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Guide to Spectral Proper Orthogonal Decomposition

We discuss the spectral proper orthogonal decomposition and its use in identifying modes, or structures, in flow data. A specific algorithm based on estimating the cross-spectral density tensor with Welch’s method is presented, and we provide guidance on selecting data sampling parameters, and understanding tradeoffs amongst them in terms of bias, variability, aliasing, and leakage. Practical implementation issues, including dealing with large datasets, are discussed and illustrated with examples involving experimental and computational turbulent flow data.

Literature:

  • [PDF] Schmidt, O. T. and T. Colonius. “Guide to spectral proper orthogonal decomposition.” Aiaa journal (2020): 1–11.
    [Bibtex]
    @article{schmidtcolonius_2020_aiaaj,
    title={Guide to Spectral Proper Orthogonal Decomposition},
    author={Schmidt, O. T. and Colonius, T.},
    journal={AIAA Journal},
    pages={1--11},
    year={2020},
    publisher={American Institute of Aeronautics and Astronautics}
    }

Code & Examples:

Code and examples from MATLAB Central File Exchange

Spectral empirical orthogonal function analysis of weather and climate data

We apply Spectral Empirical Orthogonal Function (SEOF) analysis, also known as Spectral Proper Orthogonal Decomposition (SPOD) in other fields, to educe climate patterns as dominant spatio-temporal modes of variability from reanalysis data. SEOF is a frequency-domain variant of standard Empirical Orthogonal Function (EOF) analysis, and computes modes that represent the statistically most relevant and persistent patterns from an eigendecomposition of the estimated cross-spectral density matrix (CSD). The method is applied to ERA-Interim and ERA-20C reanalysis data, demonstrating its ability to identify a number of well known spatio-temporal coherent meteorological patterns and teleconnections, including the Madden-Julian Oscillation (MJO), the Quasi-Biennial Oscillation (QBO), and the El Nino-Southern Oscillation (ENSO), i.e. a range of phenomena reoccurring with average periods ranging from months to many years.

The video shows the leading SEOF mode of the Top Thermal Radiation (TTR) with period 45.6 days computed from ERA Interim data. The mode identifies a large-scale anomaly in the Indian Ocean associated with the Madden-Julian oscillation.

Literature:

  • [PDF] [DOI] Schmidt, O. T., G. Mengaldo, G. Balsamo, and N. P. Wedi. “Spectral empirical orthogonal function analysis of weather and climate data.” Monthly weather review 147.8 (2019): 2979-2995.
    [Bibtex]
    @Article{schmidtetal_2019_mwr,
    author = {Schmidt, O. T. and Mengaldo, G. and Balsamo, G. and Wedi, N. P.},
    title = {Spectral Empirical Orthogonal Function Analysis of Weather and Climate Data},
    journal = {Monthly Weather Review},
    year = {2019},
    volume = {147},
    number = {8},
    pages = {2979-2995},
    abstract = { AbstractWe apply spectral empirical orthogonal function (SEOF) analysis to educe climate patterns as dominant spatiotemporal modes of variability from reanalysis data. SEOF is a frequency-domain variant of standard empirical orthogonal function (EOF) analysis, and computes modes that represent the statistically most relevant and persistent patterns from an eigendecomposition of the estimated cross-spectral density matrix (CSD). The spectral estimation step distinguishes the approach from other frequency-domain EOF methods based on a single realization of the Fourier transform, and results in a number of desirable mathematical properties: at each frequency, SEOF yields a set of orthogonal modes that are optimally ranked in terms of variance in the L2 sense, and that are coherent in both space and time by construction. We discuss the differences between SEOF and other competing approaches, as well as its relation to dynamical modes of stochastically forced, nonnormal linear dynamical systems. The method is applied to ERA-Interim and ERA-20C reanalysis data, demonstrating its ability to identify a number of well-known spatiotemporal coherent meteorological patterns and teleconnections, including the Madden–Julian oscillation (MJO), the quasi-biennial oscillation (QBO), and the El Niño–Southern Oscillation (ENSO) (i.e., a range of phenomena reoccurring with average periods ranging from months to many years). In addition to two-dimensional univariate analyses of surface data, we give examples of multivariate and three-dimensional meteorological patterns that illustrate how this technique can systematically identify coherent structures from different sets of data. The MATLAB code used to compute the results presented in this study, including the download scripts for the reanalysis data, is freely available online. },
    doi = {10.1175/MWR-D-18-0337.1},
    eprint = {https://doi.org/10.1175/MWR-D-18-0337.1},
    url = {https://doi.org/10.1175/MWR-D-18-0337.1},
    }

