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A brief historical digression….two years after the death of Boltzmann: Leiden, 1908: Kamerlingh Onnes succeeds in liquifying helium, an element first identified spectroscopically in India in 1868 during a total eclipse of the sun (followed by its discovery on Earth in the lava of Mount Vesuvius by the Italian physicist Luigi Palmieri). This leads to the discovery of superconductivity a few years later, superfluidity a few decades later, and many diverse applications in science and engineering. That same year in Trieste…. Trieste, 1908: Trieste resident James Joyce receives a rejection letter for what would become the first of his major works: “Dubliners.” [It was finally published 6 years later]. Discovery of superfluidity “One usually considers that superfluidity was discovered in December 1937, the submission date of the two articles on the flow of liquid helium which appeared side by side in Nature on January 8, 1938. On page 74 was the article by P. Kapitza and on page 75 the one by J.F. Allen and A.D. Misener.” S. Balibar, JLTP 2007 Helium phase diagram Three regions of interest for fluid turbulence: •Near critical gaseous helium •Helium I •Helium II thermal DeBroglie wavelength is larger than mean inter-atomic distance: T h 3mkT 8.9 A The Lambda point—resembles a Greek letter if you plot it right… Pushing Re to the limit: the case for helium Re UL U cannot be increased arbitrarily without introducing another parameter: the Mach number (=U/a). SF6 Increasing L has engineering and financial limits. Helium has the lowest kinematic viscosity of any fluid, and while some pressurized gases have comparable values (at high pressures) they have problems of their own and lack the versatility of helium. Is it worth it to explore new working fluids and environments? If high enough Reynolds numbers (meaning any Re above the point at which certain scaling properties in the inertial range cease to vary) can indeed be attained using common fluids such as air or water, there is little sense in pushing the low temperature technology. This, however, is not the case in general. Another salient fact is that the increase in the Reynolds number is measured in terms of its logarithm, and, in some respects, it makes sense to consider changes that may occur when Re changes by one or more decades. For this reason, a good experiment is one that permits Re to be explored over many decades— preferably in a single apparatus so we may neglect other effects such as changed boundary conditions, geometry and experimental protocol. This possibility can be realized using helium as the test fluid. Quantum turbulence is a new regime which may shed light on classical turbulence Zero-point energy is large thus avoiding solidification at absolute zero: x p Quantum fluid vn, vs, n s n 2 E~ 2m4 R 2 + weak interactions between closed shell atoms. Thermal DeBroglie wavelength is larger than mean inter-atomic distance: T h 3mkT 8.9 A Superfluidity exists below lambda point In analogy with Bose-Einstein condensation, some fraction of the helium atoms will condense into the zero momentum ground state (maximum around 14% at T=0). Helium II is of course a liquid but is perhaps closer to a non-ideal (interacting) gas, having a large molar volume. (Small) Density of Helium Niemela & Donnelly JLTP 1995 Clausius-Mossotti 1 2 4 M 3M Macroscopic quantization By analogy with the single particle quanum currents (orbiting electrons) it can be postulated (London) that there exists a macroscopic wave function 0 eiS (r ) where S(r) is the phase (a real function of position) and that the wave function is governed by the ordinary single particle wave equation. We have for the momentum p=m4vs p vs S S m4 The velocity is related to the gradient of the phase we can have phase coherence over macroscopic regions of the liquid. Superfluidity may qualitatively be thought of as a natural consequence of this coherence. The same argument holds for helium-3 but we need 2m3 in the velocity relation to account for the required pairing for the Fermion system. Two Fluid Model Helium II behaves as if it consisted of two separate fluids according to the two fluid theory of Tisza and Landau: a “normal” component and a “superfluid” component. The two fluids interpenetrate freely without interaction Each fluid component has its own density and velocity field, ρn and vn for the normal fluid and ρs and vs for the superfluid. The total density of Helium II is the sum of the two separate densities: ρ = ρn + ρs. The superfluid carries no entropy and experiences no flow resistance: its viscosity is identically zero and it is irrotational: vs 0 The normal fluid component carries the entire entropy and viscosity (ηn) of Helium II and is similar to a classical, viscous Navier - Stokes fluid. It can be considered