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Statistical Fluctuations of Two -d imensional Turbulence

Statistical Fluctuations of Two -d imensional Turbulence. Mike Rivera and Yonggun Jun Department of Physics & Astronomy University of Pittsburgh. Table of Contents. Introduction Experimental Setup Experimental Results • Average Behavior • Fluctuations Comparison with 3D Results

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Statistical Fluctuations of Two -d imensional Turbulence

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

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

  3. What is Turbulence? • 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 Soft-Condensed Matter Physics Group

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

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

  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. vy 7 cm Soft-Condensed Matter Physics Group Forced 2D Turbulence • Applied voltage : f = 1 Hz. • Taylor microscale Reynolds number • Rel= 110, 137, 180 and 212 • - Energy injection scale linj=0.3cm, • outer scale lo~2cm

  9. Experimental Setup Soft-Condensed Matter Physics Group

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

  11. Transitions to Turbulence Soft-Condensed Matter Physics Group

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

  13. Typical Velocity Field Soft-Condensed Matter Physics Group

  14. Evolution of Vortices Soft-Condensed Matter Physics Group

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

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

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

  18. Injection length linj Energy flux e Dissipative length ldis ………………………………….…. Energy Cascade in 3D Turbulence Soft-Condensed Matter Physics Group

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

  20. 3D 2D Energy Spectrum in 2D and 3D E(k) E(k) Ev~k-5/3 E~k-5/3 k-3 k3 k kd kd ki ki Soft-Condensed Matter Physics Group

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

  22. k l Consequence of Enstrophy Conservation k1 k0 k2 E0=E1+E2 k02E0=k12E1+k22E2 k0=k1+k2 Let k2=k0+k0/2 and k1=k0-k0/2 Soft-Condensed Matter Physics Group

  23. Energy Spectra 5/3 Urms (cm/s) 25 20 15 10 kinj Soft-Condensed Matter Physics Group

  24. Structure Functions v1 v2 l Soft-Condensed Matter Physics Group

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

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

  27. Topological Structures Soft-Condensed Matter Physics Group

  28. Vorticity and Stain-rate Fields Enstrophy Fields, w2 Squared strain-rate Fields, s2 Soft-Condensed Matter Physics Group

  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. (a) velocity fluctuations from a jet and (b) velocity fluctuationsafter high-pass filtering which shows intermittent bursts (Gagne 1980). Soft-Condensed Matter Physics Group Soft-Condensed Matter Physics Group

  31. Intermittency • From velocity time series and assuming homogeneity/isotropy of flows, e 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. The turbulent plasma in the solar corona E. Buchlin et.al A&A 436, 355-362 (2005) Soft-Condensed Matter Physics Group

  32. The PDFs of dvland Sp(l) Soft-Condensed Matter Physics Group

  33. 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) Soft-Condensed Matter Physics Group

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

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

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

  37. The Scaling Exponent zp/ z3 • Red diamonds are calculated by velocity difference vlp • ~ zp • blue circles are obtained by local energy dissipation elp • ~ p/3+tp • Solid line indicates the slope 1/3 by the classical Kolmogorov theory. • The dash line indicates the fit based on lognormal model, m~0.11 Soft-Condensed Matter Physics Group

  38. Conclusions • 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. • 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 • Mike Rivera • Yonggun Jun • Brian Martin • Jie Zhang • Pedram Roushan • Walter Goldburg • Hamid Kelley • Maarten Rutgus • Andrew Belmonte This work has been supported by NASA and NSF Soft-Condensed Matter Physics Group

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