## Frequency–time analysis, low-rank reconstruction and denoising of turbulent flows using SPOD

Four different applications of spectral proper orthogonal decomposition (SPOD) are demonstrated on large-eddy simulation data of a turbulent jet. These are: low-rank reconstruction, denoising, frequency–time analysis and prewhitening. We demonstrate SPOD-based flow-field reconstruction using direct inversion of the SPOD algorithm (frequency-domain approach) and propose an alternative approach based on projection of the time series data onto the modes (time-domain approach). We further present a SPOD-based denoising strategy that is based on hard thresholding of the SPOD eigenvalues. The proposed strategy achieves significant noise reduction while facilitating drastic data compression.

Literature:

• Nekkanti, A. and O. T. Schmidt. “Frequency–time analysis, low-rank reconstruction and denoising of turbulent flows using spod.” Journal of fluid mechanics 926 (2021): A26.
[Bibtex]
@Article{nekkantischmidt_2021_jfm,
author = {Nekkanti, A. and Schmidt, O. T.},
journal = {Journal of Fluid Mechanics},
title = {Frequency–time analysis, low-rank reconstruction and denoising of turbulent flows using SPOD},
year = {2021},
pages = {A26},
volume = {926},
doi = {10.1017/jfm.2021.681},
file = {:NekkantiSchmidt_2021_JFM.pdf:PDF},
publisher = {Cambridge University Press},
}

Code:

## A stochastic SPOD-Galerkin model for broadband turbulent flows

The use of spectral proper orthogonal decomposition (SPOD) to construct low-order models for broadband turbulent flows is explored. The choice of SPOD modes as basis vectors is motivated by their optimality and space-time coherence properties for statistically stationary flows. This work follows the modeling paradigm that complex nonlinear fluid dynamics can be approximated as stochastically forced linear systems. The proposed stochastic two-level SPOD-Galerkin model governs a compound state consisting of the modal expansion coefficients and forcing coefficients. In the first level, the modal expansion coefficients are advanced by the forced linearized Navier-Stokes operator under the linear time-invariant assumption. The second level governs the forcing coefficients, which compensate for the offset between the linear approximation and the true state. At this level, least squares regression is used to achieve closure by modeling nonlinear interactions between modes.

Literature:

• Chu, T. and O. T. Schmidt. “A stochastic spod-galerkin model for broadband turbulent flows.” Theoretical and computational fluid dynamics (2021).
[Bibtex]
@Article{chuschmidt_2021_tcfd,
author = {Chu, T. and Schmidt, O. T.},
journal = {Theoretical and Computational Fluid Dynamics},
title = {A stochastic SPOD-Galerkin model for broadband turbulent flows},
year = {2021},
issn = {1432-2250},
abstract = {The use of spectral proper orthogonal decomposition (SPOD) to construct low-order models for broadband turbulent flows is explored. The choice of SPOD modes as basis vectors is motivated by their optimality and space-time coherence properties for statistically stationary flows. This work follows the modeling paradigm that complex nonlinear fluid dynamics can be approximated as stochastically forced linear systems. The proposed stochastic two-level SPOD-Galerkin model governs a compound state consisting of the modal expansion coefficients and forcing coefficients. In the first level, the modal expansion coefficients are advanced by the forced linearized Navier-Stokes operator under the linear time-invariant assumption. The second level governs the forcing coefficients, which compensate for the offset between the linear approximation and the true state. At this level, least squares regression is used to achieve closure by modeling nonlinear interactions between modes. The statistics of the remaining residue are used to construct a dewhitening filter that facilitates the use of white noise to drive the model. If the data residue is used as the sole input, the model accurately recovers the original flow trajectory for all times. If the residue is modeled as stochastic input, then the model generates surrogate data that accurately reproduces the second-order statistics and dynamics of the original data. The stochastic model uncertainty, predictability, and stability are quantified analytically and through Monte Carlo simulations. The model is demonstrated on large eddy simulation data of a turbulent jet at Mach number $$M=0.9$$and Reynolds number $$\mathrm {Re}_D\approx 10^6$$.},
doi = {10.1007/s00162-021-00588-6},
file = {:ChuSchmidt_2021_TCFD.pdf:PDF},
refid = {Chu2021},
url = {https://doi.org/10.1007/s00162-021-00588-6},
}

## Bispectral Mode Decomposition

Triadic interactions are the fundamental mechanism of energy transfer in fluid flows. Bispectral mode decomposition (BMD) educes coherent flow structures that are associated with triadic interactions from experimental or numerical data. Triadic interactions are characterized by quadratic phase coupling which can be detected by the bispectrum. The proposed method maximizes an integral measure of this third-order statistic to compute modes associated with frequency triads, as well as a mode bispectrum that identifies resonant three-wave interactions. Unlike the classical bispectrum, the decomposition establishes a causal relationship between the three frequency components of a triad. This permits the distinction of sum- and difference-interactions, and the computation of interaction maps that indicate regions of nonlinear coupling.

