Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis
- Creators
- Towne, Aaron
- Schmidt, Oliver T.
- Colonius, Tim
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.
Additional Information
© 2018 Cambridge University Press. (Received 3 August 2017; revised 13 March 2018; accepted 30 March 2018; first published online 29 May 2018) A.T. gratefully acknowledges support from NASA grant no. NNX15AU93A and from the NNSA Predictive Science Academic Alliance Program II, grant no. DE-NA0002373. O.T.S. acknowledges support from the German science foundation (DFG) grant no. 3114/1-1, and O.T.S. and T.C. acknowledge support from ONR-N0014-11-1-0753 and ONR-N00014-16-1-2445. The authors also thank Dr K. K. Chen for supplying portions of the code used for the Ginzburg–Landau problem and Dr G. A. Brès for providing the jet LES data. The LES study was supported by NAVAIR SBIR project, under the supervision of Dr J. T. Spyropoulos. The main LES calculations were carried out on CRAY XE6 machines at DoD HPC facilities in ERDC DSRC.Attached Files
Submitted - 1708.04393
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Additional details
- Eprint ID
- 87401
- DOI
- 10.1017/jfm.2018.283
- Resolver ID
- CaltechAUTHORS:20180627-121240173
- NASA
- NNX15AU93A
- Department of Energy (DOE)
- DE-NA0002373
- Deutsche Forschungsgemeinschaft (DFG)
- 3114/1-1
- Office of Naval Research (ONR)
- N0014-11-1-0753
- Office of Naval Research (ONR)
- N00014-16-1-2445
- Created
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2018-06-27Created from EPrint's datestamp field
- Updated
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2021-11-15Created from EPrint's last_modified field