Quantum turbulence - an overview

L. Skrbek Joint Low Temperature Laboratory,

Institute of Physics ASCR and Charles University

V Holešovičkách 2, 180 00 Prague 8, Czech Republic

COSLAB, Smolenice, September 2, 2005

M. Krusius - LTL HUT (ROTA), HelsinkiV.B. Eltsov, A.P. Finne J. HosioN.B. Kopnin, G.E. Volovik

P.V.E. McClintock, G. Pickett - Lancaster UniversityD.I. Bradley, D. Charalambous, D.O. Clubb, S.N. Fisher, A.M. Guénault, P.C. Hendry, H. A. Nichol

M. Tsubota, R. Hanninen Osaka City University

and others

Thanks to collaboration and discussions with:A.V. Gordeev PragueT. ChagovetsW.F. Vinen University of BirminghamR.J. Donnelly University of OregonS.R. StalpJ.J. Niemela Trieste K.R. SreenivasanP. Skyba Košice

                                                    

                 

Turbulence - outstanding problem of classical physics that remains unsolved to this day

Leonardo Da Vinci

This talk - An experimentalist’s attempt of presenting a unified, perhaps (over)simplified, picture of quantum turbulence, based on current theoretical understanding and experiments in He II and superfluid 3He-B

Characteristic length scales in turbulence

19 2110 10 m 1010 m

Quantized vortex in He II

ReUL

3g TL

Ra

Measures of turbulence intensity

Reynolds number

For isothermal flows Rayleigh number

for thermally driven flows in a gravitational

field

Ra Re

Sun 1021 1013

Ocean 1020 109

Atmosphere 1017 109

Navy (ship) 109

Aerospace (aircraft)

108-109

  T (p) (cm2/s) /air 20 C 0,15 0,122

water 20 C 1,004x10-2 14,4

Normal 3He above Tc ~ 1, olive oil

Normal fluid of 3He B

around 0.6 Tc ~ 0.2, air

Helium I 2,25 K (VP) 1,96x10-4 3,25x10-5

Helium II 1,8 K (VP) 9,01x10-5 X

He-gas 5,5 K (2,8 bar) 3,21x10-4 1,41x108

 

•Cryogenic He Gas, He I and He II are probably the best known

working fluids for the controlled, laboratory high Re and Ra turbulence experiments

•He II and 3He B – so far the only two media where quantum turbulence has been studied under controlled laboratory conditions

T (m

K)

H (T

)

p (b

ar)

1

0

1

2

3

0.5

30

20

10

30

20

10

A

A

B

B

B NORMALN

OR

MA

L

A1

Phase diagram of 3He

Normal liquid 3HeClassical Navier-Stokesfluid

Phase diagram of 4He

T (K)

P (

kP

a)

He I – normal liquidClassical Navier-Stokes

Solid He

Cryogenic He gas

Superfluid He II Critical point

Quantum mechanical description of He II tris ,exp Macroscopic wave function

ip̂ 4ms

v 0scurl v

0 L

sdv nm

hnd

Ls

4v

Circulation –singly connected regionCirculation- multiply connected region

Quantized vortices in He II Rotating bucket of He II-thanks to the existence of rectilinear vortex linesHe II mimics solid body rotation

sm /10 27

vorticityLSN 2

Hydrodynamics of superfuid 4He - He II

Two-fluid model (Landau)He II flow is well described only in a limit of low velocities

Thermal counterflow

Second sound prediction – ( entropy or temerature waves)

Two-fluid model is also applicable to superfluid 3He phases (talk of Matti Krusius)

Very low kinematic viscosity of the normal fluid (above appr. 1 K)

Systems (simplified) to understand turbulence in

T

4He 3Henormal liquid He I Classical Navier-Stokes fluid of extremely low kinematic viscosity

normal liquid 3He Classical Navier-Stokes fluid of kinematic viscosity comparable with that of air

Superfluid transition at Tc

He II – a “mixture” of two fluids superfluid 3He Bnormal fluid of extremely normal fluid of of kinematic viscosity low kinematic viscosity comparable with that of air + +Inviscid superfluid Inviscid superfluidCirculation is quantized Circulation is quantized

