Optimal feedback control of turbulent channel owBy

Feedback control equations have been developed and tested for computing wall-normal control velocities to control turbulent ow in a channel with the objective of reducing drag. The technique used is the minimization of a \cost functional" which is constructed to represent some balance of the drag integrated over the wall and the net control eeort. A distribution of wall velocities is found which minimizes this cost functional some time shortly in the future based on current observations of the ow near the wall. Preliminary direct numerical simulations of the scheme applied to turbulent channel ow indicates it provides approximately 17% drag reduction. The mechanism apparent when the scheme is applied to a simpliied ow situation is also discussed. It is the goal of this project to study methods to counteract near-wall vortical structures in turbulent boundary layer ow using an active control system in an eeort to reduce drag. From this study, we hope to better understand the physics of drag producing events and the sensitivity of boundary layer ow to control. As a more far-reaching goal, we would like to better understand how to develop control equations for general ow control problems, utilizing the equations governing uid ow to achieve performance that is in some sense optimal for a given situation. With a well-chosen scheme using wall control only, it has been shown that a turbulent ow may be smoothed out in a near-wall region, and the drag substantially reduced. This scheme applies small amounts of wall-normal blowing and suction through the computational equivalent of holes drilled in the wall. Previous ad hoc schemes by Choi et al. (1992) have reduced the drag by as much as 20% by countering the vertical velocity slightly above the wall with an equal but opposite control velocity at the wall. The objective of this work is to derive more eeective schemes by applying optimal control theory, utilizing the equations of motion of the uid to reveal the dominant physics of the control problem and the most eecient distribution of the control energy. This work is an outgrowth of the work done by Choi et al. (1993), where optimal control theory was applied to the stochastic Burgers equation. Here, we apply the theory to the Navier-Stokes equations, which necessitates a more involved treatment of the equations and more extensive computer resources. The scheme discussed in this report depends on measurements of ow velocities …