1. Statistical Fluctuations of Two-dimensional Turbulence Mike Rivera and Yonggun Jun Department of Physics & Astronomy University of Pittsburgh

2. Table of Contents Introduction Introduction Experimental Setup Experimental Setup Experimental Results Experimental Results Average Behavior Average Behavior Fluctuations Fluctuations Comparison with 3D Results Comparison with 3D Results Conclusion Conclusion Soft-Condensed Matter Physics Group

3. What is Turbulence? Soft-Condensed Matter Physics Group Turbulence: irregularly fluctuating and unpredictable motion which is made up of a number of small eddies that travel in the fluid. Eddy: volume where the fluid move coherently. Leonardo da Vinci

4. Evolution to Turbulence At low Reynolds numbers, the flow past the rod is regular. As Reynolds number increases, the size of traveling vortices also increases. Finally, the flow becomes irregular. Soft-Condensed Matter Physics Group Re=UL/ U: typical velocity L: typical length  : viscosity Re>50

5. Freely Suspended Film is 2D Soft-Condensed Matter Physics Group *  Non-equilibrium Films: 1<h<100  m h/L ~ 10 -4 - 10 -3 L 15 o A h

6. Flows in Earth Atmosphere is 2D Soft-Condensed Matter Physics Group

7. Examples of 2D Turbulence Jupiter Great red spot Hurricane Soft-Condensed Matter Physics Group

8. Forced 2D Turbulence 7 cm vyvy - Applied voltage : f = 1 Hz. - Taylor microscale Reynolds number Re = 110, 137, 180 and 212 - Energy injection scale l inj =0.3cm, outer scale l o ~2cm

9. Experimental Setup Soft-Condensed Matter Physics Group

10. Experimental Setup Soft-Condensed Matter Physics Group Soap film frame CCD Camera Magnet array Nd-YAG Laser

11. Transitions to Turbulence Soft-Condensed Matter Physics Group

12. Particle Image Velocimetry Soft-Condensed Matter Physics Group  t=2 ms Soft-Condensed Matter Physics Group

13. Typical Velocity Field

14. Soft-Condensed Matter Physics Group Evolution of Vortices

15. Stability of the Flow Soft-Condensed Matter Physics Group

16. Fluctuations increases with Re Soft-Condensed Matter Physics Group

17. Navier-Stokes Equation v : velocity of fluid p : reduced pressure  : the viscosity  : drag coefficient between the soap film and the air f : reduced external force : incompressible condition Reynolds Number Re Soft-Condensed Matter Physics Group

18. Energy Cascade in 3D Turbulence Soft-Condensed Matter Physics Group ………………………………….…. Injection length l inj Dissipative length l dis Energy flux 

19. Vortex Stretching and Turbulence Soft-Condensed Matter Physics Group S S  X Y U(y)

20. Energy Spectrum in 2D and 3D Soft-Condensed Matter Physics Group E(k) k kdkd kiki E~k -5/3 3D kiki kdkd E(k) E v ~k -5/3 k -3 2D k3k3

21. Physics of 2D Turbulence Soft-Condensed Matter Physics Group Vorticity Equation Since no vortex stretching in 2D ( ),  is a conserved quantity when =0.

22. Consequence of Enstrophy Conservation Soft-Condensed Matter Physics Group k l k0k0 k2k2 k1k1 E 0 =E 1 +E 2 k 0 2 E 0 =k 1 2 E 1 +k 2 2 E 2 k 0 =k 1 +k 2 Let k 2 =k 0 +k 0 /2 and k 1 =k 0 -k 0 /2

23. U rms (cm/s) 25 20 15 10 Energy Spectra Soft-Condensed Matter Physics Group k inj 5/3

24. Soft-Condensed Matter Physics Group Structure Functions l v1v1 v2v2

25. U rms (cm/s) 10 8.0 5.5 4.0 3.0 Longitudinal Velocity Differences Soft-Condensed Matter Physics Group 1.9

26. 2 nd Order Structure Function Soft-Condensed Matter Physics Group

27. Topological Structures Soft-Condensed Matter Physics Group

28. Enstrophy Fields,  2 Squared strain-rate Fields,  2 Vorticity and Stain-rate Fields

29. Pressure Fields Soft-Condensed Matter Physics Group

30. Intermittency In 3D turbulence, intermittency stems from the non-uniform distribution of the energy dissipation rate by vortex stretching. Soft-Condensed Matter Physics Group (a) velocity fluctuations from a jet and (b) velocity fluctuations after high-pass filtering which shows intermittent bursts (Gagne 1980).

