Quantum turbulence: Theoretical and numerical problems

The term “quantum turbulence” (QT) unifies the wide class of phenomena where the chaotic set of one dimensional quantized vortex filaments (vortex tangles) appear in quantum fluids and greatly influence various physical features. Quantum turbulence displays itself differently depending on the physical situation, and ranges from quasi-classical turbulence in flowing fluids to a near equilibrium set of loops in phase transition. The statistical configurations of the vortex tangles are certainly different in, say, the cases of counterflowing helium and a rotating bulk, but in all the physical situations very similar theoretical and numerical problems arise. Furthermore, quite similar situations appear in other fields of physics, where a chaotic set of one dimensional topological defects, such as cosmic strings, or linear defects in solids, or lines of darkness in nonlinear light fields, appear in the system. There is an interpenetration of ideas and methods between these scientific topics which are far apart in other respects. The main purpose of this review is to bring together some of the most commonly discussed results on quantum turbulence, focusing on analytic and numerical studies. We set out a series of results on the general theory of quantum turbulence which aim to describe the properties of the chaotic vortex configuration, starting from vortex dynamics. In addition we insert a series of particular questions which are important both for the whole theory and for the various applications. We complete the article with a discussion of the hot topic, which is undoubtedly mainstream in this field, and which deals with the quasi-classical properties of quantum turbulence. We discuss this problem from the point of view of the theoretical results stated in the previous sections. We also included section, which is devoted to the experimental and numerical suggestions based on the discussed theoretical models.     

Vortex instability and the onset of superfluid turbulence

Quantized circulation, the absence of Galilean invariance due to a clamped normal component, and the vortex mutual friction are the major factors that make superfluid turbulence behave in a way different from that in classical fluids. The model is developed for the onset of superfluid turbulence that describes the initial avalanchelike multiplication of vortices into a turbulent vortex tangle.     

Vortex deconfinement in the XY model with a magnetic field

We study vortex unbinding for the classical two-dimensional XY model in a magnetic field on square and triangular lattices. A renormalization group analysis combined with duality in the model shows that at high temperature and high field, the vortices unbind as the magnetic field is lowered in a two-step process: strings of overturned spins first proliferate and then vortices unbind. The transitions are highly continuous but are not of the Kosterlitz-Thouless type. The unbound vortex fixed point is shown to inherit properties of the underlying lattice, in particular containing a set of nodal lines that reflect the lattice symmetry.     

Quantum hydrodynamics

Quantum hydrodynamics in superfluid helium and atomic Bose–Einstein condensates (BECs) has been recently one of the most important topics in low temperature physics. In these systems, a macroscopic wave function (order parameter) appears because of Bose–Einstein condensation, which creates quantized vortices. Turbulence consisting of quantized vortices is called quantum turbulence (QT). The study of quantized vortices and QT has increased in intensity for two reasons. The first is that recent studies of QT are considerably advanced over older studies, which were chiefly limited to thermal counterflow in ⁴${}^4$He, which has no analog with classical traditional turbulence, whereas new studies on QT are focused on a comparison between QT and classical turbulence. The second reason is the realization of atomic BECs in 1995, for which modern optical techniques enable the direct control and visualization of the condensate and can even change the interaction; such direct control is impossible in other quantum condensates like superfluid helium and superconductors. Our group has made many important theoretical and numerical contributions to the field of quantum hydrodynamics of both superfluid helium and atomic BECs. In this article, we review some of the important topics in detail. The topics of quantum hydrodynamics are diverse, so we have not attempted to cover all these topics in this article. We also ensure that the scope of this article does not overlap with our recent review article (arXiv:1004.5458), “Quantized vortices in superfluid helium and atomic Bose–Einstein condensates”, and other review articles.     

Modulation of homogeneous shear turbulence laden with finite-size particles

The dynamics of homogeneous shear turbulence laden with spherical finite-size particles is investigated using fully resolved numerical simulations to understand how the presence of particles modulates turbulent shear flows. We focus on a dilute flow laden with non-sedimenting particles whose diameter is slightly smaller than or comparable with those of vortex cores in turbulence. An immersed boundary method is adopted to represent a spherical finite-size particle. Numerical results show that the presence of particles augments the viscous dissipation of turbulence kinetic energy, which leads to a slower increase in the turbulence energy. Although the augmentation of energy dissipation occurs predominantly inside viscous layers surrounding particles in an initial period, the contribution from their outside becomes more significant due to the modification of turbulence structures as turbulence develops. It is found that the particles exhibit weak tendency to accumulate in vortex layers. The particles approaching and colliding with vortex layers induce large velocity fluctuations, which leads to the generation and shedding of thin vortex tubes. Newly generated vortex tubes interact with developed vortex tubes and layers, and modify the entire structure of the vorticity field.     

