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

Min, I. A. and Mezić, I. and Leonard, A. (1996) Lévy stable distributions for velocity and velocity difference in systems of vortex elements. Physics of Fluids, 8 (5). pp. 1169-1180. ISSN 1070-6631. http://resolver.caltech.edu/CaltechAUTHORS:MINpof96

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Abstract

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.


Item Type:Article
Additional Information:©1996 American Institute of Physics. Received 11 October 1995; accepted 11 January 1996.
Subject Keywords:VORTEX FLOW; DISTRIBUTION FUNCTIONS; COMPUTERIZED SIMULATION; STOCHASTIC PROCESSES; STATISTICAL MECHANICS; TURBULENT FLOW; VELOCITY DISTRIBUTION; VORTICITY
Record Number:CaltechAUTHORS:MINpof96
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:MINpof96
Alternative URL:http://dx.doi.org/10.1063/1.868908
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:6499
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:11 Dec 2006
Last Modified:26 Dec 2012 09:21

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