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

Traveling wave packets are key coherent features contributing to the dynamics of
several advective flows. This work introduces the Hilbert proper orthogonal decomposi-
tion (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
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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 spatiotempo-
ral 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