# Hydrodynamic Investigation on a Land-Fixed OWC Wave Energy Device under Irregular Waves
## 2 Numerical Model  
### Governing Equations and Boundary Conditions  
In this section, the governing equation and the boundary problems are described. Figure 1 shows a schematic layout of the numerical wave flume (NWF). The damping layer is set at the left end of the flume to absorb the reflected waves and the OWC device is placed at the other end. The Cartesian coordinate system (_Oxz_) is introduced with its origin located at the edge position of the damping layer on the stationary free surface, with the \(z\)-axis positive in the upwards direction and the \(x\)-axis positive in the wave propagation direction.  
\(B\) denotes the chamber width, \(C\) the front-wall thickness, \(d\) the front wall draught, \(e\) the air duct width, \(hc\) the chamber height and \(h\) the static water depth. Five numerical wave gauges (i.e., G1, G2, G3, G4 and G5) are used to record the continuous surface elevations at different locations. G1 is situated outside the chamber at 1.0 wavelengths from the exterior surface of the front wall. Four gauges are situated inside the chamber: G2 is at the inner side of the front wall, G3 is 0.25\(B\) from the internal side of the front wall, the third internal wave gauge, G4 is at the chamber center and G5 is at the rear wall.  
Assuming that the water is incompressible, inviscid and the fluid flow is irrotational, the fluid motion can be described by the velocity potential \(\phi\). Here, the numerical model developed by [PERSON] et al. [6] to simulate the hydrodynamic characteristics of the OWC device under regular waves is extend to model the hydrodynamic effects under irregular wave conditions. The inner-domain sources method is adopted to produce incident waves. Thus, the governing equation is modified from the Laplace equation to the Poisson equation as follows:  
\[\
abla^{2}\phi=q^{*}(x_{s},z,t) \tag{1}\]  
where \(\
abla^{2}=\partial^{2}/\partial x^{2}+\partial^{2}/\partial z^{2}\) is the 2D Laplacian operator, and \(q^{*}(x_{s},z,t)=2u\delta(x-x_{s})\) is the pulsating volume flux density, \(x_{s}\) is the horizontal position of vertical sources, \(u\) is the horizontal fluid velocity calculated from Stokes second-order analytical solution and the Dirac delta function is given by \(\delta(x-x_{s})\).  
To account for the viscosity effects, an artificial viscous damping term with coefficient \(\mu_{2}\) was introduced in free surface inside the OWC chamber [22; 23]. Then, the kinematic and dynamic free surface boundary conditions can be written as follows:  
\[\left\{\begin{array}{l}\frac{dX(x,z)}{dt}=\
abla\phi-\mu_{1}(x)(\mathbf{X}-\mathbf{X} _{0})\\ \frac{d\phi}{dt}=-g\eta+\frac{1}{2}|\
abla\phi|^{2}-\frac{p}{\rho}-\mu_{1}(x) \phi-\mu_{2}\frac{\partial\phi}{\partial n}\end{array}\right. \tag{2}\]  
where \(d/dt=\partial/\partial t+\mathbf{v}\cdot\
abla\) is the material derivative. \(\mathbf{X}\left(x,z\right)\) is the horizontal and vertical displacements of fluid particles on the free surface, respectively. \(\mathbf{X}_{0}=\left(x_{0},\,0\right)\) denotes the initial static position of the fluid particle. \(g\) is the gravity acceleration. \(p\) is the pneumatic pressure and \(\rho\) is the water density, \(\mu_{1}(x)\) and \(\mu_{2}\) are the coefficients of the damping layer and artificial viscous damping term, respectively.  
Figure 1: Sketch of the numerical wave flume with OWC-WECs.  
The boundaries at the OWC structure's surface are considered impermeable. Therefore, the zero normal velocity condition \(\partial\phi/\partial\mathbf{n}=0\) at the bottom, rear wall and front wall of the OWC is satisfied.  
The above boundary value problem can be transformed into a boundary integral equation by adopting the Green's second identity. The HOBEM is adopted here to generate the solution. Further details on the numerical model can be found in [6].