Presentation of anisotropy properties of turbulence, invariants versus eigenvalue approaches

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 barycentric map, based on the convex combination of scalar metrics dependent on eigenvalues, is proposed to provide a nondistorted 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.

Characterization of turbulence anisotropy, coherence, and intermittency at a prospective tidal energy site: Observational data analysis

“…Isotropic turbulence with uncorrelated orthogonal velocity fluctuations has the characteristic that I ¼ II ¼ III ¼ 0, and deviations away from this point describe different turbulent regimes, as illustrated by the classical Lumley Triangle [28]. The next section illustrates an updated version of the Lumley Triangle due to Banerjee et al [29]. The coordinate-system invariant scalar magnitude of the anisotropy similar to the CTKE, denoted A, is constructed from the scalars II and TKE as…”

“…Banerjee et al [29] introduce a visualization of anisotropy that contains additional information beyond A, based on the eigenvalues of the anisotropy tensor as opposed to the invariants. The "barycentric map" is a ternary diagram with vertices representing purely one-component (linear), two-component (planar), and three-component (isotropic) turbulence.…”

“…Higher-order statistics and a parameterization more grounded in turbulence theory are proposed to improve the classification of anisotropy at potential tidal energy sites. An easily understood visualization of anisotropy proposed by Banerjee et al [29] is presented as well.…”

Turbulence Anisotropy in the Near-Surface Atmosphere and the Evaluation of Multiple Outer Length Scales

“…By constructing Reynolds-stress tensors using the multiresolution spectra, the anisotropy of the motion at each spectral scale can be calculated as outlined in Sect. 3.2 by the method of Banerjee et al (2007). Using this method, Although the C 3 spectrum for each 1-h period shows the expected general behaviour of being relatively isotropic at small scales and anisotropic at large scales, there are no scales at which C 3 reaches the value of pure isotropy (Fig.…”

“…The analysis takes advantage of the property that all 3 × 3 tensors have only three independent invariants, noting that there are an infinite number of sets of three independent invariants. Choi and Lumley use a set of mathematically elegant invariants common to tensor analysis, whereas Banerjee et al (2007) use the three eigenvalues as the three invariants. Any set of invariants can be mapped to another set, therefore the two methods are transformable from one to the other.…”