Motion Of Vortex Filament Research Articles

Slender vortex filaments in the Boussinesq approximation

A model for the motion of slender vortex filaments is extended to include the effect of gravity. The model, initially introduced by Callegari and Ting [“Motion of a curved vortex filament with decaying vortical core and axial velocity,” SIAM J. Appl. Math. 35, 148–175 (1978)], is based on a matched asymptotic expansion in which the outer solution, given by the Biot–Savart law, is matched with the inner solution derived from the Navier–Stokes equations. Building on recent work by Harikrishnan et al. [“On the motion of hairpin filaments in the atmospheric boundary layer,” Phys. Fluids 35, 076603 (2023)], the Boussinesq approximation is applied such that the density variations only enter in the gravity term. However, unlike Harikrishnan et al. [“On the motion of hairpin filaments in the atmospheric boundary layer,” Phys. Fluids 35, 076603 (2023)], the density variation enters at a lower order in the asymptotic expansion and, thus, has a more significant impact on the self-induced velocity of the vortex filament. In this regime, which corresponds to the regime studied by Chang and Smith [“The motion of a buoyant vortex filament,” J. Fluid Mech. 857, R1 (2018)], the effect of gravity is given by an alteration of the core constant, which couples the motion of the filament to the motion within the vortical core, in addition to a change in the compatibility conditions (evolution equations), which determine the leading order azimuthal and tangential velocity fields in the vortex core. The results are used to explain certain properties of buoyant vortex rings, as well as qualitatively explore the impact of gravity on tornado-type atmospheric vortices.

Optical modeling for recursional Hasimoto map with normalized time fractional applications

In this paper, we construct the optical flow equations obtained with the help of a timelike curve in De Sitter 2D space S12 and with the aid of the Hasimoto map defined by this curve. First, we introduce the Hasimoto map, which we call the De-Hasimoto map, for a timelike curve in S12. Then, with the help of this map and the equation representing the vortex filament motion, we obtain some flow equations with time and arc length parameters. Finally, we compute exact solutions of time fraction of an obtained optical motion equation, and we present some physical reviews on it.

Self-similar vortex reconnection

As shown by Crow in 1970, the evolution of two almost parallel vortex filaments with opposite circulation exhibits a long-wave instability. Ultimately, the symmetric mode increases its amplitude reconnecting both filaments and ending into the formation of an almost periodic structure of vortex rings. This is a universal process, which appears in a wide range of scales: from the vortex trails behind an airplane to a microscopic scale of superfluids and Bose–Einstein condensates. In this paper, I will focus on the vortex reconnection for the latter case by employing Gross–Pitaevskii theory. Essentially, I focus on the well-known laws of interaction and motion of vortex filaments. By means of numerical simulations, as well as theoretically, I show that a self-similar finite-time dynamics manifests near the reconnection time. A self-similar profile is selected showing excellent agreement with numerical simulations.

Stochastic motion of vortex filaments in He II under random force

Langevin dynamics are applied to describe the stochastic motion of vortex filaments in He II under random force. The article describes a functional formalism, which is a modification of the method developed earlier by Migdal to deal with the stochastic dynamics of classical vortex filaments. In particular, starting with the Langevin-type equation, the functional Fokker–Planck equation for the characteristic functional was obtained. Based on this equation, and under the assumption that the random force correlator satisfies the fluctuation-dissipation theorem, thermodynamic equilibrium in a system of chaotic quantized vortices was investigated. Additionally, the case of stationary helium and the case of counterflow with the constant relative velocity of the normal and superfluid components were considered. Some physical consequences of the results obtained are also discussed.

Dynamics of Vortices in Chiral Media: The Chiral Propulsion Effect.

We study the motion of vortex filaments in chiral media and find a semiclassical analog of the anomaly-induced chiral magnetic effect. The helical solitonic excitations on vortices in a parity-breaking medium are found to carry additional energy flow along the vortex in the direction dictated by the sign of chirality imbalance; we call this new transport phenomenon the chiral propulsion effect. The dynamics of vortex filaments in the parity-breaking background is described by a modified version of the localized induction equation. We analyze the linear stability of simple vortex solutions and study the effects of chiral media on the excitation spectrum and the growth rate of the unstable modes. We also show that, if the equation of motion of the filament is symmetric under the simultaneous reversal of parity and time, planar-shape solutions cannot transport energy.

Soliton curves in three-dimensional space forms

The localized induction hierarchy is the integrable hierarchy for the localized induction equation, a geometric evolution equation for space curves modeled on the motion of vortex filaments. A solution of the n th stationary equation associated to the localized induction hierarchy is called an n th soliton curve. In this paper, we consider soliton curves in simply-connected three-dimensional space forms. In particular, we give explicit expressions of n th soliton curves in terms of the natural curvatures.

