The Method of Imbedded Lagrangian Element to Estimate Wave Power Absorption by Some Submerged Devices

Gérard C. Nihous

The Method of Imbedded Lagrangian Element to Estimate Wave Power Absorption by Some Submerged Devices

Gérard C. Nihous＊

Department of Ocean and Resources Engineering, University of Hawaii, Honolulu HI 96822, USA

A simple approach is described to estimate the wave power absorption potential of submerged devices known to cause wave focusing and flow enhancement. In particular, the presence of a flow-through power take-off (PTO) system, such as low-head turbines, can be accounted for. The wave radiation characteristics of an appropriately selected Lagrangian element (LE) in the fluid domain are first determined. In the limit of a vanishing mass, the LE reduces to a patch of distributed normal dipoles. The hydrodynamic coefficients of this virtual object are then input in a standard equation of motion where the effect of the PTO can be represented, for example, as a dashpot damping term. The process is illustrated for a class of devices recently proposed by Carter and Ertekin (2011), although in a simplified form. Favorable wave power absorption is shown for large ratios of the LE wave radiation coefficient over the LE added mass coefficient. Under optimal conditions, the relative flow reduction from the PTO theoretically lies between 0.50 and 12≈ 0.71, with lower values corresponding to better configurations. Wave power capture widths, the sensitivity of results to PTO damping and sample spectral calculations at a typical site in Hawaiian waters are proposed to further illustrate the versatility of the method.

hydrodynamics; wave power absorption; wave energy conversion; power take-off (PTO); Lagrangian element (LE); hydrodynamic coefficients

1 Introduction1

While the practicality and commercial viability of wave energy converters (WEC) remains to be unambiguously established, the extraction of some of the mechanical power transported across vast distances by surface water waves has challenged the scientific and engineering communities since the pioneering days of Salter (1974). WEC classification can be as complex and diverse as the devices themselves. In broad terms, overtopping systems rely on the wave field to supply a shallow reservoir, and power can be produced with a low-head hydraulic turbine. The mechanical energy of wave activated bodies can be partially tapped by various power take-off (PTO) mechanisms, the active component of which is in phase with the body velocity. In this case, the use of hydraulic components and gas accumulators has often been proposed because of practical advantages, such as the availability of high pressures for PTOs and power smoothing (Falc?o, 2007). Recent analyses of specific types of wave activated bodies are given, for example, in Falnes and Hals (2012) and Renzi and Dias (2013). An oscillating water column (OWC) consists of an air chamber located at the sea surface; the vertical oscillation of the internal air-water interface can drive an air turbine. Details about the state-of-the-art of WEC design and development can be found elsewhere (Cruz, 2008; Falc?o, 2010).

Carter and Ertekin (2011) were recently awarded a U.S. patent for a class of submerged wave absorption devices based on wave focusing above, and flow enhancement through an opening in the structure underneath. The preferred PTO for such systems is a low-head hydraulic turbine located in the vicinity of the central opening, and a typical structure consists of a submerged horizontal circular disc. The purported advantages of this novel idea are well documented in the patent description. Carter (2013) recently completed an extensive series of laboratory-scale experimental tests to characterize and confirm the basic hydrodynamic behavior of such systems based on surface-wave focusing and flow enhancement. Supporting calculations were also performed using linear potential theory. Carter’s work was conducted for submerged discs with central openings extending downward into short tubular sections; this geometry facilitated measurements and may actually represent a practical configuration for housing a turbine. Both experiments and calculations, however, did not explicitly account for any PTO. At about the same time, Korde and Ertekin (2014) published a comprehensive study analyzing the performance of a wave-focusing power conversion system belonging to the class described in Carter and Ertekin (2011). The effects of wave-induced motions, and provisions for several control mechanisms were included. The PTO, however, was represented by a ducted water column under the central opening connected to an underwater air chamber with an air turbine. This system therefore bears some similarity with an OWC, as far as the PTO is concerned. All work to date has demonstratedsignificant potential for the concept of Carter and Ertekin (2011).

