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 C L = 2000, 3000 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.

Riccati-less approach for optimal control and estimation: an application to two-dimensional boundary layers

“…This procedure has been used for plane channel flow by Högberg, Bewley & Henningson (2003) and later extended for weakly spatially developing flows by Chevalier et al (2007a) and Monokrousos et al (2008). Apart for the limit imposed by the hypothesis of spatial invariance of the system, a drawback of this methodology is the necessity of introducing a distributed system of sensors and actuators.…”

Modeling of transitional channel flow using balanced proper orthogonal decomposition

“…Although the body force actuation we discuss here would be desirable in applications, it is not as practically feasible as wall blowing and suction, which has been studied extensively and has shown promise both in computations and experiment [12,21]. We also note that blowing and suction could be incorporated into the control framework described here in a relatively straightforward fashion, via the lifting approach described in Ref.…”

“…Recently, Min et al [22] have achieved sub-laminar drag through open-loop control using a traveling wave actuation on the channel walls. Lee et al [20] have reported drag reduction using closed-loop controllers developed for the linearized flow, while Joshi et al [15] and Högberg et al [12] have demonstrated significant reduction in the energy of the perturbations to laminar flow using closed-loop controllers. Despite these recent successes, the adequate modeling of the fluid flow and the actuators in a framework useful for practical control design is still a challenge.…”

Closed-loop control of boundary layer streaks induced by free-stream turbulence

“…Therefore results from the control of the full spectrum are presented in this section. Figure 14 shows the kernel function, defined by (35), for controlled and uncontrolled flows, a 2 = 10 −4 and η = 2. This wall-normal location is higher than that used for the uncontrolled case in Ref.…”