Probability Distribution of Velocity Circulation in Three Dimensional Turbulence

Abstract

We elaborate the statistical field theory of Turbulence suggested in the previous paper \cite{M20a}. We clarify and simplify the basic Energy pumping equation of that theory and study mathematical properties of singular field configuration (instanton) which determine the tails of PDF for the velocity circulation around large loop $C$ in isotropic turbulence at highest Reynolds numbers. Explicit analytic solution is found for the Clebsch instanton in an Euler equation for a planar loop circulation problem. This solution for vorticity is has a term proportional to a delta function in normal direction to the minimal surface bounded by the loop. The smoothing of $\delta$ functions in the vorticity in the full Navier-Stokes equations is investigated and exponential profile of smoothed singularity is found. The PDF for circulation is now an infinite sum of decreasing exponential terms $\EXP{- n |w|}\sqrt{\frac{n}{|w|}}$, with $ w = \frac{\Gamma}{\Gamma_0[C]}$, and $ \Gamma_0[C] \sim \sqrt{A_C} $ with minimal area $A_C$. The leading term fits with adjusted $R^2 = 0.9999$ the PDF tail found in DNS over more than six orders of magnitude. The area dependence of the ratio of the circulation moments $M_8/M_6$ fits with adjusted $R^2=0.9996$ the DNS in inertial range of square loop sizes from $100 $ to $500$ Kolmogorov scales. Thus, our theory explains DNS with high degree or confidence. For a flat loop we derive two-dimensional integral equation for the dependence of the scale $\Gamma_0[C] $ of circulation as a function of the shape of the loop (aspect ratio for rectangular loop