DOI: 10.1098/rspa.1948.0127. On the theory of statistical and isotropic turbulence

On the theory of statistical and isotropic turbulence

10.1098/rspa.1948.0127

Crossref journal-article

The Royal Society

Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences (175)

Abstract

The statistical theory of turbulence, initiated by Taylor (1935) and v. Kármán & Howarth (1938), has recently been developed so far that a satisfactory explanation of the spectral distribution of energy among the turbulent eddies can be given. In fact Kolmogoroff (1941 a, b ) and independently Onsager (1945) and v. Weizsaecker (1948) have introduced a similarity hypothesis, which allows a determination of the spectrum for eddies with large Reynolds numbers, and the author (Heisenberg 1948) has extended these calculations to include those frequency components which have small Reynolds numbers. Since the distribution of energy among the largest eddies must be a geometrical and not a statistical problem, one may say that the statistical part of the spectrum is now well understood. Recently Batchelor & Townsend (1947, 1948 a, b ) have studied the decay of turbulent motion caused by a mesh grating in a wind tunnel, and the following discussions will apply the statistical theory to this problem. For the calculations the notation of Heisenberg (1948) will be used. If pF(k)dk denotes the energy contained between the wave numbers k and k + dk , the following equation for the dissipation of energy was given (Heisenberg 1948, equation (13)): S k = { μ + pk ∫( F ( k '') / k '' 3 ) dk " } ∫ k 0 2 F ( k ') k ' 2 dk '.

Authors 0

None

References 12 Referenced 165

Batchelor G. K. 1948 Proc. Roy. Soc. A 195 (in the Press).
Batchelor G. K. & TownsendA. A. 1947 Proc. Roy. A 190 534. (10.1098/rspa.1947.0095)
{'key': 'p_3', 'first-page': '539', 'article-title': 'Proc. Roy', 'volume': '193', 'author': 'Batchelor G. K.', 'year': '1948', 'journal-title': 'Soc. A'} / Soc. A / Proc. Roy by Batchelor G. K. (1948)
{'key': 'p_4', 'first-page': '527', 'article-title': '6 Proc. Roy', 'volume': '194', 'author': 'Batchelor G. K.', 'year': '1948', 'journal-title': 'Soc. A'} / Soc. A / 6 Proc. Roy by Batchelor G. K. (1948)
Heisenberg W. 1948 Z.Phys. 124 628. (10.1007/BF01668899)
{'key': 'p_6', 'first-page': '192', 'article-title': 'Proc. Roy', 'volume': '164', 'author': 'von Karman T.', 'year': '1938', 'journal-title': 'Soc. A'} / Soc. A / Proc. Roy by von Karman T. (1938)
{'key': 'p_7', 'first-page': '30', 'article-title': 'G.R', 'author': 'Kolmogoroff A. N.', 'year': '1941', 'journal-title': 'Acad. Sci. U. R. S.'} / Acad. Sci. U. R. S. / G.R by Kolmogoroff A. N. (1941)
{'key': 'p_8', 'first-page': '32', 'article-title': '6 C.R', 'author': 'Kolmogoroff A. N.', 'year': '1941', 'journal-title': 'Acad. Sci. U. R. S.'} / Acad. Sci. U. R. S. / 6 C.R by Kolmogoroff A. N. (1941)
{'key': 'p_9', 'first-page': '286', 'volume': '68', 'author': 'Onsager L.', 'year': '1945', 'journal-title': 'Phys. Rev.'} / Phys. Rev. by Onsager L. (1945)
Prandtl L. 1945 Nachr. Ges. Wiss. Gottingen p. 6.
{'key': 'p_11', 'first-page': '421', 'article-title': 'Proc. Roy', 'volume': '151', 'author': 'Taylor G. I.', 'year': '1935', 'journal-title': 'Soc. A'} / Soc. A / Proc. Roy by Taylor G. I. (1935)
{'key': 'p_12', 'first-page': '614', 'article-title': 'Z', 'volume': '124', 'author': 'von Weizsaecker C. F.', 'year': '1948', 'journal-title': 'Phys.'} / Phys. / Z by von Weizsaecker C. F. (1948)

