1. Quantum turbulence - an overview
L. Skrbek
Joint Low Temperature Laboratory,
Institute of Physics ASCR and Charles University
V Holešovičkách 2, Prague 8, Czech Republic
M. Krusius LTL HUT (ROTA), Helsinki
V.B. Eltsov, A.P. Finne J. Hosio
N.B. Kopnin, G.E. Volovik
P.V.E. McClintock, G. Pickett Lancaster University
D.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 Prague
T. Chagovets
W.F. Vinen University of Birmingham
R.J. Donnelly University of Oregon
S.R. Stalp
J.J. Niemela Trieste
K.R. Sreenivasan
P. Skyba Košice
COSLAB, Smolenice, September 2, 2005

2.                                                                       
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

3. Characteristic length scales in turbulence
Quantized vortex in He II

4. for thermally driven flows in a gravitational field
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
Navy (ship)
Aerospace (aircraft)
T (p)
 (cm2/s)
/
air
20 C
0,15
0,122
water
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

5. Phase diagram of 3He
Normal liquid 3He
Classical Navier-Stokes
fluid

6. Phase diagram of 4He Cryogenic He gas T (K) P (kPa)
Solid He
He I – normal liquid
Classical Navier-Stokes
P (kPa)
Superfluid He II
Critical point
Cryogenic He gas
T (K)

7. Quantum mechanical description of He II
Macroscopic wave function
Circulation- multiply connected region
Circulation –singly connected region
Quantized vortices in He II
Rotating bucket of He II
-thanks to the existence of rectilinear vortex lines
He II mimics solid body rotation
vorticity

8. 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)
Very low kinematic viscosity of the
normal fluid (above appr. 1 K)
Two-fluid model is also applicable to superfluid 3He phases (talk of Matti Krusius)

9. Systems (simplified) to understand turbulence in
4He
3He T
normal liquid He I
Classical Navier-Stokes fluid of extremely low kinematic viscosity 1
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 B
normal fluid of extremely normal fluid of of kinematic viscosity
low kinematic viscosity comparable with that of air
Inviscid superfluid Inviscid superfluid
Circulation is quantized Circulation is quantized
T limit
Pure superfluid Pure superfluid 3 2

10. Step 1 – classical turbulence in a viscous fluid
Grid turbulence- visualisation
Smoke
Ink or dye
Kalliroscope flakes
Hydrogen bubbles
Baker’s pH technique
Hot wire anemometry
Laser Doppler velocimetry
Particle image velocimetry …
Classical turbulence - Richardson cascade

11. 3d energy spectrum of the homogeneous isotropic turbulence
Energy containing eddies
Inertial range
Spectral energy density (log scale)
Dissipation range
Viscosity is unimportant ….. …….. up to here k
Energy containing
length scale
Dissipative (Kolmogorov)
length scale

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

13. For decaying vorticity, theree subsequent regimes predicted:
Simplest case – assume that the energy- containing scale is saturated - D
Note – on first approximation, intermittency does not change the power of the decay:

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

15. 3d energy spectrum of the superfluid homogeneous isotropic turbulence
Energy containing eddies
Inertial range, no dissipation
Spectral energy density (log scale)
Dissipation range due to phonon irradiation
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 ??? k
Energy containing
length scale
Quantum dissipative
length scale

16. Numerical evidence E(k)〜 k -n(t)
T. Araki, M.Tsubota and S.K.Nemirovskii, Phys.Rev.Lett.89, (2002): Energy Spectrum of Superfluid Turbulence with No Normal-Fluid Component
The 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.
△: 2 < k < ○: 2 < k < □: 2 < k < 16
E(k)〜 k -n(t)
E(k)
There are three works to date which studied the energy spectrum of a vortex tangle without mutual friction. One is the contribution of the Paris group, and the other two are ours.
5/3

17. Gross-Pitaevskii model by introducing the small-scale dissipation
M. Kobayashi and M. Tsubota, PRL94, (2005)obtained the Kolmogorov spectrum more clearly.
In order to overcome this difficulty, we have studied the turbulence of the Gross-Pitaevskii model in the wave number k-space by introducing the dissipative mechanism. First I will talk about the decaying turbulence, which is reported in this paper. Next, I will discuss steady turbulence of our very recent work. By introducing,..

18. Experimental evidence in He II
Davis et al, Physica B 280, 43 (2000).
It is clear that He II turbulence decays (on time scale of seconds) at very low temperature
Due to complexity of this experiment, quantitative comparison with the decay model is difficult

19. Experimental evidence in 3He B – Lancaster (courtesy of S. Fisher)
Classical spectral decay model –
universal decay

20. Step 3 – turbulence within the two-fluid model
Viscous dissipation
Phonon emission by Kelvin waves
Richardson cascade
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 together
The turbulent energy partly leaves the cascade, being transferred into heat, which makes the spectrum steeper
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
Mutual friction beyond some critical k becomes negligibly small again and Kolmogorov scaling is recovered
Phonon emission by high k Kelvin waves
D – size of our laboratory box

21. He II Could the entire energy spectrum be experimentally observed?
Typical cooling power does not exceed about 1[W] into 1 [liter] of liquid
In fluid dynamics, all energies are ment per unit volume, i.e. E [m2/s2]
Asssuming HIT, we use
Main distance between vortices
Kolmogorov dissipation length
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

