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Turbulence in cryogenic helium
Physica C 404 (2004) 354–362 www.elsevier.com/locate/physc
Turbulence in cryogenic helium L. Skrbek
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Joint Low Temperature Laboratory, Institute of Physics ASCR, Charles University, V Holesovickach 2, 180 00 Prague 8, Czech Republic
Abstract Turbulence, both classical and quantum, represents an intellectual challenge in understanding many natural phenomena. The fundamental theory of turbulence is yet to be developed and the role of experiment is therefore crucial. Cryogenic helium offers several fluids that can be used for generating turbulent flows under controlled laboratory conditions. We discuss experiments on classical and quantum cryogenic helium turbulence, pointing out similarities and differences between the two. 2004 Elsevier B.V. All rights reserved. Keywords: Turbulence; Cryogenic helium
1. Introduction Turbulence is one of the grand-challenge problems of our times: it is profound, difficult, and important in a large variety of applications. Turbulence is understood as the last unsolved problem of classical physics, involving a tremendous range of length scales from galaxy size down to angstrom size typical for the core of a quantized vortex. The fundamental theory of conventional turbulence is still absent and in developing our knowledge of turbulence, the role of experiment becomes crucial. In order to generate and study highly turbulent flows in a controlled laboratory experiment of limited size, one has to choose working fluids possessing the most favorable flow properties, most notably fluids with as low as possible kinematic viscosity. Here, low temperature physics and
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Tel.: +42-02-21912558; fax: +42-02-21912567. E-mail address: [email protected] (L. Skrbek).
cryogenics become very useful, thanks to the accumulated knowledge of physical properties of cryogenic helium. There are two stable isotopes of helium of very different behaviour at low temperature––4 He and 3 He, both commonly used in low temperature physics. 3 He is a fermion and we discuss its application for superfluid dynamics later in this paper; while 4 He atom is a boson, representing a common form and unless explicitly mentioned throughout this paper, by helium we understand 4 He. It offers three fluids of interest for experimental investigation of high Reynolds (Re) and Rayleigh (Ra) number flows under controlled laboratory conditions, primarily due to their extremely low values of kinematic viscosity, m, lowest of all known substances. The first is cryogenic helium gas near the critical point at 5.2 K and 2.26 atm. In addition to its very low viscosity it also allows an unprecedented flexibility, as its fluid properties can be easily tuned over many orders of magnitude by varying the temperature and
0921-4534/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2003.11.030
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pressure. The second is normal liquid helium, or He I, which is an ordinary Navier–Stokes fluid. He I has its normal boiling point at 4.2 K and exists (along the saturated vapor line) down to the lambda temperature Tk ffi 2:17 K. Below that liquid helium becomes superfluid and is usually referred to as He II. Properties of liquid He I and He II are well known [1] and significantly change with temperature. He II is a quantum fluid and in using it as a working fluid for experiments, special care must be taken in interpreting the data. It displays the two fluid phenomena and its flow is restricted by the fundamental requirement of quantized circulation. On the other hand, the quantum nature of He II enables application of special experimental techniques, and experiments with He II used as a working fluid can be thought both as unique and complementary to classical high Re flow research. We start with classical turbulence, then describe basic properties of He II and discuss similarities and differences between classical and quantum turbulence. For more information on this subject, see e.g., [2–7] and references therein.
2. Classical cryogenic turbulence Cryogenic turbulence experiments based on liquid normal He I and cryogenic helium gas as working fluids are not principally different from the conventional room temperature ones commonly using air or water as working fluids, although there are important practical experimental differences due to the cryogenic environment. 2.1. Cryogenic thermal convection Thermal convection occurring in a fluid layer heated from below (or cooled from above) is ubiquitous in nature and technology. The dynamical state is determined by the Rayleigh number, Ra ¼ gaDTH 3 =ðmjÞ, Prandtl number, Pr ¼ m=j and aspect ratio C ¼ D=H , where g is the acceleration of gravity, DT is the vertical temperature difference across the fluid layer of height H , D is the characteristic lateral extent, a the thermal expansion coefficient and j the thermal diffusivity.
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Turbulent thermal convection sets in at high Ra. The efficiency of the heat transport is represented by the Nusselt number, Nu, defined as the total heat flux normalized by its conductive contribution. In order to understand various processes occurring in nature, it is of particular interest to be able to predict the functional dependence Nu ¼ NuðRa; Pr; CÞ in the widest possible range of Ra, as well as for asymptotically large values. Experimentally, it is desirable to work within socalled Boussinesq approximation, which in a controlled laboratory experiment of limited size H sets an upper limit for DT , and thus the highest achievable Ra is determined by the combination a=mj for the used working fluid. For cryogenic helium gas in the vicinity of the critical point the value of a=mj exceeds values for any other known working fluid by several orders of magnitude. Moreover, as first demonstrated by Threlfall [8] and later by others [9–12], by changing temperature and pressure one can cover many orders of magnitude in Ra in a single experiment. As an example, to cover a modest range of five decades 107 < Ra < 1012 in water may easily require using cells of four different aspect ratios. Using helium gas in the range 4.3 < T < 6 K; 0.1 mbar < p <3 bar allows covering 11 orders of Ra in fully developed turbulent convection up to 1017 in a single experiment, using a convection cell of H ¼ 1 m and C ¼ 1=2 [12]. The data, over the entire huge range of Ra, can be described to the lowest order by a single power law Nu / Rab with b ffi 0:31. Even at these large Ra there is no sign of any transition to so-called ultimate scaling regime predicted long ago by Kraichnan [13] and recently claimed by others [11]. The very existence of the ultimate scaling regime thus remains an open question, although there are new ideas how to possibly resolve this issue [14]. The temperature field in various places in the convection cell has been monitored using small neutron doped germanium sensors. The temperature power spectra display both the )5/3 Kolmogorov and )7/5 Bolgiano roll-off exponents, depending on the so-called Bolgiano length scale loosely dividing ranges where the temperature field behaves as active or passive scalar. The probability density functions (PDF) of the temperature
