Hall-Vinen-Bekarevich-Khalatnikov Research Articles

Velocity circulation intermittency in finite-temperature turbulent superfluid helium

We investigate the intermittent behavior of velocity circulation statistics in quantum turbulence. We perform grid turbulence experiments in superfluid helium and numerical simulations of the two-fluid Hall-Vinen-Bekarevich-Khalatnikov (HVBK) model to show that the scaling properties of circulation moments are compatible with classical turbulence. We also show that there is no apparent temperature dependence on the intermittent behavior of the turbulent superflow.

On the relations between large-scale models of superfluid helium-4

Superfluid helium-4 is characterized by extremely small values of kinematic viscosity, and its thermal conductivity can be huge, orders of magnitude larger than that of water or air. Additionally, quantum vortices may exist within the fluid. Therefore, its behavior cannot be explained by using the classical tools of Newtonian fluid mechanics, and, over the years, a few alternative models have been proposed. In order to highlight similarities and differences between these models, we recast them within a unifying framework, the general equation for non-equilibrium reversible-irreversible coupling (GENERIC). We begin by comparing the original two-fluid model, developed by Tisza and Landau, with the Hall–Vinen–Bekarevich–Khalatnikov model, both prescribing two types of fluid motion and two fluid densities, at flow scales appreciably larger than the typical distance between quantum vortices. We find from the geometrical structure of the models that only one fluid density plays the role of state variable, which should be taken into account when choosing an adequate expression for the free energy. We also recast within the GENERIC framework the one-fluid model of superfluid helium-4, where the inviscid component of two-fluid models is replaced by a caloric quantity, such as entropy. We find that the corresponding geometrical structures are analogous, with the roles of density and entropy swapped. In short, our work demonstrates that the studied models are compatible with each other, at least when focusing on the reversible parts of the models.

Inhomogeneous distribution of particles in coflow and counterflow quantum turbulence

Particles are today the main tool to study superfluid turbulence and visualize quantum vortices. In this work, we study the dynamics and the spatial distribution of particles in co-flow and counterflow superfluid helium turbulence in the framework of the two-fluid Hall-Vinen-Bekarevich-Khalatnikov (HVBK) model. We perform three-dimensional numerical simulations of the HVBK equations along with the particle dynamics that depends on the motion of both fluid components. We find that, at low temperatures, where the superfluid mass fraction dominates, particles strongly cluster in vortex filaments regardless of their physical properties. At higher temperatures, as viscous drag becomes important and the two components become tightly coupled, the clustering dynamics in the coflowing case approach those found in classical turbulence, while under strong counterflow, the particle distribution is dominated by the quasi-two-dimensionalization of the flow.

Turbulent radial thermal counterflow in the framework of the HVBK model

We apply the coarse-grained Hall-Vinen-Bekarevich-Khalatnikov (HVBK) equations to model the statistically steady-state, turbulent, cylindrically symmetric radial counterflow generated by a moderately large heat flux from the surface of a cylinder immersed in superfluid 4He. We show that a time-independent solution exists only if a spatial non-uniformity of temperature and the dependence on temperature of the thermodynamic properties are accounted for. We demonstrate the formation of a thermal boundary layer whose thickness grows with temperature of the cylinder's surface, and analyze the properties of the flow in the radial direction, including the local average vortex line density.

Macroscopic dynamics of superfluid turbulence

The hydrodynamic problems of superfluid fluid containing chaotic tangles of quantized vortex filaments are discussed. The construction of such hydrodynamics crucially depends on the statistics of vortex tangles, and two important cases are presented. The first corresponds to a tangle that consists of entirely chaotic vortex filaments. This case is implemented in counterflowing helium, and is referred to as Vinen turbulence. In the construction of macroscopic dynamics, the system of equations is closed by the Vinen equation for the density of vortex filaments. The second, referred to as the Hall–Vinen–Bekarevich–Khalatnikov case, corresponds to a situation where the system contains bundles of polarized vortex filaments. In this instance, the system is closed by the Feynman equation that relates density of vortex filaments with the vorticity of superfluid velocity. Problems related to the application of both approaches are discussed.

Helicity conservation and twisted Seifert surfaces for superfluid vortices.

