Vortex Filaments Research Articles

Bubbles with Attached Quantum Vortices in Trapped Binary Bose–Einstein Condensates

Specific topological excitations of energetically stable “core-and-mantle” configurations of trapped two-component immiscible Bose–Einstein condensates are studied numerically within the coupled Gross–Pitaevskii equations. Nonstationary long-lived coherent structures, that consist of several quantum vortex filaments penetrating the “mantle” from outside to inside and vice versa and demonstrate quite nontrivial dynamics, are observed in simulations for the first time. The ends of filaments can remain attached to the interface between the “mantle” and the “core” if the latter is large enough while the surface tension is not small. The shapes of such “bubbles” are strongly affected by the vortices and sometimes are far from being spherical.

Vortex Tracking of Purge-Mainstream Interactions in a Rotating Turbine Stage

Abstract The use of purge flow in gas turbines allows for high turbine entry temperatures, which are essential to produce high cycle efficiency. Purge air is bled from the compressor and reintroduced in the turbine to cool vulnerable components. Wheel-spaces are formed between adjacent rotating and stationary discs, with purge air supplied at low radius before exiting into the mainstream gas-path through a rim-seal at the disc periphery. An aerodynamic penalty is incurred as the purge flow egress interacts with the mainstream. This study presents unparalleled three-dimensional velocity data from a single-stage turbine test rig, specifically designed to investigate egress–mainstream interaction using optical measurement techniques. Volumetric velocimetry is applied to the rotating environment with phase-locked measurements used to identify and track the vortical secondary flow features through the blade passage. A baseline case without purge flow is compared to experiments with a 1.7% purge mass fraction; the latter was chosen to ensure a fully sealed wheel-space. A non-localized vortex tracking function is applied to the data to identify the position of the core centroids. The strength of the secondary flow vortices was determined using a circulation criterion on rotated planes aligned to the vortex filaments. The pressure-side leg of the horseshoe vortex and a second vortex associated with the egress flow were identified by the experimental campaign. In the absence of purge flow, the two vortices merged, forming the passage vortex (PV). With the addition of purge flow, the two cores remained independent to 40% of the blade axial chord, while also demonstrating an increased radial migration and intensification of the PV. The egress core was shown to remain closer to the suction-surface with purge flow. Importantly, where the vortex filaments demonstrated strong radial or tangential components of velocity, the circulation level calculated from axial planes underpredicted the true circulation by up to 50%.

Adiabatic Approximation for the Motion of Ginzburg-Landau Vortex Filaments

In this paper, we consider the concentration property of solutions to the dispersive Ginzburg-Landau (or Gross-Pitaevskii) equation in three dimensions. On a spatial domain, it has long been conjectured that such a solution concentrates near some curve evolving according to the binormal curvature flow, and conversely, that a curve moving this way can be realized in a suitable sense by some solution to the dispersive Ginzburg-Landau equation. Some partial results are known with rather strong symmetry assumptions. Our main theorems here provide affirmative answer to both conjectures under certain small curvature assumption. The results are valid for small but fixed material parameter in the equation, in contrast to the general practice to take this parameter to its zero limit. The advantage is that we can retain precise description of the vortex filament structure. The results hold on a long but finite time interval, depending on the curvature assumption.

Properties of a quantum vortex in neutron matter at finite temperatures

We have studied systematically microscopic properties of a quantum vortex in neutron matter at finite temperatures and densities corresponding to different layers of the inner crust of a neutron star. To this end and in preparation of future simulations of the vortex dynamics, we have carried out fully self-consistent 3D Hartree-Fock-Bogoliubov calculations, using one of the latest nuclear energy-density functionals from the Brussels-Montreal family, which has been developed specifically for applications to neutron superfluidity in neutron-star crusts. By analyzing the flow around the vortex, we have determined the effective radius relevant for the vortex filament model. We have also calculated the specific heat in the presence of the quantum vortex and have shown that it is substantially larger than for a uniform system at low temperatures. The low temperature limit of the specific heat has been identified as being determined by Andreev states inside the vortex core. We have shown that the specific heat in this limit does not scale linearly with temperature. The typical energy scale associated with Andreev states is defined by the minigap, which we have extracted for various neutron-matter densities. Our results suggest that vortices may be spin-polarized in the crust of magnetars. Finally, we have obtained a lower bound for the specific heat of a collection of vortices with given surface density, taking into account both the contributions from the vortex core states and from the hydrodynamic flow.

