A NUMERICAL STUDY OF 3-DIMENSIONAL VORTEX RING INSTABILITIES - VISCOUS CORRECTIONS AND EARLY NONLINEAR STAGE

Finite-difference calculations with random and single-mode perturbations are used to study the three-dimensional instability of vortex rings. The basis of current understanding of the subject consists of a heuristic inviscid model (Widnall, Bliss and Tsai 1974) and a rigorous theory which predicts growth rates for thin-core uniform vorticity rings (Widnall and Tsai 1977). At sufficiently high Reynolds numbers the results correspond qualitatively to those predicted by the heuristic model: multiple bands of wavenumbers are amplified, each band having a distinct radial structure. However, a viscous correction factor to the peak inviscid growth rate is found. It is well described by the first term, 1 - alpha(1)(beta)/Re-s, for a large range of Re-s. Here Re-s is the Reynolds number defined by Saffman (1978), which involves the curvature-induced strain rate. It is found to be the appropriate choice since then alpha(1)(beta) varies weakly with core thickness beta. The three most nonlinearly amplified modes are a mean azimuthal velocity in the form of opposing streams, an n = 1 mode (n is the azimuthal wavenumber) which arises from the interaction of two second-mode bending waves and the harmonic of the primary second mode. When a single wave is excited, higher harmonics begin to grow successively later with nonlinear growth rates proportional to n. The modified mean flow has a doubly peaked azimuthal vorticity. Since the curvature-induced strain is not exactly stagnation-point flow there is a preference for elongation towards the rear of the ring: the outer structure of the instability wave forms a long wake consisting of n hairpin vortices whose waviness is phase shifted pi/n relative to the waviness in the core. Whereas the most amplified linear mode has three radial layers of structure, higher radial mode;s having more layers of radial structure (hairpins piled upon hairpins) are excited when the initial perturbation is large, reminiscent of visualization experiments on the formation of a turbulent ring at the generator.