6 results
Modal stability analysis of a helical vortex tube with axial flow
- Yuji Hattori, Yasuhide Fukumoto
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- Journal:
- Journal of Fluid Mechanics / Volume 738 / 10 January 2014
- Published online by Cambridge University Press:
- 05 December 2013, pp. 222-249
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The linear stability of a helical vortex tube with axial flow, which is a model of helical vortices emanating from rotating wings, is studied by modal stability analysis. At the leading order the base flow is set to the Rankine vortex with uniform velocity along the helical tube whose centreline is a helix of constant curvature and torsion. The helical vortex tube in an infinite domain, in which the free boundary condition is imposed at the surface of the tube, is our major target although the case of the rigid boundary condition is also considered in order to elucidate the effects of torsion and the combined effects of torsion and axial flow. The analysis is based on the linearized incompressible Euler equations expanded in $\epsilon $ which is the ratio of the core to curvature radius of the tube. The unstable growth rate can be evaluated using the leading-order neutral modes called the Kelvin waves with the expanded equations. At $O(\epsilon )$ the instability is a linear combination of the curvature instability due to the curvature of the tube and the precessional instability due to the axial flow, both parametric instabilities appearing at the same resonance condition. At the next order $O({\epsilon }^{2} )$ not only the effects of torsion but also the combined effects of torsion and axial flow appear, a fact which has been shown only for the short-wave limit. The maximum growth rate increases for the right-handed/left-handed helix with positive/negative helicity, in which the torsion makes the period of particle motion increase. All results converge to the previous local stability results in the short-wave limit. The differences between the two cases of different boundary conditions are due to the isolated mode of the free boundary case, whose dispersion curve depends strongly on the axial flow.
A generalized vortex ring model
- FELIX KAPLANSKI, SERGEI S SAZHIN, YASUHIDE FUKUMOTO, STEVEN BEGG, MORGAN HEIKAL
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- Journal of Fluid Mechanics / Volume 622 / 10 March 2009
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- 10 March 2009, pp. 233-258
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A conventional laminar vortex ring model is generalized by assuming that the time dependence of the vortex ring thickness ℓ is given by the relation ℓ = atb, where a is a positive number and 1/4 ≤ b ≤ 1/2. In the case in which , where ν is the laminar kinematic viscosity, and b = 1/2, the predictions of the generalized model are identical with the predictions of the conventional laminar model. In the case of b = 1/4 some of its predictions are similar to the turbulent vortex ring models, assuming that the time-dependent effective turbulent viscosity ν∗ is equal to ℓℓ′. This generalization is performed both in the case of a fixed vortex ring radius R0 and increasing vortex ring radius. In the latter case, the so-called second Saffman's formula is modified. In the case of fixed R0, the predicted vorticity distribution for short times shows a close agreement with a Gaussian form for all b and compares favourably with available experimental data. The time evolution of the location of the region of maximal vorticity and the region in which the velocity of the fluid in the frame of reference moving with the vortex ring centroid is equal to zero is analysed. It is noted that the locations of both regions depend upon b, the latter region being always further away from the vortex axis than the first one. It is shown that the axial velocities of the fluid in the first region are always greater than the axial velocities in the second region. Both velocities depend strongly upon b. Although the radial component of velocity in both of these regions is equal to zero, the location of both of these regions changes with time. This leads to the introduction of an effective radial velocity component; the latter case depends upon b. The predictions of the model are compared with the results of experimental measurements of vortex ring parameters reported in the literature.
Three-dimensional distortions of a vortex filament with axial velocity
- Yasuhide Fukumoto, Takeshi Miyazaki
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- Journal:
- Journal of Fluid Mechanics / Volume 222 / January 1991
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- 26 April 2006, pp. 369-416
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Three-dimensional motion of a thin vortex filament with axial velocity, embedded in an inviscid incompressible fluid, is investigated. The deflections of the core centreline are not restricted to be small compared with the core radius. We first derive the equation of the vortex motion, correct to the second order in the ratio of the core radius to that of curvature, by a matching procedure, which recovers the results obtained by Moore & Saffman (1972). An asymptotic formula for the linear dispersion relation is obtained up to the second order. Under the assumption of localized induction, the equation governing the self-induced motion of the vortex is reduced to a nonlinear evolution equation generalizing the localized induction equation. This new equation is equivalent to the Hirota equation which is integrable, including both the nonlinear Schrödinger equation and the modified KdV equation in certain limits. Therefore the new equation is also integrable and the soliton surface approach gives the N-soliton solution, which is identical to that of the localized induction equation if the pertinent dispersion relation is used. Among other exact solutions are a circular helix and a plane curve of Euler's elastica. This local model predicts that, owing to the existence of the axial flow, a certain class of helicoidal vortices become neutrally stable to any small perturbations. The non-local influence of the entire perturbed filament on the linear stability of a helicoidal vortex is explored with the help of the cutoff method valid to the second order, which extends the first-order scheme developed by Widnall (1972). The axial velocity is found to discriminate between right- and left-handed helices and the long-wave instability mode is found to disappear in a certain parameter range when the successive turns of the helix are not too close together. Comparison of the cutoff model with the local model reveals that the non-local induction and the core structure are crucial in making quantitative predictions.