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fluctuations measured in the centre of the cell undergo drastic changes with increasing Ra, being bimodal around 106 ––suggesting that the top and bottom plate temperatures are not yet fully mixed, mixing being essentially complete around 108 where PDF are almost gaussian, but evolving towards stretched exponential shape with still increasing Ra [12]. The asymptotic state of the temperature PDF is thus not gaussian, in accord with the notion of a strong intermittency of the turbulent temperature field. Using very long records of temperature fluctuations of order of 221 data points, the temperature structure functions in the Bolgiano regime have been computed and studied [15]. Although the direct scaling does not allow any definite conclusions about scaling exponents, application of the widely accepted extended similarity hypothesis strongly suggests that the scaling exponents do not show any tendency to saturation, in contrast to the case of passive scalar [16]. Placing two closely vertically spaced sensors in the vicinity of the vertical wall allows one to measure the velocity of the large scale motion, or the ‘‘wind’’. At relatively low Ra of order 106 the wind is unidirectional, but with increasing Ra it abruptly (i.e., within one turnover time) changes direction in a stochastic manner. The wind stays well defined up to about 1015 , above which it becomes ill-defined due to too frequent switchovers [14,17]. A feasibility study shows that a cryogenic helium gas experiment in an even larger convection cell of height H ¼ 10 m should provide the data up to Ra ffi 1020 , covering the dynamical range of convective processes occurring in the ocean or in the atmosphere. However, building such a cell represents a major technical problem and would require substantial supply of liquid helium. 2.2. Wind tunnels The advantage of using cryogenic helium gas as a working fluid in a classical wind tunnel can be illustrated by comparing the typical parameters of classical air facilities [18] with the performance of a helium gas wind tunnel built inside a conventional laboratory cryostat. Based on the test section of the wind tunnel of crosssection only 6 · 6 cm2 , by
changing the temperature 5 < T < 78 K and pressure 1 < p < 4 atm and changing the mean velocity between 2 cm/s < U < 2 m/s, a broad range of Reynolds numbers based on the Taylor microscale 30 < Rek < 8000 can be achieved in a single experiment. Such a wind tunnel has been already built at the University of Oregon. 2.3. High Re pipe flow Normal liquid He I has been used as a working fluid in a pipe flow experiment, installed into a conventional laboratory cryostat. Measurements of the friction factor in a smooth pipe cover a wide range of Re up to 3 · 106 . The physical measured quantity is the pressure drop between two ports in the pipe wall, detected by an extremely sensitive capacitive bridge. Such a high Re was previously achieved only in few classical experiments (e.g., work of Nikuradse or Princeton SuperPipe [4]) exploiting extremely large industrial size experimental facilities. It is worth mentioning that using He I with extremely low kinematic viscosity would lead to considerable saving in the power required to operate such a high Re facility. There are, in principle, no serious limitations to flow velocity for He I as the first sound velocity is high and the Mach number is likely to stay low for all technically foreseeable flow velocities. Let us point out here, that using superfluid He II would be limited by the second sound velocity about 20 m/s, when second sound shock phenomena will occur.
3. Quantum turbulence Below Tk liquid helium becomes superfluid and is referred to as He II. Quantum mechanics is needed to describe and understand most of its physical properties; it is therefore known as a quantum fluid and exhibits extraordinary flow properties. In a limit of low flow velocities, they can be largely understood within a phenomenological two fluid model [19], where He II is described as consisting of two interpenetrating fluids of independent velocity fields––the inviscid superfluid of density .s , and the normal fluid of density
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.n and dynamic viscosity g; the total density . ¼ .s þ .n . The superfluid has neither viscosity nor entropy; the entire heat content of He II is carried by the normal component. At absolute zero, He II is entirely superfluid. As the temperature increases the normal fluid fraction becomes larger until at the lambda point at which He II is entirely normal fluid. This simplified picture is reflected by the Landau two fluid equations of motion. One important outcome of these equations is the prediction of second sound, a wave described by temperature fluctuations rather than density fluctuations as is the case of ordinary sound, also referred to as first sound. The two fluid equations also explain the existence of a peculiar flow of He II called counterflow––under influence of applied heat the superfluid moves towards the heat source, becomes converted into the normal fluid and flows against the approaching superfluid in such a way that the total density of helium II stays unchanged. From the quantum-mechanical point of view (following the idea of London), the entire volume of superfluid can be described by a macroscopic pffiffiffiffi wave function of a form W ¼ .s eiu , where u is a macroscopic phase. According to quantum mechanics, the superfluid flow velocity must be proportional to the spatial gradient of this macroscopic phase (applying the impulse operator ihr on W we get the impulse p / v / ru), in close analogy with supercurrent in superconductors. It immediately follows that the superfluid velocity field must be irrotational. However, superfluid flow stays fully potential only at small velocities. In macroscopic flows involving higher velocities, typically of order cm/s, quantized vortices appear in the fluid. The superfluid flow stays irrotational everywhere except inside the core of the quantized vortex line. These lines have cores of atomic dimensions and their circulation is quantized in units of j ¼ h=m, where h is PlanckÕs constant and m is the mass of the superfluid particle. In He II j ¼ 9:997 104 cm2 /s. The quantization follows from the requirement that the macroscopic wave function is single-valued, which allows that the phase changes by any multiple of 2p while circling a path containing vortex lines. As a consequence of the Kelvin theorem, quan-
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tized vortex lines may either take shape of closed loops, or they must end at the boundary. Since the circulation is quantized and the superfluid vorticity is restricted to the vortex core, the mean superfluid vorticity is proportional to the total length of quantized vortex line per unit volume, and can be defined as hxs i ¼ x ¼ jLðtÞ. The presence of a vortex line results in a force of interaction between the superfluid and normal fluid components–– mutual friction. This interaction results in the otherwise independent normal and superfluid velocity fields becoming coupled. An instructive system to assimilate the role of quantized vortex lines in superflow is a rotating bucket of He II. In steady state, the normal fluid is in solid body rotation, with uniform vorticity x ¼ r v ¼ 2X, where X is the angular velocity of rotation. On increasing X slowly from rest, the superfluid stays irrotational (so-called Landau state), but only until we reach some critical velocity and create quantized vortices. At higher angular velocities, the fluid is threaded with a rectilinear array of quantized vortex lines that have an orientation parallel to X. Although the superfluid vorticity is restricted to vortex lines, their areal density, n0 , evolves to match the vorticity of the fluid in solid body rotation. This results in n0 j ¼ 2X. Thus on length scales larger than the inter-vortex line distance, the superfluid is also, on average, in solid body rotation, with a velocity field essentially equal to that of the normal fluid, i.e., hxs i ¼ xn and hvs i ¼ vn . The role of quantized vortex lines in a turbulent flow becomes more complex [5,6]. Computer simulations show that similar to the way vortex lines align in a rotating bucket, bundles of quantized vortices align on the cores of normal fluid eddies [20]. The superfluid vorticity is comparable to the vorticity of the normal fluid eddy and the effective fluid density is equal to the total fluid density. Turbulent flow contains a complex tangle of vortices evolving in such a way as to minimize the difference between the two otherwise independent velocity fields. The length scale of interest and the complexity of the flow determine how closely the two velocity fields become coupled together. Such a coupled flow appears to closely obey the Navier–Stokes equations and classical boundary
