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Updated: Jun 22, 2026

Simultaneous Measurement of Turbulence and Particle Kinematics Using Flow Imaging Techniques
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Published on: March 12, 2019

Time correlation functions in a similarity approximation for one-dimensional turbulence.

Makoto Okamura1, Hazime Mori

  • 1Research Institute for Applied Mechanics, Kyushu University, Kasuga 816-8580, Japan.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|June 13, 2009
PubMed
Summary

This study analyzes chaotic mode time correlation functions using projection operator formalism. It reveals three distinct decay forms for chaotic modes, offering analytical insights into their behavior.

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Area of Science:

  • Physics
  • Nonlinear Dynamics
  • Statistical Mechanics

Background:

  • Time correlation functions (Un(t)) describe the temporal evolution of chaotic systems.
  • Memory functions (Gamman(t)) are crucial for understanding the dynamics of these correlations.
  • Projection operator formalism provides a framework for deriving equations of motion for complex systems.

Purpose of the Study:

  • To derive a closed-form equation for the time evolution of chaotic mode time correlation functions (Un(t)).
  • To analytically determine the asymptotic behavior of Un(t) and its power spectrum (In(omega)).
  • To classify the decay forms of Un(t) based on wave number (kn) and compare with numerical results.

Main Methods:

  • Utilizing projection operator formalism to establish a time evolution equation for Un(t) involving Gamman(t).
  • Assuming similarity between Un(t) and Gamman(t) to derive a closed equation for Un(t).
  • Analytically solving the closed equation to obtain asymptotic behaviors and power spectra.

Main Results:

  • The time correlation function Un(t) exhibits an algebraic form 1/(1+t^2) for t→0.
  • Three distinct decay forms for Un(t) as t→∞ are identified: exponential, oscillatory exponential, and oscillatory power-law, dependent on wave number (kn).
  • Power spectra show a dual structure: Lorentzian at low frequencies (ω→0) and exponential decay at high frequencies (ω→∞).

Conclusions:

  • The derived closed equation accurately predicts the behavior of time correlation functions and power spectra for chaotic modes.
  • Solutions are consistent with numerical findings for both small and large wave numbers (kn).
  • Under specific conditions, the closed equation aligns with the direct interaction approximation in fluid turbulence.