Cylindrical linear water waves and their application to the wave-body problem

Abstract

The interaction between water waves and a floating or fixed body is bi-directional: wave forces act on and cause motion in the body, and the body alters the wave field. The impact of the body on its wave field is important to understand because: 1) it may have positive or negative consequences on the natural or built environment; 2) multiple bodies in proximity interact via the waves that are scattered and radiated by them; and 3) in ocean wave energy conversion, by conservation of energy, as a device absorbs energy, so too must the energy be removed from the wave field. Herein, the cylindrical solutions to the linear wave boundary-value problem are used to analyze the floating body wave field. These solutions describe small-amplitude, harmonic, potential-flow waves in the form of a Fourier summation of incoming and outgoing, partial, cylindrical, wave components. For a given geometry and mode of motion, the scattered or radiated waves are characterized by a particular set of complex cylindrical coefficients. A novel method is developed for finding the cylindrical coefficients of a scattered or radiated wave field by making measurements, either computationally or experimentally, over a circular-cylindrical surface that circumscribes the body and taking a Fourier transform as a function of spatial direction. To isolate evanescent modes, measurements are made on the free-surface and as a function of depth. The technique is demonstrated computationally with the boundary-element method software, WAMIT. The resulting analytical wave fields are compared with those computed directly by WAMIT and the match is found to be within 0.1%. A similar measurement and comparisons are made with experimental results. Because of the difficulty in making depth-dependent measurements, only free-surface measurements were made with a circular wave gauge array, where the gauges were positioned far from the body in order to neglect evanescent modes. The experimental results are also very good. However, both high-order harmonics and wave reflections led to difficulties. To compute efficiently the wave interactions between multiple bodies, a well-known multiple-scattering theory is employed, in which waves that are scattered and radiated by one body are considered incident to another body, which in turn radiates and scatters waves, sending energy back to the first. Wave fields are given by their cylindrical representations and unknown scattered wave amplitudes are formulated into a linear system to solve the problem. Critical to the approach is the characterization of, for each unique geometry, the cylindrical forces, the radiated wave coefficients, and the scattered waves in the form of the diffraction transfer matrix. The method developed herein for determining cylindrical coefficients is extended to new methods for finding the quantities necessary to solve the interaction problem. The approach is demonstrated computationally with WAMIT for a simple cylinder and a more complex wave energy converter (WEC). Multiple-scattering computations are verified against direct computations from WAMIT and are performed for spectral seas and a very large array of 101 WECs. The multiple-scattering computation is 1,000- 10,000 times faster than a direct computation because each body is represented by 10s of wave coefficients, rather than 100s to 1,000s of panels. A new expression for wave energy absorption using cylindrical coefficients is derived, leading to a formulation of wave energy absorption efficiency, which is extended to a nondimensional parameter that relates to efficiency, capture width and gain. Cylindrical wave energy absorption analysis allows classical results of heaving and surging point absorbers to be easily reproduced and enables interesting computations of a WEC in three-dimensions. A Bristol Cylinder type WEC is examined and it is found that its performance can be improved by flaring its ends to reduce "end effects". Finally, a computation of 100% wave absorption is demonstrated using a generalized incident wave. Cylindrical representations of linear water waves are shown to be effective for the computations of wave-body wave fields, multi-body interactions, and wave power absorption, and novel methods are presented for determining cylindrical quantities. One of the approach's greatest attributes is that once the cylindrical coefficients are found, complex representations of waves in three dimensions are stored in vectors and matrices and are manipulated with linear algebra. Further research in cylindrical water waves will likely yield useful applications such as: efficient computations of bodies interacting with short-crested seas, and continued progress in the understanding of wave energy absorption efficiency.