Thin Vortex Filament Research Articles

On a vortex filament with the axial velocity

Vortex filament, as one of the fundamental elements underlying the dynamics in high-Reynolds-number flows, has garnered researchers’ attention. Motion of the nonlinear waves on a thin vortex filament with axial velocity, embedded in an inviscid incompressible fluid, is investigated in this paper. Due to the second-order effect (SOE), the curvature and torsion of the vortex filament are governed by a disturbed Fukumoto–Miyazaki equation. Bilinear forms and nonlinear waves including the bright, multipole and double-pole solitons, are constructed. Solutions for the disturbed Fukumoto–Miyazaki equation present that: The SOE does not change the paths and shapes of the solitons on the vortex filament, while it changes the velocities of those solitons; Multipole solitons propagate around the axis periodically and their shapes change periodically, while when the SOE approaches to 0, the period of the shape change approaches to infinity, and the multipole solitons can not exist; Dipole-like solitons exist unstable and they will degenerate into two solitons because of the SOE; Breathers can also exist on the helicoidal vortex filaments, and the SOE changes the velocities of the breathers, while the amplitudes of the breathers are not affected by the SOE; Rogue waves appear suddenly on the filaments and their amplitudes reach the peaks and troughs simultaneously; Because of the SOE, the appearances of the rogue waves are accompanied by the appearances of the slight shocks.

Joukowski’s wake model for tip-splitting rotors

The interaction between a tip vortex and a solid surface is responsible for premature structural component fatigue in wind turbines and undesirable noise in helicopter rotors during low speed and descending flight. One noise reduction strategy uses a modified airfoil to split and spread the vorticity in two tip vortices. The present paper aims to provide the wake structure produced by such a rotor for wind turbine and helicopter regimes. We use a filamentary approach, such that vortices are assumed to roll-up quickly to form thin vortex filaments of finite but small size and compute the induced velocity using a cut-off method. The structure of the wake is analyzed in the near- and far-fields separately. It is found to have a dual nature and to be well-described by a twisted vortex pair locally aligned along with a larger helical structure. The linear stability of the far-wake with respect to long-wave displacements is also analyzed. Two kinds of instability modes are obtained associated with a pairing between successive turns of the large helical structure and a pairing between successive turns of the vortex pair.

Intermittency of Velocity Circulation in Quantum Turbulence

The velocity circulation, a measure of the rotation of a fluid within a closed path, is a fundamental observable in classical and quantum flows. It is indeed a Lagrangian invariant in inviscid classical fluids. In quantum flows, circulation is quantized, taking discrete values that are directly related to the number and the orientation of thin vortex filaments enclosed by the path. By varying the size of such closed loop, the circulation provides a measure of the dependence of the flow structure on the considered scale. Here we consider the scale dependence of circulation statistics in quantum turbulence, using high resolution direct numerical simulations of a generalized Gross-Pitaevskii model. Results are compared to the circulation statistics obtained from simulations of the incompressible Navier-Stokes equations. When the integration path is smaller than the mean inter-vortex distance, the statistics of circulation in quantum turbulence displays extreme intermittent behavior due to the quantization of circulation, in stark contrast with the viscous scales of classical flows. In contrast, at larger scales, circulation moments display striking similarities with the statistics probed in the inertial range of classical turbulence. This includes the emergence of the power law scalings predicted from Kolmogorov's 1941 theory, as well as intermittency deviations that closely follow the recently proposed bifractal model for circulation moments in classical flows. To date, this is the most convincing evidence of intermittency in the large scales of quantum turbulence. Moreover, our results strongly reinforce the resemblance between classical and quantum turbulence, highlighting the universality of inertial range dynamics, including intermittency, across these two a priori very different systems.