T 0 limitPure superfluid Pure superfluid0

4

3 210 [2

/ ]m

cm s

3

3 20.66 10 [ /2

2]m s

mc

1

2

3

Grid turbulence- visualisation

Step 1 – classical turbulence in a viscous fluid

Classical turbulence - Richardson cascade

SmokeInk or dyeKalliroscope flakesHydrogen bubblesBaker’s pH techniqueHot wire anemometryLaser Doppler velocimetryParticle image velocimetry…

3d energy spectrum of the homogeneous isotropic turbulence

k

Sp

ectr

al e

ner

gy d

ensi

ty (

log

scal

e)

Energy containing eddies

Energy containing length scale

Dissipative (Kolmogorov)length scale

Dissipation range

Inertial range

kkC 3/53/2

1/ 43 /disk

dtdE /

mAk

Viscosity is unimportant ….. …….. up to here

Spectral decay model for HIT Skrbek, Stalp: On the decay of homegeneous isotropic turbulence, Physics of Fluids 12, No 8, (2000) 1997

2Ak 3/53/2 kC

eek /2 D/2

1/ 43 /disk

dtdE /

0

dkkEE Diff. equation for E E t HIT

22Leff

2 Classical turbulence

Superfluid turbulence

3/53/2 kC

2Ak

The model has been recently verified by computer simulations by Touil, Bertoglio and Shao (J. Turbulence 3 (2002) 049)Direct and large eddy simulations, as well as the eddy-damped quasi-normal Markovian closure

Simplest case – assume that the energy- containing scale is saturated - D

3/53/2 kC

D/2

3 2

22/3 5/32

0 2 /

27( ) *

2 2D

C DE E k dk C k dk E t t t

2dE

dt 3

3/ 2/ 23

( ) *2

Ct t

Dt

Note – on first approximation, intermittency does not change the power of the decay:

2 3/ 22/3 5/3

0 2 /

( ) * ( ) *D

E E k dk C k Dk dk E t t t t t t

( )E k

k

For decaying vorticity, theree subsequent regimes predicted: 11/10 5/ 6 3/ 2; ;t t t

Step 2 – superfluid turbulence in a pure superfluid (T=0)

3d energy spectrum of the superfluid homogeneous isotropic turbulence

k

Sp

ectr

al e

ner

gy d

ensi

ty (

log

scal

e)

Energy containing eddies

Energy containing length scale

Quantum dissipative length scale

Dissipation range due to phonon irradiation

Inertial range, no dissipation

2/3 5/3C k

Qdissk

dtdE /

mAk

Hypothesis:The form of the superfluid energy spectrum at T=0 is of the form as that of classical turbulence

Is there any, at least partial evidence ???

Numerical evidence

△: 2 < k < 12 ○: 2 < k < 14 □: 2 < k < 16

E(k

)5/3

E(k) 〜 k -

n(t)

T. Araki, M.Tsubota and S.K.Nemirovskii, Phys.Rev.Lett.89, 145301(2002): Energy Spectrum of Superfluid Turbulence with No Normal-Fluid ComponentThe energy spectrum of a Taylor-Green vortex was obtained under the

vortex filament formulation. The absolute value with the energy dissipation rate was consistent with the Kolmogorov law.

C. Nore, M. Abid and M.E.Brachet, Phys.Fluids 9, 2644(1997)

By solving the Gross-Pitaevskii equation, they studied the energy spectrum of a Taylor-Green flow. The spectrum shows the -5/3 power on the way of the decay, but the acoustic emission makes the situation complicated.

Gross-Pitaevskii model by introducing the small-scale dissipation

M. Kobayashi and M. Tsubota, PRL94, 065302 (2005)obtained the Kolmogorov spectrum more clearly.

Experimental evidence in He IIDavis et al, Physica B 280, 43 (2000).

