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Statistical Fluctuations of Two -d imensional Turbulence

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Statistical Fluctuations of Two-dimensional Turbulence

Mike Rivera and Yonggun Jun

Department of Physics & Astronomy

University of Pittsburgh

- Introduction
- Experimental Setup
- Experimental Results
- • Average Behavior
- • Fluctuations
- Comparison with 3D Results
- Conclusion

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- 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

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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.

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h

15 oA

L

*Non-equilibrium Films: 1<h<100 m

h/L ~ 10-4 - 10-3

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Jupiter Great red spot

Hurricane

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vy

7 cm

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- 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

Experimental Setup

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CCD Camera

Nd-YAG Laser

Magnet array

Soap film frame

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Dt=2 ms

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Evolution of Vortices

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: 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

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Injection length linj

Energy flux e

Dissipative length ldis

………………………………….….

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Y

U(y)

S

X

S

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3D

2D

E(k)

E(k)

Ev~k-5/3

E~k-5/3

k-3

k3

k

kd

kd

ki

ki

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Vorticity Equation

Since no vortex stretching in 2D ( ),

, w is a conserved quantity when n=0.

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k

l

k1

k0

k2

E0=E1+E2

k02E0=k12E1+k22E2

k0=k1+k2

Let k2=k0+k0/2 and k1=k0-k0/2

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5/3

Urms (cm/s)

25

20

15

10

kinj

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Structure Functions

v1

v2

l

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Urms (cm/s)

10

8.0

5.5

4.0

3.0

1.9

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Enstrophy Fields, w2

Squared strain-rate Fields, s2

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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).

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- 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)

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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)

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In 1962, Kolmogorov suggested log-normal model.

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The el has broad tails, but log(el) is normally distributed.

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The velocity difference dvl is

correlated with the local

energy dissipation rate. But

such a dependence decreases

as l increases.

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- 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

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- 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.

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- 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

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