Eurika Kaiser

Cluster-based reduced-order modelling of a mixing layer

We propose a novel cluster-based reduced-order modelling (CROM) strategy of unsteady flows. CROM combines the cluster analysis pioneered in Gunzburgers group (Burkardt et al. 2006) and and transition matrix models introduced in fluid dynamics in Eckhardts group (Schneider et al. 2007). CROM constitutes a potential alternative to POD models and generalises the Ulam-Galerkin method classically used in dynamical systems to determine a finite-rank approximation of the Perron-Frobenius operator. The proposed strategy processes a time-resolved sequence of flow snapshots in two steps. First, the snapshot data are clustered into a small number of representative states, called centroids, in the state space. These centroids partition the state space in complementary non-overlapping regions (centroidal Voronoi cells). Departing from the standard algorithm, the probabilities of the clusters are determined, and the states are sorted by analysis of the transition matrix. Secondly, the transitions between the states are dynamically modelled using a Markov process. Physical mechanisms are then distilled by a refined analysis of the Markov process, e.g. using finite-time Lyapunov exponent and entropic methods. This CROM framework is applied to the Lorenz attractor (as illustrative example), to velocity fields of the spatially evolving incompressible mixing layer and the three-dimensional turbulent wake of a bluff body. For these examples, CROM is shown to identify non-trivial quasi-attractors and transition processes in an unsupervised manner. CROM has numerous potential applications for the systematic identification of physical mechanisms of complex dynamics, for comparison of flow evolution models, for the identification of precursors to desirable and undesirable events, and for flow control applications exploiting nonlinear actuation dynamics.

Chaos as an intermittently forced linear system

Understanding the interplay of order and disorder in chaos is a central challenge in modern quantitative science. Approximate linear representations of nonlinear dynamics have long been sought, driving considerable interest in Koopman theory. We present a universal, data-driven decomposition of chaos as an intermittently forced linear system. This work combines delay embedding and Koopman theory to decompose chaotic dynamics into a linear model in the leading delay coordinates with forcing by low-energy delay coordinates; this is called the Hankel alternative view of Koopman (HAVOK) analysis. This analysis is applied to the Lorenz system and real-world examples including Earth’s magnetic field reversal and measles outbreaks. In each case, forcing statistics are non-Gaussian, with long tails corresponding to rare intermittent forcing that precedes switching and bursting phenomena. The forcing activity demarcates coherent phase space regions where the dynamics are approximately linear from those that are strongly nonlinear.The huge amount of data generated in fields like neuroscience or finance calls for effective strategies that mine data to reveal underlying dynamics. Here Brunton et al.develop a data-driven technique to analyze chaotic systems and predict their dynamics in terms of a forced linear model.

Analysis of high speed jet flow physics with time-resolved PIV

This work focuses on a Mach 0.6 turbulent, compressible jet flow field with simultaneously sampled near and far-field pressure, as well as 10 kHz time-resolved PIV. Experiments have been conducted in the fully anechoic chamber and jet facility at Syracuse University. The PIV measurements were taken in the streamwise plane of the jet along the center plane at various downstream locations. In addition, measurements were taken off of the center plane to obtain a three-dimensional view of the jet flow. Active flow control (both open and closed-loop) was performed in order to see the effects on the potential core length and overall sound pressure levels. Various reduced-order models have been used to analyze previous experimental data sets at Syracuse University. This paper will focus on the analysis of the flow physics, using the time-resolved velocity field coupled with the simultaneously sampled pressure. Novel modeling approaches such as observable inferred decomposition and cluster-based reduced-order modeling have been implemented in an effort to link the near-field velocity with the far-field acoustics.

Cluster-based reduced-order modelling of shear flows

Cluster-based reduced-order modelling (CROM) builds on the pioneering works of Gunzburgers group in cluster analysis [1] and Eckhardts group in transition matrix models [2] and constitutes a potential alternative to reduced-order models based on a proper-orthogonal decomposition (POD). This strategy frames a time-resolved sequence of flow snapshots into a Markov model for the probabilities of cluster transitions. The information content of the Markov model is assessed with a Kullback-Leibler entropy. This entropy clearly discriminates between prediction times in which the initial conditions can be inferred by backward integration and the predictability horizon after which all information about the initial condition is lost. This approach is exemplified for a class of fluid dynamical benchmark problems like the periodic cylinder wake, the spatially evolving incompressible mixing layer, the bi-modal bluff body wake, and turbulent jet noise. For these examples, CROM is shown to distil nontrivial quasi-attr...