The submesoscale, the finescale and their interaction at a mixed layer front

The spindown of a geostrophically balanced density front in an upper-ocean mixed layer is simulated with a large eddy simulation (LES) model that resolves O(1000) m down to O(1) m scale. Our goal is to examine the interaction between the submesoscale and the turbulent finescale, and another related goal is to use the turbulence-resolving simulation to better characterize vertical transport, frontogenesis and dissipative processes. The flow passes through symmetric and baroclinic instabilities, spawns vortex filaments of O (100) m thickness as well as larger eddies with cross-front velocity as large as the along-front velocity, and develops turbulence that is spatially localized and organized. A O(100) m physical-space filter is applied to the simulated flow so that the coherent submesoscale is separated from the finescale in a decomposition that preserves the spatial organization of the flow unlike the typical practice of a split into a frontal average and a fluctuation that obscures the coherent submesoscale. The energy spectrum exhibits a change of slope at O(100) m with an approximately −5/3 slope over a subrange of the finescale. Analysis of the submesoscale vertical velocity (as large as 5mm/s) in the upper layer of the front reveals that downwelling is limited to the thin vortex filaments and the junction of the submesoscale eddies with these filaments while upwelling occurs over spatially extensive regions in the eddies. Conditional averaging shows that heavier (lighter) fluid is preferentially downwelled (upwelled) by these coherent submesoscale structures leading to an overall buoyancy flux that is restratifying. The submesoscale is unbalanced with local Rossby number as large as 5. The kinetic energy (KE) transport equations are evaluated separately for the submesoscale and the finescale to understand energy pathways in this problem. The buoyancy flux (associated with coherent motions) transfers the potential energy of the front and acts as the primary source of submesoscale KE which is then transported across the front with a fraction transferred to the finescale. The transfer, limited to thin regions of O(100) m horizontal width, is accomplished by primarily horizontal strain in the upper 10 m and by vertical shear in the rest of the 50-m deep mixed layer. Frontogenetic mechanisms are diagnosed through analysis of the transport equation for squared buoyancy gradient. Horizontal strain is the primary frontogenetic term that is especially strong in the near-surface layer. The frontogenesis is counteracted primarily by horizontal diffusion in the top 10 m while, further below, the balance is with the horizontal gradient of vertical velocity.

Tomo-PIV measurements in the wake of a tethered sphere undergoing VIV

The 3D flow field in the wake of a tethered sphere ( D=6 mm, m∗=7.77, L∗=4.1) undergoing vortex induced vibrations (VIV) in a water tunnel, was measured using tomographic particle image velocimetry (tomo-PIV). The flow field was investigated for four different reduced velocities ranging between 3.6<U∗<13.2, corresponding to Reynolds numbers ranging between 382< ReD<1,392. At U∗=3.6, the sphere did not oscillate and shed vortices resembled a train of “hairpin” heads having a vertical plane of symmetry. At U∗=5.8, sphere oscillations were highly periodic and “omega”-shaped vortices having a horizontal plane of symmetry were alternately shed, and transformed into vortex rings further downstream while still connected to upstream structures by a thin vortex filament. At U∗=7.9 and 13.2, the sphere’s transverse amplitude response was intermittent and vortices in the sphere’s wake were characterized by longitudinal and vertical “struts” as well as “arches”, with seemingly little organization.

