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

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


Abstract

Time-delayed feedback control, attributed to Pyragas (1992 Physics Letters 170(6) 421-428), is a method known to stabilise periodic orbits in low dimensional chaotic dynamical systems. A system of the form $\dot{\mathbf{x}}(t)=f(\mathbf{x})$ has an additional term $G(\mathbf{x}(t)-\mathbf{x}(t-T))$ 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 non-invasive feature makes the method attractive for stabilising 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 stabilisation 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. 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 8 additional exact coherent structures which can be stabilised 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.

Acceptance Date Jan 6, 2022
Publication Date Jan 20, 2022
Publicly Available Date Mar 28, 2024
Journal Physical Review Fluids
Print ISSN 2469-990X
Publisher American Physical Society
DOI https://doi.org/10.1103/PhysRevFluids.7.014401
Publisher URL https://journals.aps.org/prfluids/abstract/10.1103/PhysRevFluids.7.014401

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