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conditions. A vortex-coupled flow still exhibits superfluidity as evidenced by supporting the transmission of second sound, although attenuated by the presence of these lines. 3.1. He II-working fluid in fluid dynamics? Historically, superfluid turbulence was usually viewed as a tangle of quantized vortex lines and its possible relationship to conventional turbulence has not been given much thought. This resulted in disconnected fields of conventional and superfluid turbulence, and so it seemed reasonable to think that the two fields were quite different in character. Recently, however, turbulent flow in He II has been generated in a manner similar to that in studies of classical turbulence (coflow turbulence). Let highlight some experiments revealing a deep similarity between classical and superfluid flows and turbulence. Maurer and Tabeling [21] produced turbulence in a flow of liquid helium confined between counterrotating discs, and obtained the energy spectral density at T ¼ 2:3, 2.08, and 1.4 K. From the standpoint of the two fluid model, one may think of liquid helium as being largely superfluid at 1.4 K and largely normal (or classical) at 2.08 K, and entirely normal at 2.3 K. Yet, the spectra obtained at all three temperatures (at experimentally accessible length scales) were identical and possessed an inertial range of the classical Kolmogorov form. He II flow past the sphere displays the drag crisis [30] and show no appreciable difference from classical pipe flow [31]. A great advantage of using He II in the study of turbulence is that it is possible to obtain the vortex line density by measuring the attenuation of second sound [19]. This allows the direct measurement of the superfluid vorticity, averaged over some experimental volume where the second sound is excited. This method was used to study the decay of the grid turbulence, created by towing the grid through a stationary sample of He II. The sensitivity and dynamic range of this technique are enormously large and correspond to monitoring up to eight orders of magnitude of the decaying turbulent energy, clearly an impossible goal for any classical experiment––reaching such a dynamic
range would require an air wind tunnel such as in [18] with a test section more than 1000 km long. In the experiments of [22,23] turbulence was produced by pulling a grid of bars through a stationary sample of He II, and the decaying average vortex line density (i.e., length of the quantized vortex line per unit volume) over a measurement volume was obtained from the attenuation of the second sound, transmitted and detected via circular gold-plated nuclepore membranes mounted flush in the wall across the channel. The goldplated channel side of the membrane serves as one electrode of a capacitor, the dc bias being used to press it against a brass electrode beneath. The ac signal tuned to the resonance frequency of the channel––second sound resonator––sets the standing wave resonance across the channel. Moving the membrane––transducer––generates the second sound, as superfluid stays at rest passing freely through its tiny channels while normal fluid is moving with it being glued within the pores by finite viscosity. This results in a relative motion of the two fluids, i.e., in second sound. Its amplitude is measured by another membrane––receiver–– opposite the channel. Quantized vortices in a volume between the transducer and receiver attenuate the second sound, thus measuring its amplitude provides a direct information on the vortex line density. The information about the turbulent flow is therefore obtained from measuring volume of order d 3 cm3 . This natural integration bypasses tedious statistical analysis involved in any local velocity measurements in conventional turbulence, provides enormous sensitivity and unprecedented dynamical range of the method, making it very useful and complementary to classical turbulence studies. The observed values of vorticity are evaluated from measurement of the time dependent recovery amplitude of the standing second sound resonance, initially suppressed by the presence of quantized vortices created by the motion of the grid through the channel. The temporal decay of vorticity, defined as xðtÞ ¼ jLðtÞ, where L is the length of quantized vortex line per unit volume, ranges from 104 to 0.01 Hz. This unprecedented range allowed detection of several distinctly different regimes of the decay [23] and makes this system unique in the study of turbulence. It even
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became possible to develop a purely classical spectral decay model, fully applicable to classical homogeneous and isotropic turbulence [24], as it was recently proved by computer simulations [25]. Different regimes switch as the energy containing and dissipative Kolmogorov length scales gradually grow during the decay, finally both being saturated by the finite size of the channel. The classical character of the decay of the vortex line density did not change with temperature within the range 1.2 < T < 2.0 K, while the ratio of the superfluid to normal fluid density varied over this temperature range from near unity to almost zero. Still, no appreciable difference in the turbulent decay curves obtained over this wide temperature range other than a slight temperature dependence of measured effective kinematic viscosity of turbulent He II occurred. However, this similarity between classical and superfluid turbulence breaks down at small scales where the quantization effects of circulation have to be taken explicitly into account. This happens at small scales of order ðe=j3 Þ1=4 , which defines a ‘‘quantum scale’’ of order of the mean distance between quantized vortices in He II. Beyond this scale the normal and superfluid velocity fields cannot be fully matched and the spectral energy density would depend in addition on j and is likely to be of the form UðkÞ ¼ Cej1 k 3 [26], where C is the threedimensional Kolmogorov constant and e ¼ dE=dt denotes the energy decay rate. At extremely high k (up to the inverse size of the vortex core) the spectrum is likely to be of the inverse k form, consistent with the spectral form of a straight vortex line. For large scales, the spectrum is classical. Interestingly, recent computer simulations suggest similar form of the superfluid energy spectra even in the zero temperature limit, where there is no normal fluid [27] and such a form of an energy spectrum cannot occur due to the influence of the normal fluid of classical behaviour. Although quantum turbulence in He II can be generated classically, historically it was discovered [29] in a counterflow channel, where it is easily generated by applying a heat pulse. It has been emphasized by various authors [6] that counterflow turbulence has no classical analog. However, recent experiments on the decay of counterflow He
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II turbulence suggest a possibility of missing links between classical turbulent thermal convection and counterflow turbulence [28]. The efficiency of the heat transfer displays similar scaling and also the late decay of counterflow turbulence displays classical character. Further experiments are in progress and will be reported elsewhere. Thus, even though the situation in He II turbulence is almost certainly quite complex, these experiments and computer simulations suggest that the two fields of turbulence are closely related to each other and in many cases one can analyze the data obtained in turbulent He II, at least approximately, in a classical manner––treating helium II as a Navier–Stokes fluid with an extremely small kinematic viscosity of order 104 cm2 /s. This is the lowest kinematic viscosity known for any substance and provides the possibility of investigation of extremely high Re flows under controlled laboratory conditions, which can be combined with the unique possibility of using the highly sensitive second sound technique for probing such a flow. On the other hand, there are important differences between quantum and classical turbulence that are best illustrated with superfluid 3 He.