The goal of the present study is to explicitly allow for a hydraulic turbine in the fluid domain. In OWC analyses, the PTO (air turbine) is located outside of the fluid domain and only acts to modify the oscillating pressure in the air chamber. Therefore, the hydrodynamic problem can be modified by changing the boundary condition on a patch of the free surface, inside the air chamber (Evans, 1982). It is beyond the scope of this article to represent details of the flow through a water turbine. Instead, a method is proposed and described in Section 2, to represent some salient features of water turbine behavior by making pressure discontinuities within the fluid domain possible. It is a two-step process involving the solution of a straightforward wave radiation problem followed by that of a very simple equation of motion. For the sake of clarity, the geometry of the submerged structure is simplified to a horizontal annulus, which is assumed to be stationary and not subject to any controls (other than a simplified PTO). It should be noted that wave radiation from a virtual element located in the central opening of the structure is a fundamental mechanism in the proposed method, even though this particular aspect of the problem is often neglected (e.g., Korde and Ertekin, 2014). In Section 3, the method is validated for flows due only to an incident wave (no structure, no PTO), or without PTO (like in Carter, 2013). Results including a PTO are presented and discussed in Section 4, before concluding remarks are offered in Section 5.

2 The method of imbedded Lagrangian element (MILE)

The concept underlying the method of imbedded Lagrangian element (MILE) defined in this article is very simple. It consists in isolating a specific fluid cell in a Eulerian fluid domain, and to separately solve for its motion (Lagrangian element, or LE). The size and shape of the cell are arbitrary a priori, but they should fit the purpose at hand. When Lagrangian elements are meant to represent the water turbines used to generate power by some submerged wave energy converters, the goal is to allow a pressure discontinuity in the fluid while preserving flow-through continuity. In such cases, the Lagrangian element may be thin in a direction generally understood as streamwise. Any flow variation along its upstream or downstream surfaces is effectively neglected (i.e., unresolved); in other words, fluid motion there is replaced by the motion of the Lagrangian element, which can be understood as an averaging process.

To fix ideas, we consider a simple wave focusing device consisting of a fixed thin circular structure of radius R and thickness b submerged at average depth d where the vertical wave-induced flow can be magnified through a central circular opening of radius a. A Lagrangian element of choice to represent wave power absorption by a hydraulic turbine placed in the central opening consists of a very thin (water) disk of radius a located there. A vertical cross section of this basic axisymmetric problem is shown in Fig. 1. It is understood that if the incident wavelength becomes small enough, the net flow across the central opening would drop since vertical velocities would not be quasi-unidirectional any longer (internal seiching); such conditions would be very unfavorable for wave power conversion and should be avoided. A characteristic of a hydraulic turbine is the damping coefficient which linearly relates the pressure drop across the turbine to the volume flow rate, for operational conditions where such proportionality is justified. When solving for the motion of the Lagrangian element, then, the effect of pressure differences across the turbine can be described by imposing a net resistive vertical force proportional to the velocity of the Lagrangian element (linear dashpot term). The corresponding damping coefficient C (N·s·m-1) would be equal to the product of the turbine damping coefficient (N·s·m-5) by the square of the flow area (m4). The power extracted via the dashpot term, i.e., damping force times velocity, is exactly the same as the hydraulic power extracted by the turbine, i.e., pressure drop times volume flow rate. Turbine efficiency is not considered here, but as a non-causal nonlinear effect, it could easily be accounted for as a post-processing step in the time domain (see for example, Nihous, 2012).

Fig. 1 Geometry of problem illustrating MILE application

We proceed to write the equation of motion of the Lagrangian element, in the heave direction for the example under consideration. The theoretical background adopted here is linear potential theory in the frequency domain, where amplitude and phase are represented by a complex number. Linear physical quantities are understood as the real parts of these complex values. Consider an incident linear wave of amplitude A, circular frequency ω and wavenumber k propagating along the x-axis over water of finite depth h and uniform density ρ. In numerical applications, ρ is set at 1 000 kg·m-3with little loss of generality. The time dependence exp(–iωt) is chosen and generally factored out for convenience. The vertical z-axis is positive upward with its origin at the free surface; when depth or submergence is invoked, however, it is understood as an absolute value. Velocity potentials ?, such that the velocity vector V is equal toφ?, all satisfy Laplace’s equation in the fluid domain(2φ?= 0) and specified boundary conditions on the seafloor (?? /?z = 0 at z = –h) and free surface (?? /?z – ω2g–1? = 0 at z = 0). The velocity potential ?Ifor the incident wave field may be written as (e.g., Newman, 1977):

where g is the acceleration of gravity equal to 9.81 m·s-2. A dispersion relation also holds, i.e., ω2= gk×tanh(kh).

