Homogeneous Isotropic Turbulence: Geometric and Isometry Properties of the 2-point Velocity Correlation Tensor. Issue 4 (2nd October 2018)

• Record Type: Journal Article
• Authors: Grebenev, Vladimir N.; Oberlack, Martin
• Is Part Of: Journal of nonlinear mathematical physics. Volume 25:Issue 4(2018)
• Page Start: 650
• Page End: 672
• Publication Date: 2018-10-02
• Subjects: homogeneous isotropic turbulence -- two-point correlation tensor -- infinite-dimensional Lie algebra -- minimal set of differential invariants; 53B21 -- 53B30 -- 76F05; Nonlinear theories -- Periodicals; Mathematical physics -- Periodicals; 515.252
• DOI: 10.1080/14029251.2018.1503447
• Languages: English
• ISSNs: 1402-9251

Abstract

Abstract : 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 K 3 equipped with a (pseudo-) Riemannian metrics ds 2 ( t ) generated by the tensor field. The special form of this tensor field for homogeneous isotropic turbulence specifies ds 2 ( 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, ds 2 ( t )), M t ⊂ K 3 . Here the metric ds 2 ( t ), whose components are the correlation functions, evolves due to the von Kármán-Howarth equation. We review the explicit geometric realization of ds 2 ( t ) in K 3 and present symmetries (or isometric motions in K 3 ) of the metric ds 2 ( t ) which coincide with the sliding deformation of a surface arising under the geometric realization of ds 2 ( 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.