A brief historical digression….two years after the death of Boltzmann:
Leiden, 1908: Kamerlingh Onnes succeeds in liquifying
helium, an element first identified spectroscopically in India in
1868 during a total eclipse of the sun (followed by its discovery
on Earth in the lava of Mount Vesuvius by the Italian physicist
Luigi Palmieri). This leads to the discovery of superconductivity
a few years later, superfluidity a few decades later, and many
diverse applications in science and engineering.
That same year in Trieste….
Trieste, 1908: Trieste resident James
Joyce receives a rejection letter for what
would become the first of his major works:
“Dubliners.” [It was finally published 6 years
later].
Discovery of superfluidity
“One usually considers that superfluidity was discovered in December
1937, the submission date of the two articles on the flow of liquid helium
which appeared side by side in Nature on January 8, 1938. On page 74
was the article by P. Kapitza and on page 75 the one by J.F. Allen
and A.D. Misener.”
S. Balibar, JLTP 2007
Helium phase diagram
Three regions of interest for fluid turbulence:
•Near critical gaseous helium
•Helium I
•Helium II
thermal DeBroglie wavelength is larger than mean inter-atomic distance:
T
h
3mkT

8.9 A
The Lambda point—resembles a Greek letter if you plot it right…
Pushing Re to the limit: the case for helium
Re
UL
U cannot be increased arbitrarily
without introducing another
parameter: the Mach number
(=U/a).
SF6
Increasing L has engineering
and financial limits.
Helium has the lowest kinematic
viscosity of any fluid, and while
some pressurized gases have
comparable values (at high
pressures) they have problems
of their own and lack the
versatility of helium.
Is it worth it to explore new working fluids and environments?
If high enough Reynolds numbers (meaning any Re above the point at which
certain scaling properties in the inertial range cease to vary) can indeed be
attained using common fluids such as air or water, there is little sense in
pushing the low temperature technology. This, however, is not the case in
general.
Another salient fact is that the increase in the Reynolds number is measured
in terms of its logarithm, and, in some respects, it makes sense to consider
changes that may occur when Re changes by one or more decades.
For this reason, a good experiment is one that permits Re to be explored
over many decades— preferably in a single apparatus so we may neglect
other effects such as changed boundary conditions, geometry and
experimental protocol.
This possibility can be realized using helium as the test fluid.
Quantum turbulence is a new regime which may shed light on classical turbulence
Zero-point energy is large thus
avoiding solidification at
absolute zero:
x p
Quantum fluid
vn,
vs,
n
s
n

2
E~
2m4 R 2
+ weak interactions between
closed shell atoms.
Thermal DeBroglie wavelength is larger than mean inter-atomic distance:
T
h
3mkT

8.9 A
Superfluidity exists below lambda point
In analogy with Bose-Einstein condensation, some fraction of the helium
atoms will condense into the zero momentum ground state (maximum around
14% at T=0).
Helium II is of course a liquid but is perhaps closer to a non-ideal (interacting)
gas, having a large molar volume.
(Small) Density of Helium
Niemela & Donnelly JLTP 1995
Clausius-Mossotti
1
2
4
M
3M
Macroscopic quantization
By analogy with the single particle quanum currents (orbiting electrons) it can
be postulated (London) that there exists a macroscopic wave function
0
eiS (r )
where S(r) is the phase (a real function of position) and that the wave function
is governed by the ordinary single particle wave equation.
We have for the momentum p=m4vs
p
vs
 S

