F215 渦輪の曲率不安定性

概要

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We expolore the effect of curvature upon the linear stability of a thin axisymmetric vortex ring, embedded in an inviscid incompressible fluid, to three-dimensional disturbances. The basic state, a steady asymptotic solution of the Euler equations, is expressed in the form of a power series in a small parameter a, the ratio of the core radius to the ring radius. The leading-order flow is the Rankine vortex which supports the Kelvin waves. The first-order flow consists of a dipole field which originates from the effect of ring curvature. We show that the dipole field causes a parametric resonance instability between axisymmetric and bending waves at intersection points of the dispersion curves. As the wavenumber is increased, finite non-zero growth rate is maintained only at the intersection points with the frequency approaching the half of the rotation frequency of the basic flow, otherwise it decreases to zero. The energy of the Kelvin waves is calculated by invoking Cairns' formula. The instability result is compatible with Krein's theory for parametric resonance in Hamiltonian systems.
2002-07-23