Metric for attractor overlap

Abstract

We present the first general metric for attractor overlap (MAO) facilitating an unsupervised comparison of flow data sets. The starting point is two or more attractors, i.e., ensembles of states representing different operating conditions. The proposed metric generalizes the standard Hilbert-space distance between two snapshots to snapshot ensembles of two attractors. A reduced-order analysis for big data and many attractors is enabled by coarse-graining the snapshots into representative clusters with corresponding centroids and population probabilities. For a large number of attractors, MAO is augmented by proximity maps for the snapshots, the centroids, and the attractors, giving scientifically interpretable visual access to the closeness of the states. The coherent structures belonging to the overlap and disjoint states between these attractors are distilled by few representative centroids.We employ MAO for two quite different actuated flow configurations: a two-dimensional wake with vortices in a narrow frequency range and three-dimensional wall turbulence with broadband spectrum. In the first application, seven control laws are applied to the fluidic pinball, i.e., the two-dimensional flow around three circular cylinders whose centers form an equilateral triangle pointing in the upstream direction. These seven operating conditions comprise unforced shedding, boat tailing, base bleed, high-and low-frequency forcing as well as two opposing Magnus effects. In the second example, MAO is applied to three-dimensional simulation data from an open-loop drag reduction study of a turbulent boundary layer. The actuation mechanisms of 38 spanwise traveling transversal surface waves are investigated. MAO compares and classifies these actuated flows in agreement with physical intuition. For ‡ Present address : 2 R. Ishar and others instance, the first feature coordinate of the attractor proximity map correlates with drag for the fluidic pinball and for the turbulent boundary layer. MAO has a large spectrum of potential applications ranging from a quantitative comparison between numerical simulations and experimental particle-image velocimetry data to the analysis of simulations representing a myriad of different operating conditions.

Drag Reduction and Energy Saving by Spanwise Traveling Transversal Surface Waves for Flat Plate Flow

“…Instead of directly introducing spanwise velocity, the surface is wavily deflected in the wall-normal direction to generate a secondary flow field of periodic wallnormal and spanwise fluctuations. Positive drag reduction using this technique was achieved experimentally [20,47,30] and numerically for channel flow [48], boundary layer flow [26,28,27,19], and airfoil flow [2]. Tomiyama and Fukagata [48] observed a possible shielding effect of quasi-streamwise vortices from the wall by the wave-like deformations and showed that a combination of the thickness of the Stokes layer, i.e., the actuation period, and the actuation velocity amplitudes scales reasonably well with drag reduction.…”

Forecasting chaotic systems with very low connectivity reservoir computers

“…First, we can evaluate the new metric ε by comparing the output of the reservoir predictions to the true system attractor using a new metric for attractor overlap. 20 This overlap metric can also be used to quantify the qualitative observations of different failure modes in regions of our ε metric. Second, there are known results that prove that a linear network architecture with time-independent nodes, the discrete-time NARX networks, can simulate fully-connected networks.…”

Cluster-based network model

“…A + = 40 which yields the largest drag reduction of 3 % found at that wavelength. These actuation parameters correspond to case N36 in table 3 of Ishar et al (2019) and in table 2 of Albers et al (2020).…”

“…This solver is based on the finite-element method with third-order Taylor-Hood elements and implicit third-order time integration. The solver has been used for numerous configurations, like the cylinder wake (Noack et al 2016), the mixing layer (Shaqarin, Noack & Morzyński 2018) and the fluidic pinball (Ishar et al 2019), to name only a few. The red dashed curves mark the mixing-layer thickness.…”

“…This fixed point is well reproduced by CMM. In addition, CMM has provided valuable physical insights for the mixing layer and Ahmed body wake (Kaiser et al 2014), for the turbulent boundary layer (Ishar et al 2019), for combustion related mixing (Cao et al 2014) and for control design (Kaiser et al 2017;Nair et al 2019).…”