An improved boundary element method for modelling a self-reacting point absorber wave energy converter

Abstract

A numerical model based on a boundary element method (BEM) is developed to predict the performance of two-body self-reacting floating-point absorber (SRFPA) wave energy systems that operate predominantly in heave. The key numerical issues in applying the BEM are systematically discussed. In particular, some improvements and simplifications in the numerical scheme are developed to evaluate the free surface Green’s function, which is a main element of difficulty in the BEM. For a locked SRFPA system, the present method is compared with the existing experiment and the Reynolds-averaged Navier–Stokes (RANS)-based method, where it is shown that the inviscid assumption leads to substantial over-prediction of the heave response. For the unlocked SRFPA model we study in this paper, the additional viscous damping primarily induced by flow separation and vortex shedding, is modelled as a quadratic drag force, which is proportional to the square of body velocity. The inclusion of viscous drag in present method significantly improves the prediction of the heave responses and the power absorption performance of the SRFPA system, obtaining results excellent agreement with experimental data and the RANS simulation results over a broad range of incident wave periods, except near resonance in larger wave height scenarios. It is found that the wave overtopping and the re-entering impact of out-of-water floating body are observed more frequently in larger waves, where these non-linear effects are the dominant damping sources and could significantly reduce the power output and the motion responses of the SRFPA system.

Appendix

In the present BEM, all body surface boundaries are subdivided into a number of curved surfaces, which are approximated by spatial quadrilateral panels or triangular panels. As seen in Fig. 4, the centroid $P_c(x_c,y_c,z_c)$ of each panel element is represented as

$$\begin{aligned} x_c=\frac{\sum \nolimits _{k=1}^nx_k}{n},\quad y_c=\frac{\sum \nolimits _{k=1}^ny_k}{n},\quad z_c=\frac{\sum \nolimits _{k=1}^nz_k}{n}, \end{aligned}$$

(A1)

where $A_k(x_k,y_k,z_k)$ represents the k-th node of the panel element, especially $n = 3$ for triangular panels, $n = 4$ for quadrilateral panels.

Without losing generality, assuming the unit normal vector ${{\varvec{n}}}$ of the panel element directed from the fluid into bodies, for triangular panels, it can be evaluated by

$$\begin{aligned} {{\varvec{n}}}=\frac{{{\varvec{A}}}_1{{\varvec{A}}}_2\times {{\varvec{A}}}_2{{\varvec{A}}}_3}{\vert {{\varvec{A}}}_1{{\varvec{A}}}_2\times {{\varvec{A}}}_2{{\varvec{A}}}_3\vert }, \end{aligned}$$

(A2)

for quadrilateral panels,

$$\begin{aligned} {{\varvec{n}}}=\frac{{{\varvec{A}}}_1{{\varvec{A}}}_3\times {{\varvec{A}}}_2{{\varvec{A}}}_4}{\vert {{\varvec{A}}}_1{{\varvec{A}}}_3\times {{\varvec{A}}}_2{{\varvec{A}}}_4\vert }. \end{aligned}$$

(A3)

In general, the four nodes of a spatial quadrilateral panel are probably not located on the same plane, which makes it useless in the present BEM. One treatment is applied here: the whole panel element $A_1A_2A_3A_4$ is projected to a plane (perpendicular to the normal vector ${{\varvec{n}}}$ and contains the centroid $P_c$), then it is approximately replaced by the projective element $A_1'A_2'A_3'A_4'$.

For convenience, the local coordinate system $P_c-\alpha \beta \gamma$ on each panel element is established to evaluate the influence coefficient matrices. On an arbitrary panel element $\triangle S_j$, the centroid $P_c(x_c,y_c,z_c)$ is specified as the origin and let $\varvec{\tau }_\alpha$, $\varvec{\tau }_\beta$, $\varvec{\tau }_\gamma$ signify the basis vectors of $P_c-\alpha \beta \gamma$. We can choose an arbitrary node $A_k(x_k,y_k,z_k)$ of the element to determine the basis vector $\varvec{\tau }_\alpha$

$$\begin{aligned} \varvec{\tau }_\alpha =\frac{{{\varvec{P}}}_c{{\varvec{A}}}_k}{\vert {{\varvec{P}}}_c{{\varvec{A}}}_k\vert }, \end{aligned}$$

