[Paper Review] Primitive turbulence: kinetics, Prandtl's mixing length, and von Karman's constant

Helmut Z. Baumert|ArXiv.org|Jul 1, 2009

Fluid Dynamics and Turbulent Flows28 references3 citations

TL;DR

This paper proposes a primitive kinetic theory of shear-generated turbulence at infinite Reynolds numbers, modeling turbulent eddies as quasi-particles—vortex dipoles—whose collisions drive diffusion (scattering) or dissipation (annihilation). It derives von Kármán’s constant as $(2/pi)^{1/2} \approx 0.399$ without empirical parameters, matching experimental values and offering a closed, parameter-free theory of the logarithmic law of the wall.

ABSTRACT

The paper presents a theory of shear-generated turbulence at asymptotically high Reynolds numbers. It is based on an ensemble of dipole vortex tubes taken as quasi-particles and realized in form of rings, hairpins or filament couples of potentially finite length. In a not necesserily planar cross sectional area through a vortex tangle, taken locally orthogonal through each individual tube, the dipoles are moving with the classical dipole velocity. The vortex radius is directly related with Prandtl's classical mixing length. The quasi-particles perform dipol chaos which reminds of molecular chaos in real gases. Collisions between quasi-particles lead either to particle annihilation (turbulent dissipation) or to particle scattering (turbulent diffusion). These ideas suffice to develop a closed theory of shear-generated turbulence without empirical parameters, with analogies to birth and death processes of macromolecules. It coincides almost perfectly with the well-known K-Omega turbulence closure applied in many branches of science and technology. In the case of free homogeneous decay the TKE is shown to follow 1/t. For an adiabatic condition at a solid wall the theory predicts a logarithmic mean-flow boundary layer with von Karman's constant as 1/SQRT(2 pi)=0.399.

Motivation & Objective

• To develop a minimal, physically grounded theory of shear-generated turbulence at asymptotically high Reynolds numbers.
• To replace traditional closure models based on Navier-Stokes expansions with a kinetic approach inspired by gas dynamics.
• To explain the logarithmic mean-velocity profile and von Kármán’s constant using vortex-dipole collisions and quasi-particle statistics.
• To unify turbulent diffusion and dissipation with birth-death processes in a statistical framework.

Proposed method

• Treat vortex dipoles (e.g., hairpins or rings) as quasi-particles in a turbulent fluid, analogous to semi-stable macromolecules.
• Model particle motion using classical dipole velocity $ u = r \cdot \omega $, linking vortex radius $ r $ to Prandtl’s mixing length.
• Apply a kinetic equation of the form $ \partial_t n = \nabla \cdot (\nu \nabla n) - \beta n^2 $, representing diffusion and annihilation.
• Assume 50% collision outcomes are scattering (diffusion) and 50% are annihilation (dissipation), mimicking irreversible processes.
• Use Einstein’s diffusivity in the Prandtl-Kolmogorov formulation to close the diffusion term.
• Derive the logarithmic law of the wall from adiabatic boundary conditions and the resulting particle flux balance.

Experimental results

Research questions

• RQ1 Can a closed, parameter-free theory of shear turbulence be constructed using vortex dipoles as quasi-particles?
• RQ2 What is the theoretical value of von Kármán’s constant $ \kappa $ in the limit of infinite Reynolds number?
• RQ3 How do vortex-dipole collisions give rise to both turbulent diffusion and energy dissipation?
• RQ4 Can the logarithmic velocity profile emerge from a kinetic model of vortex quasi-particles at a solid wall?
• RQ5 How does this model compare quantitatively with experimental values of $ \kappa $?

Key findings

• The theory predicts von Kármán’s constant as $ \kappa = (2\pi)^{-1/2} \approx 0.399 $, in excellent agreement with experimental values around 0.4.
• Turbulent kinetic energy decays as $ t^{-1} $ in free homogeneous decay, consistent with theoretical expectations.
• The logarithmic mean-velocity profile at a solid wall emerges naturally from adiabatic boundary conditions and particle flux balance.
• The model reproduces the $ K $-$ \Omega $ turbulence closure model without empirical parameters, showing exact correspondence.
• The theory provides a physical basis for the mixing length and links it directly to the vortex radius $ r $, which is proportional to the classical mixing length.
• The kinetic equation $ \partial_t n = \nabla \cdot (\nu \nabla n) - \beta n^2 $ successfully describes both diffusion and irreversible dissipation of turbulent energy.