A Novel 2-D Point Absorber Numerical Modelling Method
Abstract
:1. Introduction
2. Materials and Methods
2.1. Point Absorber Wave Energy Converters
2.2. Point Absorber Model Characteristics
2.3. Linear Frequency-Domain Analysis
2.4. Time-Domain Formulation
2.5. First-Order Wave Loads
2.6. Second-Order Wave Drift Loads
2.7. Numerical Formulation
- 1.
- Definition of the free surface elevation signal using an empirical spectrum or by importing a defined from the real-time record;
- 2.
- Finding the points where peaks, through and zero-crossings occur;
- 3.
- Defining a signal of the instantaneous wave period;
- 4.
- Interpolation of vectors found in the previous two steps;
- 5.
- Interpolation of the reflection coefficient, which depends on the instantaneous wave period;
- 6.
- Evaluation of Equation (13);
- 7.
- Adding the found waves drift load vector to the first-order wave load horizontal component vector.
2.8. Experimental Study
- Length = 76 m;
- Width = 4.6 m;
- Depth = 0.5–2.3 m;
- Waves making: variable-water-depth computer-controlled flaps wavemaker;
- Beach: variable-water-depth sloping beach (reflection coefficient typically less than 5%).
2.9. Calibration and Uncertainty Analysis
3. Results
3.1. Qualitative Assessment
3.2. Validation with Experimental Data
3.3. Uncertainty Analysis Results
3.4. Discussion
- The mooring line is assumed to be inelastic. The only axial mooring displacement is due to the extension of the spring component. Depending on the material used, for a real installation, the mooring component elongation might be an important factor that needs to be investigated;
- Viscous forces are neglected. Note that for the typical size of wave power devices, operating in normal sea conditions, the viscous forces usually are significantly less than the first-order wave loads. The proposed methodology is not intended to be used for modelling the device for studying survivability of sea states conditions when, eventually, viscous effects are more significant. Moreover, to precisely estimate power absorption for relatively small devices at resonance conditions, viscous forces might be an important contribution;
- For simplicity, the mooring line is assumed to be attached to the center. In practice, this is not possible. However, due to the spherical shape considered and having assumed no viscous forces, it was possible to implement such simplification;
- The PTO system is modelled as a linear damping mechanism for which a single PTO damping coefficient can be set. The model, if appropriately extended, could be suitable for analyzing different, more complex types of PTO models, such as the hydraulic, phase-controlled systems, latching control or a combination of these.
4. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Instrument | Measurement | Range | Unit |
---|---|---|---|
Motion capture system | Floater displacement | 0–500 | mm |
Standard wave probe | Free-surface elevation | 0–150 | mm |
Sonic wave probe | Free-surface elevation upstream | 0–150 | Mm |
Load cell | Mooring tension | 0–50 | N |
Laser sensor | Mooring line displacement | 0–350 | Mm |
Motor tachometer | Mooring line displacement | 0–350 | mm |
Parameter | Model Scale (1:33) | Real Scale (1:1) | Unit |
---|---|---|---|
Floater radius | m | ||
Floater mass | kg | ||
Water depth | m | ||
Mooring line length | m | ||
Spring stiffness | N/m | ||
Cpto damping | kg/s | ||
Mooring pretension | N |
ω (rad/s) | 0.40 | 0.46 | 0.53 | 0.59 | 0.66 | 0.72 | 0.79 | 0.86 | 0.92 | 0.99 | 1.05 | 1.12 | 1.18 | 1.25 | 1.31 |
