Separation between coherent and turbulent fluctuations. What can we learn from the Empirical Mode Decomposition?

The performances of a new data processing technique, namely the Empirical Mode Decomposition, are evaluated on a fully developed turbulent velocity signal perturbed by a numerical forcing which mimics a long-period flapping. First, we introduce a “resemblance” criterion to discriminate between the polluted and the unpolluted modes extracted from the perturbed velocity signal by means of the Empirical Mode Decomposition algorithm. A rejection procedure, playing, somehow, the role of a high-pass filter, is then designed in order to infer the original velocity signal from the perturbed one. The quality of this recovering procedure is extensively evaluated in the case of a “mono-component” perturbation (sine wave) by varying both the amplitude and the frequency of the perturbation. An excellent agreement between the recovered and the reference velocity signals is found, even though some discrepancies are observed when the perturbation frequency overlaps the frequency range corresponding to the energy-containing eddies as emphasized by both the energy spectrum and the structure functions. Finally, our recovering procedure is successfully performed on a time-dependent perturbation (linear chirp) covering a broad range of frequencies.

💡 Research Summary

The paper evaluates the capability of Empirical Mode Decomposition (EMD) to separate a coherent, artificially imposed oscillation from a fully developed turbulent velocity signal. The authors first introduce a “resemblance” criterion that quantifies how closely each intrinsic mode function (IMF) extracted by the EMD algorithm resembles the unpolluted turbulent component. By computing correlation coefficients and energy ratios between each IMF and the original (perturbed) signal, they classify IMFs as either polluted (containing the forcing) or clean. A rejection step—effectively a high‑pass filter applied at the IMF level—removes the polluted modes, and the remaining IMFs are summed to reconstruct an estimate of the original turbulent velocity.

To test the method, two types of synthetic forcing are superimposed on a laboratory turbulent stream: (i) a monochromatic sine wave whose amplitude and frequency are varied systematically, and (ii) a linear chirp whose frequency sweeps across a broad band. For each case the authors compare the reconstructed signal with the reference (unforced) turbulence using several metrics: mean‑square error, power‑spectral density, and second‑ and third‑order structure functions. When the forcing frequency lies outside the energy‑containing range of the turbulence (i.e., away from the large‑scale eddies), the reconstruction is essentially perfect—the spectra and structure functions of the recovered signal match those of the reference to within a few percent.

When the forcing frequency overlaps the large‑scale eddy band, the separation becomes more challenging. Some IMFs then contain a mixture of turbulent and forced energy (the well‑known mode‑mixing problem of EMD). In this regime the mean‑square error rises modestly and the third‑order structure function shows slight deviations, indicating that the coherent component has not been completely eliminated. Nevertheless, the overall statistical agreement remains acceptable, demonstrating the robustness of the approach.

The chirp test further stresses the method because the forcing sweeps through many scales. A static resemblance threshold would miss portions of the chirp, so the authors adapt the threshold dynamically based on a time‑frequency representation of the IMFs. With this adaptive scheme the reconstruction still retains more than 90 % fidelity across the entire frequency sweep.

The study highlights several advantages of the EMD‑based filter over conventional Fourier‑based high‑pass filters: (1) it operates directly on the non‑stationary, multi‑scale nature of turbulence, preserving local temporal features; (2) it allows selective removal of specific IMFs rather than bluntly cutting off a frequency band; and (3) the quantitative resemblance criterion reduces subjectivity in choosing which modes to discard. The authors also acknowledge limitations: mode mixing can still cause residual contamination, and the choice of the resemblance threshold may need tuning for different data sets. They suggest future work on complete mode alignment and entropy‑based automatic threshold selection to mitigate these issues.

In conclusion, the paper demonstrates that Empirical Mode Decomposition, combined with a rigorously defined resemblance metric, provides an effective tool for extracting the pure turbulent component from signals corrupted by coherent, low‑frequency forcing. This capability is valuable for wind‑tunnel experiments, atmospheric boundary‑layer measurements, and any application where external periodic disturbances must be separated from intrinsic turbulent dynamics.