On the Discrepancies between POD and Fourier Modes on Aperiodic Domains

TL;DR

This paper investigates the differences between Proper Orthogonal Decomposition (POD) and Fourier modes on aperiodic domains, revealing that POD modes do not always converge to Fourier modes, especially at low Reynolds numbers and small domain sizes.

Contribution

The study challenges the assumption that POD modes converge to Fourier modes on increasing domain size, highlighting the influence of the Taylor macro/micro scale ratio on this discrepancy.

Findings

Discrepancies between POD and Fourier modes increase at smaller MMSRs.

Convergence rate of eigenspectrum matches the Fourier spectrum's rate.

Discrepancies influence spectral convergence on finite domains.

Abstract

The application of Fourier analysis in combination with the Proper Orthogonal Decomposition (POD) is investigated. In this approach to turbulence decomposition, which has recently been termed Spectral POD (SPOD), Fourier modes are considered as solutions to the corresponding Fredholm integral equation of the second kind along homogeneous-periodic or homogeneous coordinates. In the present work, the notion that the POD modes formally converge to Fourier modes for increasing domain length is challenged. Numerical results indicate that the discrepancy between POD and Fourier modes along \textit{locally} translationally invariant coordinates is coupled to the Taylor macro/micro scale ratio (MMSR) of the kernel in question. Increasing discrepancies are observed for smaller MMSRs, which are characteristic of low Reynolds number flows. It is observed that the asymptotic convergence rate of the eigenspectrum matches the corresponding convergence rate of the exact analytical Fourier spectrum of the kernel in question - even for extremely small domains and small MMSRs where the corresponding DFT spectra suffer heavily from windowing effects. These results indicate that the accumulated discrepancies between POD and Fourier modes play a role in producing the spectral convergence rates expected from Fourier transforms of translationally invariant kernels on infinite domains.