Starting from the continuum definition of helicity, we derive from first principles its different contributions for superfluid vortices. Our analysis shows that an internal twist contribution emerges naturally from the mathematical derivation. This reveals that the spanwise vector that is used to characterize the twist contribution must point in the direction of a surface of constant velocity potential. An immediate consequence of the Seifert framing is that the continuum definition of helicity for a superfluid is trivially zero at all times. It follows that the Gauss-linking number is a more appropriate definition of helicity for superfluids. Despite this, we explain how a quasi-classical limit can arise in a superfluid in which the continuum definition for helicity can be used. This provides a clear connection between a microscopic and a macroscopic description of a superfluid as provided by the Hall-Vinen-Bekarevich-Khalatnikov equations. This leads to consistency with the definition of helicity used for classical vortices.

Large Eddy Simulations Analysis of the Energy Spectrum Without Mutual Friction in Superfluid $$ ^{4} $$ 4 He: HVBK Model

The reliability of the filtered on the Hall–Vinen–Bekarevich–Khalatnikov (HVBK) model without mutual friction force is now investigated via some large eddy simulations of freely decaying isotropic superfluid turbulence. The filtered HVBK model is solved using a fully pseudo-spectral method, which is an extension of the classical Rogallo’s method to the two-fluid model. Furthermore, in this paper, we analyze the evolution of various terms constituting the HVBK momentum equations using the balance equation for the energy-spectrum function. Our results are presented in both cases with and without mutual friction force. LES predictions have shown that this mutual friction decreases the energy dissipation of the normal part and the energy transfer is more significant when this force is taken into account.

Relativistic formulation of the Hall-Vinen-Bekarevich-Khalatnikov superfluid hydrodynamics

The relativistic analogue of the Hall-Vinen-Bekarevich-Khalatnikov (HVBK) hydrodynamics is derived making use of the phenomenological method similar to that used by Bekarevich and Khalatnikov [1] in their derivation of HVBK hydrodynamics. The resulting equations describe a finite-temperature superfluid liquid with the distributed vorticity. The main dissipative effects, including mutual friction, are taken into account. The proposed hydrodynamics is needed for reliable modeling of the dynamical properties of superfluid neutron stars.

Analysis of the Energy Budget in Quantum Turbulence: HVBK Model

In this work, we analyze numerically the superfluid energy budget based on the Hall–Vinen–Bekarevich–Khalatnikov model, using a large eddy simulations (LES). First, LES predictions of freely decaying isotropic superfluid turbulence have shown the effect of different temperatures range and spatial resolutions. We furthermore investigate deviations from the KO-41 spectrum caused by intermittency effects. A second motivation of our work is the following: For homogeneous turbulence at very high Reynolds numbers, comparison of the terms in the spectral kinetic energy budget equation indicates, in the energy-containing range, that the production and energy-transfer effects become significant except for dissipation. In the inertial range, where the two fluids are perfectly locked, the mutual friction may be neglected with respect to other terms. Also, the LES results for the other terms of the energy balance are presented.

Spin-up of a two-component superfluid: analytic theory in arbitrary geometry

AbstractThe impulsive spin-up of a two-component superfluid and its container is solved analytically for the first time in arbitrary geometry, generalizing the extensively studied case of single-fluid spin-up. The superfluid is modelled by the Hall–Vinen–Bekarevich–Khalatnikov (HVBK) two-fluid equations, neglecting non-hydrodynamic processes like vortex tension and pinning, which are weak in certain applications (e.g. neutron stars) and confined to boundary layers in the inviscid HVBK component in other applications (e.g. helium II spin-up in smooth-walled containers). Both components of the flow are found to be columnar. The spin-up time depends on the geometry, mutual friction coefficients, $B$, ${B}^{\prime } $, the Ekman number $E$, and the superfluid density fraction ${\rho }_{n} $. For $B\sim O(1)$, the inviscid component undergoes Ekman pumping due to strong coupling to the viscous component, and the azimuthal velocities are ‘locked together’ during the spin-up. For $B\lesssim {E}^{1/ 2} $, there is no Ekman pumping in the inviscid component and the inviscid azimuthal velocity spins up via mutual friction on a combination of the mutual friction and Ekman time scales. The spin-up process is studied in spheres, cylinders (with co- and counter-rotating lids), and cones, and occurs faster in spheres and cones. Ekman pumping is (anti-) clockwise adjacent to the (lower-) upper-right boundary, and mirrored about the rotation axis. The solution obtained by Reisenegger (J. Low Temp. Phys., vol. 92, 1993, p. 77) between slowly accelerating parallel plates is recovered in the associated limit. The hydrodynamic torque on the container decays dual-exponentially with time in a cylinder but not in a sphere or cone, where it is a superposition of exponentials whose time scales vary with latitude. The torque is a good diagnostic of the flow; e.g. it is steepest initially in a sphere with strong mutual friction.