Numerical study of the effect of tip-speed ratio on hydrokinetic turbine wake recovery

The predictive capabilities of blade-resolved unsteady Reynolds averaged Navier-Stokes (URANS) and detached eddy simulation (DES), the most commonly used hybrid RANS/large eddy simulation (LES) model, are assessed for hydrokinetic turbine performance and mean and turbulent flows in the intermediate-wake region, and results for a range of tip-speed ratio encompassing design and off-design conditions are analyzed to understand the wake recovery mechanism. The performance predictions compared within 5% of the experimental data. Both URANS and DES models performed reasonably well for the near wake predictions, where the errors were <15%. DES outperformed URANS for both mean wake deficit and turbulence predictions in the intermediate-wake region and both quantities compared within 10% of the experiments. The improved prediction by DES is because of its ability to predict the tip vortex breakdown, which plays a critical role in the wake recovery, especially for higher tip speed ratios (λ). However, DES significantly underpredicted the turbulence predictions in the near-wake region, which could be partly due to the negligence of free-surface effects and partly due to modeling issues, namely modeled stress depletion. The study reveals that the tip vortex breakdown mechanism depends on λ. For lower values of λ, instabilities generated in the root vortex core are identified to be the cause of breakdown. For higher values, the breakdown occurred because of the instabilities generated during the vortex filament entanglement. Future work should focus on investigation of other hybrid RANS/LES models to address the limitations of the DES models, and extension of the study to include free-surface effects.

Non-conservation of Dimension in Divergence-Free Solutions of Passive and Active Scalar Systems

For any \(h\in (1,2]\), we give an explicit construction of a compactly supported, uniformly continuous, and (weakly) divergence-free velocity field in \({\mathbb {R}}^2\) that weakly advects a measure whose support is initially the origin but for positive times has Hausdorff dimension h. These velocities are uniformly continuous in space-time and compactly supported, locally Lipschitz except at one point and satisfy the conditions for the existence and uniqueness of a Regular Lagrangian Flow in the sense of Di Perna and Lions theory. We then construct active scalar systems in \({\mathbb {R}}^2\) and \({\mathbb {R}}^3\) with measure-valued solutions whose initial support has co-dimension 2 but such that at positive times it only has co-dimension 1. The associated velocities are divergence free, compactly supported, continuous, and sufficiently regular to admit unique Regular Lagrangian Flows. This is in part motivated by the investigation of dimension conservation for the support of measure-valued solutions to active scalar systems. This question occurs in the study of vortex filaments in the three-dimensional Euler equations.

Main Rotor Wake Interference Effects on Tail Rotor Thrust in Crosswind

Reverse pedal operational property in front crosswind flight condition is a potential hazard for accidents involving loss of tail rotor effectiveness (LTE), which is closely related to the main rotor (MR) wake interference on the tail rotor (TR). As understanding of this interaction is vital for the early warning strategy development, the MR wake influence effect on TR thrust and the effect of helicopter yaw stability are examined in this study. For this purpose, the comparison of TR thrust and flow field with wind azimuth and speed in front crosswind environment was performed by experiment and CFD simulation, respectively. Test campaign was performed at a5.5 m×4 mwind tunnel in the China Aerodynamics Research and Development Center using a high-position bottom-blade forward-rotating TR and a counterclockwise rotating MR to address the TR thrust under wind speeds of 8–22 m/s with 50°, 60°, and 70° wind azimuths. The influence of MR disc loading was also contrasted. CFD analysis was used to gain insight into the flow physics responsible for the interference effect. It was conducted with unsteady Reynolds-averaged Navier–Stokes simulations, where the MR using the actuator disk approach and the TR blade rotation was modeled via a sliding mesh method. Results indicated that the MR disc vortex has a remarkable interference effect on the TR aerodynamic performance characteristic and that the effect is sensitive to the wind speed, wind direction, and MR disc loading. The observed yaw instability is considered to be related to the lesser inflow introduced by the MR disc vortex due to the change in the relative position of the disc vortex filament and TR with the wind azimuth. The increase in TR thrust at moderate wind speeds is due to the increase in leading edge dynamic pressure caused by the opposite swirl direction of the disc vortex contrasted to the TR. The MR disc loading affects the TR thrust due to the change of disc vortex strength and position.

