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

Abstract

We consider the frequency domain form of proper orthogonal decomposition (POD), called spectral proper orthogonal decomposition (SPOD). Spectral POD is derived from a space–time POD problem for statistically stationary flows and leads to modes that each oscillate at a single frequency. This form of POD goes back to the original work of Lumley (Stochastic Tools in Turbulence, Academic Press, 1970), but has been overshadowed by a space-only form of POD since the 1990s. We clarify the relationship between these two forms of POD and show that SPOD modes represent structures that evolve coherently in space and time, while space-only POD modes in general do not. We also 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. Accordingly, SPOD modes represent structures that are dynamic in the same sense as DMD modes but also optimally account for the statistical variability of turbulent flows. Finally, we establish a connection between SPOD and resolvent analysis. The key observation is that the resolvent-mode expansion coefficients must be regarded as statistical quantities to ensure convergent approximations of the flow statistics. When the expansionmore » coefficients are uncorrelated, we show that SPOD and resolvent modes are identical. Our theoretical results and the overall utility of SPOD are demonstrated using two example problems: the complex Ginzburg–Landau equation and a turbulent jet.« less

Authors:
ORCiD logo; ORCiD logo;
Publication Date:
Research Org.:
Stanford Univ., CA (United States)
Sponsoring Org.:
USDOE National Nuclear Security Administration (NNSA)
OSTI Identifier:
1538911
DOE Contract Number:  
NA0002373
Resource Type:
Journal Article
Journal Name:
Journal of Fluid Mechanics
Additional Journal Information:
Journal Volume: 847; Journal ID: ISSN 0022-1120
Publisher:
Cambridge University Press
Country of Publication:
United States
Language:
English
Subject:
Mechanics; Physics

Citation Formats

Towne, Aaron, Schmidt, Oliver T., and Colonius, Tim. Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis. United States: N. p., 2018. Web. doi:10.1017/jfm.2018.283.
Towne, Aaron, Schmidt, Oliver T., & Colonius, Tim. Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis. United States. doi:10.1017/jfm.2018.283.
Towne, Aaron, Schmidt, Oliver T., and Colonius, Tim. Tue . "Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis". United States. doi:10.1017/jfm.2018.283.
@article{osti_1538911,
title = {Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis},
author = {Towne, Aaron and Schmidt, Oliver T. and Colonius, Tim},
abstractNote = {We consider the frequency domain form of proper orthogonal decomposition (POD), called spectral proper orthogonal decomposition (SPOD). Spectral POD is derived from a space–time POD problem for statistically stationary flows and leads to modes that each oscillate at a single frequency. This form of POD goes back to the original work of Lumley (Stochastic Tools in Turbulence, Academic Press, 1970), but has been overshadowed by a space-only form of POD since the 1990s. We clarify the relationship between these two forms of POD and show that SPOD modes represent structures that evolve coherently in space and time, while space-only POD modes in general do not. We also 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. Accordingly, SPOD modes represent structures that are dynamic in the same sense as DMD modes but also optimally account for the statistical variability of turbulent flows. Finally, we establish a connection between SPOD and resolvent analysis. The key observation is that the resolvent-mode expansion coefficients must be regarded as statistical quantities to ensure convergent approximations of the flow statistics. When the expansion coefficients are uncorrelated, we show that SPOD and resolvent modes are identical. Our theoretical results and the overall utility of SPOD are demonstrated using two example problems: the complex Ginzburg–Landau equation and a turbulent jet.},
doi = {10.1017/jfm.2018.283},
journal = {Journal of Fluid Mechanics},
issn = {0022-1120},
number = ,
volume = 847,
place = {United States},
year = {2018},
month = {5}
}

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