Real history of helical vortices: theory and applications. Annex: comments on “motion of a helical vortex by o. Velasco fuentes”

Research on helical vortices has a long history, which dates back to the famous work of Lord Kelvin in 1880 on the helical perturbations of the columnar vortex. Helical vortices are of fundamental importance to fluid mechanics because they describe one of the main states of swirling flows. An accurate estimation of velocities in the rotor plane requires knowing the velocity field induced by the helical vortices. Therefore, starting with the inspirational work of Joukowsky, the theory of helical vortices has been actively studied as a prerequisite to understand and analyze rotor aerodynamics [1]. At present the fundamentals are based on various analytical components [1, 2], which holds true for all values of the helix pitch, such as (i) the 2D Biot-Savart law for helical filaments represented by Kapteyn series or in a form with singularity separation; (ii) solutions of helical vortex tubes with finite core, governed by series expansion of helical multipoles; (iii) relations between the induction of vortex filaments and the self-induced velocity of helical vortex tubes resulting in a closed analytical solution of the helix motion; (iv) analytical representation of Goldstein’s solution for the circulation of a helical vortex sheet in equilibrium; (v) Kelvin’s N-gon stability problem of point vortices generalized to multiple helical vortices.