[Paper Review] Longitudinal and transverse velocity scaling exponents from merging of the Vortex filament and Multifractal models

K. P. Zybin, V. A. Sirota|arXiv (Cornell University)|Apr 6, 2012

Fluid Dynamics and Turbulent Flows1 references3 citations

TL;DR

This paper proposes a unified explanation for the observed difference between longitudinal and transverse velocity scaling exponents in turbulence by merging the Vortex Filament model with the Multifractal (MF) conjecture. Using physical arguments and a parabolic form for the singularity spectrum D(h), it derives analytical expressions for ζₙ∥ and ζₙ⊥ without fitting parameters, achieving excellent agreement with DNS and experimental data, particularly for high-order moments and the distinct scaling behavior in longitudinal vs. transverse structure functions.

ABSTRACT

We suggest a simple explanation of the difference between transverse and longitudinal scaling exponents observed in experiments and simulations. Based on the Vortex filament model and Multifractal conjecture, we calculate both scaling exponents for any n without any fitting parameters and ESS anzatz. The results are in very good agreement with the data of simulations.

Motivation & Objective

• To resolve the long-standing discrepancy between longitudinal and transverse velocity structure function scaling exponents observed in simulations and experiments.
• To provide a physical explanation rooted in the dynamics of vortex filaments and their geometric properties.
• To unify the Vortex Filament model with the Multifractal conjecture for a consistent derivation of scaling exponents.
• To predict ζₙ∥ and ζₙ⊥ for arbitrary n without relying on the ESS ansatz or fitting parameters.
• To explain why ζₙ∥ and ζₙ⊥ differ, especially at high n, based on distinct filament morphologies (axially symmetric vs. strongly curved).

Proposed method

• Modeling turbulent velocity structure functions using the Vortex Filament model, where vorticity is concentrated in thin filaments.
• Applying the Multifractal conjecture to describe local scaling behavior via the singularity spectrum D(h), with P ∝ l^{3−D(h)}.
• Using the Large Fluctuations Theorem to express structure functions as integrals over h: ⟨Δvⁿ⟩ ∝ ∫ l^{nh} l^{3−D(h)} dμ(h).
• Deriving scaling exponents via ζₙ = min_h (nh + 3 − D(h)), with D(h) assumed as a concave parabola D(h) = 3 − b(h − h₀)².
• Distinguishing transverse and longitudinal contributions by assigning different D(h) functions: D⊥(h) for transverse, D∥(h) for longitudinal, based on filament geometry.
• Solving for D(h) parameters using boundary conditions: D⊥(0) = 3, D⊥(h₀) = 3, D∥(0) = 0, D∥(h₀) = 3, leading to distinct h₀ and b for each case.

Experimental results

Research questions

• RQ1 Why do longitudinal and transverse velocity structure function scaling exponents differ in isotropic and homogeneous turbulence?
• RQ2 What physical mechanism in vortex filaments leads to distinct scaling behavior in longitudinal vs. transverse velocity increments?
• RQ3 Can the difference in ζₙ∥ and ζₙ⊥ be derived analytically without fitting parameters or the ESS ansatz?
• RQ4 How do the geometric properties of vortex filaments—specifically axial symmetry versus strong curvature—affect the scaling exponents?
• RQ5 What is the functional form of the singularity spectrum D(h) that reproduces observed scaling exponents for both longitudinal and transverse structure functions?

Key findings

• The model predicts ζₙ∥ = 0.367n − 1.12×10⁻²n² for n ≤ 16.3 and ζₙ∥ = 3 for n > 16.3, matching experimental and DNS data.
• The model predicts ζₙ⊥ = 0.391n − 1.91×10⁻²n² for n ≤ 10.2 and ζₙ⊥ = 2 for n > 10.2, with excellent agreement to Benzi and Gotoh’s DNS results.
• The difference in scaling exponents arises from distinct filament morphologies: axially symmetric filaments contribute to transverse structure functions, while strongly curved filaments dominate longitudinal ones.
• The model satisfies the exact condition ζ₂ = ζ₂⊥ for n=2, with a small deviation (1.6×10⁻²) that can be corrected by higher-order D(h) terms without altering the visual fit.
• The derived D(h) functions for transverse and longitudinal cases are physically distinct: D⊥(0) = 3 and D∥(0) = 0, reflecting different scaling behaviors at h=0.
• The model provides a complete analytical framework for ζₙ∥ and ζₙ⊥ across all n, with no fitting parameters, based on physical principles of vortex filament dynamics and multifractal scaling.