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Spectral proper orthogonal decomposition
Moritz Sieber, Kilian Oberleithner, Christian Oliver Paschereit
TL;DR
Existing POD, DFT, and DMD decompositions struggle to identify coherent structures in turbulent flows when structures are weak, intermittent, or span multiple frequencies. The paper introduces SPOD, which filters the POD correlation matrix to impose a tunable temporal constraint between energetic POD and spectrally pure DFT. Across three experimental PIV flow data sets, SPOD separates spatially and temporally coherent structures that conventional methods fail to assign to single modes.
Problem
Turbulent flows contain coherent motion across many scales, while existing decompositions can mix temporal scales, split frequency-modulated structures, or miss weak and intermittent structures.
Method
SPOD extends POD by filtering the correlation matrix, with filter strength controlling the shift from energetic POD toward spectrally pure DFT.
Results
Across three experimental PIV data sets, SPOD separates spatially and temporally coherent structures that conventional methods fail to assign to single modes.
Takeaways & Limitations
SPOD provides a broadly applicable decomposition for identifying coherent structures hidden in stochastic fluctuations or distributed across wide frequency ranges.
Takeaways & Limitations
SPOD requires selecting a filter timescale, and coupled mode pairs may require cumbersome manual identification through visual inspection.
Abstract
from arXiv · showhide
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 fail 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 system 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.
1. Introduction and motivation
Turbulent-flow data challenge existing decompositions because coherent motion may be weak, intermittent, variable in frequency, or distributed across temporal scales. SPOD extends POD with a filtered correlation matrix, continuously bridging energetic POD and spectrally clean DFT.
- Motivation: Turbulent-flow analysis must distinguish deterministic coherent motion from stochastic motion across broad temporal and spatial scales.
- Limitations of existing methods: POD finds energetically dominant modes but imposes no temporal constraint, so a mode can combine phenomena evolving on very different temporal scales.
- Limitations of existing methods: DFT and DMD organize modes by fixed frequencies, but variable frequency, phase jitter, and intermittency can spread one coherent structure across multiple modes.
- Limitations of existing methods: Existing methods often struggle with weak coherent structures, low signal-to-noise data, intermittent dynamics, and flows containing multiple phenomena at similar frequencies.
- SPOD: SPOD applies a filter to the POD correlation matrix, creating a continuous shift from POD’s energetic optimality toward DFT’s spectral purity.
- Scope: The paper demonstrates SPOD on three experimental PIV data sets and describes its method, interpretation through the correlation matrix, and coupled-mode identification.
2. Description and interpretation of the proposed method
SPOD extends snapshot POD by filtering its correlation matrix along diagonals, imposing temporal structure while continuously connecting energy-optimal POD with spectrally pure DFT.
- Snapshot POD: Snapshot POD decomposes fluctuating data into orthonormal spatial modes and temporal coefficients, with mode energies obtained from correlation-matrix eigenvalues.For PIV or CFD data, the temporal correlation matrix is formed between snapshots when spatial points outnumber snapshots.
- Correlation-matrix interpretation: In a forced turbulent jet, periodic vortex shedding and convection generate a diagonal wave-like correlation-matrix structure whose averaged diagonals reproduce the autocorrelation periodicity.The autocorrelation reflects spectral content but not the phase of individual frequencies.
- Correlation-matrix filtering: The SPOD method applies a low-pass filter along the diagonals of the POD correlation matrix to produce a filtered correlation matrix.The filter augments diagonal similarity and imposes clearer temporal dynamics on the decomposition.
- Limiting behavior: Extending the filter over the full time series with periodic boundaries makes SPOD equal to the DFT, enabling continuous movement from energetically optimal POD to purely spectral decomposition.In this limit, the correlation matrix becomes circulant and its eigenvectors and eigenvalues are given by the Fourier transform.
- Dynamical-systems interpretation: From a local linear time-invariant perspective, diagonal changes encode modal amplification, whereas crosswise changes encode modal frequency and phase.The local temporal extent is tied to the filter size; SPOD smoothing equalizes consecutive anti-diagonals and limits temporal variation in amplitude and frequency.
- Mode interpretation and coupling: SPOD coefficients have low-pass-filtered amplification rates and band-pass-filtered frequencies, while coupled periodic modes require subsequent identification.The paper proposes DMD-based spectral proximity as an unbiased quantitative measure for dynamic coupling, although the truncated common spectral representation is not exact.
3. Applications to experimental data
Across three experimental flows, SPOD separates coherent structures more clearly than POD or DFT, including structures obscured by turbulence, noise, similar frequencies, or differing energies.
- Experimental data: The experiments use PIV measurements of a swirl-stabilized combustor, an airfoil wake with a Gurney flap, and a sweeping jet from a fluidic oscillator.All three data sets were recorded with the same PIV measurement system.
- Swirl-stabilized combustor: In the swirling jet, SPOD identifies peaks at St = 0.09, 0.5, and 0.8 and separates distinct single- and double-helical structures.The St = 0.5 mode is linked to precessing motion of the recirculation zone, while the St = 0.8 mode is not its harmonic.
