Experimental, numerical, and analytical velocity spectra in turbulent quantum fluid

Exploring the Equivalence between Two-Dimensional Classical and Quantum Turbulence through Velocity Circulation Statistics

We study the statistics of velocity circulation in two-dimensional classical and quantum turbulence. We perform numerical simulations of the incompressible Navier-Stokes and the Gross-Pitaevskii (GP) equations for the direct and inverse cascades. Our GP simulations display clear energy spectra compatible with the double cascade theory of two-dimensional classical turbulence. In the inverse cascade, we found that circulation intermittency in quantum turbulence is the same as in classical turbulence. We compare GP data to Navier-Stokes simulations and experimental data from Zhu et al. [Phys. Rev. Lett. 130, 214001 (2023)PRLTAO0031-900710.1103/PhysRevLett.130.214001]. In the direct cascade, for nearly incompressible GP flows, classical and quantum turbulence circulation displays the same self-similar scaling. When compressibility becomes important, quasishocks generate quantum vortices and the equivalence of quantum and classical turbulence only holds for low-order moments. Our results establish the boundaries of the equivalence between two-dimensional classical and quantum turbulence.

Theory of anisotropic superfluid 4He counterflow turbulence

We develop a theory of strong anisotropy of the energy spectra in the thermally driven turbulent counterflow of superfluid ⁴He. The key ingredients of the theory are the three-dimensional differential closure for the vector of the energy flux and the anisotropy of the mutual friction force. We suggest an approximate analytic solution of the resulting energy-rate equation, which is fully supported by our numerical solution. The two-dimensional energy spectrum is strongly confined in the direction of the counterflow velocity. In agreement with the experiments, the energy spectra in the direction orthogonal to the counterflow exhibit two scaling ranges: a near-classical non-universal cascade dominated range and a universal critical regime at large wavenumbers. The theory predicts the dependence of various details of the spectra and the transition to the universal critical regime on the flow parameters. This article is part of the theme issue 'Scaling the turbulence edifice (part 2)'.

Vortex clustering, polarisation and circulation intermittency in classical and quantum turbulence

The understanding of turbulent flows is one of the biggest current challenges in physics, as no first-principles theory exists to explain their observed spatio-temporal intermittency. Turbulent flows may be regarded as an intricate collection of mutually-interacting vortices. This picture becomes accurate in quantum turbulence, which is built on tangles of discrete vortex filaments. Here, we study the statistics of velocity circulation in quantum and classical turbulence. We show that, in quantum flows, Kolmogorov turbulence emerges from the correlation of vortex orientations, while deviations-associated with intermittency-originate from their non-trivial spatial arrangement. We then link the spatial distribution of vortices in quantum turbulence to the coarse-grained energy dissipation in classical turbulence, enabling the application of existent models of classical turbulence intermittency to the quantum case. Our results provide a connection between the intermittency of quantum and classical turbulence and initiate a promising path to a better understanding of the latter.

Time-Periodic Inertial Range Dynamics

A wide class of physical systems exhibit scale invariance. While the statistical properties of such behavior can often be investigated by theoretical and experimental means, its dynamics are notoriously hard to parse. We investigate scale-invariant dynamics through an unstable periodic orbit. This orbit coexists with turbulence of an incompressible fluid and yields a significant Kolmogorov energy spectrum. We identify events of intense energy transfer across spatial scales and relate them to vortical dynamics. The results support a recently proposed mechanism for turbulent energy transfer.

Knot spectrum of turbulence

Streamlines, vortex lines and magnetic flux tubes in turbulent fluids and plasmas display a great amount of coiling, twisting and linking, raising the question as to whether their topological complexity (continually created and destroyed by reconnections) can be quantified. In superfluid helium, the discrete (quantized) nature of vorticity can be exploited to associate to each vortex loop a knot invariant called the Alexander polynomial whose degree characterizes the topology of that vortex loop. By numerically simulating the dynamics of a tangle of quantum vortex lines, we find that this quantum turbulence always contains vortex knots of very large degree which keep forming, vanishing and reforming, creating a distribution of topologies which we quantify in terms of a knot spectrum and its scaling law. We also find results analogous to those in the wider literature, demonstrating that the knotting probability of the vortex tangle grows with the vortex length, as for macromolecules, and saturates above a characteristic length, as found for tumbled strings.

