Data-driven modeling and control of oscillatory instabilities in Kolmogorov-like flow

We apply data-driven techniques to construct a nonlinear 3-mode model of a Kolmogorov-like flow transitioning from steady to periodic. Data from direct numerical simulation that include features of experimental realizations of Kolmogorov-like flow are used to build the model. Our low-order modeling methodology does not require knowledge of the underlying governing equations. The 3-mode basis for the model is determined solely from data and the sparse identification of nonlinear dynamics framework (SINDy) is used to fit a dynamical system describing modal interactions. We impose constraints within the SINDy framework to ensure the resulting model will possess energy-preserving nonlinear terms that are consistent with the underlying flow physics. We use the low-order model to determine an appropriate equilibrium solution to stabilize, thereby avoiding searching for equilibrium solutions in the full-order system. The model is linearized about the identified equilibrium solution and subsequently used to design feedback controllers that successfully suppress an oscillatory instability when applied in direct numerical simulations—a testament to the model’s ability to capture the underlying dynamics that are most relevant for flow control. Our results confirm that low-order models obtained in a purely data-driven framework can be implemented for flow control in experimentally-realizable Kolmogorov-like flow.