P. McIver

Handbook of Mathematical Techniques for Wave/Structure Interactions

INTRODUCTION The Water-Wave Problem The Linearised Equations Interaction of a Wave with a Structure Reciprocity Relations Energy of the Fluid Motion EIGENFUNCTION EXPANSIONS Introduction Construction of Vertical Eigenfunction Two-Dimensional Problems Three-Dimensional Problems Matched Eigenfunction Expansions MULTIPOLE EXPANSIONS Introduction Isolated Obstacles Multiple Bodies INTEGRAL EQUATIONS Source Distribution Greens Theorem Thin Obstacles Interior Problems Free-Surface Problems Numerical Evaluation of Greens functions Diffraction by a Gap in a Breakwater Diffraction by an Insular Breakwater Embedding Formulae Numerical Solutions THE WIENER-HOPF AND RELATED TECHNIQUES The Weiner-Hopf Technique Residue Calculus Theory ARRAYS The Wide-Spacing Approximation SMALL OBJECTS Introduction Breakwater with a Gap Vertical Cylinder Heaving Cylinder Eigenvalue Problems VARIATIONAL METHODS Scattering and Radiation Problems Eigenvalue Problems APPENDICES Bessel Functions Multipoles Principle Value and Finite Part Integrals

Some hydrodynamic aspects of arrays of wave-energy devices

Abstract A large-scale wave-power station will have a number of devices in relatively close proximity to each other. Consequently, there will be hydrodynamic interactions between neighbouring devices which may modify significantly the power absorption characteristics of a given device relative to its performance in isolation. In this paper a brief review is given of existing theory concerned with rigid-body device arrays and some new calculations presented for arrays able to absorb power through heave or surge motions. Among the points considered are the improvement of performance from the unequal spacing of devices and the effects of constraining the amplitudes of device motions.

Wave interaction with arrays of structures

Recent progress in the computation and understanding of wave interaction with arrays of offshore structures is reviewed. The article focuses on new developments in computational methods, resonant effects in infinite arrays and their consequences for finite arrays, and nonlinear effects.

Trapped modes for off-centre structures in guides

The existence of trapped modes near obstacles in two-dimensional waveguides is well established when the centerline of the guide is a line of symmetry for the geometry. In this paper we examine cases where no such line of symmetry exists. The boundary condition on the obstacle is of Neumann type and both Neumann and Dirichlet conditions on the guide walls are treated. A variety of techniques (variational methods, boundary integral equations, slender-body theory, modified residue calculus theory) are used to investigate trapped-mode phenomena in a number of different frequency bands.

The dispersion relation and eigenfunction expansions for water waves in a porous structure

The Sollitt-and-Cross model of water-wave motion in a porous structure involves a free-surface condition which contains a complex parameter. This leads to two particular difficulties when this model is used in conjunction with eigenfunction expansion techniques. First of all the roots of the dispersion relation are themselves complex and therefore difficult to locate by standard numerical methods. Secondly, the vertical eigenfunction problem is not self-adjoint and standard expansion theorems do not apply. In this paper it is shown how these two difficulties may be resolved with the aid of the theories of, respectively, complex variables and non-self-adjoint differential operators. In particular, a method is described that allows the explicit calculation of the roots of the dispersion relation, and the appropriate expansion theorem is given.

On uniqueness and trapped modes in the water-wave problem for vertical barriers

Abstract Uniqueness in the linearised water-wave problem is considered for a fluid layer of constant depth containing two, three or four vertical barriers. The barriers are parallel, of infinite length in a horizontal direction, and may be surface-piercing and/or bottom mounted and may have gaps. The case of oblique wave incidence is included in the theory. A solution for a particular geometry is unique if there are no trapped modes, that is no free oscillations of finite energy. Thus, uniqueness is established by showing that an appropriate homogeneous problem has only the trivial solution. Under the assumption that at least one barrier does not occupy the entire fluid depth, the following results have been proven: for any configuration of two barriers the homogeneous problem has only the trivial solution for any frequency within the continuous spectrum; for an arbitrary configuration of three barriers the homogeneous problem has only the trivial solution for certain ranges of frequency within the continuous spectrum; for three-barrier configurations symmetric about a vertical line, it is shown that there are no correspondingly symmetric trapped modes for any frequency within the continuous spectrum; for four-barrier configurations symmetric about a vertical line, the homogeneous problem has only the trivial solution for certain ranges of frequency within the continuous spectrum. The symmetric four-barrier problem is investigated numerically and strong evidence is presented for the existence of trapped modes in both finite and infinite depth. The trapped mode frequencies are found for particular geometries that are in agreement with the uniqueness results listed above.

Scattering of water waves by axisymmetric bodies in a channel

Methods are presented for the calculation of wave forces on a vertically axisymmetric body arbitrarily placed within a channel. Integral representations of singular solutions of the Helmholtz equation, called channel multipoles here, are derived and these allow straightforward solution of the scattering problem for a vertical cylinder extending throughout the depth. In contrast to previous methods there is no need to sum series of images. These multipoles are also used in deriving an approximate solution valid when the radius of the cylinder is small relative to the wavelength and channel width.To solve for arbitrary shaped axisymmetric bodies, a plane-wave approximation is developed based on the assumption that the wavelength is much less than the channel width. Comparisons with the accurate solution for a vertical cylinder suggest that this approximate method performs well even when this assumption is clearly violated. The results of calculations of wave forces on a truncated cylinder are also given.All of the methods described may be applied just as easily to the case of an off-centre body as to a centrally-placed body.

The scattering of water waves by an array of circular cylinders in a channel

The full linear problem of the scattering of water waves by an array of N bottom-mounted vertical circular cylinders situated in a channel of constant depth and width is solved using the method of multipoles. A simple formula is derived for the velocity potential in the vicinity of a cylinder, and in particular on the cylinder surfaces, which allows hydrodynamic quantities such as forces to be easily evaluated. The simplicity of the solution makes the evaluation of quantities of interest straightforward and extensive results are given. An approximate solution for the forces on the cylinders, based on the assumption that the wavelength of the incident wave is long compared with the cylinder radii, is also given, and this is compared with results from the ‘exact’ linear solution.

WAVE-POWER ABSORPTION BY A LINE OF SUBMERGED HORIZONTAL CYLINDERS

This paper considers the radiation of water waves by a submerged, horizontal circular cylinder in a channel. The cylinder axis is perpendicular to the channel walls and the length of the cylinder is less than the width of the channel. The method of solution combines both multipole and eigenfunction expansions. The effects of finite cylinder length and of the channel walls are discussed and application is made to a line of Bristol cylinder wave power devices.

The wave field scattered by a vertical cylinder in a narrow wave tank

Abstract When a plane wave is incident on a fixed vertical cylinder standing in a narrow wave tank, the scattered wave field will be influenced by reflections from the side walls. This situation is investigated using an approximate solution for scattering by a cylinder of arbitrary cross-section derived under the fundamental assumptions that the cylinder diameter is much less than the tank width and the wavelength. The solution is used to examine in detail the wave field around the cylinder and to improve understanding of many of the tank-confinement effects observed in other work. Particular attention is paid to the hydrodynamic forces and pressures on the cylinder surface and situations where the tank walls may have a particularly significant effect are highlighted.