DOI: 10.1017/s0022112075003047. On the geometry of homogeneous turbulence, with stress on the fractal dimension of the iso-surfaces of scalars

On the geometry of homogeneous turbulence, with stress on the fractal dimension of the iso-surfaces of scalars

10.1017/s0022112075003047

Crossref journal-article

Cambridge University Press (CUP)

Journal of Fluid Mechanics (56)

Abstract

This paper studies several geometric aspects of the Poisson and Gaussian random fields approximating Burgers k−2 and Kolmogorov $k^{-\frac{5}{3}}$ homogeneous turbulence. In particular, simulated sample scalar iso-surfaces (e.g. surfaces of constant temperature or concentration) are exhibited, and their relative degrees of wiggliness are shown to be best characterized by saying that the corresponding fractal dimensions are respectively equal to 3−½ and $3-\frac{1}{3}$.

Bibliography

Mandelbrot, B. B. (1975). On the geometry of homogeneous turbulence, with stress on the fractal dimension of the iso-surfaces of scalars. Journal of Fluid Mechanics, 72(3), 401â416.

Authors 1

Benoit B. Mandelbrot (first)

References 24 Referenced 234

Feller, W. 1968 An Introduction of Probability Theory and Its Application , vol. 1, 3rd edn.Wiley.
Mandelbrot, B. 1965 Self-similar error clusters in communications systems and the concept of conditional stationarity.I.E.E.E. Trans. Comm. Tech. COM13,71–90. (10.1109/TCOM.1965.1089090)
Mandelbrot, B. 1975a Stochastic models for the earth's relief, the shape and the fractal dimension of the coastlines, and the number-area rule for islands.Proc. Nat. Acad. Sci. U.S.A. 72 (in press). (10.1073/pnas.72.10.3825)
Mandelbrot, B. 1974 Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier.J. Fluid Mech. 62,331–358. (10.1017/S0022112074000711)
Gangolli, R. 1967 Lévy's Brownian motion of several parameters.Ann. Inst. H. Poincaré, B3,121–226.
Mandelbrot, B. 1976 Proc. June 1975 Meeting, Orsay. Journées Mathématiques sur la Turbulence. Springer.
Mandelbrot, B. & Van ness, J. W. 1968 Fractional Brownian motions, fractional noises and applications.SIAM Rev. 10,422–437. (10.1137/1010093)
Kuo, A. Y. S. & Corrsin, S. 1972 Experiments on the geometry of the fine-structure regions in fully turbulent fluid.J. Fluid Mech. 56,477–479. (10.1017/S0022112072002459)
Corrsin, S. 1951a On the spectrum of isotropic temperature fluctuations in isotropic turbulence.J. Appl. Phys. 22,469–473. (10.1063/1.1699986)
Rogers, C. A. 1970 Hausdorff Measures .Cambridge University Press.
Mckean, H. P. 1963 Brownian motion with a several dimensional time.Theory Prob. Appl. 8,357–378. (10.1137/1108042)
Neumann, J. Von 1963 Recent theories of turbulence (a report to ONR). In Collected Works , vol. 6,pp.437–472.Pergamon.
Lévy, P. 1948 Processus Stochastiques et Mouvement Brownien. Gauthier-Villars.
Mandelbrot, B. 1967 How long is the coast of Britain? Statistical self-similarity and fractional dimension.Science,155,636–638. (10.1126/science.156.3775.636)
Scheffer, V. 1975 Géométrie fractale de la turbulence. Equations de Navier-Stokes et dimension de Hausdorff.Comptes Rendus, A281 (in press).
Mandelbrot, B. 1975c Les Objets Fractals: Forme, Hasard et Dimension. Paris:Flammarion. (Revised and much augmented English trans. expected to appear in 1976, tentatively under the title Fractals: Shape, Chance and Dimension.)
Mandelbrot, B. 1972 Possible refinement of the lognormal hypothesis concerning the distribution of energy dissipation in intermittent turbulence. In Statistical Models and Turbulence (ed. M. Rosenblatt & C. Van Atta ),pp.333–351.Springer. (10.1007/3-540-05716-1_20)
Kármán, Th. Von & Howarth, L. 1938 On the statistical theory of isotropic turbulence.Proc. Roy. Soc. A151,411–478. (Reprinted in Turbulence (ed. S. K. Friedlander & L. Topper). Interscience.)
Mandelbrot, B. 1975d Géometrie fractale de la turbulence. Dimension de Hausdorff, dispersion et nature des singularités du mouvement des fluides.Comptes Rendus, A281 (in press).
Mandelbrot, B. 1975b Fonctions aléatoires pluri-temporelles: approximation poissonienne du cas brownien et généralisations.Comptes Rendus, A280,1075–1078.
Yoder, L. 1974 Variation of multiparameter Brownian motion.Proc. Am. Math. Soc. 46,302–309. (10.1090/S0002-9939-1974-0418260-4)
Yaglom, A. M. 1957 Some classes of random fields in.N-dimensional space, related to stationary random processes. (trans. R. A. Silverman). Theory Prob. Appl. 2,273–320. (10.1137/1102021)
Kolmogorov, A. N. 1940 Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum.C. R. (Doklady) Acad. Sci. URSS (N.S.),26,115–118.
Perrin, J. 1913 Les Atomes. Gallimard. (English trans. Atoms (ed. D. L. Hammick ). Constable and Norstrand.)