Curvature instability of a vortex ring
- YASUHIDE FUKUMOTO, YUJI HATTORI
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- Journal:
- Journal of Fluid Mechanics / Volume 526 / 10 March 2005
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- 25 February 2005, pp. 77-115
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A global stability analysis of Kelvin's vortex ring to three-dimensional disturbances of infinitesimal amplitude is made. The basic state is a steady asymptotic solution of the Euler equations, in powers of the ratio $\epsilon$ of the core radius to the ring radius, for an axisymmetric vortex ring with vorticity proportional to the distance from the symmetric axis. The effect of ring curvature appears at first order, in the form of a dipole field, and a local straining field, which is a quadrupole field, follows at second order. The eigenvalue problem of the Euler equations, retaining the terms to first order, is solved in closed form, in terms of the Bessel and the modified Bessel functions. We show that the dipole field causes a parametric resonance instability between a pair of Kelvin waves whose azimuthal wavenumbers are separated by 1. The most unstable mode occurs in the short-wavelength limit, under the constraint that the radial and the azimuthal wavenumbers are of the same magnitude, and the limiting value of maximum growth rate coincides with the value 165/256$\epsilon$ obtained by Hattori & Fukumoto (Phys. Fluids, vol. 15, 2003, p. 3151) by means of the geometric optics method. The instability mechanism is traced to stretching of disturbance vorticity in the toroidal direction. In the absence of viscosity, the dipole effect outweighs the straining field effect of $O(\epsilon^2)$ known as the Moore–Saffman–Tsai–Widnall instability. The viscosity acts to damp the former preferentially and these effects compete with each other.
The three-dimensional instability of a strained vortex tube revisited
- YASUHIDE FUKUMOTO
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- Journal of Fluid Mechanics / Volume 493 / 25 October 2003
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- 08 October 2003, pp. 287-318
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We revisit the Moore–Saffman–Tsai–Widnall instability, a parametric resonance between left- and right-handed bending waves of infinitesimal amplitude, on the Rankine vortex strained by a weak pure shear flow. The results of Tsai & Widnall (1976) and Eloy & Le Dizès (2001), as generalized to all pairs of Kelvin waves whose azimuthal wavenumbers $m$ are separated by 2, are simplified by providing an explicit solution of the linearized Euler equations for the disturbance flow field. Given the wavenumber $k_0$ and the frequency $\omega_0$ of an intersection point of dispersion curves, the growth rate is expressible solely in terms of the modified Bessel functions, and so is the unstable wavenumber range. Every intersection leads to instability. Most of the intersections correspond to weak instability that vanishes in the short-wave limit, and dominant modes are restricted to particular intersections. For helical waves $m=\pm 1$, the growth rate of non-rotating waves is far larger than that of rotating waves. The wavenumber width of stationary instability bands broadens linearly in $k_0$, while that of rotating instability bands is bounded. The growth rate of the stationary instability takes, in the long-wavelength limit, the value of $\varepsilon/2$ for the two-dimensional displacement instability, and, in the short-wavelength limit, the value of $9\varepsilon/16$ for the elliptical instability, being larger at large but finite values of $k_0$. Here $\varepsilon$ is the strength of shear near the core centre. For resonance between higher azimuthal wavenumbers $m$ and $m+2$, the same limiting value is approached as $k_0\,{\to}\,\infty$, along sequences of specific crossing points whose frequency rapidly converges to $m+1$, in two ways, from above for a fixed $m$ and from below for $m\,{\to}\,\infty$. The energy of the Kelvin waves is calculated by invoking Cairns' formula. The instability result is compatible with Krein's theory for Hamiltonian spectra.
Motion and expansion of a viscous vortex ring. Part 1. A higher-order asymptotic formula for the velocity
- YASUHIDE FUKUMOTO, H. K. MOFFATT
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- Journal of Fluid Mechanics / Volume 417 / 25 August 2000
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- 25 August 2000, pp. 1-45
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A large-Reynolds-number asymptotic solution of the Navier–Stokes equations is sought for the motion of an axisymmetric vortex ring of small cross-section embedded in a viscous incompressible fluid. In order to take account of the influence of elliptical deformation of the core due to the self-induced strain, the method of matched of matched asymptotic expansions is extended to a higher order in a small parameter ε = (v/Γ)1/2, where v is the kinematic viscosity of fluid and Γ is the circulation. Alternatively, ε is regarded as a measure of the ratio of the core radius to the ring radius, and our scheme is applicable also to the steady inviscid dynamics.
We establish a general formula for the translation speed of the ring valid up to third order in ε. This is a natural extension of Fraenkel–Saffman's first-order formula, and reduces, if specialized to a particular distribution of vorticity in an inviscid fluid, to Dyson's third-order formula. Moreover, it is demonstrated, for a ring starting from an infinitely thin circular loop of radius R0, that viscosity acts, at third order, to expand the circles of stagnation points of radii Rs(t) and R˜s(t) relative to the laboratory frame and a comoving frame respectively, and that of peak vorticity of radius R˜p(t) as Rs ≈ R0 + [2 log(4R0/√vt) + 1.4743424] vt/R0, R˜s ≈ R0 + 2.5902739 vt/R0, and Rp ≈ R0 + 4.5902739 vt/R0. The growth of the radial centroid of vorticity, linear in time, is also deduced. The results are compatible with the experimental results of Sallet & Widmayer (1974) and Weigand & Gharib (1997).
The procedure of pursuing the higher-order asymptotics provides a clear picture of the dynamics of a curved vortex tube; a vortex ring may be locally regarded as a line of dipoles along the core centreline, with their axes in the propagating direction, subjected to the self-induced flow field. The strength of the dipole depends not only on the curvature but also on the location of the core centre, and therefore should be specified at the initial instant. This specification removes an indeterminacy of the first-order theory. We derive a new asymptotic development of the Biot-Savart law for an arbitrary distribution of vorticity, which makes the non-local induction velocity from the dipoles calculable at third order.