希尔伯特正交分解：用于从流场数据中提取行进波包的工具
Hilbert Proper Orthogonal Decomposition: a tool for educing advective wavepackets from flow field data
AI总结：
本文提出希尔伯特正交分解（HPOD）用于从流场数据中提取行进波包，通过数学表示为调制行进波的方法，验证了两种版本在不同数据集上的有效性。
摘要：
行进波包是几种行进流中关键的相干特征。本文引入希尔伯特正交分解（HPOD）来从流场数据中提取这些特征，利用其数学表示为调制行进波。HPOD是正交分解的复数扩展，其中数据集的希尔伯特变换用于计算其解析信号。探讨并比较了两种版本：传统HPOD计算时间解析信号；新型空间仅HPOD沿对流方向计算。HPOD能提取具有时间和空间幅度和频率调制的波包。其宽带性质为瞬时、局部波特性重要时提供了替代谱纯分解的方法。空间仅版本利用行进波中空间/时间等价性，将时间操作转换为空间操作，数学上等同于传统版本。两种HPOD版本在三个按复杂性排序的数据集上进行表征和验证：一个2D-DNS的层流 bluff-body 风涡旋脱落；一个 LES 的湍流喷射，具有间歇性、高度调制的波包；以及一个 2D-PIV 的湍流喷射，具有测量误差和无时间分辨率。在行进流中，两种 HPOD 版本在实际中提供几乎相同的复数行进波包结构，特征为时空放大和衰减、波调制和间歇性现象，如在湍流喷射中。空间仅版本允许从时间分辨率不足的数据集中提取这些结构，典型于快照粒子图像测速仪。
英文摘要：
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.
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