SPECIAL FEATURE: PERSPECTIVE

SPECIAL FEATURE: PERSPECTIVE

Quantum turbulence generated by oscillating structures William F. Vinena,1 and Ladislav Skrbekb a School of Physics and Astronomy, University of Birmingham, Birmingham B15 2TT, United Kingdom; and bFaculty of Mathematics and Physics, Charles University, 121 16 Prague, Czech Republic Edited by Katepalli R. Sreenivasan, New York University, New York, NY, and approved December 12, 2013 (received for review July 22, 2013)

The paper summarizes important aspects of quantum turbulence that have been studied successfully with oscillating structures. It describes why some aspects are proving hard to interpret, and it outlines the need for new types of experiment and new developments in theoretical and computational work. superfluid helium

| quantized vortex lines

This paper is concerned with the generation of quantum turbulence (1–3) in liquid helium by various forms of oscillating structure, such as a cylinder oscillating in a direction normal to its length. Many experiments on the generation of quantum turbulence by such structures have been reported, involving not only cylinders (or wires), but also spheres, tuning forks, and grids. Many of these experiments and their possible interpretation have already been reviewed (4). We are not aiming to produce another conventional review, and we shall not consider all that has been achieved. Instead, our aim is to focus on three particular aspects of this type of work: first, on experiments that have already contributed significantly to our understanding of issues that are, as we see them, of general and fundamental importance; second, on an exposure of the difficulties in interpreting many of the experiments in any real detail, in the way that has been achieved in analogous work on classical fluids; and third, on ways in which we might overcome these difficulties, which are both theoretical and experimental. We shall be concerned ultimately with both superfluid 4He and 3He-B, but unless otherwise stated we shall be referring to 4He. There are of course connections between quantum turbulence produced by oscillating structures and more general aspects of such turbulence. The role of remanent vorticity in the nucleation and stability of quantum turbulence is universally important, and, as we discuss in a later section, it is here that experiments with oscillating structures have been particularly instructive. At a more subtle level, there is the question of the extent to which quantum turbulence is similar to classical turbulence. The observation of a Kolmogorov energy spectrum on large length scales (larger than the vortex-line spacing) in homogeneous quantum turbulence is often www.pnas.org/cgi/doi/10.1073/pnas.1312551111

quoted as evidence for this similarity. As we shall see, some aspects of the quantum turbulence produced by oscillating structures provide additional evidence. However, in contrast to homogeneous turbulence, quantum turbulence produced by an oscillating structure must be strongly influenced by the boundary conditions at a solid wall. As we discuss in a section devoted to the development of quantum turbulence, uncertainty about the boundary conditions relevant to quantum turbulence hinders interpretation of the experiments and calls for renewed experimental and theoretical study. Almost all of the experiments have involved measurements of the fluid dynamical force on the structure as a function of the amplitude of the oscillations. This force can be divided into a drag, which is in phase with the velocity and therefore dissipative, and a force that is in quadrature with the velocity and can be interpreted as a change in the effective mass of the structure. The quadrature (or inertial) force is interesting, but we do not have space to discuss it here (5–8). We shall denote the amplitude of the drag force by Fd , and that of the velocity by v. It is convenient to write the drag force in the following form:

cases, there is at low velocities a regime where Cd falls as 1=v, corresponding to a linear drag arising from laminar viscous flow of the fluid. In the classical case, the effective density in this regime is that of the whole fluid, whereas in the quantum case the effective density is that of the normal fluid only, indicating that the superfluid must be flowing irrotationally, exerting no drag. Strictly speaking, at very low temperatures, there can be a significant contribution to the linear drag from internal friction within the material of the structure, as is clear from the fact that this contribution remains present in vacuo. In the classical case at high velocities, the drag coefficient tends often to a more or less constant value of order unity, although the details are sometimes more complicated, the drag tending to exhibit maxima and minima with increasing velocity (9). The transition between the two limits is generally rather smooth. In the quantum case, there seems to be a critical velocity above which the drag coefficient undergoes a large increase, rather sharply at low temperatures, tending again at the highest velocities to what may be a constant value; this constant value can be of order unity, but often it seems to be significantly smaller, especially at low temperatures, where there is little normal fluid and where therefore the drag at low velocities is 1 2 Fd = Cd Aρv ; [1] very small (10). Again, the details may be more 2 complicated. It seems that sometimes there are two critical velocities (5, 10, 11), especially at where Cd is a dimensionless drag coefficient, low temperatures, the drag increasing only a litA is the projected area of the structure normal tle at the lower value (Fig. 1, bottom curve). to the velocity, and ρ is the total density of the fluid. The way in which Cd varies with v is Author contributions: W.F.V. and L.S. performed research; W.F.V. observed to be much the same for all types of and L.S. analyzed data; and W.F.V. and L.S. share responsibility for oscillating structure that have been studied, the article and its contents. and it is shown semischematically in Fig. 1, The authors declare no conflict of interest. in comparison with that for a classical fluid This article is a PNAS Direct Submission. [we refer the reader to the review (4) for 1To whom correspondence should be addressed. E-mail: w.f. details of the experimental results]. In all [email protected]. PNAS | March 25, 2014 | vol. 111 | suppl. 1 | 4699–4706

Fig. 1. Diagram showing semischematically how the drag coefficient for a typical oscillating structure varies with velocity amplitude. The top curve relates to a classical fluid; the middle curve, to 4He at a temperature of about 1.3 K; the bottom curve, to 4He at a temperature of about 10 mK. The kink in the bottom curve at a velocity of about 1 cm·s−1 is discussed in the text.

