The Contribution of Kawada to the Analytical Solution for the Velocity Induced by a Helical Vortex Filament

2015 ◽  
Vol 67 (6) ◽  
Author(s):  
Yasuhide Fukumoto ◽  
Valery L. Okulov ◽  
David H. Wood
Keyword(s):  
Analytical Solution ◽  
Vortex Filament ◽  
Basic Solution ◽  
Helical Vortices ◽  
Helical Vortex ◽  
Japanese Scientist

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.

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.

Vortex Breakdown in Swirling Fuel Injector Flows

2007 ◽  
Author(s):  
Kris Midgley ◽  
Adrian Spencer ◽  
James J. McGuirk
Keyword(s):  
Vortex Breakdown ◽  
Near Field ◽  
Aerodynamic Characteristics ◽  
Relative Strength ◽  
Turbulence Energy ◽  
Fuel Injector ◽  
Swirl Number ◽  
Helical Vortices ◽  
Helical Vortex ◽  
Air Mixing

It is well known that the process of vortex breakdown plays an important role in establishing the near-field aerodynamic characteristics of fuel injectors, influencing fuel/air mixing and flame stability. The precise nature of the vortex breakdown can take on several forms, which have been shown in previous papers to include both a precessing vortex core (PVC) and the appearance of multiple helical vortices formed in the swirl stream shear layer. The unsteady dynamics of these particular features can play an important role in combustion induced oscillations. The present paper reports an experimental investigation, using PIV and hot-wire-anemometry, to document variations in the relative strength of PVC and helical vortex patterns as the configuration of a generic fuel injector is altered. Examples of geometric changes which have been investigated include: • The combination of an annular swirl stream with and without a central jet; • variation in geometric details of the swirler passage, e.g. alteration in the swirler entry slots to change swirl number, and variations in the area ratio of the swirler passage. The results show that these geometric variations can influence: • the axial location of the origin of the helical vortices (from inside to outside the fuel injector); • the strength of the PVC. For example, in a configuration with no central jet (swirl number S = 0.72) the helical vortex pattern was much less coherent, but the PVC was much stronger than when a central jet was present. These changes modify the magnitude of the turbulence energy in the fuel injector near field dramatically, and hence have an important influence on fuel air mixing patterns.

Vortex Breakdown in Swirling Fuel Injector Flows

2008 ◽  
Vol 130 (2) ◽  
Author(s):  
Adrian Spencer ◽  
James J. McGuirk ◽  
Kris Midgley
Keyword(s):  
Vortex Breakdown ◽  
Near Field ◽  
Aerodynamic Characteristics ◽  
Relative Strength ◽  
Turbulence Energy ◽  
Fuel Injector ◽  
Swirl Number ◽  
Helical Vortices ◽  
Helical Vortex ◽  
Air Mixing

It is well known that the process of vortex breakdown plays an important role in establishing the near-field aerodynamic characteristics of fuel injectors, influencing fuel/air mixing and flame stability. The precise nature of the vortex breakdown can take on several forms, which have been shown in previous papers to include both a precessing vortex core (PVC) and the appearance of multiple helical vortices formed in the swirl stream shear layer. The unsteady dynamics of these particular features can play an important role in combustion induced oscillations. The present paper reports an experimental investigation, using particle image velocimetry (PIV) and hot-wire anemometry, to document variations in the relative strength of PVC and helical vortex patterns as the configuration of a generic fuel injector is altered. Examples of geometric changes that have been investigated include: the combination of an annular swirl stream with and without a central jet; variation in geometric details of the swirler passage, e.g., alteration in the swirler entry slots to change swirl number, and variations in the area ratio of the swirler passage. The results show that these geometric variations can influence: the axial location of the origin of the helical vortices (from inside to outside the fuel injector), and the strength of the PVC. For example, in a configuration with no central jet (swirl number S=0.72), the helical vortex pattern was much less coherent, but the PVC was much stronger than when a central jet was present. These changes modify the magnitude of the turbulence energy in the fuel injector near field dramatically, and hence have an important influence on fuel air mixing patterns.

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.

Helical vortices in swirl flow

1999 ◽  
Vol 382 ◽  
pp. 195-243 ◽  
Author(s):  
S. V. ALEKSEENKO ◽  
P. A. KUIBIN ◽  
V. L. OKULOV ◽  
S. I. SHTORK
Keyword(s):  
Reverse Flow ◽  
Double Helix ◽  
Cylindrical Tube ◽  
Vortex Chamber ◽  
Swirl Flow ◽  
Vortex Structures ◽  
Helical Symmetry ◽  
Swirl Flows ◽  
Helical Vortices ◽  
Helical Vortex

Helical vortices in swirl flow are studied theoretically and experimentally.A theory of helical vortices has been developed. It includes the following results: an analytical solution describing an elementary helical vortex structure – an infinitely thin filament; a solution for axisymmetrical vortices accounting for the helical shape of vortex lines and different laws of vorticity distribution; a formula for calculation of the self-induced velocity of helical vortex rotation (precession) in a cylindrical tube; an explanation of the zone with reverse flow (recirculation zone) arising in swirl flows; and the classification of vortex structures.The experimental study of helical vortices was carried out in a vertical hydrodynamical vortex chamber with a tangential supply of liquid through turning nozzles. Various vortex structures were formed owing to changing boundary conditions on the bottom and at the exit section of the chamber. The hypothesis of helical symmetry is confirmed for various types of swirl flow. The stationary helical vortex structures are described (most of them for the first time) the features of which agree with the results and predictions of the theoretical model developed. They are the following: a rectilinear vortex; a composite columnar vortex; helical vortices screwed on the right or on the left; a vortex with changing helical symmetry; a double helix – two entangled vortex filaments of the same sign.

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