Nonlinear dynamics of short-wave instabilities in a helical vortex within and beyond the weakly nonlinear regime

Abstract

The nonlinear evolution of a helical vortex with axial flow disturbed by an elliptic or curvature instability mode is investigated by direct numerical simulations. The normalized vortex core radius and reduced pitch are systematically varied to examine a wide range of linear growth rates, revealing two types of nonlinear evolution. At relatively small growth rates, both instabilities exhibit weakly nonlinear saturation, with the vortex and unstable mode structures largely preserved. Together with previous findings for other vortex configurations, these results reveal that weakly nonlinear saturation is a generic feature of short-wave instabilities. At larger growth rates, the instabilities reach higher energy levels and evolve beyond the weakly nonlinear regime, with the vortex and unstable mode structures substantially deformed. Comparison with weakly nonlinear analysis suggests that larger linear growth rates require larger disturbance amplitudes for nonlinear effects to balance linear growth. At these larger amplitudes, higher-order nonlinear effects and modal components not retained in the weakly nonlinear analysis become increasingly important, as demonstrated by the numerical results. This provides a new physical interpretation of the departure from the weakly nonlinear regime. Since helical vortices serve as a model for tip vortices shed from rotating blades, these results offer useful insights for tip-vortex control in wind turbines, which can potentially accelerate wake recovery and thereby improve turbine performance.