Time-delayed feedback technique for suppressing instabilities in time-periodic flow

A numerical method is presented that allows to compute time-periodic flow states, even in the presence of hydrodynamic instabilities. Themethod is based on filtering nonharmonic components by way of delayed feedback control, as introduced by Pyragas [Phys. Lett. A 170, 421 (1992)]. Its use in flow problems is demonstrated here for the case of a periodically forced laminar jet, subject to a subharmonic instability that gives rise to vortex pairing. The optimal choice of the filter gain, which is a free parameter in the stabilization procedure, is investigated in the context of a low-dimensional model problem, and it is shown that this model predicts well the filter performance in the high-dimensional flow system. Vortex pairing in the jet is efficiently suppressed, so that the unstable periodic flow state in response to harmonic forcing is accurately retrieved. The procedure is straightforward to implement inside any standard flow solver. Memory requirements for the delayed feedback control can be significantly reduced by means of time interpolation between checkpoints. Finally, the method is extended for the treatment of periodic problems where the frequency is not known a priori. This procedure is demonstrated for a three-dimensional cubic lid-driven cavity in supercritical conditions.