We study the dynamical process of the vortex tangle development under a spherically symmetric thermal counterflow around a heat source submerged into a bulk superfluid 4He. We reveal a peculiar vortex dynamics that is unique to this geometry, which is greatly diverse from the vortex dynamics in a homogeneous counterflow. Two types of heater are considered here, namely, a spherical heater with a solid wall and a point-like heater. In both cases, a spherical vortex tangle is formed surrounding the heater. The mechanism of vortex tangle development in the vicinity of a solid wall is strongly governed by Donnelly-Glaberson instability; while, far away from the heater or around a point heater, the mechanism is governed by the dynamics of polarized vortex loops in radial counterflow. The decay process of such localized vortex tangles is also investigated and is compared with that of homogeneous 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 ${\bf F}_{ns}$ between normal and superfluid components and to the vortex tension force $ρ_s{\bf T}$. 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 rotatin…
We numerically simulated quantum turbulence in superfluid $^4$He to investigate the emission of vortex rings from a localized vortex tangle. Turbulence is characterized by some universal statistical laws. Although there are a lot of studies on statistical laws in bulk quantum turbulence, studies in inhomogeneous or localized turbulence is scarce. We first investigate the statistical laws of localized quantum turbulence, referring to two statistical laws deduced from the vibrating wire experiments in [Yano $et$ $al.$, J. Low Temp. Phys. $\bf{196}$, $184\ (2019)$]. The first law is the Poisson p…
Quantum turbulence in superfluid He-4 in narrow channels often takes the form of moving localized vortex tangles. Such tangles, called turbulent plugs, also serve as building blocks of quantum turbulence in wider channels. We report on a numerical study of various aspects of the dynamics and structure of turbulent plugs in a wide range of governing parameters. The unrestricted growth of the tangle in a long channel provides a unique view on a natural tangle structure including superfluid motion at many scales. We argue that the edges of the plugs propagate as turbulent fronts, following the ad…
Turbulence in superfluid helium~II is a tangle of quantized vortex lines which interact via the classical Biot-Savart law. We show that vortex tangles with the same vortex line density will have different energy spectra, depending on the normal fluid which feeds energy into the superfluid component, and identify the spectral signature of two forms of superfluid turbulence: Kolmogorov tangles and Vinen tangles. By decomposing the superfluid velocity field into local and nonlocal contributions, we find that in Vinen tangles the motion of vortex lines depends mainly on the local curvature, wherea…
The theory describing the evolution of inhomogeneous vortex tangle at zero temperature is developed on the bases of kinetics of merging and splitting vortex loops. Vortex loops composing the vortex tangle can move as a whole with some drift velocity depending on their structure and their length. The flux of length, energy, momentum etc. executed by the moving vortex loops takes a place. Situation here is exactly the same as in usual classical kinetic theory with the difference that the "carriers" of various physical quantities are not the point particles, but extended objects (vortex loops), w…
A nuclear capture reaction of a single neutron by ultra-cold superfluid $^3$He results in a rapid overheating followed by the expansion and subsequent cooling of the hot subregion, in a certain analogy with the Big Bang of the early Universe. It was shown in a Grenoble experiment that a significant part of the energy released during the nuclear reaction was not converted into heat even after several seconds. It was thought that the missing energy was stored in a tangle of quantized vortex lines. This explanation, however, contradicts the expected lifetime of a bulk vortex tangle, $10^{-5}-10^{…
In the comment I develop a critical analysis of the use of the HVBK method for the study of three-dimensional turbulent flows of superfluids. The conception of the vortex bundles forming the structure of quantum turbulence is controversial and does not justify the use of the HVBK method. In addition, this conception is counterproductive, because it gives incorrect ideas about the structure of the vortex tangle as a set of bundles containing parallel lines. The only type of dynamics of vortex filaments inside these bundles is possible, namely, Kelvin waves running along the filaments. At the sa…
By injecting negative ions in superfluid 4He in the zero-temperature limit (T < 0.5 K), we generated tangles of quantized vortex line with negligible large-scale flow. For this quantum regime of superfluid turbulence, the vortex line length L was found to decay at late time t as L proportional to t^{-1}; the prefactor being independent of the initial value of L. The corresponding effective kinematic viscosity is 0.1 kappa, where kappa is the circulation quantum. At T > 0.7 K, a jet of ions generates quasi-classical tangles identical to those produced by mechanical means.