A Novel Model of Quasi-Stationary Vortices in the Earth’s Atmosphere

In the great variety of vortex motions in the atmosphere, concentrated vortices, attracting increased interest from the point of view of both fundamental research and practice, clearly stick out. A sufficiently precise definition of the concentrated vortex can be given for the case of an ideal fluid—it is an area localized in the space, surrounded by a potential flow and having a nonzero vorticity. Such vortices can be combined into a class of small-scale concentrated vortices including dust devils (DDs), waterspouts, fire vortices, and larger scale and more intense tornadoes. Unlike planetary-scale vortices (cyclones and anticyclones), DDs and tornadoes are small-scale vortices. DDs and tornadoes are generated in different environments (tornadoes occur in strong storm clouds), but they have much in common in regards to their structure. The speed of rotation in such vortices reaches the maximum value at a characteristic radius and tends to zero when approaching the center. The rotation speed in them has much in common with the rotation speed in stationary Rankine or Burgers vortices. This work is devoted to the study of a novel low-parameter model of stationary vortices. The model is most suitable for describing concentrated vortices in the Earth’s atmosphere. Within the framework of ideal hydrodynamics, a new model of thin vortex filaments is constructed at heights that are small when compared to the vertical scale of the Earth’s atmosphere. Unlike Rankine and Burgers vortices, it allows one to describe the structure limited in the radial direction. Quasi-stationary vortices in such a model arise as a result of the balance of two effects: the concentration of vertical vorticity to the center and the advection of the vortex motion in the vertical direction.

Damped solitary waves on a superfluid vortex filament

We have investigated the evolution of bright solitary wave (solitons) on a thin vortex filament under the action of time dependent damping in a superfluid at the zero temperature limit by exactly solving the damped-nonlinear Schrodinger equation (dNLSE). It is found that the dissipation of the wave amplitude depends on the functional form of the time dependency of the damping force. The damping rate is not constant but varies significantly with time. Comparison has been made between the characteristics of amplitude decay due a constant damping and a particular form of time dependent damping factor, both qualitatively and quantitatively. Stability of the solitary wave has also been studied in presence of such dissipation.

The motion of a buoyant vortex filament

We investigate the motion of a thin vortex filament in the presence of buoyancy. The asymptotic model of Moore &amp;amp; Saffman (Phil. Trans. R. Soc. Lond.A, vol. 272, 1972, pp. 403–429) is extended to take account of buoyancy forces in the force balance on a vortex element. The motion of a buoyant vortex is given by the transverse component of force balance, while the tangential component governs the dynamics of the structure in the core. We show that the local acceleration of axial flow is generated by the external pressure gradient due to gravity. The equations are then solved for vortex rings. An analytic solution for a buoyant vortex ring at a small initial inclination is obtained and asymptotically agrees with the literature.

Inhomogeneous Heisenberg spin chain and quantum vortex filament as non-holonomically deformed NLS systems

Through the Hasimoto map, various dynamical systems can be mapped to different integrodifferential generalizations of Nonlinear Schrodinger (NLS) family of equations some of which are known to be integrable. Two such continuum limits, corresponding to the inhomogeneous XXX Heisenberg spin chain [Balakrishnan, J. Phys. C 15, L1305 (1982)] and that of a thin vortex filament moving in a superfluid with drag [Shivamoggi, Eur. Phys. J. B 86, 275 (2013) 86; Van Gorder, Phys. Rev. E 91, 053201 (2015)], are shown to be particular non-holonomic deformations (NHDs) of the standard NLS system involving generalized parameterizations. Crucially, such NHDs of the NLS system are restricted to specific spectral orders that exactly complements NHDs of the original physical systems. The specific non-holonomic constraints associated with these integrodifferential generalizations additionally posses distinct semi-classical signature.

Self-similar vortex filament motion under the non-local Biot–Savart model

One type of thin vortex filament structure that has attracted interest in recent years is that which obeys self-similar scaling. Among various applications, these filaments have been used to model the motion of quantized vortex filaments in superfluid helium after reconnection events. While similarity solutions have been described analytically and numerically using the local induction approximation (LIA), they have not been studied (or even shown to exist) under the non-local Biot–Savart model. In this present paper, we show not only that self-similar vortex filament solutions exist for the non-local Biot–Savart model, but that such solutions are qualitatively similar to their LIA counterparts. This suggests that the various LIA solutions found previously should be valid physically (at least in the small amplitude regime), since they agree well with the more accurate Biot–Savart model.

