We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
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 
$F_r = U(gh^*)^{-{1}/{2}}$ where 
$g$ is gravity, and 
$U = \varGamma /(2{\rm \pi} b^*)$ with 
$\varGamma$ being the strength, 
$2{\rm \pi} b^*$ the pitch and 
$h^*$ the centre submergence of the helical vortex. In the case of 
$F_r \rightarrow 0$ 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 
$F_r > 0$, 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 
$m$th wave mode becomes resonant as 
$F_r$ approaches the critical Froude numbers 
${\mathcal {F}} (m) = (C_0^*/U)^{-1} (mh^*/b^*)^{-{1}/{2}}$, 
$m=1, 2, \ldots,$ where 
$C_0^*$ is the induced wave speed. We find that the new approximate equilibrium of the vortex exists if and only if 
$F_r < {\mathcal {F}}(1)$. Compared with the infinite fluid and 
$F_r \rightarrow 0$ 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 
$F_r$ and smaller helix angle.
