A mathematical model was built to solve the interaction across an array of WECs, arranged in a number of periodic rows facing the incident waves. The interaction model is based on the (complex-valued) reflection and transmission coefficients of each row. Assuming that those coefficients are known, and that the row-to-row distance is not too small with respect to the wavelength, the so-called wide-spacing approximation can be employed, which allows for a recursive calculation of the array reflection and transmission coefficients, whereby rows are added iteratively. In the case of identical rows, new asymptotic formulae were derived for an infinite number of rows. The interaction model was presented and validated based on experimental data.
Now that we can solve the array interaction relatively easily, there remains to be able to calculate the reflection and transmission coefficients of each row, depending on the geometrical parameters, mode of deformation and control strategy. To that end, modelling OWSCs as thin, vertical blades, a mathematical technique called “matched eigenfunction expansion” was used. Simply put, on each side of the barrier formed by the row of vertical structures, the fluid motion is written as a weighted combination of known mathematical functions (eigenfunctions) satisfying the different physical equations within the fluid, at the water bottom and at the free surface. Then, the weights on each side of the barrier can be uniquely determined by using “matching” conditions across the barrier (for example, continuity outside the structure, or zero velocity across the structure). Using that approach, the hydrodynamic problem for a row of vertical blades can be solved, taking into account the blade mode of motion (rigid or flexible) (Figure 2).
Instrumental to our modelling and optimisation approach, is a novel geometrical representation (Figure 3) for the behaviour of a row of OWSCs, which articulates the trade-offs between reflection, transmission and absorption of wave energy, depending on the OWSC shape and control parameters. More specifically, representing the transmission coefficient in the complex plane, for a given geometry and control parameter tuning, allows for visualising immediately the transmission, reflection, and hydrodynamic efficiency (i.e. the absorbed relative to incident wave power) It was found that this geometrical representation, adopted to represent a row of OWSCs, can be generalised to any other oscillating WECs with a symmetric shape, thus providing a useful and didactic tool to articulate hydrodynamics and control of WECs.
With a tool to determine and visualise the properties of individual rows, and another one to solve the interaction when those rows are combined into a large array, it is possible to start looking for optimised parameters which, for example, maximise power absorption, or minimise the wave energy reaching the coastline behind the array. Doing so, interesting effects are found: When the row-to-row distance is a multiple of half a wavelength, the waves are prevented from propagating across the array, and almost all the wave energy is reflected back to the sea. This crystallographic effect is found in other areas of physics, optics in particular.
However, another finding from those preliminary optimisation studies was that optimised parameters make the blades undergo unrealistically large motion, which could not take place in a real experiment - or, if they did, would lead to critically damaging the devices. Thus, it became clear that including losses in the fluid-structure interaction model would be instrumental for the optimisation to be realistic. Indeed, when the relative velocity between the devices and the fluid is large, vortices are shed, especially near sharp edges and corners (Figure 4). The resulting forces tend to dampen the motion, and energy is transferred to smaller scales to be eventually dissipated in the form of heat. Unfortunately, traditional approaches to model those effects onto WECs typically represent them in the form of a quadratic force onto the structure, which does not quantify the energy dissipation and how it affects the wave field (in particular, reflected and transmitted waves). Yet, in our case, it is crucial to represent how wave reflection and transmission would be affected, since it governs how the rows of WECs interact with each other. We thus represented vortex-induced dissipation in the form of a virtual porous screen near the edges of the vertical structures, and we incorporated this representation in our matched eigenfunction expansion approach. Since this type of modelling approach had never been experimentally validated, a dedicated experimental campaign was carried out (Figure 5). The influence of vortex-induced dissipation onto reflected and transmitted waves is successfully represented by the chosen modelling approach.
