Experiments on counterflow and pure superflow turbulence in ...

<strong>Experiments</strong> on counterflow and pure superflow
turbulence in superfluid 4He
New cryohall in Prague
S. Babuin, T. Chagovets, M. Rotter, M. Stammeier
and L. Skrbek
Joint Low Temperature Laboratory,
Faculty of Mathematics and Physics,
Charles University &
Institute of Physics ASCR
V Holešovičkách 2, 180 00 Prague 8, Czech Republic
Fluctuations and Coherence:
from Superfluids to Living Systems
Lancaster, UK, 13-16 July 2011

Thermal counterflow in He II
Two-fluid model (Landau)
W.F. (Joe) Vinen,
55 years ago…
V
V
N
N
N
Q
AST
V
S
s
Experimental observation: quantized vortices attenuate second sound

Counterflow tangle – courtesy of Makoto Tsubota

,
However....
Heat transport efficiency
Ra
13 /
1/3
T
Q
Nu
'
Q
2
Vortex line density
L
Q
2

Counterflow turbulence phenomenology (Vinen 1957)
2b
Vortex ring
v
t
8b
ln
4b
a
1
4
b
T 0
Finite T
In counterflow, though, if v v v V
rings with expand
t
n
s
CF
v t
b b
Dimensional analysis and analogy with classical fluid dynamics leads to the Vinen equation:
dL B n
3/ 2 2
L – voxtex line density
1 VCFL
2
L ( g(
VCF))
dt 2
m
Reproduced by Schwarz (1988) –
4
computer simulations
production decay
reconnections
V CF
For steady there is a steady value of L
Early results reviwed by J.T. Tough Turbulent states I, II, III
c
Decay of counterflow turbulence:
dL
dt
m
2
2
L
4
L
t
1 Numerous
t
t VO
experiments
(Vinen, Schwarz, Milliken…)

Turbulent states TI and TII
Allen and
Reekie rule
Two transitions
observed in counterflow in circular and
small aspect ratio rectangular channels
•temperature difference
•pressure difference
•second sound attenuation
•But only one in thin channels

Thanks to many investigators, especially to Vinen, Tough,
Simon, Donnelly, Glaberson, Schwarz, Barenghi ….
a fair level of understanding of counterflow turbulence in He
II has been achieved by the end of eighties and almost all
work has been terminated
Still, puzzles remain
•Anomalous decay of counterflow turbulence
•Nature of turbulent states TI, TII, TIII
(in particular, is the counterflowing normal fluid turbulent)
•Relationship to classical turbulence
•......

Prague second sound counterfow experiments
gold-plated
nuclepore membrane
Lock-in
A0
A
S
generator
Brass electrode
heater
Three different CF channels:
L
6
s
B
A
0
A
1
C9 S7 S10

Second sound amplitude (10 -4 V)
Second sound signal - counterflow turbulence
Can we treat our data on decaying
counterflow turbulence as isothermal
Yes
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)
heater on
heater off
time (s)
1. Initial fast decay
2. Intermediate –slow - stage
3. Late power law decay

Grid turbulence in He II -
Oregon decay experiments
Decay of counterflow
turbulence in He II
Late decay of vortex line density
(vorticity) both in grid-generated and
counterflow-generated He II turbulence
displays the classical exponent of -3/2

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)
Gordeev, Chagovets, Soukup, Skrbek,
JLTP 138 (2005) 554)
S10
S7
The late decay -consistent with
Kolmogorov – type turbulence
3/ 2
D3C
L( t) t t
*
2
eff
3/ 2
Chagovets, Gordeev, Skrbek,
Phys. Rev. E 76, 027301, 2007.

10 mm x10 mm channel
A puzzle…
Intermediate – anomalously slow
decay
…..resolved
(analytically and numerically)
Depolarization
of the vortex tangle

vortex line density (arbitrary units)
Second sound signal
Full Biot-Savart calculations
As well as LIA approximation
0.1
in a periodic box 1 cm, T= 2 K
counterflow 1 cm/s ,
vortex tangle L=190 cm
Side view of the tangle
x,y-directions
0.01
0.1 1 10
tim e since the heater is off (s)
z-direction
Time (s)

Pure superflow – early results by Tough’s group
Only one turbulent state !
Experimentally observed
quantity is T-difference
Ashton, Tough, PRL46 658(1981)

Critical velocity was found (Baehr and Tough, PRL 51 (1983) 2295)
to be 1-2 cm/s and was given in the dimensionless form of
“superfluid Reynolds number” (in analogy with thermal counterflow)
Why does pure superflow differ from thermal counterflow
How does it decay
Can we learn something more while investigating it

Prague pure superflow setup
(based on the fountain pump)
He bath level
superleaks
Q
He II

Q
Prague pure superflow setup
He II
Level
meter
and lift
system
Second
sound
sensors
v s
Fountain effect in He II
He II
Ag sintr
heater

Second sound amplitude (at resonance) – raw data
heater
On
Off
We know the mean superfluid velocity based on the power applied to
the fountain pump. Our analysis relies heavilly on this formula.