Literature:

• Schmidt, O. T.. “Bispectral mode decomposition of nonlinear flows.” Nonlinear dynamics 102(4) (2020): 2479-25013.
[Bibtex]
@Article{schmidt_2020_nody,
author = {Schmidt, O. T.},
journal = {Nonlinear Dynamics},
title = {{Bispectral mode decomposition of nonlinear flows}},
year = {2020},
issn = {0924-090X},
number = {102(4)},
pages = {2479-25013},
doi = {10.1007/s11071-020-06037-z},
file = {:Schmidt_2020_NODY.pdf:PDF},
}

Code:

## Extreme events in wall turbulence

The mechanics of extreme intensity events in the buffer and logarithmic layers of a turbulent channel at 𝑅𝑒𝜏=2000 is investigated. The 99.9th percentile of the most intense events in the dissipation of turbulent kinetic energy is analysed by means of conditional space–time proper orthogonal decomposition. The computed spatio-temporal modes are coherent in space and over the considered time frame, and optimally capture the energy of the ensemble. The most energetic mode with transverse symmetric structure describes a turbulent burst event. The underlying mechanism is a varicose instability which generates localized extrema in the dissipation and production of turbulent kinetic energy and drives the formation of a hairpin vortex. The most energetic anti-symmetric mode is related to a sinuous-type instability that is situated in the shear layer between two very-large-scale streaks. Statistical results show the energy in the symmetric mode to exceed that in the anti-symmetric mode by a near constant factor for the considered wall distances. Both mechanisms occur throughout the range of wall distances in an effectively self-similar manner that is consistent with the attached-eddy hypothesis. By analogy with transitional flows, the results suggest that the events are induced by an exponential growth mechanism.

Literature:

• Hack, M. J. P. and O. T. Schmidt. “Extreme events in wall turbulence.” Journal of fluid mechanics 907 (2021): A9.
[Bibtex]
@Article{hackschmidt_2020_jfm,
author = {Hack, M. J. P. and Schmidt, O. T.},
journal = {Journal of Fluid Mechanics},
title = {Extreme events in wall turbulence},
year = {2021},
pages = {A9},
volume = {907},
doi = {10.1017/jfm.2020.798},
file = {:HackSchmidt_2020_JFM.pdf:PDF},
publisher = {Cambridge University Press},
}

## Modal Analysis of Acoustic Directivity in Turbulent Jets

The directivity of noise from three large-eddy simulations of turbulent jets at Mach 0.7, 0.9, and 1.5 is investigated using spectral proper orthogonal decomposition (SPOD). The most energetic patterns of acoustic radiation are extracted using the far-field pressure 2-norm. Specialization of the norm to the far field is accomplished through localized spatial weighting. Radiation patterns to specific jet inlet angles are isolated by further restricting the spatial weighting to small rectangular regions in the far field. The most energetic radiation pattern for all cases and relevant frequencies is a single superdirective acoustic beam in the downstream direction. The source region of these beams is traced back to the end of the potential core for low frequencies and the shear-layer region for higher frequencies. In the sideline direction, to low angles, the acoustic patterns consist of beams that propagate upstream or perpendicular to the jet axis. The sideline radiation patterns are found to originate from the same source locations as the dominant superdirective beams. Inspection of the SPOD modes reveals that sideline radiation is directly linked to directive downstream radiation. Within the restricted radial extent of the computational domain, these results indicate that the sources of sideline and downstream radiation are intimately linked.

Literature:

• Nekkanti, A. and O. T. Schmidt. “Modal analysis of acoustic directivity in turbulent jets.” Aiaa journal (2020): 1-12.
[Bibtex]
@Article{nekkantischmidt_2020_aiaaj,
author = {Nekkanti, A. and Schmidt, O. T.},
journal = {AIAA Journal},
title = {Modal Analysis of Acoustic Directivity in Turbulent Jets},
year = {2020},
pages = {1-12},
doi = {10.2514/1.J059425},
eprint = {https://doi.org/10.2514/1.J059425},
file = {:NekkantiSchmidt_2020_AIAAJ.pdf:PDF},
url = {https://doi.org/10.2514/1.J059425},
}

## 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:

• 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:

## 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:

• 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:

## 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:

• 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:

• 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: