Hilbert Proper Orthogonal Decomposition: a tool for educing advective wavepackets from flow field data

TL;DR

This paper introduces the Hilbert proper orthogonal decomposition (HPOD), a novel method for extracting advective wavepackets from flow data, applicable to both temporal and spatial analyses, especially in under-resolved datasets.

Contribution

The work presents a new complex-valued extension of POD using the Hilbert transform, including a space-only version, validated on diverse flow datasets, enhancing wavepacket analysis in advective flows.

Findings

HPOD effectively extracts wavepackets with amplitude and frequency modulation.

The space-only HPOD is mathematically equivalent to the conventional version.

Both methods perform well on complex turbulent flow datasets.

Abstract

Travelling wavepackets are key coherent features contributing to the dynamics of several advective flows. This work introduces the Hilbert proper orthogonal decomposition (HPOD) to distil these features from flow field data, leveraging their mathematical representation as modulated travelling 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; a novel space-only HPOD, computing it along the advection direction. The HPOD is shown to extract wavepackets 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 travelling waves to swap temporal operations by spatial ones, is proved mathematically equivalent to its conventional counterpart. The two HPOD versions are characterized and validated on three datasets ordered by complexity: a 2D-DNS of a laminar bluff-body wake with periodic vortex shedding; an LES of a turbulent jet with intermittent, highly modulated wavepackets; and a 2D-PIV of a turbulent jet with measurement errors and no temporal resolution. In advecting flows, both HPOD versions deliver practically identical complex-valued advecting wavepacket 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 to extract these structures from temporally under-resolved datasets, typical of snapshot particle image velocimetry.