The Use of CFD in the Analysis of Wave Loadings Acting on Seawave Slot-Cone Generators

Abstract

The reliability of Computational Fluid Dynamics (CFD) in reproducing qualitative and quantitative features of loadings exerted by waves on Seawave Slot-cone Generators (SSG) has been investigated via 17 numerical experiments, conducted with the suite Flow 3D. The geometry of the Wave Energy Converter (WEC), as well as the characteristics of the foreshore in front of it, were identical to those used by the authors in a laboratory study, carried out on a small scale model of a pilot plant to be located along the West Norwegian coasts; the similitude of the layouts allowed an in depth comparison between the results. A good agreement has been generally found between physical and numerical experiments, apart from some aspects of the wave–structure interaction that, however, can be considered secondary for engineering purposes.

1. Introduction

In 2015, renewable energy set new records for dollar investment and amount of new capacity added, in spite of the plummeting of the fossil fuel price [1]. The reasons of this surprising result include of course the climate change policies, which encourage the de-carbonization of the energy sector, and the improved cost-competitiveness of new technologies.

Wave energy has the potential to be a particularly valuable contributor to a low-carbon energy mix since, besides being very abundant [2,3,4], it has a different geographic distribution than wind and solar, greater predictability and less intermittency. Accordingly, even in the awareness that only a small fraction of this huge resource can be exploited (see also [5]), more than 1000 Wave Energy Converters (WECs) have been patented worldwide.

WECs can be generally divided into three categories, based on the working principle. In the Oscillating Water Column (OWC) technology, a cushion of air is trapped within a partly submerged chamber where the water level rises and falls with the waves. The oscillating motion makes the air to pass through a turbine, which rotates in the same direction irrespective of the way of flow. Wave Activated Bodies (WAB) exploit the relative motion of the different parts of the device; hydraulic systems are generally employed to compress oil, air or water, which are then used to drive a generator [6,7]. In the Over Topping Devices (OTD), a sloping plate leads the waves to overtop into a reservoir. The energy is then extracted via a turbine that works with small head differences (below 3 m) and large flow volumes.

Despite most of the research efforts are addressed at solving the problem of the low efficiencies of the plants, an additional serious hurdle to the WECs development and diffusion is undoubtedly represented by the uncertainties related to their capability of resisting wave actions [8,9,10].

Due to the notable complexity of the hydrodynamic processes involved in the wave–device interaction (wave breaking, wave reflection/diffraction, wave overtopping, etc.), the structural design of wave energy converters invariably requires lengthy and expensive physical model studies, whose impact on the total cost of plants may be not negligible.

In this respect, the use of Computational Fluid Dynamics (CFD), which solves the Navier–Stokes Equations coupled with a turbulence model, could represent a valuable alternative, either as a surrogate of physical modeling [11] or as a support to reduce the number of tests and measurements required.

However, for the specific case of the structural response of WECs, there is a lack of rigorous validation studies that compare outcomes of lab experiments to those of CFD simulations. Of course this limits the use of the latter in the current design practice.

This article aims at partially filling this gap, dealing with the case of the Seawave Slot-Cone Generator (SSG, Figure 1).

Patented by WavEnergy SAS (Stavanger, Norway), this OTD includes a number of reservoirs placed on the top of each-other, which capture the water during the up-rush phase; on its way back to the sea, the fluid passes through a low-head turbine, spinning it and producing electricity [12,13,14].

The WEC is normally located at the top of a steep foreshore, which will be nicknamed “focuser” in the following, as it has the main purpose of increasing the potential run-up height.

In 2015, Buccino et al. [15] carried out a number of regular wave experiments at the LinC laboratory of the Department of Civil, Architectural and Environmental Engineering of the University of Naples “Federico II”, on a small scale model of SSG originally designed as a possible pilot plant to be located at Svåheia, along the West Norwegian coasts. The authors provided a detailed description of the nature of loadings acting onto the front face of the WEC and produced a parameterization of the obtained results.

The research above represents, then, a good reference for a detailed comparison with numerical simulations.

This article is organized as follows. Section 2 provides a brief review of the existing literature on the structural response of SSGs and summarizes the main results of Buccino et al. [15]. In Section 3, the numerical CFD experiments are described and Section 4 discusses the results of comparison with the physical model tests. In Section 5, conclusions are finally drawn.

2. Literature

In 2009, Vicinanza and Frigaard [16] carried out a physical model study to investigate the structural response of a 1:60 model of Seawave Slot-cone Generator. Since the focuser was extremely steep (1:1), the authors observed essentially non-breaking waves and quasi-static loadings. It was found out that the predictive method proposed by Takahashi et al. [17] for sloping top caisson breakwaters, could not be extended to the case of SSGs, since the peaks of horizontal force were dramatically under-predicted.

Successively, Vicinanza et al. and Buccino et al. [14,18] showed that non-impulsive wave actions induced by both non-breaking and breaking waves, could be estimated, up to a certain extent, by the Japanese design formulas for trapezoidal monolithic barriers [19].

More recently, Buccino et al. [15] have suggested a calculation method, expressly derived for SSGs, which holds for both impulsive and non-impulsive loadings. The method is essentially probabilistic and is based on a series of regular wave laboratory experiments, conducted by the authors on a model of WEC scaled down at 1:66.

The approach employs three predictive variables and namely:

(1)

The surf similarity parameter or inshore Iribarren number (ξ) [20] calculated with the incident wave height at the toe of the focuser, H_i, the wave period T and the average front slope angle, tgα_av. (Figure 2);

$$\xi =\frac{tg{\alpha }_{av.}}{\sqrt{\frac{2\pi H_i}{g{T}^2}}}$$

(1)

(2)

The (mean) slope parameter [21], i.e., the ratio between the length of the waves at the toe of the focuser (L at the depth d, Figure 2) and the mean horizontal distance between the toe of the foreshore and the shoreline (d/tanα_av.):

$$S=\frac{L·tg{\alpha }_{av.}}{2\pi d}$$

(2)

(3)

The Linear Thrust Parameter, which represents the ratio between the maximum value (over a wave period) of the wave momentum flux through the base of the focuser and the corresponding hydrostatic still water thrust:

$$L_{TP}=\frac{H_i}{d} \frac{tanh\left(kd\right)}{kd}$$

(3)

L_TP is in fact a linearized-slightly-modified form of the wave Momentum Flux Parameter (M_FP) originally introduced by Hughes [22] (see also [23]); in shallow waters it reduces to the wave height to depth ratio H_i/d.

Only two of the three variables above are independent, since it can be proven that:

$$\sqrt{\frac{L_{TP}}{2\pi }}\xi =S$$

(4)

Buccino et al. found out that the wave shapes at the wall could be efficiently distinguished using a chart where ξ is on the abscissas and L_TP is on the ordinates (Figure 3); on this plane, all the variables relevant to the breaking process (slope angle, wave steepness, and wave height to depth ratio [24]) are in fact expressly taken into account.

The Figure 3 shows that for any value of the Iribarren number, wave breaking takes place only if the momentum flux at the toe of the focuser exceeds a threshold (black solid line), which increases with growing ξ. The equation of the limit curve is:

$$L_{TP}=\frac{0.021\xi }{1+0.031\xi }$$

(5)

in which L_TP tends to 0 for small values of the Iribarren number, as on very mild slopes the waves are always expected to break.

The different breaker types can be then discriminated based on the value of S; the analysis of the experimental data suggested to choose S = 0.420 and S = 0.225 as preliminary limits for the transitions from surging to collapsing and from collapsing to plunging.

As far as wave forces are concerned, the classification introduced for vertical breakwaters within the EU funded project PROVERBS has been adopted [25]. Accordingly, a force chronogram is termed “pulsating” if it exhibits a unique smooth peak over a wave period (Figure 4, left panel); a double peak pattern is instead named either “slightly breaking” or “impact” (see also [26,27]), depending on whether the first sharp maximum (F_h,max in Figure 4) is lower or higher than 2.5 times the second “pulsating” peak (F_h,q).

For prediction purposes, Buccino et al. [15] reasoned that due to the inherent randomness of the breaking process, even under a “regular wave” attack the structural response could be best represented by a probability density function (pdf) of wave pressure, rather than a single deterministic value. Thus, it was assumed that for any H_i and T, the average hydrodynamic pressure acting onto the front face of the WEC at the instant of maximum force (${\widehat{p}}_{av.}$ in Figure 5) could be approximately described via a log-normal pdf.

Mean and standard deviation of the distribution can be calculated employing different equations, based on whether the nature of loadings is expected to be impulsive (i.e., induced by plunging breakers) or not. The authors proposed:

$$E\left(\frac{{\widehat{p}}_{av.}}{\rho gd}\right)=\{\begin{array}{c}0.751L_{TP} non impact waves\\ \\ \left[2.68{\xi }^{-2.42}\right] L_{TP} impact waves\end{array}$$

(6)

$$\sqrt{VAR\left(\frac{{\widehat{p}}_{av.}}{\rho gd}\right)}=\{\begin{array}{c}0.0012+0.0474u+0.8017u^2 non impact waves \\ \\ 0.0009\exp \left(10.39t\right) impact waves\end{array}$$

(7)

where

$$\{\begin{array}{c}u=\frac{L_{TP}}{{\xi }^{0.6}}\\ \\ t=\frac{L_{TP}^{0.3}}{\xi }\end{array}$$

(8)

Equation (6) should be applied within the ranges:

$$\{\begin{array}{c}0.03\le L_{TP}\le 0.32\\ \\ 0.015\le {\xi }^{-2.42}{L}_{TP}\le 0.108\end{array}$$

(9)

whereas the limits of Equation (7) are:

$$\{\begin{array}{c}0.0157\le u\le 0.2600\\ \\ 0.2407 \le t\le 0.4933\end{array}$$

(10)

By means of the Equations (6) and (7), the structural design of the WEC can be then performed calculating wave loadings as a function of the probability of failure the designer considers the most appropriate to the specific project.

3. Numerical Experiments

Numerical experiments have been conducted via the suite Flow 3D (Flow Science, Santa Fe, Mexico) [28], which has been proven to be particularly accurate in solving wave–structure interaction problems [29,30]. The software solves the Reynolds Averaged Navier–Stokes Equations (RANS) combined with a Volume of Fluid (VOF) method. For the present application, a Re-Nornalization Group (RNG) k-epsilon turbulence closure has been chosen.

The SSG and the focuser reproduce the Svåheia project at a full scale (Figure 6) and are geometrically identical to those employed in Buccino et al. [15] (apart from the obvious scaling up). Like in the physical model tests, the sea bottom has been kept constant seaward the focuser (Figure 6b); the distance between the latter and the wave-maker has been fixed at 277 m.

The use of a full scale CFD model allows minimizing the numerical viscosity effects induced by the turbulence closure, which may lead to significantly dampen waves and loadings on a small scale structure. On the other hand, though, the use of different scales between physical and numerical experiments could introduce some uncertainties in the comparison of results. However, since the models here analyzed are both smooth and impermeable (the SSG reservoirs have been supposed to be closed during the wave attacks), no relevant scale effects due to the dynamic viscosity of the fluid are expected to occur [31]. The main bias is rather related to the absence of air in the CFD, the consequences of which are discussed in the Section 4.

3.1. Grid Selection

The computational domain (400 m in the X direction and 60 m in the Z direction, Figure 7) is made up on three zones (Figure 8a): meshes 1 and 3 include 62,400 and 14,400 cells of size 50 cm × 50 cm, respectively. The size of the local mesh that surrounds the WEC (mesh 2, Figure 8b), has been instead selected based on an ad hoc sensitivity analysis.

In this view, three grids have been initially compared, that measured respectively 50 cm × 50 cm, 25 cm × 25 cm and 8 cm × 8 cm. Each grid has been subjected to the same regular wave attack (H = 3.78 m, T = 8.02 s) and the time history of the horizontal force exerted on the SSG has been analyzed.

The waves are generated by imposing a periodic fluctuation of the water level at the seaward end of the flume; the corresponding velocity is then calculated via the linear wave theory. It is worth noting that for each mesh size, an optimal time interval is internally selected by the software to avoid any numerical instability.

The Figure 9 shows an example of the obtained results.

The force signal associated to the grid 50 cm × 50 cm deviates significantly from that of the grid 25 cm × 25 cm (Figure 9a); conversely, the latter appears almost identical to the time history associated with 8 cm × 8 cm (Figure 9b), indicating that a convergence in the results has been achieved. This can also be quantitatively observed through the indexes reported in

Table 1. Root Mean Square Error (RMSE) and index of determination (R²) for different grid size.

GRID	RMSE	R2
50 × 50 vs. 25 × 25	6.36	0.86
25 × 25 vs. 8 × 8	0.37	0.99

. Here the coefficient of determination between the signals, R², and the relative Root Mean Square Error (RMSE) of the coarser grid results compared to finer grid ones are calculated.

It is seen that, passing from 25 cm × 25 cm to 50 cm × 50 cm, a relative error exceeding 600% is produced, whereas the correlation between the signals is only moderately high. On the other hand, between 25 cm × 25 cm and 8 cm × 8 cm, the RMSE reduces dramatically and the value of R² indicates an almost perfect coherence between the results. Thus, a 25 cm × 25 cm mesh size has been finally selected for this study.

3.2. Test Program

Consistently with Buccino et al. [15], regular wave experiments have been run with the same still water level as used in the physical lab (d = 33 m).

Table 2. Wave characteristics (prototype scale) for CFD experiments.

TEST#	Hi [m]	T [s]	Δt* [s]
1	3.01	6.70	0.03
2	3.73	6.64	0.03
3	4.34	6.65	0.03
4	4.68	6.63	0.03
5	4.75	6.63	0.03
6	2.61	8.01	0.04
7	3.78	8.02	0.04
8	4.58	8.04	0.04
9	5.25	8.03	0.04
10	6.08	8.03	0.04
11	6.95	8.01	0.04
12	7.71	8.00	0.04
13	8.29	7.99	0.04
14	1.25	17.15	0.08
15	2.32	17.05	0.08
16	6.50	17.03	0.08
17	11.55	16.40	0.08

Notes: Incident wave height (H_i), wave period (T) and sample rate (Δt*).

reports the values of the incident wave parameters H_i and T (see Section 4), along with the sampling rates (Δt*) employed for both wave and loading signals.

It is worth noticing that in all the tests there was no water at the rear of the WEC, apart that coming from the wave overtopping.

3.3. Data

A video of each experiment has been taken, to study the wave shape at the wall and the fundamentals of the wave–structure interaction. The fluctuations of the free surface were acquired at each cell seawards the focuser. Finally, the horizontal component of the total force acting on the front face of the SSG has been extracted, along with four pressure time series from transducers placed in the same positions as in Buccino et al. [15] (Figure 10 and

Table 3. The position of the sensors from the toe of the structure.

Transducer#	Transducer Centre from the SSG Toe [m]
1	0.93
2	3.70
3	6.24
4	8.75

).

4. Comparison between Results of Numerical and Physical Model Tests

4.1. The Reflection Coefficient

Like for the physical model tests, in this study, the reflection coefficient k_r (reflected to incident wave height ratio) has been measured on the flat bottom seaward the focuser; thus, unlike Zanuttigh et al. [32], it gathers the effects of both the WEC and the foreshore.

The reason for using this variable as starting point of our analysis, is that it can be considered as a global indicator of the wave–structure interaction characteristics; this in virtue of its dependence on wave run-up, wave overtopping and dissipation rate.

Incident and reflected signals have been separated via the method of Zelt and Skjelbreia [33], following the same procedure described by Buccino et al. [15]. To this aim, the signals of four wave probes have been simultaneously analyzed.

In Figure 11, numerical k_rs and data of Buccino et al. [15] are plotted vs. the Iribarren number and the relative crest freeboard R_c/H_i. As widely known, the former variable controls the breaking process, whereas the latter is strongly correlated to the average overtopping discharge. The two graphs show that an increase of reflection occurs both when the rate of energy dissipated by breaking decreases (large ξ) and when the amount of overtopping reduces (high values of R_c/H_i).

Physical and numerical data are seen to lie within the same cloud, indicating a good consistency. However, for a given R_c/H_i, CFD gives less reflection, which is likely due to a larger amount of overtopping.

4.2. Wave Shapes at the Wall

From the inspection of videos, the shape of the wave profile at the system “wall + focuser” was identified for each experiment. Like in Buccino et al. [15], the classification adopted follows the approaches described in Basco [34] and Calabrese et al. [24]. The results of the analysis are pictured in Figure 12, on the plane ξ-L_TP. Overall, the numerical simulations appear in agreement with the outcomes of the physical model tests; the majority of data refers to plunging breakers, which fall consistently within the area with S ≤ 0.225. Only in two cases, a mismatch has been observed; one refers to a plunging breaker within the “standing area”, the other is a standing wave observed in the plunging zone. Both the points, however, are quite close to the curve of incipient breaking, so that the observed discrepancy is likely ascribable to the inherent randomness which affects the breaking process even under regular waves [35]. It should be also added that in the present case the stability of waves is strongly influenced by the rate of reflection and accordingly small variations of k_r may produce different profiles at the wall.

4.3. Direct Comparison of Pressure and Force Signals

The numerical tests 6, 12 and 17 in Table 2 have wave height and period rather similar to those of three experiments presented in the study of Buccino et al. [15]. As shown in

Table 4. Relative percentage error (ERR) between wave characteristics (incident wave height H_i and period T) in numerical (CFD) and physical tests.

TEST#	CFD		PHYSICAL		ERR (%)		BREAKER TYPE
	Hi [m]	T [s]	Hi [m]	T [s]	Hi	T	Physical	Numerical
A	2.61	8.01	2.37	8.12	10	1	Standing	Standing
B	7.71	8.00	7.62	8.12	1	2	Plunging	Plunging
C	11.55	16.40	10.70	16.33	8	0	Surging	Surging

, the wave shapes observed at the wall are the same, the scatter on H_i does not exceed 10% and the difference on T is not larger than 2%. This degree of similitude allows, then, a direct comparison in terms of pressure and force chronograms.

The Figure 13 refers to the test 6, where the waves are simply reflected by the SSG. In this graph, like in the similar ones reported in the following, the loading signals are kept voluntarily out of phase to facilitate the comparison.

Although shape and magnitude of pressure signals appear consistent on average, CFD is observed to generate low impulse spikes and a slightly lower level of run-down (trough of the pressure waves in the panel p₁). It is also of interest that the pressure at the transducer p₄ is zero both in the numerical and the physical test.

The presence of spikes in the numerical pressures, which resulted almost independent of the grid size, is likely related to the features of the up-rush process. As shown in Figure 14, while climbing the WEC the run-up wedge has been observed to steepen and project a small tongue of fluid against the wall, whose impact is probably responsible of the appearance of the secondary peaks. This phenomenon, which occurs repeatedly during the up-rush phase, is likely created by the segmented shape of the SSG.

The spikes make the CFD chronogram of the horizontal force more noisy compared to that of the physical experiment; moreover, a weak increase of phase difference has been observed in the last part of the signal, as shown in the upper panels of the Figure 15. On the other side, the two chronograms appear rather consistent in magnitude and the distribution of pressures in p₁–p₄, and, at the instant of the maximum force, has been found to be bi-trapezoidal in both cases (Figure 15, lower panels).

Test 12 refers to a plunging breaker and the observed wave kinematics is generally consistent with that described in several studies conducted on vertical face breakwaters [26,27]. Both in the physical and the CFD experiments, the plunging jet [24,34] is observed to strike the lower part of the WEC (upper panels in Figure 16), with the mass of water that splashes-up and hits again the wall onto the higher plate (lower panels in Figure 16). These two impacts are clearly visible in the time history of the horizontal force (Figure 17a); it is also noteworthy that at the main peak (maximum force), the distribution of pressures results triangular with the maximum in p₁ (Figure 17b).

As mentioned earlier, the main difference in this case concerns the effect of air, not modeled in the CFD, which produces a series of oscillations of a nearly 10 Hz frequency (in prototype) both in the force and the pressure signals of the physical model test. Generally, however, the agreement between the pressure chronograms is rather satisfactory (Figure 18).

