Analytic and Numerical Study of Navier-Stokes Loop Equation in Turbulence

TL;DR

This paper develops an analytic and numerical framework for the Navier-Stokes loop equation on minimal surfaces, analyzing viscosity effects and predicting vorticity distributions for future numerical validation.

Contribution

It introduces a quadratic integral equation for vorticity on minimal surfaces and incorporates viscosity effects, providing a novel analytic and numerical approach to turbulence loop equations.

Findings

Derived a quadratic integral equation for vorticity distribution.

Analyzed viscosity effects, revealing bi-fractal behavior and symmetry breaking.

Developed a Mathematica-based numerical method for solving loop equations.

Abstract

We developed analytic approach to the non-planar loop equation, which we derived in previous papers \cite{M19a},\cite{M19b},\cite{M19c}. We found quadratic integral equation for the vorticity distribution $\Omega(r)$ we introduced on a minimal surface. There are no corrections to the minimal surface though: it is still defined by mean external curvature equal to zero, for arbitrary non-planar loop. We also analyzed the loop equations with viscosity term in Navier-Stokes equations. This term creates boundary condition for $\Omega(r\in C)$. The leading viscosity correction term mixes the moments $\left< \Gamma^p \right>$ with $\left< \Gamma^{p-1} \right>$ resembling the bi-fractal behavior observed in \cite{S19} and explicitly breaking the time reversal symmetry. We also develop numerical approach to the loop equation with arbitrary curved loop and present \Mathematica notebook building triangulated minimal surface and then numerically solving these equations. As a result we obtain predictions for future numerical experiments which will compute vorticity distribution along the loop.