Skip to main content
Springer Nature Link
Log in

The Fractal Geometry of Interfaces and the Multifractal Distribution of Dissipation in Fully Turbulent Flows

  • Chapter
Fractals in Geophysics

Part of the book series: Pure and Applied Geophysics ((PTV))

Abstract

We describe scalar interfaces in turbulent flows via elementary notions from fractal geometry. It is shown by measurement that these interfaces possess a fractal dimension of 2.35 ± 0.05 in a variety of flows, and it is demonstrated that the uniqueness of this number is a consequence of the physical principle of Reynolds number similarity. Also, the spatial distribution of scalar and energy dissipation in physical space is shown to be multifractal. We compare the f(α) curves obtained from one- and two-dimensional cuts in several flows, and examine their value in describing features of turbulence in the three-dimensional physical space.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
€34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
EUR 29.95
Price includes VAT (Germany)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
EUR 42.79
Price includes VAT (Germany)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
EUR 53.49
Price includes VAT (Germany)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

') var buybox = document.querySelector("[data-id=id_"+ timestamp +"]").parentNode var buyingOptions = buybox.querySelectorAll(".buying-option") ;[].slice.call(buyingOptions).forEach(initCollapsibles) var buyboxMaxSingleColumnWidth = 480 function initCollapsibles(subscription, index) { var toggle = subscription.querySelector(".buying-option-price") subscription.classList.remove("expanded") var form = subscription.querySelector(".buying-option-form") var priceInfo = subscription.querySelector(".price-info") var buyingOption = toggle.parentElement if (toggle && form && priceInfo) { toggle.setAttribute("role", "button") toggle.setAttribute("tabindex", "0") toggle.addEventListener("click", function (event) { var expandedBuyingOptions = buybox.querySelectorAll(".buying-option.expanded") var buyboxWidth = buybox.offsetWidth ;[].slice.call(expandedBuyingOptions).forEach(function(option) { if (buyboxWidth buyboxMaxSingleColumnWidth) { toggle.click() } else { if (index === 0) { toggle.click() } else { toggle.setAttribute("aria-expanded", "false") form.hidden = "hidden" priceInfo.hidden = "hidden" } } }) } initialStateOpen() if (window.buyboxInitialised) return window.buyboxInitialised = true initKeyControls() })()

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  • Anselmet, F., Gagne, Y., Hopfinger, E. J., and Antonia, R. A. (1984), High-order Velocity Structure Functions in Turbulent Shear Flows, J. Fluid Mech. 140, 63.

    Article  Google Scholar 

  • Chhabra, A., Jensen, R., and Sreenivasan, K. R. (1989), Multifractals, Multiplicative Processes and the Thermodynamic Formalism, Phys. Rev. A (in print).

    Google Scholar 

  • Feigenbaum, M. (1987), Some Characterizations of Strange Sets, J. Stat. Phys. 46, 919.

    Article  Google Scholar 

  • Frisch, U., and Parisi, G., On the singularity structure of fully developed turbulence, In Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics (eds. Ghil, M., Benzi, R., and Parisi, G.) (North-Holland, New York 1985).

    Google Scholar 

  • Halsey, T. C., Jensen, M. H., Kadanoff, L. P., Procaccia, I., and Shraiman, B. I. (1986), Fractal Measures and their Singularities: The Characterization of Strange Sets, Phys. Rev. A33, 1141.

    Google Scholar 

  • Hentschel, H. G. E., and Procaccia, I. (1983), The Infinite Number of Generalized Dimensions of Fractals and Strange Attractors, Physica 8D, 435.

    Google Scholar 

  • Mandelbrot, B. B. (1974), Intermittent Turbulence in Self-similar Cascades: Divergence of High Moments and Dimension of the Carrier, J. Fluid Mech. 62, 331.

    Article  Google Scholar 

  • Mandelbrot, B. B., The Fractal Geometry of Nature (Freeman, San Francisco 1982).

    Google Scholar 

  • Mandlebrot, B. B., An introduction to multifractal distribution functions, In Fluctuations and Pattern Formation (eds. Stanley, H. E., and Ostrowsky, N.) (Kluwer, Dordrecht-Boston 1988); see also this volume.

