LEVY STABLE-DISTRIBUTIONS FOR VELOCITY AND VELOCITY DIFFERENCE IN SYSTEMS OF VORTEX ELEMENTS

The probability density functions (PDFs) of the velocity and the veloc ity difference field induced by a distribution of a large number of di screte vortex elements are investigated numerically and analytically. Tails of PDFs of the velocity and velocity difference induced by a sin gle 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 usin g limit distribution theorems for stable distributions. Our results ge neralize and extend the analysis by Takayasu [Frog. Theor. Phys. 72, 4 71 (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 a nalysis is also applied to non-singular vortex blobs. We perform numer ical simulations of the system of vortex elements in two dimensions, a nd find that the results compare favorably with the theory based on th e independence assumption. These results are related to the experiment al and numerical measurements of velocity and velocity difference stat istics in the literature. In particular, the appearance of the Cauchy distribution for the velocity difference can be used to explain the ex perimental observations of Tong and Goldburg [Phys. Lett. A 127, 147 ( 1988); Phys. Rev. A 37, 2125, (1988); Phys. Fluids 31, 2841 (1988)] fo r turbulent flows. In addition, for intermediate values of the separat ion distance, near exponential tails are found. (C) 1996 American Inst itute of Physics.