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

  • Vladimir N. GrebenevEmail author
  • Martin Oberlack


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.


Beltrami surface Closure model for the von Kármán-Howarth equation Homogeneous isotropic turbulence Riemannian metric Two-point correlation tensor Length scales of turbulent motion 

Mathematics Subject Classifications (2000)

76F05 76F55 53B21 53B50 58J70 


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Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Institute of Computational TechnologiesRussian Academy of ScienceNovosibirskRussia
  2. 2.Fluid DynamicsTechnische Universität DarmstadtDarmstadtGermany

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