  Four different applications of spectral proper orthogonal decomposition
(SPOD): low-rank reconstruction, denoising, frequency-time analysis, and
prewhitening are demonstrated on large-eddy simulation data of a turbulent jet.
SPOD-based low-rank reconstruction can be performed by direct inversion of a
truncated SPOD. This spectral inversion problem, however, is ambiguous since
SPOD relies on spectral estimation. We demonstrate SPOD-based flow field
reconstruction using direct inversion of the SPOD algorithm (frequency-domain
approach) and propose an alternative approach based on projection of the time
series data onto the modes (time-domain approach). While the SPOD optimally
represents the flow in a statistical sense, the time-domain approach seeks an
optimal reconstruction of each instantaneous flow field. We further propose a
SPOD-based denoising strategy that is based on hard-thresholding of the SPOD
eigenvalues. The proposed strategy achieves significant noise reduction while
facilitating drastic data compression. In contrast to standard methods of
frequency-time analysis such as wavelet transform, a proposed SPOD-based
approach yields a spectrogram that characterizes the temporal evolution of
spatially coherent flow structures. In the frequency-domain, time-varying
expansion coefficients can be obtained by basing the SPOD on a sliding window.
This approach, however, is computationally intractable, and an alternative
strategy based on convolution in the time-domain is presented. When applied to
the turbulent jet data, SPOD-based frequency-time analysis reveals that the
intermittent occurrence of large-scale coherent structures is directly
associated with high-energy events. This work suggests that the time-domain
approach is preferable for low-rank reconstruction of individual snapshots, and
the frequency-domain approach for denoising and frequency-time analysis.
