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  • Open Access

Hilbert proper orthogonal decomposition: A tool for educing advective wave packets from flow field data

Marco Raiola*

Jochen Kriegseis

  • *Contact author: mraiola@ing.uc3m.es
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Phys. Rev. Fluids 11, 044905 – Published 6 April, 2026

DOI: https://doi.org/10.1103/bhlm-qwfh

Abstract

Traveling wave packets are key coherent features contributing to the dynamics of several advective flows. This work introduces the Hilbert proper orthogonal decomposition (HPOD) to distill these features from flow field data, leveraging their mathematical representation as modulated traveling waves. The HPOD is a complex-valued extension of the proper orthogonal decomposition, where the Hilbert transform of the dataset is used to compute its analytic signal. Two versions of the technique are explored and compared: the conventional HPOD, computing the analytic signal in time, and a novel space-only HPOD, computing it along the advection direction. The HPOD is shown to extract wave packets with amplitude and frequency modulation in time and space. Its broadband nature offers an alternative to spectrally pure decompositions when instantaneous, local wave characteristics are important. The space-only version, leveraging space/time equivalence in traveling waves to swap temporal operations with spatial ones, is proven mathematically equivalent to its conventional counterpart. The two HPOD versions are characterized and validated on three datasets ordered by complexity: a two-dimensional (2D) direct numerical simulation of a laminar bluff-body wake with periodic vortex shedding; a large eddy simulation of a turbulent jet with intermittent, highly modulated wave packets; and a 2D particle image velocimetry experiment of a turbulent jet with measurement errors and no temporal resolution. In advecting flows, both HPOD versions deliver practically identical complex-valued advecting wave-packet structures, characterized by spatiotemporal amplification and decay, wave modulation, and intermittency phenomena in turbulent flow cases, such as in turbulent jets. The space-only variant allows the extraction of these structures from temporally under-resolved datasets, typical of snapshot particle image velocimetry.

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Phys. Rev. Fluids 11, 044905– Published 6 April, 2026
  • Received 7 November 2025
  • Accepted 10 March 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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