Simulation of Counterflow Turbulence by Vortex Filaments

By using the vortex filament model with the full Biot-Savart law, we have succeeded for the first time in generating the statistically steady state of counterflow turbulence in superfluid ⁴He under periodic boundary conditions. This state exhibits the characteristic relation $L=\gamma^{2} v_{\mathit{ns}}^{2}$ between the line-length density L and the counterflow relative velocity v _ns and there is quantitative agreement between the coefficient γ and some measured values. Since we obtained the realistic state of quantum turbulence, we will use the numerical data to study the statistical property. We focus on the statistics of vortex reconnections from about 350 events in our steady counterflow turbulence simulation, and characterize the dynamics by the minimum separation distance between two reconnecting vortices, which was used in the visualization experiments using the solid hydrogen tracer particles by Paoletti et al. From our analysis we may conclude that the quantized circulation is still the dominant controlling feature and obtains the statistics of the correction factor.     

Decay of Counterflow Quantum Turbulence in Superfluid 4He

We have simulated the decay of thermal counterflow quantum turbulence from a statistically steady state at T=1.9 K, with the assumption that the normal fluid is at rest during the decay. The results are consistent with the predictions of the Vinen equation (in essence the vortex line density decays as t ^?1). For the statistically steady state, we determine the parameter c ₂, which connects the curvature of the vortex lines and the mean separation of vortices. A formula connecting the parameter χ ₂ of the Vinen equation with c ₂ is shown to agree with the results of the simulations. Disagreement with experiment is discussed briefly.     

On Flow of He II in Channels with Ends Blocked by Superleaks

We report an experimental investigation of a flow of the superfluid phase of ⁴He–He II—thermally induced by a fountain pump through vertical channels of square cross-section with ends blocked by sintered silver superleaks and its decay. We confirm the existence of a weakly temperature dependent critical velocity v _cr ^I of order 1 cm/s, which does not scale with the channel size and is therefore an intrinsic property of the self-sustained vortex tangle of vortex line density L, measured by second sound attenuation. In addition to the previously reported turbulent A-state characterized by L ^1/2=γ(T)(v?v _cr ^I ), where v is the mean superflow velocity through the channel, we have discovered a new B-state characterized by L=β(v?v _cr ^II ), where β seems only weakly temperature dependent. It poses an important question why, at higher flow velocities, the quadratic generation mechanism, so well established in thermal counterflow, ceases to work. We offer a phenomenological model assuming that in the B-state the coarse-grained superflow profile matches the classical parabolic profile, with a finite, temperature dependent slip velocity v _cr ^II of order few cm/s and that a confined viscous normal fluid flow of toroidal form is induced inside the channel due to the mutual friction force. When the fountain pump is switched off, after an initial decay, a confined quasi-viscous flow of a single-component fluid with an effective kinematic viscosity ν _eff(T) establishes, giving rise to the observed exponential decay. The values of ν _eff(T), calculated using our model from the measured decay times, are in agreement with those deduced from other experiments on decaying He II turbulence.     

A Simple Phenomenological Model for the Effective Kinematic Viscosity of Helium Superfluids

A number of experiments where quantum turbulence in helium superfluids has been generated by various means (such as towed/oscillating grids, thermal counterflow, pure superflow, spin-down, ion/vortex rings emission) displays a temporal decay of the observed vortex line density, of the power law form L=Γ t ^?3/2 at late times. The prefactor, Γ, in analogy with classical homogeneous isotropic turbulence, allows deducing the temperature dependent effective kinematic viscosity, ν_eff, for turbulent helium superfluids. It appears to be a robust quantity, independent of methods of generating quantum turbulence and detecting the decaying vortex line density. We present a simple phenomenological model to estimate ν_eff based on comparison of dissipation terms in equations of motion for classical viscous flow and vortex flow of a superfluid in a stationary normal fluid. This model leads to ν_eff≈κ q, where q=α/(1?α′); α and α′ being dimensionless mutual friction parameters. Within the temperature range where mutual friction dissipation mechanism is dominant this simple model prediction agrees well with the experimental data and with the recent theoretical estimate of Roche, Barenghi and Leveque (Europhys. Lett. 87:54006, 2009).     

