Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-18T16:08:57.020Z Has data issue: false hasContentIssue false
  • English
  • Français

The multifractal nature of turbulent energy dissipation

Published online by Cambridge University Press:  26 April 2006

Charles Meneveau  and
K. R. Sreenivasan
Charles Meneveau
Affiliation:
Mason Laboratory, Yale University, New Haven, CT 06520, USA Present address: Department of Mechanical Engineerin, Johns Hopkins University, Baltimore, MD 21218, USA.
K. R. Sreenivasan
Affiliation:
Mason Laboratory, Yale University, New Haven, CT 06520, USA
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The intermittency of the rate of turbulent energy dissipation ε is investigated experimentally, with special emphasis on its scale-similar facets. This is done using a general formulation in terms of multifractals, and by interpreting measurements in that light. The concept of multiplicative processes in turbulence is (heuristically) shown to lead to multifractal distributions, whose formalism is described in some detail. To prepare proper ground for the interpretation of experimental results, a variety of cascade models is reviewed and their physical contents are analysed qualitatively. Point-probe measurements of ε are made in several laboratory flows and in the atmospheric surface layer, using Taylor's frozen-flow hypothesis. The multifractal spectrum f(α) of ε is measured using different averaging techniques, and the results are shown to be in essential agreement among themselves and with our earlier ones. Also, long data sets obtained in two laboratory flows are used to obtain the latent part of the f(α) curve, confirming Mandelbrot's idea that it can in principle be obtained from linear cuts through a three-dimensional distribution. The tails of distributions of box-averaged dissipation are found to be of the square-root exponential type, and the implications of this finding for the f(α) distribution are discussed. A comparison of the results to a variety of cascade models shows that binomial models give the simplest possible mechanism that reproduces most of the observations. Generalizations to multinomial models are discussed.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 224 , March 1991 , pp. 429 - 484
Copyright
© 1991 Cambridge University Press

References

Andrews, L. C., Phillips, R. L., Shivamoggisc, B., Beck, J. K., 1989 A statistical theory for the distribution of energy dissipation in intermittent turbulence. Phys. Fluids A 1, 999–1006.Google Scholar
Anselmet, F., Gagne, Y., Hopfinger, E. J. & Antonia, R. A., 1984 High-order velocity structure functions in turbulent shear flows. J. Fluid Mech. 140, 63–89.Google Scholar
Antonia, R. A., Chambers, A. J. & Phan-Thien, N. 1980 Taylor's hypothesis and spectra of velocity and temperature derivatives in a turbulent shear flow. Boundary-layer Met. 19, 19–29.Google Scholar
Arneodo, A., Grasseau, G. & Kostelich, E. J., 1987 Fractal dimensions and f(a) spectrum of the Hénon attractor. Phys. Lett. 124A, 426–432.Google Scholar
Badii, R. & Politi, A., 1984 Intrinsic oscillations in measuring the fractal dimension. Phys. Lett. 104A 303–305.Google Scholar
Batchelor, G. K.: 1953 The Theory of Homogeneous Turbulence. Cambridge University Press.
Batchelor, G. K. & Townsend, A. A., 1949 The nature of turbulent motion at large wave-numbers. Proc. R. Soc. Land. A 199, 238–255.Google Scholar
Benzi, R., Paladin, G., Parisi, G. & Vulpiani, A., 1984 On the multifractal nature of fully developed turbulence and chaotic systems. J. Phys. A 17, 3521–3531.Google Scholar
Cates, M. E. & Deutsch, J. M., 1987 Spatial correlations in multifractals. Phys. Rev. A 35, 4907–4910.Google Scholar
Chhabra, A.: 1989 The thermodynamic formalism of multifractals and its applications to chaotic dynamical systems and turbulence. Ph.D. thesis, Yale University.
Chhabra, A. & Jensen, R. V., 1989 Direct determination of the f(a) singularity spectrum. Phys. Rev. Lett. 62, 1327–1330.Google Scholar
Chhabra, A., Jensen, R. V. & Sreenivasan, K. R., 1989 Extraction of underlying multiplicative processes from multifractals via the thermodynamic formalism. Phys. Rev. A 40, 4593–4611.Google Scholar
Chhabra, A., Meneveau, C., Jensen, R. & Sreenivasan, K. R., 1989 Direct determination of the f(a) singularity spectrum and its applications to fully developed turbulence. Phys. Rev. A 40, 5284–5294.Google Scholar
Chhabha, A. B. & Sreenivasan, K. R., 1990 Negative dimensions: theory, computations and experiment. Phys. Rev. A (to appear).Google Scholar
Chorin, A. J.: 1982 The evolution of a turbulent vortex. Commun. Math. Phys. 83, 517–535.Google Scholar
Chorin, A. J.: 1988a Scaling laws in the vortex lattice model of turbulence. Commun. Math. Phys. 114, 167–176.Google Scholar
Chorin, A. J.: 1988b Spectrum, dimension and polymer analogies in fluid turbulence. Phys. Rev. Lett. 60, 1947–1949.Google Scholar
Corrsin, S.: 1962 Turbulent dissipation fluctuations. Phys. Fluids 5, 1301–1302.Google Scholar
Deane, A. E. & Keefe, L. R., 1988 Multifractal spectra in homogeneous shear flows. In Proc. Summer Program 1988, Center for Turbulence Research, Stanford University.
