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Infinite dimensional Lie algebra associated with conformal transformations of the two-point velocity correlation tensor from isotropic turbulence. (English) Zbl 1350.76024

Summary: 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 [V. N. Grebenev and M. Oberlack, J. Nonlinear Math. Phys. 18, No. 1, 109–120 (2011; Zbl 1213.53024)] 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.

MSC:

76F05 Isotropic turbulence; homogeneous turbulence
76F55 Statistical turbulence modeling
76M60 Symmetry analysis, Lie group and Lie algebra methods applied to problems in fluid mechanics
53B21 Methods of local Riemannian geometry
53B50 Applications of local differential geometry to the sciences

Citations:

Zbl 1213.53024
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References:

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