4. Superfluid 3 He in superfluid dynamics At low temperature the physical properties of He and 4 He differ in a fundamental way, as 3 He atom is a fermion and obeys Fermi–Dirac statistics. 3 He therefore cannot display superfluidity based on the same physical principles as He II [7]. It becomes superfluid at much lower temperature of order 1 mK, but superfluidity of 3 He is much closer in nature to superconductivity in metals rather than to superfluidity of He II in that it includes Cooper pairs of 3 He atoms. The so-called equal spin pairing leads to the existence of Cooper pairs with both spin S ¼ 1 and orbital momentum L ¼ 1, which gives rise to a possibility of various superfluid phases. Let us limit ourselves with the experimentally most commonly observed A and B phases. For our purposes the pseudo-isotropic B-phase can be considered simply as a kind of He II possessing different flow properties. The viscosity of 3
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the normal fluid is very high, so the normal fluid could be considered almost at rest with respect to the walls of a laboratory size container of order 1 cm in size. The quantized vortices are also similar to those in He II, but the size of the vortex core is about two orders of magnitude larger. This is important from the point of view of pinning––any wall is rough for He II vortices with the atomic size core, but a smooth quartz wall can be considered hydrodynamically smooth for 3 He-B vortices. This means, in particular, that there are hardly any remnant vortices in a smooth container at rest, and by spinning such a container up the first vortex must be created by an intrinsic mechanism that requires high critical velocity. In other words, rotating 3 He-B maintains the metastable vortex free Landau state. The anizotropic 3 He-A phase is able to mimic the solid body rotation by continuously adjusting the texture (texture here having a meaning of spacial arrangement of the vector field defining the direction of the zero energy gap on the Fermi sphere), which results in the existence of so-called continuous vortices. They do not have a hard core similar to that of He II vortices, and each carries two quanta of circulation. Critical velocity for creating these continuous vortices is much lower than that in the B-phase. On first approximation, we may neglect it and consider the rotating 3 He-A sample always in solid body rotation, i.e., containing an (almost) equilibrium number of doubly quantized continuous vortices. By cooling the cylindrical 3 He sample in a smooth walls quartz cylindrical container at pressure 30 bar, first the normal to 3 He-A transition occurs slightly above 2 mK. On further cooling the whole sample would become the B-phase, but applying a strong magnetic field to the middle of the cell would prevent the B-phase from moving further up and the stable AB boundary occurs in the cell. We thus have an interesting stable situation––a cylindrical sample of 3 He-A floating over another cylindrical sample of 3 He-B. What happens if we start rotate the cryostat? Both A and B normal fluids will be, thanks to high viscosity, always in solid body rotation. So will be, on average, the entire Aphase, being threaded by an array of rectilinear continuous vortices, but the B-phase stays in a
Landau state with no quantized vortices. Vortices in the A-phase cannot end at the boundary, but in the vicinity of it (within a distance of order of their spacing) they bend away, end on the vertical cylindrical wall and form a vortex sheet parallel to the AB boundary. We therefore have a superfluid B-phase at rest, upon which the A-phase slides, exhibiting a pure shear flow [32]! But this is an ideal case considered almost two centuries ago by Lord Kelvin for a case of a wind blowing over calm water––so-called Kelvin–Helmholtz instability should occur above certain threshold, when the interface becomes corrugated at some particular wave length. It is indeed a case, and on spinning up above a threshold bursts of about 10 vortices enter the B-phase every time the threshold for instability is reached again and again. Let us mention that any experimental evidence for checking this instability criterion in a quantitative manner using classical fluids is always obscured by finite viscosity, which is not the case for a superflow. Indeed, modifying the Kelvin criterion for the described above 3 He case (it requires changing the role of gravity by magnetic anizotropy and also taking into account the two fluid hydrodynamics), good agreement between the instability theory and the experimental data is found [32]. Surprisingly, the exotic 3 He quantum fluids shed new light on the very old puzzle of classical hydrodynamics. This mechanism of injecting quantum vortex loops can be used to investigate the criteria of generating the quantum turbulence in 3 He-B. It turns out that if one or a few vortex loops are injected, e.g., at the AB interface, into rotating vortex free superfluid 3 He-B at high temperature (T > 0:6Tc ), they will evolve in a smooth way with their ends sliding the vertical wall and the resulting stage contains exactly the same number of rectilinear vortices running along X. At low temperature, however, the situation is very different in that the final observed stage represents (almost) the equilibrium solid body rotation containing many more vortices in comparison with the number of those originally injected at the interface. The nuclear magnetic resonance measurements thus indicate the existence of a sharp transition to superfluid turbulence at T =Tc ffi 0:6 [33]; this observation is supported by computer simulations.
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It appears that this transition is very different from transition to turbulence observed in conventional fluids obeying the Navier–Stokes equation. Here the transition to turbulence is determined by an interplay of the viscous and inertial terms through the Reynolds number. For a given flow geometry the critical Re at which the transition to turbulence occurs depends on the flow velocity. For superfluids, on inserting the mutual friction force in the Euler equation and assuming the normal fluid at rest one also obtains the hydrodynamic equation containing the dissipative and inertial term. The fundamental difference from conventional hydrodynamics is that the relative importance of these competing terms is determined here by an intrinsic parameter of the superfluid, q ¼ a=ð1 a0 Þ, where a and a0 are the usual mutual friction parameters. The dimensionless quantity q thus plays a role of a Reynolds number in superfluids! In the case of a single vortex line q marks crossover from the regime where Kelvin waves can propagate along a vortex core to where they are overdamped. In a particular case of 3 He-B at T =Tc ffi 0:6, where the transition to superfluid turbulence has been experimentally observed, q ffi 1:3. In 3 He-A He q about unity at extremely low temperatures, while in He II (where one has to keep in mind that viscosity of the normal fluid is extremely low) just below the kpoint.
5. Conclusion We have outlined some aspects and advantages of using cryogenic helium for experimental turbulence research. In view of the absence of a fundamental theory of turbulence based on first principles, cryogenic helium experiments open the road toward investigations of ultra-high Re and Ra flows, where applicability of any computer simulations is at the best very limited. Other important aspects involve superfluid flow and turbulence. We have shown that at some circumstances He II could be used as a working fluid in additional and complementary experiments to the conventional turbulence research. The underlying physics, however, is much more complicated and a better
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understanding of superfluid turbulence, involving superfluid phases of 3 He, represents an interesting intellectual challenge. We are convinced that experiments with superfluids and their theoretical interpretation––broadly defined as superfluid dynamics––will serve as an important tool for deeper understanding of the complex phenomenon of fluid turbulence in general.
Acknowledgements The author is deeply indebted to many colleagues with whom he had a pleasure to work or gain experience during discussions, especially to R.J. Donnelly, V.B. Eltsov, A.V. Gordeev, M. Krusius, J.J. Niemela, K.R. Sreenivasan and W.F. under Vinen. This research is supported by GACR grant 202/02/0251.
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Turbulence in cryogenic helium L. Skrbek
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Joint Low Temperature Laboratory, Institute of Physics ASCR, Charles University, V Holesovickach 2, 180 00 Prague 8, Czech Republic
Abstract Turbulence, both classical and quantum, represents an intellectual challenge in understanding many natural phenomena. The fundamental theory of turbulence is yet to be developed and the role of experiment is therefore crucial. Cryogenic helium offers several fluids that can be used for generating turbulent flows under controlled laboratory conditions. We discuss experiments on classical and quantum cryogenic helium turbulence, pointing out similarities and differences between the two. 2004 Elsevier B.V. All rights reserved. Keywords: Turbulence; Cryogenic helium
1. Introduction Turbulence is one of the grand-challenge problems of our times: it is profound, difficult, and important in a large variety of applications. Turbulence is understood as the last unsolved problem of classical physics, involving a tremendous range of length scales from galaxy size down to angstrom size typical for the core of a quantized vortex. The fundamental theory of conventional turbulence is still absent and in developing our knowledge of turbulence, the role of experiment becomes crucial. In order to generate and study highly turbulent flows in a controlled laboratory experiment of limited size, one has to choose working fluids possessing the most favorable flow properties, most notably fluids with as low as possible kinematic viscosity. Here, low temperature physics and
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Tel.: +42-02-21912558; fax: +42-02-21912567. E-mail address: [email protected] (L. Skrbek).
cryogenics become very useful, thanks to the accumulated knowledge of physical properties of cryogenic helium. There are two stable isotopes of helium of very different behaviour at low temperature––4 He and 3 He, both commonly used in low temperature physics. 3 He is a fermion and we discuss its application for superfluid dynamics later in this paper; while 4 He atom is a boson, representing a common form and unless explicitly mentioned throughout this paper, by helium we understand 4 He. It offers three fluids of interest for experimental investigation of high Reynolds (Re) and Rayleigh (Ra) number flows under controlled laboratory conditions, primarily due to their extremely low values of kinematic viscosity, m, lowest of all known substances. The first is cryogenic helium gas near the critical point at 5.2 K and 2.26 atm. In addition to its very low viscosity it also allows an unprecedented flexibility, as its fluid properties can be easily tuned over many orders of magnitude by varying the temperature and
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pressure. The second is normal liquid helium, or He I, which is an ordinary Navier–Stokes fluid. He I has its normal boiling point at 4.2 K and exists (along the saturated vapor line) down to the lambda temperature Tk ffi 2:17 K. Below that liquid helium becomes superfluid and is usually referred to as He II. Properties of liquid He I and He II are well known [1] and significantly change with temperature. He II is a quantum fluid and in using it as a working fluid for experiments, special care must be taken in interpreting the data. It displays the two fluid phenomena and its flow is restricted by the fundamental requirement of quantized circulation. On the other hand, the quantum nature of He II enables application of special experimental techniques, and experiments with He II used as a working fluid can be thought both as unique and complementary to classical high Re flow research. We start with classical turbulence, then describe basic properties of He II and discuss similarities and differences between classical and quantum turbulence. For more information on this subject, see e.g., [2–7] and references therein.
2. Classical cryogenic turbulence Cryogenic turbulence experiments based on liquid normal He I and cryogenic helium gas as working fluids are not principally different from the conventional room temperature ones commonly using air or water as working fluids, although there are important practical experimental differences due to the cryogenic environment. 2.1. Cryogenic thermal convection Thermal convection occurring in a fluid layer heated from below (or cooled from above) is ubiquitous in nature and technology. The dynamical state is determined by the Rayleigh number, Ra ¼ gaDTH 3 =ðmjÞ, Prandtl number, Pr ¼ m=j and aspect ratio C ¼ D=H , where g is the acceleration of gravity, DT is the vertical temperature difference across the fluid layer of height H , D is the characteristic lateral extent, a the thermal expansion coefficient and j the thermal diffusivity.
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Turbulent thermal convection sets in at high Ra. The efficiency of the heat transport is represented by the Nusselt number, Nu, defined as the total heat flux normalized by its conductive contribution. In order to understand various processes occurring in nature, it is of particular interest to be able to predict the functional dependence Nu ¼ NuðRa; Pr; CÞ in the widest possible range of Ra, as well as for asymptotically large values. Experimentally, it is desirable to work within socalled Boussinesq approximation, which in a controlled laboratory experiment of limited size H sets an upper limit for DT , and thus the highest achievable Ra is determined by the combination a=mj for the used working fluid. For cryogenic helium gas in the vicinity of the critical point the value of a=mj exceeds values for any other known working fluid by several orders of magnitude. Moreover, as first demonstrated by Threlfall [8] and later by others [9–12], by changing temperature and pressure one can cover many orders of magnitude in Ra in a single experiment. As an example, to cover a modest range of five decades 107 < Ra < 1012 in water may easily require using cells of four different aspect ratios. Using helium gas in the range 4.3 < T < 6 K; 0.1 mbar < p <3 bar allows covering 11 orders of Ra in fully developed turbulent convection up to 1017 in a single experiment, using a convection cell of H ¼ 1 m and C ¼ 1=2 [12]. The data, over the entire huge range of Ra, can be described to the lowest order by a single power law Nu / Rab with b ffi 0:31. Even at these large Ra there is no sign of any transition to so-called ultimate scaling regime predicted long ago by Kraichnan [13] and recently claimed by others [11]. The very existence of the ultimate scaling regime thus remains an open question, although there are new ideas how to possibly resolve this issue [14]. The temperature field in various places in the convection cell has been monitored using small neutron doped germanium sensors. The temperature power spectra display both the )5/3 Kolmogorov and )7/5 Bolgiano roll-off exponents, depending on the so-called Bolgiano length scale loosely dividing ranges where the temperature field behaves as active or passive scalar. The probability density functions (PDF) of the temperature
