Grebenev, V. N. ; Oberlack, Martin (2012)
A geometry of the correlation space and a nonlocal degenerate parabolic equation from isotropic turbulence.
In: Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM), 92 (3)
doi: 10.1002/zamm.201100021
Artikel, Bibliographie

Kurzbeschreibung (Abstract)

Considering the metric tensor ds^2(t), associated with the two-point velocity correlation tensor field (parametrized by the time variable t) in the space k^3of correlation vectors, at the first part of the paper we construct the Lagrangian system (M^t,ds^2(t)) in the extended space k^3 × R+ for homogeneous isotropic turbulence. This allows to introduce into the consideration common concept and technics of Lagrangian mechanics for the application in turbulence. Dynamics in time of (M^t,ds^2(t)) (a singled out fluid volume equipped with a family of pseudo-Riemannian metrics) is described in the frame of the geometry generated by the 1-parameter family of metrics ds^2(t) whose components are the correlation functions that evolve according to the von Kármán-Howarth equation. This is the first step one needs to get in a future analysis the physical realization of the evolution of this volume. It means that we have to construct isometric embedding of the manifold Mt equipped with metric ds^2(t) into R^3 with the Euclidean metric. In order to specify the correlation functions, at the second part of this paper we study in details an initial-boundary value problem to the closure model [19,26] for the von Kármán-Howarth equation in the case of large Reynolds numbers limit.

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