Stabilising nonlinear travelling waves in pipe flow using time-delayed feedback

Abstract

We demonstrate the first successful non-invasive stabilisation of nonlinear travelling waves in a straight cylindrical pipe using time-delayed feedback control working in various symmetric subspaces. By using an approximate linear stability analysis and by analysing the frequency-domain effect of the control using transfer functions, we find that solutions with well-separated unstable eigenfrequencies can have narrow windows of stabilising time delays. To mitigate this issue we employ a ‘multiple time-delayed feedback’ approach, where several control terms are included to attenuate a broad range of unstable eigenfrequencies. We implement a gradient descent method to dynamically adjust the gain functions in order to reduce the need for tuning a high-dimensional parameter space. This results in a novel control method where the properties of the target state are not needed in advance, and speculative guesses can result in robust stabilisation. This enables travelling waves to be stabilised from generic turbulent states and unknown travelling waves to be obtained in highly symmetric subspaces.