Stability of closed solutions to the vortex filament equation hierarchy with application to the Hirota equation

The vortex filament equation (VFE) is part of an integrable hierarchy of filament equations. Several equations in this hierarchy have been derived to describe vortex filament motion in various situations. Inspired by these results, we develop a general framework for studying the existence and the linear stability of closed solutions of the VFE hierarchy. The framework is based on the correspondence between the VFE and the nonlinear Schrödinger hierarchies. Our results establish a connection between the AKNS Floquet spectrum and the stability properties of the solutions of the filament equations. We apply our machinery to solutions of the filament equation associated to the Hirota equation. We also discuss how our framework applies to soliton solutions.

Bilge-Keel Influence on Free Decay of Roll Motion of a Realistic Hull1

The prediction of roll motion of a ship with bilge keels is particularly difficult because of the nonlinear characteristics of the viscous roll damping. Flow separation and vortex shedding caused by bilge keels significantly affect the roll damping and hence the magnitude of the roll response. To predict the ship motion, the Slender-Ship Free-Surface Random-Vortex Method (SSFSRVM) was employed. It is a fast discrete-vortex free-surface viscous-flow solver developed to run on a standard desktop computer. It features a quasi-three-dimensional formulation that allows the decomposition of the three-dimensional ship-hull problem into a series of two-dimensional computational planes, in which the two-dimensional free-surface Navier–Stokes solver Free-Surface Random-Vortex Method (FSRVM) can be applied. In this paper, the effectiveness of SSFSRVM modeling is examined by comparing the time histories of free roll-decay motion resulting from simulations and from experimental measurements. Furthermore, the detailed two-dimensional vorticity distribution near a bilge keel obtained from the numerical model will also be compared with the existing experimental Digital Particle Image Velocimetry (DPIV) images. Next, we will report, based on the time-domain simulation of the coupled hull and fluid motion, how the roll-decay coefficients and the flow field are altered by the span of the bilge keels. Plots of vorticity contour and vorticity isosurface along the three-dimensional hull will be presented to reveal the motion of fluid particles and vortex filaments near the keels.

Three-dimensional simulation of nonlinear dynamics of domain walls in films with perpendicular anisotropy

A three-dimensional computer simulation of static magnetization configurations and dynamic processes occurring in a domain wall moving in a uniaxial magnetic film with perpendicular anisotropy has been performed based on the numerical solution of Landau–Lifshitz–Gilbert equations. The calculated static states correspond to a domain wall containing Bloch lines with a surface magnetization distribution that depends on the thickness of the film. It has been shown that these structures can be characterized by particular values of the homotopy index. It has been found that the vortex and antivortex structures existing in the bulk of the film form vortex filaments. A method has been proposed for visualization of the joint motion of vortex filaments and Bloch points, which is based on the numerical calculation of the homotopy index and the winding number.

Elastohydrodynamic Synchronization of Adjacent Beating Flagella.

It is now well established that nearby beating pairs of eukaryotic flagella or cilia typically synchronize in phase. A substantial body of evidence supports the hypothesis that hydrodynamic coupling between the active filaments, combined with waveform compliance, provides a robust mechanism for synchrony. This elastohydrodynamic mechanism has been incorporated into 'bead-spring' models in which the beating flagella are represented by microspheres tethered by radial springs as they are driven about orbits by internal forces. While these low-dimensional models reproduce the phenomenon of synchrony, their parameters are not readily relatable to those of the filaments they represent. More realistic models which reflect the underlying elasticity of the axonemes and the active force generation, take the form of fourth-order nonlinear PDEs. While computational studies have shown the occurrence of synchrony, the effects of hydrodynamic coupling between nearby filaments governed by such continuum models have been theoretically examined only in the regime of interflagellar distances d large compared to flagellar length L. Yet, in many biological situations d/L ≪ 1. Here, we first present an asymptotic analysis of the hydrodynamic coupling between two extended filaments in the regime d/L ≪ 1, and find that the form of the coupling is independent of the microscopic details of the internal forces that govern the motion of the individual filaments. The analysis is analogous to that yielding the localized induction approximation for vortex filament motion, extended to the case of mutual induction. In order to understand how the elastohydrodynamic coupling mechanism leads to synchrony of extended objects, we introduce a heuristic model of flagellar beating. The model takes the form of a single fourth-order nonlinear PDE whose form is derived from symmetry considerations, the physics of elasticity, and the overdamped nature of the dynamics. Analytical and numerical studies of this model illustrate how synchrony between a pair of filaments is achieved through the asymptotic coupling.