Furthermore, the overall behavior may not be at all reproducible as between similar structures, especially at high velocities and low temperatures (10). Sometimes the behavior is observed to depend on history, with indications that turbulence does not always develop fully (12). There is little doubt that the overall behavior in the quantum case can be understood in qualitative outline as follows. As we have explained, the linear drag at low velocities is associated, at least in part, with laminar viscous flow of the normal fluid. The sudden increase in drag at a fairly welldefined critical velocity, vsc , is due to the creation of quantized vortex lines in the superfluid component. The vortex lines form a turbulent tangle, and the resulting force of mutual friction leads to a strong coupling between the two fluids. The two fluids then behave to some degree as a single turbulent fluid, giving rise to a drag coefficient that behaves to some extent like that in a classical fluid. This form of behavior presupposes that the viscous normal fluid flow does not by itself become turbulent (or at least unstable) at a velocity less than vsc ; this seems to be the case for most structures that have so far been studied, but not for all of them, as we shall see later. It should be emphasized, however, that the temperature may be so low that there is present a negligible fraction of normal fluid, so the presence of normal fluid is not essential for the behavior we have described, the linear drag being then associated with the internal friction. It should also be added that, especially at high frequencies of oscillation, there can be a significant contribution to the damping from acoustic emission, but we shall not pursue this complication (10, 13–15). This overall picture raises a number of interesting questions, most of which are 4700 | www.pnas.org/cgi/doi/10.1073/pnas.1312551111

important in the context of our general understanding of quantum turbulence. By what mechanisms and at what velocity does the vortex generation begin in the superfluid component? Can we understand the way in which the turbulence develops as the velocity increases, both fluids being involved at higher temperatures, but only the superfluid component at low temperatures? To what extent does this development mirror the behavior observed in a classical fluid, or does the quantum form of the turbulence have special characteristics? At a more detailed level, we can ask what underlies the existence of two critical velocities and the lack of reproducibility. We shall see that experiment and theory have been able to throw light on some of these questions, but that others remain as unanswered challenges, about which we can only speculate. It is relevant to emphasize that many aspects of the behavior of oscillating structures in classical fluids turn out to be very complicated (16, 17), and there is no reason to believe that the quantum case is any simpler. Progress in the classical case has often been facilitated by experiments that visualize the flow. Although techniques for the visualization of quantum turbulence have been developed recently, it has not been possible to apply them yet to oscillating structures. Furthermore, as we shall see and as we have hinted, progress on the theoretical and computational side is hampered by our being unable so far to formulate reliable boundary conditions for the superfluid component at a solid surface. The plan of this paper is as follows. In the next section, we discuss the nucleation of vorticity in a superfluid, including the allimportant role of remanent vorticity, and we focus on relevant experiments with vibrating structures. We discuss also an interesting feature of quantum turbulence (lifetime effect) that is associated with the fact that irrotational superflow, in the absence of vortex lines, is always linearly stable. In a following section, we consider the way in which quantum turbulence might develop, in practice from remanent vortices; how this development might be reflected in the observed drag coefficient; and how a second critical velocity and lack of reproducibility might arise. Here, there is as-yet no welldeveloped theory; we are forced to speculate, in a largely qualitative way, and we cannot be confident that we are right. In the penultimate section, we consider the interesting case of an oscillating grid in 3He-B, and in the final section, we discuss how new experiments and new developments in computer simulations might help to guide us toward a better understanding.

The Nucleation of Quantum Turbulence

As we have seen, for small amplitudes of oscillation and for frequencies that are too small for significant acoustic emission, the structure experiences significant drag from only the normal fluid and from losses in the material of the structure itself. The flow of the superfluid component round the structure is, in principle, irrotational, with complete slip at the boundary, and the resulting forces on the structure are nondissipative. Quite generally, for both steady and oscillatory flow, such nondissipative superflow is observed to break down above some critical velocity, and in most of the experiments with which we shall be concerned this breakdown is due to the generation of vorticity in the superfluid in the form of quantized vortex filaments (the Landau critical velocity, at which rotons can be created, is much larger than any critical velocity encountered here). As the velocity increases, the vortex lines in the vicinity of the structure form a disordered tangle, which we identify as quantum turbulence. It is easy to show that the creation of a length of quantized vortex line (circulation κ = h=m) must be opposed by a potential barrier. Consider a uniform flow of superfluid, density ρs and velocity vs , parallel to a plane solid boundary, as in Fig. 2. A stationary vortex of the appropriate sign, at a distance x from the wall and normal to the flow experiences two Magnus forces: one of magnitude ρs κvs away from the wall; and one of magnitude ρs κ 2 =4πx toward the wall, due to the image of the vortex in the wall. The resulting potential energy, U, has a maximum value equal to ðρs κ 2 =4πÞ½lnðx=ξ0 Þ − 1 at a distance from the wall equal to κ=4πvs , where ξ0 denotes the vortex core parameter. Except at temperatures very close to the

Fig. 2. Illustrating the potential barrier opposing vortex nucleation.

Vinen and Skrbek

Fig. 3. A vibrating wire.

Vinen and Skrbek

any attached remanent vortex was provided by exciting wire WB above its (small) critical velocity, which led rapidly to a large reduction in the critical velocity for wire WA . More precisely, if wire WA is first driven at a high velocity, and then wire WB is driven above its critical velocity, the response of wire WA suddenly falls by a large factor, indicating the generation of turbulence by wire WA . The excitation of a wire, such as WB , on which there is a remanent vortex leads to the emission of a beam of vortex rings (22), and the generation of turbulence around wire WA must have been nucleated by the attachment of one or more of these rings to this wire. Let us now consider what forms the nucleating vortex might take. The requirement that such a vortex be long-lived and therefore be in metastable equilibrium means that any small displacement must increase its energy, and so, probably, its length. For the case of a smooth sphere in a channel, a nucleating vortex must be in the form of a “bridging vortex,” connecting the sphere to the walls of the containing vessel, as shown in Fig. 4 as type A. In practice, vortices can be pinned at a protuberance on a rough surface (type B— note that the remnant vortex must continue beyond the sphere to the container wall to satisfy the Kelvin theorem); the small size of the core of a vortex (in 4He) means that most real surfaces are rough. Sufficient vibration can lead to the dislodging of the end of a vortex from a pinning site, so that it moves to another pinning site, on which the vortex energy is smaller. [It is interesting that what was probably the first observation of a remanent vortex and of its movement from one pinning site to another, was made with the early vibrating wire used to verify the quantization of circulation (23).] In the case of more complicated oscillating structures, such as a tuning fork, other forms of remanent vortex are possible, shown as C and D in Fig. 4 (here the Kelvin theorem is satisfied if quantized circulation is assumed around the relevant part of the structure between the pinning sites of the remanent vortices), although that shown as D is likely to disappear under the influence of vibration. A development of the experiment by Goto and coworkers (24) has revealed rather convincingly an interesting general feature of quantum turbulence. We remark first that the nucleating vortex on wire WA must have taken the form of a vortex loop, formed when a vortex ring from wire WB impinged on wire WA . If the excitation of wire WB is turned off after the turbulence produced by wire WA has been nucleated in this way, this latter turbulence is maintained for only a finite time, although this time increases without limit as the drive on wire WA is increased. After this finite time, the turbulence simply collapses, and no vortices remain

SPECIAL FEATURE: PERSPECTIVE

λ-transition or at velocities much larger than those with which we are concerned, this potential barrier cannot be overcome, either thermally or by quantum tunneling. (The processes occurring at high velocities can be observed most straightforwardly in the creation of vortex rings by moving ions in 4He; see the experiments described in ref. 18, and the corresponding theory in ref. 19.) Thus, the superflow ought to remain frictionless, up to an “intrinsic” critical velocity that is very much larger than is observed. It has long been recognized that in practice frictionless superflow usually breaks down by an “extrinsic” process in which an existing small length of vortex line expands under the influence of the superflow. This remanent vortex might have been left from an earlier experiment, or it might have been formed by the Kibble–Zurek mechanism (20) when the helium was cooled through the λ-transition. There must still be some effective barrier, because otherwise there can be no frictionless flow, but it must be small. Interestingly, direct and convincing experimental evidence for the role of remanent vortices has been reported only recently. It has been provided by Goto et al. (21) and was based on experiments with two vibrating wires. The wires are actually loops that are driven at resonance by passing an alternating current, I, through them, the loops being situated in a transverse magnetic field, B (Fig. 3). Both wires were placed in a cell, which was filled very slowly through a small orifice with helium that was already at a very low temperature, the idea being that any vortices would be filtered out by the orifice. Sometimes it was found that for one of the wires ðWA Þ, but not the other ðWB Þ, the critical velocity was unobservably high (greater than 1 ms−1). Confirmation that this high critical velocity was associated with the absence of

Fig. 4.

Illustrating various forms of remanent vortex.

attached to wire WA . It is believed that this effect is associated with a situation where the laminar flow is, intrinsically, linearly stable, i.e., where the laminar (irrotational) flow contains no vortices. This is probably the case for all transitions to quantum turbulence, in contrast to those in classical turbulence. This linear stability can also arise in special cases of classical turbulence, most notably in pipe flow through a pipe of circular cross section, where again finite lifetimes are observed (see, e.g., ref. 25). This type of finite lifetime had been observed in effect in much earlier experiments with vibrating structures, most notably by Schoepe and his colleagues (see, e.g., ref. 26) with an oscillating sphere. What was then observed was a switching of the turbulence on and off. The turbulence switches on again after it had been extinguished because the sphere still had attached to it a nucleating vortex of type A (Fig. 4), which could not have been eliminated during the extinction. Our remarks about nucleation and remanent vortices apply only to superfluid 4He. In superfluid 3He-B, the vortex core is very much larger, and this has two consequences: intrinsic nucleation can occur at quite small velocities; and vortex pinning is much less serious. Furthermore, the Landau critical velocity, at which pair breaking occurs, is quite small and often comparable with that at which vortex nucleation occurs. An example, involving an oscillating grid in 3He-B, will be discussed later. The Development of Quantum Turbulence

We consider now how a nucleating vortex might lead to a transition to fully developed turbulence in the wake of an oscillating structure. An initial clue has been provided by the simulations of Hänninen et al. (27), which relate to a smooth oscillating sphere in a superfluid at zero temperature, with a nucleating vortex of the form A in Fig. 4. The vortex configurations existing for increasing times after the oscillation starts are shown in Fig. 5. The smooth sphere means that vortices ending on the sphere are not pinned but are PNAS | March 25, 2014 | vol. 111 | suppl. 1 | 4701

free to slide without friction over the surface. The oscillation first generates Kelvin waves on the remanent vortex; these waves build up in amplitude, until self-reconnections occur, either in the bulk of the helium or at the surface of the sphere; these reconnections lead in due course to a dense, disordered, tangle of vortices, which is swept back and forth as the sphere oscillates in position. Unfortunately, the buildup of this tangle could be followed for only a limited time, which was insufficient to allow a steady state to be established; and, in any case, the simulations relate to the unrealistic case of a smooth sphere. Nevertheless, it seems reasonable to suppose that quite generally a vortex tangle forms round the structure as the result of the excitation of Kelvin waves on the remanent vortices, followed by reconnections. However, this does not in itself lead to the observed variation of drag coefficient with velocity; the drag calculated from the simulations of Hänninen et al. is much smaller than is observed, a result that is probably associated, as we shall see, with the fact that the tangle is completely disordered. Before we discuss how such a large drag might arise, we can comment on the magnitude of the critical velocity that we might expect if the initial breakdown is associated with the excitation of Kelvin waves. The excitation of Kelvin waves will in itself lead to some dissipation, but significant dissipation will not set in until there is vortex multiplication by self-reconnections. Selfreconnections will occur when, roughly, the amplitude of the Kelvin waves exceeds their wavelength. Provided there is no resonant amplification of the waves, this will occur when the amplitude of oscillation of the structure exceeds in order of magnitude the Kelvin wavelength. Given that the Kelvinwave dispersion relation is: κk2 lnð1=kξ0 Þ; ω= 4π

[2]

we see that the critical velocity is given by:   κω  ln 1=kξ0 1=2 vsc = A ≈ A′ðκωÞ1=2 ; [3] 4π

where ω is the angular frequency of oscillation of the structure, and A; A′ are numerical factors of order unity. (We have assumed that the temperature is not too high, so that the Kelvin waves are not too strongly damped by mutual friction, and that the logarithm can be taken as a constant.) This formula certainly agrees in order of magnitude and frequency dependence with many experiments (4), although the real situation must be somewhat more complicated and must depend (along with the value of A′) on the detailed forms of both the 4702 | www.pnas.org/cgi/doi/10.1073/pnas.1312551111

nucleating vortex and the pattern of irrotational flow past the structure. Other theories have produced a similar result (28), although the argument given here seems to us more plausible. We turn now to processes that can lead to a large drag, with the velocity dependence that is observed in the experiments. We remark first that the observed behavior of the drag coefficient at high velocities is similar to that found with an oscillating structure in a classical fluid (4, 5, 11, 29), and that this behavior is in turn similar to that observed with a bluff object that moves at a constant velocity through a classical fluid. These similarities are consistent with the view that classical and quantum turbulence are to some degree similar. Now we know that the classical behavior for steady flow arises because large eddies are generated in the wake of the object with a characteristic velocity similar in magnitude to the steady-flow velocity (we shall refer to a “large-scale turbulent wake”). In essence, the same type of behavior is observed to be associated with an oscillating structure in a classical fluid. For example, under appropriate conditions, an oscillating sphere is observed to throw off during each cycle a classical vortex ring, of radius similar to that of the sphere, as shown in Fig. 6. An oscillating cylinder in a classical fluid seems to behave in a more complicated way, involving the generation of classical vortices that move off sideways (17); the corresponding drag coefficient behaves in a complicated way, as we have seen, although it remains close to unity for large velocities. Independently of the details, it is generally the case that a drag coefficient of order unity in a classical fluid at high velocities is associated with the generation of a large-scale turbulent wake. We note that the instability in the laminar flow that gives rise to this wake is in turn associated with the requirement that the tangential flow velocity at the surface of the structure must vanish for a viscous fluid (the no-slip condition; see, e.g., ref. 30, chapter 5). We note also the critical velocity for the formation of a classical turbulent wake behind an oscillating structure: in the case when the viscous penetration depth is greater than the size of the structure (low frequency), the critical velocity is given by putting the Reynolds number equal to unity; in the opposite limit, it is given in order of magnitude by (31, 32):

Fig. 5. Simulations showing how a vortex tangle is generated by a smooth oscillating sphere in a channel; the sphere oscillates in the vertical direction. Adapted from ref. 27.

Let us first consider the behavior in the superfluid case at zero temperature. We have seen from the simulations of Hänninen et al. that, above a critical velocity, an oscillating structure in superfluid helium generates a disordered and dense tangle of quantized vortex lines in the superfluid. Because it is dense, the tangle has associated with it a vortex line spacing that is small compared with the size of the structure, and the disordered nature of the tangle means that the Uc = BðωνÞ1=2 ; [4] tangle itself generates no large-scale flow in the superfluid. However, we now see that a drag coefficient of order unity is likely to arise only where v is the kinematic viscosity of the fluid, if there is a large-scale turbulent wake. Now large-scale rotational motion can be and B is a dimensionless number of order unity that depends on the shape of the structure. generated in a superfluid containing a tangle of Vinen and Skrbek

Fig. 6. A sphere oscillating in water; adapted from an original photograph supplied by R. J. Donnelly and R. Hershberger (Department of Physics, University of Oregon, Eugene, OR).

