The instability of a helical vortex filament under a free surface

Abstract

We perform a theoretical investigation of the instability of a helical vortex filament beneath a free surface in a semi-infinite ideal fluid. The focus is on the leading-order free-surface boundary effect upon the equilibrium form and instability of the vortex. This effect is characterised by the Froude number Formula Presented where Formula Presented is gravity, and Formula Presented with Formula Presented being the strength, Formula Presented the pitch and Formula Presented the centre submergence of the helical vortex. In the case of Formula Presented corresponding to the presence of a rigid boundary, a new approximate equilibrium form is found if the vortex possesses a non-zero rotational velocity. Compared with the infinite fluid case (Widnall, J. Fluid Mech., vol. 54, no. 4, 1972, pp. 641-663), the vortex is destabilised (or stabilised) to relatively short- (or long-)wavelength sub-harmonic perturbations, but remains stable to super-harmonic perturbations. The wall-boundary effect becomes stronger for smaller helix angle and could dominate over the self-induced flow effect depending on the submergence. In the case of Formula Presented, we obtain the surface wave solution induced by the vortex in the context of linearised potential-flow theory. The wave elevation is unbounded when the Formula Presentedth wave mode becomes resonant as Formula Presented approaches the critical Froude numbers Formula Presented, Formula Presented where Formula Presented is the induced wave speed. We find that the new approximate equilibrium of the vortex exists if and only if Formula Presented. Compared with the infinite fluid and Formula Presented cases, the wave effect causes the vortex to be destabilised to super-harmonic and long-wavelength sub-harmonic perturbations with generally faster growth rate for greater Formula Presented and smaller helix angle.