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

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...

Outline

• Introduction/Motivation

• Experimets

• Results & Comparison

• Summary & Future Work

Richardson/K41 cascade

Intermittency

Richardson/K41 cascade

Multifractal cascade

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:

Multifractal prediction (structure function)

Multifractal prediction (dissipation)

Lagrangian turbulence?

Lagrangian turbulence

• Borgas (1993):Made connections between Eulerian and Lagrangian

multifractality for dissipation.

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.

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.

Lagrangian turbulence

Lagrangian experiments? -->Lagrangian Particle Tracking.

von Kármán swirling flow b/w counter-rotating disks

• Inertial forcing

• L = 7cm

• up to 1000

• Measurement

volume ~(2cm)3

Experiment: turbulence generator

Experiment schematic

Nd:YAG pulsed lasers532 nm, ~90W, 27 kHz

Passive tracer particles

Phantom v7.1, up to27,000 fps at 256x256

Experiment schematic

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)

Lagrangian Trajectories

: measured Lagrangian trajectories

How to measure?

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

● 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):

Experimental Data:

hincreases

hincreases

Experimental Data:

Experimental Data:

R = 200

R = 690

R = 815

Comparison with models

Relation to scaling exponents

Structure function scaling exponents are Legendre transforms of DL(h):

Frisch (1995)

Relation to scaling exponents

Direct

From DL(h)

Measure from

Multifractal spectrum is the Legendre transform of the structure finction scaling exponents:

Frisch (1995)

Measure from

Measure from

• Structure function doesn’t exist for orders

• Finite domain of -> straight line region in

Modify models?

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.

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.

Comparing to Eulerian measurements, deviation to K41 scaling is larger, suggesting stronger intermittency effect in Lagrangian frame.

Lagrangian Intermittency

Same results obtained with or without using ESS

Scaling exponents -- ESS

Velocity increment PDF

Statistical Convergence?