{
  "id": "1610.03468",
  "version": "v1",
  "published": "2016-10-11T19:03:43.000Z",
  "updated": "2016-10-11T19:03:43.000Z",
  "title": "Algebraic proof and application of Lumley's realizability triangle",
  "authors": [
    "G. A. Gerolymos",
    "I. Vallet"
  ],
  "categories": [
    "physics.flu-dyn"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-11T19:03:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "lumleys realizability triangle",
      "application",
      "general result applies",
      "positive-definite symmetric cartesian",
      "alternative formal algebraic proof"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}