Statistical Fluctuations of Two-dimensional Turbulence Mike Rivera and Yonggun Jun Department of...
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Transcript of Statistical Fluctuations of Two-dimensional Turbulence Mike Rivera and Yonggun Jun Department of...
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?
Soft-Condensed Matter Physics GroupSoft-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 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.
Soft-Condensed Matter Physics GroupSoft-Condensed Matter Physics Group
Re=UL/ U: typical velocity L: typical length: viscosity
Re>50
Freely Suspended Film is Freely Suspended Film is 2D2D
Soft-Condensed Matter Physics GroupSoft-Condensed Matter Physics Group
*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
Soft-Condensed Matter Physics GroupSoft-Condensed Matter Physics Group
Examples of 2D Examples of 2D TurbulenceTurbulence
Jupiter Great red spot Hurricane
Soft-Condensed Matter Physics GroupSoft-Condensed Matter Physics Group
Soft-Condensed Matter Physics GroupSoft-Condensed Matter Physics Group
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
Soft-Condensed Matter Physics GroupSoft-Condensed Matter Physics Group
N
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magnets
liquidinjection
lens
CCDcameraair
pump
function generatorand power supply
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magnets
liquidinjection
lens
CCDcameraair
pump
function generatorand power supply
Experimental SetupExperimental Setup
Soft-Condensed Matter Physics GroupSoft-Condensed Matter Physics Group
Soap film frame
CCD Camera
Magnet arrayNd-YAG Laser
Transitions to Transitions to TurbulenceTurbulence
Soft-Condensed Matter Physics GroupSoft-Condensed Matter Physics Group
Particle Image Particle Image VelocimetryVelocimetry
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t=2 ms
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Typical Velocity FieldTypical Velocity Field
Soft-Condensed Matter Physics GroupSoft-Condensed Matter Physics Group
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
Soft-Condensed Matter Physics GroupSoft-Condensed Matter Physics Group
Fluctuations increases Fluctuations increases with Rewith Re
Soft-Condensed Matter Physics GroupSoft-Condensed Matter Physics Group
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
Soft-Condensed Matter Physics GroupSoft-Condensed Matter Physics Group
Energy Cascade in 3D Energy Cascade in 3D TurbulenceTurbulence
Soft-Condensed Matter Physics GroupSoft-Condensed Matter Physics Group
………………………………….….
Injection length linj
Dissipative length ldis
Energy flux
Vortex Stretching and Vortex Stretching and TurbulenceTurbulence
Soft-Condensed Matter Physics GroupSoft-Condensed Matter Physics Group
S
S
X
YU(y)
Energy Spectrum in 2D Energy Spectrum in 2D and 3Dand 3D
Soft-Condensed Matter Physics GroupSoft-Condensed Matter Physics Group
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
Soft-Condensed Matter Physics GroupSoft-Condensed Matter Physics Group
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
Soft-Condensed Matter Physics GroupSoft-Condensed Matter Physics Group
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
Soft-Condensed Matter Physics GroupSoft-Condensed Matter Physics Group
kinj
5/3
Soft-Condensed Matter Physics GroupSoft-Condensed Matter Physics Group
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
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1.9
22ndnd Order Structure Order Structure FunctionFunction
Soft-Condensed Matter Physics GroupSoft-Condensed Matter Physics Group
Topological StructuresTopological Structures p2 )(
2
1 22
ji
ijji vv,
22
2
1 ji
ijji vv,
22
2
1
Soft-Condensed Matter Physics GroupSoft-Condensed Matter Physics Group
-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
Soft-Condensed Matter Physics GroupSoft-Condensed Matter Physics Group
Enstrophy Fields, 2Enstrophy Fields, 2 Squared strain-rate Fields, 2
Vorticity and Stain-rate Vorticity and Stain-rate FieldsFields
Pressure FieldsPressure Fields
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Soft-Condensed Matter Physics GroupSoft-Condensed Matter Physics Group
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.
Soft-Condensed Matter Physics GroupSoft-Condensed Matter Physics Group
(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).
Soft-Condensed Matter Physics GroupSoft-Condensed Matter Physics Group
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)
Soft-Condensed Matter Physics GroupSoft-Condensed Matter Physics Group
The PDFs of The PDFs of dvdvll and and SSpp((ll) )
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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
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The PDFs of elThe PDFs of el
Soft-Condensed Matter Physics GroupSoft-Condensed Matter Physics Group
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
||
||||
Soft-Condensed Matter Physics GroupSoft-Condensed Matter Physics Group
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
Soft-Condensed Matter Physics GroupSoft-Condensed Matter Physics Group
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
Soft-Condensed Matter Physics GroupSoft-Condensed Matter Physics Group
• 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