Crossover from interaction to driven regimes in quantum vortex reconnections

Reconnections of coherent filamentary structures play a key role in the dynamics of fluids, redistributing energy and helicity among the length scales, triggering dissipative effects, and inducing fine-scale mixing. Unlike ordinary (classical) fluids where vorticity is a continuous field, in superfluid helium and in atomic Bose-Einstein condensates (BECs) vorticity takes the form of isolated quantized vortex lines, which are conceptually easier to study. New experimental techniques now allow visualization of individual vortex reconnections in helium and condensates. It has long being suspected that reconnections obey universal laws, particularly a universal scaling with time of the minimum distance between vortices δ. Here we perform a comprehensive analysis of this scaling across a range of scenarios relevant to superfluid helium and trapped condensates, combining our own numerical simulations with the previous results in the literature. We reveal that the scaling exhibits two distinct fundamental regimes: a [Formula: see text] scaling arising from the mutual interaction of the reconnecting strands and a [Formula: see text] scaling when extrinsic factors drive the individual vortices.

Superfluid Helium in Three-Dimensional Counterflow Differs Strongly from Classical Flows: Anisotropy on Small Scales

Three-dimensional anisotropic turbulence in classical fluids tends towards isotropy and homogeneity with decreasing scales, allowing-eventually-the abstract model of homogeneous and isotropic turbulence to be relevant. We show here that the opposite is true for superfluid ^{4}He turbulence in three-dimensional counterflow channel geometry. This flow becomes less isotropic upon decreasing scales, becoming eventually quasi-two-dimensional. The physical reason for this unusual phenomenon is elucidated and supported by theory and simulations.

Mutual-friction-driven turbulent statistics in the hydrodynamic regime of superfluid ^{3}He-B

It is well known that the turbulence that evolves from the tangles of vortices in quantum fluids at scales larger than the typical quantized vortex spacing ℓ has a close resemblance with classical turbulence. The temperature-dependent mutual friction parameter α(T) drives the turbulent statistics in the hydrodynamic regime of quantum fluids that involves a self-similar cascade of energy. From a simple theoretical analysis, here we show that superfluid ^{3}He-B in the presence of mutual damping exhibits a k^{-5/3} Kolmogorov energy spectrum in the entire inertial range ℓ<r<L at temperature T≲0.2T_{c}, while at T≳0.2T_{c} dissipation begins to dominate larger eddies exhibiting a k^{-3} spectrum toward the energy pumping scale L. At T≈0.35T_{c}, eddies of all size, being highly affected by damping, exhibit a k^{-3} spectrum in the entire inertial range. The consistency of this result with the predictions of recent direct numerical simulations indicates that the present theoretical framework is applicable in quantifying the hydrodynamic regime of quantum turbulence.

Enstrophy Cascade in Decaying Two-Dimensional Quantum Turbulence

We report evidence for an enstrophy cascade in large-scale point-vortex simulations of decaying two-dimensional quantum turbulence. Devising a method to generate quantum vortex configurations with kinetic energy narrowly localized near a single length scale, the dynamics are found to be well characterized by a superfluid Reynolds number Re_{s} that depends only on the number of vortices and the initial kinetic energy scale. Under free evolution the vortices exhibit features of a classical enstrophy cascade, including a k^{-3} power-law kinetic energy spectrum, and constant enstrophy flux associated with inertial transport to small scales. Clear signatures of the cascade emerge for N≳500 vortices. Simulating up to very large Reynolds numbers (N=32 768 vortices), additional features of the classical theory are observed: the Kraichnan-Batchelor constant is found to converge to C^{'}≈1.6, and the width of the k^{-3} range scales as Re_{s}^{1/2}.

