Spectral proper orthogonal decomposition

By Moritz Sieber, Kilian Oberleithner, Christian Oliver Paschereit. Journal of Fluid Mechanics, 2015

383 citations recorded by the source corpus.

The identification of coherent structures from experimental or numerical data is an essential task when conducting research in fluid dynamics. This typically involves the construction of an empirical mode base that appropriately captures the dominant flow structures. The most prominent candidates are the energy-ranked proper orthogonal decomposition (POD) and the frequency-ranked Fourier decomposition and dynamic mode decomposition (DMD). However, these methods are not suitable when the relevant coherent structures occur at low energies or at multiple frequencies, which is often the case. To overcome the deficit of these ‘rigid’ approaches, we propose a new method termed spectral proper orthogonal decomposition (SPOD). It is based on classical POD and it can be applied to spatially and temporally resolved data. The new method involves an additional temporal constraint that enables a clear separation of phenomena that occur at multiple frequencies and energies. SPOD allows for a continuous shifting from the energetically optimal POD to the spectrally pure Fourier decomposition by changing a single parameter. In this article, SPOD is motivated from phenomenological considerations of the POD autocorrelation matrix and justified from dynamical systems theory. The new method is further applied to three sets of PIV measurements of flows from very different engineering problems. We consider the flow of a swirl-stabilized combustor, the wake of an airfoil with a Gurney flap and the flow field of the sweeping jet behind a fluidic oscillator. For these examples, the commonly used methods fail to assign the relevant coherent structures to single modes. The SPOD, however, achieves a proper separation of spatially and temporally coherent structures, which are either hidden in stochastic turbulent fluctuations or spread over a wide frequency range. The SPOD requires only one additional parameter, which can be estimated from the basic time scales of the flow. In spite of all these benefits, the algorithmic complexity and computational cost of the SPOD are only marginally greater than those of the snapshot POD.

Subfields: Computational Mathematics, Computational Mechanics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/36d2c041b342f0d0dffdb568ac041a472332b71a

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Citations in the corpus

Guide to Spectral Proper Orthogonal Decomposition (2020, methodology): “We caution that the term “ SPOD ” has been applied to a different technique proposed by Sieber et al. [7], which wedo not discuss here.”
Modal Analysis of Fluid Flows: An Overview (2017, methodology): “[71], whose approach blended between the POD and discrete Fourier transform modes by filtering the temporal correlation matrix.”
Modal Analysis of Fluid Flows: Applications and Outlook (2019, background): “i et al. [180]. The turbulent ow eld obtained from a delayed detached-eddy simulation and the extracted dominant POD modes are shown in Fig.19. In this analysis, they identied the velocity POD modes [181] that correlate with vortical structures responsible for generating the acoustic tones. Although”
Recursive dynamic mode decomposition of transient and post-transient wake flows (2016, background): “One important recent contribution is spectral POD (Sieber et al. 2015) which interpolates between POD and DMD by filtering the correlation matrix.”
On closures for reduced order models—A spectrum of first-principle to machine-learned avenues (2021, neutral): “In an effort toward the accurate identiﬁcation of coherent structures from experimental or numerical data, a spectral POD approach has been developed, 118 and its relationship to DMD and resolvent analysis has been established.”
Multi-scale proper orthogonal decomposition of complex fluid flows (2018, methodology): “This is a major difference between the mPOD and the SPOD by Sieber et al. (2016), in which the filtering procedure can eventually result in a nonorthogonal temporal basis, or the mrDMD by Kutz et al. (2016b), in which the loss of orthogonality can be produced by growing/decaying modes.”
Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis (2017, methodology): “The aforementioned method of Sieber et al. (2016) filters the temporal correlation tensor over a time horizon leading to an ad hoc interpolation between space-only POD, which is obtained when the filter width is zero, and the discrete Fourier transform, which is recovered when the filter width is…”