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A Geometric Interpretation of the Second-Order Structure Function Arising in Turbulence

Grebenev, Vladimir N. and Oberlack, Martin (2009):
A Geometric Interpretation of the Second-Order Structure Function Arising in Turbulence.
In: Mathematical Physics, Analysis and Geometry, Springer, pp. 1-18, 12, (1), ISSN 1385-0172.,
[Online-Edition: http://www.springerlink.com/content/c1k006pxg045x575/],
[Article]

Abstract

We primarily deal with homogeneous isotropic turbulence and use a closure model for the von Kármán-Howarth equation to study several geometric properties of turbulent fluid dynamics. We focus our attention on the application of Riemannian geometry methods in turbulence. Some advantage of this approach consists in exploring the specific form of a closure model for the von Kármán-Howarth equation that enables to equip a model manifold (a cylindrical domain in the correlation space) by a family of inner metrics (length scales of turbulent motion) which depends on time. We show that for large Reynolds numbers (in the limit of large Reynolds numbers) the radius of this manifold can be evaluated in terms of the second-order structure function and the correlation distance. This model manifold presents a shrinking cylindrical domain as time evolves. This result is derived by using a selfsimilar solution of the closure model for the von Kármán-Howarth equation under consideration. We demonstrate that in the new variables the selfsimilar solution obtained coincides with the element of Beltrami surface (or pseudo-sphere): a canonical surface of the constant sectional curvature equals − 1.

Item Type: Article
Erschienen: 2009
Creators: Grebenev, Vladimir N. and Oberlack, Martin
Title: A Geometric Interpretation of the Second-Order Structure Function Arising in Turbulence
Language: English
Abstract:

We primarily deal with homogeneous isotropic turbulence and use a closure model for the von Kármán-Howarth equation to study several geometric properties of turbulent fluid dynamics. We focus our attention on the application of Riemannian geometry methods in turbulence. Some advantage of this approach consists in exploring the specific form of a closure model for the von Kármán-Howarth equation that enables to equip a model manifold (a cylindrical domain in the correlation space) by a family of inner metrics (length scales of turbulent motion) which depends on time. We show that for large Reynolds numbers (in the limit of large Reynolds numbers) the radius of this manifold can be evaluated in terms of the second-order structure function and the correlation distance. This model manifold presents a shrinking cylindrical domain as time evolves. This result is derived by using a selfsimilar solution of the closure model for the von Kármán-Howarth equation under consideration. We demonstrate that in the new variables the selfsimilar solution obtained coincides with the element of Beltrami surface (or pseudo-sphere): a canonical surface of the constant sectional curvature equals − 1.

Journal or Publication Title: Mathematical Physics, Analysis and Geometry
Volume: 12
Number: 1
Publisher: Springer
Uncontrolled Keywords: Beltrami surface; Closure model for the von Kármán-Howarth equation; Homogeneous isotropic turbulence; Riemannian metric; Two-point correlation tensor; Length scales of turbulent motion
Divisions: 16 Department of Mechanical Engineering > Fluid Dynamics (fdy)
16 Department of Mechanical Engineering
Date Deposited: 24 Aug 2011 18:11
Official URL: http://www.springerlink.com/content/c1k006pxg045x575/
Additional Information:

doi:10.1007/s11040-008-9049-4

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