An acentric rotation of two helical vortices of the same circulations

The aim of this paper is to test the possibility of a secondary solution of the acentric rotation of helical vortex pairs with the same pitch, sign and strength. The investigation addresses the three-dimensional vortex dynamics of thin vortex filaments. As a result of the current investigation, this secondary solution with acentric vortex positions in the helical pairs is found. This fact was not discussed in previous studies, and the existence of the new equilibrium solution for the helical vortex pairs is an original result.

Quantum Hasimoto transformation and nonlinear waves on a superfluid vortex filament under the quantum local induction approximation.

The Hasimoto transformation between the classical LIA (local induction approximation, a model approximating the motion of a thin vortex filament) and the nonlinear Schrödinger equation (NLS) has proven very useful in the past, since it allows one to construct new solutions to the LIA once a solution to the NLS is known. In the present paper, the quantum form of the LIA (which includes mutual friction effects) is put into correspondence with a type of complex nonlinear dispersive partial differential equation (PDE) with cubic nonlinearity (similar in form to a Ginsburg-Landau equation, with additional nonlinear terms). Transforming the quantum LIA in such a way enables one to obtain quantum vortex filament solutions once solutions to this dispersive PDE are known. From our quantum Hasimoto transformation, we determine the form and behavior of Stokes waves, a standing one-soliton, traveling waves, and similarity solutions under normal and binormal friction effects. The quantum Hasimoto transformation is useful when normal fluid velocity is relatively weak, so for the case where the normal fluid velocity is dominant we resort to other approaches. We exhibit a number of solutions that exist only in the presence of the normal fluid velocity and mutual friction terms (which would therefore not exist in the limit taken to obtain the classical LIA, decaying into line filaments under such a limit), examples of which include normal fluid driven helices, stationary and propagating topological solitons, and a vortex ring whose radius varies inversely with the normal fluid magnitude. We show that, while chaos may not be impossible under the quantum LIA, it should not be expected to arise from traveling waves along quantum vortex filaments under the quantum LIA formulation.

Helical vortex filament motion under the non-local Biot–Savart model

AbstractThe thin helical vortex filament is one of the fundamental exact solutions possible under the local induction approximation (LIA). The LIA is itself an approximation to the non-local Biot–Savart dynamics governing the self-induced motion of a vortex filament, and helical filaments have also been considered for the Biot–Savart dynamics, under a variety of configurations and assumptions. We study the motion of such a helical filament in the Cartesian reference frame by determining the curve defining this filament mathematically from the Biot–Savart model. In order to do so, we consider a matched approximation to the Biot–Savart dynamics, with local effects approximated by the LIA in order to avoid the logarithmic singularity inherent in the Biot–Savart formulation. This, in turn, allows us to determine the rotational and translational velocity of the filament in terms of a local contribution (which is exactly that which is found under the LIA) and a non-local contribution, each of which depends on the wavenumber, $k$, and the helix diameter, $A$. Performing our calculations in such a way, we can easily compare our results to those of the LIA. For small $k$, the transverse velocity scales as $k^{2}$, while for large $k$, the transverse velocity scales as $k$. On the other hand, the rotational velocity attains a maximum value at some finite $k$, which corresponds to the wavenumber giving the maximal torsion.

Ideal hydrodynamics outside and inside a black hole: Hamiltonian description in Painlevé-Gullstrand coordinates

We show that when the Painlevé-Gullstrand coordinates are used in their Cartesian version, the Hamiltonian of relativistic ideal hydrodynamics in the vicinity of a nonrotating black hole differs by only one simple term from the corresponding Hamiltonian in a flat spacetime. The interior region of the black hole is also described in a unified way, because there is no singularity on the event horizon in Painlevé-Gullstrand coordinates. We present the exact solution describing the steady accretion of extremely hard matter (ɛ ∝ n 2) onto a moving black hole up to the central singularity. In the local induction approximation, we derive the equation of motion for a thin vortex filament against the background of such an accretion flow. We explicitly calculate the Hamiltonian for a fluid with an ultrarelativistic equation of state, ɛ ∝ n 4/3, and solve the problem of a centrally symmetric steady flow of such matter.

