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**Grebenev, V. N. and 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), WILEY-VCH Verlag, pp. 179-195, ISSN 1521-4001,

[Online-Edition: http://onlinelibrary.wiley.com/doi/10.1002/zamm.201100021/pd...],

[Article]

## 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.

Item Type: | Article |
---|---|

Erschienen: | 2012 |

Creators: | Grebenev, V. N. and Oberlack, Martin |

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

Language: | English |

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. |

Journal or Publication Title: | Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM) |

Volume: | 92 |

Number: | 3 |

Publisher: | WILEY-VCH Verlag |

Uncontrolled Keywords: | Two-point correlation tensor; Lagrangian; von Kármán-Howarth equation; initial-boundary value problem; solvability; asymptotic behavior |

Divisions: | 16 Department of Mechanical Engineering 16 Department of Mechanical Engineering > Fluid Dynamics (fdy) Exzellenzinitiative Exzellenzinitiative > Clusters of Excellence Zentrale Einrichtungen Exzellenzinitiative > Clusters of Excellence > Center of Smart Interfaces (CSI) |

Date Deposited: | 06 Mar 2012 09:50 |

Official URL: | http://onlinelibrary.wiley.com/doi/10.1002/zamm.201100021/pd... |

Identification Number: | doi:10.1002/zamm.201100021 |

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