Zero-Point Vacancies in Quantum Solids

A Jastrow wave function (JWF) and a shadow wave function (SWF) describe a quantum solid with Bose–Einstein condensate; i.e. a supersolid. It is known that both JWF and SWF describe a quantum solid with also a finite equilibrium concentration of vacancies x _v . We outline a route for estimating x _v by exploiting the existing formal equivalence between the absolute square of the ground state wave function and the Boltzmann weight of a classical solid. We compute x _v for the quantum solids described by JWF and SWF employing very accurate numerical techniques. For JWF we find a very small value for the zero point vacancy concentration, x _v =(1.4±0.1)×10^?6. For SWF, which presently gives the best variational energy of solid ⁴He, we find the significantly larger value x _v =(1.4±0.1)×10^?3 at a density close to melting. We also study two and three vacancies with SWF. We find that there is a strong short range attraction but the vacancies do not form a bound state, at variance with the exact finite temperature PIMC results.     

Universal Onset of Quantum Turbulence in Oscillating Flows and Crossover to Steady Flows

The critical velocity v _c for the onset of quantum turbulence in oscillatory flows of superfluid helium is universal and scales as $v_{c}\sim\sqrt{\kappa \omega}$ , where κ is the circulation quantum and ω is the oscillation frequency. This result can be derived from a general argument based on the “superfluid Reynolds number”. Only the numerical prefactor may depend somewhat on the geometry of the oscillating object because the flow velocity at the surface of the object may differ from the velocity amplitude of the body. A more detailed analysis derived from the dynamics of the turbulent state gives $v_{c}\approx\sqrt {8\kappa\,\omega/\beta}$ , where β～1 depends on the mutual friction parameters. This universality is compared with the recently discovered universality of classical oscillatory flows. We also discuss the effect of remanent vorticity on the onset of quantum turbulence. Finally, by employing the “superfluid Reynolds number” again, we argue how v _c changes when the steady case ω=0 is approached. In that case v _c scales as κ/R, where R is the size of the object.     

Propagation of Quantized Vortices Driven by an Oscillating Sphere in Superfluid 4He

We performed numerical simulation of quantum turbulence at 0 K generated from remnant vortices attached to an oscillating sphere. The remnant vortices are extended by the sphere motion and form a tangle with emitting vortex loops. As time passes, the length of vortices in a computational volume becomes statistically steady. We investigate in the statistical steady state the distribution of the length of vortex loops and anisotropy of their propagation direction caused by the sphere oscillation. The propagation direction of the emitted vortex loops is anisotropic along the oscillation direction of the sphere. The obtained results are consistent with results obtained in the experimental study using vibrating wires in superfluid ⁴He.     

The Sensitivity of the Vortex Filament Method to Different Reconnection Models

We present a detailed analysis on the effect of using different algorithms to model the reconnection of vortices in quantum turbulence, using the thin-filament approach. We examine differences between four main algorithms for the case of turbulence driven by a counterflow. In calculating the velocity field we use both the local induction approximation (LIA) and the Biot-Savart integral. We show that results of Biot-Savart simulations are not sensitive to the particular reconnection method used, but LIA results are.     

On the coupling current losses in a multifilamentary superconducting composite

The coupling current losses in a superconducting multifilamentary composite exposed to trapezoidally varying external magnetic field and carrying a small transport current are investigated for the volume pinning force density described with the dependence: F_v = α B^1?γ. Such a model allows estimation of the deviations from the solution based on Bean's (γ = 0) critical state model. Results indicate that there exists a region of small magnetic field amplitudes for which discrepancies are largest. They are also very sensitive to the rate of magnetic field change.     

The Decay of Forced Turbulent Coflow of He II Past a Grid

We present an experimental study of the decay of He II turbulence created mechanically, by a bellows-induced flow past a stationary grid in a 7×7 mm² superfluid wind tunnel. The temporal decay L(t) originating from various steady-states of vortex line length per unit volume, L ₀, has been observed based on measurements of the attenuation of second-sound, in the temperature range 1.17 K<T<1.95 K. Each presented decay curve is the average of up to 150 single decay events. We find that, independently of T and L ₀, within seconds past the sudden stop of the drive, all the decay curves show a universal behavior lasting up to 200 s, of the form L(t)∝(t?t ₀)^?3/2, where t ₀ is the virtual origin time. From this decay process we deduce the effective kinematic viscosity of turbulent He II. We compare our results with the bench-mark Oregon towed grid experiments and, despite our turbulence being non-homogeneous, find strong similarities.     

