Numerical modelling of OWCwave-power plants of the oscillating water column type A. Brito-Melo and A.J.N.A. Sarmento MARETEC, Instituto Superior Tkcnico, 1049-001Lisboa, Portugal Abstract A linear numerical model of the Boundary Element Method type has been applied in the numerical modelling of the oscillating water column (OWC) wave energy converter. The code was extended from a 3D conventional radiation-diffraction code originally developed for the hydrodynamic study of floating bodies in the frequency domain and has been adapted for generic configurations of OWC power plants using the theory of pressure distributions. This paper illustrates the application of the numerical model to the OWC pilot plant on the Island of Pico/Azores, including the discretization of the natural gully where the device is installed and to an OWC plant integrated in a semi- infinite vertical caisson breakwater is presented. 1 Introduction The utilization of wave energy for power generation was introduced in the 1970's. Since then the Oscillating Water Column (OWC) has been one of the most studied wave energy devices and one of the few that reached the stage of full-sized prototype. The OWC comprises a partly submerged structure with an opening on the front wall exposed to the incident waves, below the water free surface, thereby enclosing a column of air above the water line. The water column oscillates due to the incident waves thus forcing an alternate airflow to and from the atmosphere though a turbine coupled with an electric generator. Recently two demonstration OWC pilot plants were completed in European countries, e.g., the power plant on the Island of Pico, Azores [l],andthe LIMPET plant, at Islay, Scotland [2].
26 Bou~iw-~~ Elmcnts XXIV The research work on wave power plants of the OWC type has been based much on analytical models and experimental scale models. In firsts analytical studies the oscillating water column was modeled as rigid piston with a vertical motion. A significant contribution in the hydrodynamic modelling was given with the introduction of the spatial variation of the internal free surface byfalcfio & Sarmento [3] and generalised by Evans [4]. Several analytical studies mainly related with the OWC optimization have been performed with simple geometries. The experimental work with scale models has been carried out generally at the final phases of plant design (see e.g. 151). The use of the Boundary Element Method for the modeling of OWC devices was first showed by Lee et al. [6], extending the 3D panel code WAMIT from MIT to predict the response of a moonpool and a bottom- standing OWC either as an isolated device or integrated in a harbour. Threedimensional panel codes based on the boundary element method, extensively used in offshore engineering, revealed to be suitable to the OWC system with a required modification of the free surface boundary condition inside the chamber to account for the oscillatory pressure distribution. A numerical model was build around the WAMIT code bydelaurc & Lewis [7] for a generic bottom mounted OWC configuration and a detailed assessment of the model's accuracy was shown against experimental results. The conventional radiation-diffraction code developed for the study of floating bodies' hydrodynamics - AQUADYN - at Laboratoire de Mecanique des Fluides, Ecole Centrale de Nantes [8], has been extended to the OWC system [9] following the method presented by Lee et al. This code is able to compute the hydrodynamic coefficients for complex OWC geometries and together with a time domain model provides a very useful tool to obtain an estimation of the electrical power output for a range of realistic wave conditions [lo]. The aim of this paper is to illustrate the application of this extended code. Some difficulties linked with the highly resonance nature of the system response are pointed out. Results will be presented for the OWC Pilot plant in the Island of Pico, at Azores and for a generic OWC plant integrated in a semi-infinite breakwater. The application of the panel code in this case requires some care, since the code has been developed for isolated structures of finite length and a special arrangement has to be devised to avoid wave reflections at the downward end of the breakwater.
