Confined Vortex Surface and Irreversibility. 2. Hyperbolic Sheets and Turbulent statistics

TL;DR

This paper advances the understanding of vortex surfaces in turbulence by classifying solutions, deriving velocity fields, and exploring statistical properties like energy dissipation and velocity differences, linking vortex geometry to turbulence intermittency.

Contribution

It introduces a detailed classification of confined vortex surfaces, derives their analytical velocity fields, and connects vortex shape distributions to turbulence scaling laws.

Findings

Derived analytical velocity fields for vortex surfaces with arbitrary strain eigenvalues.

Found a universal asymmetric distribution for energy dissipation in turbulence.

Linked vortex surface shape distributions to multifractal scaling of velocity differences.

Abstract

We continue the study of Confined Vortex Surfaces (\CVS{}) that we introduced in the previous paper. We classify the solutions of the \CVS{} equation and find the analytical formula for the velocity field for arbitrary background strain eigenvalues in the stable region. The vortex surface cross-section has the form of four symmetric hyperbolic sheets with a simple equation $|y| |x|^\mu =1$ in each quadrant of the tube cross-section ($x y $ plane). We use the dilute gas approximation for the vorticity structures in a turbulent flow, assuming their size is much smaller than the mean distance between them. We introduce the Gaussian random background strain for each vortex surface as an accumulation of a large number of small random contributions coming from other surfaces far away. We compute this self-consistent background strain, relating the variance of the strain to the energy dissipation rate. We find a universal asymmetric distribution for energy dissipation. A new phenomenon is a probability distribution of the shape of the profile of the vortex tube in the $x y$ plane. This phenomenon naturally leads to imitation of the "multi-fractal" scaling of the moments of velocity difference $v(\vec r_1) - \vec v(\vec r_2)$. These moments have a nontrivial dependence of $n, \log |r_1 - r_2|$, approximating power laws with nonlinear index $\zeta(n)$. The rough estimate we provide here is not matching the observed DNS data, which may indicate necessity of the full 3D solution of the \CVS{} equations. We argue that the approximate relations for these moments suggested in a recent paper by Sreenivasan and Yakhot are consistent with the \CVS{} theory. We reinterpret their renormalization parameter $\alpha\approx 0.95$ in the Bernoulli law $ p = - \frac{1}{2}\alpha \vec v^2$ as a probability to find no vortex surface at a random point in space.