Counterflow Superfluid Turbulence Research Articles

Inertial particles in superfluid turbulence: Coflow and counterflow

We use pseudospectral direct numerical simulations to solve the three-dimensional (3D) Hall–Vinen–Bekharevich–Khalatnikov (HVBK) model of superfluid helium. We then explore the statistical properties of inertial particles, in both coflow and counterflow superfluid turbulence (ST) in the 3D HVBK system; particle motion is governed by a generalization of the Maxey–Riley–Gatignol equations. We first characterize the anisotropy of counterflow ST by showing that there exist large vortical columns. The light particles show confined motion as they are attracted toward these columns, and they form large clusters; by contrast, heavy particles are expelled from these vortical regions. We characterize the statistics of such inertial particles in 3D HVBK ST: (1) The mean angle Θ(τ) between particle positions, separated by the time lag τ, exhibits two different scaling regions in (a) dissipation and (b) inertial ranges, for different values of the parameters in our model; in particular, the value of Θ(τ), at large τ, depends on the magnitude of Uns. (2) The irreversibility of 3D HVBK turbulence is quantified by computing the statistics of energy increments for inertial particles. (3) The probability distribution function (PDF) of energy increments is of direct relevance to recent experimental studies of irreversibility in superfluid turbulence; we find, in agreement with these experiments, that, for counterflow ST, the skewness of this PDF is less pronounced than its counterparts for coflow ST or for classical fluid turbulence.

Counterflow-Induced Inverse Energy Cascade in Three-Dimensional Superfluid Turbulence.

Finite-temperature quantum turbulence is often described in terms of two immiscible fluids that can flow with a nonzero-mean relative velocity. Such out-of-equilibrium state is known as counterflow superfluid turbulence. We report here the emergence of a counterflow-induced inverse energy cascade in three-dimensional superfluid flows by performing extensive numerical simulations of the Hall-Vinen-Bekarevich-Khalatnikov model. As the intensity of the mean counterflow is increased, an abrupt transition, from a fully three-dimensional turbulent flow to a quasi-two-dimensional system exhibiting a split cascade, is observed. The findings of this work could motivate new experimental settings to study quasi-two-dimensional superfluid turbulence in the bulk of three-dimensional experiments. They might also find applications beyond superfluids in systems described by more than one fluid component.

Dynamics of the vortex line density in superfluid counterflow turbulence

Describing superfluid turbulence at intermediate scales between the inter-vortex distance and the macroscale requires an acceptable equation of motion for the density of quantized vortex lines $\cal{L}$. The closure of such an equation for superfluid inhomogeneous flows requires additional inputs besides $\cal{L}$ and the normal and superfluid velocity fields. In this paper we offer a minimal closure using one additional anisotropy parameter $I_{l0}$. Using the example of counterflow superfluid turbulence we derive two coupled closure equations for the vortex line density and the anisotropy parameter $I_{l0}$ with an input of the normal and superfluid velocity fields. The various closure assumptions and the predictions of the resulting theory are tested against numerical simulations.

Inhomogeneous vortex tangles in counterflow superfluid turbulence: flow in convergent channels

Abstract We investigate the evolution equation for the average vortex length per unit volume L of superfluid turbulence in inhomogeneous flows. Inhomogeneities in line density L andincounterflowvelocity V may contribute to vortex diffusion, vortex formation and vortex destruction. We explore two different families of contributions: those arising from asecondorder expansionofthe Vinenequationitself, andthose whichare notrelated to the original Vinen equation but must be stated by adding to it second-order terms obtained from dimensional analysis or other physical arguments.

Thermodynamic approach to vortex production and diffusion in inhomogeneous superfluid turbulence

In this paper, we use a non-equilibrium thermodynamic framework to generalize a previous nonlocal model of counterflow superfluid turbulence to incorporate some new coupled terms which may be relevant in the evolution of inhomogeneous vortex tangles. The theory chooses as fundamental fields the energy density, the heat flux, and the averaged vortex line length per unit volume. The constitutive quantities are assumed to depend on the fundamental fields and on their first spatial derivatives, allowing us to describe thermal dissipation, vortex diffusion and a new contribution to vortex formation. The restrictions on the constitutive relations are deduced from the entropy principle, using the Liu method of Lagrange multipliers. The form of the entropy and the entropy flux are compared with those obtained in the maximum-entropy formalism. The several non-local terms are discussed in detail to clarify their physical meaning, and an experiment is proposed to check their actual existence and measure their values.

