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A DMD-based approach for Spectral Proper Orthogonal Decomposition of non time-resolved multi-pulse Shake-The-Box measurements

This paper proposes a novel methodology that combines Dynamic Mode Decomposition with multi-pulse Shake-The-Box data to enable Spectral Proper Orthogonal Decomposition of non-time-resolved flow measurements, successfully extracting high-frequency modal structures from sub-10 Hz acquisition rates in turbulent jet experiments.

Original authors: Philipp Godbersen, Kilian Oberleithner, Jakob G. R. von Saldern

Published 2026-09-22

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CC BY 4.0

Original authors: Philipp Godbersen, Kilian Oberleithner, Jakob G. R. von Saldern

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

To understand the flow of air around a jet engine or a wind turbine, scientists often need to watch how the invisible currents move and twist over time. For decades, the standard way to do this has been to take a rapid-fire sequence of pictures, like a high-speed camera filming a hummingbird's wings, capturing every flicker of motion. From these continuous movies, researchers can use mathematical tools to separate the steady background wind from the swirling, energetic patterns that drive noise and vibration. However, in many real-world situations, such as inside a hot jet engine or a massive industrial fan, it is impossible to take these high-speed movies. The equipment is too expensive, or the environment is too harsh for delicate cameras. Instead, scientists have developed a clever workaround called Shake-The-Box. This technique tracks tiny particles floating in the air, but instead of filming a continuous movie, it takes a few quick snapshots in rapid succession and then waits a long time before taking the next set. The result is a collection of short, disconnected clips rather than a single long film. While this method is excellent for calculating average statistics, like how much force the wind exerts on average, it has traditionally been useless for studying the rhythmic, time-based patterns of the flow because the long gaps between clips break the continuity needed for standard analysis.

A team of researchers at the German Aerospace Center and the Technical University of Berlin has found a way to bridge this gap. They developed a new method that allows scientists to extract these rhythmic, time-based patterns from the disconnected clips produced by Shake-The-Box measurements. The core of their innovation lies in a mathematical technique called Dynamic Mode Decomposition. While traditional methods require a steady stream of data to find repeating frequencies, this new approach looks at pairs of snapshots taken just microseconds apart within each short burst. By analyzing the tiny changes between these paired moments, the method can infer the underlying rhythm of the flow, even though the bursts themselves are separated by long, silent pauses. The researchers tested this idea first on a computer simulation of a turbulent jet, where they knew the true answer. They successfully recreated the complex patterns of the flow using only the disconnected data, proving that the method works even when the data is sparse and irregular.

The team then took their method to the real world, applying it to a physical experiment involving a subsonic jet of air. In this experiment, the camera system captured snapshots at a rate of only 10 times per second, a pace far too slow to catch the rapid vibrations of the jet, which occur thousands of times per second. Despite this limitation, the new analysis successfully identified coherent structures in the flow vibrating at a frequency of 5.8 kilohertz. This is a frequency more than five hundred times faster than the rate at which the snapshots were taken. The researchers were able to visualize these high-speed patterns, revealing swirling, helical shapes in the air that twist and rotate as they move downstream. These structures, which were previously invisible to this type of measurement, appear at specific frequencies and show how energy moves through the jet. The study demonstrates that by using a different mathematical lens, scientists can now see the fast, rhythmic heartbeat of a flow even when they can only take a few slow, disconnected pictures, opening the door to studying complex fluid dynamics in environments where high-speed filming was once impossible.

Technical Summary: A DMD-based approach for Spectral Proper Orthogonal Decomposition of non time-resolved multi-pulse Shake-The-Box measurements

Problem Statement
Lagrangian particle-tracking techniques, specifically multi-pulse Shake-The-Box (MP-STB), enable flow-field measurements in regimes where high-speed time-resolved imaging is infeasible or uneconomical. However, MP-STB data consists of temporally independent snapshots separated by long intervals (e.g., 10 Hz), rendering conventional spectral analysis methods like Spectral Proper Orthogonal Decomposition (SPOD) inapplicable. SPOD requires time-resolved data to compute a Discrete Fourier Transform (DFT) for estimating cross-spectral density. While MP-STB is excellent for computing stationary statistics (e.g., Reynolds stresses), it traditionally precludes the analysis of temporal dynamics and coherent structures at frequencies higher than the snapshot acquisition rate. Existing alternatives, such as dual-time setups with variable spacing or methods that interpolate missing data, either require specialized measurement hardware or rely on filling in data gaps, which the authors aim to avoid.

