1. Institute of Computational Technologies, Russian Academy of Science, Lavrentjev ave.?6, Novosibirsk, 630090, Russia 2. Fluid Dynamics, Technische Universit?t Darmstadt, Hochschulstrasse 1, Darmstadt, 64289, Germany
Abstract:
We primarily deal with homogeneous isotropic turbulence and use a closure model for the von Kármán-Howarth equation to study
several geometric properties of turbulent fluid dynamics. We focus our attention on the application of Riemannian geometry
methods in turbulence. Some advantage of this approach consists in exploring the specific form of a closure model for the
von Kármán-Howarth equation that enables to equip a model manifold (a cylindrical domain in the correlation space) by a family
of inner metrics (length scales of turbulent motion) which depends on time. We show that for large Reynolds numbers (in the
limit of large Reynolds numbers) the radius of this manifold can be evaluated in terms of the second-order structure function
and the correlation distance. This model manifold presents a shrinking cylindrical domain as time evolves. This result is
derived by using a selfsimilar solution of the closure model for the von Kármán-Howarth equation under consideration. We demonstrate
that in the new variables the selfsimilar solution obtained coincides with the element of Beltrami surface (or pseudo-sphere):
a canonical surface of the constant sectional curvature equals − 1.