Direct numerical simulation study of the interaction between the polymer effect and velocity gradient tensor in decaying homogeneous isotropic turbulence

Direct numerical simulation of decaying homogeneous isotropic turbulence (DHIT) of a polymer solution is performed. In order to understand the polymer effect on turbulence or additive-turbulence interaction, we directly investigate the influence of polymers on velocity gradient tensor including vorticity and strain. By visualizing vortex tubes and sheets, we observe a remarkable inhibition of vortex structures in an intermediate-scale field and a small-scale field but not for a large scale field in DHIT with polymers. The geometric study indicates a strong relevance among the vorticity vector, rate-of-strain tensor, and polymer conformation tensor. Joint probability density functions show that the polymer effect can increase "strain generation resistance" and "vorticity generation resistance", i.e., inhibit the generation of vortex sheets and tubes, ultimately leading to turbulence inhibition effects.     

基于量子Fisher信息的量子计量进展

量子计量是超冷原子气体研究中的一个热点领域.超冷原子体系独特的量子性质(量子纠缠)和量子效应有助于大幅度提高待测物理量的测量精度,这已经成为量子精密测量中的共识.量子Fisher信息对该领域的发展起了非常重要的作用.本文首先介绍量子Fisher信息的基本概念和量子计量的主要内容;然后简要回顾这些理论在提高测量精度方面的应用,特别是多粒子量子纠缠态的产生及其判定;再介绍线性和非线性原子干涉仪的相关进展;最后论述量子测量过程中的统计方法的研究进展.     

Drag coefficient and mass density of moving quantum vortices in type-II superconductors

Summary The dynamics of vortex lines in type-II superconductors are discussed from a quantum electrodynamic viewpoint. When a vortex line is set into motion, electric fields are generated. These play a dual role of providing electric-field energy storage (?kinetic energy?) and also generating normal electron energy dissipation. From these considerations, the mass density and the drag coefficients for vortices can be derived. Quantum mechanisms for vortex motions and nucleation are discussed.     

On the Global Evolution of Vortex Filaments,Blobs, and Small Loops in 3D Ideal Flows

We consider a wide class of approximate models of evolution of singular distributions of vorticity in three dimensional incompressible fluids and we show that they have global smooth solutions. The proof exploits the existence of suitable Hamiltonian functions. The approximate models we analyze (essentially discrete and continuous vortex filaments and vortex loops) are related to some problem of classical physics concerning turbulence and also to the numerical approximation of flows with very high Reynolds number. Finally, we apply our strategy to discrete models for filaments used in numerical methods.     

Physics of multiple level hairpin vortex structures in turbulence

Previous experimental and numerical studies have revealed that the hairpin vortex is a basic flow element of transitional boundary layer. The hairpin vortex is believed to have legs, necks and a ring head. Based on our DNS study, the legs and the ring head are generated separately by different mechanisms. The legs function like an engine to generate low speed zones by rotation, create shear layers with surrounding high speed neighbor fluids, and further cause vortex ring formation through shear layer instability. In addition, the ring head is ?-shaped and separated from quasi-streamwise legs from the beginning. Contrary to the classical concept of "vortex breakdown", we believe transition from laminar flow to turbulence is a "buildup" process of multiple level vortical structures. The vortex rings of first level hairpins are mostly responsible for positive spikes, which cause new vorticity rollup, second level vortex leg formation and finally smaller second level vortex ring generation. The third and lower level vortices are generated following the same mechanism. In this paper, the physical process from ?-vortex to multi-level hairpin vortices is described in detail.     

