Rodney Eatock Taylor

The second order diffraction force on a horizontal cylinder in finite water depth

The problem of second order diffraction by a horizontal cylinder in finite water depth is analysed. It is found that the overpressure in the free surface condition on the second order potential tends to an oscillating function at infinity on both sides of the cylinder, instead of zero or a constant as found in the case of infinite water depth. An equation for calculating the second order diffraction force is derived, similar to that in the three dimensional case based on first order potentials alone.2,4 Numerical results are obtained for the second order diffraction force on a submerged cylinder, based on the well known multipole expansion7 for the first order potential.

Time-dependent water-wave scattering by arrays of cylinders and the approximation of near trapping

We consider the solution in the time domain of water-wave scattering by arrays of bottom-mounted cylinders. It has already been shown that near trapping occurs for certain arrangements of cylinders and we are especially focused on this phenomenon. We begin with the well-known single-frequency solution to the problem of a group of cylinders, and the extension of this solution to complex frequencies. It has been shown that singularities (scattering frequencies or resonances) occur for certain values of the complex frequency and these singularities are associated with the near-trapped mode. We show that it is possible to approximate the solution near these singularities, and produce a modal shape which is associated with the near-trapped mode. We then consider the time-dependent problem, beginning with the well-known incident plane wave packet solution. We also show how the problem of an arbitrary initial displacement can be found using the single-frequency solutions. This latter result relies on a special inner product which gives a generalized eigenfunction expansion (because the operator has a continuous spectrum). We then consider the approximation of the time-dependent motion using special mode shapes associated with the scattering frequencies. This approximation relies on the scattering frequencies lying close to the real axis. We present numerical results which show that this approximation is accurate for sufficiently large time.

A finite element model of interaction between viscous free surface waves and submerged cylinders

Abstract This paper describes the simulation of the flow of a viscous incompressible Newtonian liquid with a free surface. The Navier–Stokes equations are formulated using a streamline upwind Petrov–Galerkin scheme, and solved on a Q-tree-based finite element mesh that adapts to the moving free surface of the liquid. Special attention is given to fitting the mesh correctly to the free surface and solid wall boundaries. Fully non-linear free surface boundary conditions are implemented. Test cases include sloshing free surface motions in a rectangular tank and progressive waves over submerged cylinders.

The Effect of Directional Spreading on Interaction of Focused Wave Groups With a Cylinder

The problem of diffraction of a directionally spread focused wave group by a bottom-seated circular cylinder is considered from the view point of second-order perturbation theory. After applying the time Fourier transform and separation of vertical variable the resulting two-dimensional non-homogeneous Helmholtz equations are solved numerically using finite differences. Numerical solutions of the problem are obtained for JONSWAP amplitude spectra for the incoming wave group with various types of directional spreading. The results are compared with the corresponding results for a unidirectional wave group of the same amplitude spectrum. Finally we discuss the applicability of the averaged spreading angle concept for practical applications.