Test 17 is a large surging breaker; Buccino et al. [15] argued that under such condition, pulsating-type chronograms alternate with slightly breaking time-histories. The latter are generated by a water jet detaching from the wave profile during the up-rush phase. These features have been qualitatively observed in the CFD experiment also, but pulsating events appear less “linear”, with a rise-time significantly shorter (Figure 19a,b). Moreover, impact time-histories have been detected instead of the slightly breaking ones (Figure 20a,b); this is likely due to the fact that in the physical experiment, a large area of spray forms during the up-rush, which dampens the slam of the water tongue onto the upper plates of the WEC.

4.4. Wave Loadings Magnitude and Distribution

In addition, for the numerical simulations, the peaks of the non-dimensional average pressure ${\widehat{p}}_{av}$/ρgd resulted approximately log-normal distributed; an example is shown in Figure 21.

Moreover, the mean of the probability density functions (pdfs) has been found to attain values rather consistent with the physical model tests, both in the case of impulsive and non-impulsive wave actions. Figure 22 and Figure 23 show that the CFD data lay rather close to the Buccino et al. experiments [15], but for a single plunging breaker circled in Figure 23.

As a further point of interest, it is noteworthy that for non-impact loadings, the gathering of numerical and physical points would seem to support the hypothesis (already formulated in Buccino et al. [15]) that the relationship between the average pressure and L_TP may actually be weakly nonlinear. The following slight correction to the Equation (6) can be then proposed (Figure 21, broken line):

$$E({\widehat{p}}_{av}/\rho gd=0.7L_{{TP}^{0.93}}$$

(11)

with a R² statistics of 0.98.

As far as the standard deviation is concerned, the variables t and u resulted poorly correlated to the scatter of the numerical pdfs; in general, the CFD distribution would seem to have a variance far larger than that predicted by the Equation (7) (Figure 24 and Figure 25).

However, as shown in Figure 26, when physical and numerical data are plotted against the Iribarren number alone, a reasonable accordance is observed, with the exception of a single outlier circled in red in the graph. The trend of points can be represented by the following power type relationship:

$$\sqrt{VAR\left(\frac{{\widehat{p}}_{av.}}{\rho gd}\right)}=0.0108{\left(\xi -1\right)}^{-1.263}$$

(12)

which is suggested to be used as new design formula instead of Equation (7). As a point of strength, the new tool holds for both impact and non-impact conditions; on the other hand, it has to be pointed out that the determination index is some low (R² = 0.61) and the lack of dependence on the linear thrust parameter may lead to largely overestimate the variance of steep standing waves, which have low ξ and low L_TP. From an engineering point of view, however, the latter appears conservative, as it leads to overestimate the maximum predicted force for assigned H_i and T.

5. Conclusions

The capability of CFD simulations of reproducing the main qualitative and quantitative features of wave loadings acting onto the front face of Seawave Slot-cone Generators has been investigated through 17 numerical experiments conducted via the suite Flow 3D. The WEC modeled, and the foreshore in front of it, are geometrically identical to those employed by [15] in the frame of a physical model study conducted at the LInC laboratory of the University of Napoli “Federico II”. Numerical and physical model data resulted generally consistent with each other; the main differences found concern:

• The presence of undesired spikes in the numerical force chronogram of a standing wave, due to small local impacts provoked by the run-up wedge during the up-rush phase (Figure 13 and Figure 14).
• A weak tendency of CFD at progressively increasing in time the phase difference relative to the physical model (Figure 15 upper panels).
• The presence in the numerical tests of impact events generated by large surging breakers (Figure 19), due to the not modeling of the air that dampens the force peaks in the physical experiments.

However, the magnitude and the statistical properties of the numerically simulated wave loadings were found to be fairly in agreement with the physical model measurements. The observed consistency allowed modifying the predictive Equations (6) and (7), in order to account for the new results (Equations (11) and (12)).

On the whole, this study supports the idea that CFD could be effectively employed in the structural design of SSGs, even if the consequences of the inclusion of air within the fluid mass should be accurately investigated. Moreover, supplementary studies are needed to compare numerical and physical model outcomes in the case of random waves.

Finally, the reliability of alternative meshless advanced numerical methods, such as Smooth Particle Hydrodynamics (SPH), deserve being investigated in depth.