Stability of small-amplitude torus knot solutions of the localized induction approximation

We study the linear stability of small-amplitude torus knot solutions of the localized induction approximation equation for the motion of a thin vortex filament in an ideal fluid. Such solutions can be constructed analytically through the connection with the focusing nonlinear Schrödinger equation using the method of isoperiodic deformations. We show that these (p, q) torus knots are generically linearly unstable for p < q, while we provide examples of neutrally stable (p, q) torus knots with p > q, in contrast with an earlier linear stability study by Ricca (1993 Chaos 3 83–95; 1995 Chaos 5 346; 1995 Small-scale Structures in Three-dimensional Hydro and Magneto-dynamics Turbulence (Lecture Notes in Physics vol 462) (Berlin: Springer)). We also provide an interpretation of the original perturbative calculation in Ricca (1995), and an explanation of the numerical experiments performed by Ricca et al (1999 J. Fluid Mech. 391 29–44), in light of our results.

On Two Types of Localized Induction Equation

We are looking for motions of the strings and the filaments where they show stretch and/or shrink. For the motion of the extensible string interacting with the external field the string changes its shape by stretch not to valance the elastic force with the external force. We also considered the motion of the stretched thin vortex filament and found exact solutions of the stretched vortex filament. The thin vortex filament is one of the fundamental modes in the fluid and is described by the localized induction equation (LIE). In this short note we would like to discuss two types of LIE for the vortex filament with and without stretch from a standpoint of similarity and difference. For this purpose we review how to derive LIE. The equations of motion for the incompressible and rotational inviscid fluid are given by

Differential approximation for Kelvin wave turbulence

I present a nonlinear differential equation model (DAM) for the spectrum of Kelvin waves on a thin vortex filament. This model preserves the original scaling of the six-wave kinetic equation, its direct and inverse cascade solutions, as well as the thermodynamic equilibrium spectra. Further, I extend DAM to include the effect of sound radiation by Kelvin waves. I show that, because of the phonon radiation, the turbulence spectrum ends at a maximum frequency of ω* ∼ (ε3 c 20s /κ16)1/13, where ε is the total energy injection rate, c s is the speed of sound, and κ is the quantum of circulation.

Why the Phase Shifts for Solitons on a Vortex Filament Are So Large: A Theoretical Explanation

The phase "jumps" for solitons interacting on a vortex filament, observed in experiments, have been unaccounted for for more than 20 years. Using explicit formulas describing the interaction of two solitons on a thin vortex filament in the localized induction approximation, we show that an appropriate choice of the parameters of the solitons leads to large phase shifts. This result does not depend on the axial flow along the filament.

Turbulent dissipation near absolute zero

The superfluid component of liquid helium has exactly zero viscosity and its rotational motion is constrained by quantum mechanics to very thin vortex filaments of the same circulation. When liquid helium is made turbulent the vortex filaments form a disordered tangle. This article critically reviews recent experiments and calculations to explain the surprising observation that near absolute zero, in the absence of viscous effects, turbulent kinetic energy decays, and to highlight similarities and differences between superfluid turbulence and ordinary classical turbulence.

Applicability of the hodograph method for the problem of long-scale nonlinear dynamics of a thin vortex filament near a flat boundary

Hamiltonian dynamics of a thin vortex filament in an ideal incompressible fluid near a flat fixed boundary is considered under the conditions that at any point of the curve, determining the shape of the filament, the angle between tangent vector and the boundary plane is small. Also the distance from a point on the curve to the plane is small in comparison with the curvature radius. The dynamics is shown to be effectively described by a nonlinear system of two (1+1)-dimensional partial differential equations. The hodograph transformation reduces this system to a single linear differential equation of the second order with separable variables. Simple solutions of the linear equation are investigated for real values of spectral parameter lambda, when the filament projection on the boundary plane has shape of a two-branch spiral or a smoothed angle, depending on the sign of lambda.