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

Towne, Aaron; Schmidt, Oliver T.; Colonius, Tim

doi:10.1017/jfm.2018.283

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

Journal Article · Tue May 29 04:00:00 EDT 2018 · Journal of Fluid Mechanics

DOI:https://doi.org/10.1017/jfm.2018.283· OSTI ID:1538911

^[1]; ; Colonius, Tim

DOE/OSTI

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.

Research Organization:: Stanford Univ., CA (United States)

Sponsoring Organization:: USDOE National Nuclear Security Administration (NNSA)

DOE Contract Number:: NA0002373

OSTI ID:: 1538911

Journal Information:: Journal of Fluid Mechanics, Journal Name: Journal of Fluid Mechanics Vol. 847; ISSN 0022-1120

Publisher:: Cambridge University Press

Country of Publication:: United States

Language:: English

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