Laminar-to-Turbulence Transition Revealed Through a Reynolds Number Equivalence

Flow transition from laminar to turbulent mode (and vice versa)—that is, the initiation of turbulence—is one of the most important research subjects in the history of engineering. Even for pipe flow, predicting the onset of turbulence requires sophisticated instrumentation and/or direct numerical simulation, based on observing the instantaneous flow structure formation and evolution. In this work, a local Reynolds number equivalence γ (ratio of local inertia effect to viscous effect) is seen to conform to the Universal Law of the Wall, where γ = 1 represents a quantitative balance between the abovementioned two effects. This coincides with the wall layer thickness (y⁺ = 1, where y⁺ is the dimensionless distance from the wall surface defined in the Universal Law of the Wall). It is found that the characteristic of how the local derivative of γ against the local velocity changes with increasing velocity determines the onset of turbulence. For pipe flow, γ ≈ 25, and for plate flow, γ ≈ 151.5. These findings suggest that a certain combination of γ and velocity (nonlinearity) can qualify the source of turbulence (i.e., generate turbulent energy). Similarly, a re-evaluation of the previous findings reveals that only the geometrically narrow domain can act locally as the source of turbulence, with the rest of the flow field largely being left for transporting and dissipating. This understanding will have an impact on the future large-scale modeling of turbulence.     

Construction of kernels G l, 0 l of the nonlinear collision integral in the Boltzmann equation for arbitrary l

It was shown in [1] that kernels L _l (v, v ₁) of linear collision integral and kernels G _{l, 0} ^l (v, v ₁, v ₂) of nonlinear collision integral are related by the Laplace transform. Here, analytical expressions are derived for nonlinear kernels G _{l, 0} ^+l (v, v ₁, v ₂) with arbitrary l for models of hard spheres and pseudo-Maxwellian molecules using the Laplace transform method.     

Submillimeter surface impedance and relaxation: A case study of quasi-two-dimensional YBa2Cu3Ox

Surface impedance measurements in the normal and superconducting state are an excellent method to study the conduction electron dynamics in metals. This holds especially in the relaxation range, i.e., for distances traveled in one r.f. periods=υ _F/ω(υ_f is the Fermi velocity) being smaller or of the order of the penetration depth λ and mean free pathl. For materials withυ _F<-10⁷ cm/sec the relaxation range is easily accessible forf>0.1 THz. Then, in the normal state, relaxation defines the surface impedance with a penetration depth approaching the London penetration depth λ_L, andR≈μ ₀λ_l/2τ as surface resistance allowing a measure of λ_L and relaxation time τ(T, ω). In the superconducting state the photon interaction scales withξ _F/λ_L=l/γ (ξ _f is the dimension of Cooper pairs for l→∞) and causes at low frequencies an absorption rate growing withγ, which is decreasing withξ _F/l. The rate increase proportional toγ turns to a decrease above 0.1 THz, being accompanied by a decrease ofγ with frequency which is stronger for largeγ and smallξ _F/l. These characteristic dependences allow a measurement of material parameters, anisotropy, and dynamics of electrons. To evaluate the consequences of theâ, b, and?-direction anisotropy, the integral kernels for normal and superconducting surface impedances in its nonintegrated, angle-dependent form are presented, analyzed, and compared with impedance measurements above 0.1 THz of YBa₂Cu₃O_x.     

Non-linear Flux Flow Resistance of Type-II Superconducting Films

We analyze the flux-flow (FF) regime in type-II superconducting films exhibiting quite strong pinning. By driving the vortex lattice (VL) up to high dissipative states, the moving VL undergoes an instability, leading to an abrupt change from the FF to the normal state, which is displayed in the current-voltage characteristics as a voltage jump at a critical vortex velocity v ^?. The temperature and magnetic field dependence of v ^? is investigated in different materials, and an unpredicted low field behavior of v ^?(B) is found. Moreover, for velocities lower than v ^?, a non-linear FF resistance is observed, with a ??peak?? in the current dependence of the dynamic resistance. This is a remarkable feature of a dynamic transition from disordered to ordered VL occurring in the FF state. We suggest that both unusual effects observed in v ^?(B) and R _FF(I) can be accounted for intrinsic pinning.     

