Abstract.
The multifractal framework relates the scaling properties of turbulence to its local regularity properties through a statistical description as a collection of local singularities. The multifractal properties are moreover linked to the multiplicative cascade process that creates the peculiar properties of turbulence such as intermittency. A comprehensive estimation of the multifractal properties of turbulence from data analysis, using a tool valid for all kind of singularities (including oscillating singularities) and mathematically well-founded, is thus of first importance in order to extract a reliable information on the underlying physical processes. The wavelet leaders yield a new multifractal formalism which meets all these requests. This paper aims at describing it and at applying it to experimental turbulent velocity data. After a detailed discussion of the practical use of the wavelet leader based multifractal formalism, the following questions are carefully investigated: (1) What is the dependence of multifractal properties on the Reynolds number? (2) Are oscillating singularities present in turbulent velocity data? (3) Which multifractal model does correctly account for the observed multifractal properties? Results from several data set analysis are used to discuss the dependence of the computed multifractal properties on the Reynolds number but also to assess their common or universal component. An exact though partial answer (no oscillating singularities are detected) to the issue of the presence of oscillating singularities is provided for the first time. Eventually an accurate parameterization with cumulant exponents up to order 4 confirms that the log-normal model (with c2 = -0.025±0.002) correctly accounts for the universal multifractal properties of turbulent velocity.
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References
A.N. Kolmogorov, Dokl. Akad. Nauk SSSR 30, 299 (1941)
A. Obukhov, J. Fluid Mech. 13, 77 (1962)
A.N. Kolmogorov, J. Fluid Mech. 13, 82 (1962)
U. Frisch, Turbulence, the Legacy of A.N. Kolmogorov (Cambridge University Press, 1995)
A. Yaglom, Dokl. Akad. Nauk. SSR 166, 49 (1966)
B. Mandelbrot, J. Fluid. Mech. 62, 331 (1974)
L. Richardson, Weather prediction by numerical process (Cambridge University Press, 1922)
B. Castaing, Y. Gagne, E. Hopfinger, Physica D 46, 177 (1990)
J. Hunt, J. Vassilicos, Proc. R. Soc. Lond. A 434, 183 (1991)
N.R. Kevlahan, J. Vassilicos, Proc. R. Soc. Lond. A 447, 341 (1994)
A. Arneodo, E. Bacry, J. Muzy, Phys. Rev. Lett. 74, 4823 (1995)
A. Arneodo, E. Bacry, S. Jaffard, J. Muzy, J. Stat. Phys. 87, 179 (1997)
G. Parisi, U. Frisch, On the singularity structure of fully developed turbulence, appendix to Fully developed turbulence and intermittency by U. Frisch, in Proc. Int. Summer school Phys. Enrico Fermi, North Holland (1985)
S. Mallat, A Wavelet Tour of Signal Processing (Academic Press, San Diego, CA, 1998)
J. Muzy, E. Bacry, A. Arneodo, Phys. Rev Lett. 67, 3515 (1991)
C. Meneveau, J. Fluid Mech. 232, 469 (1991)
S. Jaffard, SIAM J. Math. Anal. 28, 944 (1997)
J. Muzy, E. Bacry, A. Arneodo, Phys. Rev. E 47, 875 (1993)
A. Arneodo, E. Bacry, J. Muzy, Physica A 213, 232 (1995)
S. Jaffard, Wavelet Techniques in Multifractal Analysis, in Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot, edited by M. Lapidus and M. van Frankenhuysen, Proc. of Symp. in Pure Mathematics (AMS, 2004)