2 Hydrodynamic formulation 2.1 Problemstatement B o m d u r - yekww~tsxxiv 27 We consider the linear interaction of monochromatic waveswithan OWC device placed in a plane horizontal seabottom at depth h. We take a Cartesian co-ordinate system with the origin on the mean free surface, x, y horizontal and z positive upwards. The fluid domain is bounded by the seabed at z=-h, the body surface, (Sh),the water free surface in the interior, (S,), and exterior, (Se), of the chamber and a cylindrical control surface sufficiently far away. The flow is considered irrotational and the fluid incompressible. The fluid motion is given by a complex velocity potential, $(x, y, z), where the harmonic time dependence, e?' has been omitted. Laplace's equation and a set of boundary conditions must be satisfied everywhere in the fluid domain. Onthemean water free surface the following linearised Bernoulli equation is imposed: where p is the complex amplitude of the oscillatory pressure acting on the internal free surface, g is the gravitational acceleration and p the water density. On solid boundaries the normal component of the fluid velocity must be equal to the velocity of the surface. Also a suitable radiation condition of outgoing waves at infinity distance is required for q3. Under the assumption of linear theory it is convenient to consider the problem of the interaction between waves and an OWC device as a superposition of two problems: a diffraction problem corresponding to the interaction of an incident wave with the device assuming the chamber open to the atmosphere (i.e. without any pressure fluctuations inside the chamber) and a radiation problem associated only with the pressure distribution, in the absence of incident waves. Thus the velocity potential may be decomposed in the following manner requiring that eachcontribution satisfy the appropriate boundary conditions: This separation of the velocity potential is made so that gd = fio +q3d is associated with the diffraction problem, whereqbd is the diffracted wave and fio represents the undisturbed incident wave. The last component is the radiation potential due to the oscillating air pressure inside the chamber in otherwise calm waters and it may be given by the expression: 2Ep& Pg (3)
28 Bou~iw-~~ Elmcnts XXIV Here the potential q+ is interpreted as the complex amplitude of the radiated velocity potential generated by a unit amplitude oscillatory pressure distribution. This concept is introduced by analogy with the one associated with the radiated waves due to each of the six body oscillation modes of a conventional floating body and variables related to this problem are denoted by the subscript 7. The decomposition into elementary velocity potentials allow us to express the problem in terms of the diffraction and radiation problems, where it is necessary to solve a boundary value problem to, respectively, ed and g7. Therefore we have as follows: the body-boundary condition: @d - &bo 297-0 ---- (4) ih an an the sea-bed boundary condition: %Lo, az ~- az the free-surface boundary condition: 347-0 (5) Also the radiation condition of out-going waves at infinite distance is required for @d and for q+. We note that in the diffraction problem, where we assume that the chamber is held fixed and open to the atmosphere, the corresponding boundary equations are the same as for a floating body. The formulation for the radiation problem in terms of a potential q+ allows us to extend a conventional code developed for the analysis of floating bodies with six degrees of freedom to the OWC system, taking the pressure distribution oscillation as a seventh degree of freedom. 3 Numerical results The results presented in this section concerns the hydrodynamic coefficients associated with the pressure distribution problem namely the added mass, A7, and damping, B7,coefficients which are defined by (see Lee et al, 1996): The normal component, n7, isdefined as n7=lon S, and n7=0on Si. According to the linear theory, the interactions between waves and the OWC device may be represented, in the frequency domain, by transfer functions. The diffraction transfer function HDis defined as
B o m d u r - yekww~tsxxiv 29 where &(W) is the complex amplitude of the volume flow displaced by the incident wave inside the OWC when the chamber is open to the atmosphere and A(@;x.,yo)is the complex amplitude of the water free-surface elevation at the point (xo,yo).the radiation transfer function is defined as the ratio of the complex amplitude of the OWC radiation flow due to the pressure distribution inside the chamber in the absence of incident waves, QR, to the derivative of the complex air pressure, iwp, so that: Q&4 4&4=-iwpo > 3.1 Comparisonbetween AQUADYN and WAMIT Here we present a comparison of the results obtained by the extended version of AQUADYN with the predictions achieved by the MIT code WAMIT for a bottom-standing OWC device with projecting sidewalls forming an harbor in front of the chamber (Fig 2). The results presented in Fig 3 concerns the damping coefficient, B7,non-dimensionalised by pl3@ (L is half the length of the chamber measured along the wave propagation direction) against dimensionless wave number kl. The body was discretized into 1132 panels, corresponding to the medium discretization used by Lee et al., whereas the internal water free surface was meshed into 414 panels. We observe a good agreement between the results from the two codes where the major differences are near the resonant wavenumber where the convergence is more difficult to achieve due to the highly resonant behavior of this system. These results were obtained using a 4-point Gaussian method in the integration performed to obtain the influence coefficients as we found that with a 1-point method, used in the initial version of AQUADYN for floating bodies, the difference between the results was appreciable. Increasing the number of Gaussian points from 4 to 9 we do not find, in this geometry, any significative difference. We verified that the l-point method is accurate for larger walls thickness but it can fail with decreasing wall thickness, thus indicating an ill-conditioned system of equations.