Dates

Type	When
Created	19 years, 4 months ago (March 29, 2006, 9:13 a.m.)
Deposited	2 months, 1 week ago (June 20, 2025, 7:57 p.m.)
Indexed	1 week, 1 day ago (Aug. 20, 2025, 8:40 a.m.)
Issued	49 years, 8 months ago (Dec. 9, 1975)
Published	49 years, 8 months ago (Dec. 9, 1975)
Published Online	19 years, 5 months ago (March 29, 2006)
Published Print	49 years, 8 months ago (Dec. 9, 1975)

Funders 0

None

BibTeX

@article{Mandelbrot_1975, title={On the geometry of homogeneous turbulence, with stress on the fractal dimension of the iso-surfaces of scalars}, volume={72}, ISSN={1469-7645}, url={http://dx.doi.org/10.1017/s0022112075003047}, DOI={10.1017/s0022112075003047}, number={3}, journal={Journal of Fluid Mechanics}, publisher={Cambridge University Press (CUP)}, author={Mandelbrot, Benoit B.}, year={1975}, month=dec, pages={401–416} }

JSON

{
  "indexed": {
    "date-parts": [
      [
        2025,
        8,
        20
      ]
    ],
    "date-time": "2025-08-20T12:40:28Z",
    "timestamp": 1755693628965,
    "version": "3.41.0"
  },
  "reference-count": 24,
  "publisher": "Cambridge University Press (CUP)",
  "issue": "3",
  "license": [
    {
      "start": {
        "date-parts": [
          [
            2006,
            3,
            29
          ]
        ],
        "date-time": "2006-03-29T00:00:00Z",
        "timestamp": 1143590400000
      },
      "content-version": "unspecified",
      "delay-in-days": 11068,
      "URL": "https://www.cambridge.org/core/terms"
    }
  ],
  "content-domain": {
    "domain": [],
    "crossmark-restriction": false
  },
  "published-print": {
    "date-parts": [
      [
        1975,
        12,
        9
      ]
    ]
  },
  "abstract": "<jats:p>This paper studies several geometric aspects of the Poisson and Gaussian random fields approximating Burgers <jats:italic>k</jats:italic><jats:sup>\u22122</jats:sup> and Kolmogorov <jats:inline-formula>$k^{-\\frac{5}{3}}$</jats:inline-formula> homogeneous turbulence. In particular, simulated sample scalar iso-surfaces (e.g. surfaces of constant temperature or concentration) are exhibited, and their relative degrees of wiggliness are shown to be best characterized by saying that the corresponding fractal dimensions are respectively equal to 3\u2212\u00bd and <jats:inline-formula>$3-\\frac{1}{3}$</jats:inline-formula>.</jats:p>",
  "DOI": "10.1017/s0022112075003047",
  "type": "journal-article",
  "created": {
    "date-parts": [
      [
        2006,
        3,
        29
      ]
    ],
    "date-time": "2006-03-29T14:13:45Z",
    "timestamp": 1143641625000
  },
  "page": "401-416",
  "source": "Crossref",
  "is-referenced-by-count": 234,
  "title": "On the geometry of homogeneous turbulence, with stress on the fractal dimension of the iso-surfaces of scalars",
  "prefix": "10.1017",
  "volume": "72",
  "author": [
    {
      "given": "Benoit B.",
      "family": "Mandelbrot",
      "sequence": "first",
      "affiliation": []
    }
  ],
  "member": "56",
  "published-online": {
    "date-parts": [
      [
        2006,
        3,
        29
      ]
    ]
  },
  "reference": [
    {
      "key": "S0022112075003047_ref002",
      "unstructured": "Feller, W. 1968 An Introduction of Probability Theory and Its Application , vol. 1, 3rd edn.Wiley."
    },
    {
      "key": "S0022112075003047_ref009",
      "doi-asserted-by": "crossref",
      "unstructured": "Mandelbrot, B. 1965 Self-similar error clusters in communications systems and the concept of conditional stationarity.I.E.E.E. Trans. Comm. Tech. COM13,71\u201390.",
