Yasuhide Fukumoto

Motion and expansion of a viscous vortex ring. Part 1. A higher-order asymptotic formula for the velocity

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 asymptotic expansions is extended to a higher order in a small parameter =( = ) 1=2 , where is the kinematic viscosity of fluid and is the circulation. Alternatively, 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 . This is a natural extension of Fraenkel{Saman’s rst-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 innitely thin circular loop of radius R0, that viscosity acts, at third order, to expand the circles of stagnation points of radii Rs(t) and ~ Rs(t) relative to the laboratory frame and a comoving frame respectively, and that of peak vorticity of radius Rp(t) as Rs R0 +[ 2 log(4R0= p t )+1 :4743424]t=R0, ~ Rs R0 +2 :5902739t=R0, and Rp R0 +4 :5902739t=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 eld. The strength of the dipole depends not only on the curvature but also on the location of the core centre, and therefore should be specied at the initial instant. This specication removes an indeterminacy of the rst-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.

Curvature instability of a vortex ring

A global stability analysis of Kelvins 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

The velocity field induced by a helical vortex tube

\epsilon

Short-wavelength stability analysis of thin vortex rings

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

STATIONARY CONFIGURATIONS OF A VORTEX FILAMENT IN BACKGROUND FLOWS

\epsilon

Stability of a hydrodynamic discontinuity

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

Global time evolution of an axisymmetric vortex ring at low Reynolds numbers

O(\epsilon^2)

Local Stability of Two-Dimensional Steady Irrotational Solenoidal Flows with Closed Streamlines

known as the Moore–Saffman–Tsai–Widnall instability. The viscosity acts to damp the former preferentially and these effects compete with each other.

Energy of hydrodynamic and magnetohydrodynamic waves with point and continuous spectra

The influence of finite-core thickness on the velocity field around a vortex tube is addressed. An asymptotic expansion of the Biot-Savart law is made to a higher order in a small parameter, the ratio of core radius to curvature radius, which consists of the velocity field due to lines of monopoles and dipoles arranged on the centerline of the tube. The former is associated with an infinitely thin core and is featured by the circulation alone. The distribution of vorticity in the core reflects on the strength of dipole. This result is applied to a helical vortex tube, and the induced velocity due to a helical filament of the dipoles is obtained in the form of the Kapteyn series, which augments Hardin’s [Phys. Fluids 25, 1949 (1982)] solution for the monopoles. Using a singularity-separation technique, a substantial part of the series is represented in a closed form for both the mono- and the dipoles. It is found from numerical calculation that the smaller the helix pitch is, the larger the relative influe...

Local instabilities in magnetized rotational flows: A short-wavelength approach

The linear stability of thin vortex rings are studied by short-wavelength stability analysis. The modified Hill–Schrodinger equation for vortex rings, which incorporates curvature effect, is derived. It is used to evaluate growth rates analytically. The growth rates are also evaluated by numerical calculation and they agree well with analytical values for small e which is the ratio of core radius to ring radius. Two types of vortex rings are considered: Kelvin’s vortex ring and a Gaussian vortex ring. For Kelvin’s vortex ring the maximum first-order growth rate is found to be 165256e. For the Gaussian vortex ring the first-order growth rate is large in the skirts of the vortex core. The first-order instability is significant for both vortex rings.