Owing to linearity, the hydrodynamic force is expressed as the sum of a radiation term FRproportional to the unknown complex heave amplitude X, and of a scattered term FScaused by the incident and diffracted wave fields. FRtraditionally is expressed as (iωλ + ω2μ), where the radiation damping coefficient λ and the added mass coefficient μ are both real functions of ω. To determine λ and μ, the radiation potential ?Rcorresponding to the unit vertical displacement of the Lagrangian element must be determined (for a given stationary submerged structure). The boundary value problem to be solved is straightforward, and in the particular case under consideration, completely axisymmetric. Additional boundary conditions are specified for ?R: on the LE surfaces, the normal fluid velocity is equal to the vertical component of the normal vector, which points out of the fluid, and at large horizontal distances R∞from the body, a Sommerfeld radiation condition of outgoing waves is imposed. Calling m the mass of the Lagrangian element of thickness ξ, i.e., m = ρπa2ξ, and collecting terms proportional to X, the equation of motion can be written:

{–ω2(m +μ) – iω (C +λ)}X = FS(2) For a heaving axisymmetric body, the modulus of FSis equal to 2A(ρgcgλ/k)1/2, where cgis the wave group speed defined as dω/dk (e.g., McIver, 1994). Replacing FSby FSin Eq. (2) simply defines the phase of FSas a reference. We then obtain:

The time-averaged power extracted by the device is equal to P= Cω2X2/2. It follows that:

The power flux in the incident wave field is PI= ρgcgA2/2. Dividing P by PIyields the absorption (or capture) width of the device W:

For a given incident wave, a necessary condition forP or W to be maximal is obtained by setting the partial derivative of the right-hand-side of Eq. (4) or (5) with respect to C to zero. This straightforward step yields. After substitution of this result in Eq. (5), we obtain:

It can be verified that the maximum absorption width expressed in Eq. (6) would never reach the theoretical limit for a heaving axisymmetric body, i.e. Wmax＜ k–1(e.g., Evans, 1976). This is due to the absence of resonance for submerged bodies (by contrast, floating bodies have a hydrostatic restoring term in their equation of motion which allows resonant conditions). It is also apparent from Eq. (6) that favorable configurations for wave power extraction correspond to large ratios of λ over μ.

An interesting point to assess is the ratio η of the motion amplitude for maximum wave power absorption conditions Xoptover that for the unconstrained device,X0when the power take-off (PTO) damping coefficient C is set to 0. This ratio indicates the reduction in flow amplification through the central opening of the structure when wave energy is optimally converted by a turbine. After some elementary algebra using Eq. (3), η is given as:

Before proceeding, it should be noted that a linear relationship between pressure drop across the turbine and volume flow rate is equivalent to Eq. (2) in the limit where m tends to zero. This is because fluid inertia in accelerating flows typically is not accounted for a priori in simple representations of turbine behavior, especially with C real. For the sake of consistency, then, the inertial mass m of the Lagrangian element should be chosen as small as possible, i.e., it should be verified that m ＜＜ μ. As a corollary, the hydrodynamic coefficients λ and μ should tend to well-defined limits as m gets small. Under these conditions, the hydrodynamic behavior of a Lagrangian element of vanishing thickness is equivalent to that of a surface patch of distributed normal dipoles, the strengths of which are proportional to the jump in velocity potential across the patch (see, for example, Martin and Farina, 1997).

3 Validation tests

MILE was tested in simple cases when no external damping force is exerted on the Lagrangian element (C = 0). Two fluid domains of very different sizes were defined. The first one, used only for the test labeled Small scale, may be representative of laboratory conditions, with a water depth of 1 m and a remote boundary R∞located at a radial distance of 10 m. The second one may correspond to at-sea field conditions, with a water depth of 50 m and a remote boundary located at a radial distance of 500 m. Three tests were considered: Small scale and Baseline both have a submerged structure defined by a, R, b at average submergence d, the values of which are listed in Table 1; No structure consists of fluid only. The targeted quantity forcomparative purposes is V, the amplitude of the average vertical fluid velocity through a circle of radius a located at depth d (central opening when a structure is present).

