Loop Equation in Turbulence

TL;DR

This paper reformulates incompressible turbulence as a loop dynamics problem, deriving a functional equation for the generating functional, finding an exact steady solution, and connecting it to Kolmogorov scaling and circulation statistics.

Contribution

It introduces a novel loop equation approach to turbulence, providing explicit solutions and linking the functional form to classical turbulence scaling laws.

Findings

Derived a functional equation for turbulence in terms of loop variables.

Found an exact steady solution exhibiting Kolmogorov scaling.

Established the circulation probability distribution as Lorentzian.

Abstract

The incompressible fluid dynamics is reformulated as dynamics of closed loops $C$ in coordinate space. This formulation allows to derive explicit functional equation for the generating functional $\Psi[C]$ in inertial range of spatial scales, which allows the scaling solutions. The requirement of finite energy dissipation rate leads then to the Kolmogorov index. We find an exact steady solution of the loop equation in inertial range of the loop sizes. The generating functional decreases as $\EXP{-A^{\tt}}$ where $A=\oint_C r \wedge dr$ is the area inside the loop. The pdf for the velocity circulation $\Gamma$ is Lorentzian, with the width $\bar{\Gamma} \propto A^{\tt} $.