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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)

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

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

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

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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 ~ TWh/year 1 (Current global demand ~ TWh/year)

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

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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?

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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?

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My scheme: model device

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1-D (surge only) idealisation

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

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

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

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Technical issues… Importance of memory effects

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

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

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

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

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Some good agreement...

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...some poor agreement...

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

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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

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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

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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 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

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Falnes' formulation 1. Falnes, J.: Ocean Waves and Oscillating Systems: linear interactions including wave-energy extraction 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. )

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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

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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

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Evaluate integrals with MATLAB symbolic math toolbox to get:

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Master equation { n.b.

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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)

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Memory integral giving good agreement for linear motion over wavelength range

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