Universal equations and constants of turbulent motion

TL;DR

This paper develops a parameter-free, two-fluids theory of high-Reynolds-number turbulence, predicting key statistical laws and constants that align with experimental and simulation data.

Contribution

It introduces a novel two-fluids model for turbulence, deriving universal equations and constants without adjustable parameters, and explains turbulence spectra and laws.

Findings

Predicts TKE decay as 1/t in free decay.

Derives von Karman's constant as 0.399.

Predicts the 3D-wavenumber spectrum prefactor as 1.8.

Abstract

This paper presents a parameter-free theory of shear-generated turbulence at asymptotically high Reynolds numbers in incompressible fluids. It is based on a two-fluids concept. Both components are materially identical and inviscid. The first component is an ensemble of quasi-rigid dipole-vortex tubes as quasi-particles in chaotic motion. The second is a superfluid performing evasive motions between the tubes. The local dipole motions follow Helmholtz' law. The vortex radii scale with the energy-containing length scale. Collisions between quasi-particles lead either to annihilation (likewise rotation, turbulent dissipation) or to scattering (counterrotation, turbulent diffusion). There are analogies with birth and death processes of population dynamics and their master equations. For free homogeneous decay the theory predicts the TKE to follow 1/t. With an adiabatic condition at the wall it predicts the logarithmic law with von Karman's constant as 1/\sqrt{2 pi} = 0.399. Likewise rotating couples form dissipative patches almost at rest ($\rightarrow$ intermittency) wherein the spectrum evolves like an "Apollonian gear" as discussed first by Herrmann, 1990. On this basis the prefactor of the 3D-wavenumber spectrum is predicted as (1/3)(4 pi)^{2/3}=1.8; in the Lagrangian frequency spectrum it is simply 2. The results are situated well within the scatter range of observational, experimental and DNS results.