English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

The probability density functions (PDFs) of the velocity and the velocity difference field induced by a distribution of a large number of discrete vortex elements are investigated numerically and analytically. Tails of PDFs of the velocity and velocity difference induced by a single vortex element are found. Treating velocities induced by different vortex elements as independent random variables, PDFs of the velocity and velocity difference induced by all vortex elements are found using limit distribution theorems for stable distributions. Our results generalize and extend the analysis by Takayasu [Prog. Theor. Phys. 72, 471 (1984)]. In particular, we are able to treat general distributions of vorticity, and obtain results for velocity differences and velocity derivatives of arbitrary order. The PDF for velocity differences of a system of singular vortex elements is shown to be Cauchy in the case of small separation r, both in 2 and 3 dimensions. A similar type of analysis is also applied to non-singular vortex blobs. We perform numerical simulations of the system of vortex elements in two dimensions, and find that the results compare favorably with the theory based on the independence assumption. These results are related to the experimental and numerical measurements of velocity and velocity difference statistics in the literature. In particular, the appearance of the Cauchy distribution for the velocity difference can be used to explain the experimental observations of Tong and Goldburg [Phys. Lett. A 127, 147 (1988); Phys. Rev. A 37, 2125, (1988); Phys. Fluids 31, 2841 (1988)] for turbulent flows. In addition, for intermediate values of the separation distance, near exponential tails are found.

Lévy stable distributions for velocity and velocity difference in systems of vortex elements