Dates

Type	When
Created	18 years, 8 months ago (Dec. 15, 2006, 1:43 p.m.)
Deposited	4 years, 6 months ago (Feb. 18, 2021, 5:17 p.m.)
Indexed	2 months, 1 week ago (June 21, 2025, 2:02 a.m.)
Issued	76 years, 8 months ago (Dec. 22, 1948)
Published	76 years, 8 months ago (Dec. 22, 1948)
Published Online	28 years, 8 months ago (Jan. 1, 1997)
Published Print	76 years, 8 months ago (Dec. 22, 1948)

Funders 0

None

BibTeX

@article{1948, volume={195}, ISSN={2053-9169}, url={http://dx.doi.org/10.1098/rspa.1948.0127}, DOI={10.1098/rspa.1948.0127}, number={1042}, journal={Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences}, publisher={The Royal Society}, year={1948}, month=dec, pages={402–406} }

JSON

{
  "indexed": {
    "date-parts": [
      [
        2025,
        6,
        21
      ]
    ],
    "date-time": "2025-06-21T06:02:54Z",
    "timestamp": 1750485774020
  },
  "reference-count": 12,
  "publisher": "The Royal Society",
  "issue": "1042",
  "license": [
    {
      "start": {
        "date-parts": [
          [
            1948,
            12,
            22
          ]
        ],
        "date-time": "1948-12-22T00:00:00Z",
        "timestamp": -663552000000
      },
      "content-version": "tdm",
      "delay-in-days": 0,
      "URL": "https://royalsociety.org/journals/ethics-policies/data-sharing-mining/"
    }
  ],
  "content-domain": {
    "domain": [],
    "crossmark-restriction": false
  },
  "published-print": {
    "date-parts": [
      [
        1948,
        12,
        22
      ]
    ]
  },
  "abstract": "<jats:p>\n            The statistical theory of turbulence, initiated by Taylor (1935) and v. K\u00e1rm\u00e1n &amp; Howarth (1938), has recently been developed so far that a satisfactory explanation of the spectral distribution of energy among the turbulent eddies can be given. In fact Kolmogoroff (1941\n            <jats:italic>a, b</jats:italic>\n            ) and independently Onsager (1945) and v. Weizsaecker (1948) have introduced a similarity hypothesis, which allows a determination of the spectrum for eddies with large Reynolds numbers, and the author (Heisenberg 1948) has extended these calculations to include those frequency components which have small Reynolds numbers. Since the distribution of energy among the largest eddies must be a geometrical and not a statistical problem, one may say that the statistical part of the spectrum is now well understood. Recently Batchelor &amp; Townsend (1947, 1948\n            <jats:italic>a, b</jats:italic>\n            ) have studied the decay of turbulent motion caused by a mesh grating in a wind tunnel, and the following discussions will apply the statis\u00adtical theory to this problem. For the calculations the notation of Heisenberg (1948) will be used. If\n            <jats:italic>pF(k)dk</jats:italic>\n            denotes the energy contained between the wave numbers\n            <jats:italic>k</jats:italic>\n            and\n            <jats:italic>k</jats:italic>\n            +\n            <jats:italic>dk</jats:italic>\n            , the following equation for the dissipation of energy was given (Heisenberg 1948, equation (13)):\n            <jats:italic>\n              S\n              <jats:sub>k</jats:sub>\n            </jats:italic>\n            = {\n            <jats:italic>\u03bc + pk</jats:italic>\n            \u222b(\n            <jats:italic>F</jats:italic>\n            (\n            <jats:italic>k</jats:italic>\n            '') /\n            <jats:italic>k</jats:italic>\n            ''\n            <jats:sup>3</jats:sup>\n            )\n            <jats:italic>dk</jats:italic>\n            \" } \u222b\n            <jats:sup>k</jats:sup>\n            <jats:sub>0</jats:sub>\n            2\n            <jats:italic>F</jats:italic>\n            (\n            <jats:italic>k</jats:italic>\n            ')\n            <jats:italic>k</jats:italic>\n            '\n            <jats:sup>2</jats:sup>\n            <jats:italic>dk</jats:italic>\n            '.\n          </jats:p>",
  "DOI": "10.1098/rspa.1948.0127",
  "type": "journal-article",
  "created": {
    "date-parts": [
      [
        2006,
        12,
