Modal Analysis of Fluid Flows: An Overview

By Kunihiko Taira, Steven L. Brunton, Scott T. M. Dawson, Clarence W. Rowley, Tim Colonius, B. McKeon, Oliver T. Schmidt, S. Gordeyev, V. Theofilis, Lawrence Ukeiley. 2017

1,507 citations recorded by the source corpus.

Simple aerodynamic configurations under even modest conditions can exhibit complex flows with a wide range of temporal and spatial features. It has become common practice in the analysis of these flows to look for and extract physically important features, or modes, as a first step in the analysis. This step typically starts with a modal decomposition of an experimental or numerical dataset of the flow field, or of an operator relevant to the system. We describe herein some of the dominant techniques for accomplishing these modal decompositions and analyses that have seen a surge of activity in recent decades. For a non-expert, keeping track of recent developments can be daunting, and the intent of this document is to provide an introduction to modal analysis that is accessible to the larger fluid dynamics community. In particular, we present a brief overview of several of the well-established techniques and clearly lay the framework of these methods using familiar linear algebra. The modal analysis techniques covered in this paper include the proper orthogonal decomposition (POD), balanced proper orthogonal decomposition (Balanced POD), dynamic mode decomposition (DMD), Koopman analysis, global linear stability analysis, and resolvent analysis.

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Citations in the corpus

Digital Twin: Values, Challenges and Enablers From a Modeling Perspective (2019, unknown): “[261] K.”
Nonlinear mode decomposition with convolutional neural networks for fluid dynamics (2019, methodology): “With both methods, the key structures embedded in the time series of flow fields can be found and visualized, although there is a difference in the sense that POD determines the optimal set of modes to represent data based on energy norm, while DMD captures dynamic modes with associated growth rates”
Resolvent-analysis-based design of airfoil separation control (2018, background): “Such an approach forms an eigenvalue problem for the global instability modes and emphasizes on the spectrum of the linearized Navier–Stokes operator (Barkley & Henderson 1996; Sipp & Lebedev 2007; Liu et al. 2016; Sun et al. 2017; Taira et al. 2017).”
Adaptive separation control of a laminar boundary layer using online dynamic mode decomposition (2020, background): “…interactions inherent in fluid flows, modal decomposition approaches aid in the design and implementation of active flow control since the instabilities targeted often consist of discrete frequency, spatio-temporal coherent flow structures (modes) (Kutz et al. 2016; Taira et al. 2017, 2019).”
Guide to Spectral Proper Orthogonal Decomposition (2020, background): “Alongside other operator- and data-driven decompositions employed in fluid mechanics [1], the resulting modes can be used for a variety of purposes, from classification to reduced-order modeling to control.”
Aeroacoustic resonance and self-excitation in screeching and impinging supersonic jets – A review (2019, methodology): “contains some more recent developments.(161) Application of global-mode analysis to supersonic jets is still in its relative infancy.”
Multi-scale proper orthogonal decomposition of complex fluid flows (2018, methodology): “This approach consists in computing the POD structures as eigenvectors of the cross-spectral density matrix (see Gordeyev & Thomas 2000; Citriniti & George 2000; Gudmundsson & Colonius 2011) and is referred by several authors (e.g Picard & Delville 2000; Taira et al. 2017; Towne et al. 2018) to as S”
Machine Learning for Fluid Mechanics (2019, background): “Advances in automation have also facilitated their application in experimental (Strom et al. 2017; Martin & Gharib 2018) and industrial settings (Bueche et al. 2002).”
Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis (2017, methodology): “An efficient algorithm for computing space-only POD modes from snapshots of discrete flow data using the method of snapshots (Sirovich, 1987) is well-known and described in detail in numerous publications (e.g., Rowley & Dawson, 2017; Taira et al., 2017).”