Measuring Entropy Rate Fluctuations in Compressible Turbulent Flow

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Measuring Entropy Rate Fluctuations in
Compressible Turbulent Flow
Mahesh M. Bandi Department of Physics
Astronomy, University of Pittsburgh. Walter I.
Goldburg Department of Physics Astronomy,
University of Pittsburgh. John R. Cressman
Jr. Krasnow Institute, George Mason University.
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Turbulence on a free surface.
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Surface Compressibility
Incompressible fluid (such as water)
Particles floating on the surface
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• Experiment 1 0n dS/dt

Start with
Falkovich Fouxon, New J Phys. 6, 11 (2004)
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alternatively
local divergence
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where ni(t) is the instantaneous concentration in
ith cell, interpreted here as a probability for
calculation of the instantaneous Entropy.
At 8 pixels/cell, 10000 pixels
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Pump
laser
1 m
Work station
High speed video camera
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Dimensionless compressibility
C 0.5
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Results
Instantaneous Entropy ltS(t)gt
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• Entropy production rate dS/dt in compressible
  turbulence.
• Goal Compare with dS/dt ?1?2
• 2nd experiment
• Fluctuations in dS/dt in lagrangian frame
• Goal Test Fluctuation Relation of
• Gallavotti and Cohen and others -in SS

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Area Term (lt0)
Boundary Term
The term of interest
SS reached in 200 ms
- 0.76 Hz
- 1.8 Hz
200 ms
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Results for dS/dt

• Simulations of Boffetta, Davoudi, Eckhardt,
  Schumacher, PRL 2004
• ?1 ?2 -2.0 0.25 -1.75 Hz

From FF
Also from FF
?
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Experiment 2
Test for the Fluctuation Relation -lagrangian
frame (FR)
Thermal Eq Fluctuations about the mean are
related to dissipation FDT (see any text on
Stat. Mech)
What about fluctuations for driven system in
steady state The local entropy rate ? is a
r.v. that can be pos neg
Coagulation implies that mainly ? is negative
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An equation concerning the entropy current dS/dt
- in the lagrangian frame Recall that Falkovich
and Fouxon showed that
all x,y in A

Velocity divergence is thus a local entropy rate
or entropy current
We measure the fluctuations in local entropy rate
(in lagrangian frame) - dimensionless units s
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In the lagrangian frame
For each initial r, one evaluates the divergence
?(r,t) of the turbulently moving floater. This
quantity fluctuates from on trajectory to another
and from one instant t to another Define a
dimensionless time-averaged entropy rate ??
t0
0.2s
Steady state
Trans. state
1.8 s
uniform dist at t0
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Introduce a dimensionless time- averaged ?
t gt 80tc
For each track starting at r
Dominantly negative
(neg)
?Hz ??dimensionless entropy rate
or entropy current
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• The Steady State Fluctuation Relation.
• The Result of Cohen and Gallavotti.

coag. more likely
O is the average of entropy rate. It is negative
(coagulation) -0.37 Hz t is a short time over
which you average the system.
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coagulation
dispersal
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Th fails
Theory works
saturation
Theory fails
Theory works
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Summary of FR Expt

• Turbulent flow is a special case of chaotic
  dynamics
• -skip NSE
• Prob of coag only slightly exceeds prob of
  dispersal
• The FR (steady state) holds macroscopic systems
• (e.g. turbulent compressible flow) - limited
  range of t