Code & Examples:

Code and examples from MATLAB Central File Exchange

A conditional space–time POD formalism for intermittent and rare events: example of acoustic bursts in turbulent jets

We present a conditional space–time proper orthogonal decomposition (POD) formulation that is tailored to the eduction of rare or intermittent events. By construction, the resulting spatio-temporal modes are coherent in space and over a finite time horizon, and optimally capture the energy of the flow. For the example of intermittent acoustic radiation from a turbulent jet, we extract the statistically loudest event from high-fidelity simulation data. Our formulation identifies the statistically most significant ‘prototype’ burst event and tracks its evolution over time. We investigate the underlying mechanism using linear stability theory and find that its structure and evolution are accurately predicted by optimal transient growth theory.

The empirical ‘prototype’ burst event and the optimal transient growth prediction are compared side-by-side in the video.

Literature:

  • [PDF] [DOI] Schmidt, O. T. and P. J. Schmid. “A conditional space–time pod formalism for intermittent and rare events: example of acoustic bursts in turbulent jets.” Journal of fluid mechanics 867 (2019): R2.
    [Bibtex]
    @Article{SchmidtSchmid_2019_JFMR,
    author = {Schmidt, O. T. and Schmid, P. J.},
    title = {A conditional space–time POD formalism for intermittent and rare events: example of acoustic bursts in turbulent jets},
    journal = {Journal of Fluid Mechanics},
    year = {2019},
    volume = {867},
    pages = {R2},
    doi = {10.1017/jfm.2019.200},
    file = {:SchmidtSchmid_2019_JFM.pdf:PDF},
    publisher = {Cambridge University Press},
    }

An efficient streaming algorithm for spectral proper orthogonal decomposition

A streaming algorithm to compute the spectral proper orthogonal decomposition (SPOD) of stationary random processes is presented. As new data becomes available, an incremental update of the truncated eigenbasis of the estimated cross-spectral density (CSD) matrix is performed. The algorithm requires access to only one temporal snapshot of the data at a time and converges orthogonal sets of SPOD modes at discrete frequencies that are optimally ranked in terms of energy. We define measures of error and convergence, and demonstrate the algorithm’s performance on two datasets. The first example considers a high-fidelity numerical simulation of a turbulent jet, and the second uses optical flow data obtained from high-speed camera recordings of a stepped spillway experiment. For both cases, the most energetic SPOD modes are reliably converged. The algorithm’s low memory requirement enables real-time deployment and allows for the convergence of second-order statistics from arbitrarily long streams of data.

Literature:

  • [PDF] [DOI] Schmidt, O. T. and A. Towne. “An efficient streaming algorithm for spectral proper orthogonal decomposition.” Computer physics communications (2018).
    [Bibtex]
    @Article{SchmidtTowne_2018_CPC,
    author = {Schmidt, O. T. and Towne, A.},
    title = {An efficient streaming algorithm for spectral proper orthogonal decomposition},
    journal = {Computer Physics Communications},
    year = {2018},
    issn = {0010-4655},
    abstract = {A streaming algorithm to compute the spectral proper orthogonal decomposition (SPOD) of stationary random processes is presented. As new data becomes available, an incremental update of the truncated eigenbasis of the estimated cross-spectral density (CSD) matrix is performed. The algorithm requires access to only one temporal snapshot of the data at a time and converges orthogonal sets of SPOD modes at discrete frequencies that are optimally ranked in terms of energy. We define measures of error and convergence, and demonstrate the algorithm’s performance on two datasets. The first example considers a high-fidelity numerical simulation of a turbulent jet, and the second example uses optical flow data obtained from high-speed camera recordings of a stepped spillway experiment. For both cases, the most energetic SPOD modes are reliably converged. The algorithm’s low memory requirement enables real-time deployment and allows for the convergence of second-order statistics from arbitrarily long streams of data. A MATLAB implementation of the algorithm along with a test database for the jet example, can be found in the Supplementary material.},
    doi = {https://doi.org/10.1016/j.cpc.2018.11.009},
    file = {:SchmidtTowne_2018_CPC.pdf:PDF},
    keywords = {Proper orthogonal decomposition, Principal component analysis, Spectral analysis},
    url = {http://www.sciencedirect.com/science/article/pii/S0010465518304016},}

