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

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. Periodic features such as vortex shedding can appear as POD mode pairs in strongly periodic flows, but detecting propagating structures in more complex flows is challenging. Hilbert proper orthogonal decomposition (HPOD) addresses this by applying POD to the analytic signal of the turbulent fluctuations, which yields complex modes with a $\pi /2$ phase shift between the real and imaginary components. These modes capture propagating structures effectively but introduce spectral leakage from the Hilbert transform 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 explores 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 determined more efficiently using the Hilbert transform directly on the POD modes. This method enhances the interpretive power of POD, enabling more detailed analysis of the turbulent dynamics without the need to compute the analytic signal of the entire turbulent fluctuation data.