Fourier Modes in the SPOD Formalism on Aperiodic Domains

In the current work the effect of the use of Fourier analysis in combination with the Proper Orthogonal Decomposition (POD) is investigated. This approach to turbulence decomposition was introduced in [Lumley, 1967] and has more recently become known as Spectral POD (SPOD), [Towne et al., 2018]. The Fourier decomposition in the SPOD method is conventionally applied in the temporal coordinate direction for stationary flows as well as along any homogeneous directions. We reconsider the derivations underlying the conclusions that the Fourier basis is a solution to the POD integral eigenvalue problem along homogeneous (non-periodic)/stationary coordinate directions. We show theoretically that the POD modes do not reduce to Fourier modes along aperiodic coordinates, meaning that Fourier modes cannot formally be considered as solutions to the POD integral eigenvalue problem in this case. This fact is substantiated by numerical analyses of numerical POD modes related to various operators with displacement invariant kernels on aperiodic domains. Furthermore, we quantify the deviation between the numerical modes and Fourier modes for various kernels of the POD integral operator in order to evaluate the approximation accuracy of the Fourier bases in those cases.

[Lumley, 1967] Lumley, J. L. (1967). The structure of inhomogeneous turbulent flows. In Atmospheric turbulence and radio wave propagation, pages 166–178. Nauka, Moscow.

[Towne et al., 2018] Towne, A., Schmidt, O. T., and Colonius, T. (2018). Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis. Journal of Fluid Mechanics, 847:821–867.