Code & Examples:

Code and examples from MATLAB Central File Exchange

Modal Analysis of Fluid Flows: An Overview

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.

The figure shows the modal decomposition of two-dimensional incompressible flow over a flat-plate wing. This example shows complex nonlinear separated flow being well represented by only two POD modes and the mean flowfield. Visualized are the streamwise velocity profiles.

Literature:

  • [PDF] [DOI] Taira, K., S. L. Brunton, S. Dawson, C. W. Rowley, T. Colonius, B. J. McKeon, O. T. Schmidt, S. Gordeyev, V. Theofilis, and L. S. Ukeiley. “Modal analysis of fluid flows: an overview.” Aiaa journal (2017).
    [Bibtex]
    @Article{TairaEtAl_2017_AIAAJ,
    author = {Taira, K. and Brunton, S. L. and Dawson, S. and Rowley, C. W. and Colonius, T. and McKeon, B. J. and Schmidt, O. T. and Gordeyev, S. and Theofilis, V. and Ukeiley, L. S.},
    title = {Modal analysis of fluid flows: An overview},
    journal = {AIAA Journal},
    year = {2017},
    doi = {https://doi.org/10.2514/1.J056060},
    file = {:TairaEtAl_2017_AIAAJ.pdf:PDF},
    }

Spectral Analysis of Jet Turbulence

Informed by LES data and resolvent analysis of the mean flow, we examine the structure of turbulence in jets in the subsonic, transonic, and supersonic regimes. Spectral (frequency-space) proper orthogonal decomposition is used to extract energy spectra and decompose the flow into energy-ranked coherent structures. The educed structures are generally well predicted by the resolvent analysis. Over a range of low frequencies and the first few azimuthal mode numbers, these jets exhibit a low-rank response characterized by Kelvin-Helmholtz (KH) type wavepackets associated with the annular shear layer up to the end of the potential core and that are excited by forcing in the very-near-nozzle shear layer. These modes too the have been experimentally observed before and predicted by quasi-parallel stability theory and other approximations–they comprise a considerable portion of the total turbulent energy. At still lower frequencies, particularly for the axisymmetric mode, and again at high frequencies for all azimuthal wavenumbers, the response is not low rank, but consists of a family of similarly amplified modes. These modes, which are primarily active downstream of the potential core, are associated with the Orr mechanism. They occur also as sub-dominant modes in the range of frequencies dominated by the KH response. Our global analysis helps tie together previous observations based on local spatial stability theory, and explains why quasi-parallel predictions were successful at some frequencies and azimuthal wavenumbers, but failed at others.

The video shows, in quick succession, the raw LES data, a single Fourier component for m=0, and the leading SPOD mode at the same frequency. The streamwise perturbation velocity is visualized.