Dynamics, properties and spectrum of reconnecting vortex loops in superfluid helium (Review article)

The quantum turbulence is a collection of the interacting quantum vortex loops and filaments. The main goal of this work is to systematize the information on the current state of the research on the reconnecting quantized vortex loops in the superfluid helium: the geometry, the dynamics, the properties of the vortex loops, the energy spectrum before and after the reconnections. The paper discusses the possible role of the reconnections in the formation of the turbulent spectrum. In addition, in this paper the main methods and approaches to the study of the reconnecting vortex loops and quantum turbulence are discussed.

Numerical investigation of the effect of towers and nacelles on the near wake of a horizontal-axis wind turbine model

Near wake flow has a great effect on the technology of horizontal-axis wind turbines. In this paper, the effect of the nacelle and tower on the near wake of a small-scale horizontal-axis wind turbine is studied by a numerical method. A large eddy simulation is used to obtain the time-varying wake flow. There are tip vortices, hub vortices, and blade and tower shedding vortices active in the near wake. Dynamic mode decomposition (DMD) is employed to analyse the features of these unsteady vorticities in both an absolute frame and a frame static relative to the rotor. The DMD results in the absolute frame illustrate the downstream motion of the tip vortices. In the relative frame, the results indicate that the disturbances distort the helical vortex filament and finally lead to a breakdown of the tip vortices in the case without a nacelle and tower. However, considering the effect of the nacelle and tower, the shedding vortices of the nacelle and tower directly break the tip vortices. The interaction between the tip vortices and the tower shedding vortices not only dominates the breakdown of the tip vortices but also promotes the instability of the tip vortices. This condition leads to an earlier wake recovery and influences the energy extraction and fatigue loads of the downstream turbines.

Validation of a free vortex filament wake module for the integrated simulation of multi-rotor wind turbines

Multi wind tUrbine Simulation Tool (MUST) is a new CENER's in-house code, based on NREL's OpenFAST code, designed to allow the aero-hydro-servo-elastic modelling of novel multi-rotor wind turbine concepts. MUST also includes a new aerodynamic module, called AeroVIEW (Aerodynamic Vortex fIlamEnt Wake) based on an implementation of the Free Vortex filament Method (FVM) combined with an unsteady Lifting Line (LL). This model allows to represent the influence existing between all the rotors, in each one of them. In this article, a validation of AeroVIEW performance for single and multi-rotor configurations is presented. First, a study of the wake properties (wake geometry, core radius and circulation of tip vortex filament) obtained with AeroVIEW has been successfully compared with several MexNext III cases. Then, an analysis of the rotor wakes interaction, in different arrays of wind turbines, has been made, obtaining power and thrust increases, due to this rotor-wake interaction, for different conditions. The increases obtained in power are very similar to results of high fidelity tools found in the literature. Moreover, this beneficial effect on the power generated, has a counterpart in the average thrust, whose increase is around half of the power production increase.

Turbulent energy cascade associated with viscous reconnection of two vortex rings

Collision of two vortex rings (VR) initially arranged in axis-offset and orthogonal configurations at Reynolds numbers (ReΓ) in the range of 5000–200 000 was simulated to investigate turbulent energy cascade associated with their reconnection. Two elliptical VRs are generated by joining each part of the first VR with another part of the second VR for the axis-offset collision, while two VRs associate to form a double U-shaped vortex, and this vortex reconnects itself at two points to form three elliptical VRs linked by the vortex filaments for the orthogonal collision. Many vortex structures in various scales and shapes, including small-scale VRs and horseshoe vortices, are observed in connection regions for both cases. As ReΓ increases, the energy of formed small vortices raises and their wavenumber (k) range enlarges. The flow energy spectrum approaches a k−5/3 slope of the Kolmogorov hypotheses at low wavenumbers. For the axis-offset collision, the energy spectrum at medium wavenumbers continuously changes from k−3.0 at ReΓ= 5000 to k−1.8 at ReΓ= 200 000, and the exponent (α) of the wavenumber is determined by a function as α=0.3304 ln(ReΓ)−5.6538. Meanwhile, the energy spectrum at two medium-wavenumber subranges for the orthogonal collision with ReΓ≥ 20 000 approaches the slopes of k−3.0 and k−2.6. Turbulent mixing performance due to the axis-offset collision of two vortex rings is better than that with the orthogonal one.