31. Soft-Condensed Matter Physics Group Intermittency From velocity time series and assuming homogeneity/isotropy of flows,  can be calculated. From velocity time series and assuming homogeneity/isotropy of flows,  can be calculated. In 2D turbulence, it is generally believed that it is immune to intermittency because the statistics of the velocity difference are close to Gaussian. In 2D turbulence, it is generally believed that it is immune to intermittency because the statistics of the velocity difference are close to Gaussian. From velocity time series and assuming homogeneity/isotropy of flows,  can be calculated. From velocity time series and assuming homogeneity/isotropy of flows,  can be calculated. In 2D turbulence, it is generally believed that it is immune to intermittency because the statistics of the velocity difference are close to Gaussian. In 2D turbulence, it is generally believed that it is immune to intermittency because the statistics of the velocity difference are close to Gaussian. The turbulent plasma in the solar corona E. Buchlin et.al A&A 436, 355-362 (2005)

32. Soft-Condensed Matter Physics Group The PDFs of dv l and S p ( l )

33. Soft-Condensed Matter Physics Group The Scaling Exponents Red: Our data; Blue: 2D turbulence by Paret and Tabeling (Phys. of Fluids, 1998) Green: 3D turbulence by Anselmet et. al. (J. of Fluid Mech. 1984) Red: Our data; Blue: 2D turbulence by Paret and Tabeling (Phys. of Fluids, 1998) Green: 3D turbulence by Anselmet et. al. (J. of Fluid Mech. 1984)

34. Log-Normal Model In 1962, Kolmogorov suggested log-normal model. Soft-Condensed Matter Physics Group

35. The PDFs of el Soft-Condensed Matter Physics Group The  l has broad tails, but log(  l ) is normally distributed.

36. Cross-correlation Function between dv l and  l Soft-Condensed Matter Physics Group The velocity difference dv l is correlated with the local energy dissipation rate. But such a dependence decreases as l increases.

37. The Scaling Exponent  p /  3 Red diamonds are calculated by velocity difference v l p ~  p blue circles are obtained by local energy dissipation  l p ~ p/3+ p Solid line indicates the slope 1/3 by the classical Kolmogorov theory. The dash line indicates the fit based on lognormal model, ~0.11 Red diamonds are calculated by velocity difference v l p ~  p blue circles are obtained by local energy dissipation  l p ~ p/3+ p Solid line indicates the slope 1/3 by the classical Kolmogorov theory. The dash line indicates the fit based on lognormal model, ~0.11 Soft-Condensed Matter Physics Group

38. ConclusionsConclusions We demonstrated that it is possible to conduct fluid flow and turbulence studies in freely suspended soap films that behave two dimensionally. We demonstrated that it is possible to conduct fluid flow and turbulence studies in freely suspended soap films that behave two dimensionally. The conventional wisdom suggests that turbulence in 2D and 3D are very different. Our experiment shows that this difference exists only for the mean quantities such as the average energy transfer rate. As far as fluctuations are concerned, they are very similar. The conventional wisdom suggests that turbulence in 2D and 3D are very different. Our experiment shows that this difference exists only for the mean quantities such as the average energy transfer rate. As far as fluctuations are concerned, they are very similar. Intermittency exists and can be accounted for by non- uniform distribution of saddle points similar to 3D turbulence. Intermittency exists and can be accounted for by non- uniform distribution of saddle points similar to 3D turbulence. Soft-Condensed Matter Physics Group

39. Acknowledgement Walter Goldburg Hamid Kelley Maarten Rutgus Andrew Belmonte This work has been supported by NASA and NSF Mike Rivera Yonggun Jun Brian Martin Jie Zhang Pedram Roushan