2020/06/22 by Alexander Migdal, Migdal, Alexander
Engineering · Physics and Astronomy · #Chaotic Dynamics (nlin.CD) #Cosmology and Gravitation Theories #FOS: Physical sciences #Fluid Dynamics (physics.flu-dyn) #Fluid Dynamics and Turbulent Flows #High Energy Physics - Theory (hep-th) #Pattern Formation and Solitons (nlin.PS) #Solar and Space Plasma Dynamics #Statistical Mechanics (cond-mat.stat-mech)
paper · pdf · doi:10.48550/arxiv.2006.12008
openalex publication_date 2020/06/22 · openalex created_date 2020/06/25 · openalex updated_date 2026/07/28
We elaborate the statistical field theory of Turbulence suggested in the previous paper \citeM20a. 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 δ 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|√((n)/(|w|)), with w = \fracΓΓ0[C], and Γ0[C] ∼ √(AC) with minimal area AC. The leading term fits with adjusted R2 = 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 M8/M6 fits with adjusted R2=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 Γ0[C] of circulation as a function of the shape of the loop (aspect ratio for rectangular loop