as a gas of thermal excitations (phonons and rotons). [Note: There is constant confusion about the term “superfluid” which is sometimes used to refer to helium II and sometimes to the component of helium II] For the superfluid component the simple two-fluid model (without mutual friction) gives: vs t vs 1 vs p s T where s is the specific entropy. This looks like the classical Euler equation except for the last term. This term allows for the thermo-mechanical effect; i.e. pressure gradients can result from gradients in temperature. This equation is only valid at low velocities--otherwise we will need to add a mutual friction term between the two components. The simple two-fluid equations are: n t ( S) t vs t vn t vn s Sv n vs vn vs vn vs 0 (All entropy flows with the normal fluid) 0 1 1 p S T p s n S T 2 n vn Demonstrating the two fluid nature Measuring the density of the two fluids: Andronikashvili’s pendulum of oscillating disks Spacing between plates is much lerss than the viscous penetration depth for the normal fluid (“skin depth”): z 2 n The normal fluid was entrained and contributed to the moment of inertia of the pendulum bob while the superfluid remained stationary. Since the total density is easily measured separately, the superfluid density could then be found. Total density: s n Consequences of the two-fluid model Counterflow In a channel open at one end to a helium II bath, a heater placed at the closed end causes a counterflow. Here the normal fluid flows away from the heater carrying the heat and to conserve mass the superfluid component flows toward the heater. This is a common method of producing turbulence in the superfluid but it has no obvious classical analogue. All the entropy flows with the normal fluid ( s) t Second sound Perturbations in heat obey a wave equation rather than a diffusion equation. Second sound occurs for constant total density but varying fraction of normal and superfuid densities. Can be produced by pulsed heaters and detected mechanically or vice versa. svn c2 0 2 s s T nCv First sound: c1 ~ 200 m/s Second sound: c2 ~ 20 m/s Demonstrating Potential flow A curious observation Rotating containers of helium II were observed to have a parabolic meniscus (Osborne, 1950). The shape of the meniscus was independent of temperature which was surprising since it was assumed that the superfluid component would not rotate as a solid body. In fact, this was resolved by considering the fluid to be threaded with an array of quantized vortices whose number obeyed Feynman’s rule: n 2 Note, here the angular velocity is denoted by Ω, rather than the vorticity as we used before. The vorticity is equal to 2Ω in solid body rotation, hence Feynman’s rule says that a sufficient number of vortices will be produced to mimic solid body rotation in the superfluid. Clearly this only works well for n large. Regular arrays and irregular tangles of quantized vortices Visualizing indirectly the regular array of vortices in a rotating bucket The simulated tangle of quantized vortices on the left corresponds to 1.6K, while that on the right is at 0K. After Tsubota, et al (2000). Yarmchuk, et al. 1978 As we shall see later, turbulent flows in the Kolmogov sense can mimic eddies on all scales through partial polarization of vortex bundles. NLSE for the condensate A condensate of weakly interacting Bose particles is described by a single particle wavefunction of N bosons of mass m which obeys the NLSE (or Gross-Pitaevski equation) where V0 is the strength of the repulsive interaction between the bosons and Ev is the energy increase on adding one boson (chemical potential). For fluid dynamic applications we can apply the so-called Madelung transformation where R is the amplitude and S is phase of . If we substitute this into the NLSE we obtain the continuity equation: s ( t s vs S vs ) 0 m S m where * mR 2 The irrotational condition follows by taking the curl of the superfluid velocity: S m vs vs 0 Momentum s vsi t vsj vsi xj p xj quantum stress: pressure: p ij xj ij V0 2m 2 2m 2 2 s ln s xi x j 2 s Without the quantum stress term the equations reduce to the Euler equations Vortices and vortex reconnections Koplik and Levine (PRL 1993) considered a single-quantum rectilinear vortex along r =0 as a two dimensional solution to the NSLE in cylindrical polar coordinates and described by the function f (r )ei where the function f (r) goes to zero at r = 0 and tends to a constant value (f0) for r large compared with the coherence length a, given by a 2 2 2mV0 f 02 where f0 is the value of f at large r. In this model the vortex core can be identified with the coherence length and the density is identified with the superfluid density. Estimates for the core: The coherence length is approximately 0.5 Angstrom for helium 4. Assuming a cylindrical “hole” in the fluid in which the Bernoulli