论文标题
使用时间延迟的反馈稳定二维湍流中确切的相干结构
Stabilisation of exact coherent structures in two-dimensional turbulence using time-delayed feedback
论文作者
论文摘要
延迟的反馈控制归因于Pyragas（1992 Physics Letters 170（6）421-428），是一种已知的方法，可在低维混沌动力学系统中稳定周期性轨道。 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.延迟期限的形式使得它在任何时期$ t的轨道上都会消失，因此也使其成为不受控制的系统的轨道。这种非侵入性特征使该方法具有稳定流体湍流中精确相干结构的吸引力。在这里，我们首先验证Kolmogorov流中基本流的方法。二维不可压缩的Navier-Stokes与正弦体力流动。直接数值模拟可以很好地捕获稳定的线性预测。通过应用自适应方法来调整延迟的流式翻译，已知的行驶波解决方案可以稳定到相对较高的雷诺数。我们发现，通过使用系统的对称性，可以克服流体问题中著名的“奇数”限制。这导致发现了8个其他确切的一致结构，可以通过这种方法稳定。这意味着可以通过简单地定位一组方程，从而规避通常的收敛算法来获得某些不稳定的确切相干结构。
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.