S. Jaffard, Ann. Fac. Scien. Toul. 15, 3 (2006)
B. Lashermes, Ph.D. thesis, École Normale Supérieure de Lyon (2005)
B. Lashermes, S. Jaffard, P. Abry, Wavelet Leaders Based Multifractal Analysis, in ICASSP 2005 Conference, Philadelphia, USA (2005)
S. Jaffard, B. Lashermes, P. Abry, Wavelet Analysis and Applications, edited by T. Quian et al., in Applied and Numerical Harmonic Analysis (Springer, 2006)
M. Schroeder, Fractals, Chaos, Power Laws. Minutes from an Infinite Paradise (W.H. Freeman and Company, 1991)
A.N. Kolmogorov, Dokl. Akad. Nauk SSSR 32, 16 (1941)
A. Arneodo, S. Manneville, J. Muzy, Eur. Phys. J. B 1, 129 (1998)
J. Delour, J. Muzy, A. Arneodo, Eur. Phys. J. B 23, 243 (2001)
V. Venugopal, S. Roux, E. Foufoula-Georgiou, A. Arneodo, Water Resour. Res. 42, W06D14 (2006)
B. Lashermes, E. Foufoula-Georgiou, Water Resour. Res. 43, W09405 (2007)
S. Jaffard, Y. Meyer, Memoirs of the AMS. 123, 1 (1996)
A. Arneodo, E. Bacry, S. Jaffard, J. Muzy, J. Four. Anal. Appl. 4, 159 (1998)
S. Roux, Ph.D. thesis, Université d'Aix-Marseille II (1996)
A. Arneodo, E. Bacry, J. Muzy, J. Math. Phys. 39, 4142 (1998)
S. Jaffard, J. Math. Phys. 39, 4129 (1998)
C. Melot, J. Math. Pures Appl. 83, 367 (2004)
Z.S. She, E. Lévêque, Phys. Rev. Lett. 72, 336 (1994)
B. Castaing, B. Dubrulle, J. Phys. II France 5, 895 (1995)
E. Novikov, Phys. Rev. E 50, R3303 (1994)
B. Dubrulle, Phys. Rev. Lett. 73, 959 (1994)
B. Lashermes, P. Abry, P. Chainais, Int. J. Wavelets, Multiresolution and Information Processing 2, 497 (2004)
B. Lashermes, C. Baudet, P. Abry, P. Chainais, Limitation of Scaling Exponent Estimation in Turbulence, in Advances in Turbulence X, Proc. of ETC10 Conference, Trondheim, Norway, edited by H.I. Anderson, P.-A. Krogstad (CIMNE, Barcelona, 2004)
O. Chanal, B. Chabaud, B. Castaing, B. Hébral, Eur. Phys. J. B 17, 309 (2000)
Y. Malécot, C. Auriault, H. Kahalerras, Y. Gagne, O. Chanal, B. Chabaud, B. Castaing, Eur. Phys. J. B 16, 549 (2000)
T. von Kármán, L. Howarth, Proc. Roy. Soc. A164, 917 (1938)
A. Arneodo, C. Baudet, F. Belin, R. Benzi, B. Castaing, B. Chabaud, R. Chavarria, S. Ciliberto, R. Camussi, F. Chillà et al., Europhys. Lett. 34, 411 (1996)
E. Lindborg, Phys. Fluids 11, 510 (1999)
L. Danaila, F. Anselmet, T. Zhou, R. Antonia, J. Fluid Mech. 391, 359 (1999)
F. Moisy, P. Tabeling, H. Willaime, Phys. Rev. Lett. 82, 3994 (1999)
T. Lundgren, Phys. Fluids 14, 638 (2002)
Y. Gagne, B. Castaing, C. Baudet, Y. Malécot, Phys. Fluids 16, 482 (2004)
N.R. Kevlahan, J. Alam, O. Vasilyev, J. Fluid Mech. 570, 217 (2007)
H. Tennekes, J. Lumley, A First Course in Turbulence (MIT Press, 1972)
I. Daubechies, Comm. Pure App. Math. 41, 909 (1988)
P. Abry, P. Flandrin, M. Taqqu, D. Veitch, Wavelets for the analysis, estimation and synthesis of scaling data, in Self-similar Network Traffic and Performance Evaluation (Wiley, 2000)
P. Mattila, Geometry of sets and measures in Euclidean spaces (Cambridge University Press, 1995)
E. Novikov, Prikl. Math. Mekh. 35, 266 (1970)
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Lashermes, B., Roux, S., Abry, P. et al. Comprehensive multifractal analysis of turbulent velocity using the wavelet leaders. Eur. Phys. J. B 61, 201–215 (2008). https://doi.org/10.1140/epjb/e2008-00058-4
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DOI: https://doi.org/10.1140/epjb/e2008-00058-4