Vortex dynamics in nonlinear free surface flows

The two-dimensional motion of point vortices in an inviscid fluid with a free surface and an impenetrable bed is investigated. The work is based on forming a closed system of equations for surface variables and vortex positions using a variant of the Ablowitz, Fokas, and Musslimani formulation [M. J. Ablowitz, A. S. Fokas, and Z. H. Musslimani, J. Fluid Mech. 562, 313-343 (2006)] of the water-wave free-surface problem. The equations are approximated with a dealiased spectral method making use of a high-order approximation of the Dirichlet-Neumann operator and a high-order time-stepping scheme. Numerical simulations reveal that the combination of vortex motion and solid bottom boundary yields interesting dynamics not seen in the case of vortex motion in an infinitely deep fluid. In particular, strong deformations of the free surface, including non-symmetric surface profiles and regions of large energy concentration, are observed. Our simulations also uncover a rich variety of vortex trajectories including orbiting and nearly parallel patterns of motion. The dynamics of the free surface and of the point vortices are strongly influenced by the initial placement and polarity of the vortices. The method put forward here is flexible enough to handle a large number of vortices and may easily be extended to include the effects of varying bathymetry, stratification, and background shear currents. Published by AIP Publishing.