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Nested polyhedra model of turbulence.

Ö D Gürcan1

  • 1CNRS, Laboratoire de Physique des Plasmas, Ecole Polytechnique, 91120 Palaiseau, and Sorbonne Universités, UPMC Université Paris 06, 75005 Paris, France.

Physical Review. E
|July 16, 2017
PubMed
Summary

A novel discretization of wave-number space using nested polyhedra simplifies fluid dynamics simulations. This method reduces computational complexity, enabling high Reynolds number calculations and yielding the Kolmogorov spectrum.

Area of Science:

  • Computational physics
  • Fluid dynamics
  • Applied mathematics

Background:

  • Simulating fluid dynamics, particularly at high Reynolds numbers, requires significant computational resources.
  • Traditional methods often struggle with the vast number of degrees of freedom involved in turbulent flows.

Purpose of the Study:

  • To propose a novel discretization of wave-number space for efficient fluid dynamics simulations.
  • To reduce the computational complexity of solving the Navier-Stokes equation.

Main Methods:

  • Utilizing nested polyhedra (dodecahedra and icosahedra) with golden ratio scaling for wave-number space discretization.
  • Representing the discretization as logarithmically spaced, nested dodecahedron-icosahedron compounds.
  • Transforming the Navier-Stokes convolution integral into a sum over interacting wave vector triads.

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Main Results:

  • The proposed grid allows wave vectors to form triangles with other discretized wave vectors.
  • The convolution integral is reduced to a sum over 9 or 15 interaction pairs.
  • The numerical model reproduces the Kolmogorov spectrum (k^{-5/3}).

Conclusions:

  • The nested polyhedra grid offers an efficient reduction of Fourier space for fluid dynamics.
  • This method enables simulations at very high Reynolds numbers with fewer degrees of freedom.
  • The model can be used to derive shell models under specific assumptions.