Linear proportional-integral control of turbulent channel flow for drag reduction

Choi, Moin \& Kim (1994, JFM) applied an opposition control, $v_w = - v_{y^+\approx 10}$, to turbulent channel flow and obtained about 25 \% drag reduction, where $v_w$ is the blowing and suction at the wall, and $v$ is the wall-normal velocity. The idea in that study was to provide a distributed blowing/suction at the wall opposite to the induced motion by the near-wall streamwise vortices and to reduce their strength, resulting in drag reduction. In the present study, we reconsider this control problem from the view point of linear proportional- integral-differential control. The opposition control by Choi et al. (1994) is a proportional control and thus contains steady- state errors. In other words, the target sensing velocity does not go to zero ($v_{y^+\approx 10} \ne 0$) even after control. To reduce this steady-state errors, we introduce a proportional- integral (PI) control, $v_w = - \alpha~ v_{y^+_s} - \beta \int v_ {y^+_s} dt$, where $\alpha$ and $\beta$ are the feedback gains, and $y^+_s$ is the sensing location above the wall. As a result of applying the PI control, the steady-state errors are significantly reduced and the effective sensing region becomes wide. The detailed results by varying the feedback gains and sensing location will be shown in the presentation.