Hydrodynamics and turbulence in classical and quantum fluids

V. Quantum turbulence experiments

flow

grid

(Approximately)

homogeneous

turbulence

E(k) = C 2/3 k-5/3Energy Cascade

RECALL

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

the density goes to zero) gives

44

s lvm

hn

md

circulation:

rv

2

Recall quantized vortices

He II

Superfluid grid flow

Theses: M.R. Smith, S.R. Stalp

Measure decay of L = length of vortex line per unit volume

Pocket-size! 1-cm square channel

Original grid: robust, 65% open brass

monoplanar grid with tines 1.5 mm

thick and mesh spacing of 0.167 cm

Newer grid: 28 rectangular tines of

width 0.012 cm forming 13 full meshes

across the channel of approximate

dimension 0.064 cm.

And with all its components: still a relatively simple

experiment, with one moving part

The entire apparatus sits in this 1-m diameter rotating rig at the University of

Oregon, which was originally a lathe chuck from General Motors now turned

on its side.

Second sound is excited and detected using

vibrating nuclepore membranes 9 mm in diameter

mounted flush on opposing walls of the channel.

The 6 micro-meter thick polycarbonate membranes

have a dense distribution of 0.1 micro-meter holes

and on one side is evaporated a think layer of gold

which makes contact with the channel wall.

The gold layer forms one electrode of a capacitor transducer, the other being

a brass electrode as shown. An ac signal of about 0.5 V peak to peak (in

addition to a 100 V DC bias) results in an oscillatory motion of the membrane.

Exciting second sound

The channel acts as a second sound resonator.

Typically a high harmonic n=50 is used to ensure

plane waves, which corresponds to about 20-40

kHz. A Lorenzian resonance peak is obtained have

a FWHM that is temperature dependent and

typically reaches values of Hz without

quantized vortices in the channel

This oscillation of the membrane thus creates a variation of the relative density

between normal and superfluid components. Because this density ratio is strongly

temperature dependent the resulting wave is also a temperature or entropy wave

and can be detected using either a similar mechanical transducer or a thermometer!

Second sound standing

wave resonance

In second sound the two fluid components move in antiphase (above right) such that

0nnss vv and the overall density and pressure remain constant.

Vinen and Hall: in experiments with a rotating container of He II they observed an

excess attentuation of second sound in direction perpendicular to rotation axis.

This extra attenuation resulted from scattering of the elementary excitations—

normal fluid– by the vortex lines and was absent for second sound propagating

parallel to the rotation axis.

The vorticity in the container was known: = 2 = L, where was the

angular velocity of the container, the quantum of circulation, and L the length of

vortex line per unit volume.

Calibration

The extra attenuation was found to be given by:

where B is a mutual friction coefficient, u2 the speed of second sound, and A, A0

are the amplitudes of the second sound resonance with and without vortices

present.

We can extend this to the case of a homogeneous vortex tangle. Taking into

account that vortices oriented parallel to the second sound propagation do

not contribute to the excess attenuation. Then we have for the total length

of quantized vortex line per unit volume:

116 00

A

A

BL

The length of quatized vortex line per unit volume L is

obtained from the second sound measurements through

the relation

Here’s the experimental procedure:

•Park the grid at the top of the channel, establish a

second sound standing wave and fit it to a

Lorentzian function (make sure it’s really parked!)

•Slowly lower the grid to the bottom and wait a bit.

•Pull the grid such that the velocity profile is linear

over most the the channel and, most of all, through

the test section

•Monitor the recovery of the second sound

resonance peak

Where, again, A and A0 are respectively the amplitudes

of the second sound standing wave resonance peak with

and without vortices present, B is the mutual friction

coefficient, and is the FWHM (see figure at right). A

more complicated formula applies more generally which

we shall ignore for now.

Why the grid should be against the TOP

wall when taking the reference

measurement…

The observed decay of L can be related to classical decaying turbulence

if we make the following assumptions:

1) In classical fluid turbulence the energy dissipation rate per unit mass is

related to the rms vorticity by the relation In the superfluid we assume that the energy dissipation per unit mass is given by ‘ L2,

where is the quantum of circulation and the coefficient ‘ is an effective

kinematic viscosity; i.e., we assume that the quantity L2 ~ , the total

rms vorticity.

