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A geometry of the correlation space and a nonlocal degenerate parabolic equation from isotropic turbulence
Author(s) -
Grebenev V.N.,
Oberlack M.
Publication year - 2012
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.201100021
Subject(s) - mathematics , tensor (intrinsic definition) , mathematical analysis , turbulence , metric (unit) , euclidean space , metric tensor , homogeneous isotropic turbulence , dimension (graph theory) , isotropy , physics , geometry , reynolds number , pure mathematics , direct numerical simulation , quantum mechanics , operations management , economics , thermodynamics , geodesic
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 3 of correlation vectors, at the first part of the paper we construct the Lagrangian system (M t ,ds 2 (t)) in the extended space 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 M t 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.