Table 1 Parameter values for MILE computations (see text for details)

for Small scale and Baseline were obtained by solving for the three-dimensional diffraction potential ?Din the presence of the incident wave field defined by Eq. (1) with A = 1 (unit amplitude); the vertical velocity from the sum of incident and diffracted wave fields was then averaged through the central opening. Additional boundary conditions specified for ?Dare that on the LE surfaces, the normal derivative of ?Dis opposite the normal derivative of ?I, and at large horizontal distances R∞from the body, a Sommerfeld radiation condition of outgoing waves holds.

In the absence of structure (No structure), no diffraction of the incident wave takes place and the reference for Vcan be analytically determined from Eq. (1). We find in this case that:

where J1is the well-known Bessel function of the first kind, order 1. The first zero of Voccurs for ka ≈ 3.83. From the perspective of wave energy conversion, meaningful calculations should only consider smaller values of ka. It should be noted that the heave radiation of a submerged horizontal circular disk of zero thickness in deep water has been studied analytically before Martin and Farina (1997). The solution procedure is elegant, and only involves a Fredholm integral equation of the second kind. Notwithstanding the influence of water depth, such a study provides a useful benchmark for the hydrodynamic coefficients estimated in the absence of structure (No structure).

MILE computations require the solution of unit-heave axisymmetric radiation problems to determine the added mass and damping coefficients μ and λ. The Lagrangian element of radius a and thickness ξ located at depth d is treated as a solid disk oscillating vertically with unit amplitude, while the structure (if any) is fixed. Eq. (3) then yields X (with A = 1 and C = 0 here), and thus,V= ωX.

All velocity potentials were numerically evaluated using the commercial finite-element solver FEMLAB?3, Version 3.1 (COMSOL, Inc., 2003). The diffraction potentials ?Dused in this section to determine references for Vrequired a three-dimensional domain since the boundary condition on the LE surfaces is not axisymmetric. The radiation potentials ?Rin all MILE computations could be obtained with a two-dimensional domain (radial and vertical coordinates), as long as the boundary value problem was recast accordingly. This resulted in great numerical efficiency while allowing high accuracy. Typically, about half a minute per wave frequency was necessary on a modest single-processor machine with a speed of 1.86 GHz, for unstructured finite-element grids of the order of 40 000 elements. To verify energy conservation, the wave radiation damping coefficient λ was always computed following two methods, either from direct pressure integration on the LE surfaces (according to the definition of λ and μ), or using the relationship between λ and the calculated far-field (r = R∞) radiated wave amplitude in the Sommerfeld boundary condition (e.g., McIver, 1994).

Fig. 2 shows Vas a function of ka. The references, i.e., numerically calculated three-dimensional flow for Small scale and Baseline or Eq. (8) for No structure, compare very well to MILE computations. Note that two ordinate scales have been used for clarity since a unit amplitude wave (A = 1 m) in the small fluid domain is only theoretical (amplitudes of centimeters would be more likely in a real situation). Furthermore, it can be seen in Fig. 3, where quantities have been non-dimensionalized by ρa(bǔ)3, that the inertial masses m of the Lagrangian elements tested here (from 0.013 to 0.126) are much smaller than the hydrodynamic added mass μ. Once more, two ordinate scales were used so that the convergence of the results to some limit as m gets smaller can better be appreciated for No structure. In this case, deep water calculations based on the solution of Martin and Farina (1997) when m = 0 are also presented; following the Nystr?m method, the kernel of the key Fredholm integral equation in their analysis was estimated at Gauss-Legendre quadrature nodes over the interval [-1, 1]; node coordinates and weights were determined using an algorithm from Press et al. (1992). Wenote that water depth effects are hardly discernible, with h = 50 m, and that the LE submergence (d = 10 m) is sufficient for μ to approach the unbounded fluid limit of 8/3 ≈ 2.67 (Lamb, 1932). Fig. 4 illustrates similar results for the damping coefficient. It can be seen from Figs. 3 and 4 that the ratio of LE damping coefficient over LE added mass coefficient is much larger with the Baseline structure than with No structure. As suggested earlier from Eq. (6), this indicates a superior wave power absorption potential when the structure is present.