    Google Scholar 

  • Marstrand, J. M. (1954), Some Fundamental Geometrical Properties of Plane Sets of Fractal Dimensions, London Math. Soc. 3, 257.

    Google Scholar 

  • Mattila, P. (1975), Hausdorff Dimension, Orthogonal Projections and Intersections with Planes, Ann. Acad. Sci. Fenn. Ser. A I Math. 1, 227.

    Google Scholar 

  • Meneveau, C. (1989), The Multifractal Nature of Turbulence, Ph.D. Thesis, Yale University.

    Google Scholar 

  • Meneveau, C., and Sreenivasan, K. R. (1987a), The Multifractal Dissipation Field in Turbulent Flows, Nuclear Physics B (Proc. Suppl.) 2, 49.

    Article  Google Scholar 

  • Meneveau, C., and Sreenivasan, K. R. (1987b), Simple Multifractal Cascade Model for Fully Developed Turbulence, Phys. Rev. Lett. 59, 1424.

    Article  Google Scholar 

  • Meneveau, C., and Sreenivasan, K. R. (1989), Measurement of f (α) from Scaling of Histograms, and Application to Dynamical Systems and Fully Developed Turbulence, Phys. Lett. A (in print).

    Google Scholar 

  • Prasad, R. R., Meneveau, C., and Sreenivasan, K. R. (1988), Multifractal Nature of the Dissipation Field of Passive Scalars in Fully Turbulent Flows, Phys. Rev. Lett. 61, 74.

    Article  Google Scholar 

  • Prasad, R. R., and Sreenivasan, K. R. (1989), Scalar Interfaces in Digital Images of Turbulent Flows, Experiments in Fluids 7, 259.

    Article  Google Scholar 

  • Ramshankar, R. (1988), The Dynamics of Countercurrent Mixing Layers, Ph.D. Thesis, Yale University.

    Google Scholar 

  • Richardson, L. F., Weather Prediction by Numerical Process (Cambridge University Press, Cambridge, U.K. 1922).

    Google Scholar 

  • Sreenivasan, K. R., and Meneveau, C. (1986), The Fractal Facets of Turbulence, J. Fluid Mech. 173, 357.

    Article  Google Scholar 

  • Sreenivasan, K. R., and Meneveau, C. (1988), Singularities of the Equations of Fluid Motion, Phys. Rev. A38, 6287.

    Google Scholar 

  • Sreenivasan, K. R., Ramshankar, R., and Meneveau, C. (1989), Mixing, Entrainment, and Fractal Dimension of Interfaces in Turbulent Flows, Proc. Roy. Soc. Lond. A421, 79.

    Google Scholar 

  • Taylor, G. I. (1938), The Spectrum of Turbulence, Proc. Roy. Soc. Lond. A164, 476.

    Google Scholar 

Download references

Author information

Author notes

  1. R. Ramshankar

    Present address: Department of Physics, Haverford College, Haverford, PA, 19041, USA

Authors and Affiliations

  1. Mason Laboratory, Yale University, New Haven, CT, 06520, USA

    K. R. Sreenivasan, R. R. Prasad, C. Meneveau & R. Ramshankar

Authors
  1. K. R. Sreenivasan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. R. Prasad
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. C. Meneveau
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. R. Ramshankar
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Lamont-Doherty Geological Observatory, Columbia University, Palisades, NY, 10964, USA

    Christopher H. Scholz

  2. T. J. Watson Research Center, IBM, P.O. Box 218, Yorktown Heights, NY, 10598, USA

    Benoît B. Mandelbrot

Rights and permissions

Reprints and permissions

Copyright information

© 1989 Springer Basel AG

About this chapter

Cite this chapter

Sreenivasan, K.R., Prasad, R.R., Meneveau, C., Ramshankar, R. (1989). The Fractal Geometry of Interfaces and the Multifractal Distribution of Dissipation in Fully Turbulent Flows. In: Scholz, C.H., Mandelbrot, B.B. (eds) Fractals in Geophysics. Pure and Applied Geophysics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-6389-6_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-6389-6_3

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-6391-9

  • Online ISBN: 978-3-0348-6389-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics