Visualization of two-fluid flows of superfluid helium-4 SPECIAL FEATURE
Wei Guoa,b, Marco La Mantiac, Daniel P. Lathropd, and Steven W. Van Scivera,b,1
aMechanical Engineering Department, Florida State University, Tallahassee, FL 32303; bNational High Magnetic Field Laboratory, Florida State University, Tallahassee, FL 32310; cDepartment of Low Temperature Physics, Faculty of Mathematics and Physics, Charles University, 180 00 Prague, Czech Republic; and dDepartments of Physics and Geology, Institute for Research in Electronics and Applied Physics, and Institute for Physics Science and Technology, University of Maryland, College Park, MD 20742
Edited by Katepalli R. Sreenivasan, New York University, New York, NY, and approved December 13, 2013 (received for review July 17, 2013) Cryogenic flow visualization techniques have been proved in not kept pace, in part due to the extremely low temperature and recent years to be a very powerful experimental method to study low density of the fluid. A number of early efforts were devoted superfluid turbulence. Micron-sized solid particles and metastable to producing macroscopic particles for qualitative investigations helium molecules are specifically being used to investigate in (10–12) and the challenge of producing neutrally buoyant par- detail the dynamics of quantum flows. These studies belong to ticles that faithfully follow the complex flow fields has been the a well-established, interdisciplinary line of inquiry that focuses on main impediment to quantitative advancement. In addition, sev- the deeper understanding of turbulence, one of the open problem eral attempts have been made to visualize fluid dynamics in su- of modern physics, relevant to many research fields, ranging from perfluid helium with microscopic tracers, which include neutron fluid mechanics to cosmology. Progress made to date is discussed, absorption tomography, using 3He particles (13), and acoustic to highlight its relevance to a wider scientific community, and cavitation imaging, using electron bubbles (14). These small par- future directions are outlined. The latter include, e.g., detailed ticles are expected to follow the fluid motion, because Stokes drag, studies of normal-fluid turbulence, dissipative mechanisms, and from the normal-fluid flow, is deemed to dominate other forces. unsteady/oscillatory flows. However, these methods have specific challenges. Neutron ab- sorption tomography requires a finely collimated neutron beam uantum fluids have been studied experimentally for many and the ability to raster the neutron beam through the region of years and have by now become a major focus of low-tem- interest. The electron-bubble cavitation method relies on the Q PHYSICS perature physics (ref. 1 and references therein). Applications of generation of strong ultrasonic sound waves in helium that in- the subject are widely ranged, from engineering, where super- evitably disturb the flow to be studied. Recently, the groups rep- fluid 4He is used as a coolant for superconducting magnets and resented by the present authors have successfully developed a infrared detectors (2), to astrophysics, where superfluidity is in- number of liquid helium flow-visualization techniques: particle voked to explain glitches in the rotation of neutron stars (3, 4) image velocimetry (PIV) and particle tracking velocimetry (PTV) and the formation of cosmic strings (5, 6). More recently, su- techniques, using micron-sized solid particles (15–21), and a laser- perfluidity has been used to describe the collective behavior of induced fluorescence imaging technique, using angstrom-sized p birds (7) and a cosmological model has been used to obtain He2 excited molecules (22, 23). results relevant to superfluid turbulence (8). The latter form of turbulence, occurring in quantum fluids, is indeed an especially PIV and PTV Techniques. PIV and PTV are valuable, quantitative interesting topic because of its quantum peculiarities and its tools that have been applied to study many scientific and in- similarity to classical turbulence. Superfluids, in which turbu- dustrial problems (24). PIV can estimate the fluid velocity in lence can be directly visualized and studied, include superfluid a section of the flow field, by assuming a single, smoothly varying 4He and atomic Bose–Einstein condensates (9). Due to the limit velocity field, whereas PTV allows the measurement of La- of small sample volumes, the experimental study of turbulence in grangian quantities, i.e., the local velocity and its derivatives. Bose–Einstein condensates has hardly begun. The development With both techniques the particles are suspended in the fluid and of visualization techniques applicable to superfluid 4He is thus reflect the light from a laser sheet that illuminates the flow field essential, if our understanding of quantum turbulence is to make of interest. The time-dependent positions of the particles are significant progress in the near future. thus captured and analyzed by a suitable digital imaging system. Superfluid 4He is viewed as consisting of two interpenetrating The particles for liquid helium experimentation can be broadly fluids. The gas of thermal excitations forms the normal compo- classified into two categories: solid particles, as are often used in nent, which can be considered as a viscous fluid. The superfluid classical fluid dynamics experiments, and solidified particles, component is inviscid and its rotational motion is possible only in produced by injecting gases (usually hydrogen or deuterium) into the presence of topological defects, in the form of quantized liquid helium. Micron-sized solid particles have been successfully vortex filaments. Turbulence in the superfluid component there- used in conjunction with the PIV technique to observe broad, fore takes the form of a tangle of quantized vortex lines. Turbu- average properties of the turbulent state of superfluid helium lence in the normal fluid is more conventional, although the (15, 25, 26). However, such particles have proved to be too dense interaction between the normal fluid and the vortices leads to the to explore the detailed structure of quantum turbulence. As a nonclassical force of mutual friction between the two fluids. result, most recent experiments have used solidified hydrogen (or Turbulence in such a system can exhibit a behavior that is similar deuterium) particles (16–21). To produce these particles, a gas- to that found in a classical fluid; but it may take forms that are eous mixture of helium and hydrogen in a volume ratio of ∼100:1 unknown in classical fluid mechanics: for example, forms rele- is injected directly into liquid helium. A cloud of solid particles vant to a fluid in which there is no viscous dissipation, and those with diameters typically of a few microns can be produced. that depend on the coexistence of the two fluids. Study of quan- tum turbulence can therefore enrich our knowledge of turbulence in general, as well as being interesting in its own right. Author contributions: W.G., M.L.M., D.P.L., and S.W.V.S. designed research; W.G. and M.L.M. performed research; W.G., M.L.M., and D.P.L. analyzed data; and W.G., M.L.M., D.P.L., and Visualization Techniques S.W.V.S. wrote the paper. Flow visualization techniques have been developed to a high The authors declare no conflict of interest. degree of precision and speed for classical fluid dynamics This article is a PNAS Direct Submission. investigations. However, for liquid helium, such techniques have 1To whom correspondence should be addressed. E-mail: [email protected].
www.pnas.org/cgi/doi/10.1073/pnas.1312546111 PNAS Early Edition | 1of6 Downloaded by guest on September 26, 2021 Although solids from other gases, such as argon, methane, ni- trogen, and propane, have been tested (27), hydrogen and deu- ABC terium produce particles that are close to neutrally buoyant.
p He2 Fluorescence Imaging Technique. Recently, a new visualization * technique, using excited He2 triplet molecules, was developed (22, 23). These molecules can be produced in large numbers in liquid helium, following the ionization or excitation of ground state helium atoms (28, 29). The singlet state molecules radia- tively decay in a few nanoseconds (30), but the triplet state Fig. 1. Intensity inverted images showing hydrogen ice particles trapped on molecules are metastable with a radiative lifetime of about 13 s quantized vortex lines in superfluid helium (16). The concentration of hy- (31). These triplet molecules form bubbles in liquid helium with drogen ice can be varied such that (A) only isolated particles are trapped on vortices, (B) multiple particles form dotted lines on vortices, or finally (C) a radius of about 6 Å (32) and can be used as tracers. To image * solid hydrogen skeletons perturb the dynamics of the vortices and stabilize the He2 triplet molecules, a cycling-transition laser-induced branches and crossings. The natural state is for crossings to reconnect. fluorescence technique, first developed by McKinsey et al. (33) and McKinsey and coworkers ( 34), has been used. A laser pulse − at 905 nm can excite helium molecules from their triplet ground which is characterized by the predicted v 3 power-law distribu- 3Σ+ 3Σ+ state a u to the excited electronic state d u . Over 90% of the tion (17, 42). This clearly distinguishes quantum turbulence from 3 molecules in the d state quickly decay to an intermediate b Πg classical turbulent flows, as the velocity distribution of the latter state, emitting detectable fluorescent photons at 640 nm (35). A has a nearly Gaussian shape (43). Such a power-law shape of the filter can be used to block unwanted laser light, to achieve low tails of the velocity distributions was later confirmed in two-fluid 3 background. From the b Πg state, molecules quench back to the flow experiments (20, 21) and in superflow numerical simu- 3Σ+ a u state, and the process can be repeated so that each mole- lations (44–47). Note, however, that the reasons why the tails of cule produces many fluorescence photons. the velocity distributions obtained in two-fluid flow (17, 20, 21, 42) are consistent with those computed in the absence of normal- Particle Motion in Superfluid Helium. In superfluid helium-4, vor- fluid flow (44–47) are still not entirely understood and further ticity is concentrated along the filamentary cores of quantized investigations are consequently required to address the issue. vortex lines, and the velocity circulation around any such line is Besides visualizing vortex reconnection, PTV has also been κ = = ’ −3 2= ’ equal to h m4 10 cm s, where h is Planck s constant used to study Kelvin waves excited on vortex lines, which have 4 and m4 indicates the mass of a He atom. A particle positioned long been discussed as important in the energetics and dynamics on a quantized vortex line in superfluid helium displaces liquid of the quantum fluid state (48). Recent visualization experiments that has high kinetic energy, and, as a result, there is an energy at the University of Maryland strongly support the existence of binding the particle to a vortex line. The micron-sized particles Kelvin waves generated by vortex reconnection (49). These used in the PIV and PTV experiments can consequently get waves exhibit a self-similar, traveling helical perturbation to the trapped on quantized vortex lines, besides tracing normal-fluid vortex lines following a line reconnection, as was theoretically flows (to date these experiments have been performed at tem- predicted by Schwarz (50). Vortex rings, loaded by particles, peratures above about 1.5 K, mainly due to heat load concerns). have also been visualized (51) and the corresponding particle Moreover, as detailed below, imaging trapped particles allows dynamics theoretically addressed (36). the study of interesting vortex-line dynamics. However, in flows where the normal fluid, the superfluid, and the vortices have Thermal Counterflow Experiments. Thermal counterflow is a unique different velocity fields, the behavior of these particles might flow mode that exists only in the superfluid phase of helium, * become difficult to interpret (19, 36, 37). On the other hand, He2 even though its study might be also relevant to the understanding molecular tracers are entrained solely by the normal-fluid com- of heat transport, e.g., in the form of turbulent convection (52, ponent above 1 K, due to their small binding energy on vortices 53). The condition can be easily established by applying heat at (38). This makes them suitable for unambiguously probing var- ious normal-fluid flows; and, as revealed in recent experiments, * He2 molecules can attach to quantized vortex lines below 0.2 K 100 microns (39), which has the potential to allow for vortex-line imaging at low temperatures. More generally, it is clear that special care should be taken when choosing the tracer particles, keeping es- pecially in mind that particle size should always be smaller than relevant flow length scales (19, 36). Progress on Flow Visualization in Helium Vortex-Line Imaging Experiments. Bewley et al. (16) first observed solid hydrogen particles trapped on quantized vortices, using the PTV facilities at University of Maryland (Fig. 1). The particle tracking technique was then further developed to investigate vortex reconnection in superfluid helium. Reconnection was first -37.5 ms -12.5 ms 12.5 ms 37.5 ms predicted by Feynmann in 1955 as a process by which dissipation could be allowed even in pure superfluid (40). Experiments, which strongly suggest the occurrence of vortex reconnection, (t) were performed by Bewley et al. (41) and Paoletti et al. (17) in (t) 2008, supporting both the existence of this topological change and the details of the predicted vortex motions. Fig. 2 shows an t
2of6 | www.pnas.org/cgi/doi/10.1073/pnas.1312546111 Guo et al. Downloaded by guest on September 26, 2021 3∑ + 1 To unambiguously examine the normal-fluid motion in ther- A d u 0 B * SPECIAL FEATURE
925 nm mal counterflow, the He molecule visualization technique was 2 *
905 nm recently used (23). The He tracers were produced by a tungsten 640 nm 2 910 nm field-emission source in a glass counterflow channel. A focused 925 nm 5 mm 3∑+ 1 pump laser pulse at 910 nm was used to tag a line of molecules c g 0 1099 nm across the channel by driving the molecules to a long-lived vi- 925 nm
1073 nm t=0 ms t=40 ms t=80 ms
905 nm 3
910 nm ∏ brational level a(1) (Fig. 3A). This tagged line was imaged sub- b g sequently, using a probe laser pulse at 925 nm. Up to 40 images 2 1 were superimposed at each given pump–probe delay time to 3∑+ a u 0 achieve a good image quality. Typical summed images are shown in Fig. 3B, suggesting a flat averaged normal-fluid velocity profile Fig. 3. (A) Schematic diagram showing the optical transitions for imaging * that should be expected for turbulent flow, in a long enough the He2 triplet molecules. The levels, labeled 0, 1, 2 for each electronic state, are the vibrational levels of the corresponding state. (B) Averaged images of channel. The observed rapid growth of the averaged line width a line of tagged helium molecules via the tagging fluorescence method with time further supports the claim that the normal-fluid flow is across a square channel (5 mm side width) in thermal counterflow with turbulent (23). Note that, due to the mutual friction between the a heat flux of 640 mW/cm2. two fluid components, dissipation occurs at all length scales in the normal fluid, which contrasts with the situation in classical turbulence, where dissipation is deemed to take place only below the closed end of a flow channel containing superfluid helium. The a small length scale, called