Despite the fundamentally different dissipation mechanisms, many laws and phenomena of classical turbulence equivalently manifest in quantum turbulence. The Reynolds law of dynamical similarity states that two objects of the same geometry across different length scales are hydrodynamically equivalent under the same Reynolds number, leading to a universal drag coefficient law. In this Letter we confirm the existence of a universal drag law in a superfluid wake, facilitated by the nucleation of quantized vortices. We numerically study superfluid flow across a range of Reynolds numbers for the paradigmatic classical hard wall and the Gaussian obstacle, popular in experimental quantum hydrodynamics. In addition, we provide a feasible method for measuring superfluid drag forces in an experimental environment using control volumes.
Turbulence, the irregular motion of fluids, is a challenging problem in physics. Yet some properties of turbulence appear to be universal, independent of the underlying host fluid supporting the motion. Recent studies have found that turbulence in superfluid helium, a quantum fluid, exhibits two of the most fundamental laws of classical fluid turbulence: the Kolmogorov −5/3 law, and the dissipation anomaly. These laws appear despite the fluid being highly constrained by quantum mechanical effects, and completely lacking kinematic viscosity. Such findings suggest further insight into the universal features of turbulent phenomena can be gained by studying analogies between classical and quantum turbulence.\nAtomic Bose-Einstein condensates (BECs) offer a new platform for the study of quantum turbulence; the geometric control available in BEC experiments offers the possibility of studying quantum turbulence in effectively two-dimensional fluids. As two-dimensional turbulence exhibits dramatically different features from its 3D counterpart, BEC systems allow for further study of the analogies between classical and quantum turbulence. In this thesis we numerically and theoretically study 2D quantum turbulence in BECs within the framework of the Gross-Pitaevskii model. We focus on analogies with classical 2D turbulence, with the aim of identifying common or universal features.\nFirst we investigate coherent vortex structures in negative temperature equilibria via an experimentally accessible flow-field measure. Coherent vortices are shown to produce a clear signal in this measure that is independent of the confinement geometry, and we demonstrate that it can be observed in dynamical simulations.\nSecond, studying a quantum analogue of the two-dimensional cylinder wake, we investigate the phenomenon of Strouhal oscillations. We find that the Strouhal number obeys a universal relation, similar to the classical form, upon introducing a modified superfluid Reynolds number that accounts for the critical velocity for vortex nucleation. Like the classical Reynolds number, the superfluid Reynolds number is found to govern the transition from laminar to turbulent behaviour in the quantum fluid.\nFinally, simulating decaying 2D quantum turbulence for very large vortex numbers, we show that quantum fluids are capable of supporting the direct enstrophy cascade, a fundamental feature of two-dimensional turbulent flows. The quantum fluid manifests key features of the classical cascade, including Batchelor’s −3 law of the inertial range, scaling of the inertial range against the superfluid Reynolds number, and the value of the Kraichnan-Batchelor constant.\nThe findings from this work thus provide some new insight into the universality of fundamental turbulence concepts, and their applicability to quantum fluids.
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.
ABSTRACT We theoretically study the line density of quantum vortices in a superfluid turbulent wake based on the conjecture of quantum turbulence of a pure superfluid in the zero temperature limit. The conjecture states that a collection of many quantum vortices can mimic a classical eddy in the continuum approximation. By straightforwardly relating the Kolmogorov's similarity to this conjecture, we can estimate that the effective viscosity of a pure superfluid turbulence is comparable to the circulation quantum , the minimum circulation of quantum vortices. This result naturally leads to a definition of the superfluid Reynolds number, with the characteristic size and velocity , supported by the fact that the equation of motion of a pure superfluid reduces to the Euler equation for high . By using the classical description of the rate of energy injection or dissipation, we quantitatively evaluate a characteristic density of vortex lines in a superfluid turbulent wake and its spatial profile of grid turbulence in terms of the superfluid Reynolds number.
