Experimental Measurements of the Multifractal Dimension of Lagrangian Turbulence

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Experimental Measurements of the Multifractal
Dimension of Lagrangian Turbulence
Haitao Xu, Nicholas T. Ouellette, and Eberhard
Bodenschatz
Max Planck Institute for Dynamics and
Self-Organization, Göttingen
Laboratory of Atomic and Solid State Physics,
Cornell University
International Collaboration for Turbulence
Research
August 28, 2006, Stirring Mixing, Leiden
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Thanks to...

• 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

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Outline

• Introduction/Motivation
• Experimets
• Results Comparison
• Summary Future Work

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Richardson/K41 cascade
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Intermittency
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Richardson/K41 cascade
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Multifractal cascade
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Multifractal description of intermittency

• Cascade need not be space filling
• -- fractal dimension
• Allow a multiplicity of self-similarity scaling
  exponents
• -- multifractalism
• Correct probablity by space filling factor

Eulerian
Lagrangian
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Multifractal prediction (structure function)
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Multifractal prediction (dissipation)
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Lagrangian turbulence?
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Lagrangian turbulence

• Borgas (1993)
• Made connections between Eulerian and Lagrangian
  multifractality for dissipation.

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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.

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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.

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Lagrangian turbulence
Lagrangian experiments? --gt Lagrangian Particle
Tracking.
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Experiment turbulence generator
von Kármán swirling flow b/w counter-rotating
disks
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Experiment schematic
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Experiment schematic
NdYAG pulsed lasers 532 nm, 90W, 27 kHz
Passive tracer particles
Phantom v7.1, up to 27,000 fps at 256x256
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Particle Tracking Algorithm

• Particle finding
• particle center on image plane
• Stereoscopic matching
• 2D image plane --gt 3D laboratory frame
• Tracking in time
• Lagrangian trajectories

Ouellette et al. (2006)
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Lagrangian Trajectories
measured Lagrangian trajectories
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How to measure?
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How to measure -- Probabilistic formulation

• Frisch (1995) Measure cumulative probability
  by integrating velocity increment PDF

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How to measure -- Probabilistic formulation
Frisch (1995)

• Integrate PDF about to find probability
  that increment scales with h
• Scaling

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Experimental Data
h increases
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Experimental Data
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Experimental Data
R? 200
R? 690
R? 815
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Comparison with models
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Relation to scaling exponents
Structure function scaling exponents are Legendre
transforms of DL(h)
Frisch (1995)
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Relation to scaling exponents
Direct
From DL(h)
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Measure from
Multifractal spectrum is the Legendre transform
of the structure finction scaling exponents
Frisch (1995)
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Measure from
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Measure from

• Structure function doesnt exist for orders
• Finite domain of -gt straight line region
  in

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Modify models?
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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.

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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.

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Lagrangian Intermittency
Comparing to Eulerian measurements, deviation to
K41 scaling is larger, suggesting stronger
intermittency effect in Lagrangian frame.
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Scaling exponents -- ESS
Same results obtained with or without using ESS
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Velocity increment PDF
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Statistical Convergence?