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

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

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

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

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

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

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

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

Experimental Setup Soft-Condensed Matter Physics Group

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

Transitions to Turbulence Soft-Condensed Matter Physics Group

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

Typical Velocity Field

Soft-Condensed Matter Physics Group Evolution of Vortices

Stability of the Flow Soft-Condensed Matter Physics Group

Fluctuations increases with Re Soft-Condensed Matter Physics Group

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

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

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

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

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

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

U rms (cm/s) Energy Spectra Soft-Condensed Matter Physics Group k inj 5/3

Soft-Condensed Matter Physics Group Structure Functions l v1v1 v2v2

U rms (cm/s) Longitudinal Velocity Differences Soft-Condensed Matter Physics Group 1.9

2 nd Order Structure Function Soft-Condensed Matter Physics Group

Topological Structures Soft-Condensed Matter Physics Group

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

Pressure Fields Soft-Condensed Matter Physics Group

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).

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, (2005)

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

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)

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

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

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.

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

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

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