the Kolmogorov length scale (54). normal-fluid component carries the heat and moves away from the The experiment revealed a unique normal-fluid turbulence in heater at a mean velocity v = q=ρST,whereq is the heat flux, ρ is n counterflow (57). the helium density, T denotes the temperature, and S represents the specific entropy of helium (54). The superfluid component Normal-Fluid Turbulence in Counterflow. The unique type of tur- moves toward the heat source, serving to eliminate any net mass bulence just discussed obviously calls for further attention. flow. It has been known for many years that, above a (small) critical Studying it not only will likely broaden our understanding of value of heat flux, the superfluid component in counterflow turbulence in general, but also might have practical significance becomes turbulent. This results in a tangle of quantized vortex PHYSICS because the turbulent normal-fluid flow could, e.g., alter our lines, whose dynamical behavior is an essential ingredient of understanding of heat transfer. An experiment has been specif- quantum turbulence (55). Counterflow allows a controlled forcing ically designed at Florida State University to examine the nor- of the superfluid state away from equilibrium. mal-fluid velocity field in counterflow. A thin line of He* tracers A number of flow visualization experiments have been per- 2 formed in thermal counterflow. Early PIV experiments indicated is created via laser-field ionization in helium. To achieve the that average particle velocities were typically less than the nor- required high electric field for ionizing ground state helium atoms, laser intensity as high as 1013 W/cm2 is needed (58). Such mal-fluid velocity vn (26). Subsequently, using PTV techniques, Paoletti et al. showed that, at low relative velocity, the particle a high instantaneous laser intensity can be achieved by focusing velocity distribution is indeed bimodal (56). Some particles move a femtosecond laser pulse through a tiny cross-section. The in the opposite direction to the heat current. These particles are molecule density so created is high enough to allow high-quality single-shot imaging of the tracer line. Fig. 4 shows fluorescence interpreted as being trapped in the tangle of quantized vortices * generated by the counterflow [note that the vortex tangle moves images of He2 tracer lines that have been successfully generated toward the heat source with a velocity that is generally different and imaged in counterflow, at 1.85 K, with a 35-fs laser pulse, at μ from the superfluid velocity (48)]. The rest of the particles are 55 J. At low heat fluxes, a straight tracer line deforms into mainly influenced by Stokes drag, from the normal-fluid flow, a parabolic shape, indicating the Poiseuille laminar velocity and their velocity agrees with the prediction of Landau and profile of the normal fluid. At large heat fluxes, the tracer line Lifshitz (54). Note, however, that particle trapping is generally distorts, possibly due to the turbulent eddies in the normal fluid. a dynamical phenomenon; i.e., particles can also escape from The local normal-fluid velocity could then be estimated by di- vortices, depending on the experimental conditions. More re- viding the center displacement of a small line segment by the cently, Chagovets and Van Sciver (19) also used PTV to show drift time. Structure functions of the turbulent flow could be that the bimodal velocity distribution occurs only at low relative computed based on the derived velocities (59), which should velocities and that above a critical velocity, associated with the allow us to gain information on the turbulent energy spectrum. particles being untrapped from vortex lines, the velocity distri- By creating multiple lines to include crosses or grid tracer bution becomes monovalued, similar to that observed by Zhang structure, measurements of normal-fluid vorticity and other and Van Sciver, using PIV (26). complex velocity derivatives can be made.
T=1.85K Laminar flow Turbulent flow
80 um
1 cm Heat flux: 10 mW/cm2 Heat flux: 215 mW/cm2 No heat flux Drift time: 900 ms Drift time: 15 ms
* Fig. 4. Fluorescence images showing the motion of a thin line of He2 tracers in thermal counterflow. The tracer line is created via laser-field ionization by focusing a femtosecond laser pulse into superfluid helium. The drift time denotes the time between the creation and the imaging of the tracer line. The second image was taken in steady-state flow, whereas the third image was taken by the time the heater was turned off.
Guo et al. PNAS Early Edition | 3of6 Downloaded by guest on September 26, 2021 ð − meanÞ= sd mean sd Fig. 5. Probability density function (PDF)of az az az , where az and az are the mean and SD of the instantaneous vertical acceleration az,re- spectively (trajectories with at least five points, total number of points of each dataset larger than 105, and the area below the curves normalized to 1). Solid squares, τ = 0:13; solid circles, τ = 0:21; solid triangles, τ = 0:28; open squares, τ = 0:14; and solid line, log-normal fit, calculated by using equation 1 in ref. 61, with s = 1, the employed variable being the particle normalized acceleration shown on the horizontal axis.