We present a combined numerical and experimental study of Andreev scattering from quantum turbulence in superfluid $^3$He-B at ultralow temperatures. We simulate the evolution of moderately dense, three-dimensional, quasiclassical vortex tangles and the Andreev reflection of thermal quasiparticle excitations by these tangles. This numerical simulation enables us to generate the two-dimensional map of local Andreev reflections for excitations incident on one of the faces of a cubic computational domain, and to calculate the total coefficient of Andreev reflection as a function of the vortex lin…
By assuming a self-similar structure for Kelvin waves along vortex loops with successive smaller scale features, we model the fractal dimension of a superfluid vortex tangle in the zero temperature limit. Our model assumes that at each step the total energy of the vortices is conserved, but the total length can change. We obtain a relation between the fractal dimension and the exponent describing how the vortex energy per unit length changes with the length scale. This relation does not depend on the specific model, and shows that if smaller length scales make a decreasing relative contributio…
An equation previously proposed to describe the evolution of vortex line density in rotating counterflow turbulent tangles in superfluid helium is generalized to incorporate nonvanishing barycentric velocity and velocity gradients. Our generalization is compared with an analogous approach proposed by Lipniacki, and with experimental results by Swanson et al. in rotating counterflow, and it is used to evaluate the vortex density in plane Couette and Poiseuille flows of superfluid helium.
Observations of quantum turbulence in pure superfluid 4He in a rotating container are reported. New techniques of large-scale forcing (rotational oscillations of the cubic container) and detecting (monitoring ion transport along the axis of rotation) turbulence were implemented. Near the axial walls, with increasing forcing the vortex tangle grows without an observable threshold. This tangle gradually develops into bulk turbulence at a characteristic amplitude of forcing that depends on forcing frequency and rotation rate. At higher amplitudes, the total vortex line length increases rapidly. R…
Superfluid $^3$He-B in the zero-temperature limit offers a unique means of studying quantum turbulence by the Andreev reflection of quasiparticle excitations by the vortex flow fields. We validate the experimental visualization of turbulence in $^3$He-B by showing the relation between the vortex-line density and the Andreev reflectance of the vortex tangle in the first simulations of the Andreev reflectance by a realistic 3D vortex tangle, and comparing the results with the first experimental measurements able to probe quantum turbulence on length scales smaller than the inter-vortex separatio…
A complete study of the propagation of waves (namely longitudinal density and temperature waves, longitudinal and transversal velocity waves and heat waves) in turbulent superfluid helium is made in three situations: a rotating frame, a thermal counterflow, and the simultaneous combination of thermal counterflow and rotation. Our analysis aims to obtain as much as possible information on the tangle of quantized vortices from the wave speed and attenuation factor of these different waves, depending on their relative direction of propagation with respect to the rotation vector.
In this study, we numerically investigate the internal structure of localized quantum turbulence in superfluid $^4$He at zero temperature with the expectation of self-similarity in the real space. In our previous study, we collected the statistics of vortex rings emitted from a localized vortex tangle. As a result, the power law between the minimum size of detectable vortex rings and the emission frequency is obtained, which suggests that the vortex tangle has self-similarity in the real space [Nakagawa $et$ $al.$, Phys. Rev. B $\boldsymbol{101}$, 184515 (2020)]. In this work, we study the fra…
The interaction between vortex density waves and high-frequency second sound in counterflow superfluid turbulence is examined, incorporating diffusive and elastic contributions of the vortex tangle. The analysis is based on a set of evolution equations for the energy density, the heat flux, the vortex line density, and the vortex flux, the latter being considered here as an independent variable, in contrast to previous works. The latter feature is crucial in the transition from diffusive to propagative behavior of vortex density perturbations, which is necessary to interpret the details of hig…
In two previous papers two evolution equations for the vortex line density $L$, proposed by Vinen, were generalized to rotating superfluid turbulence and compared with each other. Here, the already generalized alternative Vinen equation is extended to the case in which counterflow and rotation are not collinear. Then, the obtained equation is considered from the viewpoint of non-equilibrium thermodynamics. According with this formalism, the compatibility between this evolution equation for $L$ and that one for the velocity of the superfluid component is studied. The compatibility condition req…
We report computer simulations of the interaction of seeding particles with quantized vortices and with the normal fluid flow in thermal counterflow of superfluid $^4$He. We show that if the particle concentration is too large, the vortex tangle is significantly affected, posing problems in the interpretation of visualization experiments. The main effects are an increase in vortex line density and a change in polarization of the vortex tangle, caused by the action of the Stokes drag of the viscous normal fluid on the trapped particles. We argue that in the case of large particle concentration,…
