{
  "annotation_id": "82e384dd-1374-4d39-a1cd-e44f563d9cf6",
  "date_created": "2026-03-02T18:00:35.184000Z",
  "date_modified": "2026-03-02T18:00:35.184000Z",
  "file_hash": "9cca0d77e3d066ff0f4f3a52644b2f7ee9f6c681bc6b946e7cafa800a6a046cf",
  "private": false,
  "record": {
    "abstract": "The hierarchy of exact equations is given that relates two-spatial-point\nvelocity structure functions of arbitrary order with other statistics. Because\nno assumption is used, the exact statistical equations can apply to any flow\nfor which the Navier-Stokes equations are accurate, and they apply no matter\nhow small the number of samples in the ensemble. The exact statistical\nequations can be used to verify DNS computations and to detect their\nlimitations. For example,if DNS data are used to evaluate the exact statistical\nequations, then the equations should balance to within numerical precision,\notherwise a computational problem is indicated. The equations allow\nquantification of the approach to local homogeneity and to local isotropy.\nTesting the balance of the equations allows detection of scaling ranges for\nquantification of scaling-range exponents. The second-order equations lead to\nKolmogorov\u0027s equation. All higher-order equations contain a statistic composed\nof one factor of the two-point difference of the pressure gradient multiplied\nby factors of velocity difference. Investigation of this\npressure-gradient-difference statistic can reveal much about two issues: 1)\nwhether or not different components of the velocity structure function of given\norder have differing exponents in the inertial range, and 2) the increasing\ndeviation of those exponents from Kolmogorov scaling as the order increases.\nFull disclosure of the mathematical methods is in\nxxx.lanl.gov/list/physics.flu-dyn/0102055.",
    "arxiv_id": "physics/0102063",
    "authors": [
      "Reginald J. Hill"
    ],
    "categories": [
      "physics.flu-dyn"
    ],
    "doi": "10.1017/S0022112001003949",
    "journal_ref": "J. Fluid Mech. v.430, pp.1-10, 2001",
    "license": "http://creativecommons.org/publicdomain/zero/1.0/",
    "title": "Equations relating structure functions of all orders",
    "url": "https://arxiv.org/abs/physics/0102063"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "367da2c2-835c-4e57-860d-9ca7b6be0867",
    "id": "arXiv Dataset IDs",
    "type": "Model",
    "variant": "snapshot-2026-03-01",
    "version": "0.1.0"
  },
  "user_id": 1000002
}