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The directivity of noise from three large-eddy simulations of turbulent jets at Mach 0.7, 0.9, and 1.5 is investigated using spectral proper orthogonal decomposition (SPOD). The most energetic patt...

Experimental spatio-temporal flow data often contain gaps or other types of undesired artifacts. To reconstruct flow data in the compromised or missing regions, a data completion method based on spectral proper orthogonal decomposition (SPOD) is developed. The algorithm leverages the temporal correlation of the SPOD modes with preceding and succeeding snapshots, and their spatial correlation with the surrounding data at the same time instant. For each gap, the algorithm first computes the SPOD of the remaining, unaffected data. In the next step, the compromised data are projected onto the basis of the SPOD modes. This corresponds to a local inversion of the SPOD problem and yields expansion coefficients that permit the reconstruction in the affected regions. This local reconstruction is successively applied to each gap. After all gaps are filled in, the procedure is repeated in an iterative manner until convergence. This method is demonstrated on two examples: direct numerical simulation of laminar flow around a cylinder, and time-resolved PIV data of turbulent cavity flow obtained by Zhang et al. (2019). Randomly added gaps correspond to 1%, 5%, and 20% of data loss. Even for 20% data corruption, and in the presence of measurement noise in the experimental data, the algorithm recovers 97% and 80% of the original data in the corrupted regions of the simulation and PIV data, respectively. These values are higher than those achieved by established methods like gappy POD and Kriging.

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.

The mechanisms by which near-nozzle forcing alters the turbulence structure and far-field sound of turbulent jets are not well understood. We perform large-eddy simulations of subsonic and supersonic axisymmetric isothermal turbulent jets subjected to an axisymmetric periodic forcing. The triple decomposition framework and spectral proper orthogonal decomposition are used to study the effect of the forcing on the underlying turbulence spectrum and determine how the periodic forcing alters coherent non-phase-locked structures. For subsonic jets, high-amplitude (1% of the jet velocity) low-frequency, Stf = 0.3, forcing is required to achieve a small change to the underlying turbulence spectrum and most energetic modes of the subsonic jet, despite producing a very energetic, phase-locked (tonal) response. The changes in the spectrum and the phase-locked structures are predicted well via resolvent analysis and linearized Navier-Stokes computations performed on the new turbulent mean, respectively. This suggests that there is little nonlinear interaction between the phase-locked structures and natural turbulence. High-frequency forcing, Stf = 1.5, shows similarly linear behavior except for around St≈0.75, which appears to be associated with vortex pairings triggered by the natural turbulence. Results of a similar analysis performed on a Mach 1.5 ideally expanded isothermal jet will also be presented.