Homogeneous Isotropic Turbulence: Geometric and Isometry Properties of the 2-point Velocity Correlation Tensor

DOI

10.1080/14029251.2018.1503447How to use a DOI?

Keywords

homogeneous isotropic turbulence; two-point correlation tensor; infinite-dimensional Lie algebra; minimal set of differential invariants

Abstract

The emphasis of this review is both the geometric realization of the 2-point velocity correlation tensor field B_ij(x,x′,t) and isometries of the correlation space K³ equipped with a (pseudo-) Riemannian metrics ds²(t) generated by the tensor field. The special form of this tensor field for homogeneous isotropic turbulence specifies ds²(t) as the semi-reducible pseudo-Riemannian metric. This construction presents the template for the application of methods of Riemannian geometry in turbulence to observe, in particular, the deformation of length scales of turbulent motion localized within a singled out fluid volume of the flow in time. This also allows to use common concepts and technics of Lagrangian mechanics for a Lagrangian system (M^t, ds²(t)), M^t ⊂ K³. Here the metric ds²(t), whose components are the correlation functions, evolves due to the von Kármán-Howarth equation. We review the explicit geometric realization of ds²(t) in K³ and present symmetries (or isometric motions in K³) of the metric ds²(t) which coincide with the sliding deformation of a surface arising under the geometric realization of ds²(t). We expose the fine structure of a Lie algebra associated with this symmetry transformation and construct the basis of differential invariants. Minimal generating set of differential invariants is derived. We demonstrate that the well-known Taylor microscale λ_g is a second-order differential invariant and show how λ_g can be obtained by the minimal generating set of differential invariants and the operators of invariant differentiation. Finally, we establish that there exists a nontrivial central extension of the infinite-dimensional Lie algebra constructed wherein the central charge is defined by the same bilinear skew-symmetric form c as for the Witt algebra which measures the number of internal degrees of freedom of the system. For turbulence, we give the asymptotic expansion of the transversal correlation function for the geometry generated by a quadratic form.

Copyright

© 2018 The Authors. Published by Atlantis and Taylor & Francis

Open Access

This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - Vladimir N. Grebenev
AU  - Martin Oberlack
PY  - 2021
DA  - 2021/01/06
TI  - Homogeneous Isotropic Turbulence: Geometric and Isometry Properties of the 2-point Velocity Correlation Tensor
JO  - Journal of Nonlinear Mathematical Physics
SP  - 650
EP  - 672
VL  - 25
IS  - 4
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2018.1503447
DO  - 10.1080/14029251.2018.1503447
ID  - Grebenev2021
ER  -