Steady-state data
1.92 K
1.73 K
1.58 K
1.49 K
•On increasing power,
there are three states
•Pure superflow
L=0
•A - state (known)
•We have discovered
a new B - state
Agrees within error bars with
the data of Baehr and Tough

Critical velocity
(cm/s)
(K)
K.W. Schwarz: Phys. Rev. B 18, 245 - 262 (1978)
Our observation of the critical velocity in
two channels of sizes almost two orders
of magnitude wider agrees within the
error bars with that of Baehr and Tough
and shows that for pure superflow (or,
via Galilean transformation also for
homogeneous counterflow) “superfluid
Re” scaling does not hold and this
critical velocity is an intrinsic
property of the self-sustaining vortex
tangle, in accord with the early
considerations and calculations of
Schwarz

How to understand the B - State
We believe the key point is to realize the ability of superfluid to mimic classical flows.
•Consider a classical viscous laminar flow through a circular pipe of the size of our
channel, with parabolic Poiseuille flow profile
•It results in the solenoidal vorticity profile, with the mean
•Assuming that superflow mimics this classical behavior, the required vortex line
density that would be observed by second sound must exceed
•This simple physical picture is in qualitative agreement with our observations.
Allowing for the ”slip velocity”, it correctly predicts the linear behavior in the B – state
with temperature independent
Could we verify such
a scenario
What about the
decay data

Heater off
Decay data
Viscous decay of
such a flow structure
2k
1 2
With no fitting parameters these calculated values of effective kinematic
viscosity agree with those deduced from (i) decaying towed grid turbulence;
(ii) decaying counterflow turbulence; (iii) spin-down experiments (Golov);
(iv) theoretical estimates of Vinen and Niemela

Generation of QT
mechanical vs thermal

L 1/2 (cm -1 )
3000
2500
2000
Steady-state vortex line density
T (K)
1.35
1.45
1.65
1.75
1.95
1500
1000
500
300
200
100
0
0 2 4 6 8 10 12 14 16 18 20 22 24
v (cm/s)
0
0.0 0.5 1.0 1.5 2.0
Critical velocity

350
300
250
A: present work
B: TC & LS (2008)
C: Ashton et al. (1981)
D: Schwarz theory (1978)
E: Adachi et al. theory (2010)
200
s/cm 2
150
100
50
0
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1
T (K)

L (10 5 cm -2 )
14
12
10
8
A
T(K)
1.35
1.45
1.65
1.75
1.95
6
4
2
1.92 K
1.73 K 1.58 K
1.49 K
B
0
0 2 4 6 8 10 12 14 16 18
v (cm/s)

L (cm -2 )
L (cm -2 )
L (cm -2 )
10 6
T = 1.35 K, 5 samples at flow speed = 15.6 cm/s
Decay data
10 7 steady state
10 6
flow speed
(cm/s)
15.6
10.4
6.9
4.3
10 5
10 4
10 5
10 4
10 3
-3/2
power
10 2
0.1 1 10 100
time (s)
curves smoothened by 25 point averaging
10 5
T=1.35 K
(1e6)*exp(-(x-1.2)/0.05) + (1e5)*(x^(-1.5))
10 6 T = 1.35 K, 5 samples at flow speed = 15.6 cm/s
Fast initial exponential
decay, then crossover
to (quasi)classical -3/2
power law decay
10 3
T = 1.35 K
10 2
0.1 1 10 100
t (s)
t -3/2
But: it does not work nearly as
nicely at higher temperatures
0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4
time (s)

Summary
• We have experimentally investigated flow of superffluid 4He restricted to pure
net superflow through the channel, generated both thermally and mechanically
• We confirm the existence of weakly temperature dependent critical velocity of order 1
cm/s, which does not scale with the channel size and is therefore an intrinsic property of selfsustained
vortex tangle.
• A new B - state is discovered, characterized by where seems
temperature independent, however, only when QT is generated thermally. Mechanical
generation produces quadratic dependence of vortex line density on flow velocity (called A-
state), as in thermal counterflow.
•Two different states of developed QT can exist in pure superflow !!
•In steady state, they display different vortex line density as well as its functional dependence
on mean superflow velocity (linear versus quadratic). Moreover, their decays differ drastically
(late exponential versus classical-like power law decay). Accepted values of effective kinematic
viscosity can be extracted from both these decays, based on very different theoretical models.
•A lot of new data measured, only partly analysed and not yet published....
Half a century after the discovery of thermally generated quantum turbulence
in He II there is still plenty of interesting physics to play with…

L (cm -2 )
25 pts adjacent averaging
10 7 t -1 steady state
10 6
10 5
flow speed (cm/s)
9.1
7.8
6.5
5.2
3.9
10 4
10 3
t -3/2
10 2
T = 1.65 K
10 1
0.1 1 10 100
t (s)