Randomized methods to characterize large-scale vortical flow network

We demonstrate the effective use of randomized methods for linear algebra to perform network-based analysis of complex vortical flows. Network theoretic approaches can reveal the connectivity structures among a set of vortical elements and analyze their collective dynamics. These approaches have recently been generalized to analyze high-dimensional turbulent flows, for which network computations can become prohibitively expensive. In this work, we propose efficient methods to approximate network quantities, such as the leading eigendecomposition of the adjacency matrix, using randomized methods. Specifically, we use the Nystrom method to approximate the leading eigenvalues and eigenvectors, achieving significant computational savings and reduced memory requirements. The effectiveness of the proposed technique is demonstrated on two high-dimensional flow fields: two-dimensional flow past an airfoil and two-dimensional turbulence. We find that quasi-uniform column sampling outperforms uniform column sampling, while both feature the same computational complexity.

Network community-based model reduction for vortical flows

A network community-based reduced-order model is developed to capture key interactions amongst coherent structures in high-dimensional unsteady vortical flows. The present approach is data-inspired and founded on network-theoretic techniques to identify important vortical communities that are comprised of vortical elements that share similar dynamical behavior. The overall interaction-based physics of the high-dimensional flow field is distilled into the vortical community centroids, considerably reducing the system dimension. Taking advantage of these vortical interactions, the proposed methodology is applied to formulate reduced-order models for the inter-community dynamics of vortical flows, and predict lift and drag forces on bodies in wake flows. We demonstrate the capabilities of these models by accurately capturing the macroscopic dynamics of a collection of discrete point vortices, and the complex unsteady aerodynamic forces on a circular cylinder and an airfoil with a Gurney flap. The present formulation is found to be robust against simulated experimental noise and turbulence due to its integrating nature of the system reduction.