## Simulation Progress

### Simulation Background

The CycWEC and wave-induced ﬂow ﬁeld are modeled using potential ﬂow theory. For an inviscid, incompressible, and irrotational ﬂow, the governing continuity equation simpliﬁes to the Laplace equation:

$\nabla^2\Phi=0$

The linearized free surface boundary condition is used to model the water surface. Subject to these kinematic and dynamic boundary conditions, the complex potential for a vortex moving under a free surface in the complex plane is:

$F(z,t)&=&\frac{\Gamma(t)}{2\pi i} ln\left(\frac{z-c(t)}{z-\bar{c}(t)}\right) &+&\frac{g}{\pi i}\int^{t}_{0}\int^{\infty}_{0} \frac{\Gamma(\tau)}{\sqrt{gk}}e^{-ik(z-\bar{c}(\tau))} &\times&sin\left[\sqrt{gk}(t-\tau)\right]dkd\tau$

Note: Please see scientific publications and references for full explanation and derivation

Besides a powerful numerical tool for the validation and evaluation of the novel CycWEC design, the simulation also provides:

• A means to compare and validate model vs. full scale behavior
• Exact solution of governing equations
• Energy conservation
• Spatial grid not needed
• Nominal solution times
• Powerful tool for WEC kinematic and dynamic design
• Radius
• Submergence depth
• Number of blades and azimuthal positions
• Blade pitch scheduling
• Hydrofoil shape and design
• Tool for control algorithm design
• Sensor placement
• Estimation algorithm
• Pitch angle to wave height relationship

### Results

An animation of typical simulation is shown below. Here an upwave sensor measuring surface elevation is used to estimated the instantaneous frequency and amplitude of the incoming wave. The WEC which is modelled by the above equations rotates by the constant excitation of the incoming wave.

The water surface is visualized (note: the surface elevation is amplified by a factor of 3 for visualization ease). The wave is completely being terminated by the CycWEC. By a control volume analysis the energy differential between up and downwave locations is provided directly to the shaft of CycWEC; thus converting wave energy directly into shaft power.

The efficiency of the device for harmonic and irregular waves is presented in the below chart. Efficiencies at the design point are greater than 99%. The efficiency envelope remains at this value for a wide range of incoming wavelengths. As the device becomes small compared to the incoming wavelength, the efficiency drops off faster than as the device being oversized for the incoming wavelength. A North Atlantic sea state probability is also plotted to show that the efficiency envelope of the CycWEC fully encompasses all sea states experienced at a given location throughout the duration of an entire year. The yearly efficiency for unidirectional, irregular wave environment is computed to be 98.3% in the inviscid limit.