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fluctuations measured in the centre of the cell undergo drastic changes with increasing Ra, being bimodal around 106 ––suggesting that the top and bottom plate temperatures are not yet fully mixed, mixing being essentially complete around 108 where PDF are almost gaussian, but evolving towards stretched exponential shape with still increasing Ra [12]. The asymptotic state of the temperature PDF is thus not gaussian, in accord with the notion of a strong intermittency of the turbulent temperature field. Using very long records of temperature fluctuations of order of 221 data points, the temperature structure functions in the Bolgiano regime have been computed and studied [15]. Although the direct scaling does not allow any definite conclusions about scaling exponents, application of the widely accepted extended similarity hypothesis strongly suggests that the scaling exponents do not show any tendency to saturation, in contrast to the case of passive scalar [16]. Placing two closely vertically spaced sensors in the vicinity of the vertical wall allows one to measure the velocity of the large scale motion, or the ‘‘wind’’. At relatively low Ra of order 106 the wind is unidirectional, but with increasing Ra it abruptly (i.e., within one turnover time) changes direction in a stochastic manner. The wind stays well defined up to about 1015 , above which it becomes ill-defined due to too frequent switchovers [14,17]. A feasibility study shows that a cryogenic helium gas experiment in an even larger convection cell of height H ¼ 10 m should provide the data up to Ra ffi 1020 , covering the dynamical range of convective processes occurring in the ocean or in the atmosphere. However, building such a cell represents a major technical problem and would require substantial supply of liquid helium. 2.2. Wind tunnels The advantage of using cryogenic helium gas as a working fluid in a classical wind tunnel can be illustrated by comparing the typical parameters of classical air facilities [18] with the performance of a helium gas wind tunnel built inside a conventional laboratory cryostat. Based on the test section of the wind tunnel of crosssection only 6 · 6 cm2 , by
changing the temperature 5 < T < 78 K and pressure 1 < p < 4 atm and changing the mean velocity between 2 cm/s < U < 2 m/s, a broad range of Reynolds numbers based on the Taylor microscale 30 < Rek < 8000 can be achieved in a single experiment. Such a wind tunnel has been already built at the University of Oregon. 2.3. High Re pipe flow Normal liquid He I has been used as a working fluid in a pipe flow experiment, installed into a conventional laboratory cryostat. Measurements of the friction factor in a smooth pipe cover a wide range of Re up to 3 · 106 . The physical measured quantity is the pressure drop between two ports in the pipe wall, detected by an extremely sensitive capacitive bridge. Such a high Re was previously achieved only in few classical experiments (e.g., work of Nikuradse or Princeton SuperPipe [4]) exploiting extremely large industrial size experimental facilities. It is worth mentioning that using He I with extremely low kinematic viscosity would lead to considerable saving in the power required to operate such a high Re facility. There are, in principle, no serious limitations to flow velocity for He I as the first sound velocity is high and the Mach number is likely to stay low for all technically foreseeable flow velocities. Let us point out here, that using superfluid He II would be limited by the second sound velocity about 20 m/s, when second sound shock phenomena will occur.
3. Quantum turbulence Below Tk liquid helium becomes superfluid and is referred to as He II. Quantum mechanics is needed to describe and understand most of its physical properties; it is therefore known as a quantum fluid and exhibits extraordinary flow properties. In a limit of low flow velocities, they can be largely understood within a phenomenological two fluid model [19], where He II is described as consisting of two interpenetrating fluids of independent velocity fields––the inviscid superfluid of density .s , and the normal fluid of density
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.n and dynamic viscosity g; the total density . ¼ .s þ .n . The superfluid has neither viscosity nor entropy; the entire heat content of He II is carried by the normal component. At absolute zero, He II is entirely superfluid. As the temperature increases the normal fluid fraction becomes larger until at the lambda point at which He II is entirely normal fluid. This simplified picture is reflected by the Landau two fluid equations of motion. One important outcome of these equations is the prediction of second sound, a wave described by temperature fluctuations rather than density fluctuations as is the case of ordinary sound, also referred to as first sound. The two fluid equations also explain the existence of a peculiar flow of He II called counterflow––under influence of applied heat the superfluid moves towards the heat source, becomes converted into the normal fluid and flows against the approaching superfluid in such a way that the total density of helium II stays unchanged. From the quantum-mechanical point of view (following the idea of London), the entire volume of superfluid can be described by a macroscopic pffiffiffiffi wave function of a form W ¼ .s eiu , where u is a macroscopic phase. According to quantum mechanics, the superfluid flow velocity must be proportional to the spatial gradient of this macroscopic phase (applying the impulse operator ihr on W we get the impulse p / v / ru), in close analogy with supercurrent in superconductors. It immediately follows that the superfluid velocity field must be irrotational. However, superfluid flow stays fully potential only at small velocities. In macroscopic flows involving higher velocities, typically of order cm/s, quantized vortices appear in the fluid. The superfluid flow stays irrotational everywhere except inside the core of the quantized vortex line. These lines have cores of atomic dimensions and their circulation is quantized in units of j ¼ h=m, where h is PlanckÕs constant and m is the mass of the superfluid particle. In He II j ¼ 9:997 104 cm2 /s. The quantization follows from the requirement that the macroscopic wave function is single-valued, which allows that the phase changes by any multiple of 2p while circling a path containing vortex lines. As a consequence of the Kelvin theorem, quan-
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tized vortex lines may either take shape of closed loops, or they must end at the boundary. Since the circulation is quantized and the superfluid vorticity is restricted to the vortex core, the mean superfluid vorticity is proportional to the total length of quantized vortex line per unit volume, and can be defined as hxs i ¼ x ¼ jLðtÞ. The presence of a vortex line results in a force of interaction between the superfluid and normal fluid components–– mutual friction. This interaction results in the otherwise independent normal and superfluid velocity fields becoming coupled. An instructive system to assimilate the role of quantized vortex lines in superflow is a rotating bucket of He II. In steady state, the normal fluid is in solid body rotation, with uniform vorticity x ¼ r v ¼ 2X, where X is the angular velocity of rotation. On increasing X slowly from rest, the superfluid stays irrotational (so-called Landau state), but only until we reach some critical velocity and create quantized vortices. At higher angular velocities, the fluid is threaded with a rectilinear array of quantized vortex lines that have an orientation parallel to X. Although the superfluid vorticity is restricted to vortex lines, their areal density, n0 , evolves to match the vorticity of the fluid in solid body rotation. This results in n0 j ¼ 2X. Thus on length scales larger than the inter-vortex line distance, the superfluid is also, on average, in solid body rotation, with a velocity field essentially equal to that of the normal fluid, i.e., hxs i ¼ xn and hvs i ¼ vn . The role of quantized vortex lines in a turbulent flow becomes more complex [5,6]. Computer simulations show that similar to the way vortex lines align in a rotating bucket, bundles of quantized vortices align on the cores of normal fluid eddies [20]. The superfluid vorticity is comparable to the vorticity of the normal fluid eddy and the effective fluid density is equal to the total fluid density. Turbulent flow contains a complex tangle of vortices evolving in such a way as to minimize the difference between the two otherwise independent velocity fields. The length scale of interest and the complexity of the flow determine how closely the two velocity fields become coupled together. Such a coupled flow appears to closely obey the Navier–Stokes equations and classical boundary
