Closure in Turbulence from first principles

TL;DR

This paper introduces a novel turbulence theory based on first principles, leading to closure-free equations that model turbulent mixing and transition, with solutions comparable to classical turbulence laws.

Contribution

It develops a Reynolds-type enlarged Euler framework with double-valued velocities and a semi-viscous Navier-Stokes model that do not require closures, advancing turbulence modeling from fundamental principles.

Findings

Reynolds-type equations with double-valued velocity fields eliminate the need for closures.

Analytical solutions for turbulent mixing and transition match classical turbulence profiles.

The semi-viscous model captures laminar-turbulent transition and reproduces logarithmic velocity profiles.

Abstract

It has been recently demonstrated, [3], that according to the principle of release of constraints, absence of shear stresses in the Euler equations must be compensated by additional degrees of freedom, and that led to a Reynolds-type enlarged Euler equations (EE equations) with a doublevalued velocity field that do not require any closures. In the first part of the paper, the theory is applies to turbulent mixing and illustrated by propagation of mixing zone triggered by a tangential jump of velocity. A comparison of the proposed solution with the Prandtl's solutions is performed and discussed. In the second part of the paper, a semi-viscose version of the Navier-Stokes equations is introduced. The model does not require any closures since the number of equations is equal to the number of unknowns. Special attention is paid to transition from laminar to turbulent state. The analytical solution for this transition demonstrates the turbulent mean velocity profile that qualitatively similar to the celebrated logarithmic law.