        15
      ]
    ],
    "date-time": "2006-12-15T18:43:34Z",
    "timestamp": 1166208214000
  },
  "page": "402-406",
  "source": "Crossref",
  "is-referenced-by-count": 165,
  "title": "On the theory of statistical and isotropic turbulence",
  "prefix": "10.1098",
  "volume": "195",
  "member": "175",
  "published-online": {
    "date-parts": [
      [
        1997,
        1
      ]
    ]
  },
  "reference": [
    {
      "key": "p_1",
      "unstructured": "Batchelor G. K. 1948 Proc. Roy. Soc. A 195 (in the Press)."
    },
    {
      "key": "p_2",
      "doi-asserted-by": "crossref",
      "unstructured": "Batchelor G. K. & TownsendA. A. 1947 Proc. Roy. A 190 534.",
      "DOI": "10.1098/rspa.1947.0095"
    },
    {
      "key": "p_3",
      "first-page": "539",
      "article-title": "Proc. Roy",
      "volume": "193",
      "author": "Batchelor G. K.",
      "year": "1948",
      "journal-title": "Soc. A"
    },
    {
      "key": "p_4",
      "first-page": "527",
      "article-title": "6 Proc. Roy",
      "volume": "194",
      "author": "Batchelor G. K.",
      "year": "1948",
      "journal-title": "Soc. A"
    },
    {
      "key": "p_5",
      "doi-asserted-by": "crossref",
      "unstructured": "Heisenberg W. 1948 Z.Phys. 124 628.",
      "DOI": "10.1007/BF01668899"
    },
    {
      "key": "p_6",
      "first-page": "192",
      "article-title": "Proc. Roy",
      "volume": "164",
      "author": "von Karman T.",
      "year": "1938",
      "journal-title": "Soc. A"
    },
    {
      "key": "p_7",
      "first-page": "30",
      "article-title": "G.R",
      "author": "Kolmogoroff A. N.",
      "year": "1941",
      "journal-title": "Acad. Sci. U. R. S."
    },
    {
      "key": "p_8",
      "first-page": "32",
      "article-title": "6 C.R",
      "author": "Kolmogoroff A. N.",
      "year": "1941",
      "journal-title": "Acad. Sci. U. R. S."
    },
    {
      "key": "p_9",
      "first-page": "286",
      "volume": "68",
      "author": "Onsager L.",
      "year": "1945",
      "journal-title": "Phys. Rev."
    },
    {
      "key": "p_10",
      "unstructured": "Prandtl L. 1945 Nachr. Ges. Wiss. Gottingen p. 6."
    },
    {
      "key": "p_11",
      "first-page": "421",
      "article-title": "Proc. Roy",
      "volume": "151",
      "author": "Taylor G. I.",
      "year": "1935",
      "journal-title": "Soc. A"
    },
    {
      "key": "p_12",
      "first-page": "614",
      "article-title": "Z",
      "volume": "124",
      "author": "von Weizsaecker C. F.",
      "year": "1948",
      "journal-title": "Phys."
    }
  ],
  "container-title": "Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences",
  "original-title": [],
  "language": "en",
  "link": [
    {
      "URL": "https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.1948.0127",
      "content-type": "application/pdf",
      "content-version": "vor",
      "intended-application": "text-mining"
    },
    {
      "URL": "https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.1948.0127",
      "content-type": "unspecified",
      "content-version": "vor",
      "intended-application": "similarity-checking"
    }
  ],
  "deposited": {
    "date-parts": [
      [
        2021,
        2,
        18
      ]
    ],
    "date-time": "2021-02-18T22:17:28Z",
    "timestamp": 1613686648000
  },
  "score": 1,
  "resource": {
    "primary": {
      "URL": "https://royalsocietypublishing.org/doi/10.1098/rspa.1948.0127"
    }
  },
  "subtitle": [],
  "short-title": [],
  "issued": {
    "date-parts": [
      [
        1948,
        12,
        22
      ]
    ]
  },
  "references-count": 12,
  "journal-issue": {
    "issue": "1042",
    "published-print": {
      "date-parts": [
        [
          1948,
          12,
          22
        ]
      ]
    }
  },
  "alternative-id": [
    "10.1098/rspa.1948.0127"
  ],
  "URL": "http://dx.doi.org/10.1098/rspa.1948.0127",
  "relation": {},
  "ISSN": [
    "0080-4630",
    "2053-9169"
  ],
  "subject": [],
  "container-title-short": "Proc. R. Soc. Lond. A",
  "published": {
    "date-parts": [
      [
        1948,
        12,
        22
      ]
    ]
  }
}