Literature:

  • [PDF] [DOI] Schmidt, O. T., A. Towne, G. Rigas, T. Colonius, and G. A. Brès. “Spectral analysis of jet turbulence.” Journal of fluid mechanics 855 (2018): 953–982.
    [Bibtex]
    @Article{SchmidtEtAl_2018_JFM,
    author = {Schmidt, O. T. and Towne, A. and Rigas, G. and Colonius, T. and Br{\`e}s, G. A.},
    title = {Spectral analysis of jet turbulence},
    journal = {Journal of Fluid Mechanics},
    year = {2018},
    volume = {855},
    pages = {953–982},
    doi = {10.1017/jfm.2018.675},
    file = {:SchmidtEtAl_2018_JFM.pdf:PDF},
    publisher = {Cambridge University Press},
    }

Wavepackets and trapped acoustic modes in turbulent jets

Coherent features of a turbulent Mach 0.9, Reynolds number 1M jet are educed from a high-fidelity large eddy simulation. Besides the well-known Kelvin-Helmholtz instabilities of the shear-layer, a new class of trapped acoustic waves is identified in the potential core. In two parallel studies, we investigate these trapped acoustic waves using different techniques.

The video shows a trapped acoustic mode in the potential core of a Mach 0.9 turbulent jet obtained from a global stability analysis of the mean flow.

Literature:

  • [PDF] [DOI] Schmidt, O. T., A. Towne, T. Colonius, A. V. G. Cavalieri, P. Jordan, and G. A. Brès. “Wavepackets and trapped acoustic modes in a turbulent jet: coherent structure eduction and global stability.” Journal of fluid mechanics 825 (2017): 1153-1181.
    [Bibtex]
    @Article{SchmidtEtAl_2017_JFM,
    author = {Schmidt, O. T. and Towne, A. and Colonius, T. and Cavalieri, A. V. G. and Jordan, P. and Br{\`e}s, G. A.},
    title = {Wavepackets and trapped acoustic modes in a turbulent jet: coherent structure eduction and global stability},
    journal = {Journal of Fluid Mechanics},
    year = {2017},
    volume = {825},
    pages = {1153-1181},
    doi = {10.1017/jfm.2017.407},
    file = {:SchmidtEtAl_2017_JFM.pdf:PDF},
    publisher = {Cambridge University Press},
    }
  • [PDF] [DOI] Towne, A., A. V. G. Cavalieri, P. Jordan, T. Colonius, O. T. Schmidt, V. Jaunet, and G. A. Brès. “Acoustic resonance in the potential core of subsonic jets.” Journal of fluid mechanics 825 (2017): 1113-1152.
    [Bibtex]
    @Article{TowneEtAl_2017_JFM,
    author = {Towne, A. and Cavalieri, A. V. G. and Jordan, P. and Colonius, T. and Schmidt, O. T. and Jaunet, V. and Br{\`e}s, G. A.},
    title = {Acoustic resonance in the potential core of subsonic jets},
    journal = {Journal of Fluid Mechanics},
    year = {2017},
    volume = {825},
    pages = {1113-1152},
    doi = {10.1017/jfm.2017.346},
    file = {:TowneEtAl_2017_JFM.pdf:PDF},
    publisher = {Cambridge University Press},
    }

Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis

We consider the frequency domain form of proper orthogonal decomposition (POD) called spectral proper orthogonal decomposition (SPOD). SPOD is derived from a space-time POD problem for stationary flows and leads to modes that each oscillate at a single frequency. We establish a relationship between SPOD and dynamic mode decomposition (DMD); we show that SPOD modes are in fact optimally averaged DMD modes obtained from an ensemble DMD problem for stationary flows. Finally, we establish a connection between SPOD and resolvent analysis. The key observation is that the resolvent-mode expansion coefficients, which are usually treated as deterministic quantities described by an amplitude and phase, should be regarded as statistical quantities, described by their cross-spectral density, in order for the resolvent-mode expansion to properly capture the flow statistics. When the expansion coefficients are uncorrelated, we show that SPOD and resolvent modes are identical.

Literature:

  • [PDF] [DOI] Towne, A., O. T. Schmidt, and T. Colonius. “Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis.” Journal of fluid mechanics 847 (2018): 821–867.
    [Bibtex]
    @Article{TowneSchmidtColonius_2018_JFM,
    author = {Towne, A. and Schmidt, O. T. and Colonius, T.},
    title = {Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis},
    journal = {Journal of Fluid Mechanics},
    year = {2018},
    volume = {847},
    pages = {821–867},
    doi = {10.1017/jfm.2018.283},
    file = {:TowneSchmidtColonius_2018_JFM.pdf:PDF},
    publisher = {Cambridge University Press},
    }

Stability, Receptivity and Transition of Compressible Corner Flows

The corner flow problem is a generic model for wing-fuselage intersections, rotor-hub junctions, and supersonic engine inlets. Corner flows have been subject to more than six decades of extensive study, in particular because of their importance to the aeronautical industry.

