Session L27: Tutorial: Modal Analysis Methods for Fluid Flows

Modal decompositions in fluid mechanics: an overview

We provide an overview of modal decompositions that are used for educing coherent structures in unsteady flows, and we discuss ways in which the modes can be exploited for data compression, compressed sensing, reduced-order modeling, and controller development. We compare and contrast a number of popular data-driven techniques for modal decomposition including POD, DMD/Koopman, DFT and spectral POD. We highlight connections between modes educed from data and modes associated with the spectrum of linear operators associated to disturbances to basic flows, including both laminar flows (i.e. stability analysis) and turbulent mean flow fields (i.e. resolvent analysis). For stationary turbulent flows, recently established connections between DMD, resolvent analysis and spectral POD are discussed. Applications to turbulent wall-bounded and free-shear flows are highlighted.   

Modal analysis of fluid flows using variants of proper orthogonal decomposition

This talk gives an overview of several methods for analyzing fluid flows, based on variants of proper orthogonal decomposition. These methods may be used to determine simplified, approximate models that capture the essential features of these flows, in order to better understand the dominant physical mechanisms, and potentially to develop appropriate strategies for model-based flow control. We discuss balanced proper orthogonal decomposition as an approximation of balanced truncation, and explain connections with system identification methods such as the eigensystem realization algorithm. We demonstrate the methods on several canonical examples, including a linearized channel flow and the flow past a circular cylinder.   

Dynamic Mode Decomposition of numerical and experimental data

DMD extracts dynamic information from a sequence of flow fields generated by numerical simulations or physical experiments. It can be used to reconstruct a low-dimensional inter-snapshot map whose spectral properties describe the underlying fluid behavior contained in the processed flow fields. This tutorial gives a brief introduction to the method, demonstrates its applicability to a variety of flow situations and discusses extensions and generalizations. Examples will be drawn from numerical and experimental data of a wide range of applications.   

Koopman operator theory: Past, present, and future

Koopman operator theory has emerged as a dominant method to represent nonlinear dynamics in terms of an infinite-dimensional linear operator. The Koopman operator acts on the space of all possible measurement functions of the system state, advancing these measurements with the flow of the dynamics. A linear representation of nonlinear dynamics has tremendous potential to enable the prediction, estimation, and control of nonlinear systems with standard textbook methods developed for linear systems. Dynamic mode decomposition has become the leading data-driven method to approximate the Koopman operator, although there are still open questions and challenges around how to obtain accurate approximations for strongly nonlinear systems. This talk will provide an introductory overview of modern Koopman operator theory, reviewing the basics and describing recent theoretical and algorithmic developments. Particular emphasis will be placed on the use of data-driven Koopman theory to characterize and control high-dimensional fluid dynamic systems. This talk will also address key advances in the rapidly growing fields of machine learning and data science that are likely to drive future developments.   

A tutorial on resolvent methods

Arising from the interaction between control theory and more traditional fluid dynamics, resolvent analysis methods focus on the response of the fluid to dynamic disturbances. Many styles of this approach now exist, which variously consider that disturbance as a stochastic forcing, as an externally applied forcing, as internally generated Reynolds stresses, or as some combination thereof. In this tutorial paper, we shall introduce the theoretical viewpoint advanced by the authors in previous work. The approach is a complete and self-consistent restatement of the Navier-Stokes equations, which permits a series of progressively lower-dimensional approximations to be made. This viewpoint has a number of interesting characteristics that relate to other techniques. For example, the approach offers a more defensible alternative to eigenvalue stability calculations about a temporal-mean flow. The presentation will also highlight the natural links to the Dynamic Mode Decomposition and Koopman modes approaches. We shall illustrate the approach by reviewing some recent applications to passive and active flow control strategies, flow estimation, flow structure prediction, and flow spectra.   

An overview of global stability analysis

Approaches employed to access a subset of the eigenspectrum of the linearized Navier-Stokes equations operator describing global instability of laminar flows over or through geometries with multiple inhomogeneous spatial directions are overviewed. The observation is made that in the incompressible limit numerical methods developed over the last two decades have now reached a state of maturity that permits employing any of the well-documented time-stepping or matrix-forming approaches to predict (in)stability of a given flow. A particular wall-boundary closure that permits accurate recovery of 2D and 3D eigenspectra on collocated grids will be highlighted. Far less attention has been paid to-date to global instability of compressible flows. The most promising recent developments in methods for the efficient extraction of quasi-3D compressible flow global instabilities will be outlined. Finally, issues pertaining to laminar hypersonic flow instability, relating with large slip velocities at tips of cones and leading edges of lifting surfaces, will be discussed and novel instability analysis methodologies based on linearization of the probability distribution function will be discussed.