statistical fluctuations of two -d imensional turbulence

Statistical Fluctuations Statistical Fluctuations of of

TwoTwo-d-dimensional imensional TurbulenceTurbulence

Mike Rivera and Yonggun JunMike Rivera and Yonggun Jun

Department of Physics & AstronomyDepartment of Physics & Astronomy

University of PittsburghUniversity of Pittsburgh

Table of ContentsTable of Contents IntroductionIntroduction Experimental SetupExperimental Setup Experimental ResultsExperimental Results • • Average BehaviorAverage Behavior • • FluctuationsFluctuations Comparison with 3D ResultsComparison with 3D Results ConclusionConclusion

Soft-Condensed Matter Physics GroupSoft-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

Evolution to TurbulenceEvolution to TurbulenceAt 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.


Re=UL/ U: typical velocity L: typical length: viscosity

Re>50

Freely Suspended Film is Freely Suspended Film is 2D2D


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

h/L ~ 10-4 - 10-3

L

15 oA

h

Flows in Earth Flows in Earth Atmosphere is 2DAtmosphere is 2D


Examples of 2D Examples of 2D TurbulenceTurbulence

Jupiter Great red spot Hurricane



Forced 2D TurbulenceForced 2D Turbulence

7 cm

vy

- Applied voltage : f = 1 Hz.

- Taylor microscale Reynolds number Re= 110, 137, 180 and 212

- Energy injection scale linj=0.3cm, outer scale lo~2cm

Experimental SetupExperimental Setup


N

S

N

S

N

S

N

S

S

N

N

S

S

N

S

N

S

N

S

N

magnets

liquidinjection

lens

CCDcameraair

pump

function generatorand power supply

N

S

N

S

N

S

N

S

N

S

N

S

N

S

N

S

S

N

S

N

N

S

N

S

S

N

S

N

S

N

S

N

S

N

S

N

S

N

S

N

magnets

liquidinjection

lens

CCDcameraair

pump

function generatorand power supply

Experimental SetupExperimental Setup


Soap film frame

CCD Camera

Magnet arrayNd-YAG Laser

Transitions to Transitions to TurbulenceTurbulence


Particle Image Particle Image VelocimetryVelocimetry


t=2 ms



Typical Velocity FieldTypical Velocity Field


Evolution of VorticesEvolution of Vortices

0

2

4

6

8

10

(1

03

s-2)

0 100 200t (s)

0

2

4

6

8

10

12

14

urm

s(c

m/s

)

Stability of the FlowStability of the Flow


Fluctuations increases Fluctuations increases with Rewith Re


Navier-Stokes EquationNavier-Stokes Equation )v(vvv

v 2

fpt

0v

v : velocity of fluidp : reduced pressure: the viscosity: drag coefficient between the soap film and the airf : reduced external force

: incompressible condition

LV

|v|

|vv|Re

2

Reynolds Number Re


Energy Cascade in 3D Energy Cascade in 3D TurbulenceTurbulence


………………………………….….

Injection length linj

Dissipative length ldis

Energy flux

Vortex Stretching and Vortex Stretching and TurbulenceTurbulence


S

S

X

YU(y)

Energy Spectrum in 2D Energy Spectrum in 2D and 3Dand 3D


E(k)

kkdki

E~k-5/33D

ki kd

E(k)Ev~k-5/3

k-3

2D

0

2 )(v2

1dkkE

k3

Physics of 2D TurbulencePhysics of 2D Turbulence


vp-

t

v 2

D

D

Vorticity Equation

v

2vt

D

D

Since no vortex stretching in 2D ( ),

2

t

D

D

0v

is a conserved quantity when =0.

Consequence of Enstrophy Consequence of Enstrophy ConservationConservation

)()( 2 kEkk

221

20

20

22

1 Ekk

kkE


0

2 )(2

1dkk

kl

k0 k2k1

E0=E1+E2

k02E0=k1

2E1+k22E2 k0=k1+k2

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

21 3

5EE

Urms (cm/s)25201510

Energy SpectraEnergy Spectra


kinj

5/3


Structure FunctionsStructure Functions

plvlvvlS pl

pp

~||]ˆ)[()( 12 plvlvvlS p

lp

p ~||]ˆ)[()( 12

l

v1

v2

Urms (cm/s)108.05.54.03.0

Longitudinal Velocity Longitudinal Velocity DifferencesDifferences


1.9

22ndnd Order Structure Order Structure FunctionFunction


Topological StructuresTopological Structures p2 )(

2

1 22

ji

ijji vv,

22

2

1 ji

ijji vv,

22

2

1


-1 -0.5 0 0.5 1X

-1

-0.5

0

0.5

1

Y

-1 -0.5 0 0.5 1X

-1

-0.5

0

0.5

1

Y

ji

ijji vv,

22

2

1 ji

ijji vv,

22

2

1


Enstrophy Fields, 2Enstrophy Fields, 2 Squared strain-rate Fields, 2

Vorticity and Stain-rate Vorticity and Stain-rate FieldsFields

Pressure FieldsPressure Fields



IntermittencyIntermittency In 3D turbulence, intermittency stems from the non-uniform

