Hostname: page-component-6bf8c574d5-pdxrj Total loading time: 0 Render date: 2025-03-08T02:18:40.730Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  29 March 2006

Benoit B. Mandelbrot
Benoit B. Mandelbrot
Affiliation:
General Sciences Department, IBM Thomas J. Watson Research Center, Yorktown Heights, New York 10598
Get access Rights & Permissions [Opens in a new window]

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 $3-\frac{1}{3}$.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 72 , Issue 3 , 9 December 1975 , pp. 401 - 416
Copyright
© 1975 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Corrsin, S. 1951a On the spectrum of isotropic temperature fluctuations in isotropic turbulence. J. Appl. Phys. 22, 469–473.Google Scholar
Feller, W. 1968 An Introduction of Probability Theory and Its Application, vol. 1, 3rd edn. Wiley.
Gangolli, R. 1967 Lévy's Brownian motion of several parameters. Ann. Inst. H. Poincaré, B 3, 121–226.Google Scholar
Kármán, Th. Von & Howarth, L. 1938 On the statistical theory of isotropic turbulence. Proc. Roy. Soc. A 151, 411–478. (Reprinted in Turbulence (ed. S. K. Friedlander & L. Topper). Interscience.)Google Scholar
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.Google Scholar
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.Google Scholar
Lévy, P. 1948 Processus Stochastiques et Mouvement Brownien. Gauthier-Villars.
Mckean, H. P. 1963 Brownian motion with a several dimensional time. Theory Prob. Appl. 8, 357–378.Google Scholar
Mandelbrot, B. 1965 Self-similar error clusters in communications systems and the concept of conditional stationarity. I.E.E.E. Trans. Comm. Tech. COM 13, 71–90.Google Scholar
Mandelbrot, B. 1967 How long is the coast of Britain? Statistical self-similarity and fractional dimension. Science, 155, 636–638.Google Scholar
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.
Mandelbrot, B. 1974 Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier. J. Fluid Mech. 62, 331–358.Google Scholar
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).Google Scholar
Mandelbrot, B. 1975b Fonctions aléatoires pluri-temporelles: approximation poissonienne du cas brownien et généralisations. Comptes Rendus, A 280, 1075–1078.Google Scholar
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. 1975d Géometrie fractale de la turbulence. Dimension de Hausdorff, dispersion et nature des singularités du mouvement des fluides. Comptes Rendus, A 281 (in press).Google Scholar
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.Google Scholar
Neumann, J. Von 1963 Recent theories of turbulence (a report to ONR). In Collected Works, vol. 6, pp. 437–472. Pergamon.
Perrin, J. 1913 Les Atomes. Gallimard. (English trans. Atoms (ed. D. L. Hammick). Constable and Norstrand.)
Rogers, C. A. 1970 Hausdorff Measures. Cambridge University Press.
Scheffer, V. 1975 Géométrie fractale de la turbulence. Equations de Navier-Stokes et dimension de Hausdorff. Comptes Rendus, A281 (in press).Google Scholar
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.Google Scholar
Yoder, L. 1974 Variation of multiparameter Brownian motion. Proc. Am. Math. Soc. 46, 302–309.Google Scholar