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Spectral analysis of fluid flows using sub-Nyquist-rate PIV data

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Abstract

Spectral methods are ubiquitous in the analysis of dynamically evolving fluid flows. However, tools like Fourier transformation and dynamic mode decomposition (DMD) require data that satisfy the Nyquist–Shannon sampling criterion. In many fluid flow experiments, such data are impossible to acquire. We propose a new approach that combines ideas from DMD and compressed sensing to accommodate sub-Nyquist-rate sampling. Given a vector-valued signal, we take measurements randomly in time (at a sub-Nyquist rate) and project the data onto a low-dimensional subspace. We then use compressed sensing to identify the dominant frequencies in the signal and their corresponding modes. We demonstrate this method using two examples, analyzing both an artificially constructed dataset and particle image velocimetry data from the flow past a cylinder. In each case, our method correctly identifies the characteristic frequencies and oscillatory modes dominating the signal, proving it to be a capable tool for spectral analysis using sub-Nyquist-rate sampling.

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Notes

  1. Compressed sensing is also known as “compressive sampling” or “compressive sensing” and is closely related to “sparse approximation”/“sparse reconstruction”/“sparse recovery” methods.

  2. Though technically not a norm, the cardinality of a vector is often referred to as its \(\ell _0\) norm.

  3. We note that the convention in fluid mechanics is for each column of the snapshot matrix to correspond to an instant in time. For compressed sensing, the convention is reversed: each row of the signal matrix corresponds to a particular instant.

  4. Again, we use the compressed sensing convention: rows of \(A_r\) correspond to instants in time.

  5. We find that the POD modes do not differ much when computed from the time-resolved signal \(F\) versus the sampled signal \(G\).

  6. In practice, we would compute the POD modes using only the sampled data, and not the time-resolved data. However, for this flow, we do not expect the POD basis to change much if computed from the sampled data, due to the strong attraction of the dynamics onto the low-dimensional POD subspace.

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Acknowledgments

The authors acknowledge funding from the AFOSR and the NSF and thank Steve Brunton for many insightful discussions regarding compressed sensing in the context of dynamical systems.

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Authors and Affiliations

  1. Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ, 08544, USA

    Jonathan H. Tu, Clarence W. Rowley & Jessica K. Shang

  2. Department of Applied Mathematics, University of Washington, Seattle, WA, 98195, USA

    J. Nathan Kutz

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  1. Jonathan H. Tu
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  2. Clarence W. Rowley
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  3. J. Nathan Kutz
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Correspondence to Jonathan H. Tu.

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Tu, J.H., Rowley, C.W., Kutz, J.N. et al. Spectral analysis of fluid flows using sub-Nyquist-rate PIV data. Exp Fluids 55, 1805 (2014). https://doi.org/10.1007/s00348-014-1805-6

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  • DOI: https://doi.org/10.1007/s00348-014-1805-6

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