CFD Simulation of a Passively Controlled Point

CFD Simulation of a Passively Controlled Point Absorber

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Minghao Wu, Weizhi Wang, Johannes Palm and Claes Eskilsson

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Department of Shipping and Marine Technology, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden

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Abstract Recently CorPower Ocean AB presented laboratory tests of a point absorber wave energy converter equipped with a novel technique for passive phase control (Todalshaug et al, 2015). The technique, known as WaveSpring, widens the response bandwidth by a negative spring arrangement, and in the tank experiment an up to three-fold increase in delivered power as compared to pure linear damping was observed. As previously reported, for point absorbers close to resonance the use of standard radiation-diffraction models can become unreliable while CFD simulations accurately captures the nonlinear wave height dependent response. Thus, in the present study a module representing the WaveSpring technology was implanted in the OpenFOAM framework and CFD simulations of the buoy were performed both with and without the WaveSpring module. Good agreement between simulated and experimental results was observed, and the WaveSpring behavior was well captured in the numerical simulation. The CFD model can be used for further tuning of the WaveSpring/buoy design as well as providing validation data for radiation-diffraction models.

Keywords Wave Energy · Point Absorber · Passive control · CFD · Nonlinear response

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Introduction

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Point absorbers are a special class of wave energy converters (WECs), defined by having a geometry that is small in relation to the wavelength. Point absorbers are oscillating bodies and typically subjected to large amplitude motions in the resonance region. Seminal analytical studies of the power capture of point absorbers include Budal and Falnes (1975) and Evans (1976). In order to increase the power capture different control strategies have been proposed, such as reactive control, latching, declutching, etc; see e.g. Falnes (2002), Hals et al (2011) and Korde and Ringwood (2016) for discussions and comparisons of hydrodynamic control of WECs. Recently a new passive control strategy based on negative spring arrangement was presented: The WaveSpring technology (Toldalshaug, 2014). The WaveSpring technology is

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today developed and used in the CorPower Ocean buoy, soon to be deployed at EMEC’s scale test site in Orkney, Scotland (EMEC, 2017). It is well known that the motion response of point absorbers in the resonance region is subjected to nonlinear effects and that linear radiation-diffraction methods over-predicts the response amplitude for steeper waves. CFD studies of point absorbers in the resonance region have conformed this. Yu and Li (2013) investigated and showed that the linear model over-predicted the power production of a self-reacting point absorber in heave. Palm et al (2016) illustrated that this also was true with a coupled mooring analysis using CFD of a point absorber undergoing 6 degree-offreedom (DoF) motion. These effects are even more highlighted for point absorbers using phase control. Indeed, in a recent study Giorgio and Ringwood (2016) implemented latching control of a heaving sphere in a CFD-based numerical wave tank and compared to a standard linear method. The results show that the linear model overestimates the amplitude of motion, and, as a result, the extracted power. Importantly, the optimal latching duration given by the linear model will not maximize the performance of the actual device. So while the advantage of CFD is obvious – nonlinear and viscous effects, overtopping, large amplitude waves and motions, etc. are all included – the major disadvantage of using CFD for modelling WECs is that it is very computationally expensive, especially when irregular sea states is simulated. Eskilsson et al (2015) computed the overtopping discharge of the Wave Dragon wave energy converter and reported a computational effort in the order of 150000 CPU hours for a 3-hour sea state simulation using a computational mesh of 20 million cells. However, in a recent technical review regarding the applicability of the CFD-based numerical tank test of offshore floating structure design (Kim et al, 2016), it is concluded that the CFD-based numerical tank test shows the benefits of lower cost, shorter project period and higher flexibility compared with physical tank test. In the light of findings of Giorgio and Ringwood (2016) we expect that the passively controlled CorPower buoy also will require CFD simulations in order to get reliable values for motion response and power production. In this paper, the main objective is to establish a three dimensional numerical wave tank based on the opensource CFD solver OpenFOAM and test its feasibility of modelling the CorPower WEC motion in waves. The simulation will consider both the pretension mooring force and the nonlinear PTO force and will support 6 DoF motion. The nonlinear effects due to the viscous flow and the nonlinear motion excited by WaveSpring phase control system are to be investigated. The results of motions and forces will be compared with the laboratory test data under the same settings.