force was just balanced by surface tension, Feynman was able to “estimate” a core size approximately 0.3 Angstrom (where of course the concept of a surface tension would not be valid). The circulation in a multiply-connected region (around the core where the density goes to zero) gives vs dl circulation: m4 n h m4 Each vortex has a tension (energy per unit length): 2 Ev s 4 ln reff a0 (~10-7erg cm-1, a0~0.1 nm, reff ~ l) There is no evidence for multiply-quantized vortices (n=1 only). Since the energy of a vortex line goes as it is energetically favorable to have vortices with a single quantum of circulation. Experiments by Vinen (1957) have confirmed this. The quantum stress we saw earlier is important only for distances from r=0 of ≈10-8 cm. This is much smaller than the typical vortex separation in turbulent superfluid flows which is between 10-3 and 10-4 cm. Hence away from the core we have the classical Euler equations. Near the core, where the stress term is important, reconnections can occur which are not allowed in an Euler fluid (unless viscosity is somehow restored). Vortices and vortex reconnections in the NLSE Superfluid helium has been termed a “reconnecting Euler fluid” (Barenghi), this being an important difference. To mimic classical turbulence with structures (eddies) on all scales, we require bundling of quantized vortices. Do these bundles also coherently reconnect? Alamri, Youd & Barenghi, 2008 Reconnection of vortex bundles Turbulence in helium II We will consider in lecture 5 details of the quantum turbulence problem. Phase diagram revisited: convective flows Ra 4.4 K , 2 mbar: / 5.8 10 Ra ~ ( 2 CP). Ra increases as 2 T H3 g 3 In cgs units 5.25 K, 2.4 bar: / 6.5 109 Water: 10 Air: 0.1 away from critical point and as CP in its vicinity. Working near critical point different experiments need to use the same temperature scale…. It is clear that the small kinematic viscosity of helium can help make Ra=( large, as we saw for Re. g L3 In addition, for helium gas, the thermal diffusivity =kf / CP , where kf is the thermal conductivity of the fluid, and CP is the constant pressure specific heat, can be both very small or very large depending on the the distance from the critical point where CP diverges. The isobaric thermal expansion coefficient: α≡−ρ−1∂ρ/∂T For non-interacting gases, α =1/T , and so, low temperatures themselves have a particular advantage for buoyancy-driven flows. For helium near its critical point, α is thermodynamically related to the specific heat and therefore also diverges, ie, CP CV Tv 2 / T where CV is the specific heat at constant volume (weak divergence), v is the molar volume, and T is the isothermal compressibility. We will return to this when we consider in more detail thermal convection in cryogenic helium. Dynamical similarity For given initial and boundary conditions, the state of the flow is the same if the relevant dynamical parameters are the same. e.g., Re = UL/ , Ra =g( / L3, … These parameters vary by many orders of magnitude in practice Example Ra Re Sun 1022 1013 Ocean 1019 1010 Atmosphere 1015 109 Naval applications --- 109 Aerospace applications --- 5 108 Material properties An indirect benefit of the use of low temperature helium comes from the relation between the fluid properties and those of the solid material used to confine it. Boundary conditions play an important role in fluid mechanics and, in the case of thermal turbulence, for instance, the usual boundary condition at the horizontal surfaces is one of constant temperature. In practice, this condition is effected by using highly conductive plates with constant heating or with good temperature control. Annealed and oxygenf ree copper has a thermal conductivity of order 1kWm−1 K−1 to be compared with the nominal thermal conductivity of helium, which is of order 10−2 W m−1 K−1. The effective thermal conductivity of the fluid at very high Ra can be quite large because of turbulence, in some experiments as high as 10,000 times that of the quiescent fluid. This brings the effective conductivity considerably closer to that of the plates and necessitates some correction. As we shall see later in more detail, the correction is negligible for helium except at the very highest Ra where it is of the order of a few percent. On the other hand, it is typically of the order of 10-20% for conventional fluids even when the Ra is moderately high and orders of magnitude lower than that achieved with helium. Note on corrections due to finite conductivity of the plates 0.020 Biot <<1 ~ 0.1 = constant temperature B.C. 0.018 •X =Rf/Rp= kpH/(kf*Nu*e) 0.016 kp= thermal conductivity of plates kf= thermal conductivity of fluid e= thickness of the plates. Biot number 0.014 0.012 0.010 0.008 0.006 0.004 0.002 0.000 -0.002 7 10 8 10 9 10 10 10 11 10 12 10 13 10 Ra 14 10 10 15 16 10 17 10