Fig. 2 Average free-flow (C = 0) vertical velocity Vthrough central opening as a function of wavenumber in MILE validation tests

Fig. 3 Sample calculations of the heave added mass coefficient for the disc-shaped Lagrangian element as a function of wavenumber

Fig. 4 Sample calculations of the heave radiation damping coefficient for the disc-shaped Lagrangian element as a function of wavenumber

4 Results and discussion

To illustrate its versatility and robustness, the MILE methodology was implemented for simple structures of the type described in section 2, i.e., stationary horizontal annuli with a focus on field scale dimensions. The different scenarios envisioned here do not define a thorough parametric investigation. The few cases considered in Table 1 are merely meant to confirm rough trends. In the absence of a PTO, more details can be found in Carter (2013). To highlight the wave focusing and flow enhancement capabilities of the structures, results for a No structure scenario are generally provided.

The first set of calculations concerns the free-flow (C = 0) characteristics of various configurations. Fig. 5 showsV= ωX, and all structures generate significant flow enhancement trough the central opening of radius a. The outer radius of the structure R has a strong effect, with greater values ofVfor wider structures as long as the wavelengths roughly exceed 4R (two diameters); in other words, the Baseline and Wide cases exhibit differences for wavelengths greater than about 60 m (ka ≈ 0.5 to 0.6), while the Baseline, Wide and Narrow cases are distinct for yet smaller wavelengths (ka ≈ 0.8 to 0.9). The relative velocity boost due to shallower structure submergence is very pronounced (Shallow versus Baseline), which highlights both stronger wave making by – and wave forcing on – the LE element at reduced water depths. The positive effect of an outer vertical ring wall (bringthick and ξringtall) was postulated as Claim 17 in Carter and Ertekin (2011), and is confirmed here (Ring versus Baseline). As can be seen for scenarios Small, Baseline and Big, the size of the central opening (radius a) greatly impacts V, with higher values for smaller a. Caution should be exercised in this case, however, since the trend is actually inverted if one considers average flow rate through the opening (πa2V). Moreover, using the abscissa ka in Fig. 5 artificially shifts the three corresponding sets of values even though the three configurations would show good dimensional frequency alignment.

Fig. 5 Average free-flow (C = 0) vertical velocity V through central opening as a function of wavenumber for various configurations (cf. Table 1)

So far, no provision has been made for wave power conversion, and MILE would be utterly unnecessary with just C = 0. In fact, free-flow situations theoretically correspond to P= 0, even if they provide a strong indication of possibilities, since flow enhancement with C = 0 corresponds to vanishing pressure drops for the LE. By contrast, Figs. 6, 7 and 8 respectively show the optimum motion amplitude ratio η, the maximum absorption width Wmaxand the maximum power output Pmaxin unit-amplitude (A = 1 m) incident waves. All results are plotted as a function of wavelength. The correspondence between wavelength and wave period is given for convenience in Table 2.

Fig. 6 Motion amplitude ratio η for maximum wave power absorption as a function of wavelength for various configurations (cf. Table 1)

Fig. 7 Maximum absorption width Wmaxas a function of wavelength for various configurations (cf. Table 1)

Fig. 8 Maximum power extracted from unit amplitude (A = 1 m) incident waves as a function of wavelength for various configurations (cf. Table 1)

