Measuring Entropy Rate Fluctuations in Compressible Turbulent Flow

1. 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.

2. Turbulence on a free surface.

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

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

5. alternatively local divergence

6. 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

7. Pump laser 1 m Work station High speed video camera

9. Dimensionless compressibility C = 0.5

10. Results Instantaneous Entropy <S(t)>

12. Entropy production rate dS/dt in compressible turbulence. Goal: Compare with dS/dt =1+2 2nd experiment Fluctuations in dS/dt inlagrangian frame: Goal: Test Fluctuation Relation of Gallavotti and Cohen and others -in SS

14. Area Term (<0) Boundary Term The term of interest SS reached in ~ 200 ms - 0.76 Hz - 1.8 Hz ~200 ms

15. 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 ?

16. 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 rateω is a r.v. that can be pos & neg Coagulation implies that mainly ω is negative

17. 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 σ

18. 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  t=0 0.2s Steady state Trans. state 1.8 s uniform dist at t=0

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

20. The Steady State Fluctuation Relation. • The Result of Cohen and Gallavotti. coag. more likely Ω is the average of entropy rate. It is negative (coagulation)= -0.37 Hz τis a short time over which you average the system.

21. coagulation dispersal

22. Th fails Theory works saturation Theory fails Theory works

23. 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 τ