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Robert A. Van Gorder - One of the best experts on this subject based on the ideXlab platform.
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self similar Vortex Filament motion under the non local biot savart model
Journal of Fluid Mechanics, 2016Co-Authors: Robert A. Van GorderAbstract: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.
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the biot savart description of kelvin waves on a quantum Vortex Filament in the presence of mutual friction and a driving fluid
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2015Co-Authors: Robert A. Van GorderAbstract:We study the dynamics of Kelvin waves along a quantum Vortex Filament in the presence of mutual friction and a driving fluid while taking into account non-local effects due to Biot–Savart integrals. The Schwarz model reduces to a nonlinear and non-local dynamical system of dimension three, the solutions of which determine the translational and rotational motion of the Kelvin waves, as well as the amplification or decay of such waves. We determine the possible qualitative behaviours of the resulting Kelvin waves. It is well known from experimental and theoretical studies that the Donnelly–Glaberson instability plays a role on the amplification or decay of Kelvin waves in the presence of a driving normal fluid velocity, and we obtain the relevant stability criterion for the non-local model. While the stability criterion is the same for local and non-local models when the wavenumber is sufficiently small, we show that large differences emerge for the large wavenumber case (tightly coiled helices). The results demonstrate that non-local effects have a stabilizing effect on the Kelvin waves, and hence larger normal fluid velocities are required for amplification of large wavenumber Kelvin waves. Additional qualitative differences between the local and non-local models are explored.
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quantum hasimoto transformation and nonlinear waves on a superfluid Vortex Filament under the quantum local induction approximation
Physical Review E, 2015Co-Authors: Robert A. Van GorderAbstract:The Hasimoto transformation between the classical LIA (local induction approximation, a model approximating the motion of a thin Vortex Filament) and the nonlinear Schrodinger 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 and a standing 1-soliton solution under normal and binormal friction effects. The soliton solution on a quantum Vortex Filament is a natural generalization of the classical 1-soliton solution constructed mathematically by Hasimoto (which motivated subsequent real-world experiments). We also find that self-similar solutions are possible under this quantum Hasimoto transform, and these solutions demonstrate that even in the presence of dissipative effects, there can be strong local increases in curvature or torsion even though asymptotically the solutions are smoothed through dissipation. More general traveling wave solutions propagating along the quantum Filament with fixed wave speed are also studied, with such solutions holding the 1-soliton as a special case. 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 consider the dynamics of the tangent vector to the Vortex Filament directly from the quantum LIA, and this approach, while less elegant than the quantum Hasimoto transformation, enables us to study waves primarily driven by the normal fluid velocity. 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 can also view the dynamics of a traveling wave solution for the tangent vector in terms of a dynamical system on the unit sphere, and this formulation gives us another framework by which to obtain solutions. 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
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helical Vortex Filament motion under the non local biot savart model
Journal of Fluid Mechanics, 2015Co-Authors: Robert A. Van GorderAbstract:The 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, , and the helix diameter, . Performing our calculations in such a way, we can easily compare our results to those of the LIA. For small , the transverse velocity scales as , while for large , the transverse velocity scales as . On the other hand, the rotational velocity attains a maximum value at some finite , which corresponds to the wavenumber giving the maximal torsion.
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quantum hasimoto transformation and nonlinear waves on a superfluid Vortex Filament under the quantum local induction approximation
arXiv: Fluid Dynamics, 2014Co-Authors: Robert A. Van GorderAbstract:The Hasimoto transformation between the classical LIA (local induction approximation, a model approximating the motion of a thin Vortex Filament) and the nonlinear Schr\"odinger 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 and a standing 1-soliton solution under normal and binormal friction effects. The soliton solution on a quantum Vortex Filament is a natural generalization of the classical 1-soliton solution constructed mathematically by Hasimoto (which motivated subsequent real-world experiments). 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 consider the dynamics of the tangent vector to the Vortex Filament directly from the quantum LIA, and this approach, while less elegant than the quantum Hasimoto transformation, enables us to study waves primarily driven by the normal fluid velocity.
Luis Vega - One of the best experts on this subject based on the ideXlab platform.
