Haitao Xu, Nicholas T. Ouellette, and Eberhard Bodenschatz August 28, 2006, Stirring & Mixing,...
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![Page 1: Haitao Xu, Nicholas T. Ouellette, and Eberhard Bodenschatz August 28, 2006, Stirring & Mixing, Leiden Experimental Measurements of the Multifractal Dimension.](https://reader035.fdocuments.us/reader035/viewer/2022062322/5697c0221a28abf838cd37fe/html5/thumbnails/1.jpg)
Haitao Xu, Nicholas T. Ouellette, and Eberhard Bodenschatz
August 28, 2006, Stirring & Mixing, Leiden
Experimental Measurements of the Multifractal Dimension of Lagrangian
Turbulence
International Collaboration for Turbulence Research
Laboratory of Atomic and Solid State Physics, Cornell University
Max Planck Institute for Dynamics and Self-Organization, Göttingen
![Page 2: Haitao Xu, Nicholas T. Ouellette, and Eberhard Bodenschatz August 28, 2006, Stirring & Mixing, Leiden Experimental Measurements of the Multifractal Dimension.](https://reader035.fdocuments.us/reader035/viewer/2022062322/5697c0221a28abf838cd37fe/html5/thumbnails/2.jpg)
M. Bourgoin, J. Berg, L. Collins, Z. Warhaft, S. Ayyalasomayajula, A. Gylfason
U.S. National Science Foundation, grants PHY-9988755 and PHY-0216406
Max Planck Society, Germany
Thanks to...
![Page 3: Haitao Xu, Nicholas T. Ouellette, and Eberhard Bodenschatz August 28, 2006, Stirring & Mixing, Leiden Experimental Measurements of the Multifractal Dimension.](https://reader035.fdocuments.us/reader035/viewer/2022062322/5697c0221a28abf838cd37fe/html5/thumbnails/3.jpg)
Outline
• Introduction/Motivation
• Experimets
• Results & Comparison
• Summary & Future Work
![Page 4: Haitao Xu, Nicholas T. Ouellette, and Eberhard Bodenschatz August 28, 2006, Stirring & Mixing, Leiden Experimental Measurements of the Multifractal Dimension.](https://reader035.fdocuments.us/reader035/viewer/2022062322/5697c0221a28abf838cd37fe/html5/thumbnails/4.jpg)
Richardson/K41 cascade
![Page 5: Haitao Xu, Nicholas T. Ouellette, and Eberhard Bodenschatz August 28, 2006, Stirring & Mixing, Leiden Experimental Measurements of the Multifractal Dimension.](https://reader035.fdocuments.us/reader035/viewer/2022062322/5697c0221a28abf838cd37fe/html5/thumbnails/5.jpg)
Intermittency
![Page 6: Haitao Xu, Nicholas T. Ouellette, and Eberhard Bodenschatz August 28, 2006, Stirring & Mixing, Leiden Experimental Measurements of the Multifractal Dimension.](https://reader035.fdocuments.us/reader035/viewer/2022062322/5697c0221a28abf838cd37fe/html5/thumbnails/6.jpg)
Richardson/K41 cascade
![Page 7: Haitao Xu, Nicholas T. Ouellette, and Eberhard Bodenschatz August 28, 2006, Stirring & Mixing, Leiden Experimental Measurements of the Multifractal Dimension.](https://reader035.fdocuments.us/reader035/viewer/2022062322/5697c0221a28abf838cd37fe/html5/thumbnails/7.jpg)
Multifractal cascade
![Page 8: Haitao Xu, Nicholas T. Ouellette, and Eberhard Bodenschatz August 28, 2006, Stirring & Mixing, Leiden Experimental Measurements of the Multifractal Dimension.](https://reader035.fdocuments.us/reader035/viewer/2022062322/5697c0221a28abf838cd37fe/html5/thumbnails/8.jpg)
Multifractal description of intermittency
EulerianEulerian LagrangianLagrangian
• Cascade need not be space filling
-- fractal dimension;
• Allow a multiplicity of self-similarity scaling exponents
-- multifractalism;
• Correct probablity by space filling factor:
![Page 9: Haitao Xu, Nicholas T. Ouellette, and Eberhard Bodenschatz August 28, 2006, Stirring & Mixing, Leiden Experimental Measurements of the Multifractal Dimension.](https://reader035.fdocuments.us/reader035/viewer/2022062322/5697c0221a28abf838cd37fe/html5/thumbnails/9.jpg)
Multifractal prediction (structure function)
![Page 10: Haitao Xu, Nicholas T. Ouellette, and Eberhard Bodenschatz August 28, 2006, Stirring & Mixing, Leiden Experimental Measurements of the Multifractal Dimension.](https://reader035.fdocuments.us/reader035/viewer/2022062322/5697c0221a28abf838cd37fe/html5/thumbnails/10.jpg)
Multifractal prediction (dissipation)
![Page 11: Haitao Xu, Nicholas T. Ouellette, and Eberhard Bodenschatz August 28, 2006, Stirring & Mixing, Leiden Experimental Measurements of the Multifractal Dimension.](https://reader035.fdocuments.us/reader035/viewer/2022062322/5697c0221a28abf838cd37fe/html5/thumbnails/11.jpg)
Lagrangian turbulence?