Everson, R., Sirovich, L. & Sreenivasan, K. R., 1990 Wavelet analysis of the turbulent jet. Phys. Lett. A 145, 314–322.Google Scholar
Farmer, J. D., Ott, E. & Yorke, J. A., 1983 The dimension of chaotic attractors. Physica 7D, 153–180.Google Scholar
Feigenbaum, M. J., Jensen, M. H. & Procaccia, I., 1986 Time ordering and the thermodynamics of strange sets: theory and experimental tests. Phys. Rev. Lett. 57, 1503–1506.Google Scholar
Finn, J. M. & Ott, E., 1988 Chaotic flows and fast magnetic dynamos. Phys. Fluids 31n, 2992–3011.Google Scholar
Frenkiel, F. N. & Klebanoff, P. S., 1975 On the lognormality of the small-scale structure of turbulence. Boundary-layer Met. 8, 173–200.Google Scholar
Frisch, U. & Parisi, G., 1985 On the singularity structure of fully developed turbulence. In Turbulence and Predictability in Geophysical Fluid Dynamics (ed. M. Gil, R. Benzi & G. Parisi), pp. 84–88. North-Holland.
Frisch, U., Sulem, P.-L. & Nelkin, M. 1978 A simple dynamical model of intermittent fully developed turbulence. J. Fluid Mech. 87, 719–736.Google Scholar
Gagne, Y.: 1987 Etude expérimentale de 1′intermittence et des singularités dans le plan complexe en turbulence développée. Ph.D. thesis, Université de Grenoble.
Gibson, C. H., Stegen, G. R. & Mcconnell, S., 1970 Measurements of the universal constant in Kolmogorov's third hypothesis for high Reynolds number turbulence. Phys. Fluids 13, 2448–2451.Google Scholar
Grossmann, A. & Morlet, J., 1984 Decomposition of Hardy functions into square integrable wavelets of constant shape. SIAM J. Math. Anal. 15, 723–736.Google Scholar
Gurvich, A. S. & Yaglom, A. M., 1967 Breakdown of eddies and probability distributions for small-scale turbulence, boundary layers and turbulence. Phys. Fluids Suppl. 10, S59–65.Google Scholar
Halsey, T. C., Jensen, M. H., Kadanoff, L. P., Procaccia, I. & Shraiman, B. I., 1986 Fractal measures and their singularities: the characterization of strange sets. Phys. Rev. 33, 1141–1151.Google Scholar
Hentschel, H. G. E. & Procaccia, I. 1982 Intermittency exponent in fractally homogeneous turbulence. Phys. Rev. Lett. 49, 1158–1161.Google Scholar
Hentschel, H. G. E. & Procaccia, I. 1983 The infinite number of generalized dimensions of fractals and strange attractors. Physica 8D, 435–444.Google Scholar
Hosokawa, I.: 1989 An advanced model of dissipation cascade in locally isotropic turbulence. Phys. Fluids A 1, 186–189.Google Scholar
Hosokawa, I. & Yamamoto, K., 1990 Intermittency exponents and generalized dimensions of a directly simulated fully developed turbulence. Phys. Fluids A 2, 889–892.Google Scholar
Kanane, J.-P.: 1974 Sur le modele de turbulence de Benoit Mandelbrot. CR Acad. Sci. Paris 278, 621–623.Google Scholar
Kahane, J.-P. & Peyriere, J. 1976 Sur certaines martingales de Benoit Mandelbrot. Adv. Maths 22, 131–145.Google Scholar
Kolmogorov, A. N.: 1941 The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. CR Acad. Sci. USSR 30, 299–303.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, 82–85.Google Scholar
Kraichnan, R. H.: 1974 On Kolmogorov's inertial-range theories. J. Fluid Mech. 62, 305–330.Google Scholar
Kraichnan, R. H.: 1990 Models of intermittency in hydrodynamic turbulence. In New Perspectives in Turbulence (ed. L. Sirovich). Springer (to appear).