vortex lines if that tangle becomes at least partially polarized. It is believed that this polarization produces flow of the superfluid, on scales larger than the quantized vortex spacing, that behaves in an essentially classical way. It is for this reason that superfluid helium can, as we believe, support a Richardson inertialrange cascade, with a Kolmogorov energy spectrum, on these large scales (33). Furthermore, dissipation per unit mass in the bulk, responsible for terminating such a Richardson cascade, is given by (2, 34): ɛ = ν′κ 2 L2 ;

[5]

where ν′ is an effective kinematic viscosity (35) (of order κ) and L is the vortex line density; this expression is closely similar to the classical formula ɛ = νhω2 i, because κ 2 L2 is an effective mean square vorticity hω2 i. However, boundary conditions also play an important role in classical fluid mechanics, and, as we have seen, the generation of large-scale eddy motion in the wake of a moving structure is closely associated with the no-slip condition at a solid boundary. For a smooth surface, no such boundary condition restricts the flow of the superfluid component, even when it is threaded by quantized vortices, although the normal component of the superfluid velocity must still vanish, as in the classical case. It is probably for this reason that, in the simulations of Hänninen et al., no large-scale turbulent motion was generated. However, as we have emphasized, real surfaces are not smooth and they can pin the end of any vortex that is attached to them. Unfortunately, it is not immediately obvious what effect this pinning has on the flow near a solid boundary. Suppose that it were the case that pinning of quantum vortices at a solid boundary leads in some adequate approximation to a no-slip condition for the superfluid velocity at a solid boundary. Then we might expect that, once a dense tangle of quantized vortices has been Vinen and Skrbek

must result in a coupling between the two fluids, but there can be no transition to a fully turbulent wake until U > Uct . We guess that in the intermediate regime the drag coefficient would increase a little, but that there would be no sharp rise and flattening off toward a value of order unity. It is tempting to suppose that this is the origin of the two critical velocities. We note that according to this idea the two critical velocities could still be present at a very low temperature, where there is no normal fluid. Our assumption that Ucn is large may not be true at higher temperatures. If it were the case that Ucn < vsc , turbulence would appear in the normal fluid at a velocity, U, that is smaller than that required for turbulence in the superfluid component. This seems to be the case in the recent experiments of Zemma and Luzuriaga (36) with an oscillating paddle. A mathematical formulation of these ideas for the case when vsc > Uct was presented by Blazková et al. (29), and, in a slightly different form, by Bradley et al. (5, 11). This formulation allowed the limiting value of the drag coefficient at high velocities to be adjustable, and other adjustable parameters allowed for the fact that transitions must be to some extent gradual. Comparison with experiment, particularly for vibrating tuning forks, showed that a reasonable agreement with experiment could be achieved with reasonable values of the adjustable parameters. However, the fitted parameters seemed to vary from fork to fork in a way that had no obvious explanation. This was particularly the case at low temperatures, where there is little or no normal fluid, and where the damping at small amplitudes is caused largely by losses in the material of the structure. It is especially worrying that the limiting value of the drag coefficient at high velocities seemed to vary significantly from fork to fork; one might have expected that high vortex densities might lead to behavior that is more reproducible. These effects are examples of the lack of reproducibility to which we have already referred. We can speculate about how such a lack of reproducibility might arise. We have emphasized that the generation of a large-scale turbulent wake in a classical fluid depends on the no-slip condition at a solid boundary. It follows that any relaxation of this boundary condition is likely to have a serious effect on the structure of this turbulent wake, and therefore, in particular, on the dependence of drag coefficient on velocity at high velocities. As we have emphasized, we do not really know the boundary condition for a turbulent superfluid, except that it must depend on the extent of vortex pinning at the boundary. The extent of this pinning must depend on the state of roughness of the surface, and PNAS | March 25, 2014 | vol. 111 | suppl. 1 | 4703

SPECIAL FEATURE: PERSPECTIVE

created in the vicinity of an oscillating structure, the superfluid would behave like a classical fluid with kinematic viscosity of order ν′ (or κ), with a no-slip condition at a solid boundary. The evolution of the largescale flow round an oscillating structure should then be classical, leading to classical behavior of the drag coefficient. The assumption that the random vortex tangle behaves like a classical fluid with kinematic viscosity of order κ is consistent with the plausible idea that this tangle has an eddy kinematic viscosity given by multiplying the characteristic length ℓ, equal to the vortex-line spacing, by the characteristic velocity on this scale, which is κ=ℓ. At a finite temperature, this picture must be modified by the presence of the normal fluid. The presence of a dense tangle of vortices implies that there is a large force of mutual friction between the two fluids, so that motion in these fluids is strongly coupled (2, 33). At the same time, the boundary condition on the normal fluid is that of no slip, exactly as we have just assumed for the superfluid component. It is reasonable then to suppose that the two fluids act as a single fluid with density equal to the total density, and with the classical boundary condition, so we can still expect overall classical behavior. The effective kinematic viscosity, νt , of this single fluid is presumably of order κ, because both the eddy kinematic viscosity of the superfluid component and the ratio of the viscosity of the normal fluid to the total fluid density are of order κ. To this picture, we must now add the idea that the dense tangle of quantized vortices (without, initially, any large-scale motion) can form only at a velocity greater than that given by Eq. 3. There are then two other relevant critical velocities, related to Eq. 4: Ucn = Bðωνn Þ1=2 , and Uct = Bðωνt Þ1=2 , where νn is the kinematic viscosity of the normal fluid (ηn =ρn ) (the forms of Uc assume the high-frequency limit). Suppose first that ρn =ρ is small compared with unity, which is true if the temperature is fairly low, so that Ucn  Uct . Suppose also that vsc is greater than Uct , but smaller than Ucn . Let the velocity of the structure be U. Then if U < vsc , there can be no turbulence in either fluid and the drag must be due to laminar flow of the normal fluid only. When U > vsc a vortex tangle is produced in the superfluid component, the two fluids become coupled, and the coupled fluids make a transition to a state in which there is a developed turbulent wake; the drag coefficient rises from a small value corresponding to a drag from laminar flow of the normal fluid only to a value approaching unity. If, however, vsc < Uct , then at a velocity a little larger than vsc a dense tangle of vortex lines must form around the structure, which

therefore it could well vary from structure to structure in an unpredictable way. A corresponding variation in the boundary condition may well follow. It is interesting that the lack of reproducibility seems to be most serious at very low temperatures, suggesting that the presence of the normal fluid is helpful in reducing irreproducibility; perhaps this is to be expected because the boundary condition on the normal fluid, combined with mutual friction, ought indeed to help bring the superfluid component to rest close to a solid boundary. It should be added that in some cases a lack of reproducibility may have been enhanced by the small size of the oscillating structure; some experiments have been carried out on wires as thin as 4 μm. In this case, the size of the structure might not have been large compared with the important characteristic length in quantum turbulence arising from the spacing between the discrete vortex lines. The ideas that we have just been outlining are obviously speculative. Without more detailed evidence, absent at the present time, it is hard to know whether they are right or wrong. This is our problem. Later, we shall discuss how such evidence might be acquired. Vibrating Grids

So far, we have confined our discussion to oscillating structures, in superfluid 4He, in the form of spheres, cylinders (wires), and tuning forks. Many experiments on oscillating grids have also been reported. In the case of grids in superfluid 4He, the results have often exhibited the same (or worse) lack of reproducibility as between apparently similar grids that we have seen with other structures (ref. 37 and references therein). Furthermore we know very little about the detailed flow patterns generated by motion of the grid. Therefore, it does not seem profitable to discuss this case any further at this stage. Instead, we shall describe interesting results obtained with oscillating grids in 3He-B. As we have already commented, experiments on oscillating structures in superfluid 3 He-B are often hard to interpret because vortex nucleation velocities are comparable with the Landau critical velocity for pair breaking. Nevertheless, it has turned out that experiments with a vibrating grid in 3He-B at very low temperatures have produced especially interesting results, which are relevant to some important issues of general interest in quantum turbulence, and which we shall now discuss (38, 39). The grid, with a mesh size of about 50 μm, is mounted on a wire loop and situated in a transverse magnetic field, as shown in Fig. 7, so that an alternating current through the loop serves to drive a vibration of the grid (in a flapping motion) at a natural frequency of 1,250 Hz. Vortex lines produced by the 4704 | www.pnas.org/cgi/doi/10.1073/pnas.1312551111

vibrating grid are detected, and their density measured, by a method involving the Andreev reflection of a low density of thermal quasiparticles from the superfluid velocity field; the quasiparticles are detected by their effect on the damping of two vibrating wire resonators (not shown in the figure). The method is described in detail in another article in this volume (40). At very small velocity amplitudes, it is probable that no vortex lines are produced by the grid. In a range of velocity amplitudes between about 1.9 and 3.5 mm·s−1, vortices are detected, but after the grid drive is removed they disappear from the region in front of the grid in less than 0.1 s. At velocity amplitudes greater than 3.5 mm s−1, the observed vortices in front of the grid persist for much longer times, decaying over a period of more than 10 s. This observation is interpreted as follows. At velocity amplitudes less than 3.5 mm s−1, vortex rings are emitted from the grid at a low density, so that interactions between the rings can be neglected. The rings propagate away from the grid at a velocity that is sufficiently large that they disappear from its neighborhood in less than 0.1 s, which means that the radii of the rings must be less than about 5 μm. The mechanism of production of these rings may be associated with remanent vortices and may be similar to that operating in the simulations of Hänninen et al., or it may be intrinsic. (The critical velocity for the production of vortex rings at 1,250 Hz, given by Eq. 3, is about 28 mm·s−1, compared with the grid velocity of 1.9 mm·s−1 at which rings are observed to be produced. There must be some enhancement of the superfluid velocity near sharp corners of the grid, but it may not be large enough.) At velocity amplitudes greater than 3.5 mm·s−1, the density of rings in front of the grid becomes large enough for the rings to interact, to undergo reconnections, and to produce a tangle of vortex lines, initially with a line spacing of order 5 μm. This tangle decays only slowly. These observations are of interest for two reasons. First, the process by which irrotational flow first breaks down—the production of small vortex rings—is very similar to the nucleation process that we have already described in connection with other types of oscillating structure in 4He. We see this as evidence that this type of process is common in oscillatory flows that lead to quantum turbulence. However, two reservations must be added: as we have noted, there is the possibility that the nucleation in 3 He-B is intrinsic; and in the case of an oscillating grid in 4He, where the nucleation must be extrinsic, we have as-yet no evidence that nucleation leads first to the

Fig. 7.

