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. 
G.A. Gerolymos and I. Vallet
physics.flu-dyn