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CorPower Ocean AB is a Swedish technology company focusing on the researching and developing the high-efficiency wave energy converter. The CorPower WEC prototype adopts the point-absorber type and the phase control technology to minimize the size of the buoy and maximum the efficiency. The shape and the inner structure of

The CorPower Point Absorber

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the buoy is shown in Figure 1. The inside power-take-off (PTO) gear extracts the kinematics energy and convert to electricity.

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Fig. 1 CorPower Ocean WEC prototype. From CorPower (2017)

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The application of the WaveSpring phase control system invented at Norwegian University of Science and Technology (Toldalshaug, 2014) gives a more efficient wave energy conversion. WaveSpring is a passive control system and, in contrast to conventional latching control, requires less sensors and active loops since the incident wave information is not utilized by the control system. The working principles of the WaveSpring system is described in Todalshuag et al (2015). In short, pressurized pneumatic cylinders provide a spring force which is contrary to the heave hydrostatic force causing the motion to be amplified when the buoy is off the equilibrium position. The net vertical force along the central rod, FWS,z is expressed as 𝐹"# 𝑧 =   𝑁( 𝐴( 𝑃 𝑧

𝑧 𝑙,-

−

𝑧-

 ,

(1)

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where Nc is the number of cylinders, Ac is the area of a cylinder, l0 is the initial length of the cylinder at its horizontal position and P(z) is the cylinder pressure. Here z is the relative WEC excursion along the center rod. This negative spring arrangement is able to amplify the power capture of the buoy and increase the power absorption by widening the response bandwidth. In the experimental study described in the next section it was found that the average power increased with a factor 3 by passive control provided by the WaveSpring system (Toldalshaug et al, 2015).

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The tank tests of the CorPower WEC model were carried out in the Hydrodynamic and Ocean Engineering Tank at Ecole Centrale de Nantes in France during November 2014. A detail description of this series tank test is presented in Todalshung et al (2015) The system consists of a buoy model in scale 1:16, a rope, two pulleys and the

Physical Experiments

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motor rig, as shown in Figure 2. The rope connected to the buoy is used as mooring line to restrain the buoy, while the bottom mooring lines are used to fix the submerged pulley. The mooring line is pre-tensioned to keep the buoy in its equilibrium position. The motor rig is connected to a load control system and can approximate the on-board machine force as long as the angle between the buoy axis and the mooring line is small. The experimental parameters are listed in Table 1.

Fig. 2 Tank test configuration (left) and photo of experiment (right).

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Various sensors are used for capturing the transient motion, wave height and forces. The wave surface elevation is measured by five resistive wave probes. The sixdegree-of-freedom motion of the buoy is measured by Qualisys optical position sensors. Two force sensors are installed at both the buoy end and the motor rig end of the mooring line. In the experiments a wide array of different regular and irregular waves is tested, but here only one wave is investigated: a regular wave with amplitude 0.156 m and a wave period of 2.25 s. This is a fairly linear incident wave, having a steepness of 0.02.

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Table 1 Experimental parameters

Item Buoy diameter Buoy height Buoy mass Equ. displacement Equ. waterline Moment of inertia

Value 0.525 m 1.125 m 19.75 kg 0.0708 m3 0.98 m 2.38 kgm2

Item Water depth Rope length Depth at mooring point Rope and sheave mass Mooring line stiffness Pretension force

Value 5m 15.26 m 3.09 m 1.85 kg 202 N/m 500 N

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The numerical model is based on incompressible Reynolds Averaged Navier-Stokes (RANS) equations with the single fluid (mixture) approximation. The two-phase problem is handled using the volume of fluid (VOF) approach. The VOF-RANS equations are solved using the interDyMFoam solver in the open-source finite volume framework OpenFOAM (OpenFOAM 2016; Weller et al 1997). The interDyMFoam solver supports dynamic mesh motion to allow for the motion of the WEC and waves are generated/absorbed by the waves2Foam package (Jacobsen et al, 2012). This combination is a fairly popular choice of CFD-based numerical wave tank in the wave energy community, see Davidson et al (2015) and the references therein. In order to model the CorPower buoy the WaveSpring force (1) was implanted as a restraint inside body motion library of OpenFOAM. Also, due to the set-up of the motor rig, a simplified mooring made up of a linear pre-tensioned spring was used in the simulations.