A Kolmogorov like energy spectrum applies: k C k-5/3

Quasi-classical analysis the decay of the vortex line density

3/23/23/53/23/53/2

1 1

2

3dCdkkCdkkCE

dk

d d

Integrating (2) we have for the energy:

Here d is the size of the channel (the largest dimension of the measured volume)

dt

ddC

dt

dE 3/23/1

This total energy is decreasing slowly with time, which can described, at

least approximately, by allowing to be time-dependent. Therefore we can

write

Intergrating we get

32327 tdC

Substituting for

23

21

233 /

/

/

'

)(t

dCL

The only unknown quantity then is the effective kinematic viscosity

The Kolmogorov constant C can be taken equal to 1.5 which is its approximate

value in classical fluid turbulence.

In fact, we observe precisely a -3/2 roll-off of the line density vs time

T=1.3K

We see that this is indeed the case, even though for the experimental data shown

here at 1.3K the normal fluid fraction is nearly negligible ( roughly one percent).

By fitting the decay curves to the expression for L(t) we determine the value of the

only unknown: the effective kinematic viscosity ’

The black points are from the thesis of S.R.

Stalp using a robust but rather ―odd‖ grid:

The red points (Niemela, Sreenivasan and Donnelly, 2004) were taken using a

more conventional, albeit delicate, grid with 13 full meshes across the channel.

The dashed line is the kinematic viscosity of the total fluid defined as the ratio of the

shear viscosity of the normal component to the total density

All data are for a mesh Reynolds

number, ReM=150k, corresponding to

typical grid velocities of order 1 m s-1.

We note that values of the effective viscosity have the same order of magnitude as n , but a

different temperature dependence. The order-of-magnitude agreement with n is probably an

accident, arising from the fact that and n happen to have similar magnitudes.

The effective kinematic viscosity

Note that with the assumed expression for the energy dissipation

together with the numerical equivalence between and , we can estimate the average inter-vortex line spacing l:

i.e., the dissipation scale is of the same order of

magnitude as the average spacing between vortex lines

~

The line connecting plusses is a theoretical result for the effective kinematic viscosity

(Vinen & Niemela, 2002), which is proportional to the quantum of circulation:

dissL

413

21

21

2121

//

/

//

~

There sources of dissipation are the viscosity of the normal fluid and mutual

friction mutual friction force between the two fluids, which arises from a frictional

interaction between the normal fluid and the cores and nearby flow fields of the

quantized vortices.

The latter occurs only on length scales less than, or of order, l since otherwise the

two turbulent velocity fields are coupled. The former occurs at the dissipation

scale diss

Finally, how good was the assumption that there was a -3/2 power law

rather than something else?

Clearly, curve ―c‖, corresponding to the power 3/2, best represents

horizontality in this normalized plot.

Note: we can derive the expression for the decaying line density without

explicitly invoking Kolmogorov. We take the expression for the energy

dissipation rate that we considered before (lecture 2):

3uC where the constant 50.C

2

2

3u

dt

d

dt

dE

We assume that the length scale grows with time just as in classical turbulence

becoming comparable to the channel width d. We then can write:

3

2

2

C

du

Taking we havedt

d

C

d 31

32

/

/

Integrating and using 2)( L we obtain

23

21

2127 /

/

/

't

C

dL equivalent to what we found before with 50.C

Recall: the ―classical‖ French Washing Machine (Tabeling’s apparatus)

cryogenic hot wire

7 micron size

PDF of velocity increments Vr =V (x +r)−V (x)

showing intermittency at small scales

Superfluid ―washing machine‖ (Maurer and Tabeling 1998)

Counter-rotating disks

Hot wires don’t work because of the counterflow which would be set up

(giving rise to a large thermal conductivity in the fluid. Instead Maurer

and Tabeling made pressure fluctuation measurements which were

sensitive to the turbulent kinetic energy directly

(a)Helium I

(b)Helium II n ~ s

(c) Helium II s/

Strong evidence of classical energy cascade

Presumably we have that, in large-scale

turbulent flow of the superfluid phase, the

two fluids move with the same velocity

field, identical with that expected in a

classical fluid with density ( n + s) flowing

at high Reynolds number.

These experiments had at least two things in common: the fraction of normal

nonsuperfluid was small but not negligible, and the measurements were sensitive

to scales much larger than that of individual vortex lines in the turbulent state.

About the first, note that motion of a quantized vortex relative to the normal fluid

produces a mutual friction force, coupling the two fluids at large scales (as well as

providing dissipation at small ones), so it is not unthinkable then that both

normal and superfluid act together to produce a Kolmogorov spectrum.

This may take place as a result of a partial or complete polarization, or local alignment

of spin axes, of a large number of vortex filaments that mimics the range of eddies we

see in classical flows.

