{"paper":{"title":"Algebraic proof and application of Lumley's realizability triangle","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"physics.flu-dyn","authors_text":"G.A. Gerolymos, I. Vallet","submitted_at":"2016-10-11T19:03:43Z","abstract_excerpt":"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 ma"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.03468","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}