An efficient streaming algorithm for spectral proper orthogonal decomposition

Schmidt, Oliver T.; Towne, Aaron

doi:10.1016/j.cpc.2018.11.009

Cited by 60 publications

(27 citation statements)

References 21 publications

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“…A Matlab implementation is available at https://github.com/SpectralPOD/spod_matlab. We also note that Schmidt (2017) recently formulated a streaming version of the algorithm that can reduce computational cost for large data sets.…”

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confidence: 99%

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

2018

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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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confidence: 99%

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

2018

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“…From the raw, experimental data, SPOD calculates energy-ranked modes which are orthogonal to one another and oscillate at a single frequency. The SPOD modes are computed using the procedure described in Towne et al (2018) and the code developed by Schmidt & Towne (2019). A brief summary of the steps is presented below.…”

Section: Spectral Proper Orthogonal Decompositionmentioning

confidence: 99%

A tale of two airfoils: resolvent-based modelling of an oscillator versus an amplifier from an experimental mean

2019

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The flows around a NACA 0018 airfoil at a chord-based Reynolds number of Re = 10250 and angles of attack of α = 0 • and α = 10 • are modelled using resolvent analysis and limited experimental measurements obtained from particle image velocimetry. The experimental mean velocity profiles are data-assimilated so that they are solutions of the incompressible Reynolds-averaged Navier-Stokes equations forced by Reynolds stress terms which are derived from experimental data. Spectral proper orthogonal decompositions of the velocity fluctuations and nonlinear forcing suggest different modelling approaches should be taken based on the angle of attack under consideration. For the α = 0 • case, the cross-spectral density tensors of both the velocity fluctuations and nonlinear forcing are low-rank at the shedding frequency and its higher harmonics. In the α = 10 • case, low-rank behaviour is observed for the velocity fluctuations in two bands of frequencies. Resolvent analysis of the data-assimilated means identifies lowrank behaviour only in the vicinity of the shedding frequency for α = 0 • and none of its harmonics. The resolvent operator for the α = 10 • case, on the other hand, identifies two linear mechanisms whose frequencies are a close match with those identified by spectral proper orthogonal decomposition. It is also shown that the second linear mechanism, corresponding to the Kelvin-Helmholtz instability in the shear layer, cannot be identified just by considering the time-averaged experimental measurements as a mean flow for resolvent analysis. This is due to the fact that experimental data are missing near the leading edge of the airfoil. The α = 0 • case is classified as an oscillator where the flow is organized around an intrinsic instability mechanism while the α = 10 • case behaves like an amplifier whose forcing is unstructured. For both cases, resolvent modes resemble those from spectral proper orthogonal decomposition when the operator is low-rank. To model the higher harmonics where this is not the case, we add parasitic resolvent modes, as opposed to classical resolvent modes which are the most amplified, by approximating the nonlinear forcing from limited triadic interactions of known resolvent modes. The amplifier case is modelled without parasitic modes at frequencies where the resolvent is low-rank. The two cases suggest that resolvent-based modelling can be achieved for more complex flows with limited experimental measurements and the nonlinear forcing need not be approximated unless the flow behaves like an oscillator.

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“…We define the data matrix , The rows of are measurements of points along the beam, and the columns are the time series for each point with size . Assuming that the system is stationary and consistent with the procedure in Towne et al. (2018) and Schmidt & Towne (2019), the discrete Fourier transform of each row of our is carried out using Welch's method (Welch 1967). In the procedure, each discrete time series is segmented into 50 % overlapping blocks of size , Fourier transformed, and assembled into a Fourier domain data matrix at each discrete frequency , where , is the total number of blocks in Welch's method, is a Fourier realization of the data and block number index.…”

Section: Tablementioning

confidence: 99%