Presentation of anisotropy properties of turbulence, invariants versus eigenvalue approaches

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 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 normalized 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 map in terms of (II, III) are one-to-one uniquely interdependent, and as a result satisfies the requirement of realizability.