Stabilization of exact coherent structures in two-dimensional turbulence using time-delayed feedback

Time-delayed feedback control, attributed to Pyragas [Phys. Lett. A 170, 421 (1992)], is a method known to stabilize periodic orbits in low-dimensional chaotic dynamical systems. A system of the form ẋ (t) = f (x) has an additional term G(x(t - T) - x(t)) introduced where G is some "gain matrix" and T a time delay. The form of the delay term is such that it will vanish for any orbit of period T, therefore making it also an orbit of the uncontrolled system. This noninvasive feature makes the method attractive for stabilizing exact coherent structures in fluid turbulence. Here we begin by validating the method for the basic flow in Kolmogorov flow; a two-dimensional incompressible Navier-Stokes flow with a sinusoidal body force. The linear predictions for stabilization are well captured by direct numerical simulation. By applying an adaptive method to adjust the streamwise translation of the delay, a known traveling wave solution is able to be stabilized up to relatively high Reynolds number. We discover that the famous "odd-number" limitation of this time-delayed feedback method can be overcome in the fluid problem by using the symmetries of the system. This leads to the discovery of eight additional exact coherent structures which can be stabilized with this approach. This means that certain unstable exact coherent structures can be obtained by simply time stepping a modified set of equations, thus circumventing the usual convergence algorithms.