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 $|y||x{|}^{\hat{I}}¼=1$ in each quadrant of the tube cross-section ($xy$ 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 $xy$ 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 $\hat{I}\P (n)$. 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 $\hat{I}\pm \approx 0.95$ in the Bernoulli law $p=-\frac{1}{2}\hat{I}\pm {\vec{v}}^2$ as a probability to find no vortex surface at a random point in space.

54 pages, 17 figures. The new solution was rejected by further analysis, so the old hyperbolic solution was advanced and improved. The multifractal model is studied in some detail