Optimized Feedback Control of Vortex Shedding on an Inclined Flat Plate

This thesis examines flow control and the potentially favorable effects of feedback, associated with unsteady actuation in separated flows over airfoils. The objective of the flow control is to enhance lift at post-stall angles of attack by changing the dynamics of the wake vortices. We present results from a numerical study of unsteady actuation on a two-dimensional flat plate at post-stall angles of attack at Reynolds number (Re) of 300 and 3000. At Re=300, the control waveform is optimized and a feedback strategy is developed to optimize the phase of the control relative to the lift with either a sinusoidal or the optimized waveform, resulting in a high-lift limit cycle of vortex shedding. Also at Re=3000, we show that certain frequencies and actuator waveforms lead to stable (high-lift) limit cycles, in which the flow is phase locked to the actuation. First, a two-dimensional flat plate model at a high angle of attack at a Re of 300 is considered. We design the feedback to slightly adjust the frequency and/or phase of actuation to lock it to a particular phase of the lift, thus achieving a phase-locked flow with the maximal period-averaged lift over every cycle of actuation. With the sinusoidal forcing and feedback, we show that it is possible to optimize the phase of the control relative to the lift in order to achieve the highest possible period-averaged lift in a consistent fashion. However, continuous sinusoidal forcing could be adding circulation when it is unnecessary, or undesirable. Thus we employ an adjoint-based optimization in order to find the waveform (time history of the jet velocity) that maximizes the lift for a given actuation amplitude. The adjoint of the linearized perturbed equations is solved backwards in time to obtain the gradient of the lift to changes in actuation (the jet velocity), and this information is used to iteratively improve the controls. Optimal control provides a periodic control waveform, resulting in high lift shedding cycle with minimal control input. However, if applied in open loop, the flow fails to phase lock onto the optimal waveform, degrading the lift performance. Thus, the optimized waveform is also implemented in a closed-loop controller where the control signal is shifted or deformed periodically to adjust to the (instantaneous) frequency of the lift fluctuations. The feedback utilizes a narrowband filter and an Extended Kalman Filter to robustly estimate the phase of vortex shedding and achieve phase-locked, high lift flow states. Feedback control of the optimized waveform is able to reproduce the high-lift limit cycle from the optimization, but starting from an arbitrary phase of the baseline limit cycle. Finally, we apply the tools developed and knowledge gained at Re=300 to a Re of 3000 on a thin airfoil with a thickness-to-chord ratio of 4%, which were chosen to match the experimental studies of Greenblatt et al. (2008). We consider more detailed time-dependent aspects of the lift and corresponding flow fields, particularly the flow structures at the minimum and maximum lift, and the phase of pulses relative to the lift, in order to more precisely compare different actuated flow fields and distinguish the differences responsible for higher or lower instantaneous lift, along with identifying different vortex evolutions. We consider two representative angles of attack, 10 and 20 degrees, and investigate the lift enhancement and which combinations of forcing frequency and duty cycle lead to phase-locked flow. Finally, we show that for certain frequencies and actuator waveforms, there occur stable limit cycles in which the flow is phase locked to the actuation.