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4.1 Mesh and Numerical Settings

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The fluid domain is described in a Cartesian coordinate system with the origin point at the bottom of the WEC buoy. The positive x-direction is to the wavepropagation direction, the y-direction points to aside and the z-direction is vertically upward. The length of the domain is 48 m. The bottom as well as the side walls are modelled as slip conditions. To avoid the reflection from the bank, the distance from the floating buoy to the side wall is set to 4m, which gives an 8.6m full width. Two relaxation zones are set at two ends of the domain for wave generation (inlet) and cancellation (outlet). Each relaxation zones are set to cover two wavelengths. Figure 3 outline the computational domain used.

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Numerical Modelling

Fig. 3 Numerical wave tank dimensions: side view (top) and top view (bottom).

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The grids are generated with the snappyHexMesh utility. In order to capture the near-body flow and the wave surface, several levels of grid refinement are used to increase the grid resolution near the buoy surface and the air-water interface. To keep a good grid quality, the background mesh is created as uniform hexahedral cells, with a cell aspect ratio approximately equal to unity after refinement. The total cell number is 10.9 million cells. There are roughly 40 cells per wave length along the x-direction and 30 cells per wave height in the z-direction. By adding five layers to the near-wall region, the y+ value is roughly 50, which is appropriate for using standard wall functions for the turbulence equations. Figure 4 shows the computational mesh. Although not discussed in this paper a verification and validation study has been performed indicating that this mesh is associated with less than 5 percent numerical uncertainty in the heave amplitude. The second-order van Leer scheme is used for the divergence terms in the NavierStokes equations while the first-order upwind scheme is used in the turbulence equations. The diffusion terms are treated with second-order central difference scheme. The solution is advanced in time using the implicit Crank-Nicholson scheme.

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Fig. 4 Computational mesh: global view (left) and close up at the buoy (right).

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5.1 Surge Decay Test

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The surge decay of the buoy in calm water is evaluated numerically and compared to tank test experiment. The mooring line is simplified as a spring with a linearized stiffness coefficient, which is calculated by dividing the pretension by the mooring line length. The anchor of the mooring line is fixed at the submerged pulley in the test. The resulting time series are plotted in Figure 5 and the obtained surge natural period and damping ratios in Table 2. The computed and measured natural surge period are in general good agreement. The initial release position is a bit uncertain from the experimental tests, which explains the amplitude difference in the first oscillation and also the larger difference in damping ratio.

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Test Cases

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Fig. 5 Surge decay test. Experimental data is presented in dashed black line while computa-

tional results are printed in solid red lines

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Table 2 Surge decay test. Average surge decay period and non-dimensional linear damping

factor.

Average period Damping ratio

Experimental 3.98 s 9.55E-02

Numerical 4.03 s 8.75E-02

Relative error 1.3 % 8.3 %

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5.2 Regular Waves with Linear Damper Only

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In this case, the motion with only the linear damper PTO system is simulated. The linear damping coefficient is set to 628 kg/s, which is the optimum value obtained for unconstrained heave. The set-up of this case corresponds to a Keulegan–Carpenter number (KC) of 1.1 and a Reynolds number of 1.1E05, both values based on the buoy diameter. The incident waves are given by 5th order Stokes theory. The simulation is run for 50 s, by which time a steady periodic motion of the WEC has been established. Figure 6 shows the perspective global view of the wave surface and the side view of local mesh motion around the buoy at simulation time t=48s. Under this relatively large PTO damping coefficient, the buoy heave amplitude is small so that the wave surface is well captured in the mesh refined region near the waterline. No overtopping is observed according to the post-processing. The motion and force logs of both the CFD simulation and experiments are plotted in Figure 7, including surge, pitch, heave and the PTO-mooring system force. The response is seen to be linear, there is no higher harmonics visible in the time series. Generally, there is good agreement between numerical and experimental response amplitude. The problem is the heave (the motion peak is over-predicted) and the pitch (negative pitch is under-predicted) responses. These two modes are coupled and it was found that these motion is sensitive to the value of the pre-tension, which is a bit uncertain from the experimental tests. In addition, the moment of inertia is estimated

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based on approximate gyration radius data, which might explain some of the error in the pitch motion.

