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Large eddy simulations (LES) reveal that finer grid spacing in turbulence models leads to faster error growth in small-scale structures. This limits the predictability horizon, regardless of Reynolds number, when normalized by Kolmogorov units.

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Area of Science:

  • Fluid dynamics
  • Computational physics
  • Turbulence modeling

Background:

  • The Navier-Stokes equations describe fluid motion, but solving them directly is computationally expensive for turbulent flows.
  • Large eddy simulations (LES) offer a compromise by modeling small-scale turbulence, but their accuracy depends on grid resolution.
  • Lyapunov exponents quantify the rate of separation of infinitesimally close trajectories in dynamical systems, indicating sensitivity to initial conditions.

Purpose of the Study:

  • To investigate the relationship between grid resolution, Reynolds number, and the maximal Lyapunov exponent in turbulent flows.
  • To assess the impact of fine-scale structures on error growth in large eddy simulations (LES) and direct numerical simulations (DNS).
  • To determine the theoretical limits on prediction accuracy imposed by numerical resolution in simulating turbulent systems.

Main Methods:

  • Performing large eddy simulations (LES) and direct numerical simulations (DNS) of sinusoidally driven three-dimensional Navier-Stokes equations.
  • Estimating the maximal Lyapunov exponent at various resolutions and Reynolds numbers.
  • Analyzing the scaling behavior of the Lyapunov exponent with respect to grid spacing and Kolmogorov units.

Main Results:

  • The LES Lyapunov exponent diverges as an inverse power of the effective grid spacing, independent of the Reynolds number when nondimensionalized by Kolmogorov units.
  • Fine-scale structures exhibit significantly faster error growth rates compared to larger scales.
  • This behavior implies an inherent upper bound on the prediction horizon achievable by refining initial conditions or measurement grids.

Conclusions:

  • The resolution of turbulence simulations, particularly in LES, fundamentally limits prediction accuracy due to amplified error growth in sub-grid scales.
  • Improving initial condition precision through finer grids has diminishing returns on predictability beyond a certain scale.
  • The findings highlight the importance of understanding and modeling small-scale turbulence accurately for reliable long-term predictions in fluid dynamics.