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

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.