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Vortex Filament Equation for a regular polygon in the hyperbolic plane.
arXiv: Numerical Analysis, 2020Co-Authors: Francisco De La Hoz, Sandeep Kumar, Luis VegaAbstract:The aim of this article is twofold. First, we show the evolution of the Vortex Filament equation (VFE) for a regular planar polygon in the hyperbolic space. Unlike in the Euclidean space, the planar polygon is open and both of its ends grow exponentially, which makes the problem more challenging from a numerical point of view. However, with fixed boundary conditions, a finite difference scheme and a fourth-order Runge--Kutta method in time, we show that the numerical solution is in complete agreement with the one obtained from algebraic techniques. Second, as in the Euclidean case, we claim that, at infinitesimal times, the evolution of VFE for a planar polygon as the initial datum can be described as a superposition of several one-corner initial data. As a consequence, not only can we compute the speed of the center of mass of the planar polygon, but the relationship also allows us to compare the time evolution of any of its corners with that in the Euclidean case.
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on the evolution of the Vortex Filament equation for regular m polygons with nonzero torsion
Siam Journal on Applied Mathematics, 2020Co-Authors: Francisco De La Hoz, Sandeep Kumar, Luis VegaAbstract:In this paper, we consider the evolution of the Vortex Filament equation (VFE): $X_t = X_s \wedge X_{ss},$ taking $M$-sided regular polygons with nonzero torsion as initial data. Using algebraic te...
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on the energy of critical solutions of the binormal flow
Communications in Partial Differential Equations, 2020Co-Authors: Valeria Banica, Luis VegaAbstract:The binormal flow is a model for the dynamics of a Vortex Filament in a 3-D inviscid incompressible fluid. The flow is also related with the classical continuous Heisenberg model in ferromagnetism,...
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on the evolution of the Vortex Filament equation for regular m polygons with nonzero torsion
arXiv: Numerical Analysis, 2019Co-Authors: Francisco De La Hoz, Sandeep Kumar, Luis VegaAbstract:In this paper, we consider the evolution of the Vortex Filament equation (VFE): \begin{equation*} \mathbf X_t = \mathbf Xs \wedge \mathbf Xss, \end{equation*} taking $M$-sided regular polygons with nonzero torsion as initial data. Using algebraic techniques, backed by numerical simulations, we show that the solutions are polygons at rational times, as in the zero-torsion case. However, unlike in that case, the evolution is not periodic in time; moreover, the multifractal trajectory of the point $\mathbf X(0,t)$ is not planar, and appears to be a helix for large times. These new solutions of VFE can be used to illustrate numerically that the smooth solutions of VFE given by helices and straight lines share the same instability as the one already established for circles. This is accomplished by showing the existence of variants of the so-called Riemann's non-differentiable function that are as close to smooth curves as desired, when measured in the right topology. This topology is motivated by some recent results on the well-posedness of VFE, which prove that the selfsimilar solutions of VFE have finite renormalized energy.
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on the relationship between the one corner problem and the m corner problem for the Vortex Filament equation
Journal of Nonlinear Science, 2018Co-Authors: Francisco De La Hoz, Luis VegaAbstract:In this paper, we give evidence that the evolution of the Vortex Filament equation (VFE) for a regular M-corner polygon as initial datum can be explained at infinitesimal times as the superposition of M one-corner initial data. This fact is mainly sustained with the calculation of the speed of the center of mass; in particular, we show that several conjectures made at the numerical level are in agreement with the theoretical expectations. Moreover, due to the spatial periodicity, the evolution of VFE at later times can be understood as the nonlinear interaction of infinitely many Filaments, one for each corner; and this interaction turns out to be some kind of nonlinear Talbot effect. We also give very strong numerical evidence of the transfer of energy and linear momentum for the M-corner case; and the numerical experiments carried out provide new arguments that support the multifractal character of the trajectory defined by one of the corners of the initial polygon.
Yasuhide Fukumoto - One of the best experts on this subject based on the ideXlab platform.