On the energy spectrum of the 3D velocity field, generated by an ensemble of vortex loops

The energy spectra of a three dimensional velocity field, induced by a set of vortex loops with various configurations are reviewed. This problem is closely related to the actual question of whether a chaotic set of vortex filaments can reproduce the real hydrodynamic turbulence. In the paper we discuss several cases that allow evaluating spectra in an exact form. The research was made for an ensemble of vortex rings of different sizes as well as for vortex loops with fractal Hausdorf dimension equal to 5/3, which corresponds to Flory’s vortex model, the so-called self-avoid lines. The results obtained are discussed.

Clebsch gauge fluid

We propose a novel gauge fluid solver based on Clebsch wave functions to solve incompressible fluid equations. Our method combines the expressive power of Clebsch wave functions to represent coherent vortical structures and the generality of gauge methods to accommodate a broad array of fluid phenomena. By evolving a transformed wave function as the system's gauge variable enhanced by an additional projection step to enforce pressure jumps on the free boundaries, our method can significantly improve the vorticity generation and preservation ability for a broad range of gaseous and liquid phenomena. Our approach can be easily implemented by modifying a standard grid-based fluid simulator. It can be used to solve various fluid dynamics, including complex vortex filament dynamics, fluids with different obstacles, and surface-tension flow.

Magnetic helicity and normal electromagnetic vortex filament flows under the influence of Lorentz force in MHD

In this paper, we attempt to introduce a new approach to determine the magnetic helicity and energy dissipation of the normal electromagnetic vortex filaments in magnetohydrodynamics (MHD). We successfully compute the vector potential and magnetic helicity of Frenet–Serret vectors and associated electromagnetic fields in the existence of the Lorentz force. We further describe a set of differential equations relating the magnetic vector potential of the electromagnetic vector fields and the Godbillon–Vey helicity model. Then, we focus on the time evolution of the normal electromagnetic field lines in MHD to evaluate the evolution of the magnetic helicity and energy dissipation theoretically. As a result, we obtain analytical solutions and numerical simulations of the time evolution equations of both geometric quantities as well as the magnetic and electric field vectors.

Lagrangian three-dimensional transport and dispersion by submesoscale currents at an upper-ocean front

The three-dimensional transport pathways, the time and space scales of vertical transport, and the dispersion characteristics (single-, pair- and multi-particle statistics) of submesoscale currents at an upper-ocean front are investigated using material points (tracer particles) that advect with the local fluid velocity. Coherent vortex filaments and eddies, generated and sustained during the evolution of baroclinic instability, dominate submesoscale (0.1 - 10 km) dynamics. These coherent structures play a crucial role in the particle transport and dispersion, which we quantify here. Particles in the central frontal region organize into vertically inclined lobes with anticyclonic circulation, each associated with a coherent eddy with cyclonic rotation. Furthermore, the coherent filaments associated with heavy/light edges of the front transfer particles from the edges to the lobes by downwelling/upwelling. This flux of new particles into the central region causes the particles circulating in the lobes to adjust, which leads to slumping of the front. The particle motion in the vertical shows multiple time scales – a fast time scale with O(10m) vertical displacement within an hour and a slower near-inertial time scale, comparable to the intrinsic time scale of the growing instability. The fast O(1h) time scale is also seen in the decorrelation of vertical velocity and the tapering of the initial ballistic growth of relative particle dispersion. The overall slumping process is slower than what one might anticipate from the large magnitude of vertical velocity in the filaments since it requires a sustained correlation over time between the lateral and the vertical motion. By tracking clouds of particles, we show that their centers of mass downwell/upwell over 1−2 inertial time periods, after which an adjustment follows with a sub-inertial time scale. The shape change in clusters of four particles reveals filamentogenesis, i.e. deformation into thin, needle-like structures by the coherent, anisotropic motions of the submesoscale, which occurs as a rapid process that is complete within approximately an hour.

