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

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.