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 and PHY  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 Eulerian Lagrangian 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 lasers 532 nm, ~90W, 27 kHz Passive tracer particles Phantom v7.1, up to 27,000 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 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: h increases h increases

Experimental Data:

R = 200 R = 690 R = 815

Comparison with models

Relation to scaling exponents Structure function scaling exponents are Legendre transforms of D L (h): Frisch (1995)

Relation to scaling exponents Direct From D L (h)

Measure from Multifractal spectrum is the Legendre transform of the structure finction scaling exponents: Frisch (1995)

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?