Kolmogorov constants for the second-order structure function and the energy spectrum

Area of Science:

• Fluid dynamics
• Turbulence theory
• Statistical physics

Background:

• The Kolmogorov theory provides a framework for understanding turbulence.
• Kolmogorov constants are crucial for characterizing energy transfer in turbulent flows.
• Previous studies assumed these constants are independent of the Reynolds number.

Purpose of the Study:

• To investigate the behavior of Kolmogorov constants C(2), C(k), and C(k1).
• To determine the dependence of these constants and their ratios on the microscale Reynolds number R(λ).
• To re-evaluate the validity of commonly used relationships between these constants.

Main Methods:

• Analysis of the second-order longitudinal structure function.
• Examination of the three-dimensional and one-dimensional longitudinal energy spectra.
• Correlation of constant ratios with the microscale Reynolds number R(λ).

Main Results:

• The ratios C(2)/C(k1) and C(k)/C(k1) show a clear dependence on R(λ).
• A specific relationship was found: (C(k1)/C(2)-0.25)=1.95R(λ)(-0.68).
• The widely accepted relation C(2)=4.02C(k1) is only asymptotically valid for R(λ) ≥ 10^5.

Conclusions:

• Kolmogorov constants are not universally independent of the Reynolds number.
• The dependence varies among C(2), C(k), and C(k1), with C(2) showing stronger R(λ) dependence.
• Differences in real-space and wave-number space inertial ranges explain the varying R(λ) dependence.