30 Bou~iw-~~ Elmcnts XXIV Figure 2: Discretization of an OWC mounted on the sea bottom (Draft: 5 m; Horizontal dimensions of the interior chamber: 20 x 20 m'; Walls thickness: 0.5 m; Front wall submerged aperture: 3 m; Length of the projecting sidewalls: 20 m. Figure 3: Non-dimensional damping coefficient B7 (k 1 0 m) computed by AQUADYN and WfMIT. 3.2 Pico power plant The OWC Pico Plant in the Azores is a concrete structure with a chamber of 12x12m' square cross-section, standing on a small gully on the rocky sea bottom. The influence of the coastline contours in shoreline OWC plants, in comparison to the case of an isolated structure in a horizontal bottom is presented. On Fig 4 we can see two body meshes: the one of the isolated OWC device (on left) and the one with the surrounding coast included based on data of the bathymetry map of the area around the plant. In the remaining part of the domain (not discretized) a uniform depth of 7.8 meters (mean water level in situ) was assumed. The convergence was obtained with 920 panels at the OWC structure, 671 panels on the surrounding coast and 196 panels at the internal water free surface. In this case a l-point Gaussin Method has been used after checking the convergence with increasing the number of points. Fig. 4 shows the behaviour of the hydrodynamic coefficients A7 and B7 non- dimensionalised, respectively, by pl3 and pl3co, for the plant with the nearby coastline compared with those for an isolated device. Kt
~~~~~. ~~~~ 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. Figure 4: Discretization of the Pico Power Plant. Horizontal internal dimensions of the chamber: 12x 12 m2; Walls thickness: 2 m and 2.5 m (front wall); Front wall submerged aperture: 3 m. 16.0 14.0 12.0 10.0 2% 8.0 6.0 & 4.0 ppw 2.0 -B77 (isolated) ------A77(onshore) -B77 (onshore) 0.0-2.0-4.0 v l l ~~ v -6.0 0.0 0.5 1.0 1.5 2.0 2.5 Frequency (radk) Figure 5: Non-dimensional Added Mass and Damping coefficients (k12m) We observe that for the case of the isolated device, the B7 curve is characterized byan intense peak at CO = 0.56 radh and a much more less signiikant peak at W = 1.68radk When the coastline is considered, two more peaks appear at CO = 0.26 radh and OI = 1.00radk, while the already existing peak frequencies are slightly shifted. Fig 6 shows the modulus of radiation and diffraction transfer functions. As expected, from the precedent analysis of the hydrodynamic coefficients, the coastline cause, two more peaks, which are at about CO=0.3 rads, W = l.o m k The already existing peaks appear again, with slight shifts of the resonance frequencies as we noticed in the B7 curve. In the diffraction problem the amplitude of the higher resonance peak is relatively close in both cases presented in Fig. 6. In the radiation problem, however we observe that the amplitude of the higher resonance peak is significantly higher in the case of the isolated plant.
32 Bou~iw-~~ Elmcnts XXIV 0.0 0.5 1.0 1S Frequency (radk) 20 2.5 0.07 0.06.--. - 0.05 $ 'v1 0.04 "E v - 0.03 S - 0.02 0.01 0 0.0 0.5 1.0 1.5 2.0 2.5 Frequency (rads) Figure 6: Transfer functions modulus of Diffraction Problem (on top) and Radiation Problem (on bottom) for the Pico OWC isolated and in the coastline. 3.3 OWC integrated in a Caisson breakwater We consider an OWC integrated in the head region of a semi-infinite vertical caisson breakwater extending from the coast into the ocean in waters of 9 m depth. In the simulations a breakwater of finite length has to be considered. In order to check the influence of this parameter in the OWC hydrodynamic curves, thee breakwater lengths have been considered: k40.5, LF 67.5 and k94. 5m. The discretization of the breakwater with a length L equal to 67.5 m is shown in Fig 7 with a mesh of 901 panels. For k40.5 and k94.5m the number of panels was, respectively, 751 and 951, being the internal water free surface discretized into 231 panels. The modulus of the transfer function of the OWC diffraction problem, where strong reflections are expected to occur, is represented in Fig. 8 (on the left) for the thee meshed breakwater lengths in order to analyse how the limited breakwater extension can affect the results. We observe that the length of the breakwater affects the diffraction transfer function particularly in the resonant zone. This means that the incident wave reflects significantly on the