Sparsity enabled cluster reduced-order models for control

Abstract Characterizing and controlling nonlinear, multi-scale phenomena are central goals in science and engineering. Cluster-based reduced-order modeling (CROM) was introduced to exploit the underlying low-dimensional dynamics of complex systems. CROM builds a data-driven discretization of the Perron–Frobenius operator, resulting in a probabilistic model for ensembles of trajectories. A key advantage of CROM is that it embeds nonlinear dynamics in a linear framework, which enables the application of standard linear techniques to the nonlinear system. CROM is typically computed on high-dimensional data; however, access to and computations on this full-state data limit the online implementation of CROM for prediction and control. Here, we address this key challenge by identifying a small subset of critical measurements to learn an efficient CROM, referred to as sparsity-enabled CROM. In particular, we leverage compressive measurements to faithfully embed the cluster geometry and preserve the probabilistic dynamics. Further, we show how to identify fewer optimized sensor locations tailored to a specific problem that outperform random measurements. Both of these sparsity-enabled sensing strategies significantly reduce the burden of data acquisition and processing for low-latency in-time estimation and control. We illustrate this unsupervised learning approach on three different high-dimensional nonlinear dynamical systems from fluids with increasing complexity, with one application in flow control. Sparsity-enabled CROM is a critical facilitator for real-time implementation on high-dimensional systems where full-state information may be inaccessible.

Data-Driven Methods in Fluid Dynamics: Sparse Classification from Experimental Data

This work explores the use of data-driven methods, including machine learning and sparse sampling, for systems in fluid dynamics. In particular, camera images of a transitional separation bubble are used with dimensionality reduction and supervised classification techniques to discriminate between an actuated and an unactuated flow. After classification is demonstrated on full-resolution image data, similar classification performance is obtained using heavily subsampled pixels from the images. Finally, a sparse sensor optimization based on compressed sensing is used to determine optimal pixel locations for accurate classification. With 5–10 specially selected sensors, the median cross-validated classification accuracy is ≥ 97 %, as opposed to a random set of 5–10 pixels, which results in classification accuracy of 70–80 %. The methods developed here apply broadly to high-dimensional data from fluid dynamics experiments. Relevant connections between sparse sampling and the representation of high-dimensional data in a low-rank feature space are discussed.

Visualizing vortex clusters in the wake of a high-speed train

Visualization of fluid flows at a high-Reynolds number (Re ∼ 105) presents difficulties for user comprehension due to density and ambiguous interactions between vortices. Prior work has used cluster-based reduced-order modelling (CROM) to analyze the wake of a High-Speed Train (HST) with Re = 86,000. In this paper, we present a novel surface visualization to convey the spatiotemporal changes undergone by clustered vortices in the HST wake. This visualization is accomplished through dimensional reduction of 3D volumetric vortices into 1D ridges, and physics-based feature tracking. The result is 3D surfaces visualizing the behavior of the vortices in the HST wake. Compared to conventional still-image representations, these surfaces allow the user to quickly compare and analyze the two shedding cycles identified via CROM. The spatiotemporal differences of the primary vortices in these shedding cycles provide analytic insight to influence the aerodynamics of the HST.

Bayesian cyclic networks, mutual information and reduced-order Bayesian inference

We examine Bayesian cyclic networks, here defined as complete directed graphs in which the nodes, representing the domains of discrete or continuous variables, are connected by directed edges representing conditional probabilities between all pairs of variables. The prior probabilities associated with each domain are also included as probabilistic edges into each domain. Such networks provide a graphical representation of the inferential connections between variables, and substantially extend the standard definition of “Bayesian networks”, usually defined as one-directional (acyclic) directed graphs. In a binary system, the proposed representation provides a graphical expression of Bayes’ theorem. In higher-dimensional systems, further probabilistic relations can be recovered from the network cyclic properties and the joint probability of all variables. In particular, adopting a Markovian assumption leads to the theorem that the mutual information between any pair of variables on the network must be equiv...