22. 3He B
Typical cooling power does not exceed about 1[nW] into 1 [cm3] of liquid
Asssuming HIT, we get
Main distance between vortices
Kolmogorov dissipation length
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.
We need a theory for superfluid turbulence in a stationary normal fluid

23. Classical flow - Navier-Stokes equation Re depends on the geometry of flow in question
Dissipation:
Superfluid flow –by averaging the Euler equation over distance >> the intervortex spacing
assuming the normal fluid is at rest:
After rescaling the time (Volovik)
plays a role of inverse Reynolds number
and does NOT depend on the geometry
Dissipation is distinctly different, depends on the large-scale velocity U:

24. Classical or Kolmogorov regime quantum cutoff
energy in
energy in
Dissipation due to mutual friction
at large scales
viscous dissipation
Continuous approach is justified if circulation at the smallest scale >>circulation quantum
Classical or Kolmogorov regime quantum cutoff
L’vov, Nazarenko, Volovik:
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 recovered

25. Vinen considers the effect of mutual friction, which
can be written in the form
Considering the effect of mutual friction at different length scales Vinen obtains (for all length scales) a characteristic decay time time
The Kolmogorov spectrum will survive if this time exceeds the corresponding eddy turn-over time.
This is satisfied only for eddies smaller than
Bigger eddies cannot exist – they become fast dissipated by mutual friction
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

26. He II - experiment Inertial range Spectral energy density (log scale)
Energy containing eddies
Inertial range
Spectral energy density (log scale)
dissipation
Maurer, Tabeling:
Europhysics Lett. 43 (1998) 29
Flow between counterrotating discs
U=80cm/s; Re=2x106 a-2.3K; b-2.08K; c-1.4 K k
Energy containing
length scale
Dissipative (Kolmogorov)
length scale

27. NS HeII Local pressure fluctuations are measured
using a small total-head pressure tube
local pressure upstream flow
After removing the mean parts NS
HeII

28. 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 over
channel crossection )

29. Decay of the grid generated turbulence in He II – main features
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
There are four different regimes of the decay of vortex line density (vorticity – if defined as ) in the finite channel, characterized by:
Stalp, Skrbek, Donnelly: Decay of Grid Turbulence in a Finite Channel, Phys. Rev. Lett. 82 (1999) 4831
Skrbek, Niemela,Donnelly: Four Regimes of Decaying Grid Turbulence in a Finite Channel, Phys. Rev. Lett. 85 (2000) 2973

30. Energy spectrum – He II grid turbulence
Skrbek, Niemela, Sreenivasan: The Energy Spectrum of Grid – Generated Turbulence, Phys. Rev. E 64 (2001)
Quantum length scale
Is the classically generated He II coflow turbulence the same as classical?
Yes – on large length scales, exceeding the quantum length
No – on length scales comparable or smaller

31. Conclusions (1)
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….)
Richardson cascade
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 together
The turbulent energy partly leaves the cascade, being transferred into heat, which makes the spectrum steeper
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
Mutual friction beyond some critical k becomes negligibly small again and Kolmogorov scaling is recovered
Phonon emission by high k Kelvin waves
D – size of our laboratory box

32. 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)
Vortex ring
T 0 2b
Finite T
In counterflow, though, if rings with expand
Dimensional analysis and analogy with classical fluid dynamics leads to the Vinen equation:
Reproduced by Schwarz (1988) –
computer simulations
Local induction approximation
Importance of reconnections
production decay

33. Experiments on counterflow turbulence in superfluid He II
Joint Low Temperature Laboratory, Institute of Physics and Charles University, Prague
Detection method:
Second sound attenuation
Second sound is generated and detected by oscillating gold-plated porous nuclepore membranes
Steady-state data
superfluid
normal fluid
Vinen equation predicts the decay
of vortex line density of the form

34. 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
Note: without knowing anything of quantized vortices,
this simple model predicts that the extra attenuation
decays linearly with time!
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

35. Estimating the effective volume (i.e., V +half of the
counterflow channel volume) to be 7.5 cm3
Isothermal decaying turbulence
Thermally
driven decaying
turbulence

36. 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 ?

37. L Depolarization of the vortex tangle?
Wang, Swanson, Donnelly, PRB 36 (1987) 5236
vortex tangle generated in the steady state counterflow turbulence is polarized L
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
“randomization” - transverse second sound would indicate a net growth by a factor
up to i.e., up to about 23 %

38. Decay of counterflow turbulence in He II above 1K
Stalp, Skrbek, Donnelly:
Phys. Rev. Lett. 82 (1999) 4831
Skrbek, Gordeev, Soukup: Phys. Rev. E (2003)
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

39. After saturation of the energy-containing length scale – universal decay law
Dependence on the channel size experimentally confirmed
for the first time (even for classical turbulence)
S10
The late decay -consistent with
Kolmogorov – type turbulence S6

40. Turbulence in He II generated by pure superflow
(there is no net normal fluid flow through the channel)
He II bath level
Sintered silver superleaks
The vortex line density decays exponentially
Superfluid flow profile is flat – the channel width is no more
relevant parameter

41. 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?