Particle Acceleration Measurements. The Lagrangian dynamics of however, that, as suggested in ref. 21, the classical fit seems not solid deuterium particles, at length scales comparable to the to agree with the experimental data at large accelerations, for the mean distance ℓ between quantized vortices, have been recently smallest times. The distribution tails of the particle acceleration studied in steady-state thermal counterflow at Charles University in the horizontal direction display indeed more noticeable by using the PTV technique (20, 21). It was unexpectedly found departures from the classical shapes, as the particle dynamics are that the normalized distribution of the particle acceleration dominated by the imposed vertical velocities (56). The outcome, appears to follow a classical-like behavior (21). Fig. 5 displays whose details will be reported elsewhere, represents unique ex- the normalized probability density function (PDF)ofthein- perimental evidence that the mean distance between quan- stantaneous particle acceleration in the vertical direction, that of tized vortices can be seen as the length scale distinguishing the imposed counterflow, in different experimental conditions, at classical-like from quantum behavior. length scales about one order of magnitude smaller than those reported previously (21). The result was obtained by collecting Flow-Across-Obstacle Experiments. An important, classical fluid images at frame rates that allow the study of the particle dy- dynamical condition is that of the flow past solid objects, such as namics at scales smaller than ℓ. The latter scales can be quanti- cylinders and spheres, with the associated drag crisis (62). Initial fied by introducing the nondimensional time τ = t1=t2, where t1 is studies using solid particles and the PIV technique showed the the time used for the calculation of accelerations along the existence of large-scale turbulent eddies, both behind and in tracks, t2 = ℓ=Vabs, and Vabs denotes the mean particle velocity. It front of a cylinder in thermal counterflow (15). Such a behavior can be seen that the agreement with a log-normal fit used for has no classical analog. More recently, Chagovets and Van Sciver classical turbulent flows (60, 61) to hold also for τ ’ 0:1, i.e., at discussed the existence of two distinct velocity fields in coun- length scales about one order of magnitude smaller than ℓ. Note, terflow around a cylinder (63). At low relative velocities, typically
AB
q q y (mm)
)mm( x )mm( x )mm(
Fig. 6. Particle trajectories around a 2-mm cylinder obtained in counterflow by the PTV technique with hydrogen particles. (A)Theparticlesthatmovewiththe 2 normal fluid velocity. (B) The motion of particles that interact with quantized vortex lines (q = 50 mW/cm , T = 1.93 K, vn = 0.23 cm/s, Reynolds number Re = 485).
4of6 | www.pnas.org/cgi/doi/10.1073/pnas.1312546111 Guo et al. Downloaded by guest on September 26, 2021 cross-section channel, of side W = 20 mm. The channel is hori-
1.2 SPECIAL FEATURE zontal, in a specially designed cryostat that allows optical access 1.1 to the flow field. The results from these experiments showed a velocity distribution that is essentially classical in character, 1.0 with a measurable velocity boundary layer that scales with Re 0.9 (Fig. 7). Further investigations are consequently needed to Re=9.00x10 clarify whether a parameter range exists where forced flow su- 0.8 1.26x105 perfluid dynamics are different from classical hydrodynamics and 1.80x105 also to assess in which conditions the influence of the physical ave0.7 max 2.34x105
U /U 5 boundaries on quantum flows can be neglected. 0.6 2.70x10 3.06x105 5 Conclusion and Future Work 0.5 3.60x10 4.50x105 The study of quantum turbulence has in recent years benefited 0.4 n=7 from the use of flow visualization techniques, which led researchers n=8.8 to appreciate more clearly similarities and differences between 0.3 classical and quantum flows (69). These very powerful tools pro- 0.0 0.2 0.4 0.6 0.8 1.0 2y/W duced, at the same time, results that are posing more questions than giving clear answers, showing thus that the probed phenomena are indeed worth investigating. Thermal counterflow is a well-know Fig. 7. Normalized velocity profile near the wall of a channel containing forced flow superfluid helium at 2.1 K. The lines correspond to the fit quantum flow, characterized by a unique form of heat transport, 1=n ðUave=Umax Þ = ð2y=WÞ , where Uave is the flow velocity at a distance y from whose links to classical turbulent convection (52, 53) are yet to the channel wall of side W; Umax represents the maximum velocity, which occurs be explored in detail. Similarly, the newly discovered normal- at the center of the channel; and n, specified on the curve, shows the best fit fluid turbulence (23) deserves further attention, as it might lead between 7 and 8.8, depending on the value of the Reynolds number Re (67). to broadening our comprehension of turbulence in general. Vortex reconnection (41) and Kelvin waves (49) are dissipative