It is clear that He II turbulence decays (on time scale of seconds) at very low temperatureDue to complexity of this experiment, quantitative comparison with the decay model is difficult

5/ 6t

3/ 2t 3/ 2

3/ 23( ) *

2 eff

CL t t t

D

Classical spectral decay model – universal decay

Experimental evidence in 3He B – Lancaster (courtesy of S. Fisher)

Step 3 – turbulence within the two-fluid

model2/3 5/3C k 2/3 5/3

SC k

1/ 43 /dis nk

1/ 43 /dis nk

2/3 5/3SC k

2/3 5/3C k

E k E k

E k

k k

k

3k

Energy spectrum is steeper here, due to action of the dissipative mutual friction, not any more strong enough to lock the N and S eddies togetherThe turbulent energy partly leaves the cascade, being transferred into heat, which makes the spectrum steeper

Mutual friction beyond some critical k becomes negligibly small again and Kolmogorov scaling is recovered

Phonon emission by high k Kelvin waves

Richardson cascade

Up to here the normal and superfluid eddies are locked together by the mutual friction force. They evolve together and there is negligibly small energy dissipation – hence Kolmogorov scaling

2Dk

D

D – size of our laboratory box

Viscous dissipation Phonon emission by Kelvin waves

Could the entire energy spectrum be experimentally observed?

He II Typical cooling power does not exceed about 1[W] into 1 [liter] of liquidIn fluid dynamics, all energies are ment per unit volume, i.e. E [m2/s2]

3 3 3 2 3[ / ] [10 ; 145 / ]7] [ /W m kg m m s Asssuming HIT, we use

22 11 21. [1/ ]9 10eff eff mL L Main distance between vortices

Kolmogorov dissipation length

1/ 2.3[ ]L m 1/ 43

[2

2 6.5 ]dissdissk

m

Using a typical-size laboratory cells or flow channels (of order 1 cm) we can, in principle, observe the entire energy spectrum of He II turbulence.

Note that during the decay the energy containing length scale, as well as the relevant dissipation scales grow.

We can therefore use decaying turbulence to study them more closely

3He B Typical cooling power does not exceed about 1[nW] into 1 [cm3] of liquid

3 3 353 2[ / ]10 ; 10 [ / ]0 1 [ ]0 /W m kg m m s Asssuming HIT, we get

Main distance between vortices

Kolmogorov dissipation length

1/ 0.5[ ]mL m 1/ 432

2 ]18[dissdissk

cm

Using a typical-size laboratory cells or flow channels (of order 1 cm) we cannot observe the entire energy spectrum of He II turbulence, the normal fluid will always stay stationary.

6 23.4 10 [1/ ]L m

We need a theory for superfluid turbulence in a stationary normal fluid

t

v

v ω v

ˆqt

v

v ω ω ω v

1q

plays a role of inverse Reynolds number and does NOT depend on the geometry

Superfluid flow –by averaging the Euler equation over distance >> the intervortex spacing

Classical flow - Navier-Stokes equation Re depends on the geometry of flow in question

vωωωvv

ˆ)1( t

)(ˆ)(: NNtSonin vvωωωvvωv

v

assuming the normal fluid is at rest:

After rescaling the time (Volovik) (1 )t t

Dissipation is distinctly different, depends on the large-scale velocity U:

2ˆ( ) vdE

q qdt t

v

v v ω ω v ω

Dissipation:2ω

dt

dE

1/ R 1/ 43/disk

energy in

viscous dissipation

2/3 5/3C k

1/ R

energy in

Dissipation due to mutual friction at large scales

3k

5/3k

0

Vq

R

Continuous approach is justified if circulation at the smallest scale >>circulation quantumClassical or Kolmogorov regime quantum cutoff

L’vov, Nazarenko, Volovik:

t

v

v ω v ˆqt

v

v ω ω ω v

They describe cascade in the simplest possible manner, using the differential form of the energy transfer term in the energy budget equation Ek is the one-dimensional density of the turbulent kinetic energy in the k-space

If then the Komogorov scaling is recoveredk

dE

dt

5/3k

3k

The flow of energy toward larger wave numbers can be described as a diffusion of energy in k space. It can be shown that this diffusion must obey a nonlinear diffusion equation, which can be written, for the case of homogeneous, isotropic turbulence, in the form

Vinen considers the effect of mutual friction, which can be written in the form

sf Lu Considering the effect of mutual friction at different length scales Vinen obtains (for all length scales) a characteristic decay time time 3

2 L

The Kolmogorov spectrum will survive if this time exceeds the corresponding eddy turn-over time.