      "DOI": "10.1109/TCOM.1965.1089090"
    },
    {
      "key": "S0022112075003047_ref013",
      "doi-asserted-by": "crossref",
      "unstructured": "Mandelbrot, B. 1975a Stochastic models for the earth's relief, the shape and the fractal dimension of the coastlines, and the number-area rule for islands.Proc. Nat. Acad. Sci. U.S.A. 72 (in press).",
      "DOI": "10.1073/pnas.72.10.3825"
    },
    {
      "key": "S0022112075003047_ref012",
      "doi-asserted-by": "crossref",
      "unstructured": "Mandelbrot, B. 1974 Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier.J. Fluid Mech. 62,331\u2013358.",
      "DOI": "10.1017/S0022112074000711"
    },
    {
      "key": "S0022112075003047_ref003",
      "unstructured": "Gangolli, R. 1967 L\u00e9vy's Brownian motion of several parameters.Ann. Inst. H. Poincar\u00e9, B3,121\u2013226."
    },
    {
      "key": "S0022112075003047_ref017",
      "unstructured": "Mandelbrot, B. 1976 Proc. June 1975 Meeting, Orsay. Journ\u00e9es Math\u00e9matiques sur la Turbulence. Springer."
    },
    {
      "key": "S0022112075003047_ref018",
      "doi-asserted-by": "crossref",
      "unstructured": "Mandelbrot, B. & Van ness, J. W. 1968 Fractional Brownian motions, fractional noises and applications.SIAM Rev. 10,422\u2013437.",
      "DOI": "10.1137/1010093"
    },
    {
      "key": "S0022112075003047_ref006",
      "doi-asserted-by": "crossref",
      "unstructured": "Kuo, A. Y. S. & Corrsin, S. 1972 Experiments on the geometry of the fine-structure regions in fully turbulent fluid.J. Fluid Mech. 56,477\u2013479.",
      "DOI": "10.1017/S0022112072002459"
    },
    {
      "key": "S0022112075003047_ref001",
      "doi-asserted-by": "crossref",
      "unstructured": "Corrsin, S. 1951a On the spectrum of isotropic temperature fluctuations in isotropic turbulence.J. Appl. Phys. 22,469\u2013473.",
      "DOI": "10.1063/1.1699986"
    },
    {
      "key": "S0022112075003047_ref021",
      "unstructured": "Rogers, C. A. 1970 Hausdorff Measures .Cambridge University Press."
    },
    {
      "key": "S0022112075003047_ref008",
      "doi-asserted-by": "crossref",
      "unstructured": "Mckean, H. P. 1963 Brownian motion with a several dimensional time.Theory Prob. Appl. 8,357\u2013378.",
      "DOI": "10.1137/1108042"
    },
    {
      "key": "S0022112075003047_ref019",
      "unstructured": "Neumann, J. Von 1963 Recent theories of turbulence (a report to ONR). In Collected Works , vol. 6,pp.437\u2013472.Pergamon."
    },
    {
      "key": "S0022112075003047_ref007",
      "unstructured": "L\u00e9vy, P. 1948 Processus Stochastiques et Mouvement Brownien. Gauthier-Villars."
    },
    {
      "key": "S0022112075003047_ref010",
      "doi-asserted-by": "crossref",
      "unstructured": "Mandelbrot, B. 1967 How long is the coast of Britain? Statistical self-similarity and fractional dimension.Science,155,636\u2013638.",
      "DOI": "10.1126/science.156.3775.636"
    },
    {
      "key": "S0022112075003047_ref022",
      "unstructured": "Scheffer, V. 1975 G\u00e9om\u00e9trie fractale de la turbulence. Equations de Navier-Stokes et dimension de Hausdorff.Comptes Rendus, A281 (in press)."
    },
    {
      "key": "S0022112075003047_ref015",
      "unstructured": "Mandelbrot, B. 1975c Les Objets Fractals: Forme, Hasard et Dimension. Paris:Flammarion. (Revised and much augmented English trans. expected to appear in 1976, tentatively under the title Fractals: Shape, Chance and Dimension.)"
