1. The Power of Many?..... Coupled Wave Energy Point Absorbers Paul Young MSc candidate, University of Otago Supervised by Craig Stevens (NIWA), Pat Langhorne & Vernon Squire (Otago)

2. Motivation The big idea The physics Results Where to next? Talk outline WECs… WTF?

5. World resource Wave energy flux magnitude (kW per metre of wavefront) Source: Pelamis Wave Power website

6. Source: Smith et al (NIWA), Analysis for Marine Renewable Energy: Wave Energy, 2008

8. 1. Estimate by UK Carbon Trust Advantages: High energy density Low social & environmental impact (?) Reliability & predictability (c.f. wind) Low EROEI (?) Direct desalination AND... Practical worldwide resource ~ 2000-4000 TWh/year 1 (Current global demand ~ 17000 TWh/year)

9. Motivation The big idea The physics Results Where to next? Talk outline

12. Point absorbers Pros: Suitable for community scale Less disruption in event of device failure Cheaper per kW/h? Cons: Non-resonant in typical sea conditions Lower efficiency Maybe a linked chain of point absorbers will 'see' long wavelengths better than a lone device?

13. Key questions Is it possible to obtain better power output (per unit) with a linked chain? (Can we improve peak efficiency and/or widen bandwidth?) How are the mooring forces affected? (Survivability) What is the interplay between the device spacing and the wavelength?

14. My scheme: model device

15. 1-D (surge only) idealisation

16. Motivation The big idea The physics Results Where to next? Talk outline

17. Further assumptions/simplifications Small-body approximation Linear, small amplitude waves Neglect hydrodynamic interaction between devices

18. Forces Mooring forces Hydrodynamic forces: excitation, drag and radiation Master equation: (not including power take-off)

19. Technical issues… Importance of memory effects

20. Motivation The big idea The physics Results Where to next? Talk outline

21. Validating numerical code For lone device with zero drag, easy to solve equation of motion analytically.

22. Discrepancy between models with and without memory effects noticeable when nonlinear drag introduced, but small.

23. HOT OFF THE PRESS: Things get interesting with multiple devices.

24. Some good agreement...

25. ...some poor agreement...

27. Motivation The big idea The physics Results Where to next? Talk outline

29. Mooring and linkage forces { Chacterise as tension-only spring Spring stiffness (Linkage force on device J from device K) Device spacing Position of device K

30. Hydrodynamic forces (The tricky part...) Inline force on small(ish) bodies in oscillatory flow often described by Morison equation: BUT added mass depends on the oscillation frequency... Drag coefficient Area 'seen' by fluid Fluid density Fluid velocity Added mass Submerged volume

31. But under nonlinear conditions, device response may be over much broader range of frequencies... Data from Hulme, A.: The wave forces acting on a floating hemisphere undergoing forced periodic oscillations. 1982. How big is the effect? Semi-submerged sphere moving in surge For device with a ≈ 2m, energy-bearing wavelengths in typical sea state are 0.056ka0.126

32. Falnes' formulation 1. Falnes, J.: Ocean Waves and Oscillating Systems: linear interactions including wave-energy extraction. 2002. Wave forces are decomposed in frequency domain into excitation and radiation forces. For surge, under small-body approximation, these are 1 : ( + damping term) (c.f. )

33. Added mass at infinite frequency Impulse response function Added damping This expression is exact, but added mass and damping depend on body geometry. Radiation force in time domain

34. Thankfully......can fit an analytic function that isn't horrible Data from Hulme, A.: The wave forces acting on a floating hemisphere undergoing forced periodic oscillations. 1982.

35. Evaluate integrals with MATLAB symbolic math toolbox to get:

36. Master equation { n.b.

37. Solution method Solve numerically with 4 th order Runge-Kutta procedure on MATLAB Cast as 1 st order vector equation for (n.b. will be 4n entries with internal mass included)

38. Memory integral giving good agreement for linear motion over wavelength range