A numerical procedure to study the stability of helical vortices

Abstract

A numerical approach is proposed for the study of instabilities in helical vortex systems as found in the near-wake of turbines or propellers. The methodology has a high degree of generality, yet the present paper focusses on the case of one unique helical vortex. First, a method based on helical symmetry aimed at computing a three-dimensional base flow with prescribed parameters—helical pitch, helical radius, vortex circulation, core size and inner jet component—is presented. Second, the linear instability of this base flow is examined by reducing the three-dimensional instability problem to two-dimensional simulations with wavenumbers prescribed along the helix axis. Each simulation converges towards an exponentially growing or decaying complex state from which eigenfunctions, growth rate and frequency are extracted. This procedure is validated against a standard method based on direct three-dimensional numerical simulations of the Navier–Stokes equations linearized in the vicinity of the same helical base flows. Three illustrative base flows are presented with or without inner jet component, the instability of which is dominated, at the prescribed axial wavenumber, by unstable modes of three different types: long-wave instability, short-wave elliptic and curvature instabilities. Results from the new procedure and from the fully three-dimensional one are found in excellent agreement, which validates the new methodology. The gain in computational time is typically the one that is achieved while going from three-dimensional to two-dimensional simulations.