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.

Dynamic modeling and analysis of vortex filament motion using a novel curve-fitting method

Applications of a novel curve-fitting technique are presented to efficiently predict the motion of the vortex filament, which is trailed from a rigid body such as wings and rotors. The governing equations of the motion, when a Lagrangian approach with the present curve-fitting method is applied, can be transformed into an easily solvable form of the system of nonlinear ordinary differential equations. The applicability of Bézier curves, B-spline, and Lagrange interpolating polynomials is investigated. Local Lagrange interpolating polynomials with a shift operator are proposed as the best selection for applications, since it provides superior system characteristics with minimum computing time, compared to other methods. In addition, the Gauss quadrature formula with local refinement strategy has been developed for an accurate prediction of the induced velocity computed with the line integration of the Biot–Savart law. Rotary-wing problems including a vortex ring problem are analyzed to show the efficiency, accuracy, and flexibility in the applications of the proposed method.

Non-local dynamics governing the self-induced motion of a planar vortex filament

While the Hasimoto planar vortex filament is one of the few exact solutions to the local induction approximation (LIA) approximating the self-induced motion of a vortex filament, it is natural to wonder whether such a vortex filament solution would exist for the non-local Biot-Savart dynamics exactly governing the filament motion, and if so, whether the non-local effects would drastically modify the solution properties. Both helical vortex filaments and vortex rings are known to exist under both the LIA and non-local Biot-Savart dynamics; however, the planar filament is a bit more complicated. In the present paper, we demonstrate that a planar vortex filament solution does exist for the non-local Biot-Savart formulation, provided that a specific non-linear integral equation (governing the spatial structure of such a filament) has a non-trivial solution. By using the Poincaré–Lindstedt method, we are able to obtain an accurate analytical approximation to the solution of this integral equation under physically reasonable assumptions. To obtain these solutions, we approximate local effects near the singularity of the integral equation using the LIA and non-local effects using the Biot-Savart formulation. Mathematically, the results constitute an analytical solution to an interesting nonlinear singular integro-differential equation in space and time variables. Physically, these results show that planar vortex filaments exist and maintain their forms under the non-local Biot-Savart formulation, as one would hope. Due to the regularization approach utilized, we are able to compare the structure of the planar filaments obtained under both LIA and Biot-Savart formulations in a rather straightforward manner, in order to determine the role of the non-locality on the structure of the planar filament.

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.

Motion of a helical vortex filament in superfluid 4He under the extrinsic form of the local induction approximation

Very recently, Shivamoggi [“Vortex motion in superfluid 4He: Reformulation in the extrinsic vortex-filament coordinate space,” Phys. Rev. B 84, 012506 (2011)]10.1103/PhysRevB.84.012506 studied the extrinsic form of the local induction approximation (LIA) for the motion of a Kelvin wave on a vortex filament in superfluid 4He, and obtained some results in a cubic approximation. Presently, we study the motion of helical vortex filaments in superfluid 4He under the exact fully nonlinear LIA considered in potential form by Van Gorder [“Fully nonlinear local induction equation describing the motion of a vortex filament in superfluid 4He,” J. Fluid Mech. 707, 585 (2012)]10.1017/jfm.2012.308 and obtained from the Biot-Savart law through the equations of Hall and Vinen [“The rotation of liquid helium II. I. Experiments on the propagation of second sound in uniformly rotating helium II,” Proc. R. Soc. London, Ser. A 238, 204 (1956)]10.1098/rspa.1956.0214 including superfluid friction terms. Nonlinear dispersion relations governing the helical Kelvin wave on such a vortex filament are derived in exact form, from which we may exactly calculate the phase and group velocity of the Kelvin wave. With this, we classify the motion of a helical Kelvin wave on a vortex filament under the LIA. The dispersion relations and results, which follow are exact in nature, in contrast to most results in the literature, which are usually numerical approximations. As such, our results accurately capture the qualitative behavior of the Kelvin waves under the LIA. Extensions to other frameworks are discussed.

