Theoretical Analysis of the Aerodynamic Stability of Multiple, Interdigitated Helical Vortices

Abstract

A small perturbation stability analysis of a doubly infinite array of interdigitated, right circular helical vortices has been formulated. This array corresponds to the vortices trailed from the tips of the blades of a helicopter rotor or propeller in static thrust or axial flight condition and at great distance from the plane of rotation of the blades. A continuum of instability modes has been found associated with all values of wave numbers; only modes with wave numbers 0 and 1 are so much as neutrally stable, and for the case of a single helix. The most unstable modes involve the most axial motion of adjacent vortex segments relative to each other. Maximum divergence rates increase as the helix pitch decreases, increase as the number of helices increase and decrease as the number of cycles of deformations in one turn of the helix (i.e., wave number) increases. The helix filament core diameter can have substantial effect on the stability of a single helix, but not for multiple arrays. The larger the core diameter, the more sensitive the analysis is to the means by which the singularities in the self-induction integrals are eliminated. Increasing core diameters reduces the maximum divergence rates in all cases.