A variational approach to probing extreme events in turbulent dynamical systems

Extreme events are ubiquitous in a wide range of dynamical systems, including turbulent fluid flows, nonlinear waves, large scale networks and biological systems. Here, we propose a variational framework for probing conditions that trigger intermittent extreme events in high-dimensional nonlinear dynamical systems. We seek the triggers as the probabilistically feasible solutions of an appropriately constrained optimization problem, where the function to be maximized is a system observable exhibiting intermittent extreme bursts. The constraints are imposed to ensure the physical admissibility of the optimal solutions, i.e., significant probability for their occurrence under the natural flow of the dynamical system. We apply the method to a body-forced incompressible Navier--Stokes equation, known as the Kolmogorov flow. We find that the intermittent bursts of the energy dissipation are independent of the external forcing and are instead caused by the spontaneous transfer of energy from large scales to the mean flow via nonlinear triad interactions. The global maximizer of the corresponding variational problem identifies the responsible triad, hence providing a precursor for the occurrence of extreme dissipation events. Specifically, monitoring the energy transfers within this triad, allows us to develop a data-driven short-term predictor for the intermittent bursts of energy dissipation. We assess the performance of this predictor through direct numerical simulations.Comment: Minor revisions, generalized the constraints in Eq. (2

Flow control design inspired by linear stability analysis

In the recent literature, a growing number of research papers have been dedicated to applying the techniques of global stability and sensitivity analysis to the design of flow controls. The controls that are designed in this way are mainly passive or open-loop controls. Among those, we consider here controls that are aimed at linearly stabilizing flow configurations which would be otherwise globally unstable. In particular, a review of the literature on flow controls designed on the basis of stability and sensitivity analysis is presented. The mentioned methods can be rigorously applied to relatively simple flow regimes, typically observed at low values of the Reynolds number. In this respect, the recent literature also demonstrates a large interest in the application of the same methods for the control of coherent large-scale flow structures in turbulent flows, as, for instance, the quasiperiodic shedding of vortices in turbulent wakes. The papers dedicated to this subject are also reviewed here. Finally, all the described methods imply the solution of eigenvalue problems which are at the state-of-the-art for computational complexity. On the one hand, there are attempts to reduce the complexity of the involved computational problems by applying local stability analysis, and some examples are illustrated. On the other hand, recent advances in numerical methods, also concisely reviewed here, allow the manipulation of large eigenvalue problems and greatly simplify the development of numerical tools for stability and sensitivity analysis of complex flow models, often built using existing fluid dynamics codes