- Swirl-stabilized combustor: Compared with POD and DFT, SPOD separates coherent and stochastic fluctuations while retaining interpretable spatial structures and frequency information.POD partially mixes structures, whereas DFT produces noisy modes and does not identify frequencies of interest.
- Method scope: Alternative POD strategies can recover some structures but require prior knowledge of their spatial extent or symmetry, whereas SPOD requires neither.The limitation concerns approaches based on restricting the domain or exploiting known spatial symmetries.
- Airfoil with Gurney flap: For the Gurney-flap airfoil wake, SPOD separates vortex shedding, its modulation, and a higher harmonic that POD misses and DFT does not capture reliably.The SPOD filter length equals three shedding periods, approximately the traveling time through the measurement domain.
- Fluidic oscillator: For the sweeping jet, SPOD resolves the fundamental and higher harmonics despite frequency variation, whereas DFT fails on the weak seventh harmonic.The DFT splits peaks into several modes and cannot reproduce the seventh-harmonic structure when frequency jitter is substantial and mode energy is low.
4. Summary and conclusion
SPOD extends POD with a temporal filter that continuously shifts modal representations toward DFT-like spectral purity while preserving interpretable coherent structures.
- Method: SPOD extends POD for time-resolved data by filtering the snapshot correlation matrix with negligible additional computational cost.The method was developed for broad applicability to turbulent-flow data with minimal user input.
- Method: The filter width controls modal spectral bandwidth, constraining amplitude, frequency, and growth-rate variations; maximum width yields strictly periodic DFT modes.Increasing filter width progressively limits temporal variations until a stable limit cycle is reached.
- Applications: Across swirl-stabilized combustor, Gurney-flap airfoil, and fluidic-oscillator flows, SPOD showed advantageous features and greater versatility than alternative methods.Other methods performed equally well in individual cases, but none was as versatile overall.
- Capabilities: SPOD separates individual fluid-dynamic phenomena into single modes more effectively than POD or DFT, which can mix or spread them across modes.This soft spectral constraint supports clearer separation of structures with different temporal behavior.
- Capabilities: SPOD can reject noise, recover dynamics below the overall noise level, complete partially recorded temporal dynamics, and produce smooth coefficients with adjustable frequency and amplitude variation.The adjustable variations are set by the filter size.
- Applications: SPOD modes support linked-mode identification, comparisons with simultaneous measurements, reduced-order modeling, and improved modal inputs for DMD.These characteristics make the modes useful for downstream analysis beyond the initial decomposition.
- Scope: The method requires one flow-dynamics assumption: a filter timescale chosen from the dominant frequency or convective timescale.This timescale acts as inertia limiting the rate of change of modal frequency and amplitude.
Appendix A. The spatial correlation version of SPOD
The spatial-correlation SPOD formulation enables efficient decomposition when measurements contain many samples relative to measured spatial positions.
- Motivation: The spatial-correlation formulation is computationally more efficient than the temporal formulation when the number of snapshots greatly exceeds the number of grid points.For very large time series, the temporal correlation can become impractical because of memory and eigenvalue-solution costs.
- Applicability: For simultaneous multipoint pressure measurements, the spatial formulation is appropriate when measured positions M are much fewer than samples N.The number of samples may reach a million or more in this setting.
- Computation: The correlation tensor is reshaped into a matrix for numerical implementation and decomposed into eigenvalues and eigenvectors.The eigenvalues are ordered from largest to smallest and are nonnegative.
- Modal construction: Each eigenvector acts as a discrete convolution filter applied to the time series to obtain modal coefficients, while its zero-delay part defines the spatial mode.The full eigenvectors form a data-driven filter bank for decomposing time series into modal contributions.
- Applications: The spatial approach supports single- or multisensor signals, including phase reconstruction of dominant oscillations from pressure measurements.Its modal interpretation is compared with empirical mode decomposition and multitime-delay POD phase estimation.
- Computational cost: The spatial-version computational cost scales with filter size and is more efficient than the snapshot approach only when M(2Nf + 1) < N.This condition compares the number of spatial positions, filter width, and samples.
Appendix B. Properties of the SPOD modes
SPOD modes are orthonormal only when each spatial mode is considered together with all temporally shifted instances. The snapshot-based zero-delay representation introduces nonorthogonality, while mode norms indicate how well the data capture each mode.
- Including all temporally shifted instances makes the SPOD modes orthonormal.
- The snapshot calculation uses only the zero-delay part of each spatial mode, limiting modal decomposition to time-independent spatial modes.This restriction introduces imperfections in the resulting representation.
- The selected snapshot-based modes are neither normal nor orthogonal.
- A mode norm indicates how well that single mode is represented by the investigated data set.
- Filtering constructs an idealized correlation matrix whose mode norms ζ_i deviate from one according to how completely the initial data capture each mode.
- For the sweeping jet, SPOD completes missing data in the partially captured fundamental mode and produces an equal-energy mode pair with levels μ_i.