Linear feedback control and estimation of transition in plane channel flow

Abstract

Modern linear control theory has recently been established as a viable tool for developing effective, spatially localized convolution kernels for the feedback control and estimation of linearized Navier–Stokes systems. In the present paper, the effectiveness of these kernels for significantly expanding the basin of attraction of the laminar state in a subcritical nonlinear channel flow system is quantified using direct numerical simulations for a range of Reynolds numbers ($Re_{\CL}=2000, 3000 {\rm and} 5000$) and for a variety of initial conditions of physical interest. This is done by quantifying the change in the transition thresholds (see Reddy et al. 1998) when feedback control is applied. Such transition thresholds provide a relevant measure of performance for transition control strategies even in the nonlinear regime. Initial flow perturbations with streamwise vortices, oblique waves, and random excitations over an array of several Fourier modes are considered. It is shown that the minimum amplitude of these initial flow perturbations that is sufficient to excite nonlinear instability, and thereby promote transition to turbulence, is significantly increased by application of the control feedback. The kernels used to apply the feedback are found to decay exponentially with distance far from the origin, as predicted by the analysis of Bamieh, Paganini & Dahleh (2002). In the present paper, it is demonstrated via numerical simulation that truncation of these spatially localized convolution kernels to spatially compact kernels with finite non-zero support does not significantly degrade the effectiveness of the control feedback. In addition to the new state-feedback control results, exponential convergence of a localized physical-space state estimator with wall measurements is also demonstrated. The estimator and the full-state feedback controller are then combined to obtain a wall-information-based linear compensator. The compensator performance is also quantified, and key issues related to improving the performance of this compensator, which is degraded compared with the full-state feedback controller, are discussed.