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.

Turbulence on a free surface.

Surface Compressibility Incompressible fluid (such as water): Particles floating on the surface:

Experiment #1 0n dS/dt Start with Falkovich & Fouxon, New J Phys. 6, 11 (2004)

local divergence alternatively

At 8 pixels/cell, pixels where n i (t) is the instantaneous concentration in i th cell, interpreted here as a probability for calculation of the instantaneous Entropy.

1 m Work station High speed video camera Pump laser

Dimensionless compressibility C = 0.5

Instantaneous Entropy Results

Entropy production rate dS/dt in compressible turbulence. Goal: Compare with dS/dt = nd experiment Fluctuations in dS/dt in lagrangian frame: Goal: Test Fluctuation Relation of Gallavotti and Cohen and others -in SS

Area Term (<0) Hz 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 = = Hz Also from FF 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 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 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   0.2s uniform dist at t=0 t=0 1.8 s Steady state Trans. state In the lagrangian frame

[Ω]=Hz [   ]=dimensionless entropy rate or entropy current Introduce a dimensionless time- averaged  For each track starting at r τ > 80τ c (neg) Dominantly negative

The Steady State Fluctuation Relation. The Result of Cohen and Gallavotti. Ω is the average of entropy rate. It is negative (coagulation)= 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