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the contribution of kawada to the analytical solution for the velocity induced by a helical Vortex Filament and modern applications of helical vortices
International Conference on Mathematical Analysis of Continuum Mechanics and Industrial Applications, 2017Co-Authors: Yasuhide Fukumoto, Valery Okulov, D H WoodAbstract:Currently, the analytical form of the velocity field induced by a helical Vortex Filament is well known as Hardin’s solution (1982). But essentially the same result had been obtained by a Japanese scientist Sandi Kawada, which predates Hardin by as long as 46 years. Kawada (1936) provided a comprehensive treatment of deriving the induced velocity by helical vortices with a view to applying it to the propeller theory. This paper recollects Kawada’s contribution, together with his life devoted to lead the Japanese aeronautical engineering in the time of its dawning.
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the contribution of kawada to the analytical solution for the velocity induced by a helical Vortex Filament
Applied Mechanics Reviews, 2015Co-Authors: Yasuhide Fukumoto, Valery Okulov, D H WoodAbstract:The basic solution for the velocity induced by helical Vortex Filament is well known as Hardin's solution, published in 1982. A study of early publications on helical vortices now shows that the Japanese scientist Kawada from Tokyo Imperial University also produced many of these results in 1936, which predates Hardin by 46 years. Consequently, in order to honor both, we have studied their derivations to establish the originality of both solutions.
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Three-dimensional motion of a Vortex Filament and its relation to the localized induction hierarchy
The European Physical Journal B - Condensed Matter, 2002Co-Authors: Yasuhide FukumotoAbstract:Three-dimensional motion of a slender Vortex tube, embedded in an inviscid incompressible fluid, is investigated under the localized induction approximation for the Euler equations. Using the method of matched asymptotic expansions in a small parameter e, the ratio of core radius to curvature radius, the velocity of a Vortex Filament is derived to O(e3), whereby the influence of elliptical deformation of the core due to the self-induced strain is taken into account. It is found that there is an integrable line in the core whose evolution obeys a summation of the first and third terms of the localized induction hierarchy.
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motion of a curved Vortex Filament higher order asymptotics
2001Co-Authors: Yasuhide FukumotoAbstract:Three-dimensional motion of a slender Vortex tube, embedded in an inviscid incompressible fluid, is investigated based on the Euler equations. Using the method of matched asymptotic expansions in a small parameter ∈, the ratio of core radius to curvature radius, the velocity of a Vortex Filament is derived to O(∈ 3), whereby the influence of elliptical deformation of the core due to the self-induced strain is taken into account. In the localized induction approximation, this is reducible to a completely integrable evolution equation among the localized induction hierarchy.
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stationary configurations of a Vortex Filament in background flows
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 1997Co-Authors: Yasuhide FukumotoAbstract:We investigate the three-dimensional configurations of a thin Vortex Filament embedded in background flows of an inviscid incompressible fluid, based on the localized induction approximation. An analogy is found between stationary configurations of a Vortex Filament in a steady flow and trajectories of a charged particle in a steady magnetic field. An analogy with trajectories of a sound ray in a steady low-Mach number flow is drawn as well. These analogies allow us to use the Lagrangian and Hamiltonian formalism of classical mechanics for calculating fully nonlinear forms of a Vortex Filament. The equations for the Filament curve are integrable when the background flow has two spatial symmetries. Only integrable cases are explored in some detail. To illustrate the advantages of use of the analogy, we re-examine the invariant shapes of a Vortex Filament moving through a still fluid obtained by Kida (1981). Analogies of the Kida class with the motions of a heavy symmetrical top and a charged spherical pendulum in the field of a magnetic monopole are discussed. As next examples, we consider a point source (sink) and a line source (sink) flows. A Filament takes the form of a non-uniform helix wound around a cone. The last example is a circular jet of a parabolic velocity profile. We construct a closed Filament with azimuthal waves, lying on a plane orthogonal to the flow direction. Among them, an elliptic Vortex ring is found to be distinct from higher waves.
Valeria Banica - One of the best experts on this subject based on the ideXlab platform.
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on the energy of critical solutions of the binormal flow
Communications in Partial Differential Equations, 2020Co-Authors: Valeria Banica, Luis VegaAbstract:The binormal flow is a model for the dynamics of a Vortex Filament in a 3-D inviscid incompressible fluid. The flow is also related with the classical continuous Heisenberg model in ferromagnetism,...
