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

Time-delayed feedback control (Pyragas 1992 \textit{Phys. Letts. A} \textbf{170}(6) 421-428), is a method known to stabilise periodic orbits in chaotic dynamical systems. A system $\dot{\mathbf{x}}(t)=f(\mathbf{x})$ is supplemented with $G(\mathbf{x}(t)-\mathbf{x}(t-T))$ where $G$ is a `gain matrix' and $T$ a time delay. The form of the delay term is such that it will vanish for any orbit of period $T,$ making it an orbit of the uncontrolled system. This non-invasive feature makes the method attractive for stabilising exact coherent structures in fluid turbulence. Here we validate the method using the basic flow in Kolmogorov flow; a two-dimensional incompressible viscous flow with a sinusoidal body force. Linear predictions are well captured by direct numerical simulation. By applying an adaptive method to adjust the streamwise translation of the delay, a known travelling wave solution is able to be stabilised up to relatively high Reynolds number. Finally an adaptive method to converge the period $T$ is also presented to enable periodic orbits to be stabilised in a proof of concept study at low Reynolds numbers. These results demonstrate that unstable ECSs may be found by timestepping a modified set of equations, thus circumventing the usual convergence algorithms.