Fig. 8 is merely shown to provide an additional dimensional perspective (power units); it is obtained from Fig. 7 via a multiplication of the ordinate by the incident wave power flux PI(the two sets of curves are very similar, with a stretching of the ordinate range in Fig. 8 at more energetic longer wavelengths). The theoretical limit for any heaving axisymmetric system (single degree-of-freedom), i.e. wavelength divided by 2π, is also plotted in Fig. 7 to illustrate the excellent performance of some configurations at small to moderate wavelengths. The relative effect of R is definitely confirmed, with greater maximum power predicted for wider structures, which also peaks at greater wavelengths. Of course, the practicality of greater overall structure size ultimately would require a complete system analysis and cost estimation. The outstanding advantage of shallower submergence d is also reaffirmed, with the configuration Shallow nearly following the theoretical limit up to wavelengths exceeding 100 m. The most interesting variance between inferences from free-flow results (C = 0) and optimal power absorption (C = Copt≠ 0) relates to the influence of the central opening radius a. Even though a relatively smaller value of a resulted in a high free-flow average velocity (see Small in Fig. 5), maximum wave power absorption performance for the same scenario is relatively poor. This primarily illustrates the preponderance of volume flow rate, rather than average velocity, in power output calculations. Fig. 6 shows that the relative motion amplitude reduction with an optimized PTO lies in a narrow range for all cases. Because of the form of Eqs. (6) and (7) where the ratio of λ over μ is prominent, however, lower values of η also correspond to higher power outputs.

Table 2 Correspondence between wavelength (m) and wave period (s) for selected values

The results in Fig. 8 correspond to systems where the coefficient C is optimal for each wave frequency (C = Copt). The choice of PTO damping in real seaways should be based, in principle, on predicted power output when the system is simultaneously exposed to a range of spectral wave components representative of the seaway. This methodology may even extend to a particular site’s wave climate (Duclos et al., 2006). It turns out that the simple devices under consideration here are not very sensitive to C. Fig. 9 shows little differences between power outputs when the PTO damping has been optimized for each wave, or when C has been picked for a specific wavelength of interest (e.g., 126 m corresponding to ka = 0.25 and C/ρ = 510 m3·s-1in the Baseline scenario). To be more specific, power output is relatively insensitive to overdamping; in other words, it decreases slowly as C exceeds Copt. Moreover, the power maxima are quite broad in the neighborhood of Copt, especially for smaller wavelengths.These features can be appreciated in Fig. 10, where the variation of P with C/ρ is displayed for several wavelengths in the Baseline scenario.

Fig. 9 Maximum power versus power when output is maximized for 126 m wavelength (Baseline configuration)

Fig. 10 Sensitivity of power output to PTO damping coefficient for several wavelengths (Baseline configuration)

With these points in mind, typical spectral calculations were carried out for the Baseline scenario and a fixed value of the PTO damping, i.e. C/ρ = 510 m3·s-1. A representative site was selected in Hawaiian waters off of Kaneohe, on the northeast (Windward) coast of the island of Oahu, where a water depth of 58 m is comparable to the value chosen in MILE calculations (h = 50 m). Twelve monthly average wave spectra were constructed from hourly local values obtained from hindcast simulations with the shallow-water wave propagation model SWAN in a nested domain; the boundary conditions were provided by the global model WaveWatch3 (WW3), and high-resolution wind forcing by the weather research and forecast (WRF) model. Details can be found in Stopa et al. (2011). Although such hindcast wave data is available over a period of ten years, only Year 2009 was used, with little loss of generality. Fig. 11 shows the twelve monthly plots of wave power spectral density S(ω) at the Kaneohe site. The winter months, November through March exhibit significant northern swell contributions, e.g., with wave periods often exceeding 12 s (ω ＜ 0.5 rad·s-1). Easterly to Northeasterly Trade Winds are present most of the year, but more consistently through the summer; they locally generate waves of smaller periods, e.g. between 6 and 9 s (ω in a range of 0.7 to 1.0 rad·s-1). Overall, the Kaneohe site can be considered to have modest to moderate wave resources since the monthly average significant wave heights range between 1.15 m and 2.2 m. Note that the Baseline configuration would absorb a maximum time-averaged power PA=1of nearly 150 kW in unit-ampitude waves when ω = 0.7 rad·s-1(ka = 0.25). Single unit-amplitude waves correspond to an average energy density of ρg/2 (since A2= 1), while the spectral band of the same frequency has an energy density equal to ρgS(ω)dω. Therefore, the spectral output of the device is given as:

Fig. 11 2009 monthly average wave spectra at a typical site in Hawaiian waters (Kaneohe, island of Oahu)

Fig. 12 Monthly average power at Kaneohe when output is maximized for 126 m wavelength

