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Advances in Mathematical Physics

Volume 2013 (2013), Article ID 469654, 11 pages

http://dx.doi.org/10.1155/2013/469654

## The Extended Symmetry Lie Algebra and the Asymptotic Expansion of the Transversal Correlation Function for the Isotropic Turbulence

^{1}Institute of Computational Technologies, Russian Academy of Science, Lavrentjev Avenue 6, Novosibirsk 630090, Russia^{2}Institute of Mathematics and Statistics, University of Sao Paulo, Rua do Matao 1010, 66281 Sao Paulo, SP, Brazil^{3}Technical University Darmstadt, Department of Mechanical Engineering, Petersenstrasse 30, 64287 Darmstadt, Germany

Received 4 January 2013; Accepted 12 February 2013

Academic Editor: Rutwig Campoamor-Stursberg

Copyright © 2013 V. N. Grebenev et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The extended symmetry of the functional of length determined in an affine space of the correlation vectors for homogeneous isotropic turbulence is studied. The two-point velocity-correlation tensor field (parametrized by the time variable ) of the velocity fluctuations is used to equip this space by a family of the pseudo-Riemannian metrics (Grebenev and Oberlack (2011)). First, we observe the results obtained by Grebenev and Oberlack (2011) and Grebenev et al. (2012) about a geometry of the correlation space and expose the Lie algebra associated with the equivalence transformation of the above-mentioned functional for the quadratic form generated by which is similar to the Lie algebra constructed by Grebenev et al. (2012). Then, using the properties of this Lie algebra, we show that there exists a nontrivial central extension wherein the central charge is defined by the same bilinear skew-symmetric form as for the Witt algebra which measures the number of internal degrees of freedom of the system. For the applications in turbulence, as the main result, we establish the asymptotic expansion of the transversal correlation function for large correlation distances in the frame of .