Post on 21-Jan-2016
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?