[Paper Review] Nonlocal interactions versus viscosity in turbulence

A. Bershadskii|ArXiv.org|Apr 15, 2007

Fluid Dynamics and Turbulent Flows32 references3 citations

TL;DR

This paper proposes that nonlocal interactions—where large and small scales couple dynamically—dominate energy spectrum shaping in isotropic turbulence at low and moderate Reynolds numbers, replacing the traditional local cascade model. It derives a perturbation framework linking nonlocal strain to viscosity, showing that the bottleneck effect and energy spectrum shape arise from this nonlocal mechanism, with quantitative agreement to DNS and experiments.

ABSTRACT

It is shown that nonlocal interactions determine energy spectrum in isotropic turbulence at small Reynolds numbers. It is also shown that for moderate Reynolds numbers the bottleneck effect is determined by the same nonlocal interactions. Role of the large and small scales covariance at the nonlocal interactions and in energy balance has been investigated. A possible hydrodynamic mechanism of the nonlocal solution instability at large scales has been briefly discussed. A quantitative relationship between effective strain of the nonlocal interactions and viscosity has been found. All results are supported by comparison with the data of experiments and numerical simulations.

Motivation & Objective

• To resolve the lack of scaling behavior in low-Reynolds-number isotropic turbulence, where traditional cascade models fail.
• To identify the hydrodynamic mechanism behind the bottleneck effect in moderate-Reynolds-number turbulence.
• To establish a quantitative relationship between nonlocal interaction strain and viscous effects.
• To demonstrate that large-scale and small-scale covariance at nonlocal interactions maintains energy flux balance.
• To validate the nonlocal model against high-resolution DNS and experimental data

Proposed method

• Derives a dimensionless function α(k) from the energy spectrum E(k) to detect scaling deviations.
• Uses a perturbation expansion in k/kd (normalized by dissipation wavenumber) to correct power-law scaling.
• Introduces a nonlocal interaction model where two short-wave modes and one long-wave mode form a triadic interaction.
• Establishes a dynamical relationship between effective strain s and viscosity ν via s ∝ ν⁻¹/², linking nonlocal strain to viscous scale η.
• Applies scale covariance (ηs/η ≈ const) to unify large-scale and small-scale dynamics in nonlocal interactions.
• Fits energy spectra from DNS and experiments to a nonlocal power-law model with exponential decay, validating α₁ ≈ 6.0

Experimental results

Research questions

• RQ1 What mechanism explains the absence of scaling in energy spectra at low Reynolds numbers, where viscous effects dominate?
• RQ2 How do nonlocal interactions contribute to the bottleneck effect in moderate-Reynolds-number turbulence?
• RQ3 What is the quantitative relationship between nonlocal interaction strain and viscosity in isotropic turbulence?
• RQ4 How does large-scale and small-scale covariance at nonlocal interactions maintain energy flux balance?
• RQ5 Why is the position of the bottleneck hump universal across Reynolds numbers?

Key findings

• Nonlocal interactions dominate energy spectrum shaping at low Reynolds numbers, even in isotropic turbulence, contradicting local cascade assumptions.
• The bottleneck effect in moderate-Reynolds-number turbulence is explained by nonlocal interactions, not by reduced cascade efficiency.
• A quantitative relationship s ∝ ν⁻¹/² is derived, linking effective strain of nonlocal interactions to viscosity.
• The ratio ηs/η ≈ const (≈ 18/c)3/2 ≈ 18.5 is universal across Reynolds numbers, explaining the universality of the bottleneck hump position.
• The model fits DNS and experimental data with α₁ ≈ 6.0 and c ≈ 2, yielding kₛ/kd ≈ 0.037, consistent with observations.
• Passive scalar spectra also follow the same nonlocal model, confirming universal scale covariance across scalar and velocity fields.