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.

A brief historical digression….two years after the death of Boltzmann:

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].

That same year in Trieste….

“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

Discovery of superfluidity

Helium phase diagram

Three regions of interest for fluid turbulence:

•Near critical gaseous helium

•Helium I

•Helium II

A9.83mkT

hTthermal DeBroglie wavelength is larger than mean inter-atomic distance:

The Lambda point—resembles a Greek letter if you plot it right…

ReUL

U cannot be increased arbitrarily

without introducing another

parameter: the Mach number

(=U/a).

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.

Pushing Re to the limit: the case for helium

SF6

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.

Is it worth it to explore new working fluids and environments?

Quantum turbulence is a new regime which may shed light on classical turbulence

Quantum fluid

vn, n

vs, s

n

A9.83mkT

hTThermal DeBroglie wavelength is larger than mean inter-atomic distance:

Zero-point energy is large thus

avoiding solidification at

absolute zero:

px

2

4

2

2 RmE

~

+ weak interactions between

closed shell atoms.

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.

M

M

3

4

2

1

Clausius-Mossotti

(Small) Density of Helium

Niemela & Donnelly JLTP 1995

Macroscopic quantization

By analogy with the single particle quanum currents (orbiting electrons) it can

be postulated (London) that there exists a macroscopic wave function

)(riSe0

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

Sp

Sm

s

4

v

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.

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).

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.

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 two fluids interpenetrate freely without interaction

The superfluid carries no entropy and experiences no flow resistance: its

viscosity is identically zero and it is irrotational:

0sv

[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]

Two Fluid Model

For the superfluid component the simple two-fluid model (without mutual friction) gives:

Tspt

sss 1

vvv

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

2

nnn v

1vv

vn

n

s TSpt

TSpt

1vv

vss

s

0vv sn snt

0vnSt

S )((All entropy flows with the normal fluid)

nz 2

Measuring the density of the two fluids: Andronikashvili’s pendulum of oscillating disks

ns

Total density:

Spacing between plates is much lerss than the viscous penetration depth for the

normal fluid (“skin depth”):

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.

Demonstrating the two fluid nature

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.

0nsvt

s)(

All the entropy flows with

the normal fluid

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.

v

2

2C

Tsc

n

s

First sound: c1 ~ 200 m/s

Second sound: c2 ~ 20 m/s

Demonstrating Potential flow

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:

2n

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.

A curious observation

Regular arrays and irregular tangles of quantized vortices

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).

Visualizing indirectly the regular array of

vortices in a rotating bucket

As we shall see later, turbulent flows in the Kolmogov sense can mimic eddies

on all scales through partial polarization of vortex bundles.

Yarmchuk, et al. 1978

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:

0)v( sSs

t

2mRms *

Sm

s

v

where

j

ij

jj

sisj

sis

xx

p

x

vv

t

v

ji

ssij

xxm

ln

2

22

2

2

0

2s

m

Vppressure:

quantum stress:

Momentum

Without the quantum stress term the equations reduce to the Euler equations

The irrotational condition follows by taking the curl of the superfluid velocity:

0svSm

s

v

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

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

2

00

22

2 fmVa

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.

Vortices and vortex reconnections

ierf )(

The circulation in a multiply-connected region (around the core where

the density goes to zero) gives

44

s lvm

hn

md

circulation:

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.

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).

Each vortex has a tension (energy per unit length):

(~10-7erg cm-1, a0~0.1 nm, reff ~ l)0

2

v ln4

Ea

reffs

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

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.

Superfluid helium has been termed a “reconnecting Euler fluid” (Barenghi),

this being an important difference.

Alamri, Youd

& Barenghi,

2008

Reconnection of vortex bundles

To mimic classical turbulence with structures (eddies) on all scales, we

require bundling of quantized vortices. Do these bundles also

coherently reconnect?

Turbulence in helium II

We will consider in lecture 5 details of the quantum turbulence problem.

Phase diagram revisited: convective flows

4.4 K , 2 mbar:3/ 5.8 10

5.25 K, 2.4 bar:9/ 6.5 10

3HTgRa

Ra ~ ( 2 CP). Ra increases as 2 away from critical point and as CP in its vicinity.

Working near critical point different experiments need to use the same

temperature scale….

In cgs units

Water: 10

Air: 0.1

It is clear that the small kinematic viscosity of helium can help make Ra=( g L3

large, as we saw for Re.

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,

T2

VP TvCC /

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.

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

Dynamical similarity

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.

•X =Rf/Rp= kpH/(kf*Nu*e)

kp= thermal conductivity of plates

kf= thermal conductivity of fluid

e= thickness of the plates.

Note on corrections due to finite conductivity of the plates

107

108

109

1010

1011

1012

1013

1014

1015

1016

1017

-0.002

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

0.018

0.020Biot <<1 ~ 0.1 = constant temperature B.C.

Bio

t num

ber

Ra