Analysis of Two Infrared Bands of CH2D2

Two infrared absorption bands of CH₂D₂ have been analyzed in the semirigid rotor approximation. These are the A-type band at 2671.67 cm⁻¹ and the C-type band at 4425.61 cm⁻¹. The A-type band has previously been assigned as v₃+v₉, and the C-type band is tentatively assigned as v₃+v₆ The upper state of the A-type band is perturbed presumably by the close lying level 2v₅. This interaction has not been investigated. The following values were found for the rotational constants of the ground vibrational state: A₀=4.303 cm⁻¹, B₀= 3.504 cm⁻¹, C₀= 3.049 cm⁻¹.     

Gapless state of bismuth-type semimetals

It is shown that in semimetals of the bismuth type under certain conditions the direct energy gap in the vicinity of theL point of the Brillouin zine can be closed. The energy spectrum is found for values of the parameters near the gap annihilation point (γ=γ_t0): $$\varepsilon = p_z^2 /2m_1 \pm \sqrt {\left[ {\gamma - \gamma _0 + ({\text{p}}_{\text{z}}^{\text{2}} /2m_2 )} \right]^2 + (v_x p_x )^2 + (v_y p_y )^2 + \lambda ^2 } $$ where the parameter γ?γ₀ can be altered by variation of the external pressure or of the alloy concentration;m ₁>m ₂; thez axis corresponds to the direction of “elongation” of electron “ellipsoid”; and γ～(γ₀?γ)^3/2 for γ₀?γ>0. It is shown that it is permissible to assume γ?γ₀≈0 for pure bismuth. The qualitative and quantitative characteristics of the spectrum, including the optical properties, are well explained in this case. The role of the interaction of electrons in the gapless state is analyzed (the energy maximum at theT point is below the band contact level in the vicinity of theL point). The carrier mobility in the gapless and semiconducting states is found as a function of the temperature, the parameter γ?γ₀, and the number of carriers injected by alloying.     

Determination of the pressurising period for stored cryogenic fluids

It is a normal operating condition for a lot of cryogenic storage vessels that no boil-off gas is vented over long periods, which leads to a simultaneous pressure increase of the stored fluid. One main reason therefore is to avoid product losses during transport or between withdrawals. At transport conditions the mixing of the fluid can be assumed to be ideally, which results in a maximum reachable pressurising period. At stationary conditions the pressurising period is expected to be shorter, because a stratification is rising up, so that the heat capacity of the stored fluid cannot be used completely.In a thermodynamic view, an isochore change of state takes place and the heat flux into the vessel rises the internal energy of the fluid. For the representation of the isochore change of state a new developed Δu/v-v-diagram with the 1 × 10⁵ Pa reference as a basis line is introduced. The basis line is linear for the filling rate and functionally connected with the specific volume.For the fluids He and H₂ e.g., Δu/v-v-diagrams are pointed out, using the u- and v-values on the saturation lines for the two phase region and those of some isobar lines for the region above the critical pressure.     

Heterogeneity of spiral wear patterns produced by local heating on amorphous polymers

We report on spiral wear patterns produced at constant angular velocity by hot tip atomic force microscopy (HT-AFM) on surfaces of two common amorphous polymers: polystyrene (PS) and polymethylmethacrylate (PMMA). Topography of these patterns is obtained with regular AFM cantilevers. Topography cross-sections taken from a center of each spiral at a given azimuthal angle Θ relate changes of surface corrugation h_corr with tangential velocity v of a thermal cantilever. Polymer wear is characterized by a power law h_corr(v) = α(v/v_max)^−β, which yields a pre-factor α and an exponent β. Below the glass transition temperature T_g, α is polymer specific and β varies weakly between similar conditions and samples. Variations of β are hypothesized to reflect polymer relaxation processes, which are expected to vary only weakly between amorphous polymers. At and above T_g, α approaches initial thermal tip indentation depth within a polymer, β plummets, and a power law relation of h_corr with v diverges. These results are explained by heterogeneous wear around T_g due to a local nature of glass transition. At all studied temperatures, additional wear heterogeneities are found as due to position on the polymer and Θ. Variations of α and β with position on the polymer are found to be only marginally larger then uncertainties of the thermal tip–polymer interface temperature. Variations of α and β with Θ are found to be largely influenced by buckling of thermal cantilevers traveling in a spiral pattern.