Upload
others
View
6
Download
0
Embed Size (px)
Citation preview
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.