Methodology
The authors propose a novel methodology to compute SPOD modes from MP-STB data by leveraging Dynamic Mode Decomposition (DMD) derived from snapshot pairs. The approach involves the following steps:

Snapshot Pair Generation: Instead of sampling the continuous particle tracks (fitted via polynomial) at their midpoints, the method queries the tracks at two distinct time instances ( $t_1$ and $t_2$ ) separated by a fixed, short interval $\Delta t$ . This generates snapshot pairs $(x_i, y_i)$ where $y_i = A x_i$ , representing the flow evolution over $\Delta t$ .
DMD Application: Dynamic Mode Decomposition is applied to these pairs. Unlike standard DMD which assumes sequential data, this approach utilizes the non-sequential DMD formulation (based on the Moore-Penrose pseudoinverse) where the linear operator $A$ is estimated from independent pairs. This yields DMD modes with associated eigenvalues $\mu_k$ , from which frequencies are derived via $f_k = \text{Im}(\ln(\mu_k)) / (2\pi \Delta t)$ .
Modified SPOD Construction:
- The time-resolved SPOD workflow (Towne et al., 2018) is adapted. The standard DFT is replaced by the DMD modes.
- Since DMD modes from MP-STB data have variable, data-dependent frequencies rather than fixed DFT bins, the modes are binned into frequency intervals ( $b_k$ ) matching the desired resolution.
- For each frequency bin, DMD modes from multiple independent blocks (realizations) are assembled to estimate the cross-spectral density (CSD) tensor.
- Eigenvalue decomposition of the CSD matrix yields the SPOD modes and their energies.

The method utilizes a QR-decomposition-based DMD algorithm to ensure the output modes are directly scaled, avoiding the need for additional amplitude normalization steps required by SVD-based approaches.

Key Contributions

Extension of SPOD to Non-Time-Resolved Data: The paper demonstrates that SPOD modes can be computed from MP-STB data without requiring time-resolved acquisition or specialized dual-time measurement setups.
Frequency Resolution Beyond Nyquist: The method enables the extraction of flow dynamics at frequencies significantly higher than the global snapshot acquisition rate (e.g., 10 Hz), limited only by the short inter-pulse interval within the MP-STB burst (microseconds).
Validation on Synthetic and Real Data: The approach is validated against a Large Eddy Simulation (LES) of a turbulent jet and applied to a real 3D MP-STB measurement of a subsonic turbulent jet.

Results

Synthetic Validation: Using LES data of a subsonic turbulent jet, the authors generated synthetic MP-STB datasets by subsampling and randomizing sequential data. The modified SPOD approach successfully recovered the SPOD spectrum and mode shapes, showing good agreement with conventional SPOD results for the first few modes and low-to-mid frequencies.
- Limitations: The method showed degradation at higher frequencies. This is attributed to the non-uniform distribution of DMD frequencies, which leads to fewer modes contributing to high-frequency bins, thereby reducing the quality of the CSD estimation.
- Spacing Sensitivity: The method remained robust even when the spacing between snapshot pairs was increased (up to 10 simulation time steps), though the spectra became noisier due to reduced data availability.
Experimental Application: The method was applied to a real 3D, three-component velocity dataset of a subsonic turbulent jet ( $Re \approx 1.7 \times 10^5$ $R e \approx 1.7 \times 1 0^{5}$ , $M \approx 0.5$ $M \approx 0.5$ ).
- High-Frequency Extraction: The authors successfully extracted coherent structures at a Strouhal number of $St = 0.17$ and up to $St \approx 0.5$ .
- Frequency Breakthrough: For this experiment, $St = 0.5$ corresponds to a physical frequency of 5.8 kHz, which is orders of magnitude higher than the 10 Hz global snapshot acquisition rate of the MP-STB system.
- Mode Structures: The analysis revealed helical modes with various azimuthal wavenumbers ( $m=1, 2, 3$ ), visualized as counter-rotating pairs. The dominant modes were clearly resolved, while higher frequencies ( $St > 0.5$ ) were dominated by noise.

Significance and Claims
The paper claims that this approach makes the examination of temporal dynamics and coherent structures possible for MP-STB measurements, a capability previously restricted to time-resolved techniques. By utilizing the short temporal separation within multi-pulse bursts, the method bypasses the Nyquist limit imposed by the global repetition rate of the measurement system.

The authors maintain a modest stance regarding the method's limitations, acknowledging that the quality of the SPOD estimation degrades at higher frequencies due to the statistical scarcity of modes in those bins. They note that while the current implementation is computationally feasible (processed in a few hours on a 128-core system), future work could explore streaming DMD algorithms or advanced DMD variants (e.g., forward-backward DMD, optimized DMD) to improve robustness against noise and reduce memory requirements. The paper does not claim to replace time-resolved methods where available but positions itself as a powerful tool for extracting spectral information from data types where such methods are otherwise inaccessible.

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