A geometry of the correlation space and a nonlocal degenerate parabolic equation from isotropic turbulence

Abstract

Key words Two-point correlation tensor, Lagrangian, von Kármán-Howarth equation, initial-boundary value problem, solvability, asymptotic behavior.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 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 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 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.

The Extended Symmetry Lie Algebra and the Asymptotic Expansion of the Transversal Correlation Function for the Isotropic Turbulence

“…This paper is a continuation of [1][2][3] wherein we investigated both the geometry and the group of transformations of an affine space 3 of the correlation vectors. In [1], we used the two-point velocity-correlation tensor to equip the correlation space 3 by the structure of a pseudo-Riemannian manifold of a variable signature and gave the geometric realization of the two-point velocity-correlation tensor which presents a metric tensor in the case of homogeneous isotropic turbulence. This construction presents the template for embedding the couple ( 3 , 2 ( )) into the Euclidean space R 3 with the standard metric.…”

“…This construction presents the template for embedding the couple ( 3 , 2 ( )) into the Euclidean space R 3 with the standard metric. The Lagrangian system in the extended space 3 × + was introduced in [3] that allowed us to attract common concept and technics of the Lagrangian mechanics for the application in turbulence. Dynamics in time of a singled out fluid volume equipped with a family of pseudo-Riemannian metrics was described in the frame of the geometry generated by 2 ( ) whose components are the correlation functions that evolve according to the von Kármán-Howarth equation [4].…”

“…Dynamics in time of a singled out fluid volume equipped with a family of pseudo-Riemannian metrics was described in the frame of the geometry generated by 2 ( ) whose components are the correlation functions that evolve according to the von Kármán-Howarth equation [4]. Notice here that the first integrals of the equations of geodesic curves form the "kinematic" conservation laws; see, for more details, [3]. In [2], we considered the functional of length and studied the infinitesimal transformations admitted by this functional.…”

“…The main result of the paper is the derivation of the asymptotic expansion of the transversal correlation function for large correlation distances. Notice that this expansion is obtained in the case when the correlation space 3 is equipped with the metric generated by the twopoint velocity-correlation tensor. This derivation is based on the algebraic constructions as in Conformal Field Theory (CFT).…”

“…The paper is organized as follows. Section 1 contains the results obtained (see [1][2][3] for more details) in a compressed form about the geometry of the correlation space 3 and the extended group of the transformations admitted by the Lagrangian system. In Section 2, we consider the energymomentum tensor associated with the metric 2 2 ( ) and show that the components ( ) and ( ) of this tensor are holomorphic functions.…”