Algebraic proof and application of Lumley's realizability triangle

TL;DR

This paper provides an algebraic proof that the anisotropy invariants of positive-definite symmetric tensors in 3D always lie within Lumley's realizability triangle, extending its applicability to various tensors in turbulence analysis.

Contribution

It offers a new algebraic proof of Lumley's realizability triangle, broadening its relevance beyond Reynolds-stress tensors to any positive-definite symmetric tensor in 3D.

Findings

Anisotropy invariants of positive-definite tensors lie within the triangle.

The proof applies to various tensors in turbulent flow analysis.

DNS data confirms the theoretical results.

Abstract

Lumley [Lumley J.L.: Adv. Appl. Mech. 18 (1978) 123--176] provided a geometrical proof that any Reynolds-stress tensor $\overline{u_i'u_j'}$ (indeed any tensor whose eigenvalues are invariably nonnegative) should remain inside the so-called Lumley's realizability triangle. An alternative formal algebraic proof is given that the anisotropy invariants of any positive-definite symmetric Cartesian rank-2 tensor in the 3-D Euclidian space $\mathbb{E}^3$ define a point which lies within the realizability triangle. This general result applies therefore not only to $\overline{u_i'u_j'}$ but also to many other tensors that appear in the analysis and modeling of turbulent flows. Typical examples are presented based on DNS data for plane channel flow.