Linear proportional–integral control for skin-friction reduction in a turbulent channel flow

In the present study, we apply a proportional (P)–integral (I) feedback control to a turbulent channel flow for skin-friction reduction. The instantaneous 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 PI control. The performance of PI controls is estimated by the change in the skin friction while varying the sensing plane location $y_{s}$ and the proportional and integral feedback gains ($\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ respectively). The opposition control proposed by Choi et al. (J. Fluid Mech., vol. 262, 1994, pp. 75–110) corresponds to a P control with $\unicode[STIX]{x1D6FC}=1$. When the sensing plane is located close to the wall ($y_{s}^{+}\lesssim 10$), PI controls result in greater skin-friction reductions than corresponding P controls. The root-mean-square (r.m.s.) sensing velocity fluctuations, considered as the control error, approach zero with successful PI controls, but do not with P controls. Successful PI controls reduce the strength of near-wall coherent structures and the r.m.s. velocity fluctuations above the wall apart from those near the wall due to the control input. The frequency spectra of the sensing velocity show that the I component of PI controls significantly reduces the energy at low frequencies, much more than P controls do. Proportional–integral controls are also applied to a linearized flow model having transient growth of disturbances. The performance of PI controls for a linearized flow model is very similar to that for a turbulent channel flow, i.e. the low-frequency components of disturbances are significantly reduced by the I component of PI controls, and the transient energy growth is suppressed more than by P controls.