mechanisms, crucial to the energy transfer in quantum flows, and PHYSICS vn < 1 cm/s, the normal-fluid flow appears as a laminar flow, whereas the particles tracking the superflow are more typically should be extensively investigated, also to assess their relevance trapped on the vortex lines (Fig. 6). At higher relative velocity, the to other research fields, such as magnetohydrodynamics (70). fields of the two fluid components are no longer separable, and the The complex interactions between particles, quantized vortices, particle motion begins to display the large-scale eddies first ob- and macroscopic eddies in quantum flows represent still an open served using solid particles (15). As with channel flow (19), this problem of cryogenic flow visualization (37) and its relations with transition is deemed to occur as a result of Stokes drag, due to the the behavior of particles in classical turbulent flows (71) should normal-fluid flow, being sufficient to dislodge the trapped particles be fully exploited (21). The study by visualization of unsteady/ from the superfluid vortex lines. Other related visualization studies oscillating flows is a new line of inquiry (64) that could, for ex- include preliminary results on the flow past an oscillating sphere ample, give valuable contributions to the popular study of the (64) and additional investigations on the occurrence of macroscopic temporal and spatial decay of turbulent flows (1, 72). The in- vortices in the proximity of cylinders in counterflow (65). fluence of the boundaries of the experimental volume on the studied quantum flows is also an open issue (68) and it might be Forced Flow Experiments. Another basic, classical problem that interesting to know how vortex tangles behave close to solid can be used to determine the extent to which superfluid dy- walls. The latter possible, future directions of cryogenic flow namics display classical behavior is the pressure gradient in visualization are further evidence that the discussed techniques turbulent pipe flow. Unlike counterflow, in forced flow of tur- are very valuable tools for the analysis of puzzling natural phe- bulent superfluid helium the two fluid components are believed nomena, whose understanding can lead to useful insights into the to be coupled together by the mutual friction force such that the underlying physics of many, interconnected research fields. flow velocity Uave equals the velocities of the two fluids. This leads to a turbulent friction factor that scales with the classical ACKNOWLEDGMENTS. W.G. acknowledges his collaborators D. N. McKinsey, Reynolds number Re = UaveW=νn, where W is a relevant flow W. F. Vinen, G. Ihas, A. Golov, P. V. E. McClintock, and S. Fisher. M.L.M. thanks D. Duda, M. Rotter, and L. Skrbek for fruitful discussions and valuable scale and νn denotes the kinematic viscosity of the normal-fluid help. D.P.L. acknowledges his collaborators G. Bewley, M. Paoletti, K. Gaff, component (66). To display such a behavior, the fluid would be E. Fonda, D. Meichle, and M. E. Fisher. S.W.V.S. acknowledges collaborators expected to have a velocity boundary layer similar to that seen in T. Chagovets, S. Fuzier, T. Xu, and T. Zhang. W.G. acknowledges the startup classical fluids (67). At Florida State University, Xu and Van support from Florida State University and the National High Magnetic Field Laboratory. M.L.M. acknowledges the support from the Czech Science Sciver made PIV measurements of the velocity profile near the ≈ 5 Foundation under Grant GACR P203/11/0442. D.P.L. acknowledges support wall in forced flow of superfluid helium at Re 10 (68). The from the Center for Nanophysics and Advanced Materials and from the Na- experimental apparatus used calibrated bellows pumps driven by tional Science Foundation (DMR-0906109). S.W.V.S. acknowledges the support linear actuators to produce a known average flow rate in a square from the US Department of Energy under Grant DE-FG02-96ER40952.
1. Skrbek L, Sreenivasan KR (2012) Developed quantum turbulence and its decay. Phys 7. Attanasi A, et al. (2013) Superfluid transport of information in turning flocks of Fluids 24(1):011301. starlings. arXiv:1303.7097. 2. Kessler MF, et al. (1996) The Infrared Space Observatory (ISO) mission. Astron As- 8. Chesler PM, Liu H, Adams A (2013) Holographic vortex liquids and superfluid turbu- trophys 315:L27–L31. lence. Science 341(6144):368–372. 3. Packard RE (1972) Pulsar speedups related to metastability of the superfluid neutron- 9. Tsubota M, Kobayashi M, Takeuchi H (2013) Quantum hydrodynamics. Phys Rep 522: star core. Phys Rev Lett 28(16):1080–1082. 191–238. 4. Anderson PW, Itho N (1975) Pulsar glitches and restlessness as a hard superfluidity 10. Chopra KL, Brown JB (1957) Suspension of particles in liquid helium. Phys Rev phenomenon. Nature 256:25–27. 108(1):157. 5. Zurek WH (1985) Cosmological experiments in superfluid helium? Nature 317:505–508. 11. Murakami M, Ichikawa N (1989) Flow visualization study of thermal counterflow jet in 6. Hendry PC, Lawson NS, Lee RAM, McClintock PVE, Williams CDH (1994) Generation of He II. Cryogenics 29(4):438–443. defects in superfluid 4He as an analogue of the formation of cosmic strings. Nature 12. Bielert F, Stamm G (1993) Visualization of Taylor-Couette flow in superfluid helium. 368:315–317. Cryogenics 33(10):938–940.