BULM~UI-~ Elemcwts XXIV 33 downward end of the breakwater. This reflection is produced due to the abrupt change of the reflection coefficient of the structure, which changes from 1 to zero at the downward end of the breakwater. To avoid this the breakwater is now assumed to be composed by two parts: the breakwater itself of length L with some permeability, including the plant at his head, and an artificial extension at the opposite end of the breakwater, whose permeability increases with distance in such a way that the reflection coefficient goes to zero at its extremity. The purpose of this artificial part is to create a zone of slowly varying boundary condition. The same three different lengths were considered and the length of the artificial part of the breakwater was taken to be 100 m. It was assumed a reflection coefficient equal l in the OWC walls, equal to 0.7 in the breakwater and varying from 0.7 to 0 with distance at the artificial rear part of the breakwater. We observe in Fig 8 (on the right) that the three curves merge together, which is interpreted as an indication that no strong reflections of the incident wave occur at the end of the breakwater (i.e. including the artificial permeable part). Therefore it is assumed that any of the breakwaters (with k40.5 m, 67.5 m or 94.5 m) represents a semi-infinite breakwater. Figure 7: Discretization of the vertical caisson breakwater with the OWC plant at its head. Plan interior area of the chamber: 10x 14m2, front wall submergence depth: 2.0 m, front wall thickness: 2.8 m. xnc, 1 7031 I 0.2 0.4 n.6 0.8 1.0 1.2 1.4 1.6 LR 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 FnxluencY@W Frequency (rads) Figure 8: Modulus of the diffraction transfer function. Reflection coefficient equal 1 over the entire breakwater surface (on the left). Reflection coefficient equal 1 on the OWC, 0.7 on the breakwater walls and going to 0 with distance in the artificial part (on the rigth).
34 Bou~iw-~~ Elmcnts XXIV 4 Conclusion A BEM code used in the analysis of conventional floating bodies was adapted to the OWC system. Thee main conclusions were illustrated in this paper: i) the convergence of the panel code may depend critically on the thickness of the structure, for a given discretization; thus the convergence should be carefully tested not only with the discretization but also with the number of points in the Gaussian Method usedin the numerical integration; ii) when simulating an OWC shoreline plant, the nearby coast needs to be included in the simulation to provide realistic results; iii) strong unrealistic reflections may result when an OWC device is in the vicinity of reflection walls such as nearby coastlines or breakwaters, which require special care in the handling of the reflection coefficient at these boundaries. References [l]falciio, A. F. de 0. The shoreline OWC wave power plant at the Azores. Proc. of the 4th European Wave Energy Conference, Aalborg, Denmark, pp. 42-48,2000. [2] Heath, T., Whittaker, T.J.T., Boake, C.B. The Design, Construction and Operation of the LIMPET Wave Energy Converter (Islay/Scotland). Proc. of the 4th European Wave Energy Conference, Aalborg, Denmark, pp. 49-55, 2000. [3] FalcZo, J., & Sarmento, A.J.N.A. Wave generation by a periodic surface pressure and its application in wave-energy extraction. Proc. of the 15th Int. Cong. Theor. Appl. Mech., Toronto, 1980. [4] Evans, D. V. Wave-power absorption by systems of oscillating surface pressure distributions..l. Fluid Mech. 114,481-99, 1982. [5] Sarmento, A.J.N.A. ModeLtest otptimization of an OWC wave power plant. International Journal of Offshore and Polar Engineering, 3, pp. 66-72, 1993. [6] Lee, C.-H., Newman, J.N., &L Nielsen, F.G. Wave Interactions with an Oscillating Water Column. Proc. of the 6th Int. Offshore and Polar Eng. Con$, Los Angeles, ISOPE, I, pp. 82-90, 1996. [7] DelaurC, Y.M.C., & Lewis, A. An Assessment of 3D Boundar Element Methods for Response Perediction of Generic OWCs. Proc. of the 10th Int. Offshore and Polar Eng. Cor$, Seattle, ISOPE, I, pp. 387-393, 2000. [8] Delhommeau, G. Les problkmes de diffraction-radiation et de rtsistance de vagues: Ctude thkorique et rtsolution numcrique par la mtthode des singularit&. Thbse de Docteur 2s Sciences, E.N.S.M. Nantes, 1987. [9] Brito-Melo, A., Sarmento, A.J.N.A., ClCment, A. H., & Delhommeau, G. A 3D Boundary Element Code for the Analysis of OWC wave-power Plants. Proc of the 9th Int Offshore and Polar Eng Cor$ Brest, France, ISOPE, I, pp. 188-195, 1999. [lo]sarmento, A.J.N.A., & Brito-Melo, A. An Experiment-Based Time- Domain Mathematical Model of OWC Power Plants, Int. J. Offshore and Polar Engng., 6(3),pp. 227-233, 1995.