Anisotropy Properties of Turbulence SANKHA BANERJEE Massachusetts Institute of Technology Department of Mechanical Engineering U.S.A OEZGUER ERTUNC & FRANZ DURST University of Erlangen-Nueremberg Institute of Chemical Engineering Germany Abstract: In the literature, anisotropy-invariant maps are being proposed to represent a domain within which all realizable Reynolds stress invariants must lie. It is shown that the representation proposed by Lumley and Newman has disadvantages owing to the fact that the anisotropy invariants (II,III ) are nonlinear functions of stresses. In the current work, it is proposed to use an equivalent linear representation of the anisotropy invariants in terms of eigenvalues. A novel barycentric map, based on the convex combination of scalar metrics dependent on eigenvalues, is proposed to provide a non-distorted visual representation of anisotropy in turbulent quantities. This barycentric map provides the possibility of viewing the Reynolds stress and any anisotropic stress tensor. Additionally the barycentric map provides the possibility of quantifying the weighting for any point inside it in terms of the limiting states (one component, two component, three component). The mathematical basis for the barycentric map is derived using the spectral decomposition theorem for second-order tensors. In this way, an analytical proof is provided that All turbulence lies along the boundaries or inside the barycentric map. It is proved that the barycentric map and the anisotropy-invariant maps are one to one uniquely interdependent and as a result satisfy the requirement of realizability. Key–Words: Nonlinearity, Barycentric map, Quantification of anisotropy. 1 Introduction The momentum transport at a point in a complex three-dimensional turbulent flows is not uniform in all directions, since, this process is characterized by a tensor, namely Reynolds stress tensor. This tensor u i u j can be identified as normal (when i = j ) and tangential (when i = j ) turbulent stresses. It is very difficult to gain complete information contained in the Reynolds stress tensor using only one scalar, so, it is necessary, to extract information about, the different aspects of the tensor using different scalars. The Reynolds stress tensor carry information about the componentality of the turbulence (i.e the rel- ative strengths of different velocity fluctuation compo- nents). From DNS studies of turbulent flows, it can be concluded that each large scale structure tends to or- ganize spatially the fluctuating motion in its vicinity, and in so doing, it tries to eliminate the gradients of the fluctuation fields in certain directions (the direc- tions in which the spatial structure is significant), and it enhances the gradients of fluctuations in other direc- tions. Thus associated with each eddy in a turbulent momentum transport are local axes of dependence and independence. In the undeformed state of isotropic turbulence the various axes of dependence and inde- pendence due to individual eddies are oriented ran- domly. The velocity fluctuation field which can be thought to be ensemble of all the eddies has gradients in all three directions. In turbulent flows, the mean flow gradient cre- ates structural anisotropy, since it stretches and aligns the energy-containing turbulent eddies. This in turn aligns the axes of dependence and indepen- dence of individual eddies and creates directions in which, in a statistical sense, the gradients of energy- containing fluctuations are weak. Thus by acting on the energy-containing structure mean flow gradient creates axes of independence, which are reflected in auto-correlations of gradients of the fluctuation fields. Thus there are regions in a turbulent flow, where there is one such direction of independence, and the turbu- lence becomes two-dimensional. However the direc- tion of independence, says nothing about the intensity of the fluctuations in that direction relative to the to- tal intensity. Thus the anisotropy in the dimension- ality of the turbulence is in general distinct from the anisotropy of its componentality. So characterization of the more general state of the turbulence requires information about the dimen- sionality of the turbulence (i.e the relative unifor- mity of structures in different directions) as well as the componentality of the turbulence (i.e the rela- tive strengths of different velocity fluctuation com- Proceedings of the 13th WSEAS International Conference on APPLIED MATHEMATICS (MATH'08) ISSN: 1790-2769 26 ISBN: 978-960-474-034-5