![Page 12: Haitao Xu, Nicholas T. Ouellette, and Eberhard Bodenschatz August 28, 2006, Stirring & Mixing, Leiden Experimental Measurements of the Multifractal Dimension.](https://reader035.fdocuments.us/reader035/viewer/2022062322/5697c0221a28abf838cd37fe/html5/thumbnails/12.jpg)
Lagrangian turbulence
• Borgas (1993):Made connections between Eulerian and Lagrangian
multifractality for dissipation.
![Page 13: Haitao Xu, Nicholas T. Ouellette, and Eberhard Bodenschatz August 28, 2006, Stirring & Mixing, Leiden Experimental Measurements of the Multifractal Dimension.](https://reader035.fdocuments.us/reader035/viewer/2022062322/5697c0221a28abf838cd37fe/html5/thumbnails/13.jpg)
Lagrangian turbulence
• Borgas (1993):Made connections between Eulerian and Lagrangian
multifractality for dissipation.
• Chevillard et al. (2003):Proposed model for Lagrangian multifractal spectrum
for velocity increments;
Mapped Eulerian multifractal spectrum to Lagrangian.
![Page 14: Haitao Xu, Nicholas T. Ouellette, and Eberhard Bodenschatz August 28, 2006, Stirring & Mixing, Leiden Experimental Measurements of the Multifractal Dimension.](https://reader035.fdocuments.us/reader035/viewer/2022062322/5697c0221a28abf838cd37fe/html5/thumbnails/14.jpg)
Lagrangian turbulence
• Borgas (1993):Made connections between Eulerian and Lagrangian
multifractality for dissipation.
• Chevillard et al. (2003):Proposed model for Lagrangian multifractal spectrum
for velocity increments;
Mapped Eulerian multifractal spectrum to Lagrangian.
• Biferale et al. (2004):Predicted scaling exponents of Lagrangian velocity
structure functions based on multifractalism and
compared with DNS.
![Page 15: Haitao Xu, Nicholas T. Ouellette, and Eberhard Bodenschatz August 28, 2006, Stirring & Mixing, Leiden Experimental Measurements of the Multifractal Dimension.](https://reader035.fdocuments.us/reader035/viewer/2022062322/5697c0221a28abf838cd37fe/html5/thumbnails/15.jpg)
Lagrangian turbulence
Lagrangian experiments? -->Lagrangian Particle Tracking.