distribution of the energy dissipation rate by vortex stretching. In 3D turbulence, intermittency stems from the non-uniform

distribution of the energy dissipation rate by vortex stretching.


(a) velocity (a) velocity flfluctuations from a jet and (b) velocity uctuations from a jet and (b) velocity flfluctuationsuctuations after high-pass after high-pass fifiltering which shows ltering which shows intermittent bursts (Gagne 1980).intermittent bursts (Gagne 1980).


IntermittencyIntermittency

From velocity time series and assuming From velocity time series and assuming homogeneity/isotropy of flows, homogeneity/isotropy of flows, can be calculated. can be calculated.

In 2D turbulence, it is generally believed that it is In 2D turbulence, it is generally believed that it is immune to intermittency because the statistics of immune to intermittency because the statistics of the velocity difference are close to Gaussian.the velocity difference are close to Gaussian.

From velocity time series and assuming From velocity time series and assuming homogeneity/isotropy of flows, homogeneity/isotropy of flows, can be calculated. can be calculated.

In 2D turbulence, it is generally believed that it is In 2D turbulence, it is generally believed that it is immune to intermittency because the statistics of immune to intermittency because the statistics of the velocity difference are close to Gaussian.the velocity difference are close to Gaussian.

The turbulent plasma in the solar coronaE. Buchlin et.al A&A 436, 355-362 (2005)


The PDFs of The PDFs of dvdvll and and SSpp((ll) )


The Scaling ExponentsThe 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 ModelLog-Normal Model

2/'||2')'(

4where

lxxl dxxl

2/'||2')'(

4where

lxxl dxxl

3/

3

3/

~,~||,~

pp

pl

pll

l

p

llvl

v pp

3/

3

3/

~,~||,~

pp

pl

pll

l

p

llvl

v pp

In 1962, Kolmogorov suggested log-normal model. In 1962, Kolmogorov suggested log-normal model.

parameterncy intermitte :)3(18/3/

d,distriburey lognormall isn dissipatioenergy local If2 pppp parameterncy intermitte :)3(18/3/

d,distriburey lognormall isn dissipatioenergy local If2 pppp


The PDFs of elThe PDFs of el


Thel has broad tails, but log(l) is normally distributed.

Cross-correlation Cross-correlation Function Function

between between dvdvll and and ll ll εv

llll εεvvC

||

||||

ll εv

llll εεvvC

||

||||


The velocity difference dvl iscorrelated with the localenergy dissipation rate. Butsuch a dependence decreasesas l increases.

The Scaling Exponent The Scaling Exponent pp/ / 33

The Scaling Exponent The Scaling Exponent pp/ / 33

• Red diamonds are calculated by velocity difference vl

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 vl

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

parameterncy intermitte :)3(18/3/

model Lognormalon Based2 pppp parameterncy intermitte :)3(18/3/

model Lognormalon Based2 pppp


ConclusionsConclusionsConclusionsConclusions We demonstrated that it is possible to conduct We demonstrated that it is possible to conduct

fluid flow and turbulence studies in freely fluid flow and turbulence studies in freely suspended soap films that behave two suspended soap films that behave two dimensionally.dimensionally.

The conventional wisdom suggests that The conventional wisdom suggests that turbulence in 2D and 3D are very different. turbulence in 2D and 3D are very different. Our experiment shows that this difference Our experiment shows that this difference exists only for the mean quantities such as the exists only for the mean quantities such as the average energy transfer rate. As far as average energy transfer rate. As far as fluctuations are concerned, they are very fluctuations are concerned, they are very similar.similar.

Intermittency exists and can be accounted for Intermittency exists and can be accounted for by non-uniform distribution of saddle points by non-uniform distribution of saddle points similar to 3D turbulence.similar to 3D turbulence. Soft-Condensed Matter Physics GroupSoft-Condensed Matter Physics Group

AcknowledgementAcknowledgement


• 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

statistical fluctuations of two -d imensional turbulence

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