Rupture of an Amphiphile on airwater interface.

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PV2/R
R
V
2
Fma (N-S Eq.)
acceleration
force/Kg
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The turbulent cascade in the right and in the
wrong direction
Walter Goldburg (goldburg_at_pitt.edu)

• Mahesh M. Bandi Pitt
• John R. Cressman Jr GMU
• Alain Pumir, CNRS Nice.

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• Often one measures the time average of
  fluctuations in velocity, energy dissipation,
  etc. at a point in the flow
• Here we measure the spatial average of a property
  of the flow, the average being over the the scale
  of the largest eddies, lo
• The trick new technology

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Homogeneous isotropic turbulence
r ?
r l0
r
e
e
e
e
No energy dissipation in this inertial range
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Integral scale l0 and dissipative scale ?
Ref U. Frisch, Turbulence, (Cambridge 1995)
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Experimental Parameters.
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the set-up
Pump
laser
1 m
Work station
High speed video camera
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r
v(xr)
v(x)
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• Because the energy flux is just about the only
  variable for which there is an exact calculation
  of its average value.
• In 1941 A. N. Kolmogorov derived the4/5 law.

Assumptions isotropy homogeniety (l0 infinite)
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S3,being a random variable, fluctuates in time.
. For every r
We measure its PDF - averaged over all eddy sizes
r in the energy cascade range. Thus the
measurements represent a global average (over a
fraction of the total volume of the turbulent
fluid) made at many instants of time.
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Visual Capture. Velocity Field
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• Tabelings Measurements of the third-moment.
• Zocchi et. al. PRE 50, 3693, (1994) Tabeling
  et. al. PRE 53, 1613 (1996).
• Liquid Helium driven by counter-rotating disks.
  Re? 174 5040.

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• Sreenivasans Measurements of the third-moment.
• Sreenivasan et. al. http//arxiv.org/abs/chao-dyn
  /9906041 (1999)
• Atmospheric turbulence measurements using hot
  wire probe 35 m above ground.

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Experimental Procedure.
Movie capture with CCD Camera. 5 second movie
broken into 2040 TIFF images.
Velocity field reconstruction. Input Pair of
TIFF Images. Output Velocity field.
Calculation of Moments. S3(r)
Spatial averaging PDF -e(t) S3(r)/(4/5)r
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Verifying the four-fifths law.
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How wild are the third-moment fluctuations?
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PDF of Spatially Averaged Energy Transfer Rate.
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PDF of Spatially Averaged Energy Transfer Rate.
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How often does the Energy Cascade reverse
direction?
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• Simulation (No theory).
• Enter Smooth Particle Hydrodynamics.
• - ?t, a dimensionless constant, not to be
  confused with kinematic viscosity.
• - vij, velocity difference between two points
  separated by distance rij.
• - h, the smoothing distance over which all
  fluctuations are averaged out.
• Pumir Shraiman. Lagrangian particle approach
  to large eddy simulations of hydrodynamic
  turbulence, J. Stat. Phys. 113, 693 (2003).

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When computed over entire volume of the
Simulation Box.
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When computed over a sub-domain (1/4th size of
Simulation Box).
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• Summary.
• A study the PDF of spatially-averaged
  fluctuations in the flux of energy through a
  turbulent 3D fluid
• The energy-cascade frequently reverses its
  direction (25 of the time)
• the energy in small scales is momentarily being
  returned to large spatial scales.
• Numerical Simulations also demonstrate momentary
  energy-cascade reversals.
• A theory exists for the average flux large to
  small scales but not for flux PDF.

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Talk ends with previous slide
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• Outline.
• Motivation
• Past Measurements
• Whats different about our Experiment?
• Experimental Procedure.
• Description of the Experimental Setup.
• The spatially and temporally averaged
  third-moment.
• Time trace and PDF of energy transfer rate.

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The Energy Cascade Process. Direction of energy
cascade.
Kolmogorovs four-fifths law
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• the average velocity at a point
• the nth moment of velocity differences between
  two closely spaced points

• The mean rate at which energy is dissipated in
• J / Kg s

Kinematic viscosity
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• Dan Lathrops measurements in 3D.
• Zeff, B. et. al. Nature 421, 146 (2003).

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Visual Capture. CCD Movie
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• Re-expressing the Third-moment.
• The spatially averaged steady-state Energy
  Transfer Rate.
• It has also been re-derived by Andersen et. al.
  to hold for Lagrangian measurements.
• Note e is positive, making S3(r) negative
• Mann, J., Ott, S. Andersen, J. S. Experimental
  study of relative, turbulent diffusion.

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• In the study of turbulence, one usually measures
  the average value of various quantities, A.
• These quantities are random variables and hence
  have a PDF associated with them
• Here we focus on the flux of energy through a
  turbulent system. Why?

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S.T. Bramwell et al. Nature, 396, 552 (1998)
P.Tabeling et al., Phys. Rev. E .53, 1613 (1996)
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• The third-moment in all past measurements was
  obtained from time-averages.
• Provides mean energy transfer rate of fluid, but
  says nothing about fluctuations.
• We are interested in fluctuations about the mean
  Energy Flux.
• Can measure third-moment at various spatial
  scales at every instant of time.