Linear proportional-integral control of turbulent channel ow for skin-friction reductionLinear proportional-integral control of turbulent channel ow for skin-friction reduction

Abstract

In the present study, proportional (P) and proportional-integral (PI) feedback control methods are applied to a turbulent channel flow when Reτ = 140 as a means of drag reduction. The control strategy comes from the opposition control method proposed by Choi, Moin & Kim (J. Fluid Mech., Vol. 262, 1994, pp. 75-110) which is a proportional control method with a fixed control gain. The wall-normal velocity at a sensing plane above the wall is measured as a sensing parameter, and blowing/suction is provided at the wall based on the control strategies. The performance of the control methods is investigated by the direct measurement of the drag in a direct numerical simulation while varying the sensing plane location y+ s , the proportional gain α, and the integral gain β. For the P control, as ys+ increases, the drag decreases, reaches the minimum at an optimum sensing position and increases significantly. The effects of α are also investigated. As α increases the sensing velocity fluctuations decrease as 1/(1 + α), resulting in drag decreases. With smaller α, the amount of drag reduction becomes smaller while the range of ys+ reducing drag becomes wider. With large α (α > 1), the drag increases significantly. Thus, other control strategies, such as I control, are needed for more drag reduction. The PI control results in greater drag reduction than the P control when the sensing plane locates very close to the wall (ys+ < 10). The sensing velocity fluctuations, considered as an error in the control, approach zero with the PI control, while they do not go to zero with the P control. From the frequency spectra of sensing velocity fluctuations, it is found that the P control reduces the fluctuations at all frequency range, furthermore the I component of the control effectively reduces the sensing velocity fluctuations at low frequency range. The performance of the control methods is also investigated in a linearized flow model. From the frequency response of the system, it is found that the I component of the control effectively reduces the sensing velocity fluctuations at low frequency. Furthermore, the performance of control methods is investigated by testing the ability of suppressing the transient energy growth of disturbances in the linearized flow model. The variation of the maximum transient energy growth by the control methods in a linearized flow model is very similar to the drag variation in a turbulent channel flow. When the sensing plane locates very near the wall (ys+ < 10), the PI control shows better performance than P control. When the sensing plane moves away from the wall (ys+ > 10), the maximum transient growth ratio increases significantly with the PI control. This indicates that the linearized flow model can be used as a guideline for control designs for drag reduction.