Stability of a system of multiple helical vortices

Abstract

The system of vortices created by the hub and tip vortices of rotors and propellers is composed of two subsystems of helical vortices that have different radii and pitches. A system of external and internal vortices can also be created by some blade devices proposed to destabilize the tip vortices of helicopters and a similar system was observed in an idealized simulation of two in-line wind turbines. The steady solution of these systems of vortices was recently described. However, their stability was not studied. The stability of a system of multiple helical vortices is studied in this work using two methods: a complex-step technique combined with a time-stepping algorithm used to linearize the Biot-Savart law and the vorticity transport equations; and an approximate analytical method. It was noted that the hub and tip vortices do not interact and their linear stability can be treated separately if the velocity field induced by one system is considered in the stability analysis of the other. Strong interaction between the vortices was observed for internal vortices closer to the external vortices, with an out-of-phase mechanism appearing as one of the main phenomena.QC 20221024</p