Infinite dimensional Lie algebra associated with conformal transformations of the two-point velocity correlation tensor from isotropic turbulence

  • V. N. GrebenevEmail author
  • M. Oberlack
  • A. N. Grishkov


We deal with homogeneous isotropic turbulence and use the two-point velocity correlation tensor field (parametrized by the time variable t) of the velocity fluctuations to equip an affine space K 3 of the correlation vectors by a family of metrics. It was shown in Grebenev and Oberlack (J Nonlinear Math Phys 18:109–120, 2011) that a special form of this tensor field generates the so-called semi-reducible pseudo-Riemannian metrics ds 2(t) in K 3. This construction presents the template for embedding the couple (K 3, ds 2(t)) into the Euclidean space \({\mathbb{R}^3}\) with the standard metric. This allows to introduce into the consideration the function of length between the fluid particles, and the accompanying important problem to address is to find out which transformations leave the statistic of length to be invariant that presents a basic interest of the paper. Also we classify the geometry of the particles configuration at least locally for a positive Gaussian curvature of this configuration and comment the case of a negative Gaussian curvature.

Mathematics Subject Classification (2010)

Primary 76F05 Secondary 76F55 53B21 53B50 


Isotropic turbulence Two-point correlation tensor Pseudo-Riemannian metric Functional of length Equivalencetransformation Witt algebra Differential invariants 


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Authors and Affiliations

  1. 1.Russian Academy of ScienceInstitute of Computational TechnologiesNovosibirskRussia
  2. 2.Department of Mechanical EngineeringTechnische Universität DarmstadtDarmstadtGermany
  3. 3.Institute of Mathematics and StatisticsUniversity of Sao PauloSao PauloBrazil

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