Dan Lucas
Stabilisation of exact coherent structures in two-dimensional turbulence using time-delayed feedback
Lucas, Dan; Yasuda, Tatsuya
Authors
Tatsuya Yasuda
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.
Citation
Lucas, D., & Yasuda, T. (2022). Stabilisation of exact coherent structures in two-dimensional turbulence using time-delayed feedback. Physical Review Fluids, https://doi.org/10.1103/PhysRevFluids.7.014401
Acceptance Date | Jan 6, 2022 |
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Publication Date | Jan 20, 2022 |
Publicly Available Date | Jan 25, 2022 |
Journal | Physical Review Fluids |
Print ISSN | 2469-990X |
Publisher | American Physical Society |
DOI | https://doi.org/10.1103/PhysRevFluids.7.014401 |
Public URL | https://keele-repository.worktribe.com/output/422109 |
Publisher URL | https://journals.aps.org/prfluids/abstract/10.1103/PhysRevFluids.7.014401 |
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https://creativecommons.org/licenses/by-nc/4.0/
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