Fourier limit in fluid mechanics, and link to the turbulence

Abstract

Macroscopic fluids are considered continuous down to microscopic scales where they fail to be dense enough for separate particles to have sufficiently large number of collisions (interactions) over that scale. We argue that a similar limit applies to the characteristic fluid length scales in the Fourier space, defined by wavenumber k. At $\frac{\Delta k}{k}\ll 1$, the Fourier space cannot be regarded continuous, and at these scales the energy-density spectrum becomes discrete. We further note that in physical space, the particle centres cannot come closer to each other than their diameter, and within this volume the physical forces are of a completely different nature. Similarly, very different interactions can be expected if wave modes have wavenumbers closer than a threshold in the Fourier space. It is argued that the wavenumber space $\frac{\Delta k}{k}<1/400$ can be considered as a Fourier volume of wave mode k. It is the hypothesis of this paper that this volume of the Fourier modes is filled with background seed turbulence, - like the volume of solid particles in the physical space is filled with the matter. The wave energy between modes in the discrete wave spectrum is not missing, it is the energy of background turbulence, like internal energy of molecules of the moving solid particle. This turbulence seed energy is about 0.1 % of the energy of neighbouring wave modes as argued in this paper, and it can grow subject to instabilities. The unidirectional flow can be treated as a half-wave with wavelength twice the size of the phenomenon where the mean velocity defines kinetic energy, part of which goes to the seed turbulence. The depth (cross-length) of this flow is not an independent property and is proportional to ratio of the length and velocity scales all squared. In case of surface present, this depth is perpendicular to the surface and defines the length scale of the boundary layer.