Entropy Current in Compressible Turbulence

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Entropy Current in Compressible Turbulence
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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One need not start with the Navier-Stokes
equation The alternative Treat the Turbulence
as a general Dynamical System
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Lorenz Attractor
http//www.google.com/search?hlenieISO-8859-1q
chaos3dimensionsdiagrampdf
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Pump
laser
1 m
Work station
High speed video camera
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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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• Two experiments
• Entropy production rate dS/dt in compressible
  turbulence.
• Goal Compare with dS/dt ?1?2
• Fluctuations in dS/dt in lagrangian frame
• Goal Test Fluctuation Relation of
• Gallavotti and Cohen and others -in SS

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• Experiment 1 0n dS/dt

Start with
And derive
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Theory of Falkovich Fouxon.

• where

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If system is highly chaotic (obeys Sinai-Ruelle
Bowen Statistics, SRB)
Re?t
Local dS/dt
These simulated
Boffeta et al. PRL 93 134501 (2004)
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Also
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Results
Instantaneous Entropy ltS(t)gt
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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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Lagrangian Velocity Divergence Correlation
Function
Measured Area -0.3 Hz.
tc 20 ms
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Three expressions for dS/dt

• simulations of Boffetta et al.
• ?1 ?2 -2.0 0.25 -1.75 Hz

From FF
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.
• 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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pos st gt neg ? (O -0.37 Hz lt0)
?(?) gt ?(-?)
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Summary

• Turbulent flow is a special case of chaotic
  dynamics
• -skip NSE
• The FR (steady state) holds macroscopic systems
• (e.g. turbulent compressible flow) - limited
  range of t
• In this turbulence expt. ?? takes on both signs
• with almost equal probability but more likely
  positive
• (divergence more likely lt0) - coagulation

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Theory of Falkovich Fouxon.

• Lagrangian velocity divergence
  correlation function
• G. Falkovich A. Fouxon, New J. Phys, 6 (2004).
• G. Falkovich A. Fouxon nlin/CD0312033 (2003).

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• Previous Experiments Ciliberto et. al. 2004.
• Force Fluctuations of an obstacle in Turbulent
  Flow.
• S. Ciliberto et. al., Physica A340, 240 (2004).

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r
v(xr)
v(x)
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• -1.8 Hz -1.77 Hz
  -1.75 Hz
• -2.96 Hz -1.8 Hz -0.76 Hz
• Results are in good agreement with theory of
  Falkovich Fouxon.

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Along the many particle paths evaluate the
time-average entropy rate In the transinet state
or transient state
?, ?? random variables make a histogram
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Theory of Falkovich Fouxon.

• If system is very chaotic (follows
  Sinai-Ruelle-Bowen statistics)
• where are systems Lyapunov exponents.
• G. Falkovich A. Fouxon, New J. Phys, 6 (2004).
• G. Falkovich A. Fouxon nlin/CD0312033 (2003).

In this experiment RHS ?1 ?2
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Notation and Definitions. Global Entropy
Rate Local Entropy Rate (Lagrangian
frame)
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Turbulence on a free surface.
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Lyapunov Exponents from simulations (similar flow
conditions). Simulations by G. Boffetta, J.
Davoudi, B. Eckhardt J. Schumacher. PRL, 93
134501 (2004)
From simulations ?1 ?2 -1.75 Hz
Expt from SRB statistics -0.3 Hz
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• Background.
• The Classical Fluctuation Dissipation Theorem.
• - External forces and random fluctuations.
• - System at Thermal Equilibrium.
• - Tiny fluctuations, short correlations
  Gaussian Distributions.
• What about fluctuations in deep Non-equilibrium
  state?
• - Recall Non-equilibrium Steady State
• Rate of Energy Injection Rate of Dissipation.
• - Wild fluctuations ? Non-Gaussian
  Distributions with fat tails.
• Comparison of Phase Space Dynamics.

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• The Steady State Fluctuation Relation.
• Consider fluctuations in Entropy Production Rate
  s

Time
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How do particles floating on surface differ from
particles in the bulk flow?

• The floaters are confined to the surface if a
  water molecule ducks down (vz gt0), a floating
  particle will not follow that motion.
• Floaters share energy with the bulk no energy
  conservation for them.
• At surface dimensional arguments involving ?
  (J/Kg-s) fail at the surface