Regimes of turbulence without an energy cascade

Experiments and numerical simulations of turbulent 4He and 3He-B have established that, at hydrodynamic length scales larger than the average distance between quantum vortices, the energy spectrum obeys the same 5/3 Kolmogorov law which is observed in the homogeneous isotropic turbulence of ordinary fluids. The importance of the 5/3 law is that it points to the existence of a Richardson energy cascade from large eddies to small eddies. However, there is also evidence of quantum turbulent regimes without Kolmogorov scaling. This raises the important questions of why, in such regimes, the Kolmogorov spectrum fails to form, what is the physical nature of turbulence without energy cascade, and whether hydrodynamical models can account for the unusual behaviour of turbulent superfluid helium. In this work we describe simple physical mechanisms which prevent the formation of Kolmogorov scaling in the thermal counterflow, and analyze the conditions necessary for emergence of quasiclassical regime in quantum turbulence generated by injection of vortex rings at low temperatures. Our models justify the hydrodynamical description of quantum turbulence and shed light into an unexpected regime of vortex dynamics.

Multiscaling in superfluid turbulence: A shell-model study

We examine the multiscaling behavior of the normal- and superfluid-velocity structure functions in three-dimensional superfluid turbulence by using a shell model for the three-dimensional (3D) Hall-Vinen-Bekharevich-Khalatnikov (HVBK) equations. Our 3D-HVBK shell model is based on the Gledzer-Okhitani-Yamada shell model. We examine the dependence of the multiscaling exponents on the normal-fluid fraction and the mutual-friction coefficients. Our extensive study of the 3D-HVBK shell model shows that the multiscaling behavior of the velocity structure functions in superfluid turbulence is more complicated than it is in fluid turbulence.

Vortex clustering and universal scaling laws in two-dimensional quantum turbulence

We investigate numerically the statistics of quantized vortices in two-dimensional quantum turbulence using the Gross-Pitaevskii equation. We find that a universal -5/3 scaling law in the turbulent energy spectrum is intimately connected with the vortex statistics, such as number fluctuations and vortex velocity, which is also characterized by a similar scaling behavior. The -5/3 scaling law appearing in the power spectrum of vortex number fluctuations is consistent with the scenario of passive advection of isolated vortices by a turbulent superfluid velocity generated by like-signed vortex clusters. The velocity probability distribution of clustered vortices is also sensitive to spatial configurations, and exhibits a power-law tail distribution with a -5/3 exponent.

Dissipation of Quasiclassical Turbulence in Superfluid ^{4}He

We compare the decay of turbulence in superfluid ^{4}He produced by a moving grid to the decay of turbulence created by either impulsive spin-down to rest or by intense ion injection. In all cases, the vortex line density L decays at late time t as L∝t^{-3/2}. At temperatures above 0.8 K, all methods result in the same rate of decay. Below 0.8 K, the spin-down turbulence maintains initial rotation and decays slower than grid turbulence and ion-jet turbulence. This may be due to a decoupling of the large-scale superfluid flow from the normal component at low temperatures, which changes its effective boundary condition from no-slip to slip.

Visualizing Pure Quantum Turbulence in Superfluid 3He: Andreev Reflection and its Spectral Properties

Superfluid 3He-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 3He-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 intervortex separation.

Hot-wire anemometry for superfluid turbulent coflows

We report the first evidence of an enhancement of the heat transfer from a heated wire to an external turbulent coflow of superfluid helium. We used a standard Pt-Rh hot-wire anemometer and overheat it up to 21 K in a pressurized liquid helium turbulent round jet at temperatures between 1.9 K and 2.12 K. The null-velocity response of the sensor can be satisfactorily modeled by the counterflow mechanism, while the extra cooling produced by the forced convection is found to scale similarly as the corresponding extra cooling in classical fluids. We propose a preliminary analysis of the response of the sensor and show that-contrary to a common assumption-such sensor can be used to probe local velocity in turbulent superfluid helium.

Introduction to quantum turbulence

The term quantum turbulence denotes the turbulent motion of quantum fluids, systems such as superfluid helium and atomic Bose-Einstein condensates, which are characterized by quantized vorticity, superfluidity, and, at finite temperatures, two-fluid behavior. This article introduces their basic properties, describes types and regimes of turbulence that have been observed, and highlights similarities and differences between quantum turbulence and classical turbulence in ordinary fluids. Our aim is also to link together the articles of this special issue and to provide a perspective of the future development of a subject that contains aspects of fluid mechanics, atomic physics, condensed matter, and low-temperature physics.