Guo et al. PNAS Early Edition | 5of6 Downloaded by guest on September 26, 2021 13. Hayden ME, et al. (2004) Neutron-detected tomography of impurity-seeded super- 43. Paoletti MS, Lathrop DP (2011) Quantum turbulence. Annu Rev Cond Mat Phys 2: fluid helium. Phys Rev Lett 93(10):105302. 213–234. 14. Guo W, Jin D, Seidel GM, Maris HJ (2009) Experiments with single electrons in liquid 44. White AC, Barenghi CF, Proukakis NP, Youd AJ, Wacks DH (2010) Nonclassical velocity helium. Phys Rev B 79(5):054515. statistics in a turbulent atomic Bose-Einstein condensate. Phys Rev Lett 104(7):075301. 15. Zhang T, Van Sciver SW (2005) Large scale turbulent flow around a cylinder in 45. Adachi H, Tsubota M (2011) Numerical study of velocity statistics in steady counter- 4 counterflow superfluid He (He II). Nat Phys 1:36–38. flow quantum turbulence. Phys Rev B 83(13):132503. 16. Bewley GP, Lathrop DP, Sreenivasan KR (2006) Superfluid helium: Visualization of 46. Baggaley AW, Barenghi CF (2011) Quantum turbulent velocity statistics and quasi- quantized vortices. Nature 441(7093):588. classical limit. Phys Rev E Stat Nonlin Soft Matter Phys 84(6 Pt 2):067301. 17. Paoletti MS, Fisher ME, Sreenivasan KR, Lathrop DP (2008) Velocity statistics distin- 47. Proment D, Nazarenko S, Onorato M (2012) Sustained turbulence in the three- guish quantum turbulence from classical turbulence. Phys Rev Lett 101(15):154501. dimensional Gross-Pitaevskii model. Physica D 241(3):304–314. 18. Bewley GP, Sreenivasan KR, Lathrop DP (2008) Particles for tracing turbulent liquid 48. Donnelly RJ (1991) Quantized Vortices in Helium II (Cambridge Univ Press, Cambridge, helium. Exp Fluids 44(6):887–896. UK). 19. Chagovets TV, Van Sciver SW (2011) A study of thermal counterflow using particle 49. Fonda E, et al. (2013) Visualization of Kelvin waves on quantum vortices. arXiv: tracking velocimetry. Phys Fluids 23(10):107102. 1210.5194. 20. La Mantia M, Chagovets TV, Rotter M, Skrbek L (2012) Testing the performance of 50. Schwarz KW (1977) Theory of turbulence in superfluid 4He. Phys Rev Lett 38(10): a cryogenic visualization system on thermal counterflow by using hydrogen and 551–554. deuterium solid tracers. Rev Sci Instrum 83(5):055109. 51. Bewley GP, Sreenivasan KR (2009) The decay of a quantized vortex ring and the in- 21. La Mantia M, Duda D, Rotter M, Skrbek L (2013) Lagrangian accelerations of particles fluence of tracer particles. J Low Temp Phys 156:84–94. in superfluid turbulence. J Fluid Mech 717:R9. 52. Skrbek L, Gordeev AV, Soukup F (2003) Decay of counterflow He II turbulence in 22. Guo W, Wright JD, Cahn SB, Nikkel JA, McKinsey DN (2009) Metastable helium molecules as tracers in superfluid 4He. Phys Rev Lett 102(23):235301. a finite channel: Possibility of missing links between classical and quantum turbu- 23. Guo W, Cahn SB, Nikkel JA, Vinen WF, McKinsey DN (2010) Visualization study of lence. Phys Rev E Stat Nonlin Soft Matter Phys 67(4 Pt 2):047302. counterflow in superfluid 4He using metastable helium molecules. Phys Rev Lett 53. Chillà F, Schumacher J (2012) New perspectives in turbulent Rayleigh-Bénard con- 105(4):045301. vection. Eur Phys J E Soft Matter 35(7):58. 24. Raffel M, Villert CE, Werely ST, Kompenhans J (2007) Particle Image Velocimetry - A 54. Landau LD, Lifshitz EM (1987) Fluid Mechanics (Elsevier Butterworth-Heinemann, Practical Guide (Springer, Berlin), 2nd Ed. New York), 2nd Ed. 25. Zhang T, Celik D, Van Sciver SW (2004) Tracer particles for application to PIV studies of 55. Vinen WF (2006) An introduction to quantum turbulence. J Low Temp Phys 145:7–24. liquid helium. J Low Temp Phys 134(3-4):985–1000. 56. Paoletti MS, Fiorito RB, Sreenivasan KR, Lathrop DP (2008) Visualization of superfluid 26. Zhang T, Van Sciver SW (2005) The motion of micron-sized particles in He II coun- helium flow. J Phys Soc Jpn 77(11):111007. terflow as observed by the PIV technique. J Low Temp Phys 138(3-4):865–870. 57. Melotte DJ, Barenghi CF (1998) Transition to normal fluid turbulence in helium II. Phys 27. Fonda E, Sreenivasan KR, Lathrop DP (2012) Liquid nitrogen in fluid dynamics: Visu- Rev Lett 80(19):4181–4184. alization and velocimetry using frozen particles. Rev Sci Instrum 83(8):085101. 