This is satisfied only for eddies smaller than

max 3RL

Bigger eddies cannot exist – they become fast dissipated by mutual friction

He II - experiment

k

Sp

ectr

al e

ner

gy d

ensi

ty (

log

scal

e)

Energy containing eddies

Energy containing length scale

Dissipative (Kolmogorov)length scale

dissipation

Inertial range kkC 3/53/2

1/ 43 /disk

dtdE /

Maurer, Tabeling:Europhysics Lett. 43 (1998) 29Flow between counterrotating discsU=80cm/s; Re=2x106 a-2.3K; b-2.08K; c-1.4 K

mAk

3k

Local pressure fluctuations are measured using a small total-head pressure tube

2( ) ( ) ( )measp t p t U t

local pressure upstream flowAfter removing the mean parts

( ) ( ) ( )fluct fluctmeasp t p t Uu t

( ) ( ) ( ) ( )fluct fluctmeas s s s n n np t p t U u t U u t

NS

HeII

Grid turbulence in He II -detection scheme used in the Oregon decay experiments

Detection method:Second sound attenuation

Second sound is generated and detected by oscillating gold-plated porous membranes

Quantity detected:

quantized vortex line density, L(its projection to a plane perpendicular to the direction of the second sound propagation, averaged overchannel crossection )

0.1 1 10 1000.01

0.1

1

10

100

1000

(t)t-5/6

(t)t-3/2

(t)t-3/2

t (s)

>(t)

(H

z)

0.1 1 10 10030

100

200

t3/

2 (s1/

2 )

0.1 1 10 100

20

30

40

50

60

708090

100

(1

/s)

-3/2

-5/6

-1.1

3/

2 (s1/

2 ) (s)

0 20 40 60 80 100

0.1

1

(s)

Decay of the grid generated turbulence in He II – main features

There are four different regimes of the decay of vortex line density (vorticity – if defined as ) in the finite channel, characterized by:

Character of the decay does not change with temprature (1,1 K < T < 2,1 K), although the normal to superfluid density ratio does by more than order of magnitude

11/10t 5/ 6t 3/ 2t exp /t

Stalp, Skrbek, Donnelly: Decay of Grid Turbulence in a Finite Channel, Phys. Rev. Lett. 82 (1999) 4831Skrbek, Niemela,Donnelly: Four Regimes of Decaying Grid Turbulence in a Finite Channel, Phys. Rev. Lett. 85 (2000) 2973

L

Energy spectrum – He II grid turbulence Skrbek, Niemela, Sreenivasan: The Energy Spectrum of Grid – Generated Turbulence, Phys. Rev. E 64 (2001) 067301

3/53/2 kC2Ak

3C k

k/14/1

3

quant

D/2

e

Is the classically generated He II coflow turbulence the same as classical?

Yes – on large length scales, exceeding the quantum lengthNo – on length scales comparable or smaller

Quantum length scale

1/3

3 4dimensionless

k

Conclusions (1)

1/ 43 /dis nk

2/3 5/3SC k

2/3 5/3C k E k

k

3k

Energy spectrum is steeper here, due to action of the dissipative mutual friction, not any more strong enough to lock the N and S eddies togetherThe turbulent energy partly leaves the cascade, being transferred into heat, which makes the spectrum steeper

Mutual friction beyond some critical k becomes negligibly small again and Kolmogorov scaling is recovered

Phonon emission by high k Kelvin waves

Richardson cascade

Up to here the normal and superfluid eddies are locked together by the mutual friction force. They evolve together and there is negligibly small energy dissipation – hence Kolmogorov scaling

2Dk

D

D – size of our laboratory box

•It seems that co-flow quantum turbulence in He II and 3He B can be, at least approximately, understood, based on the suggested form of the energy spectrum•Depending on the energy decay rate and size of the sample, various parts of the spectrum can be probed

•Does this form of the energy spectrum apply to other systems?•(BEC, cosmic strings….)

Does this model also work for counterflow (pure superflow) turbulence in He II ?