Vortex motion in superfluid 4He: effects of normal fluid flow

The motion of a vortex filament in superfluid 4He is considered by using the Hall-Vinen-Bekarevich-Khalatnikov (HVBK) [H.E. Hall, W.F. Vinen, Proc. Roy. Soc. Lond. A 238, 204 (1956); H.E. Hall, W.F. Vinen, Proc. Roy. Soc. Lond. A 238, 215 (1956); I.L. Bekarevich, I.M. Khalatnikov, Sov. Phys. J. Exp. Theor. Phys. 13, 643 (1961)] phenomenological model for the scattering process between the vortex and thermal excitations in liquid 4He. The HVBK equations are analytically formulated first in the intrinsic geometric parameter space to obtain insights into the physical implications of the friction terms, associated with the friction coefficients α and α′ (in the Hall-Vinen notation) as well as the previous neglect of the friction term associated with the friction coefficient α′. The normal fluid velocity components both along and transverse to the vortex filament are included. This analytical development also serves to highlight the difficulties arising in making further progress on this route. A reformulation of the HVBK equation in the extrinsic vortex filament coordinate space is then given which is known [B.K. Shivamoggi, Phys. Rev. B 84, 012506 (2011)] to provide a useful alternative analytical approach in this regard. A nonlinear Schrodinger equation for the propagation of nonlinear Kelvin waves on a vortex filament in a superfluid is given taking into account the generalized normal fluid flow. The friction term associated with α′, even in the presence of the normal fluid velocity components transverse to the vortex filament, is shown to produce merely an algebraic growth of the Kelvin waves hence providing further justification for the neglect of this term. On the other hand, the instability produced by the friction term associated with α via the normal fluid velocity component along the vortex filament is shown to manifest itself as a parametric amplification on considering the problem of a rotating planar vortex filament in a superfluid.

Fluctuations of the vortex line density in turbulent flows of quantum fluids

We present an analytical study of fluctuations of the vortex line density (VLD) $\ensuremath{\langle}\ensuremath{\delta}\mathcal{L}(\ensuremath{\omega})\ensuremath{\delta}\mathcal{L}(\ensuremath{-}\ensuremath{\omega})\ensuremath{\rangle}$ in turbulent flows of quantum fluids. Two cases are considered. The first is the counterflowing (Vinen) turbulence, where the vortex lines are disordered, and the evolution of quantity $\mathcal{L}(t)$ obeys the Vinen equation. The second case is the fluctuations of the VLD in a single vortex bundle, which develops inside the domain of the concentrated normal-fluid vorticity. The dynamics of the vortex bundle is described by the Hall-Vinen-Bekarevich-Khalatnikov (HVBK) equations. The latter case is of special interest, because the set of the quantum vortex bundles is believed to mimic classical hydrodynamic turbulence. In steady states the VLD is related to the normal velocity as $\mathcal{L}={(\ensuremath{\rho}\ensuremath{\gamma}/{\ensuremath{\rho}}_{s})}^{2}{v}_{n}^{2}$ for the Vinen case. In the vortex bundle case, which appears inside the domain of a concentrated vorticity of normal fluid, the stationary quantity $\mathcal{L}$ can be found from the matching of velocities and is described by $\mathcal{L}=|\ensuremath{\nabla}\ifmmode\times\else\texttimes\fi{}{\mathbf{v}}_{n}|/\ensuremath{\kappa}$. In nonstationary situations, and particularly in the fluctuating turbulent flow, there is a retardation between the instantaneous value of the normal velocity and the quantity $\mathcal{L}$. This retardation tends to decrease in accordance with the inner dynamics, which has a relaxation character. In both cases, the relaxation dynamics of the VLD is related to fluctuations of the relative velocity. However, for the Vinen case the rate of temporal change for $\mathcal{L}(t)$ is directly dependent upon $\ensuremath{\delta}{\mathbf{v}}_{ns}$, whereas for HVBK dynamics it depends on $\ensuremath{\nabla}\ifmmode\times\else\texttimes\fi{}\ensuremath{\delta}{\mathbf{v}}_{ns}$. Therefore, for the disordered case the spectrum $\ensuremath{\langle}\ensuremath{\delta}\mathcal{L}(\ensuremath{\omega})\ensuremath{\delta}\mathcal{L}(\ensuremath{-}\ensuremath{\omega})\ensuremath{\rangle}$ coincides with the spectrum ${\ensuremath{\omega}}^{\ensuremath{-}5/3}$. In the case of the bundle arrangement, the spectrum of the VLD varies (at different temperatures) from ${\ensuremath{\omega}}^{1/3}$ to ${\ensuremath{\omega}}^{\ensuremath{-}5/3}$ dependencies. This conclusion may serve as a basis for the experimental determination of what kind of turbulence is implemented in different types of generation.

