measuring entropy rate fluctuations in compressible turbulent flow
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
Measuring Entropy Rate Fluctuations in Compressible Turbulent Flow
Mahesh M. BandiDepartment of Physics & Astronomy, University of Pittsburgh.
Walter I. GoldburgDepartment of Physics & Astronomy, University of Pittsburgh.
John R. Cressman Jr.Krasnow Institute, George Mason University.
Turbulence on a free surface.
)r,t(n ln )r,t(n rd)t(S
Surface Compressibility
Incompressible fluid (such as water): 0)t,r(v.3
2
u (r , t) ux (x, y,z 0, t)
xuy (x,y,z 0, t)
y
uz (x,y,z 0,t)z
0
Particles floating on the surface:
Experiment #1 0n dS/dt
u
untn
2
0)(
)r,t(n ln )r,t(n rd)t(S
Start with
Falkovich & Fouxon, New J Phys. 6, 11 (2004)
t
A
ttdrtrtnrddtdS )()(),(),(
AA
SdvrtnrtnrtrtnrddtdS .),(ln),(),(),(
local divergence
alternatively
At 8 pixels/cell, 10000 pixels
i
ii )t(nln)t(n)t(S
where ni(t) is the instantaneous concentration in ith cell,interpreted here as a probability for calculation ofthe instantaneous Entropy.
1 m
Work station
High speed video camera
Pump
laser
Dimensionless compressibility
2222
22
zzxyyxxx uuuu
uC
C = 0.5
Instantaneous Entropy <S(t)>
Results
Entropy production rate dS/dt in compressible turbulence.
Goal: Compare with dS/dt =1+2
2nd experimentFluctuations in dS/dt in lagrangian frame: Goal: Test Fluctuation Relation of Gallavotti and Cohen and others -in SS
AA
SdurtnrtnrtrtnrddtdS ),(ln),(),(),(
Area Term (<0)
- 1.8 Hz
- 0.76 Hz
Boundary Term
~200 ms
The term of interest
SS reached in ~ 200 ms
Results for dS/dt Simulations of Boffetta, Davoudi, Eckhardt, &Schumacher, PRL 2004
1 + 2 = -2.0 + 0.25 = -1.75 Hz
Hz 07.082.1- :),(),( dxdytrtrnA
Also from FF
d (t )(t) 0.290.02 Hz
From FF
?
Test for the Fluctuation Relation -lagrangian frame (FR)
Experiment #2
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 ω is a r.v. that can be pos & neg
Coagulation implies that mainly ω is negative
An equation concerning the entropy current dS/dt - in the lagrangian frame
Recall that Falkovich and Fouxon showed that
A
dxdytyxtyxnS ),,(),,(
Velocity divergence is thus a local entropy rate or entropy current
We measure the fluctuations in local entropy rate (in lagrangian frame) - dimensionless units σ
all x,y in A
For each initial r, one evaluates the divergence (r,t) of theturbulently moving floater.
This quantity fluctuates from on trajectory to another and from one instant t to another
Define a dimensionless time-averaged entropy rate
0.2s
uniform dist at t=0
t=0
1.8 s
Steady state
Trans. state
In the lagrangian frame
0),(1lim trdt
[Ω]=Hz []=dimensionless entropy rate or entropy current
Introduce a dimensionless time- averaged
For each track starting at r
0
),(1 trdt
τ > 80τc
(neg)
Dominantlynegative
ln
e
The Steady State Fluctuation Relation.
• The Result of Cohen and Gallavotti.
Ω is the average of entropy rate. It is negative (coagulation)= -0.37 Hz τ is a short time over which you average the system.
coag. more likely
coagulation
dispersal
saturation
Theory works Theory fails
Theory works
Th fails
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 τ
Summary of FR Expt