Fig. 12 displays the predicted monthly average power outputs from the Baseline configuration at Kaneohe. The asymmetry of available wave power resources between the summer (lower) and the winter (higher) is very clear. The yearly average power output is 32 kW, with monthly values ranging from 16 to 56 kW. Such results may seem disappointing when compared to the maximum of PA=1for the device (if this maximum were used for power rating, the capacity factor at Kaneohe would average 22%, and range between 10% and nearly 40%). Of course, the Kaneohe site is not exceptional and the Baseline configuration is certainlynot an engineering optimum; its submergence d of 10 m, in particular, is rather conservative. To test this point, the same calculations were repeated for the Shallow configuration ( d= 5 m), and also plotted in Fig. 12. As expected, power output nearly tripled, with a yearly average of 90 kW; note that the PTO damping in this case was set at its power maximizing value of C/ρ = 718 m3·s-1for the same wave condition (126 m wavelength).

Although the consideration of real seaways is important since it involves the bandwidth response characteristics of a wave energy converter, the behavior of real PTO systems generally imposes very critical design constraints. A number of air turbines, for example, have been developed for oscillating water column devices. In these instances, the air chamber and the PTO (turbine) are external to the water domain, so that the sizes of both can be quite different. In addition, several turbine design options are a priori available, such as radial turbines (e.g., Falc?o et al., 2013). Axial flow machines like Wells turbines remain a common choice for OWCs; their blades are symmetric across the plane of rotation to allow unidirectional rotation in bi-directional air flow. Deniss-Auld turbines are similar, but rely on variable-pitch blades (Curran et al., 2000; see also Cruz, 2008). For submerged WECs of the type discussed in this article, the dimension of the central opening where flow enhancement occurs would define the size of axial-flow water turbines. In other words, it is consistent to interpret the diameter of the central opening as the turbine rotor diameter. If any ducting is considered, the stationary constriction of the flow path leading to the turbine rotor would belong to the structure, and the hydrodynamic behavior of the LE should be re-evaluated accordingly. Similarly, an annular Lagrangian element rather than a disc-shaped one should be considered if the turbine hub diameter (2ahub) is significant.

To illustrate the sensitivity of power production to PTO selection, we consider here a Deniss-Auld air turbine recently used in a study of OWC arrays (Nihous, 2012). The behavior of the machine is characterized in non-dimensional terms, and the laws of similitude are invoked to use it in water. Calling Ω the rotational speed of the turbine, the non-dimensional pressure drop δP*=δP/(4ρΩ2a2)is linearly related to the non-dimensional volume flow rate Q*= Q/(πΩa3), and in this case, the coefficient of proportionality was found to be of order 1. Recall that the PTO damping C used in MILE computations is equal to the turbine damping δP/Q multiplied by the square of the flow area π(a2–ah2ub), and that Q = π(a2–ah2ub)V as well, where V is the instantaneous average velocity (note that the provision has been made here for large turbine hubs and annular Lagrangian elements). Combining definitions and using the linearity between δP*and Q*to eliminate Ω yields the following expression:

Another characteristic of the turbine is its efficiency ε in converting hydraulic power into mechanical power. For the machine under consideration, ε is moderately high (＞ 70%) for a relatively narrow range of flows (Q*roughly between 1.3 and 2). In oscillating flows, this results in a power production significantly less than P (see for example, Nihous, 2012; Falc?o et al., 2013). Detailed and fairly straightforward calculations can be done in the time domain to account for the non-causal effect of ε, but this is not our focus here. While a drop in power production caused by ε cannot be avoided in time-varying environments, a mismatch between turbine and flow at all times would be extremely detrimental. This represents a design selection problem more than an operational issue. One should ensure that turbine efficiency and flow somewhat peak together, especially for wave conditions of particular interest. To probe this matter, we may replace V and C in Eq. (10) by Vopt= ηVC=0and Copt, respectively, for a wavelength of 126 m. Since the turbine size is primarily determined by a, we first examine the scenarios Baseline, Small and Big before considering the effect of a turbine hub. We find that a=2.5 m (Small) corresponds to a very good alignment between maximum flow and turbine efficiency, with Q*= 1.63. As a increases to 5 m (Baseline) and 7.5 m (Big), however, Q*drops to 0.47 and 0.22, respectively. As seen in Fig. 8, more (hydraulic) power is available for larger values of a (75, 148 and 205 kW for the three scenarios in a 126 m long unit-amplitude wave). Yet, this ranking would not be maintained in terms of mechanical power since ε drops very sharply when Q*＜ 1. The effect of turbine hub was probed by considering an annular LE and a structure defined for radii not only between a and R as before, but also less than ahub. In the Baseline scenario (a = 5 m), it was found that little change occurred with ahub= 2 m, but that for larger hubs, both Q*and P drop significantly (e.g., with