Vortex diffusion and vortex-line hysteresis in radial quantum turbulence

We study the influence of vortex diffusion on the evolution of inhomogeneous quantized vortex tangles. A simple hydrodynamical model to describe inhomogeneous counterflow superfluid turbulence is used. As an illustration, we obtain solutions for these effects in radial counterflow of helium II between two concentric cylinders at different temperatures. The vortex diffusion from the inner hotter cylinder to the outer colder cylinder increases the vortex length density everywhere as compared with the non-diffusive situation. The possibility of hysteresis in the vortex line density under cyclical variations of the heat flow is explored.

Nonlocal and rotational effects in quantum turbulence

We discuss phenomenological equations for the evolution of vortex tangle in counterflow superfluid turbulence, which takes into account the influence of the non local effects, both in absence and in the presence of rotation.

Alternative Vinen equation and its extension to rotating counterflow superfluid turbulence

Two alternative Vinen's evolution equations for the vortex line density L in counterflow superfluid turbulence are physically admissible and lead to analogous results in steady states. In Jou and Mongiovı` [Phys. Rev. B 69 (2004) 094513] the most used of them was generalized to counterflow superfluid turbulence in rotating containers. Here, the analogous generalization for the alternative Vinen equation is proposed. Both generalized Vinen's equations are compared with the experimental results, not only in steady states but also in some unsteady situations. From this analysis follows that the stationary solutions of the alternative Vinen equation fit better the experimental data and that its unsteady solutions tend faster to the corresponding final steady-state values than the solutions of the usual Vinen's equation.

A mathematical model of counterflow superfluid turbulence describing heat waves and vortex-density waves

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 high-frequency second sound.

Nonequilibrium effective temperature of superfluid vortex tangle

An effective nonequilibrium temperature in counterflow superfluid turbulence is proposed, as a parameter characterizing a canonical probability distribution function of vortex orientation, and relating the diffusion coefficient of vortex lines to the vortex friction through an Einstein relation.

Superfluid turbulence in rotating containers: Phenomenological description of the influence of the wall

In this paper a previous equation for the evolution of vortex line density L in counterflow superfluid turbulence in rotating containers is generalized, in order to take into account the influence of the walls. This model incorporates the effects of counterflow velocity V and of angular velocity {omega} of the container, and introduces corrective terms depending on {delta}/d, {delta} being the intervortex spacing, of the order L{sup -1/2}, and d the diameter of the channel. The stability of the solutions for L, for several regimes of averaged counterflow velocity V and angular velocity {omega}, is analyzed. Our mathematical analysis reveals that qualitative consistency allows us to reduce the four coefficients characterizing the dependence on {delta}/d to only one additional independent coefficient, linked to the critical angular velocity {omega}{sub c} needed for the appearance of vortex lines in a rotating superfluid.

Nonlocal effects in superfluid turbulence: Application to the low-density- to high-density-state transition and to vortex decay

We discuss a phenomenological equation for the evolution of vortex tangle in counterflow superfluid turbulence, which takes into account the influence of the nonlocal effects, introducing into the original equation of Vinen two simple corrective terms dependent on a nonvanishing ratio between the average separation between vortex lines and the diameter of the channel. The equation allows one to describe, in relatively good agreement with experimental results, the two turbulent regimes present in counterflow superfluid turbulence and the transition between them. The decay rate of the vortex line density $L$, when the heat flux is suddenly turned off, is also investigated; due to the simplicity of the model, which does not take into account the coupling of the line density $L$ with the superfluid velocity, this decay agrees with experiments in the initial and intermediate stages, but does not describe the full slow-down observed at very long times.

Phenomenological description of counterflow superfluid turbulence in rotating containers

In this paper a simple equation for the vortex line density describing some of the most relevant observed effects in counterflow superfluid turbulence in ${}^{4}\mathrm{He}$ in the presence of rotation is proposed. This model is based on a generalization of Vinen's equation which incorporates as additional quantity the angular velocity \ensuremath{\Omega}.