<tex>$H_infty $</tex>Control of Nonperiodic Two-Dimensional Channel Flow

Abstract

This paper deals with finite-dimensional boundary control of the two-dimensional (2-D) flow between two infinite parallel planes. Surface transpiration along a few regularly spaced sections of the bottom wall is used to control the flow. Measurements from several discrete, suitably placed shear-stress sensors provide the feedback. Unlike other studies in this area, the flow is not assumed to be periodic, and spatially growing flows are considered. Using spatial discretization in the streamwise direction, frequency responses for a relevant part of the channel are obtained. A low-order model is fitted to these data and the modeling uncertainty is estimated. An controller is designed to guarantee stability for the model set and to reduce the wall-shear stress at the channel wall. A nonlinear Navier-Stokes PDE solver was used to test the designs in the loop. The only assumption made in these simulations is that the flow is two dimensional. The results showed that, although the problem was linearized when designing the controller, the controller could significantly reduce fundamental 2-D disturbances in practice. Index Terms-control, flow control, model validation, nonperiodic flows. I. INTRODUCTION F LOW control is currently attracting considerable interest in the fluids research community. A major motivation for this is the possibility of reducing drag on a body by preventing or delaying transition from laminar to turbulent flow. The systems dealt with in these problems are, in control terms, very complex, nonlinear, and infinite dimensional, even if, in fluid mechanical terms, the structure of the flow field is simple in nature. Plane Poiseuille flow, i.e., flow between two infinite parallel plates is one of the simplest and best understood cases of fluid dynamics. Controlling this flow is, however, still an extremely challenging problem, even if it is assumed that deviations from the steady state are small enough for the governing equations to be linearized. It has become a benchmark problem for developing control algorithms for fluid flows and was considered in [4], [6], [9], [14]-[16] among others. All these references make a fundamental assumption that the flow is spatially periodic in the streamwise direction, and hence that a Fourier-Galerkin decomposition can be used to obtain independent dynamics for Manuscript

Localised estimation and control of linear instabilities in two-dimensional wall-bounded shear flows

“…The application of linear control theory to fluid flows is therefore considered as a viable route to suppress instabilities and delay transition for reducing skin-friction drag (Joshi et al 1997;Cortelezzi & Speyer 1998;Bewley & Liu 1998;Högberg et al 2003a;Baramov et al 2004;Chevalier et al 2007;Bagheri et al 2009b;Semeraro et al 2013;Jones et al 2015). In particular, optimal multivariable control strategies (LQG/H 2 , H ∞ ) (Zhou et al 1996;Skogestad & Postlethwaite 2005) have been successfully applied, see Kim & Bewley (2007); ; for an in-depth review on this subject.…”

Dynamically correct formulations of the linearised Navier–Stokes equations

“…The different formulations of the governing equations are discretised in space on computational meshes with grid spacing δ in both the x and y directions. Second-order accurate centred finite difference schemes are used for spatial derivatives as this yields simple interconnections between neighbouring node subsystems, which enables efficient evaluation of system frequency response by chaining nodes together, as demonstrated in [17,15]. Such finite difference schemes are of the form…”

Modelling for robust feedback control of fluid flows

“…Given the importance of suppressing turbulence for reducing skin-friction drag, much attention has been focussed on designing controllers for wall-bounded flows, particularly plane channel-flows (Bewley & Liu 1998;Lee et al 2001;Hogberg et al 2003;Baramov et al 2004;Hoepffner et al 2005;Kim & Bewley 2007). Kim (2003) examined different types of Linear Quadratic Regulator (LQR), also for turbulent channel flow, to minimise (1) wall-shear stress fluctuations, (2) turbulent kinetic energy, and (3), the linear coupling term.…”

“…Although the ability to reduce the effects of uncertainty is an inherent feature of any control system employing feedback, certain branches of control theory handle the effects of uncertainty in a more rigorous fashion than others. Robust control (Zhou et al 1996;Zhou & Doyle 1998;Dullerud & Paganini 2000), comprising a family of H ∞ design methods (Glad & Ljung 2000;Skogestad & Postlethwaite 2005) are of particular importance in this respect, and successful application of these methods has been demonstrated upon fluid flows (Bewley & Liu 1998;Baramov et al 2004;Luaga & Bewley 2004;Bobba 2004). An attractive feature of robust control is its ability to provide a priori guarantees concerning the degree of stability of the closed-loop system, subject to model uncertainty and exogenous disturbances.…”