No Why ?Due to counterflow, large normal and superfluid eddies cannot be locked together, so the dissipationless inertial range cannot exist(Steady state) counterflow turbulence = thermally driven turbulence,similarly as thermal convection

Counterflow turbulence phenomenology (Vinen 1957)

2b

Vortex ring

ba

b

bt

4

18ln

4v T 0

tv

Finite T

In counterflow, though, if rings with expandCFsnt V vvvcbb

Dimensional analysis and analogy with classical fluid dynamics leads to the Vinen equation:

))((2

2

42

2/31 CFCF

n VgLm

LVB

dt

dL

production decay

Reproduced by Schwarz (1988) –computer simulationsLocal induction approximationImportance of reconnections

1/3 /31/3 1Nu R Qa T T

3g TL

Ra

Experiments on counterflow turbulence in superfluid He IIJoint Low Temperature Laboratory, Institute of Physics and Charles University, Prague

Detection method:Second sound attenuationSecond sound is generated and detected by oscillating gold-plated porous nuclepore membranes

superfluid

normal fluid

s s n nv v n

qv

ST

Vinen equation predicts the decay of vortex line density of the form

VOtt

tL

1

1/3Q T

Steady-state data

CFL Q V

Steady state data

When heater is switched off, the following amount of heat must flow away through the channel (neglecting the channel volume)

Assuming quasi-equilibrium, for decaying temperature gradient we get

We can use this simple model to estimate the time over which the temperature difference disappears

-It is easy to take into account -the volume of the channel itself, assuming linear T gradient along it (i.e., add half of its volume)

During this time, although the heater is switched off, the turbulence is NOT isothermal, but is driven by the (decaying) temperature gradient

Note: without knowing anything of quantized vortices, this simple model predicts that the extra attenuation decays linearly with time!

1/3' ' tq T

Estimating the effective volume (i.e., V +half of thecounterflow channel volume) to be 7.5 cm3

Thermallydriven decayingturbulence

Isothermal decaying turbulence

' t

Can we treat our data on decaying counterflow turbulence as isothermal ?

Yes (all our second sound data within experimentally accessible time domain)•Direct measurement of the temperature difference across the channel•Simple model for decaying temperature gradient when the heater is switched off(Gordeev, Chagovets, Soukup, Skrbek, JLTP 138 (2005) 554)

A net increase of the vortex line density during the decay ?

Depolarization of the vortex tangle?

Wang, Swanson, Donnelly, PRB 36 (1987) 5236vortex tangle generated in the steady state counterflow turbulence is polarized

(i) The tangle is fully polarized vortices lying randomly in planes perpendicular counterflow

velocity. Second sound is effectively attenuated by their projection to

the plane perpendicular to the second sound wave, i.e., by

(ii) The tangle is random in 3D Again, the second sound is effectively attenuated by the

projection of these vortices to the plane perpendicular to the second sound

wave, this time by

L

“randomization” - transverse second sound would indicate a net growth by a factor up to i.e., up to about 23 %

2 / 8

q

LL D

22

43LL D

Decay of counterflow turbulence in He II above 1K

for t > tsat the decay of vortex line density (vorticity) both in grid-generated and counterflow-generated He II turbulence displays the classical exponent of -3/2

Skrbek, Gordeev, Soukup: Phys. Rev. E (2003)

Stalp, Skrbek, Donnelly:Phys. Rev. Lett. 82 (1999) 4831

3/ 2

3/ 23( ) *

2 eff

CL t t t

D

After saturation of the energy-containing length scale – universal decay law

S10

S6

Dependence on the channel size experimentally confirmed for the first time (even for classical turbulence)

The late decay -consistent withKolmogorov – type turbulence

He II bath level

Q

sv

The vortex line density decays exponentially

Turbulence in He II generated by pure superflow(there is no net normal fluid flow through the channel)

Sintered silver superleaks

Superfluid flow profile is flat – the channel width is no more relevant parameter

Conclusions (2)

•Developed turbulence in He superfluids displays features consistent with two different turbulent states•Kolmogorov state (classical-like regime)•Vinen state (quantum regime)

•Transitions between these states most likely occur and have been observed (T1 to TII transition as classified by Tough, decay of thermal counterflow)

•There are experimental data for He II and 3He-B, consistent with the existence of the flow phase diagram predicted by Volovik

Do quantum fluids hold the key to unlocking the underlying physics of fluid turbulence?