Schematic diagram of oscillating grid.

production of vortex rings. Second, we see that the oscillating grid serves to produce, initially, a particular type of quantum turbulence, in which the vortex lines are disordered, with negligible flow on scales much larger than the line spacing. The way in which this type of turbulence evolves and decays is of great interest (see, for example, ref. 41). The Future

Although experiments on the generation of quantum turbulence have produced a wealth of experimental results, many of them are, as we have seen, hard to interpret in a convincing way. The skeptic could well claim that we have indulged in too much speculation. Our view is that speculation is justified if, as we now argue, it can lead to new and informative experiments, theory, and computation. The most obvious experiment would be to develop some form of visualization of the turbulent flow round an oscillating structure. The technique based on Andreev reflection of quasiparticles, applicable to 3He-B and mentioned in the preceding section, provides some degree of visualization, but it has poor spatial resolution and is not easy to interpret (40, 42). Methods based on a seeding of the flow with either micrometer-sized particles of solid hydrogen/deuterium or triplet-state He2 excimer molecules are being successfully developed, as described in another article (43), and the former technique has revealed some strange flow patterns in the neighborhood of a solid cylinder immersed in a steady thermal counterflow in 4He above 1 K. However, experiments with these hydrogen particles are hard to interpret because at high temperatures the particles undergo complicated interactions with both the normal fluid and the vortex lines, whereas loading is difficult at low temperatures. The excimer molecules might be Vinen and Skrbek

Vinen and Skrbek

in some simpler way. In any case, it is clear that simulations of the effect of surface roughness on the evolution of a vortex tangle attached to an oscillating structure are urgently required. Indeed, this problem of surface roughness, and the associated problem of the boundary conditions for the superfluid component at a solid surface, is relevant to a wider range of quantum turbulent flows; pipe flow is an obvious and important example. Of course, it may turn out that the boundary condition depends very much on the precise state of roughness of the surface, in which case it would be very useful to discover experimentally whether a particular surface treatment can lead to a reproducible boundary condition.

Turning to computational work, we have already mentioned that simulations of the behavior of a vortex tangle close to a rough boundary pose difficulties. An ambitious simulation of the effect of wall roughness on thermal counterflow in a straight channel was carried out by Schwarz (46). The wall roughness was simulated by an array of well separated hemispherical protuberances. A vortex attached to the wall was allowed to slide along the wall. When it came close to a protuberance, it interacted in a way that had been simulated in an earlier paper (47); a surface reconnection led to its jumping onto the protuberance; its end then moved over the protuberance; and a second surface reconnection led to its jumping off the protuberance. Applying this procedure to an oscillating structure would be a formidable task, and it has not to our knowledge been attempted. It would be interesting to examine whether the effect of roughness could be simulated to an adequate extent

ACKNOWLEDGMENTS. We are grateful to many of our friends and colleagues for stimulating discussions relating to the subject of this article. At the same time, we are conscious that much interesting work on oscillating structures in a superfluid has not been included in this article. L.S. acknowledges the support of the Czech Science Foundation under Project GACR 203/11/0442.

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19 Muirhead CM, Vinen WF, Donnelly RJ (1984) The nucleation of vorticity by ions in superfluid He-4. 1. Basic theory. Philos Trans R Soc A 311(1518):433–467. 20 Zurek WH (1985) Cosmological experiments in superfluidhelium. Nature 317(6037):505–508. 21 Goto R, et al. (2008) Turbulence in boundary flow of superfluid 4He triggered by free vortex rings. Phys Rev Lett 100(4): 045301. 22 Nago Y, et al. (2011) Time-of-flight experiments of vortex rings propagating from turbulent region of superfluid He-4 at high temperature. J Low Temp Phys 162(3-4):322–328. 23 Vinen WF (1961) Detection of single quanta of circulation in liquid helium 2. Proc R Soc A 260(130):218–236. 24 Yano H, et al. (2010) Critical behavior of steady quantum turbulence generated by oscillating structures in superfluid 4He. Phys Rev B 81(22):220507. 25 Avila M, Willis AP, Hof B (2010) On the transient nature of localized pipe flow turbulence. J Fluid Mech 646:127–136. 26 Niemetz M, Schoepe W (2004) Stability of laminar and turbulent flow of superfluid He-4 at mK temperatures around an oscillating microsphere. J Low Temp Phys 135(5-6):447–469. 27 Hänninen R, Tsubota M, Vinen WF (2007) Generation of turbulence by oscillating structures in superfluid helium at very low temperatures. Phys Rev B 75(6):064502. 28 Hänninen R, Schoepe W (2008) Universal critical velocity for the onset of turbulence of oscillatory superfluid flow. J Low Temp Phys 153(5-6):189–196. 29 Blazková M, Schmoranzer D, Skrbek L, Vinen WF (2009) Generation of turbulence by vibrating forks and other structures in superfluid He-4. Phys Rev B 79(5):054522. 30 Batchelor GK (1967) An Introduction to Fluid Dynamics (Cambridge Univ Press, Cambridge, UK). 31 Schmoranzer D, Král’ová M, Pilcová V, Vinen WF, Skrbek L (2010) Experiments relating to the flow induced by a vibrating quartz tuning fork and similar structures in a classical fluid. Phys Rev E Stat Nonlin Soft Matter Phys 81(6 Pt 2):066316. 32 Blazková M, Schmoranzer D, Skrbek L (2007) Transition from laminar to turbulent drag in flow due to a vibrating quartz fork. Phys Rev E Stat Nonlin Soft Matter Phys 75(2 Pt 2):025302. 33 Vinen WF (2000) Classical character of turbulence in a quantum liquid. Phys Rev B 61(2):1410–1420. 34 Stalp SR, Skrbek L, Donnelly RJ (1999) Decay of grid turbulence in a finite channel. Phys Rev Lett 82(24):4831–4834. 35 Chagovets TV, Gordeev AV, Skrbek L (2007) Effective kinematic viscosity of turbulent He II. Phys Rev E Stat Nonlin Soft Matter Phys 76(2 Pt 2):027301. 36 Zemma E, Luzuriaga J (2012) Measurements of turbulence onset and dissipation in superfluid helium with a silicon double paddle oscillator. J Low Temp Phys 166(3-4):171–181.

