Flow control in laminar-separated flows such as bluff-body wakes or separation induced by an adverse pressure gradient over a flat-plate boundary layer remains a challenge for modern algorithms. Separation induced by the geometry of the flow induces recirculation regions whose dynamics is not yet fully understood. Depending on their characteristics, recirculation regions can be subject to self-excited instabilities also known as resonator dynamics and transient growth associated with the amplification of perturbations. Recent progress has been made in modeling and control of either the amplified dynamics or the resonator dynamics (Marquet et al. 2008) but control strategies capable of suppressing both dynamics simultaneously and drive the flow back to its steady state remain an open issue.

We address this problem using augmented Lagrangian optimization procedures to control a separated boundary layer over a bump. This particular flow geometry is known to exhibit a self-excited low-frequency flapping instability, characterized by a large scale oscillation of the recirculation region, while simultaneously amplifying perturbations localized upstream the separation region (Ehrenstein & Gallaire 2008, Passaggia et al. 2012). The control of this particular flow was already investigated using model reduction (Ehrenstein et al. 2011) and adjoint-based optimization methods (Passaggia & Ehrenstein 2013). However, neither of these approaches proved to be capable to control the flow back to its steady state and proved to lack robustness.

We circumvent this problem by controlling the baseflow and the instability dynamics using complementary approaches. Starting from a steady state forced by a suction actuator located near the separation point, the baseflow modification is shown to be controlled by an exponentially vanishing suction strategy. For weakly unstable flow regimes, this control law can be further optimized by means of direct-adjoint iterations of the nonlinear Navier-Stokes equations. In the absence of external noise, this novel approach proves to be capable of controlling the transient dynamics and the baseflow modification simultaneously. The present strategy allows of controlling the flow from a fully developed nonlinear state back to the steady state using a single actuator located at the separation point (Passaggia & Ehrenstein 2018).

Finally we will introduce methods to compute control kernels using the adjoint method. The aim is to make the link between optimization and reduced-order controllers capable of performing feedback control.

References:

O. Marquet, D. Sipp, J.-M. Chomaz, and L. Jacquin. J. Fluid Mech., 605:429–443, 2008.

U. Ehrenstein and F. Gallaire. J. Fluid Mech., 614:315--327, 2008.

U. Ehrenstein, P-Y. Passaggia, and F. Gallaire. Theor. Comput. Fluid Dyn., 25:195–207, 2011.

P.-Y. Passaggia and U. Ehrenstein. Eur. J. Mech. B/Fluids, 41:169–177, 2013.

P.-Y. Passaggia, T. Leweke, and U. Ehrenstein. J. Fluid Mech., 703:363–373, 7 2012.

P.-Y. Passaggia and U. Ehrenstein. J. Fluid Mech., 840:238–265, 2018.