dorsal/arxiv
View SchemaEstimates for the two-dimensional Navier-Stokes equations in terms of the Reynolds number
| Authors | J. D. Gibbon, G. A. Pavliotis |
|---|---|
| Categories | |
| ArXiv ID | physics/0605086 |
| URL | https://arxiv.org/abs/physics/0605086 |
| DOI | 10.1063/1.2356912 |
Abstract
The tradition in Navier-Stokes analysis of finding estimates in terms of the Grashof number $\bG$, whose character depends on the ratio of the forcing to the viscosity $\nu$, means that it is difficult to make comparisons with other results expressed in terms of Reynolds number $\Rey$, whose character depends on the fluid response to the forcing. The first task of this paper is to apply the approach of Doering and Foias \cite{DF} to the two-dimensional Navier-Stokes equations on a periodic domain $[0,L]^{2}$ by estimating quantities of physical relevance, particularly long-time averages $\left<\cdot\right>$, in terms of the Reynolds number $\Rey = U\ell/\nu$, where $U^{2}= L^{-2}\left<\|\bu\|_{2}^{2}\right>$ and $\ell$ is the forcing scale. In particular, the Constantin-Foias-Temam upper bound \cite{CFT} on the attractor dimension converts to $a_{\ell}^{2}\Rey(1 + \ln\Rey)^{1/3}$, while the estimate for the inverse Kraichnan length is $(a_{\ell}^{2}\Rey)^{1/2}$, where $a_{\ell}$ is the aspect ratio of the forcing. Other inverse length scales, based on time averages, and associated with higher derivatives, are estimated in a similar manner. The second task is to address the issue of intermittency : it is shown how the time axis is broken up into very short intervals on which various quantities have lower bounds, larger than long time-averages, which are themselves interspersed by longer, more quiescent, intervals of time.
{
"annotation_id": "823b6fde-57ca-404c-9fa7-f5b9a2fc7ebc",
"date_created": "2026-03-02T18:01:07.201000Z",
"date_modified": "2026-03-02T18:01:07.201000Z",
"file_hash": "a3b66ee793c7e23a0d3fd2abaf1b363188e15ec1fcb481991401d5b0334155ff",
"private": false,
"record": {
"abstract": "The tradition in Navier-Stokes analysis of finding estimates in terms of the\nGrashof number $\\bG$, whose character depends on the ratio of the forcing to\nthe viscosity $\\nu$, means that it is difficult to make comparisons with other\nresults expressed in terms of Reynolds number $\\Rey$, whose character depends\non the fluid response to the forcing. The first task of this paper is to apply\nthe approach of Doering and Foias \\cite{DF} to the two-dimensional\nNavier-Stokes equations on a periodic domain $[0,L]^{2}$ by estimating\nquantities of physical relevance, particularly long-time averages\n$\\left\u003c\\cdot\\right\u003e$, in terms of the Reynolds number $\\Rey = U\\ell/\\nu$, where\n$U^{2}= L^{-2}\\left\u003c\\|\\bu\\|_{2}^{2}\\right\u003e$ and $\\ell$ is the forcing scale. In\nparticular, the Constantin-Foias-Temam upper bound \\cite{CFT} on the attractor\ndimension converts to $a_{\\ell}^{2}\\Rey(1 + \\ln\\Rey)^{1/3}$, while the estimate\nfor the inverse Kraichnan length is $(a_{\\ell}^{2}\\Rey)^{1/2}$, where\n$a_{\\ell}$ is the aspect ratio of the forcing. Other inverse length scales,\nbased on time averages, and associated with higher derivatives, are estimated\nin a similar manner. The second task is to address the issue of intermittency :\nit is shown how the time axis is broken up into very short intervals on which\nvarious quantities have lower bounds, larger than long time-averages, which are\nthemselves interspersed by longer, more quiescent, intervals of time.",
"arxiv_id": "physics/0605086",
"authors": [
"J. D. Gibbon",
"G. A. Pavliotis"
],
"categories": [
"physics.flu-dyn",
"nlin.CD"
],
"doi": "10.1063/1.2356912",
"title": "Estimates for the two-dimensional Navier-Stokes equations in terms of the Reynolds number",
"url": "https://arxiv.org/abs/physics/0605086"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "c104eb2a-9a55-471a-a623-ab00021eb3a4",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}