maxrespective values of 0.14 and 81 kW when ahub= 4.5 m). In the Small scenario (a = 2.5 m), a hub of 1 m radius had little effect. Although it is probable that the particular turbines and submerged geometric structures considered here are far from optimal designs, the difficulty in matching their characteristics unambiguously demonstrates the type of trade-offs typical of wave power conversion. It should also be added that low turbine rotational speeds, while offering some definite advantages, also present a challenge from the points of view of power conditioning and transmission.

5 Conclusions

A straightforward method (designated as MILE) was proposed to include the behavior of flow-through PTO systems, such as water turbines, when modeling submerged structures known to locally cause significant flow enhancement from incident surface waves. The determination of the wave radiation characteristics of a Lagrangian element in the fluid domain, presumably located where the PTO would operate, represents a fundamentalstep of the solution procedure. The corresponding radiation problem is traditionally defined, i.e., for the unit motion of the LE along the oscillating-flow direction of interest in otherwise calm water. With axisymmetric configurations, the scattering force on the LE, from incident and diffracted wave fields when the structure is stationary, conveniently is known from the wave radiation damping coefficient. This basic hydrodynamic information can be input into the LE equation of motion when the PTO is present. Linear water turbines are found to be equivalent to an external dashpot force proportional to the LE velocity when the mass m of the LE tends to zero. The wave radiation behavior of an LE of vanishing mass theoretically corresponds to that of a patch of distributed normal dipoles. In practice, the LE is adequate as long as m is much smaller than the hydrodynamic added mass.

MILE was first validated in the absence of any structure (incident wave only), or with a structure but no PTO (free flow). The method was then applied to the most simplified representation of a class of submerged flow-enhancing wave power absorbers recently proposed by Carter and Ertekin (2011). These can be reduced to shallow horizontal annuli, with LEs consisting of very thin horizontal discs placed in the central opening. Optimal or off-optimal wave power absorption performance could be determined, and the influence of salient geometric features of the structures was probed, albeit without attempting any extensive parametric analysis. The relative flow reduction resulting from the PTO was found to theoretically lie between 0.5 and 1/2≈ 0.71 under optimal PTO conditions. An important criterion to rank configurations in terms of hydraulic power was identified as the ratio of LE wave damping coefficient over LE added mass coefficient. Larger values of this ratio correspond to better wave energy conversion capabilities, as well as greater relative flow reductions.

Sample calculations were also performed at a typical site in Hawaiian waters using monthly average spectra. As is commonly found with wave power conversion devices at most sites, the output is very sensitive to the prevailing wave climate, reflecting both the device’s limited bandwidth and substantial seasonal variations in the available wave power flux. Finally, a short discussion was offered to illustrate the paramount importance of turbine design and selection. It was shown that hydraulic power calculations alone do not suffice to properly evaluate this type of WECs, and that any mismatch between turbine characteristics and WEC geometry could severely limit mechanical power output.

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Author’s biography

Gérard Nihous is an Associate Professor in the Department of Ocean and Resources Engineering at the University of Hawaii. Since graduating with a Ph.D. in Ocean Engineering from the University of California at Berkeley in 1983, the focus of his teaching and research activities has remained the development of marine renewable energy.

1671-9433(2014)02-0134-09

date: 2014-01-04.

Accepted date: 2014-03-28.

Unsponsored (cost share) contribution to the U.S. Department of Energy through the Hawaii National Marine Renewable Energy Center (Hawaii Natural Energy Institute, University of Hawaii), Account No. 6658090.

＊Corresponding author Email: nihous@hawaii.edu

? Harbin Engineering University and Springer-Verlag Berlin Heidelberg 2014