A-233 渦輪のパラメータ共鳴不安定に対する曲率の効果

概要

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We investigate a global stability of Kelvin's vortex ring to three-dimensional disturbances of infinitesimal amplitude. The basic state is a steady asymptotic solution of the Euler equations in powers of the ratio ε, the core radius to the ring radius, retaining the terms to first order. The leadning-oder flow is the Rankine vortex. The effect of ring curvature makes its appearance, as the first-order perturbation to it, in the form of a dipole field, and a local straning field, a quadrupole field, follows at second order. 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 modes occur in the short wavelength limit, under the condtraint that the radial and the azimuthal wavenumbers are of the same magnitude, and the maximum growth rate is 165/256ε. Over a wide range of ε, the dipole effect overweighs the straning field effect of O(ε^2), being well known as the Moore-Saffman-Tsai-Widnall instability. The instability mechanism is traced to stretching of disturbance vortex lines not only in the poloidal plane but also in the toroidal direction.
2003-07-28