The velocity field induced by a helical vortex filament

1982 ◽  
Vol 25 (11) ◽  
pp. 1949 ◽  
Author(s):  
Jay C. Hardin
Keyword(s):  
Velocity Field ◽  
Vortex Filament ◽  
Helical Vortex

scholarly journals The velocity field induced by a helical vortex tube

2005 ◽  
Vol 17 (10) ◽  
pp. 107101 ◽  
Author(s):  
Y. Fukumoto ◽  
V. L. Okulov
Keyword(s):  
Velocity Field ◽  
Vortex Tube ◽  
Helical Vortex

Long-wave instability of a helical vortex

2015 ◽  
Vol 780 ◽  
pp. 687-716 ◽  
Author(s):  
Hugo Umberto Quaranta ◽  
Hadrien Bolnot ◽  
Thomas Leweke
Keyword(s):  
Growth Rate ◽  
Water Channel ◽  
Vortex Filament ◽  
Linear Stability Theory ◽  
Wave Instability ◽  
Characteristic Distance ◽  
Long Wave ◽  
Instability Modes ◽  
Small Pitch ◽  
Helical Vortex

We investigate the instability of a single helical vortex filament of small pitch with respect to displacement perturbations whose wavelength is large compared to the vortex core size. We first revisit previous theoretical analyses concerning infinite Rankine vortices, and consider in addition the more realistic case of vortices with Gausssian vorticity distributions and axial core flow. We show that the various instability modes are related to the local pairing of successive helix turns through mutual induction, and that the growth rate curve can be qualitatively and quantitatively predicted from the classical pairing of an array of point vortices. We then present results from an experimental study of a helical vortex filament generated in a water channel by a single-bladed rotor under carefully controlled conditions. Various modes of displacement perturbations could be triggered by suitable modulation of the blade rotation. Dye visualisations and particle image velocimetry allowed a detailed characterisation of the vortex geometry and the determination of the growth rate of the long-wave instability modes, showing good agreement with theoretical predictions for the experimental base flow. The long-term (downstream) development of the pairing instability leads to a grouping and swapping of helix loops. Despite the resulting complicated three-dimensional structure, the vortex filaments surprisingly remain mostly intact in our observation interval. The characteristic distance of evolution of the helical wake behind the rotor decreases with increasing initial amplitude of the perturbations; this can be predicted from the linear stability theory.

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

2014 ◽  
Vol 762 ◽  
pp. 141-155 ◽  
Author(s):  
Robert A. Van Gorder
Keyword(s):  
Logarithmic Singularity ◽  
Transverse Velocity ◽  
Rotational Velocity ◽  
Vortex Filament ◽  
Maximal Torsion ◽  
Helical Vortex ◽  
Local Contribution ◽  
Induced Motion ◽  
Non Local ◽  
Local Induction Approximation

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

2017 ◽  
Vol 836 ◽  
Author(s):  
Oscar Velasco Fuentes
Keyword(s):  
Velocity Field ◽  
Cross Section ◽  
Single Point ◽  
Small Circle ◽  
Inviscid Incompressible Fluid ◽  
Helical Vortex ◽  
Angular Velocities ◽  
Infinite Tube ◽  
Range Of Values ◽  
Theoretical Results

This paper deals with the motion of a single helical vortex in an unbounded inviscid incompressible fluid. The vortex is an infinite tube whose centreline is a helix and whose cross-section is a small circle where the vorticity is uniform and parallel to the centreline. Ever since Joukowsky (Trudy Otd. Fiz. Nauk Mosk. Obshch. Lyub. Estest., vol. 16, 1912, pp. 1–31) deduced that this vortex translates and rotates steadily without change of form, numerous attempts have been made to compute the velocities. Here, Hardin’s (Phys. Fluids, vol. 25, 1982, pp. 1949–1952) solution for the velocity field is used to find new expressions for the linear and angular velocities of the vortex. The theoretical results are verified by numerically computing the velocity at a single point using the Helmholtz integral and the Rosenhead–Moore approximation to the Biot–Savart law, and by numerically simulating the vortex evolution, under the Euler equations, in a triple-periodic cube. The new formulae are also shown to be more accurate than previous results over the whole range of values of the vortex pitch and cross-section.

The stability of a helical vortex filament

1972 ◽  
Vol 54 (4) ◽  
pp. 641-663 ◽  
Author(s):  
Sheila E. Widnall
Keyword(s):  
Short Wave ◽  
Mutual Inductance ◽  
The Self ◽  
Vortex Filament ◽  
Wave Instability ◽  
Long Wave ◽  
Amplification Rate ◽  
Helical Vortex ◽  
Induced Motion ◽  
The Stability

The stability of a helical vortex filament of finite core and infinite extent to small sinusoidal displacements of its centre-line is considered. The influence of the entire perturbed filament on the self-induced motion of each element is taken into account. The effect of the details of the vorticity distribution within the finite vortex core on the self-induced motion due to the bending of its axis is calculated using the results obtained previously by Widnall, Bliss & Zalay (1970). In this previous work, an application of the method of matched asymptotic expansions resulted in a general solution for the self-induced motion resulting from the bending of a slender vortex filament with an arbitrary distribution of vorticity and axial velocity within the core.The results of the stability calculations presented in this paper show that the helical vortex filament has three modes of instability: a very short-wave instability which probably exists on all curved filaments, a long-wave mode which is also found to be unstable by the local-induction model and a mutual-inductance mode which appears as the pitch of the helix decreases and the neighbouring turns of the filament begin to interact strongly. Increasing the vortex core size is found to reduce the amplification rate of the long-wave instability, to increase the amplification rate of the mutual-inductance instability and to decrease the wavenumber of the short-wave instability.

scholarly journals The effect of torsion on the motion of a helical vortex filament

1994 ◽  
Vol 273 ◽  
pp. 241-259 ◽  
Author(s):  
Renzo L. Ricca
Keyword(s):  
Integral Formula ◽  
Vortex Filament ◽  
Vortex Filaments ◽  
Small Pitch ◽  
Helical Vortex ◽  
Limit Behaviour ◽  
Induced Velocity ◽  
First Time ◽  
Limit Form

In this paper we analyse in detail, and for the first time, the rôle of torsion in the dynamics of twisted vortex filaments. We demonstrate that torsion may influence considerably the motion of helical vortex filaments in an incompressible perfect fluid. The binormal component of the induced velocity, asymptotically responsible for the displacement of the vortex filament in the fluid, is closely analysed. The study is performed by applying the prescription of Moore & Saffman (1972) to helices of any pitch and a new asymptotic integral formula is derived. We give a rigorous proof that the Kelvin régime and its limit behaviour are obtained as a limit form of that integral asymptotic formula. The results are compared with new calculations based on the re-elaboration of Hardin's (1982) approach and with results obtained by Levy & Forsdyke (1928) and Widnall (1972) for helices of small pitch, here also re-elaborated for the purpose.

Regular and chaotic particle motion near a helical vortex filament

1998 ◽  
Vol 111 (1-4) ◽  
pp. 179-201 ◽  
Author(s):  
Igor Mezić ◽  
Anthony Leonard ◽  
Stephen Wiggins
Keyword(s):  
Particle Motion ◽  
Vortex Filament ◽  
Helical Vortex
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