Our research focuses on the stability and transition of compressible corner flows using direct numerical simulation and linear theory. The key contributions to the understanding of corner flows are a detailed study of the effect of compressibility on their stability that lead to the discovery of a new modal instability mechanism at supersonic speeds, and an in-depth investigation of a non-modal mechanism that explains the flow’s sensitivity and early transition. The relevance of this mechanism became apparent in the first direct numerical simulations of laminar-turbulent transition in corner flows.

The video shows a direct numerical simulation of the symmetric transition scenario in terms of isosurfaces of the Lambda2 vortex criterion.

Literature:

  • [PDF] [DOI] Schmidt, O. T. and U. Rist. “Linear stability of compressible flow in a streamwise corner.” Journal of fluid mechanics 688 (2011): 569-590.
    [Bibtex]
    @Article{SchmidtRist_2011_JFM,
    Title = {Linear stability of compressible flow in a streamwise corner},
    Author = {Schmidt, O. T. and Rist, U.},
    Journal = {Journal of Fluid Mechanics},
    Year = {2011},
    Pages = {569-590},
    Volume = {688},
    Doi = {10.1017/jfm.2011.405},
    File = {:./SchmidtRist_2011_JFM.pdf:PDF},
    Publisher = {Cambridge Univ Press}
    }
  • [PDF] [DOI] Schmidt, O. T., B. Selent, and U. Rist. “Direct numerical simulation of boundary layer transition in streamwise corner-flow.” High performance computing in science and engineering (2013): 337-348.
    [Bibtex]
    @Article{SchmidtRistSelent_2013_HPCSE,
    author = {Schmidt, O. T. and Selent, B. and Rist, U.},
    title = {Direct numerical simulation of boundary layer transition in streamwise corner-flow},
    journal = {High Performance Computing in Science and Engineering},
    year = {2013},
    pages = {337-348},
    doi = {10.1017/S0022112095003284},
    file = {:SchmidtRistSelent_2013_HPCSE.pdf:PDF},
    owner = {iagoschm},
    publisher = {Cambridge Univ Press},
    }
  • [PDF] [DOI] Schmidt, O. T. and U. Rist. “Viscid–inviscid pseudo-resonance in streamwise corner flow.” Journal of fluid mechanics 743 (2014): 327–357.
    [Bibtex]
    @Article{SchmidtRist_2014_JFM,
    author = {Schmidt, O. T. and Rist, U.},
    title = {Viscid--inviscid pseudo-resonance in streamwise corner flow},
    journal = {Journal of Fluid Mechanics},
    year = {2014},
    volume = {743},
    pages = {327--357},
    doi = {10.1017/jfm.2014.31},
    file = {:SchmidtRist_2014_JFM.pdf:PDF},
    publisher = {Cambridge Univ Press},
    }
  • [PDF] [DOI] Schmidt, O. T., S. M. Hosseini, Ulrich Rist, A. Hanifi, and D. S. Henningson. “Optimal wavepackets in streamwise corner flow.” Journal of fluid mechanics 766 (2015): 405–435.
    [Bibtex]
    @Article{SchmidtEtAl_2015_JFM,
    Title = {Optimal wavepackets in streamwise corner flow},
    Author = {Schmidt, O. T. and Hosseini, S. M. and Rist, Ulrich and Hanifi, A. and Henningson, D. S.},
    Journal = {Journal of Fluid Mechanics},
    Year = {2015},
    Pages = {405--435},
    Volume = {766},
    Doi = {10.1017/jfm.2015.18},
    File = {:SchmidtEtAl_2015_JFM.pdf:PDF},
    Publisher = {Cambridge Univ Press}
    }