A simple example of such polarization under nonturbulent conditions is the mimicking of

solid body rotation in a rapidly rotating container filled with superfluid helium, which

results from the alignment of a large array of quantized vortices all along the axis of

rotation

At the scale of individual vortices, Schwarz (1985) developed numerical simulations of superfluid turbulence, based on the assumption that vortex filaments approachingeach other too closely will reconnect

Using entirely classical analysis, he was able to account for most of the experimental observations in the commonly studied thermal counterflow.

Koplik and Levine (1993), using the nonlinear Schrödinger equation, showedthat Schwarz’ assumptions about reconnections were correct.

A reconnection event (from Barenghi)

ReconnectionsWhat happens at smaller scales?

Vortex reconnections should be frequent in superfluid turbulence and this is a

fundamental difference from the classical case.

At absolute zero, where there is neither viscosity nor mutual friction to dissipate energy,

reconnections between vortices are expected to lead to Kelvin waves along the cores

allowing the energy cascade to proceed beyond the level of the intervortex line spacing.

Kelvin waves are defined as helical displacements of a rectilinear vortex line propagating

along the core. When a vortex reconnection occurs, the cusps or kinks at the crossing

point (see above) can relax into Kelvin waves and subsequent reconnections in the

turbulent regime generate more waves whose nonlinear interactions lead to a wide

spectrum of Kelvin waves extending to high frequencies.

At the highest frequencies (wave numbers) these waves can generate phonons, thus

dissipating the turbulent kinetic energy. The bridge between classical and quantum

regimes of turbulence it seems, must be provided by numerous reconnection events.

phonons

Classical Richardson cascade on

scales greater than vortex line

spacing.

Kelvin wave cascade on scales

less than .

energy flow

Phonons

energy flow

Dissipation of turbulent energy at T=0

T>0 T=0

The intermediate step between the cascades requires

reconnections of quantized vortices

sphere is trapped by vortex

simulations of C. Barenghi and colleagues

To observe vortices and vortex reconnections we

need to dress them with light scattering particles

particles

hydrogen, of the order of a micron in size

For example G.P. Bewley, D.P. Lathrop & K.R. Sreenivasan, Nature 441, 558 (2006). See also work by van Sciver’s group in FSU.

• Particles are produced by injecting a mixture of 2% H2 and 98% helium-4 into the liquid helium above the superfluid transition temperature.

• The volume fraction of hydrogen is 10-8–10-7 so that each vortex has only a few trapped particles

hollow glass spheres

White, Karpetis & Sreenivasan, J. Fluid Mech. 452, 189 (2002)Donnelly, Karpetis, Niemela, Sreenivasan, Vinen, White, J. Low Temp. Phys. 126, 327 (2002)

For a discussion of interaction between the fluid and particles in He II, see Sergeev, Barenghi & Kivotides, Phys. Rev. B 74,184506 (2006)

Panel (a) shows a suspension of hydrogen particles just above the transition

temperature. Panel (b) shows similar hydrogen particles after the fluid is cooled

below the lambda point. Some particles have collected along branching filaments,

while other are randomly distributed as before. Fewer free particles are apparent

in (b) only because the light intensity is reduced to highlight the brighter filaments in

the image. Panel (c) shows an example of particles arranged along vertical lines

when the system is rotating steadily about the vertical axis. G.P. Bewley, D.P.

Lathrop & K.R. Sreenivasan, Nature 441, 558 (2006)

just above T

just below T

R.P. Feynman (1955)

Prog. Low Temp. Phys. 1, 17

The cores of reconnecting vortices at the moment of reconnection, t0, and after

reconnection, t > to. The small circles mark the positions of particles trapped on

the cores of the vortices. The arrows indicate the motion of the vortices and

particles. The reconnected vortices recoil rapidly due to their large curvature

(local induction).

Each series of frames in (a), (b) and (c) are images of hydrogen particles suspended in liquid helium,

taken at 50 ms intervals. Some of the particles are trapped on quantized vortex cores, while others are

randomly distributed in the fluid. Before reconnection, particles drift collectively with the background flow

in a configuration similar to that shown in the first frames of (a), (b) and (c). Subsequent frames show

reconnection as the sudden motion of a group of particles. In (a), both vortices participating in the

reconnection have several particles along their cores. In projection, the approaching vortices in the first

frame appear crossed. In (b), particles make only one vortex visible, the other vortex probably has not

yet trapped any particles. In (c), we infer the existence of a pair of reconnecting vortices from the sudden

motion of pairs of particles recoiling from each other.