A phenomenological model is developed to explain a new set of detailed torsional oscillator data for hcp 4He. The model is based on Anderson's idea of the vortex fluid (vortex tangle) in solid 4He. Utilizing a well-studied treatment of dynamics of quantized vortices we describe how the "local superfluid component" is involved in rotation (torsional oscillation) via a polaried vortices tangle. The polarization in the tangle appears both due to aligning the remnant or thermal vortices and due to penetration of additional vortices into volume. Both are supposed to occur in a relaxation manner, an…
Historically, there is little faith in particle tracking velocimetry (PTV) as a tool to make quantitative measurements of thermal counterflow in He II, since tracer particle motion is complicated by influences from the normal fluid, superfluid, and quantized vortex lines, or a combination thereof. Recently, we introduced a scheme for differentiating particles trapped on vortices (G1) from particles entrained by the normal fluid (G2). In this paper, we apply this scheme to demonstrate the utility of PTV for quantitative measurements of vortex dynamics in He II counterflow. We estimate $\ell$, t…
We present a comprehensive statistical study of free decay of the quantized vortex tangle in superfluid $^4$He at low and ultra-low temperatures, $0\leqslant T \leqslant 1.1\,$K. Using high resolution vortex filament simulations with full Biot-Savart vortex dynamics, we show that for ultra-low temperatures $T\lesssim 0.5 \,$K, when the mutual friction parameters $α\simeq α' < 10^{-5}$, the vortex reconnections excite Kelvin waves with wave lengths $λ$ of the order of the inter-vortex distance $\ell$. These excitations cascade down to the resolution scale $Δξ$ which in our simulations is of the…
Generic scaling laws, such as the Kolmogorov's 5/3-law, are milestone achievements of turbulence research in classical fluids. For quantum fluids such as atomic Bose-Einstein condensates, superfluid helium, and superfluid neutron stars, turbulence can also exist in the presence of a chaotic tangle of evolving quantized vortex lines. However, due to the lack of suitable experimental tools to directly probe the vortex-tangle motion, so far little is known about possible scaling laws that characterize the velocity correlations and trajectory statistics of the vortices in quantum-fluid turbulence …
We perform a numerical analysis of superfluid turbulence produced by thermal counterflow in He II by using the vortex filament model. Counterflow in a low aspect ratio channel is known to show the transition from laminar flow to the two turbulent states TI and TII. The present understanding is that the TI has the turbulent superfluid and the laminar normal fluid but both fluids are turbulent in the TII state. This work studies the vortex tangle in the TI state. Solid boundary condition is applied to walls of a square channel, and the velocity field of the normal fluid is prescribed to be a lam…
The Hydrodynamics of Superfluid Turbulence (HST) describes the flows (or counterflows) of HeII in the presence of a chaotic set of vortex filaments, so called superfluid turbulence. The HST equations govern both a slow variation of the hydrodynamic variables due to dissipation related to the vortex tangle and fast processes of the first and second sound propagation. This circumstance prevents an effective numerical simulations of the problems of unsteady heat transfer in HeII. By virtue of a pertinent multi-scale perturbation analysis we show how one can eliminate the fast processes to derive …
Vortex flow remains laminar up to large Reynolds numbers (Re~1000) in a cylinder filled with 3He-B. This is inferred from NMR measurements and numerical vortex filament calculations where we study the spin up and spin down responses of the superfluid component, after a sudden change in rotation velocity. In normal fluids and in superfluid 4He these responses are turbulent. In 3He-B the vortex core radius is much larger which reduces both surface pinning and vortex reconnections, the phenomena, which enhance vortex bending and the creation of turbulent tangles. Thus the origin for the greater s…
We perform a numerical analysis of counterflow quantum turbulence of superfluid 4He with nonuni- form flows by using the vortex filament model. In recent visualization experiments nonuniform laminar flows of the normal fluid, namely, Hagen-Poiseuille flow and tail-flattened flow, have been observed. Tail-flattened flow is a novel laminar flow in which the outer part of the Hagen-Poiseuille flow becomes flat. In our simulation, the velocity field of the normal fluid is prescribed to be two nonuniform profiles. This work addresses a square channel to obtain important physics not revealed in the …
The interpretation of data obtained from particle image/tracking velocimetry in the study of superfluid flows has been so far a challenging task. Tracking particles (as solid hydrogen or deuterium) are attracted to the cores of quantized vortices, so that their dynamics can be strongly affected by the surrounding vortex tangle. Previous phenomenological arguments indicate that tracking particles and micro-sized vortex rings could form bound states (denoted here as VRP states). While a comprehensive description of the vortex ring-particle bonding mechanism has to deal with somewhat involved flo…