58. Benderskii AV, Zadoyan R, Schwentner N, Apkarian VA (1999) Photodynamics in su- 28. Surko CM, Reif F (1968) Investigation of a new kind of energetic neutral excitation in perfluid helium: Femtosecond laser-induced ionization, charge recombination, and superfluid helium. Phys Rev 175(1):229–241. preparation of molecular Rydberg states. J Chem Phys 110(3):1542–1557. 29. Dennis WS, Durbin E, Fitzsimmons WA, Heybey O, Walters GK (1969) Spectroscopic 59. Guo W, et al. (2013) Visualization technique for determining the structure functions identification of excited atomic and molecular states in electron-bombarded liquid of normal-fluid turbulence in superfluid helium-4. J Low Temp Phys 171(5-6):497–503. helium. Phys Rev Lett 23(19):1083–1086. 60. Mordant N, Crawford AM, Bodenschatz E (2004) Three-dimensional structure of the – 30. Hill PC (1989) Ultraviolet continua of helium molecules. Phys Rev A 40(9):5006 5016. Lagrangian acceleration in turbulent flows. Phys Rev Lett 93(21):214501. *ð 3Σ+Þ 31. McKinsey DN, et al. (1999) Radiative decay of the metastable He2 a u molecule in 61. Qureshi NM, et al. (2007) Turbulent transport of material particles: an experimental – liquid helium. Phys Rev A 59(1):200 204. study of finite size effects. Phys Rev Lett 99(18):184502. 32. Benderskii AV, Eloranta J, Zadoyan R, Apkarian VA (2002) A direct interrogation of 62. Schlichting H (1979) Boundary Layer Theory (McGraw Hill, New York), 7th Ed. – superfluidity on molecular scales. J Chem Phys 117(3):1201 1213. 63. Chagovets TV, Van Sciver SW (2013) Visualization of He II counterflow around a cyl- 33. McKinsey DN, Lippincott WH, Nikkel JA, Rellergert WG (2005) Trace detection of inder. Phys Fluids 25(10):105104. metastable helium molecules in superfluid helium by laser-induced fluorescence. Phys 64. Zemma E, Luzuriaga J (2013) Anomalous trajectories of H2 solid particles observed Rev Lett 95(11):111101. near a sphere oscillating in superfluid turbulent 4He. J Low Temp Phys 173:71–79. 34. Rellergert WG, et al. (2008) Detection and imaging of He2 molecules in superfluid 65. Duda D, La Mantia M, Rotter M, Skrbek L (2013) On the visualization of thermal helium. Phys Rev Lett 100(2):025301. counterflow of He II past a circular cylinder. J Low Temp Phys, 10.1007/s10909-013- 35. Rellergert WG (2008) Detecting and imaging He molecules in superfluid helium by 2 0961-z. laser-induced fluorescence. PhD thesis (Yale University, New Haven, CT). 66. Walstrom PL, Weisend JG II, Maddocks JR, Van Sciver SW (1988) Turbulent flow 36. Barenghi CF, Sergeev YA (2009) Motion of vortex ring with tracer particles in su- pressure drop in various He II transfer system components. Cryogenics 28:101–109. perfluid helium. Phys Rev B 80(2):024514. 67. Nikuradse J (1932) Gesetzmässigkeiten der turbulenten Strömung in glatten Röhren 37. Sergeev YA, Barenghi CF (2009) Particle-vortex interactions and flow visualization in 4He. J Low Temp Phys 157(5-6):429–475. [Laws of turbulent flow in smooth pipes]. Forschg Arb Ing Wes 356. German. 38. Vinen WF (2006) Quantum turbulence: Where do we go from here? AIP Conf Proc 68. Xu T, Van Sciver SW (2007) Particle image velocimetry measurements of the velocity 850:169–172. boundary layer in forced flow He II. Phys Fluids 19(7):071703. * 69. Vinen WF (2013) Quantum turbulence: Aspects of visualization and homogeneous 39. Zmeev DE, et al. (2013) Excimers He2 as tracers of quantum turbulence in 4He in the t = 0 limit. Phys Rev Lett 110(17):175303. turbulence. J Low Temp Phys, 10.1007/s10909-013-0911-9. 40. Feynman RP (1955) Progress in Low Temperature Physics, ed Gorter CJ (North Holland, 70. Cassak PA, Drake JF, Shay MA, Eckhardt B (2007) Onset of fast magnetic reconnection. Amsterdam), Vol 1, pp 17–53. Phys Rev Lett 98(21):215001. 41. Bewley GP, Paoletti MS, Sreenivasan KR, Lathrop DP (2008) Characterization of re- 71. Toschi F, Bodenschatz E (2009) Lagrangian properties of particles in turbulence. Annu connecting vortices in superfluid helium. Proc Natl Acad Sci USA 105(37):13707–13710. Rev Fluid Mech 41:375–404. 42. Paoletti MS, Fisher ME, Lathrop DP (2010) Reconnection dynamics for quantized 72. Nedic J, Vassilicos JC, Ganapathisubramani B (2013) Axisymmetric turbulent wakes vortices. Physica D 239(14):1367–1377. with new nonequilibrium similarity scalings. Phys Rev Lett 111(14):144503.
6of6 | www.pnas.org/cgi/doi/10.1073/pnas.1312546111 Guo et al. Downloaded by guest on September 26, 2021