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)

z0

Particles floating on the surface:

Experiment #1 0n dS/dt

u

unt

n

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