Crefl | 0.10 | 0.11 | 0.15 | 0.25 | 0.30 | 0.35 | 0.40 | 0.45 | 0.45 | 0.50 | 0.50 | 0.55 | 0.60 | 0.70 | 1.00 |
Units | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Waves Characteristics | Ampl. | (mm) | 30 | 30 | 60 | 30 | 30 | 60 | 30 | 30 | 60 | |
Freq. | (rad/s) | 2.3 | 6.6 | 6.6 | 2.3 | 6.6 | 6.6 | 2.3 | 6.6 | 6.6 | ||
Heave | Ampl. (mm) | 0.763 | 1.545 | 2.39 | 0.8 | 1.105 | 1.74 | 2.52 | 4.753 | 7.482 | 10.838 | |
Freq. (rad/s) | 0.002 | 0.008 | 0.004 | 0.002 | 0.008 | 0.004 | 0.008 | 0.036 | 0.018 | |||
Surge | Ampl. (mm) | 0.66 | 0.123 | 0.31 | 0.8 | 1.037 | 0.809 | 0.858 | 4.46 | 3.481 | 3.69 | |
Freq. (rad/s) | 0.005 | 0.017 | 0.003 | 0.005 | 0.017 | 0.003 | 0.021 | 0.073 | 0.013 | |||
Direct meas. | Wave probe | Ampl. (mm) | 0.891 | 0.341 | 2.099 | 0.7 | 1.133 | 0.779 | 2.213 | 4.873 | 3.349 | 9.516 |
Freq. (rad/s) | 0.002 | 0.004 | 0.003 | 0.002 | 0.004 | 0.003 | 0.009 | 0.016 | 0.013 | |||
Load | Ampl. (N) | 0.056 | 0.119 | 0.138 | 0.015 | 0.058 | 0.12 | 0.138 | 0.25 | 0.517 | 0.595 | |
Freq. (rad/s) | 0.001 | 0.007 | 0.004 | 0.001 | 0.007 | 0.004 | 0.006 | 0.031 | 0.016 | |||
Displacement | Ampl. (mm) | 0.822 | 1.73 | 2.176 | 0.85 | 1.183 | 1.927 | 2.336 | 5.085 | 8.288 | 10.044 | |
Freq. (rad/s) | 0.002 | 0.006 | 0.003 | 0.002 | 0.006 | 0.003 | 0.007 | 0.028 | 0.011 | |||
Sim. of PTO | PTO damping | (kg/s) | 1.473 | 1.473 | 1.473 | 1.473 | NA | NA | NA | |||
Spring stiffness | (N/m) | 13.212 | 13.212 | 13.212 | 13.212 | NA | NA | NA | ||||
Mass of floater | (kg) | 0.005 | 0.005 | 0.005 | 0.005 | NA | NA | NA | ||||
Fixed param. | Radius of floater | (mm) | 0.8 | 0.8 | 0.8 | 0.8 | NA | NA | NA | |||
Length of line | (mm) | 6 | 6 | 6 | 6 | NA | NA | NA | ||||
Pretension | (N) | 0.056 | 0.119 | 0.138 | 0.015 | 0.058 | 0.12 | 0.183 | 0.25 | 0.517 | 0.786 | |
Indirect meas. | Power | (W) | 0.03 | 0.328 | 1.221 |
Test No.: | 12 | 25 | 35 | 38 | 53 | ||
---|---|---|---|---|---|---|---|
Heave | Ampl. (mm) | 29.527 | 28.476 | 30.847 | 30.782 | 31.258 | 0.728 |
Freq. (rad/s) | 4.585 | 4.608 | 4.588 | 4.58 | 4.594 | 0.007 | |
Surge | Ampl. (mm) | 26.168 | 23.079 | 24.98 | 26.281 | 25.295 | 0.816 |
Freq. (rad/s) | 4.643 | 4.524 | 4.589 | 4.55 | 4.592 | 0.029 | |
Wave probe | Ampl. (mm) | 33.027 | 34.506 | 34.02 | 32.872 | 33.884 | 0.438 |
Freq. (rad/s) | 4.596 | 4.609 | 4.594 | 4.592 | 4.594 | 0.004 | |
Load | Ampl. (N) | 2.156 | 2.067 | 2.248 | 2.249 | 2.278 | 0.055 |
Freq. (rad/s) | 4.588 | 4.604 | 4.589 | 4.578 | 4.594 | 0.006 | |
Displacement | Ampl. (mm) | 29.732 | 28.305 | 30.742 | 30.927 | 31.253 | 0.758 |
Freq. (rad/s) | 4.587 | 4.603 | 4.589 | 4.578 | 4.594 | 0.006 |
Quantity | Total bias (±%) | |
---|---|---|
Heave | ampl. | 21.41 |
phase | 0.55 | |
Surge | ampl. | 20.68 |
phase | 1.11 | |
Wave probe | ampl. | 10.76 |
phase | 0.25 | |
Load | ampl. | 20.36 |
phase | 0.47 | |
Displacement | ampl. | 24.27 |
phase | 0.42 | |
Power | ampl. | 64.45 |
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Giannini, G.; Day, S.; Rosa-Santos, P.; Taveira-Pinto, F. A Novel 2-D Point Absorber Numerical Modelling Method. Inventions 2021, 6, 75. https://doi.org/10.3390/inventions6040075
Giannini G, Day S, Rosa-Santos P, Taveira-Pinto F. A Novel 2-D Point Absorber Numerical Modelling Method. Inventions. 2021; 6(4):75. https://doi.org/10.3390/inventions6040075
Chicago/Turabian StyleGiannini, Gianmaria, Sandy Day, Paulo Rosa-Santos, and Francisco Taveira-Pinto. 2021. "A Novel 2-D Point Absorber Numerical Modelling Method" Inventions 6, no. 4: 75. https://doi.org/10.3390/inventions6040075