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von Kármán swirling flow b/w counter-rotating disks
• Inertial forcing
• L = 7cm
• up to 1000
• Measurement
volume ~(2cm)3
Experiment: turbulence generator
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Experiment schematic
![Page 18: Haitao Xu, Nicholas T. Ouellette, and Eberhard Bodenschatz August 28, 2006, Stirring & Mixing, Leiden Experimental Measurements of the Multifractal Dimension.](https://reader035.fdocuments.us/reader035/viewer/2022062322/5697c0221a28abf838cd37fe/html5/thumbnails/18.jpg)
Nd:YAG pulsed lasers532 nm, ~90W, 27 kHz
Passive tracer particles
Phantom v7.1, up to27,000 fps at 256x256
Experiment schematic
![Page 19: Haitao Xu, Nicholas T. Ouellette, and Eberhard Bodenschatz August 28, 2006, Stirring & Mixing, Leiden Experimental Measurements of the Multifractal Dimension.](https://reader035.fdocuments.us/reader035/viewer/2022062322/5697c0221a28abf838cd37fe/html5/thumbnails/19.jpg)
Particle Tracking Algorithm
1. Particle finding:
particle center on image plane
2. Stereoscopic matching:
2D image plane --> 3D laboratory frame
3. Tracking in time:
Lagrangian trajectories
Ouellette et al. (2006)
![Page 20: Haitao Xu, Nicholas T. Ouellette, and Eberhard Bodenschatz August 28, 2006, Stirring & Mixing, Leiden Experimental Measurements of the Multifractal Dimension.](https://reader035.fdocuments.us/reader035/viewer/2022062322/5697c0221a28abf838cd37fe/html5/thumbnails/20.jpg)
Lagrangian Trajectories
: measured Lagrangian trajectories
![Page 21: Haitao Xu, Nicholas T. Ouellette, and Eberhard Bodenschatz August 28, 2006, Stirring & Mixing, Leiden Experimental Measurements of the Multifractal Dimension.](https://reader035.fdocuments.us/reader035/viewer/2022062322/5697c0221a28abf838cd37fe/html5/thumbnails/21.jpg)
How to measure?
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Frisch (1995): Measure cumulative probability by integrating velocity increment PDF
Frisch (1995): Measure cumulative probability by integrating velocity increment PDF
How to measure -- Probabilistic formulation
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● Integrate PDF about to find
probability that increment scales with h
● Scaling:
● Integrate PDF about to find
probability that increment scales with h
● Scaling:
How to measure -- Probabilistic formulation
Frisch (1995):
![Page 24: Haitao Xu, Nicholas T. Ouellette, and Eberhard Bodenschatz August 28, 2006, Stirring & Mixing, Leiden Experimental Measurements of the Multifractal Dimension.](https://reader035.fdocuments.us/reader035/viewer/2022062322/5697c0221a28abf838cd37fe/html5/thumbnails/24.jpg)
Experimental Data:
hincreases
hincreases
![Page 25: Haitao Xu, Nicholas T. Ouellette, and Eberhard Bodenschatz August 28, 2006, Stirring & Mixing, Leiden Experimental Measurements of the Multifractal Dimension.](https://reader035.fdocuments.us/reader035/viewer/2022062322/5697c0221a28abf838cd37fe/html5/thumbnails/25.jpg)
Experimental Data:
![Page 26: Haitao Xu, Nicholas T. Ouellette, and Eberhard Bodenschatz August 28, 2006, Stirring & Mixing, Leiden Experimental Measurements of the Multifractal Dimension.](https://reader035.fdocuments.us/reader035/viewer/2022062322/5697c0221a28abf838cd37fe/html5/thumbnails/26.jpg)
Experimental Data:
R = 200
R = 690
R = 815
![Page 27: Haitao Xu, Nicholas T. Ouellette, and Eberhard Bodenschatz August 28, 2006, Stirring & Mixing, Leiden Experimental Measurements of the Multifractal Dimension.](https://reader035.fdocuments.us/reader035/viewer/2022062322/5697c0221a28abf838cd37fe/html5/thumbnails/27.jpg)
Comparison with models
![Page 28: Haitao Xu, Nicholas T. Ouellette, and Eberhard Bodenschatz August 28, 2006, Stirring & Mixing, Leiden Experimental Measurements of the Multifractal Dimension.](https://reader035.fdocuments.us/reader035/viewer/2022062322/5697c0221a28abf838cd37fe/html5/thumbnails/28.jpg)
Relation to scaling exponents
Structure function scaling exponents are Legendre transforms of DL(h):
Frisch (1995)
