Nested polyhedra model of turbulence

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.

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.