Identified as a unique regime of fluid flow some 500 years ago, turbulence has fascinated and confounded for centuries. Indeed, a fundamental understanding of the turbulence phenomenon remains one of the most important unsolved problems in classical physics. Interestingly, superfluid atomic Bose-Einstein condensates (BECs) can also be agitated into highly excited states of chaotic vortex motion, suggestive of turbulence. However, in contrast to classical fluids, the vortices in these quantum systems exhibit topological stability, the restriction of quantized circulation, and a well defined length scale characterized by the so-called healing length of the superfluid. Such simplifying aspects motivate the study of turbulence in these systems, which may shed new light on the longstanding classical problem. \n\nWhilst a considerable number of studies have been conducted on turbulence in three-dimensional BEC systems, and such studies have uncovered similarities between the classical and quantum counterparts, including the presence of the Kolmogorov
k−5/3 law in the kinetic energy spectrum and the associated direct energy cascade, comparatively few studies have been conducted on two-dimensional quantum turbulence. An important question for such systems is whether the inverse energy cascade, a defining feature of classical two-dimensional turbulent fluid flows, may occur in these systems. In addition to a Kolmogorov kinetic energy spectrum, this process is signified by the clustering of same-circulation vortices, which results in the emergence of macroscopic vortex structures from the turbulent flow. \n\nAlthough intermittent few-vortex clustering has been observed in the break-down of superfluid flow around an obstacle, current attempts to observe a sustained inverse cascade have been unsuccessful, largely due to complications which arise due to the compressibility of the dilute Bose gas - a feature not considered in the classical theory.\n\nWe present a study of two-dimensional quantum turbulence in atomic BECs within the framework of damped Gross-Pitaevskii theory. With the motivation of determining the existence of the inverse cascade phenomenon in such systems, we focusing on experimentally realistic forcing procedures, and study the details of kinetic energy spectra, the statistics of vortex clustering, and the transport of kinetic energy through scale space in the resulting turbulent flows.\nWe investigate a range of vortex injection regimes produced through the circular stirring of a harmonically confined system with a repulsive Gaussian obstacle potential, identifying distinct regimes of turbulence. In the incompressible regime relevant to the inverse cascade, we identify numerous problems with the system and forcing procedure. Motivated by these findings, we investigate turbulence in a minimal model of the spindown of a toroidal persistent current. In this system, we identify a regime of forcing and damping in which the inverse cascade is corroborated by three independent measures: I) merging of vortex clusters; II) a Kolmogorov scaling in the incompressible kinetic energy spectrum; and III) Spectral condensation, associated with the emergence of large scale rotating structures from the turbulent flow. We provide clear evidence that the inverse cascade phenomenon, previously observed in a range of classical fluid systems, can also occur in quantum turbulence.
The typical spacing between superfluid vortices in an isothermal turbulent tangle is proportional to the integral scale H rescaled by the quantum Reynolds number Re_K=H.V/K to the power of 3/4, where K is the quantum of circulation around of single vortex [Salort et al.,EPL 2011]. This empirical relation can be seen as the quantum-turbulence version of the corresponding well-know equation giving Kolmogorov dissipative scale in classical turbulence. In 2014, we studied the temperature dependence of the numerical factor (d/H).Re_K^{3/4} in 4He by joint numerical and experimental analysis of steady state turbulence over a wide temperature interval (1.2 - 2.16 K) [Babuin et al., EPL 2014]. Agreement between the two analyses was found good except at the very ends of this temperature interval. We will discuss this issue by presenting additional experimental data obtained by post-processing of superfluid experiments published between 1975 and 1998.
We report on the experimental observation of vortex cluster shedding from a moving obstacle in an oblate atomic Bose-Einstein condensate.At low obstacle velocities v above a critical value, vortex clusters consisting of two like-sign vortices are generated to form a regular configuration like a von Kármán street, and as v is increased, the shedding pattern becomes irregular with many different kinds of vortex clusters.In particular, we observe that the Stouhal number associated with the shedding frequency exhibits saturation behavior with increasing v.The regular-to-turbulent transition of the vortex cluster shedding reveals remarkable similarities between a superfluid and a classical viscous fluid.Our work opens a new direction for experimental investigations of the superfluid Reynolds number characterizing universal superfluid hydrodynamics.