Hydrodynamics and turbulence in classical and quantum fluids V. Quantum turbulence experiments

Transcription

1 Hydrodynamics and turbulence in classical and quantum fluids V. Quantum turbulence experiments

2 grid RECALL flow (Approximately) homogeneous turbulence Energy Cascade E(k) = C 2/3 k -5/3

3 Recall quantized vortices The circulation in a multiply-connected region (around the core where the density goes to zero) gives circulation: v s dl m 4 n h m 4 v 2 r

4 Superfluid grid flow Theses: M.R. Smith, S.R. Stalp Pocket-size! 1-cm square channel He II Original grid: robust, 65% open brass monoplanar grid with tines 1.5 mm thick and mesh spacing of cm Newer grid: 28 rectangular tines of width cm forming 13 full meshes across the channel of approximate dimension cm. Measure decay of L = length of vortex line per unit volume

5 And with all its components: still a relatively simple experiment, with one moving part

6 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.

7 Exciting second sound 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.

8 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! In second sound the two fluid components move in antiphase (above right) such that s v s n v n 0 and the overall density and pressure remain constant. The channel acts as a second sound resonator. Typically a high harmonic n=50 is used to ensure plane waves, which corresponds to about 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 Second sound standing wave resonance

9 Calibration 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.

10 The extra attenuation was found to be given by: where B is a mutual friction coefficient, u 2 the speed of second sound, and A, A 0 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: L 16 0 A0 B A 1

11 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 Why the grid should be against the TOP wall when taking the reference measurement The length of quatized vortex line per unit volume L is obtained from the second sound measurements through the relation Where, again, A and A 0 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.

12 Quasi-classical analysis the decay of the vortex line density 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 L 2, where is the quantum of circulation and the coefficient is an effective kinematic viscosity; i.e., we assume that the quantity L 2 ~, the total rms vorticity. A Kolmogorov like energy spectrum applies: k C k -5/3 Integrating (2) we have for the energy: E d k d C 2/3 k 5/3 dk 1 1 d C 2/3 k 5/3 dk 3 C 2 2/3 d 2/3 Here d is the size of the channel (the largest dimension of the measured volume)

13 This total energy is decreasing slowly with time, which can described, at least approximately, by allowing to be time-dependent. Therefore we can write de 1/3 2/ d C d 3 dt dt Intergrating we get C d t 3 Substituting for L ( 3/ 2 3 C) d 3/ 2 t 1/ 2 ' The Kolmogorov constant C can be taken equal to 1.5 which is its approximate value in classical fluid turbulence. The only unknown quantity then is the effective kinematic viscosity

14 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

15 The effective kinematic viscosity All data are for a mesh Reynolds number, Re M =150k, corresponding to typical grid velocities of order 1 m s -1. 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 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.

16 The line connecting plusses is a theoretical result for the effective kinematic viscosity (Vinen & Niemela, 2002), which is proportional to the quantum of circulation: 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: ~ 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 L 1/ 2 1/ 2 1/ 2 1/ 2 ~ 3 1/ 4 diss i.e., the dissipation scale is of the same order of magnitude as the average spacing between vortex lines

17 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.

18 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): 3 u C where the constant C 0. 5 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: u 2 d C 2 3 Taking de dt d dt 3 u 2 2 we have 2/ 3 d 1/ 3 C d dt Integrating and using 2 ( L) we obtain 27 d / ' C 1/ 2 L 3 2 t 1/ 2 equivalent to what we found before with C 0. 5

19 Recall: the classical French Washing Machine (Tabeling s apparatus) cryogenic hot wire 7 micron size PDF of velocity increments V r =V (x +r) V (x) showing intermittency at small scales

20 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

21 Strong evidence of classical energy cascade (a) Helium I (b) Helium II n ~ s (c) Helium II s/ 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.

22 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

23 What happens at smaller scales? Reconnections A reconnection event (from Barenghi) At the scale of individual vortices, Schwarz (1985) developed numerical simulations of superfluid turbulence, based on the assumption that vortex filaments approaching each 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, showed that Schwarz assumptions about reconnections were correct.

24 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.

25 Dissipation of turbulent energy at T=0 T>0 T=0 Classical Richardson cascade on scales greater than vortex line spacing. energy flow Kelvin wave cascade on scales less than. energy flow phonons Phonons

26 The intermediate step between the cascades requires reconnections of quantized vortices

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28 To observe vortices and vortex reconnections we need to dress them with light scattering particles sphere is trapped by vortex simulations of C. Barenghi and colleagues

29 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% H 2 and 98% helium-4 into the liquid helium above the superfluid transition temperature. The volume fraction of hydrogen is 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, (2006)

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31 just below T just above T 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)

32 R.P. Feynman (1955) Prog. Low Temp. Phys. 1, 17

33 The cores of reconnecting vortices at the moment of reconnection, t 0, and after reconnection, t > t o. 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).

34 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.

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