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conditions. A vortex-coupled flow still exhibits superfluidity as evidenced by supporting the transmission of second sound, although attenuated by the presence of these lines. 3.1. He II-working fluid in fluid dynamics? Historically, superfluid turbulence was usually viewed as a tangle of quantized vortex lines and its possible relationship to conventional turbulence has not been given much thought. This resulted in disconnected fields of conventional and superfluid turbulence, and so it seemed reasonable to think that the two fields were quite different in character. Recently, however, turbulent flow in He II has been generated in a manner similar to that in studies of classical turbulence (coflow turbulence). Let highlight some experiments revealing a deep similarity between classical and superfluid flows and turbulence. Maurer and Tabeling [21] produced turbulence in a flow of liquid helium confined between counterrotating discs, and obtained the energy spectral density at T ¼ 2:3, 2.08, and 1.4 K. From the standpoint of the two fluid model, one may think of liquid helium as being largely superfluid at 1.4 K and largely normal (or classical) at 2.08 K, and entirely normal at 2.3 K. Yet, the spectra obtained at all three temperatures (at experimentally accessible length scales) were identical and possessed an inertial range of the classical Kolmogorov form. He II flow past the sphere displays the drag crisis [30] and show no appreciable difference from classical pipe flow [31]. A great advantage of using He II in the study of turbulence is that it is possible to obtain the vortex line density by measuring the attenuation of second sound [19]. This allows the direct measurement of the superfluid vorticity, averaged over some experimental volume where the second sound is excited. This method was used to study the decay of the grid turbulence, created by towing the grid through a stationary sample of He II. The sensitivity and dynamic range of this technique are enormously large and correspond to monitoring up to eight orders of magnitude of the decaying turbulent energy, clearly an impossible goal for any classical experiment––reaching such a dynamic
range would require an air wind tunnel such as in [18] with a test section more than 1000 km long. In the experiments of [22,23] turbulence was produced by pulling a grid of bars through a stationary sample of He II, and the decaying average vortex line density (i.e., length of the quantized vortex line per unit volume) over a measurement volume was obtained from the attenuation of the second sound, transmitted and detected via circular gold-plated nuclepore membranes mounted flush in the wall across the channel. The goldplated channel side of the membrane serves as one electrode of a capacitor, the dc bias being used to press it against a brass electrode beneath. The ac signal tuned to the resonance frequency of the channel––second sound resonator––sets the standing wave resonance across the channel. Moving the membrane––transducer––generates the second sound, as superfluid stays at rest passing freely through its tiny channels while normal fluid is moving with it being glued within the pores by finite viscosity. This results in a relative motion of the two fluids, i.e., in second sound. Its amplitude is measured by another membrane––receiver–– opposite the channel. Quantized vortices in a volume between the transducer and receiver attenuate the second sound, thus measuring its amplitude provides a direct information on the vortex line density. The information about the turbulent flow is therefore obtained from measuring volume of order d 3 cm3 . This natural integration bypasses tedious statistical analysis involved in any local velocity measurements in conventional turbulence, provides enormous sensitivity and unprecedented dynamical range of the method, making it very useful and complementary to classical turbulence studies. The observed values of vorticity are evaluated from measurement of the time dependent recovery amplitude of the standing second sound resonance, initially suppressed by the presence of quantized vortices created by the motion of the grid through the channel. The temporal decay of vorticity, defined as xðtÞ ¼ jLðtÞ, where L is the length of quantized vortex line per unit volume, ranges from 104 to 0.01 Hz. This unprecedented range allowed detection of several distinctly different regimes of the decay [23] and makes this system unique in the study of turbulence. It even
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became possible to develop a purely classical spectral decay model, fully applicable to classical homogeneous and isotropic turbulence [24], as it was recently proved by computer simulations [25]. Different regimes switch as the energy containing and dissipative Kolmogorov length scales gradually grow during the decay, finally both being saturated by the finite size of the channel. The classical character of the decay of the vortex line density did not change with temperature within the range 1.2 < T < 2.0 K, while the ratio of the superfluid to normal fluid density varied over this temperature range from near unity to almost zero. Still, no appreciable difference in the turbulent decay curves obtained over this wide temperature range other than a slight temperature dependence of measured effective kinematic viscosity of turbulent He II occurred. However, this similarity between classical and superfluid turbulence breaks down at small scales where the quantization effects of circulation have to be taken explicitly into account. This happens at small scales of order ðe=j3 Þ1=4 , which defines a ‘‘quantum scale’’ of order of the mean distance between quantized vortices in He II. Beyond this scale the normal and superfluid velocity fields cannot be fully matched and the spectral energy density would depend in addition on j and is likely to be of the form UðkÞ ¼ Cej1 k 3 [26], where C is the threedimensional Kolmogorov constant and e ¼ dE=dt denotes the energy decay rate. At extremely high k (up to the inverse size of the vortex core) the spectrum is likely to be of the inverse k form, consistent with the spectral form of a straight vortex line. For large scales, the spectrum is classical. Interestingly, recent computer simulations suggest similar form of the superfluid energy spectra even in the zero temperature limit, where there is no normal fluid [27] and such a form of an energy spectrum cannot occur due to the influence of the normal fluid of classical behaviour. Although quantum turbulence in He II can be generated classically, historically it was discovered [29] in a counterflow channel, where it is easily generated by applying a heat pulse. It has been emphasized by various authors [6] that counterflow turbulence has no classical analog. However, recent experiments on the decay of counterflow He
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II turbulence suggest a possibility of missing links between classical turbulent thermal convection and counterflow turbulence [28]. The efficiency of the heat transfer displays similar scaling and also the late decay of counterflow turbulence displays classical character. Further experiments are in progress and will be reported elsewhere. Thus, even though the situation in He II turbulence is almost certainly quite complex, these experiments and computer simulations suggest that the two fields of turbulence are closely related to each other and in many cases one can analyze the data obtained in turbulent He II, at least approximately, in a classical manner––treating helium II as a Navier–Stokes fluid with an extremely small kinematic viscosity of order 104 cm2 /s. This is the lowest kinematic viscosity known for any substance and provides the possibility of investigation of extremely high Re flows under controlled laboratory conditions, which can be combined with the unique possibility of using the highly sensitive second sound technique for probing such a flow. On the other hand, there are important differences between quantum and classical turbulence that are best illustrated with superfluid 3 He.