Cascades and reconnection in interacting vortex filaments

At high Reynolds number, the interaction between two vortex tubes leads to intense velocity gradients, which are at the heart of fluid turbulence. This vorticity amplification comes about through two different instability mechanisms of the initial vortex tubes, assumed anti-parallel and with a mirror plane of symmetry. At moderate Reynolds number, the tubes destabilize via a Crow instability, with the nonlinear development leading to strong flattening of the cores into thin sheets. These sheets then break down into filaments which can repeat the process. At higher Reynolds number, the instability proceeds via the elliptical instability, producing vortex tubes that are perpendicular to the original tube directions. In this work, we demonstrate that these same transition between Crow and Elliptical instability occurs at moderate Reynolds number when we vary the initial angle $\beta$ between two straight vortex tubes. We demonstrate that when the angle between the two tubes is close to $\pi/2$, the interaction between tubes leads to the formation of thin vortex sheets. The subsequent breakdown of these sheets involves a twisting of the paired sheets, followed by the appearance of a localized cloud of small scale vortex structures. At smaller values of the angle $\beta$ between the two tubes, the breakdown mechanism changes to an elliptic cascade-like mechanism. Whereas the interaction of two vortices depends on the initial condition, the rapid formation of fine-scales vortex structures appears to be a robust feature, possibly universal at very high Reynolds numbers.

Differential geometric aspects of nonlinear Schrödinger equation

The main scope of this paper is to examine the smoke ring (or vortex filament) equation which can be viewed as a dynamical system on the space curve in E³. The differential geometric properties the soliton surface accociated with Nonlinear Schrödinger (NLS) equation, which is called NLS surface or Hasimoto surface, are investigated by using Darboux frame. Moreover, Gaussian and mean curvature of Hasimoto surface are found in terms of Darboux curvatures k_{n}, k_{g} and τ_{g.}. Then, we give a different proof of that the s- parameter curves of NLS surface are the geodesics of the soliton surface. As applications we examine two NLS surfaces with Darboux Frame.

Local flow dynamics in the motion of slug bubbles in a flowing mini square channel

For a slug flow flowing in a square capillary, the surrounding liquid flows through a thin film between the slug bubble and the wall and the corners of the channel and generates an asymmetric two-phase flow field with significant spatial variations. The current literature mainly discusses the bulk features of such a flow field, while local phenomena in these flows are not well known yet. In this study, the local dynamics of the liquid film and corner flows in a flowing square channel with a width of 3 mm were investigated by experiments and numerical simulations. Instantaneous velocity fields in the liquid flow were measured at the channel's central plane using particle tracking velocimetry (PTV). The obtained data were utilized to calculate the axial variations of the film and corner velocities. Three-dimensional numerical simulations with adaptive mesh refinement of the interface were performed for a series of flow conditions. Comparing the film velocity obtained from PTV and simulations demonstrates an overall good agreement between the results, with discrepancies close to the solid walls, where the PTV data is sporadic. Film and corner flows show rapid velocity changes adjacent to the minimum film area. Both experiments and simulations elucidate the backflow phenomena in the analyzed flow fields, where the liquid locally flows in the opposite streamwise direction. The backflow region shrinks as the liquid flow rate and speed of the bubble increase. Film flow shows significant pressure drops as it passes through the minimum film area. On the contrary, the corner flow does not demonstrate any rapid pressure fluctuations. The vortex structures’ identification revealed dominant vortex loops inside the gas and small-scale vortex filaments along the channel walls, generated from the liquid film rupture.

Dynamics of nearly parallel vortex filaments for the Gross–Pitaevskii equation

Klein et al. (J Fluid Mech 288:201–248, 1995) have formally derived a simplified asymptotic motion law for the evolution of nearly parallel vortex filaments in the context of the three dimensional Euler equation for incompressible fluids. In the present work, we rigorously derive the corresponding asymptotic motion law in the context of the Gross–Pitaevskii equation.

On the Hausdorff dimension of Riemann’s non-differentiable function

Recent findings show that the classical Riemann's non-differentiable function has a physical and geometric nature as the irregular trajectory of a polygonal vortex filament driven by the binormal flow. In this article, we give an upper estimate of its Hausdorff dimension. We also adapt this result to the multifractal setting. To prove these results, we recalculate the asymptotic behavior of Riemann's function around rationals from a novel perspective, underlining its connections with the Talbot effect and Gauss sums, with the hope that it is useful to give a lower bound of its dimension and to answer further geometric questions.