Hydrodynamic equations of anisotropic, polarized and inhomogeneous superfluid vortex tangles

We include the effects of anisotropy and polarization in the hydrodynamics of inhomogeneous vortex tangles, thus generalizing the well known Hall–Vinen–Bekarevich–Khalatnikov equations, which do not take them in consideration. These effects contribute to the mutual friction force F n s between normal and superfluid components and to the vortex tension force ρ s T . These equations are complemented by an evolution equation for the vortex line density L , which takes into account these contributions. These equations are expected to be more suitable than the usual ones for rotating counterflows, or turbulence behind a cylinder, or turbulence produced by a grid of parallel thin cylinders towed across a superfluid, because in these situations polarization is expected to play a relevant role.

Superfluid spherical Couette flow

We solve numerically for the first time the two-fluid Hall–Vinen–Bekarevich–Khalatnikov (HVBK) equations for an He-II-like superfluid contained in a differentially rotating spherical shell, generalizing previous simulations of viscous spherical Couette flow (SCF) and superfluid Taylor–Couette flow. The simulations are conducted for Reynolds numbers in the range 1 × 102≤Re≤3 × 104, rotational shear 0.1≤ΔΩ/Ω≤0.3, and dimensionless gap widths 0.2≤δ≤0.5. The system tends towards a stationary but unsteady state, where the torque oscillates persistently, with amplitude and period determined by δ and ΔΩ/Ω. In axisymmetric superfluid SCF, the number of meridional circulation cells multiplies as Re increases, and their shapes become more complex, especially in the superfluid component, with multiple secondary cells arising for Re > 103. The torque exerted by the normal component is approximately three times greater in a superfluid with anisotropic Hall–Vinen (HV) mutual friction than in a classical viscous fluid or a superfluid with isotropic Gorter–Mellink (GM) mutual friction. HV mutual friction also tends to ‘pinch’ meridional circulation cells more than GM mutual friction. The boundary condition on the superfluid component, whether no slip or perfect slip, does not affect the large-scale structure of the flow appreciably, but it does alter the cores of the circulation cells, especially at lower Re. As Re increases, and after initial transients die away, the mutual friction force dominates the vortex tension, and the streamlines of the superfluid and normal fluid components increasingly resemble each other. In non-axisymmetric superfluid SCF, three-dimensional vortex structures are classified according to topological invariants. For misaligned spheres, the flow is focal throughout most of its volume, except for thread-like zones where it is strain-dominated near the equator (inviscid component) and poles (viscous component). A wedge-shaped isosurface of vorticity rotates around the equator at roughly the rotation period. For a freely precessing outer sphere, the flow is equally strain- and vorticity-dominated throughout its volume. Unstable focus/contracting points are slightly more common than stable node/saddle/saddle points in the viscous component, but not in the inviscid component. Isosurfaces of positive and negative vorticity form interlocking poloidal ribbons (viscous component) or toroidal tongues (inviscid component) which attach and detach at roughly the rotation period.

Mechanism of “rigid-body” rotation of the superfluid and normal components during phase separation of a supersaturated He3–4He solution

It is shown that unstable hydrodynamic vortices can form inside subcritical domains in the normal component of a supersaturated decomposing He3–4He solution. A mechanism for the entrainment of the superfluid component by the normal component of the He3–4He solution into “rigid-body” rotation due to Hall–Vinen–Bekarevich–Khalatnikov forces in the equations of two-fluid hydrodynamics, which leads to the creation of quantized vortices, is considered. An increase of the average density of quantized vortices can increase the rate of heterogeneous decomposition of the He3–4He solution.