# On linear and nonlinear wave-ship interactions

NWWW01 - Nonlinear water waves

Dynamics of buoy and ship motion in linear and nonlinear water waves will be considered. The starting point will be a “pre-Luke” variational principle (see Cotter & B. [1] for the case without ship) for the combined water-wave motion and the dynamics of a rigid buoy/ship. This principle reduces to Luke's variational principle when a scaled density D is set to unity, in the case without buoy/ship dynamics. The coupling between waves and ship is straightforward: that the shape of the water surface equals the shape of the moving buoy/ship is imposed via a Lagrange multiplier. Pre-Luke's principle has the advantage that the boundary conditions on the Lagrange multiplier emerge directly from the variational principle due to the weak imposition of the density constraint, while in Luke's form of the variational principle of the coupled dynamics the boundary condition needs to be imposed on the Lagrange multiplier. The consequences for a completely space-time variational numerical discretisation will be discussed. Attempts to find an exact/reduced test solution in the shallow water limit of buoy-fluid motion may be considered as well. Movies of a wave-energy device and preliminary numerical solutions [2,3,4] will be used throughout the presentation.

[1] C. Cotter and O.B. 2010: Water wave model with accurate dispersion and vertical vorticity. Peregrine Commemorative Issue J. Eng. Maths. 67, 33-54.

[2] Youtube channel Anna Kalogirou: Movies of simulations and scaled wave-energy device: https://www.youtube.com/user/anna9k

[3] A. Kalogirou and O.B 2016: Mathematical and numerical modelling of wave impact on wave-energy buoys. Proc. ASME 2016 35th International Conference on Ocean, Offshore and Arctic Engineering (OMAE 2016), Busan, South Korea.

[4] A. Kalogirou, O.B. and D. Ham 2017: Modelling of nonlinear wave-buoy dynamics using constrained variational methods. Proc. ASME 2017 36th Int. Conf. Ocean, Offshore and Arctic Engineering (OMAE 2017), Trondheim, Norway.

This talk is part of the Isaac Newton Institute Seminar Series series.