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Closely spaced co-rotating helical vortices: Long-wave instability

We consider as base flow the stationary vortex filament solution obtained by Castillo- Castellanos et al. (2021) in the far-wake of a rotor with tip-splitting blades. The cases of a single blade and of two blades with a hub vortex are studied. In these solutions, each blade generates two closely spaced co-rotating tip vortices that form a braided helical pattern in the far-wake. The long-wave stability of these solutions is analysed using the same vortex filament framework. Both the linear spectrum and the linear impulse response are considered. We demonstrate the existence of different types of instability modes. A first type corresponds to the local pairing of consecutive turns of the helical pattern, which is well described by the instability of an uniform helical vortex with a core size given by the mean separation distance of the vortices in the pair. A second type corresponds to the pairing of consecutive turns of vortex pair and is observed only for densely braided patterns, which is well described by the instability of two interlaced helical vortices by straightening out the baseline helix. A third type of unstable modes modifies the separation distance between the vortices in each pair and amplifies specific (linear) wavelengths. These unstable modes also spread spatially with a weaker rate.