A discretization of the wave-number space using a self-similar, alternating, dodecahedral/icosahedral basis for the Navier-Stokes equation

Ö. D. Gürcan

Published 2016-03-04Version 1

A discretization of the wave-number space of the Navier-Stokes equation, using a logarithmically spaced chain of alternating icosa-dodeca-hedral spheres is proposed. This strange choice allows the possibility of forming triangles using only discretized wave-vectors when the scaling between two consecutive dodecahedra is equal to the golden ratio, and the icosahedron between the two dodecahedra is the dual of the inner dodecahedron. Alternatively, the same discretization can be described as a logarithmically spaced (with a scaling equal to the golden ratio) dodecahedron-icosahedron compounds. A wave-vector which points from the origin to a vertex of such a mesh, can always find two other discretized wave-vectors that are also on the vertices of the mesh (which is not true for an arbitrary mesh). For each vertex (i.e. discretized wave-vector) in this space, there are either 9 or 15 pairs of vertices (i.e. wave-vectors) with which the initial vertex can interact to form a triangle. This allows the reduction of the convolution integral in the Navier-Stokes equation to a sum over 9 or 15 interaction pairs. Transforming the equation in Fourier space, to a network of interacting nodes, that can be constructed as a numerical model, which evolves each component of the velocity vector on each node of the network. Such a model gives the usual Kolmogorov spectrum of k^(-5/3). Since the scaling is logarithmic, and the number of nodes for each scale is constant, a very large inertial range (i.e. very high Reynolds numbers) can be considered with a much lower number of degrees of freedom in such a model.