dorsal/arxiv
View SchemaExact Second-Order Structure-Function Relationships
| Authors | Reginald J. Hill |
|---|---|
| Categories | |
| ArXiv ID | physics/0209027 |
| URL | https://arxiv.org/abs/physics/0209027 |
| DOI | 10.1017/S0022112002001696 |
| Journal | Journal of Fluid Mechanics 468, 317-326 (2002) |
Abstract
Equations that follow from the Navier-Stokes equation and incompressibility but with no other approximations are "exact.". Exact equations relating second- and third-order structure functions are studied, as is an exact incompressibility condition on the second-order velocity structure function. Opportunities for investigations using these equations are discussed. Precisely defined averaging operations are required to obtain exact averaged equations. Ensemble, temporal, and spatial averages are all considered because they produce different statistical equations and because they apply to theoretical purposes, experiment, and numerical simulation of turbulence. Particularly simple exact equations are obtained for the following cases: i) the trace of the structure functions, ii) DNS that has periodic boundary conditions, and iii) an average over a sphere in r-space. The last case (iii) introduces the average over orientations of r into the structure function equations. The energy dissipation rate appears in the exact trace equation without averaging, whereas in previous formulations energy dissipation rate appears after averaging and use of local isotropy. The trace mitigates the effect of anisotropy in the equations, thereby revealing that the trace of the third-order structure function is expected to be superior for quantifying asymptotic scaling laws. The orientation average has the same property.
{
"annotation_id": "4281716e-6b25-4d81-8b9b-e77dd92fc5ec",
"date_created": "2026-03-02T18:00:39.548000Z",
"date_modified": "2026-03-02T18:00:39.548000Z",
"file_hash": "a92072d18a9fcac603319e8a93cf8a4dafc58ad75c8b677d2434bedda915227a",
"private": false,
"record": {
"abstract": "Equations that follow from the Navier-Stokes equation and incompressibility\nbut with no other approximations are \"exact.\". Exact equations relating second-\nand third-order structure functions are studied, as is an exact\nincompressibility condition on the second-order velocity structure function.\nOpportunities for investigations using these equations are discussed. Precisely\ndefined averaging operations are required to obtain exact averaged equations.\nEnsemble, temporal, and spatial averages are all considered because they\nproduce different statistical equations and because they apply to theoretical\npurposes, experiment, and numerical simulation of turbulence. Particularly\nsimple exact equations are obtained for the following cases: i) the trace of\nthe structure functions, ii) DNS that has periodic boundary conditions, and\niii) an average over a sphere in r-space. The last case (iii) introduces the\naverage over orientations of r into the structure function equations. The\nenergy dissipation rate appears in the exact trace equation without averaging,\nwhereas in previous formulations energy dissipation rate appears after\naveraging and use of local isotropy. The trace mitigates the effect of\nanisotropy in the equations, thereby revealing that the trace of the\nthird-order structure function is expected to be superior for quantifying\nasymptotic scaling laws. The orientation average has the same property.",
"arxiv_id": "physics/0209027",
"authors": [
"Reginald J. Hill"
],
"categories": [
"physics.flu-dyn"
],
"doi": "10.1017/S0022112002001696",
"journal_ref": "Journal of Fluid Mechanics 468, 317-326 (2002)",
"title": "Exact Second-Order Structure-Function Relationships",
"url": "https://arxiv.org/abs/physics/0209027"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "6b5ff9f9-f7a9-4123-95db-95907a4fe1cd",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}