Algebraic proof and application of Lumley's realizability triangle

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.