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 𝐹𝑟=𝑈(𝑔ℎ∗)−1/2 where 𝑔 is gravity, and 𝑈=𝛤/(2𝜋𝑏∗) with 𝛤 being the strength, 2𝜋𝑏∗ the pitch and ℎ∗ the centre submergence of the helical vortex. In the case of 𝐹𝑟→0 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 𝐹𝑟>0, 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 mth wave mode becomes resonant as Fr approaches the critical Froude numbers F(m) = (C∗0/U)−1(mh∗/b∗)−1/2, m = 1, 2, . . . , where C∗0 is the induced wave speed. We find that the new approximate equilibrium of the vortex exists if and only if Fr < F(1). Compared with the infinite fluid and Fr → 0 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 Fr and smaller helix angle.