Vortex motion in superfluid 4He: effects of normal fluid flow

The motion of a vortex filament in superfluid 4He is considered by using the Hall-Vinen-Bekarevich-Khalatnikov (HVBK) [H.E. Hall, W.F. Vinen, Proc. Roy. Soc. Lond. A 238, 204 (1956); H.E. Hall, W.F. Vinen, Proc. Roy. Soc. Lond. A 238, 215 (1956); I.L. Bekarevich, I.M. Khalatnikov, Sov. Phys. J. Exp. Theor. Phys. 13, 643 (1961)] phenomenological model for the scattering process between the vortex and thermal excitations in liquid 4He. The HVBK equations are analytically formulated first in the intrinsic geometric parameter space to obtain insights into the physical implications of the friction terms, associated with the friction coefficients α and α′ (in the Hall-Vinen notation) as well as the previous neglect of the friction term associated with the friction coefficient α′. The normal fluid velocity components both along and transverse to the vortex filament are included. This analytical development also serves to highlight the difficulties arising in making further progress on this route. A reformulation of the HVBK equation in the extrinsic vortex filament coordinate space is then given which is known [B.K. Shivamoggi, Phys. Rev. B 84, 012506 (2011)] to provide a useful alternative analytical approach in this regard. A nonlinear Schrodinger equation for the propagation of nonlinear Kelvin waves on a vortex filament in a superfluid is given taking into account the generalized normal fluid flow. The friction term associated with α′, even in the presence of the normal fluid velocity components transverse to the vortex filament, is shown to produce merely an algebraic growth of the Kelvin waves hence providing further justification for the neglect of this term. On the other hand, the instability produced by the friction term associated with α via the normal fluid velocity component along the vortex filament is shown to manifest itself as a parametric amplification on considering the problem of a rotating planar vortex filament in a superfluid.

Scaling laws and accurate small-amplitude stationary solution for the motion of a planar vortex filament in the Cartesian form of the local induction approximation

We provide a formulation of the local induction approximation (LIA) for the motion of a vortex filament in the Cartesian reference frame (the extrinsic coordinate system) which allows for scaling of the reference coordinate. For general monotone scalings of the reference coordinate, we derive an equation for the planar solution to the derivative nonlinear Schrödinger equation governing the LIA. We proceed to solve this equation perturbatively in small amplitude through an application of multiple-scales analysis, which allows for accurate computation of the period of the planar vortex filament. The perturbation result is shown to agree strongly with numerical simulations, and we also relate this solution back to the solution obtained in the arclength reference frame (the intrinsic coordinate system). Finally, we discuss nonmonotone coordinate scalings and their application for finding self-intersections of vortex filaments. These self-intersecting vortex filaments are likely unstable and collapse into other structures or dissipate completely.

A numerical study of vortex filament motion around an ellipse by a conformal mapping technique

The paper presents a numerical study of the vortex motion around an ellipse using the conformal mapping of the circle exterior on the ellipse exterior. The vortex motion velocity and the stream function were obtained taking into account Routh’s correction. The trajectories of the vortex filament motion were constructed by integrating the vortex velocity differential equation and according to the condition of the stream function equality to a constant magnitude.

Quasi‐Classical Approximation in Vortex Filament Dynamics. Integrable Systems, Gradient Catastrophe, and Flutter

Quasi‐classical approximation in the intrinsic description of the vortex filament dynamics is discussed. Within this approximation, the governing equations are given by elliptic system of quasi‐linear partial differential equations of the first order. Dispersionless Da Rios system and dispersionless Hirota equation are among them. They describe motion of vortex filament with slow‐varying curvature and torsion without or with axial flow. Gradient catastrophe for governing equations is studied. It is shown that geometrically this catastrophe manifests as a fast oscillation of a filament curve around the rectifying plane which resembles the flutter of airfoils. Analytically, it is the elliptic umbilic singularity in the terminology of the catastrophe theory. It is demonstrated that its double scaling regularization is governed by the Painlevé I equation.

On the stability of self-similar solutions of 1D cubic Schrödinger equations

In this paper we will study the stability properties of self-similar solutions of $$1$$ D cubic NLS equations with time-dependent coefficients of the form 0.1 $$\begin{aligned} \displaystyle { iu_t+u_{xx}+\frac{u}{2} \left(|u|^2-\frac{A}{t}\right)=0, \quad A\in \mathbb{R }. } \end{aligned}$$ The study of the stability of these self-similar solutions is related, through the Hasimoto transformation, to the stability of some singular vortex dynamics in the setting of the Localized Induction Equation (LIE), an equation modeling the self-induced motion of vortex filaments in ideal fluids and superfluids. We follow the approach used by Banica and Vega that is based on the so-called pseudo-conformal transformation, which reduces the problem to the construction of modified wave operators for solutions of the equation $$\begin{aligned} iv_t+ v_{xx} +\frac{v}{2t}(|v|^2-A)=0. \end{aligned}$$ As a by-product of our results we prove that Eq. (0.1) is well-posed in appropriate function spaces when the initial datum is given by $$u(0,x)= z_0 \mathrm p.v \frac{1}{x}$$ for some values of $$z_0\in \mathbb{C }\setminus \{ 0\}$$ , and $$A$$ is adequately chosen. This is in deep contrast with the case when the initial datum is the Dirac-delta distribution.