    },
    {
      "key": "S0022112075003047_ref011",
      "doi-asserted-by": "crossref",
      "unstructured": "Mandelbrot, B. 1972 Possible refinement of the lognormal hypothesis concerning the distribution of energy dissipation in intermittent turbulence. In Statistical Models and Turbulence (ed. M. Rosenblatt & C. Van Atta ),pp.333\u2013351.Springer.",
      "DOI": "10.1007/3-540-05716-1_20"
    },
    {
      "key": "S0022112075003047_ref004",
      "unstructured": "K\u00e1rm\u00e1n, Th. Von & Howarth, L. 1938 On the statistical theory of isotropic turbulence.Proc. Roy. Soc. A151,411\u2013478. (Reprinted in Turbulence (ed. S. K. Friedlander & L. Topper). Interscience.)"
    },
    {
      "key": "S0022112075003047_ref016",
      "unstructured": "Mandelbrot, B. 1975d G\u00e9ometrie fractale de la turbulence. Dimension de Hausdorff, dispersion et nature des singularit\u00e9s du mouvement des fluides.Comptes Rendus, A281 (in press)."
    },
    {
      "key": "S0022112075003047_ref014",
      "unstructured": "Mandelbrot, B. 1975b Fonctions al\u00e9atoires pluri-temporelles: approximation poissonienne du cas brownien et g\u00e9n\u00e9ralisations.Comptes Rendus, A280,1075\u20131078."
    },
    {
      "key": "S0022112075003047_ref024",
      "doi-asserted-by": "crossref",
      "unstructured": "Yoder, L. 1974 Variation of multiparameter Brownian motion.Proc. Am. Math. Soc. 46,302\u2013309.",
      "DOI": "10.1090/S0002-9939-1974-0418260-4"
    },
    {
      "key": "S0022112075003047_ref023",
      "doi-asserted-by": "crossref",
      "unstructured": "Yaglom, A. M. 1957 Some classes of random fields in.N-dimensional space, related to stationary random processes. (trans. R. A. Silverman). Theory Prob. Appl. 2,273\u2013320.",
      "DOI": "10.1137/1102021"
    },
    {
      "key": "S0022112075003047_ref005",
      "unstructured": "Kolmogorov, A. N. 1940 Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum.C. R. (Doklady) Acad. Sci. URSS (N.S.),26,115\u2013118."
    },
    {
      "key": "S0022112075003047_ref020",
      "unstructured": "Perrin, J. 1913 Les Atomes. Gallimard. (English trans. Atoms (ed. D. L. Hammick ). Constable and Norstrand.)"
    }
  ],
  "container-title": "Journal of Fluid Mechanics",
  "original-title": [],
  "language": "en",
  "link": [
    {
      "URL": "https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112075003047",
      "content-type": "unspecified",
      "content-version": "vor",
      "intended-application": "similarity-checking"
    }
  ],
  "deposited": {
    "date-parts": [
      [
        2025,
        6,
        20
      ]
    ],
    "date-time": "2025-06-20T23:57:13Z",
    "timestamp": 1750463833000
  },
  "score": 1,
  "resource": {
    "primary": {
      "URL": "https://www.cambridge.org/core/product/identifier/S0022112075003047/type/journal_article"
    }
  },
  "subtitle": [],
  "short-title": [],
  "issued": {
    "date-parts": [
      [
        1975,
        12,
        9
      ]
    ]
  },
  "references-count": 24,
  "journal-issue": {
    "issue": "3",
    "published-print": {
      "date-parts": [
        [
          1975,
          11,
          11
        ]
      ]
    }
  },
  "alternative-id": [
    "S0022112075003047"
  ],
  "URL": "http://dx.doi.org/10.1017/S0022112075003047",
  "relation": {},
  "ISSN": [
    "0022-1120",
    "1469-7645"
  ],
  "subject": [],
  "container-title-short": "J. Fluid Mech.",
  "published": {
    "date-parts": [
      [
        1975,
        12,
        9
      ]
    ]
  }
}