Homogeneous Isotropic Turbulence: Geometric and Isometry Properties of the 2-point Velocity Correlation Tensor
- 10.1080/14029251.2018.1503447How to use a DOI?
- homogeneous isotropic turbulence; two-point correlation tensor; infinite-dimensional Lie algebra; minimal set of differential invariants
The emphasis of this review is both the geometric realization of the 2-point velocity correlation tensor field Bij(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 (Mt, ds2(t)), Mt ⊂ 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 λg is a second-order differential invariant and show how λg 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.
- © 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/).
Cite this article
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 -