Generalizing the wavelet-based multifractal formalism to random vector fields: application to three-dimensional...

Area of Science:

• * Applied Mathematics
• * Fluid Dynamics
• * Statistical Physics

Background:

• * Traditional multifractal analysis often focuses on scalar fields.
• * Analyzing vector-valued random fields, like those in turbulence, presents unique challenges.
• * Existing methods may not fully capture the complex intermittency of turbulent flows.

Purpose of the Study:

• * To generalize the wavelet transform modulus maxima method for multifractal analysis of vector-valued random fields.
• * To develop and calibrate a novel method using singular value decomposition (SVD).
• * To investigate the multifractal properties of velocity and vorticity fields in 3D isotropic turbulence.

Main Methods:

• * Singular Value Decomposition (SVD) techniques.
• * Generalization of the wavelet transform modulus maxima method.
• * Calibration using synthetic multifractal 2D vector measures and monofractal 3D fractional Brownian vector fields.
• * Application to velocity and vorticity fields from 3D isotropic turbulence simulations.

Main Results:

• * Successful calibration of the generalized method on synthetic data.
• * Application revealed a significant relationship between singularity spectra of velocity and vorticity fields.
• * Identified higher intermittency in these vector fields than previously estimated from velocity increment statistics.

Conclusions:

• * The SVD-generalized wavelet method provides a robust tool for multifractal analysis of vector fields.
• * Turbulence exhibits a more intricate intermittent structure in its velocity and vorticity components than previously understood.
• * This finding has implications for turbulence modeling and understanding energy dissipation.