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SPECIAL FEATURE: PERSPECTIVE

a better bet in 4He, because loading is easy; furthermore, above 1 K they track only the normal fluid, whereas well below 1 K they are expected to be trapped by, and therefore decorate, the vortex lines. We look forward to the development and application of these techniques to oscillating turbulent flows, although we recognize that this will take time. In the meantime, we suggest that further experiments of the type already carried out might be fruitful, provided that they involve structures with simple geometries and a range of well-defined surface conditions, carefully chosen to throw light on specific questions. We describe one example. We have suggested that a remanent vortex, attached to an oscillating structure, can lead to the generation of a random tangle of vortex lines over the surface of the structure, and that, in the presence of surface pinning, this small-scale tangle then allows large-scale turbulent motion to develop in the superfluid component around the structure. It would be very helpful to have clear evidence in an appropriately simple geometry for the existence and characteristics of this small-scale tangle as a precursor to the development of large-scale turbulence. The appropriate geometry is one in which the complication of a transition to large-scale turbulence is, as far as possible, inhibited, or at least delayed, and also one for which the forms of remanent vortex can be guessed with some confidence. The instability that leads to large-scale turbulence is often associated with flow over a surface with convex curvature: the production of Taylor–Görtler vortices in oscillating flows (see, e.g., ref. 9), and the separation of a steady flow behind a bluff obstacle, are examples. Thus, our proposal is as follows. Take a vessel in the form of a closed pillbox, which can be suspended from a torsion fiber so that it can oscillate about its axis of circular symmetry. The pillbox is filled with superfluid helium, the outside being in vacuo, and the damping of the torsional oscillations can be observed at all temperatures as a function of the amplitude of oscillation. With suitable design, the formation of a small-scale tangle will lead to measurable contributions to the damping and to the moment of inertia. A transition due to large-scale turbulence can be expected at a high enough velocity, corresponding to the transition to turbulence in the viscous penetration depth associated with oscillatory flow of a classical fluid over a plane surface (44), but this can be estimated to occur only at velocities much higher than those expected for the formation of small-scale quantum turbulence. The form and density of the remanent vortices that are likely to be stretched between the sides of the pillbox can be guessed plausibly from the work of Awschalom and Schwarz (45).

37 Charalambous D, Skrbek L, Hendry PC, McClintock PVE, Vinen WF

quantum turbulence in 3 He-B. Proc Natl Acad Sci USA

(2006) Experimental investigation of the dynamics of a vibrating grid in

111:4659–4666. 41 Baggaley AW, Barenghi CF, Sergeev YA (2012) Quasiclassical and

superfluid 4He over a range of temperatures and pressures. Phys Rev E Stat Nonlin Soft Matter Phys 74(3 Pt 2):036307. 38 Bradley DI, et al. (2005) Emission of discrete vortex rings by a vibrating grid in superfluid 3He-B: A precursor to quantum turbulence. Phys Rev Lett 95(3):035302. 39 Bradley DI, et al. (2006) Decay of pure quantum turbulence in superfluid 3He-B. Phys Rev Lett 96(3):035301. 40 Fisher SN, Jackson MJ, Sergeev YA, Tsepelin V (2014) Andreev reflection, a tool to investigate vortex dynamics and

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ultraquantum decay of superfluid turbulence. Phys Rev B 85(6): 060501. 42 Suramlishvili N, Baggeley AW, Barenghi CF, Sergeev YA (2012) Cross-sections of Andreev scattering by quantized vortex rings in He3-B. Phys Rev B 85(17):174526. 43 Guo W, La Mantia M, Lathrop DP, Van Sciver SW (2014) Visualization of two-fluid flows of superfluid helium-4. Proc Natl Acad Sci USA 111:4653–4658.

44 Jensen BL, Sumer BM, Fredsoe J (1989) Turbulent oscillatory boundary-layers at high Reynolds-numbers. J Fluid Mech 206:265–297. 45 Awschalom DD, Schwarz KW (1984) Observation of a remanent vortex-line density in superfluid-helium. Phys Rev Lett 52(1):49–52. 46 Schwarz KW (1992) Effect of surface roughness on the critical velocities of superfluid 4He. Phys Rev Lett 69(23): 3342–3345. 47 Schwarz KW (1985) Three-dimensional vortex dynamics in superfluid 4He: Line-line and line-boundary interactions. Phys Rev B Condens Matter 31(9):5782–5804.

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