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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-dimensional Turbulence

Mike Rivera and Yonggun Jun

Department of Physics & Astronomy

University of Pittsburgh


Table of contents
Table of Contents

  • Introduction

  • Experimental Setup

  • Experimental Results

  • • Average Behavior

  • • Fluctuations

  • Comparison with 3D Results

  • Conclusion

Soft-Condensed Matter Physics Group


What is turbulence
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

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Evolution to turbulence
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


Freely suspended film is 2d

h

15 oA

Freely Suspended Film is 2D

L

*Non-equilibrium Films: 1<h<100 m

h/L ~ 10-4 - 10-3

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Flows in earth atmosphere is 2d
Flows in Earth Atmosphere is 2D

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Examples of 2d turbulence
Examples of 2D Turbulence

Jupiter Great red spot

Hurricane

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Forced 2d turbulence

vy

7 cm

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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


Experimental Setup

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Experimental setup
Experimental Setup

CCD Camera

Nd-YAG Laser

Magnet array

Soap film frame

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Transitions to turbulence
Transitions to Turbulence

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Particle image velocimetry
Particle Image Velocimetry

Dt=2 ms

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Soft-Condensed Matter Physics Group


Typical velocity field
Typical Velocity Field

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Evolution of Vortices

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Stability of the flow
Stability of the Flow

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Fluctuations increases with re
Fluctuations increases with Re

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Navier stokes equation
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

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Energy cascade in 3d turbulence

Injection length linj

Energy flux e

Dissipative length ldis

………………………………….….

Energy Cascade in 3D Turbulence

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Vortex stretching and turbulence

Y

U(y)

S

X

S

Vortex Stretching and Turbulence

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Energy spectrum in 2d and 3d

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

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Physics of 2d turbulence
Physics of 2D Turbulence

Vorticity Equation

Since no vortex stretching in 2D ( ),

, w is a conserved quantity when n=0.

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Consequence of enstrophy conservation

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

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Energy spectra
Energy Spectra

5/3

Urms (cm/s)

25

20

15

10

kinj

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Structure Functions

v1

v2

l

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Longitudinal velocity differences
Longitudinal Velocity Differences

Urms (cm/s)

10

8.0

5.5

4.0

3.0

1.9

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2 nd order structure function
2nd Order Structure Function

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Topological structures
Topological Structures

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Vorticity and stain rate fields
Vorticity and Stain-rate Fields

Enstrophy Fields, w2

Squared strain-rate Fields, s2

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Pressure fields
Pressure Fields

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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).

Soft-Condensed Matter Physics Group

Soft-Condensed Matter Physics Group


Intermittency
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)

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The pdfs of dv l and s p l
The PDFs of dvland Sp(l)

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The scaling exponents
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)

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Log normal model
Log-Normal Model

In 1962, Kolmogorov suggested log-normal model.

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The pdfs of el
The PDFs of el

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

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Cross correlation function between dv l and e l
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.

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The scaling exponent z p z 3
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

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Conclusions
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


Acknowledgement
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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