{
  "id": "physics/0605086",
  "version": "v3",
  "published": "2006-05-10T14:20:51.000Z",
  "updated": "2006-09-13T15:37:19.000Z",
  "title": "Estimates for the two-dimensional Navier-Stokes equations in terms of the Reynolds number",
  "authors": [
    "J. D. Gibbon",
    "G. A. Pavliotis"
  ],
  "comment": "21 pages, 1 figure, accepted for publication from J. Math. Phys. for the special issue on mathematical fluid mechanics",
  "doi": "10.1063/1.2356912",
  "categories": [
    "physics.flu-dyn",
    "nlin.CD"
  ],
  "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.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2006-09-13T15:37:19.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "two-dimensional navier-stokes equations",
      "reynolds number",
      "character depends",
      "inverse length scales",
      "inverse kraichnan length"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}