4. Superfluid 3 He in superfluid dynamics At low temperature the physical properties of He and 4 He differ in a fundamental way, as 3 He atom is a fermion and obeys Fermi–Dirac statistics. 3 He therefore cannot display superfluidity based on the same physical principles as He II [7]. It becomes superfluid at much lower temperature of order 1 mK, but superfluidity of 3 He is much closer in nature to superconductivity in metals rather than to superfluidity of He II in that it includes Cooper pairs of 3 He atoms. The so-called equal spin pairing leads to the existence of Cooper pairs with both spin S ¼ 1 and orbital momentum L ¼ 1, which gives rise to a possibility of various superfluid phases. Let us limit ourselves with the experimentally most commonly observed A and B phases. For our purposes the pseudo-isotropic B-phase can be considered simply as a kind of He II possessing different flow properties. The viscosity of 3
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the normal fluid is very high, so the normal fluid could be considered almost at rest with respect to the walls of a laboratory size container of order 1 cm in size. The quantized vortices are also similar to those in He II, but the size of the vortex core is about two orders of magnitude larger. This is important from the point of view of pinning––any wall is rough for He II vortices with the atomic size core, but a smooth quartz wall can be considered hydrodynamically smooth for 3 He-B vortices. This means, in particular, that there are hardly any remnant vortices in a smooth container at rest, and by spinning such a container up the first vortex must be created by an intrinsic mechanism that requires high critical velocity. In other words, rotating 3 He-B maintains the metastable vortex free Landau state. The anizotropic 3 He-A phase is able to mimic the solid body rotation by continuously adjusting the texture (texture here having a meaning of spacial arrangement of the vector field defining the direction of the zero energy gap on the Fermi sphere), which results in the existence of so-called continuous vortices. They do not have a hard core similar to that of He II vortices, and each carries two quanta of circulation. Critical velocity for creating these continuous vortices is much lower than that in the B-phase. On first approximation, we may neglect it and consider the rotating 3 He-A sample always in solid body rotation, i.e., containing an (almost) equilibrium number of doubly quantized continuous vortices. By cooling the cylindrical 3 He sample in a smooth walls quartz cylindrical container at pressure 30 bar, first the normal to 3 He-A transition occurs slightly above 2 mK. On further cooling the whole sample would become the B-phase, but applying a strong magnetic field to the middle of the cell would prevent the B-phase from moving further up and the stable AB boundary occurs in the cell. We thus have an interesting stable situation––a cylindrical sample of 3 He-A floating over another cylindrical sample of 3 He-B. What happens if we start rotate the cryostat? Both A and B normal fluids will be, thanks to high viscosity, always in solid body rotation. So will be, on average, the entire Aphase, being threaded by an array of rectilinear continuous vortices, but the B-phase stays in a
Landau state with no quantized vortices. Vortices in the A-phase cannot end at the boundary, but in the vicinity of it (within a distance of order of their spacing) they bend away, end on the vertical cylindrical wall and form a vortex sheet parallel to the AB boundary. We therefore have a superfluid B-phase at rest, upon which the A-phase slides, exhibiting a pure shear flow [32]! But this is an ideal case considered almost two centuries ago by Lord Kelvin for a case of a wind blowing over calm water––so-called Kelvin–Helmholtz instability should occur above certain threshold, when the interface becomes corrugated at some particular wave length. It is indeed a case, and on spinning up above a threshold bursts of about 10 vortices enter the B-phase every time the threshold for instability is reached again and again. Let us mention that any experimental evidence for checking this instability criterion in a quantitative manner using classical fluids is always obscured by finite viscosity, which is not the case for a superflow. Indeed, modifying the Kelvin criterion for the described above 3 He case (it requires changing the role of gravity by magnetic anizotropy and also taking into account the two fluid hydrodynamics), good agreement between the instability theory and the experimental data is found [32]. Surprisingly, the exotic 3 He quantum fluids shed new light on the very old puzzle of classical hydrodynamics. This mechanism of injecting quantum vortex loops can be used to investigate the criteria of generating the quantum turbulence in 3 He-B. It turns out that if one or a few vortex loops are injected, e.g., at the AB interface, into rotating vortex free superfluid 3 He-B at high temperature (T > 0:6Tc ), they will evolve in a smooth way with their ends sliding the vertical wall and the resulting stage contains exactly the same number of rectilinear vortices running along X. At low temperature, however, the situation is very different in that the final observed stage represents (almost) the equilibrium solid body rotation containing many more vortices in comparison with the number of those originally injected at the interface. The nuclear magnetic resonance measurements thus indicate the existence of a sharp transition to superfluid turbulence at T =Tc ffi 0:6 [33]; this observation is supported by computer simulations.
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It appears that this transition is very different from transition to turbulence observed in conventional fluids obeying the Navier–Stokes equation. Here the transition to turbulence is determined by an interplay of the viscous and inertial terms through the Reynolds number. For a given flow geometry the critical Re at which the transition to turbulence occurs depends on the flow velocity. For superfluids, on inserting the mutual friction force in the Euler equation and assuming the normal fluid at rest one also obtains the hydrodynamic equation containing the dissipative and inertial term. The fundamental difference from conventional hydrodynamics is that the relative importance of these competing terms is determined here by an intrinsic parameter of the superfluid, q ¼ a=ð1 a0 Þ, where a and a0 are the usual mutual friction parameters. The dimensionless quantity q thus plays a role of a Reynolds number in superfluids! In the case of a single vortex line q marks crossover from the regime where Kelvin waves can propagate along a vortex core to where they are overdamped. In a particular case of 3 He-B at T =Tc ffi 0:6, where the transition to superfluid turbulence has been experimentally observed, q ffi 1:3. In 3 He-A He q about unity at extremely low temperatures, while in He II (where one has to keep in mind that viscosity of the normal fluid is extremely low) just below the kpoint.
5. Conclusion We have outlined some aspects and advantages of using cryogenic helium for experimental turbulence research. In view of the absence of a fundamental theory of turbulence based on first principles, cryogenic helium experiments open the road toward investigations of ultra-high Re and Ra flows, where applicability of any computer simulations is at the best very limited. Other important aspects involve superfluid flow and turbulence. We have shown that at some circumstances He II could be used as a working fluid in additional and complementary experiments to the conventional turbulence research. The underlying physics, however, is much more complicated and a better
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understanding of superfluid turbulence, involving superfluid phases of 3 He, represents an interesting intellectual challenge. We are convinced that experiments with superfluids and their theoretical interpretation––broadly defined as superfluid dynamics––will serve as an important tool for deeper understanding of the complex phenomenon of fluid turbulence in general.
Acknowledgements The author is deeply indebted to many colleagues with whom he had a pleasure to work or gain experience during discussions, especially to R.J. Donnelly, V.B. Eltsov, A.V. Gordeev, M. Krusius, J.J. Niemela, K.R. Sreenivasan and W.F. under Vinen. This research is supported by GACR grant 202/02/0251.
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