The instability of a helical vortex filament under a free surface
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Published version
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Author(s)
Li, Cheng
Liu, Yuming
Wan, Minping
Chen, Shiyi
Yue, Dick KP
Date Issued
April 25, 2022
Journal
Journal of Fluid Mechanics
Publisher
Cambridge University Press (CUP)
Citation
Li, Cheng, Liu, Yuming, Wan, Minping, Chen, Shiyi and Yue, Dick KP. 2022. "The instability of a helical vortex filament under a free surface." Journal of Fluid Mechanics, 937.
Version
Final published version
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.
MIT Department
Massachusetts Institute of Technology. Department of Mechanical Engineering
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DOI of Published Version
https://doi.org/10.1017/jfm.2022.112