![Page 29: Haitao Xu, Nicholas T. Ouellette, and Eberhard Bodenschatz August 28, 2006, Stirring & Mixing, Leiden Experimental Measurements of the Multifractal Dimension.](https://reader035.fdocuments.us/reader035/viewer/2022062322/5697c0221a28abf838cd37fe/html5/thumbnails/29.jpg)
Relation to scaling exponents
Direct
From DL(h)
![Page 30: Haitao Xu, Nicholas T. Ouellette, and Eberhard Bodenschatz August 28, 2006, Stirring & Mixing, Leiden Experimental Measurements of the Multifractal Dimension.](https://reader035.fdocuments.us/reader035/viewer/2022062322/5697c0221a28abf838cd37fe/html5/thumbnails/30.jpg)
Measure from
Multifractal spectrum is the Legendre transform of the structure finction scaling exponents:
Frisch (1995)
![Page 31: Haitao Xu, Nicholas T. Ouellette, and Eberhard Bodenschatz August 28, 2006, Stirring & Mixing, Leiden Experimental Measurements of the Multifractal Dimension.](https://reader035.fdocuments.us/reader035/viewer/2022062322/5697c0221a28abf838cd37fe/html5/thumbnails/31.jpg)
Measure from
![Page 32: Haitao Xu, Nicholas T. Ouellette, and Eberhard Bodenschatz August 28, 2006, Stirring & Mixing, Leiden Experimental Measurements of the Multifractal Dimension.](https://reader035.fdocuments.us/reader035/viewer/2022062322/5697c0221a28abf838cd37fe/html5/thumbnails/32.jpg)
Measure from
• Structure function doesn’t exist for orders
• Finite domain of -> straight line region in
![Page 33: Haitao Xu, Nicholas T. Ouellette, and Eberhard Bodenschatz August 28, 2006, Stirring & Mixing, Leiden Experimental Measurements of the Multifractal Dimension.](https://reader035.fdocuments.us/reader035/viewer/2022062322/5697c0221a28abf838cd37fe/html5/thumbnails/33.jpg)
Modify models?
![Page 34: Haitao Xu, Nicholas T. Ouellette, and Eberhard Bodenschatz August 28, 2006, Stirring & Mixing, Leiden Experimental Measurements of the Multifractal Dimension.](https://reader035.fdocuments.us/reader035/viewer/2022062322/5697c0221a28abf838cd37fe/html5/thumbnails/34.jpg)
Summary
• Measured Lagrangian multifractal
dimension spectrum from experiment data.
• Confirmed Legendre transform relation
between multifractal dimension spectrum
and structre function scaling exponents.
• Experiment data agree with modified
existing multifractal models.
![Page 35: Haitao Xu, Nicholas T. Ouellette, and Eberhard Bodenschatz August 28, 2006, Stirring & Mixing, Leiden Experimental Measurements of the Multifractal Dimension.](https://reader035.fdocuments.us/reader035/viewer/2022062322/5697c0221a28abf838cd37fe/html5/thumbnails/35.jpg)
Future work
• Measure the right side of spectrum from dissipation range and compare with models.
• Measure Eulerian multifractal dimension spectrum and test the mapping between Eulerian and Lagrangian framework.
• Measure dissipation dimension spectrum and test its relation to the multifractal dimension spectrum measured from velocity.
![Page 36: Haitao Xu, Nicholas T. Ouellette, and Eberhard Bodenschatz August 28, 2006, Stirring & Mixing, Leiden Experimental Measurements of the Multifractal Dimension.](https://reader035.fdocuments.us/reader035/viewer/2022062322/5697c0221a28abf838cd37fe/html5/thumbnails/36.jpg)
Comparing to Eulerian measurements, deviation to K41 scaling is larger, suggesting stronger intermittency effect in Lagrangian frame.
Lagrangian Intermittency
![Page 37: Haitao Xu, Nicholas T. Ouellette, and Eberhard Bodenschatz August 28, 2006, Stirring & Mixing, Leiden Experimental Measurements of the Multifractal Dimension.](https://reader035.fdocuments.us/reader035/viewer/2022062322/5697c0221a28abf838cd37fe/html5/thumbnails/37.jpg)
Same results obtained with or without using ESS
Scaling exponents -- ESS
![Page 38: Haitao Xu, Nicholas T. Ouellette, and Eberhard Bodenschatz August 28, 2006, Stirring & Mixing, Leiden Experimental Measurements of the Multifractal Dimension.](https://reader035.fdocuments.us/reader035/viewer/2022062322/5697c0221a28abf838cd37fe/html5/thumbnails/38.jpg)
Velocity increment PDF
![Page 39: Haitao Xu, Nicholas T. Ouellette, and Eberhard Bodenschatz August 28, 2006, Stirring & Mixing, Leiden Experimental Measurements of the Multifractal Dimension.](https://reader035.fdocuments.us/reader035/viewer/2022062322/5697c0221a28abf838cd37fe/html5/thumbnails/39.jpg)
Statistical Convergence?