Quantum mechanics places significant restrictions on the hydrodynamics of superfluid flows. Despite this it has been observed that turbulence in superfluids can, in a statistical sense, share many of the properties of its classical brethren; coherent bundles of superfluid vortices are often invoked as an important feature leading to this quasi-classical behavior. A recent experimental study [E. Rusaouen, B. Rousset, and P.-E. Roche, EPL, {\bf 118}, 1, 14005, (2017)] inferred the presence of these bundles through intermittency in the pressure field, however direct visualization of the quantized…
There are two commonly discussed forms of quantum turbulence in superfluid $^4$He above 1K: in one there is a random tangle of quantizes vortex lines, existing in the presence of a non-turbulent normal fluid; in the second there is a coupled turbulent motion of the two fluids, often exhibiting quasi-classical characteristics on scales larger than the separation between the quantized vortex lines in the superfluid component. The decay of vortex line density, $L$, in the former case is often described by the equation $dL/dt=-χ_2 (κ/2π)L^2$, where $κ$ is the quantum of circulation, and $χ_2$ is a…
Turbulence in a superfluid in the zero temperature limit consists of a dynamic tangle of quantized vortex filaments. Different types of turbulence are possible depending on the level of correlations in the orientation of vortex lines. We provide an overview of turbulence in superfluid $^4$He with a particular focus on recent experiments probing the decay of turbulence in the zero temperature regime below 0.5 K. We describe extensive measurements of the vortex line density during the free decay of different types of turbulence: ultraquantum and quasiclassical turbulence in both stationary and r…
Heat is carried in superfluid He-4 by the motion of the normal fluid$^{1}$, a counterflowing superfluid component serving to eliminate any net mass flow. It has been known for many years that above a critical heat current the superfluid component in this counterflow becomes turbulent. This turbulence takes the form of a disorganized tangle of quantized vortex lines and is maintained by the relative motion of the two fluids$^{2-3}$. It has been suspected that the normal fluid may also become turbulent$^{4}$, but experimental verification is difficult without a technique for visualizing the flow…
The Hydrodynamics of Superfluid Turbulence (HST) describes the flows (or counterflows) of HeII in the presence of a chaotic set of vortex filaments. The HST equations govern both a slow variation of the hydrodynamic variables due to dissipation related to the vortex tangle and fast processes of the first and second sound propagation. This circumstance prevents effective numerical simulations of the problems of unsteady heat transfer in HeII. By virtue of a pertinent multi-scale perturbation analysis we show how one can eliminate the fast processes to derive the evolution equation for the slow …
We present a numerical method to compute the evolution of vortex filaments in superfluid helium. The method is based on a tree algorithm which considerably speeds up the calculation of Biot-Savart integrals. We show that the computational cost scales as Nlog{(N) rather than N squared, where $N$ is the number of discretization points. We test the method and its properties for a variety of vortex configurations, ranging from simple vortex rings to a counterflow vortex tangle, and compare results against the Local Induction Approximation and the exact Biot-Savart law.
We perform numerical simulations of finite temperature quantum turbulence produced through thermal counterflow in superfluid $^4$He, using the vortex filament model. We investigate the effects of solid boundaries along one of the Cartesian directions, assuming a laminar normal fluid with a Poiseuille velocity profile, whilst varying the temperature and the normal fluid velocity. We analyze the distribution of the quantized vortices, reconnection rates and quantized-vorticity production as a function of the wall-normal direction. We find that the quantized vortex lines tend to concentrate close…
Visualization of thermal counterflow in He II using PIV (particle image velocimetry) and PTV (particle tracking velocimetry) is difficult because tracer particle motion can be influenced by both the normal fluid and superfluid components of He II as well as the quantized vortex tangle. For instance, an early PTV experiment observed particles moving at the normal fluid velocity $v_n$, while a PIV experiment observed particles moving at $v_n/2$, though the range of heat flux applied in these experiments differed by an order of magnitude. To resolve this apparent discrepancy and explore statistic…
We report on studies of quantum turbulence with second-sound in superfluid 4He in which the turbulence is generated by the flow of the superfluid component through a wide square channel, the ends of which are plugged with sintered silver superleaks, the flow being generated by compression of a bellows. The superleaks ensure that there is no net flow of the normal fluid. In an earlier paper (Phys. Rev. B, 86, 134515 (2012)) we have shown that steady flow of this kind generates a density of vortex lines that is essentially identical with that generated by thermal counterflow, when the average re…
We study pinning and unpinning of superfluid vortices in the inner crust of a neutron star using 3-dimensional dynamical simulations. Strong pinning occurs for certain lattice orientations of an idealized, body-centered cubic lattice, and occurs generally in an amorphous or impure nuclear lattice. The pinning force per unit length is $\sim 10^{16}$ dyn cm$^{-1}$ for a vortex-nucleus interaction that is repulsive, and $\sim 10^{17}$ dyn cm$^{-1}$ for an attractive interaction. The pinning force is strong enough to account for observed spin jumps (glitches). Vortices forced through the lattice m…
