Skip to main content Accessibility help
×
Home
Hostname: page-component-768ffcd9cc-9th95 Total loading time: 0.325 Render date: 2022-12-03T23:32:38.233Z Has data issue: true Feature Flags: { "useRatesEcommerce": false } hasContentIssue true

Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier

Published online by Cambridge University Press:  29 March 2006

Benoit B. Mandelbrot
Affiliation:
Mathematical Sciences Department, IBM Thomas J. Watson Research Center, Yorktown Heights, New York 10598

Abstract

Kolmogorov's ‘third hypothesis’ asserts that in intermittent turbulence the average ε of the dissipation ε, taken over any domain D, is ruled by the lognormal probability distribution. This hypothesis will be shown to be inconsistent, save under assumptions that are extreme and unlikely. Further, a widely used justification of lognormality, due to Yaglom and based on probabilistic argument involving a self-similar cascade, will be discussed. In this model, lognormality indeed applies strictly when D is ‘an eddy’, typically a three-dimensional box embedded in a self-similar hierarchy, and may perhaps remain a reasonable approximation when D consists of a few such eddies. On the other hand, the experimental situation is described better by considering averages taken over essentially one-dimensional domains D. The first purpose of this paper is to carry out Yaglom's cascade argument, labelled as ‘microcanonical’, for such averaging domains. The second is to replace Yaglom's model by a different, less constrained one, based upon the concept of ‘canonical cascade’. It will be shown, both for one-dimensional domains in a microcanonical cascade, and for all domains in canonical cascades, that in every non-degenerate caJe the distribution of ε differs from the lognormal distribution. Depending upon various parameters, the discrepancy may be either moderate, or considerable, or even extreme. In the latter two cases, high-order moments of E turn out to be infinite. This avoids various paradoxes (to be explored) that are present in Kolmogorov's and Yaglom's approaches. The third purpose is to note that high-order moments become infinite only when the number of levels of the cascade tends to infinity, meaning that the internal scale η tends to zero. Granted the usual value of η, this number of levels is actually small, so the representativity of the limit is questionable. This issue was investigated through computer simulation. The results bear upon the question of the extent to which Kolmogorov's second hypothesis applies in the face of intermittency. The fourth purpose is as follows. Yaglom noted that the cascade model predicts that dissipation only occurs in a portion of space of very small total volume. In order to describe the structure of this portion of space, it will be shown useful to introduce the concept of the ‘intrinsic fractional dimension’ A of the carrier of intermittent turbulence. The fifth purpose is to study the relations between the parameters ruling the distribution of η, and those ruling its spectral and dimensional properties. Both conceptually and numerically, these various parameters turn out to be distinct, which opens up several problems for empirical study.

Type
Research Article
Copyright
© 1974 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

Billingsley, P. 1965 Ergodic Theory and Information. Wiley.
De Wijs, H. J. 1951 Statistics of ore distribution. Geol. en Mijnbouw (Netherlands), 13, 365375.Google Scholar
De Wijs, H. J. 1953 Statistics of ore distribution. Geol. en Mijnbouw (Netherlands), 15, 1224.
Doob, J. L. 1953 Stochastic Processes. Wiley.
Feller, W. 1971 An Introduction to the Theory of Probability and its Applications. Wiley.
Harris, T. 1963 Branching Processes. Springer.
Kahane, J. P. 1973 Sur le modèle de turbulence de Benoit Mandelbrot, Comptes Rendus, in press.Google Scholar
Kolmogorov, A. N. 1962 A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number J. Fluid Mech. 13, 8285. (See also 1962 Mécanique de la Turbulence, pp. 447–458 (in French and Russian).)Google Scholar
Mandelbrot, B. 1967 How long is the coast of Britain? Statistical self-similarity and fractional dimension. Science, 156, 636638.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. Rosenblatt & Van Atta), pp. 333351. Springer.
Mandelbrot, B. 1973 Multiplications aléatoires iterées, et distributions invariantes par moyenne pondérée. Comptes Rendus, in press.Google Scholar
Mandelbrot, B. 1974 Physical objects with fractional dimension: seacoasts, galaxy clusters, turbulence and soap. IBM Res. Ext. Report. (Submitted to Bull. Inst. Math. Applics.)Google Scholar
Novikov, E. A. 1969 Scale similarity for random fields Dokl. Akad. Nauk SSSR, 184, 10721075. (English trans. Sov. Phys. Dokl. 14, 104–107.)Google Scholar
Novikov, E. A. 1971 Intermittency and scale similarity in the structure of a turbulent flow Prikl. Mat. Mekh. 35, 266277. (See also P.M.M. Appl. Math. Mech. 35, 231–241.)Google Scholar
Novikov, E. A. & Stewart, R. W. 1964 Intermittency of turbulence and the spectrum of fluctuations of energy dissipation Isv. Akad. Nauk SSSR, Seria Geofiz. 3, 408.Google Scholar
Oboukhov, A. M. 1962 Some specific features of atmospheric turbulence J. Fluid Mech. 13, 7781.Google Scholar
Orszag, S. A. 1970 Indeterminacy of the moment problem for intermittent turbulence Phys. Fluids, 13, 22112212.Google Scholar
Yaglom, A. M. 1966 The influence of the fluctuation in energy dissipation on the shape of turbulent characteristics in the inertial interval Dokl. Akad. Nauk SSSR, 166, 4952. (English trans. Sov. Phys. Dokl. 2, 26–29.)Google Scholar
1420
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *