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). 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...
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| Published in: | Journal of fluid mechanics Vol. 847; pp. 821 - 867 |
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| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Cambridge, UK
Cambridge University Press
25.07.2018
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| Subjects: | |
| ISSN: | 0022-1120, 1469-7645 |
| Online Access: | Get full text |
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| 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 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. |
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| AbstractList | 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. 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. |
| Author | Colonius, Tim Towne, Aaron Schmidt, Oliver T. |
| Author_xml | – sequence: 1 givenname: Aaron orcidid: 0000-0002-7315-5375 surname: Towne fullname: Towne, Aaron email: atowne@stanford.edu organization: Center for Turbulence Research, Stanford University, Stanford, CA 94305, USA – sequence: 2 givenname: Oliver T. orcidid: 0000-0002-7097-0235 surname: Schmidt fullname: Schmidt, Oliver T. organization: California Institute of Technology, Pasadena, CA 91125, USA – sequence: 3 givenname: Tim surname: Colonius fullname: Colonius, Tim organization: California Institute of Technology, Pasadena, CA 91125, USA |
| BackLink | https://www.osti.gov/biblio/1538911$$D View this record in Osti.gov |
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| ContentType | Journal Article |
| Copyright | 2018 Cambridge University Press |
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| DOI | 10.1017/jfm.2018.283 |
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| DocumentTitleAlternate | Spectral POD, DMD and resolvent analysis A. Towne, O. T. Schmidt and T. Colonius |
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| Snippet | We consider the frequency domain form of proper orthogonal decomposition (POD), called spectral proper orthogonal decomposition (SPOD). Spectral POD is derived... |
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| SubjectTerms | Coefficients Decomposition Fluid dynamics JFM Papers Landau-Ginzburg equations Mechanics Modes Physics Proper Orthogonal Decomposition Spectra Statistical analysis Statistical methods Statistics Steady flow Thermal expansion Turbulence Turbulent jets |
| Title | Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis |
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