Turbulent drag reduction by feedback: a Wiener-filtering approach

Modern control theory has recently been employed in the design of linear controllers[1], state estimators[2], and compensators[3; 4], in an attempt to devise control laws for reducing drag in turbulent wall flows. These approaches led to encouraging results, revealing the potential of linear control in targeting significant dynamics in wall turbulence[5]. All the aforementioned works, however, rely on an approximate state-space representation of the system dynamics, obtained by linearization of the governing equations. The statespace formulation reduces the compensator design problem to the solution of two matrix Riccati equations, a procedure that becomes computationally cumbersome for high dimensional systems. Effects of nonlinearities and modeling errors are accounted for by introducing state and measurement noises with known (approximately modeled) statistics. In contrast to previously proposed approaches, in this work we employ a linearized model of the wall-forced turbulent channel flow system in the form of an average impulse response function; such model can be directly measured with DNS using the procedure proposed in Luchini et al.[6]. These authors introduced small velocity perturbations at the channel walls in the form of a space-time white noise, and computed runtime the cross-correlation between the flow state and the wall forcing. Leveraging a well known result in linear system theory, they used the computed correlation function to define a linear impulse response function, representing the average linear dynamics of a turbulent channel flow when impulsive wall forcing is applied. A model given in the form of impulse response function would require first a statespace realization in order for standard Riccati-based control techniques to be applied. Instead of performing such realization – which would be impractical in the present very highdimensional setting – we employ a frequency domain formulation of the optimal compensator design problem, that allows us to directly use the Fourier transform of the impulse response function (i.e. the frequency response function) in the compensator design procedure. The block diagram of the feedback problem at hand is shown in fig. 1, where the feedback compensator having frequency response function K(ω) feeds the system whose frequency response is H(ω) with a signal u determined on the basis of real-time measurements y, obtained with the sensor C(ω). Note that both the measurement and the state x are corrupted by disturbances d and noise n, respectively, that are supposed to be uncorrelated. The spectral density functions of the disturbances and noise will be denoted by φdd(ω) and φnn(ω), respectively, and may be functions of the frequency. The goal is the design of an optimal feedback compensator such that the usual LQG expectation functional J = E{xQx + uRu} (1)