Kuo, A. Y.-S. & Corrsin, S. 1972 Experiment on the geometry of the fine-structure regions in fully turbulent fluid. J. Fluid Mech. 56, 447–479.Google Scholar
Landau, L. D. & Lifshitz, E. M., 1959 Fluid Mechanics. Addison-Wesley.
Levich, E.: 1987 Certain problems in the theory of developed hydrodynamic turbulence. Phys. Rep. 151, 129–238.Google Scholar
Lumley, J. L.: 1965 Interpretation of time spectra measured in high-intensity shear flows. Phys. Fluids 8, 1056–1062.Google Scholar
Mandelbrot, B. 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. 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. B.: 1976 Intermittent turbulence and fractal dimension: kurtosis and the spectral exponent 5/3 + 5. In Turbulence and the Navier-Stokes Equations (ed. R. Temam), pp. 121–145. Springer.
Mandelbrot, B. B.: 1982 The Fractal Geometry of Naturep. Freeman.
Mandelbrot, B. B.: 1984 Fractals in physics: squid clusters, diffusions, fractal measures and the unicity of fractal dimensionality. J. Statist. Phys. 34, 895–930.Google Scholar
Mandelbbot, B. B.: 1989 Multifractal measures, especially for the geophysicist. Pure Appl. Geophys. 131, 5–42.Google Scholar
McConnell, S. O.: 1976 The fine structure of velocity and temperature measured in the laboratory and the atmospheric marine boundary layer. Ph.D. thesis, University of California, San Diego.
Meneveau, C.: 1989 The multifractal nature of turbulence. Ph.D. thesis, Yale University.
Meneveau, C.: 1990 Turbulence dynamics in the wavelet representation. Phys. Rev. Lett. (submitted).Google Scholar
Meneveau, C. & Chhabra, A., 1990 Two-point statistics of multifractal measures. Physica A 164, 564–574.Google Scholar
Meneveau, C. & Nelkin, M., 1989 Attractor size in intermittent turbulence. Phys. Rev. A 39, 3732–3733.Google Scholar
Meneveau, C. & Sreenivasan, K. R., 1987a The multifractal spectrum of the dissipation field in turbulent flows. Nucl. Phys. B (Proc. Suppl. p) 2, 49–76.Google Scholar
Meneveau, C. & Sreenivasan, K. R., 1987b Simple multifractal cascade model for fully developed turbulence. Phys. Rev. Lett. 59, 1424–1427.Google Scholar
Meneveau, C. & Sreenivasan, K. R., 1989 Measurement of f(a) from scaling of histograms and applications to dynamical systems and fully developed turbulence. Phys. Lett. A 137, 103–112.Google Scholar
Meneveau, C. & Sreenivasan, K. R., 1990 Interface dimension in intermittent turbulence. Phys. Rev. A 41, 2246–2248.Google Scholar
Meneveau, C., Sreenivasan, K. R., Kailasnath, P. & Fan, S., 1989 Joint multifractal measures: theory and applications to turbulence. Phys. Rev. A 40, 894–913.Google Scholar
Monin, A. S. & Yaglom, A. M., 1971 Statistical Fluid Mechanics, Vol. II. MIT Press.
Mori, H.: 1980 Anomalous diffusion of vorticity in fully-developed turbulence. Suppl. Prog. Theor. Phys. 69, 111–121.Google Scholar
Nakano, T.: 1988a Determination of a dynamical scaling function in a cascade model of turbulence. Prog. Theor. Phys. 79, 569–580.Google Scholar
Nakano, T.: 1988b Generalized dimensions of three-dimensional turbulence. Preprint.
Nakano, T. & Nelkin, M., 1985 Crossover model for the scaling exponents of intermittent fully developed turbulence. Phys. Rev. A 31, 1980–1982.Google Scholar
Nelkin, M.: 1973 Intermittency in fully developed turbulence as a consequence of the Navier-Stokes equations. Phys. Rev. Lett. 30, 1029–1031.Google Scholar
Nelkin, M.: 1989 What do we know about self-similarity in fluid turbulence? J. Statist. Phys. 54, 1–15.Google Scholar
Novikov, E. A.: 1969 Scale similarity for random fields. 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. Mech. 35, 266–277.Google Scholar
Novikov, E. A.: 1990 The effects of intermittency on statistical characteristics of turbulence and scale similarity of breakdown coefficients. Phys. Fluids A 2, 814.Google Scholar
Novikov, A. A. & Stewart, R. W., 1964 Intermittency of turbulence and the spectrum of fluctuations of energy dissipation. Izv. Akad. Nauk. SSSR, Geofiz. 3, 408.Google Scholar
Obukhov, A. M.: 1962 Some specific features of atmospheric turbulence. J. Fluid Mech. 13, 77–81.Google Scholar
Orszag, S. A.: 1970 Indeterminacy of the moment problem for intermittent turbulence. Phys. Fluids 13, 2211–2212.Google Scholar
Ott, E. & Antonson, T. M., 1988 Chaotic fluid convection and the fractal nature of passive scalar gradients. Phys. Rev. Lett. 61, 2839–2842.Google Scholar
Paladin, G. & Vulipiani, A., 1987 Anomalous scaling laws in multifractal objects. Phys. Rep. 156, 148–225.Google Scholar
Park, J. T.: 1976 Inertial subrange measurements in the marine boundary layer. Ph.D. thesis, University of California, San Diego.
