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Numerical study of chaos based on a shell model.

M. Yagi1, S.-I. Itoh, K. Itoh

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

Chaos (Woodbury, N.Y.)
|June 5, 2003
PubMed
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A new shell model simulates turbulence from thermal instability (Rayleigh-Benard convection). The study finds maximum Lyapunov exponent is insensitive to numerical errors in strong turbulence simulations.

Area of Science:

  • Fluid Dynamics
  • Nonlinear Dynamics
  • Computational Physics

Background:

  • Turbulence driven by thermal instability, specifically Rayleigh-Benard convection, is a complex phenomenon.
  • Understanding the cascade and chaotic behavior in strong turbulence requires robust computational models.

Purpose of the Study:

  • Introduce and analyze a novel shell model for simulating turbulence.
  • Investigate the characteristics of turbulence, including chaos, in high Rayleigh number regimes.
  • Compare the shell model's performance and results against the established Gledzer-Ohkitani-Yamada (GOY) model.

Main Methods:

  • Development of a shell model equation to describe turbulent cascade and chaos.
  • Numerical simulation of the shell model for strong turbulence.

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  • Analysis of turbulence characteristics: energy, spectra, probability distributions, and Lyapunov exponents.
  • Examination of the impact of numerical integration errors on simulation accuracy.

    • Main Results:

    • The shell model successfully describes cascade and chaos in strong turbulence.
    • Evaluated quantities include energy, standard deviation, wave and frequency spectra, and Lyapunov exponent distributions.
    • A key finding is the insensitivity of the maximum Lyapunov exponent to truncation errors.
    • No clear correlation was found between numerical accuracy and the accuracy of evaluated quantities due to error-nonlinear interactions.

    Conclusions:

    • The introduced shell model provides a viable framework for studying turbulent phenomena.
    • The insensitivity of the maximum Lyapunov exponent to numerical errors is a significant observation for computational fluid dynamics.
    • Further research can explore the model's applicability to other turbulent systems.