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Collisions of Vortex Filament pairs
Journal of Nonlinear Science, 2014Co-Authors: Valeria Banica, Erwan Faou, Evelyne MiotAbstract:We consider the problem of collisions of Vortex Filaments for a model introduced by Klein, Majda and Damodaran, and Zakharov to describe the interaction of almost parallel Vortex Filaments in three-dimensional fluids. Since the results of Crow examples of collisions are searched as perturbations of antiparallel translating pairs of Filaments, with initial perturbations related to the unstable mode of the linearized problem; most results are numerical calculations. In this article we first consider a related model for the evolution of pairs of Filaments and we display another type of initial perturbation leading to collision in finite time. Moreover we give numerical evidence that it also leads to collision through the initial model. We finally study the self-similar solutions of the model.
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Stability of the Self-similar Dynamics of a Vortex Filament
Archive for Rational Mechanics and Analysis, 2013Co-Authors: Valeria Banica, Luis VegaAbstract:In this paper we continue our investigation of self-similar solutions of the Vortex Filament equation, also known as the binormal flow or the localized induction equation. Our main result is the stability of the self-similar dynamics of small perturbations of a given self-similar solution. The proof relies on finding precise asymptotics in space and time for the tangent and the normal vectors of the perturbations. A main ingredient in the proof is the control of the evolution of weighted norms for a cubic one-dimensional Schrödinger equation, connected to the binormal flow by Hasimoto’s transform.
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stability of the selfsimilar dynamics of a Vortex Filament
arXiv: Analysis of PDEs, 2012Co-Authors: Valeria Banica, Luis VegaAbstract:In this paper we continue our investigation about selfsimilar solutions of the Vortex Filament equation, also known as the binormal flow (BF) or the localized induction equation (LIE). Our main result is the stability of the selfsimilar dynamics of small pertubations of a given selfsimilar solution. The proof relies on finding precise asymptotics in space and time for the tangent and the normal vectors of the perturbations. A main ingredient in the proof is the control of the evolution of weighted norms for a cubic 1-D Schr\"odinger equation, connected to the binormal flow by Hasimoto's transform.
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selfsimilar solutions of the binormal flow and their stability
arXiv: Analysis of PDEs, 2011Co-Authors: Valeria Banica, Luis VegaAbstract:We review some recent results concerning the evolution of a Vortex Filament and its relation to the cubic non-linear Schrodinger equation. Selfsimilar solutions and questions related to their stability are studied.
Valery Okulov - One of the best experts on this subject based on the ideXlab platform.
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the contribution of kawada to the analytical solution for the velocity induced by a helical Vortex Filament and modern applications of helical vortices
International Conference on Mathematical Analysis of Continuum Mechanics and Industrial Applications, 2017Co-Authors: Yasuhide Fukumoto, Valery Okulov, D H WoodAbstract:Currently, the analytical form of the velocity field induced by a helical Vortex Filament is well known as Hardin’s solution (1982). But essentially the same result had been obtained by a Japanese scientist Sandi Kawada, which predates Hardin by as long as 46 years. Kawada (1936) provided a comprehensive treatment of deriving the induced velocity by helical vortices with a view to applying it to the propeller theory. This paper recollects Kawada’s contribution, together with his life devoted to lead the Japanese aeronautical engineering in the time of its dawning.
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the contribution of kawada to the analytical solution for the velocity induced by a helical Vortex Filament
Applied Mechanics Reviews, 2015Co-Authors: Yasuhide Fukumoto, Valery Okulov, D H WoodAbstract:The basic solution for the velocity induced by helical Vortex Filament is well known as Hardin's solution, published in 1982. A study of early publications on helical vortices now shows that the Japanese scientist Kawada from Tokyo Imperial University also produced many of these results in 1936, which predates Hardin by 46 years. Consequently, in order to honor both, we have studied their derivations to establish the originality of both solutions.
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self induced motion and asymptotic expansion of the velocity field in the vicinity of a helical Vortex Filament
Physics of Fluids, 1998Co-Authors: P A Kuibin, Valery OkulovAbstract:A derivation of an expression for the self-induced velocity of a helical Vortex Filament with moderate and small pitch is presented for the cases of an infinite space as well as a space bounded by a cylindrical tube. This result supplements the known Kelvin’s formula for helix with large pitch in an infinite space.
