallaire 2011) and 'proper orthogonal modes' (e.g. Aubry et al 1988;Podvin & Lumley 1998;Graham, Peraire & Tang 1999;Ravindran 2000aRavindran ,b, 2002Ravindran , 2006Prabhu, Collis & Chang 2001;Ma & Karniadakis 2002;Noack et al 2003;Gloerfelt 2008;Siegel et al 2008;Barbagallo et al 2009Barbagallo et al , 2012Tadmor et al 2010) can yield successful ROMs, but 'balanced modes' have been shown to lead to superior performance in terms of robustness and required model order in a number of studies (e.g. Willcox & Peraire 2002;Ilak & Rowley 2008;Bagheri et al 2009c;Barbagallo et al 2009;Dergham et al 2011).…”

“…For instance, the linear-quadratic-Gaussian (LQG) framework has been successfully used in numerous studies (e.g. Åkervik et al 2007;Huang & Kim 2008;Bagheri et al 2009a,c;Bagheri, Brandt & Henningson 2009b;Barbagallo, Sipp & Schmid 2009Semeraro et al 2011;Illingworth, Morgans & Rowley 2011Barbagallo et al 2012;Dadfar et al 2013;Juillet, Schmid & Huerre 2013;Fabbiane et al 2014Fabbiane et al , 2015. It is appealing for its theoretical optimality and intuitive structure, based on the design of a dynamic observer (Kalman filter), which optimally estimates the state of the system, and a linear-quadratic regulator (LQR), which minimises a chosen cost function.…”

On the wave-cancelling nature of boundary layer flow control

“…Differently from adaptive schemes, which have been more recently applied due to their increased robustness to plant uncertainties [17,45], LQG is computed off-line, using the reduced-order model, and remains unchanged during its operation on the nonlinear system. Examples of applications of LQG designed using reduced-order models may be found in the works of Barbagallo et al [6,7] for an oscillator flow and Barbagallo et al [5] for an amplifier flow. The reader is referred to the work of Schmid and Sipp for an overview of optimal control applied to flow problems [41].…”