Studying propagating turbulent structures in the near wake of a sphere using Hilbert proper orthogonal decomposition

TL;DR

This study investigates propagating turbulent structures in the near wake of a sphere using Hilbert Proper Orthogonal Decomposition (HPOD). By comparing POD and HPOD outputs for velocity fluctuations at $Re_D=7780$, the authors show that HPOD emphasizes propagating structures and yields mode pairs that correspond to the same wake features, with leading modes carrying more energy in the analytic signal than in the original data. They further demonstrate that the analytic signal of POD modes, obtained via the Hilbert transform directly on the POD modes, reproduces the same POD mode pairings as HPOD with substantially lower computational cost, albeit with the usual caveat of spectral leakage. Phase-averaged analyses based on these paired modes reveal coherent propagating flapping and pulsating wake structures, highlighting the practical utility and trade-offs of HPOD and Hilbert-transform-based POD pairing for interpreting turbulent wake dynamics. The work provides a practical pathway to identify and analyze propagating coherent structures in non-time-resolved wake measurements, facilitating reduced-order representations and phase-averaged diagnostics in complex turbulent flows.

Abstract

Turbulent flows, despite their apparent randomness, exhibit coherent structures that underpin their dynamics. Proper orthogonal decomposition (POD) has been widely used to extract these structures from experimental data. While periodic features like vortex shedding can be identified using POD mode pairs when periodicity dominates the flow, detecting such structures in complex flows is more challenging. The Hilbert proper orthogonal decomposition (HPOD) addresses this by applying POD to the analytic signal of the turbulent fluctuations, yielding complex modes with a $90^\circ$ phase shift between the real and imaginary components. These modes capture propagating structures effectively but introduce filtering artefacts from the Hilbert transform that is used to derive the analytic signal. The current work investigates the relationship between the modes of the POD and those of the HPOD on the velocity fluctuations in the wake of a sphere. By comparing their outputs, POD mode pairs that correspond to the same propagating structures revealed by HPOD are identified. Furthermore, this study explored whether computing the analytic signal of the POD modes can replicate the HPOD modes, offering a more computationally efficient method for determining the pairs of POD modes that represent propagating structures. The results show that the pairs of POD modes identified by the HPOD can be more efficiently determined using the Hilbert transform directly on the POD modes. This method enhances the interpretive power of POD, enabling more detailed analysis of turbulent dynamics without introducing the filtering from the Hilbert transform.

Figures (22)

• Figure 1: (a) Vertical water tunnel facility, (b) close-up of the mounting structure and sphere, and (c) 3-D printed sphere design davey2025experimental.
• Figure 2: (a) Individual and (b) cumulative TKE contribution of the POD and HPOD modes.
• Figure 3: (i) Streamwise and (ii) transverse components of the first ten POD modes.
• Figure 4: (i) Streamwise and (ii) transverse components of the first four HPOD modes. Real parts are shown in (a), (c), (e), and (g) and corresponding imaginary parts are shown in (b), (d), (f), and (h)
• Figure 5: Correlation coefficient between each paired POD mode and the corresponding HPOD mode at the best match phase angle as a function of streamwise position. Superscripts $^u$ and $^v$ denote the streamwise and transverse components, respectively.