S
m4
The velocity is related to the gradient of the phase we can have phase coherence
over macroscopic regions of the liquid. Superfluidity may qualitatively be thought of
as a natural consequence of this coherence.
The same argument holds for helium-3 but we need 2m3 in the velocity relation to
account for the required pairing for the Fermion system.
Two Fluid Model
Helium II behaves as if it consisted of two separate fluids according to the two fluid theory of Tisza and Landau: a “normal” component and a “superfluid”
component.
The two fluids interpenetrate freely without interaction
Each fluid component has its own density and velocity field, ρn and vn for the
normal fluid and ρs and vs for the superfluid. The total density of Helium II is
the sum of the two separate densities: ρ = ρn + ρs.
The superfluid carries no entropy and experiences no flow resistance: its
viscosity is identically zero and it is irrotational:
vs 0
The normal fluid component carries the entire entropy and
viscosity (ηn) of Helium II and is similar to a classical,
viscous Navier - Stokes fluid. It can be considered as a gas
of thermal excitations (phonons and rotons).
[Note: There is constant confusion about the term
“superfluid” which is sometimes used to refer to helium II
and sometimes to the component of helium II]
For the superfluid component the simple two-fluid model (without mutual friction) gives:
vs
t
vs
1
vs
p s T
where s is the specific entropy. This looks like the classical Euler equation except for
the last term. This term allows for the thermo-mechanical effect; i.e. pressure
gradients can result from gradients in temperature. This equation is only valid at low
velocities--otherwise we will need to add a mutual friction term between the two
components. The simple two-fluid equations are:
n
t
( S)
t
vs
t
vn
t
vn
s
Sv n
vs
vn
vs
vn
vs
0
(All entropy flows with the normal fluid)
0
1
1
p S T
p
s
n
S T
2
n
vn
Demonstrating the two fluid nature
Measuring the density of the two fluids: Andronikashvili’s pendulum of oscillating disks
Spacing between plates is much lerss than the viscous penetration depth for the
normal fluid (“skin depth”):
z
2
n
The normal fluid was entrained and contributed to the moment of inertia of the
pendulum bob while the superfluid remained stationary. Since the total density is
easily measured separately, the superfluid density could then be found.
Total density:
s
n
Consequences of the two-fluid model
Counterflow
In a channel open at one end to a helium II bath, a heater placed at the closed end
causes a counterflow. Here the normal fluid flows away from the heater carrying
the heat and to conserve mass the superfluid component flows toward the heater.
This is a common method of producing turbulence in the superfluid but it has no
obvious classical analogue.
All the entropy flows with
the normal fluid
( s)
t
Second sound
Perturbations in heat obey a wave equation rather than
a diffusion equation. Second sound occurs for constant
total density but varying fraction of normal and
superfuid densities. Can be produced by pulsed
heaters and detected mechanically or vice versa.
svn
c2
0
2
s
s T
nCv
First sound: c1 ~ 200 m/s
Second sound: c2 ~ 20 m/s
Demonstrating Potential flow
A curious observation
Rotating containers of helium II were observed to have a
parabolic meniscus (Osborne, 1950). The shape of the
meniscus was independent of temperature which was surprising
since it was assumed that the superfluid component would not
rotate as a solid body.
In fact, this was resolved by considering the fluid to be
threaded with an array of quantized vortices whose
number obeyed Feynman’s rule:
n
2
Note, here the angular velocity is denoted by Ω, rather than the vorticity as we used before.
The vorticity is equal to 2Ω in solid body rotation, hence Feynman’s rule says that a sufficient
number of vortices will be produced to mimic solid body rotation in the superfluid. Clearly this
only works well for n large.
Regular arrays and irregular tangles of quantized vortices
Visualizing indirectly the regular array of
vortices in a rotating bucket
The simulated tangle of quantized vortices on
the left corresponds to 1.6K, while that on the
right is at 0K. After Tsubota, et al (2000).
Yarmchuk, et al. 1978
As we shall see later, turbulent flows in the Kolmogov sense can mimic eddies
on all scales through partial polarization of vortex bundles.
NLSE for the condensate
A condensate of weakly interacting Bose particles is described by a single particle
wavefunction of N bosons of mass m which obeys the NLSE (or Gross-Pitaevski
equation)
where V0 is the strength of the repulsive interaction between the bosons and
Ev is the energy increase on adding one boson (chemical potential).
For fluid dynamic applications we can apply the so-called Madelung transformation
where R is the amplitude and S is phase of .
If we substitute this into the NLSE we obtain the continuity equation:
s
(
t
s
vs
S
vs ) 0
m

S
m
where
* mR 2
The irrotational condition follows by taking the curl of the superfluid velocity:

S
m
vs
vs
0
Momentum
s
vsi
t
vsj
vsi
xj
p
xj
quantum stress:
pressure:
p
ij
xj
ij
V0
2m 2

2m
2
2
s
ln s
xi x j
2
s
Without the quantum stress term the equations reduce to the Euler equations
Vortices and vortex reconnections
Koplik and Levine (PRL 1993) considered a single-quantum rectilinear
vortex along r =0 as a two dimensional solution to the NSLE in cylindrical
polar coordinates and described by the function
f (r )ei
where the function f (r) goes to zero at r = 0 and tends to a constant value (f0)
for r large compared with the coherence length a, given by
a
2
2
2mV0 f 02
where f0 is the value of f at large r. In this model the vortex core can be identified
with the coherence length and the density is identified with the superfluid density.
Estimates for the core:
The coherence length is approximately 0.5 Angstrom for helium 4.
Assuming a cylindrical “hole” in the fluid in which the Bernoulli force was
just balanced by surface tension, Feynman was able to “estimate” a core
size approximately 0.3 Angstrom (where of course the concept of a
surface tension would not be valid).
The circulation in a multiply-connected region (around the core where
the density goes to zero) gives
vs dl
circulation:

m4
n
h
m4
Each vortex has a tension (energy per unit length):
2
Ev
s
4
ln
reff
a0
(~10-7erg cm-1, a0~0.1 nm, reff ~ l)
There is no evidence for multiply-quantized vortices (n=1 only). Since the energy
of a vortex line goes as it is energetically favorable to have vortices with a
single quantum of circulation. Experiments by Vinen (1957) have confirmed this.
The quantum stress we saw earlier is important only for distances from r=0 of
≈10-8 cm. This is much smaller than the typical vortex separation in turbulent
superfluid flows which is between 10-3 and 10-4 cm. Hence away from the core
we have the classical Euler equations.
Near the core, where the stress term is important, reconnections can occur
which are not allowed in an Euler fluid (unless viscosity is somehow restored).
Vortices and vortex reconnections in the NLSE
Superfluid helium has been termed a “reconnecting Euler fluid” (Barenghi),
this being an important difference.
To mimic classical turbulence with structures (eddies) on all scales, we
require bundling of quantized vortices. Do these bundles also
coherently reconnect?
Alamri, Youd
& Barenghi,
2008
Reconnection of vortex bundles
Turbulence in helium II
We will consider in lecture 5 details of the quantum turbulence problem.
Phase diagram revisited: convective flows
Ra
4.4 K , 2 mbar:
/
5.8 10
Ra ~ (
2
CP). Ra increases as
2
T H3
g
3
In cgs units
5.25 K, 2.4 bar:
/
6.5 109
Water: 10
Air:
0.1
away from critical point and as CP in its vicinity.
Working near critical point different experiments need to use the same
temperature scale….
It is clear that the small kinematic viscosity of helium can help make Ra=(
large, as we saw for Re.
g
L3
In addition, for helium gas, the thermal diffusivity =kf / CP , where kf is the thermal
conductivity of the fluid, and CP is the constant pressure specific heat, can be both very
small or very large depending on the the distance from the critical point where CP
diverges.
The isobaric thermal expansion coefficient: α≡−ρ−1∂ρ/∂T
For non-interacting gases, α =1/T , and so, low temperatures themselves have a
particular advantage for buoyancy-driven flows. For helium near its critical point, α is
thermodynamically related to the specific heat and therefore also diverges, ie,
CP
CV
Tv
2
/
T
where CV is the specific heat at constant volume (weak divergence), v is the molar volume,
and T is the isothermal compressibility. We will return to this when we consider in more
detail thermal convection in cryogenic helium.
Dynamical similarity
For given initial and boundary conditions, the state of the flow is
the same if the relevant dynamical parameters are the same.
e.g., Re = UL/ , Ra =g( /
L3, …
These parameters vary by many orders of magnitude in practice
Example
Ra
Re
Sun
1022
1013
Ocean
1019
1010
Atmosphere
1015
109
Naval applications
---
109
Aerospace applications
---
5 108
Material properties
An indirect benefit of the use of low temperature helium comes from the relation between
the fluid properties and those of the solid material used to confine it.
Boundary conditions play an important role in fluid mechanics and, in the case of thermal
turbulence, for instance, the usual boundary condition at the horizontal surfaces is one of
constant temperature. In practice, this condition is effected by using highly conductive
plates with constant heating or with good temperature control. Annealed and oxygenf ree
copper has a thermal conductivity of order 1kWm−1 K−1 to be compared with the nominal
thermal conductivity of helium, which is of order 10−2 W m−1 K−1.
The effective thermal conductivity of the fluid at very high Ra can be quite large because
of turbulence, in some experiments as high as 10,000 times that of the quiescent fluid.
This brings the effective conductivity considerably closer to that of the plates and
necessitates some correction. As we shall see later in more detail, the correction is
negligible for helium except at the very highest Ra where it is of the order of a few
percent.
On the other hand, it is typically of the order of 10-20% for conventional fluids even when
the Ra is moderately high and orders of magnitude lower than that achieved with helium.
Note on corrections due to finite conductivity of the plates
0.020
Biot <<1 ~ 0.1 = constant temperature B.C.
0.018
•X =Rf/Rp= kpH/(kf*Nu*e)
0.016
kp= thermal conductivity of plates
kf= thermal conductivity of fluid
e= thickness of the plates.
Biot number
0.014
0.012
0.010
0.008
0.006
0.004
0.002
0.000
-0.002
7
10
8
10
9
10
10 10
11
10
12
10
13
10
Ra
14
10
10
15
16
10
17
10