Optimal dynamical systems of Navier-Stokes equations based on generalized helical-wave bases and the fundamental elements of turbulence

In this paper, we present the theory of constructing optimal generalized helical-wave coupling dynamical systems. Applying the helical-wave decomposition method to Navier-Stokes equations, we derive a pair of coupling dynamical systems based on optimal generalized helical-wave bases. Then with the method of multi-scale global optimization based on coarse graining analysis, a set of global optimal generalized helical-wave bases is obtained. Optimal generalized helical-wave bases retain the good properties of classical helical-wave bases. Moreover, they are optimal for the dynamical systems of Navier-Stokes equations, and suitable for complex physical and geometric boundary conditions. Then we find that the optimal generalized helical-wave vortexes fitted by a finite number of optimal generalized helical-wave bases can be used as the fundamental elements of turbulence, and have important significance for studying physical properties of complex flows and turbulent vortex structures in a deeper level.     

On the axiomatic approach to quasi-classical field theory

We study the connection between quasi-classical field theory and axiomatic statements of the quantum field theory, Schwinger source theory, and the Lehmann-Symanzik-Zimmermann (LSZ) formalism. The classical Schwinger source is connected with the classical field; the LSZ R-function is connected with the quantum field operator. The axioms of the quantum field theory are written in the context of the quasi-classical expansion. In the considered approach, the stationary action principle and canonical commutation relations for field operators are obtained as corollaries and are not postulated as initial statements of the theory.     

A numerical investigation of vortex-coupled superfluidity

We have performed numerical simulations of quantized vortex lines in a model of normal fluid turbulence. The results are used to discuss the idea, put forward to explain some recent experiments, that in isothermal turbulent helium flow the high density of vortex lines locks the two fluid components together.     

Classical and quantum regimes of superfluid turbulence

We argue that turbulence in superfluids is governed by two dimensionless parameters. One of them is the intrinsic parameter q which characterizes the friction forces acting on a vortex moving with respect to the heat bath, with q^?1 playing the same role as the Reynolds number Re=UR/ν in classical hydrodynamics. It marks the transition between the “laminar” and turbulent regimes of vortex dynamics. The developed turbulence described by Kolmogorov cascade occurs when Re?1 in classical hydrodynamics, and q?1 in superfluid hydrodynamics. Another parameter of superfluid turbulence is the superfluid Reynolds number Re_s=UR/κ, which contains the circulation quantum κ characterizing quantized vorticity in superfluids. This parameter may regulate the crossover or transition between two classes of superfluid turbulence: (i) the classical regime of Kolmogorov cascade where vortices are locally polarized and the quantization of vorticity is not important; (ii) the quantum Vinen turbulence whose properties are determined by the quantization of vorticity. A phase diagram of the dynamical vortex states is suggested.     

Dynamics of relativistic vortex lines and their relation to dual theory

We analyse the dynamics of relativistic vortex lines that occur in some classical field theories. We prove that in the zero-width limit they move like Nambu strings. We also discuss the relevance of vortex lines to the corresponding quantum field theory.     

Polarization of superfluid turbulence

We show that normal-fluid eddies in turbulent helium II polarize the tangle of quantized vortex lines present in the flow, thus inducing superfluid vorticity patterns similar to the driving normal-fluid eddies. We also show that the polarization is effective over the entire inertial range. The results help explain the surprising analogies between classical and superfluid turbulence which have been observed recently.     

A Closure for Isotropic Turbulence Based on Extended Scale Similarity Theory in Physical Space

The closure of a turbulence field is a longstanding fundamental problem, while most closure models are introduced in spectral space. Inspired by Chou's quasi-normal closure method in spectral space, we propose an analytical closure model for isotropic turbulence based on the extended scale similarity theory of the velocity structure function in physical space. The assumptions and certain approximations are justified with direct numerical simulation. The asymptotic scaling properties are reproduced by this new closure method, in comparison to the classical Batchelor model.     

时均流诱导声振荡的数值计算研究

时均流诱导的声振荡可以为热声制冷提供驱动源或驱动发电机和换能器发电,为风能利用提供了新思路,是热声领域的最新研究方向之一。本文基于计算流体动力学(CFD)方法,建立了正十字型时均流激声发动机的三维数学模型,采用大涡模拟湍流模型计算。计算结果验证了时均流诱导声振荡效应,揭示出谐振管内声场分布和谐振管内部压力与开口处涡的关系,为后续的实验研究奠定了理论基础。