Multifractal Dimension of Lagrangian Turbulence

Haitao Xu, Nicholas T. Ouellette, and Eberhard Bodenschatz (International Collaboration for Turbulence Research)

Phys. Rev. Lett. 96, 114503 – Published 24 March 2006

Abstract

We report experimental measurements of the Lagrangian multifractal dimension spectrum in an intensely turbulent laboratory water flow by the optical tracking of tracer particles. The Legendre transform of the measured spectrum is compared with measurements of the scaling exponents of the Lagrangian velocity structure functions, and excellent agreement between the two measurements is found, in support of the multifractal picture of turbulence. These measurements are compared with three model dimension spectra. When the nonexistence of structure functions of order less than $- 1$ is accounted for, the models are shown to agree well with the measured spectrum.

Received 5 December 2005

DOI:https://doi.org/10.1103/PhysRevLett.96.114503

Authors & Affiliations

Haitao Xu^1,2, Nicholas T. Ouellette¹, and Eberhard Bodenschatz^1,2,* (International Collaboration for Turbulence Research)

¹Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853, USA
²Max Planck Institute for Dynamics and Self-Organization, 37077 Göttingen, Germany

^*Electronic address: eberhard.bodenschatz@ds.mpg.de

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Vol. 96, Iss. 11 — 24 March 2006

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Images

Figure 1
Sketch of the experimental facility. The trajectories of tracer particles were recorded by three high-speed digital cameras in a $2.5 \times 2.5 \times 2.5 {cm}^{3}$ subvolume in the middle of the cylindrical tank. The tracers were illuminated using two pulsed $Nd ∶ YAG$ lasers with intensities of roughly 60 and 90 W. Three Phantom v 7.1 CMOS cameras manufactured by Vision Research, Inc. were arranged in a single plane in the forward scattering direction from both beams with an angular separation of roughly 45°.
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Figure 2
Probabilities ${\bar{P}}_{h} (τ)$ plotted as a function of $τ / T_{L}$ for seven values of $h$ in logarithmic coordinates. The curves have been offset for clarity. From top to bottom, the data correspond to $h$ values of 0.12, 0.25, 0.50, 0.75, 1.00, 1.25, and 1.50. The vertical lines show the region in which we fit power laws to the data. The exponents of these power laws are given by $1 - D^{L} (h)$ .
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Figure 3
Direct measurement of the Lagrangian multifractal dimension spectrum. The symbols denote our experimental measurements at three different Reynolds numbers: the ■ correspond to $R_{λ} = 200$ , the ● to $R_{λ} = 690$ , and the ▲ to $R_{λ} = 815$ . The measured multifractal dimension spectra agree well for all three Reynolds numbers, suggesting that $D^{L} (h)$ has at most a weak Reynolds number dependence. The three curves correspond to models: the dashed line is the model due to Chevillard et al. [9], the solid line is Kolmogorov’s log-normal model [13], and the dot-dashed line is the log-Poisson model of She and Lévêque [7].
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Figure 4
Scaling exponents $ζ_{p}^{L}$ of the Lagrangian structure functions as a function of order. The ● denote direct measurements of the $ζ^{L} (p)$ at $R_{λ} = 690$ . The ■ show the exponents extracted from our measured $D^{L} (h)$ data via Eq. (4). The two experimental measurements agree very well with each other. The curves are models: the dashed line is again the model of Chevillard et al. [9], the solid curved line is Kolmogorov’s log-normal model [13], and the dot-dashed line is the model of She and Lévêque [7]. The solid straight line shows Kolmogorov’s 1941 prediction for the $ζ_{p}^{L}$ [12].
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Figure 5
Comparison of the direct and indirect measurements of the multifractal dimension spectrum. The ● show again the direct measurement of the multifractal dimension spectrum at $R_{λ} = 690$ , while the ▲ show the inverse Legendre transform of the measured $ζ_{p}^{L}$ data. The two experimental curves again are consistent with each other. The lines are the same three models shown in Fig. 3 but modified so that past $h^{*}$ , defined as the point where $p = - 1$ minimizes $h p + 1 - ζ_{p}^{L}$ , the spectrum is given by $D^{L} (h) = - h + 1 - ζ_{- 1}^{L}$ . The agreement between the modified model spectra and our measured spectra is excellent.
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