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Fig. 6 Buoy and mesh motion response in regular waves with linear damper: At t = XX s (left)

and t = XX s (right)

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Fig. 7 WEC motion response in regular waves with linear damper: Heave (top left), surge (top

right), pitch (bottom left) and PTO force (bottom right). Experimental data is presented in dashed black line while computational results are printed in solid red lines

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5.3 Regular Waves with WaveSpring System

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As mentioned above, in order to increase the efficiency of the power capture the WaveSpring system is operated together with the linear damper PTO system. This introduces highly nonlinear factors in the numerical modelling. The wave case of section 5.2 is repeated with the WaveSpring system included. The WaveSpring system in this test is made up of 3 pistons, each with cross section of 5.82E-04 m2, initial length 0.25 m and initial volume 1.59E-04 m3. Figure 8 illustrate the buoy and mesh deformation at four instances, one second apart. The motion is highly exaggerated compared to the linear damper only case. Please note that there are instances where it is very close that the water surface becomes positioned outside the refinement zone, due to the large motion of the buoy. The motion and forces time series are shown in Figure 9. It is clear that the experimental test has not reached a steady periodic behavior. The experimental test is aborted after 50 s in order to avoid waves reflected from the basin walls interfering with the WEC. Clearly 50 s is too short time for the initial transients in pitch and surge to stabilize. The numerical simulations on the other hand is run until a steady periodic behavior is reached. As the experimental case is not fully developed it is hard to say anything conclusive of the validation of the WaveSpring modules. However, please note that there is a good fit in heave response as well as for the forces in PTO and WaveSpring forces. The numerical pitch and surge motion after 60 s do have clear resemblance to the experimental motion after some 40 s until the wave forcing

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Fig. 7 WEC motion response in regular waves with WaveSpring system. At t = XX s (top left),

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t = XX s (top right), t = XX s (bottom left) and t = XX s (bottom right)

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ends. This indicates that the numerical results are sound. The magnitudes of the motion response are far larger than the linear damper case. It is also clear that higher order terms are present in all heave, pitch and surge motions. The passive phase control clearly enhances the nonlinearity and amplifies the motion response.

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Fig. 9 WEC motion response in regular waves with WaveSpring system: Heave (top left), surge

(top right), pitch (middle left), PTO force (middle right) and WaveSpring force (bottom). Experimental data is presented in dashed black line while computational results are printed in solid red lines

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Concluding Remarks

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In this paper, a CFD-based numerical wave tank is used to compute the WEC motion and forces of a passively controlled point-absorber designed by CorPower Ocean AB. The negative spring arrangement of the WaveSpring system has been implemented into the 6 degrees-of-freedom solver and three computational cases are carried out: Ÿ   WEC surge decay test with a 0.1m initial horizontal offset. Ÿ   WEC motion in waves when only PTO system operates. Ÿ   WEC motion in waves when both PTO system and WaveSpring system operates. The simulated results showed general good agreement with the laboratory data and the relative errors between CFD and experimental results are acceptable, albeit there are differences of course. There are typical large problems associated with validation of CFD results. As CFD lacks semi-empirical tuning parameters it is more sensitive to input data than traditional radiation-diffraction methods. For example, Palm et al (2016) illustrated that by shifting the centre of gravity just 3 mm for a 0.5 m diameter cylinder (well inside the uncertainty range of the experimental data) the decay period was significantly changed. In the case of the CorPower buoy tests there are uncertainties associated with the pre-temsion, with the moment of inertia as well as the WaveSpring case was not run until a steady periodic state was established. This makes detailed validation very hard, but this problem is often occurring as WEC experiments are typically not performed with enough accuracy to be used as validation cases – it is simply not the main purpose of the experimental test campaigns. For the strongly nonlinear motion excited by the WaveSpring technology, the simulation appears to capture the key system characteristics, the exaggerated motion response as well as the high-order motion responses are clearly visible. This provides a confidence in the continued use of the CFD-based wave tank for further tuning of the WaveSpring/buoy design as well as providing validation data for radiationdiffraction models.

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Acknowledgements

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The experimental data was gratefully provided by CorPower Ocean AB. We especially thank Dr. Jørgen Hals Todalshaug at CorPower for the the invaluable assistance with regard to the WaveSpring implementation. Support for this work was given by the Swedish Energy Agency under grant no. P40428-1. The simulations were made on resources from Chalmers Centre for Computational Science and Engineering, provided by the Swedish National Infrastructure for Computing.

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References

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