Behaviour of eigenmodes of an array of oscillating water column devices

Abstract

The eigenmodes of an array of wave energy converters is derived using the self-consistent method, which circumvents some of the problems of multiple scattering. The aim is a rapid, intuitive method of estimating the optimum array configuration. The equations of motion are derived in the time domain for an array of oscillating-water-column devices. The eigenmodes under the self-consistent method are solutions for the relative displacements of all devices. The interactions of the devices owing to wave radiation are accounted for by coupling-magnitude and time-delay constants relating the response of a device and that of an interacting partner. This gives a system of delay differential equations which is solved with and without time delays. The resulting coupled-oscillator model is used to study the eigenmodes of linear and triangular arrays. It is found that the inter-device spacing is the dominant control on the complex eigenfrequencies, the ordering of which can reverse as spacing varies. Identification of the dominant eigenmode of an array may lead to smaller individual machines, which are more economical to construct, install and maintain.

Introduction

It has been theoretically established for nearly four decades that the hydrodynamic interactions between ocean waves and multiple wave energy converters (WECs) makes it possible, by an appropriate arrangement of the WECs, to extract more power than from the same number of single isolated devices [1]. Conversely, an inappropriate arrangement can lead to less power than the same number of single isolated devices. This has led to substantial research into array design: where a number of WECs are arranged into a configuration with the aim of delivering maximum power. The optimisation of the array usually involves calculation of the $q$ factor [1], which is the ratio of the power actually delivered by the array to the power delivered by the same number of isolated devices. The majority of analyses are based on linear, inviscid wave theory, so that problems are reduced to potential flow.

Exact calculations of interactions between waves and WEC arrays [2], and thus determination of the $q$ factor, require an understanding of some principles of the complex problem of multiple scattering which is a well-known problem in many branches of physics [3]. Instead of treating the multiple scattering problem, alternative techniques are known [1], [4], [5], [6], but are not always accurate enough over all the different parameters with which an array is likely to operate [7]. Hence, the efficient calculation of problems involving multiple scattering is an important research topic in modelling arrays of WECs, as it is in other fields [8].

The self-consistent method is an established approach for dealing with multiple scattering in other areas of wave physics such as acoustics [8], [9]. One of the preliminary implementations of this approach in the case of many floating bodies in water waves can be found in the work of Kagemoto and Yue [10]; in the ocean-wave-power community, it is known as the Direct Inversion method, although it is nonetheless a self-consistent method. The essence of the self-consistent method lies in the assumption of a hypothetical situation at infinite time, after all orders of scattering have occurred, where one can express the equivalence between the incident and scattered quantity (i.e., wave potential) for a given scatterer in the array. The Direct Inversion method concentrated on evaluating the diffraction transfer matrix for the evaluation of unknown wave potentials. However, starting from the same first principles, one can construct a system of equations which would predict the hypothetical self-consistent displacement of all the devices. Retaining these coupled equations for the entire array in the time domain, one can in principle identify the possible eigenmodes which mark the mode of operation of the entire array. Then the hydrodynamic calculation can be simplified by associating the dominant eigenmode with a hypothetical machine which can represent the entire array as a whole, and whose natural frequency and damping coefficients are that of the eigenmode. This eigenmode approach has not hitherto been tried for WEC arrays.

In this paper, we construct equations in the time domain based on the self-consistent approach for an array of fixed Oscillating Water Column (OWC) devices. The OWC is a key technology which has received significant attention in the theory [11], the numerical investigation [12], [13] and the practice [14] of harnessing wave power. In the context of the present paper, the simplest possible form of OWC has the advantage of a very simple geometry and flow. The archetypal OWC is simply a rigid vertical tube fixed in inertial space, open at both ends, with a length $a$ immersed in water and the top open to the atmosphere. For infinitesimal, ‘piston-like’ displacements of water inside the tube and assuming the tube diameter is very small compared to the wavelength of any wave it creates, it is easy to show that the water inside oscillates with a radian natural frequency of $\sqrt{(\mathrm{g}∕a)}$, where g is the acceleration due to gravity. Once wave propagation is significant, as in the present paper, this natural frequency is modified.

Our aim is to identify the eigenmodes of this coupled set of differential equations. Moreover, the effects on a given device of wave radiation from the other devices in the array are represented by complex coupling constants that represent both the magnitude of each interaction and a finite time delay. The overall coupled Delay Differential Equations (DDEs) for the entire array yields a tool for the study of the behaviour of the eigenmodes.

In the following section, we firstly describe the coupled oscillator model for a generic WEC array. The solution strategies for these coupled DDEs both with and without finite time delays are then provided in Section 3. The developed model is then applied in Section 4 to investigate various possible eigenmodes of an array of OWCs. Three different array layouts are considered: one with two and another with three devices. We further investigate the behaviour of the eigenmodes as one changes the relative separations of the devices within the array: This has been performed for (i) Three devices in a line and (ii) Three devices at the apices of a triangle. The significance of the various eigenmodes on power and on the design of arrays is discussed.