Lagrangian single-particle turbulent statistics through the Hilbert-Huang transform

Area of Science:

• Fluid Dynamics
• Turbulence Research
• Nonlinear Dynamics

Background:

• Analyzing single-particle Lagrangian velocity data is crucial for understanding hydrodynamic turbulence.
• Traditional methods like Fourier transforms and structure functions have limitations in capturing turbulent energy transfer dynamics.

Purpose of the Study:

• To apply the Hilbert-Huang transform (HHT) for analyzing Lagrangian velocity data.
• To introduce and evaluate Hilbert spectra (HS) as a novel statistical tool for turbulence analysis.
• To compare the scaling properties of HS with traditional methods.

Main Methods:

• Decomposition of velocity trajectories into intrinsic mode functions (IMFs) and instantaneous frequencies using HHT.
• Definition of q-order Hilbert spectra (HS) based on ω-conditioned statistical moments of IMFs.
• Empirical analysis of scaling properties of HS and comparison with structure functions.

Main Results:

• Hilbert spectra (HS) exhibit enhanced scaling properties compared to Fourier transform and structure functions.
• The second-order HS demonstrates linear scaling in time within the inertial range, aligning with dimensional analysis.
• Direct measurement of high-order moment scaling exponents provides estimates consistent with multifractal predictions.

Conclusions:

• The Hilbert-Huang transform and derived Hilbert spectra offer superior insights into turbulent energy transfer.
• HS provide a more effective tool for analyzing scaling exponents in Lagrangian turbulence.
• The findings support multifractal models of turbulence in the Lagrangian frame.