Perry, A. E. & Abell, C. J., 1975 Scaling laws for pipe flows. J. Fluid Mech. 67, 257–271.Google Scholar
Peyriere, M. J.: 1974 Turbulence et dimension de Hausdorff. CR Acad. Sci. Paris 278, 567–569.Google Scholar
Prasad, R. R., Meneveau, C. & Sreenivasan, K. R., 1988 The multifractal nature of the dissipation field of passive scalars in fully turbulent flows. Phys. Rev. Lett. 61, 74–77.Google Scholar
Prasad, R. R. & Sreenivasan, K. R., 1990a Quantitative three-dimensional imaging and the structure of passive scalar fields in fully turbulent flows. J. Fluid Mech. 216, 1–34.Google Scholar
Pbasad, R. R. & Sreenivasan, K. R., 1990b Scaling ranges in turbulence spectra. Preprint.
Ramshankar, R.: 1988 The dynamics of countercurrent mixing layers. Ph.D. thesis, Yale University.
Richardson, L. F.: 1922 Weather Prediction by Numerical Process. Cambridge University Press.
Schertzer, D. & Lovejoy, S., 1985 The dimension and intermittency of atmospheric dynamics. In Turbulent Shear Flows 4 (ed. L. J. S. Bradbury, F. Durst, B. Launder, F. W. Schmidt & J. H. Whitelaw), pp. 7–33. Springer.
Siggia, E. D.: 1978 Model of intermittency in three-dimensional turbulence. Phys. Rev. A 17, 1166–1176.Google Scholar
Siggia, E. D.: 1981 Invariants for the one-point vorticity and strain rate correlation functions. Phys. Fluids 24, 1934–1936.Google Scholar
Smith, L. A., Fournier, J.-D. & Spiegel, E. A. 1986 Lacunarity and intermittency in fluid turbulence. Phys. Lett. 114A, 465–468.Google Scholar
Sreenivasan, K. R.: 1985 On the fine-scale intermittency of turbulence. J. Fluid Mech. 151, 81–103.Google Scholar
Sreenivasan, K. R., Antonia, R. A. & Danh, H. Q., 1977 Temperature dissipation fluctuations in a turbulent boundary layer. Phys. Fluids 20, 1238–1249.Google Scholar
Sreenivasan, K. R. & Fan, M. S., 1989 Kinematics and dynamics of turbulent vorticity fluctuations. Paper EC5, Bull. Am. Phys. Soc. 34, 2293 (abstract only).Google Scholar
Sreenivasan, K. R. & Meneveau, C., 1986 The fractal facets of turbulence. J. Fluid Mech. 173, 357–386.Google Scholar
Sreenivasan, K. R. & Meneveau, C., 1988 Singularities of the equations of fluid motion. Phys. Rev. A 38, 6287–6295.Google Scholar
Tennekes, H.: 1968 Simple model for the small-scale structure of turbulence. Phys. Fluids 11, 669–671.Google Scholar
Tennekes, H. & Wyngaard, J. C., 1972 The intermittent small-scale structure of turbulence; data-processing hazards. J. Fluid Mech. 55, 93–103.Google Scholar
Tong, P. & Goldburg, W. I., 1988 Experimental study of relative velocity fluctuations in turbulence. Phys. Lett. A 127, 147–150.Google Scholar
Van Atta, C. W. & Antonia, R. A. 1980 Reynolds number dependence of skewness and flatness factors of turbulent velocity derivates. Phys. Fluids 23, 252–257.Google Scholar
Yaglom, A. M.: 1966 The influence of fluctuations in energy dissipation on the shape of turbulent characteristics in the inertial interval. Sov. Phys. Dokl. 11, 26–29.Google Scholar
Yakhot, V., She, Z-S. & Orszag, S. A. 1989 Deviations from the classical Kolmogorov theory of the inertial range of homogeneous turbulence. Phys. Fluids A 1, 289–293.Google Scholar