(A4)

and

$$\begin{aligned} \varvec{\tau }_\gamma ={{\varvec{n}}},\quad \varvec{\tau }_\beta =\varvec{\tau }_\gamma \times \varvec{\tau }_\alpha . \end{aligned}$$

(A5)

Let $\varvec{\tau }_\alpha$, $\varvec{\tau }_\beta$, $\varvec{\tau }_\gamma$ be column vectors, thus an orthogonal coordinate-transformation matrix can be obtained

$$\begin{aligned} {{\varvec{C}}}_t=(\varvec{\tau }_\alpha , \varvec{\tau }_\beta , \varvec{\tau }_\gamma )^\mathrm{T}, \end{aligned}$$

(A6)

then the transformation relation between the coordinates (x, y, z) in $O-xyz$ and $(\alpha ,\beta ,\gamma )$ in $P_c-\alpha \beta \gamma$ is represented as

$$\begin{aligned} \begin{pmatrix} \alpha \\ \beta \\ \gamma \end{pmatrix} = {{\varvec{C}}}_t\begin{pmatrix} x-x_c \\ y-y_c \\ z-z_c \end{pmatrix},\qquad \begin{pmatrix} x \\ y \\ z \end{pmatrix} = {{\varvec{C}}}_t^\mathrm{T}\begin{pmatrix} \alpha \\ \beta \\ \gamma \end{pmatrix} + \begin{pmatrix} x_c \\ y_c \\ z_c \end{pmatrix}, \end{aligned}$$

(A7)

On the element $\triangle S_j$, a bilinear transformation between the coordinates $(\alpha ,\beta )$ in the local system $P_c-\alpha \beta \gamma$ and (u, v) on the standard square $E^s_j$ is used here

$$\begin{aligned} \alpha (u,v)&={\bar{\alpha }}+a_{\alpha }u+b_{\alpha }v+c_{\alpha }uv, \end{aligned}$$

(A8)

$$\begin{aligned} \beta (u,v)&={\bar{\beta }}+a_{\beta }u+b_{\beta }v+c_{\beta }uv, \end{aligned}$$

(A9)

where

$$\begin{aligned}&{\bar{\alpha }}=\frac{1}{4}\sum \limits _{k=1}^4\alpha _k,\quad a_{\alpha }={\bar{\alpha }}-\frac{1}{2}(\alpha _2+\alpha _3), \\&b_{\alpha }={\bar{\alpha }}-\frac{1}{2}(\alpha _3+\alpha _4),\quad c_{\alpha }={\bar{\alpha }}-\frac{1}{2}(\alpha _2+\alpha _4), \end{aligned}$$

(A10)

$$\begin{aligned}&{\bar{\beta }}=\frac{1}{4}\sum \limits _{k=1}^4\beta _k,\quad a_{\beta }={\bar{\beta }}-\frac{1}{2}(\beta _2+\beta _3), \\&b_{\beta }={\bar{\beta }}-\frac{1}{2}(\beta _3+\beta _4),\quad c_{\beta }={\bar{\beta }}-\frac{1}{2}(\beta _2+\beta _4), \end{aligned}$$

(A11)

where $(\alpha _k,\beta _k)$ represents the local coordinates of the k-th node on the element $\triangle S_j$. The Jacobian determinant associated with the transformation of the element $\triangle S_j$ to its image $E^s_j$ is evaluated by

$$\begin{aligned}&J(u,v) = \left| \frac{D(\alpha ,\beta )}{D(u,v)} \right| = a_\alpha b_\beta -b_\alpha a_\beta \\&\qquad \quad \qquad +\,(a_\alpha c_\beta -c_\alpha a_\beta )u+(c_\alpha b_\beta -b_\alpha c_\beta )v, \end{aligned}$$

(A12)

For triangular panel elements, the transformation Eqs. (A8) and (A9) are no longer applicable. Without losing generality, assuming the nodes $A_3$ and $A_4$ of the element coinciding, we apply a linear transformation

$$\begin{aligned} \alpha (u,v)&={\bar{\alpha }}+a_{\alpha }u+b_{\alpha }v, \end{aligned}$$

(A13)

$$\begin{aligned} \beta (u,v)&={\bar{\beta }}+a_{\beta }u+b_{\beta }v, \end{aligned}$$

(A14)

where

$$\begin{aligned} {\bar{\alpha }}&=\frac{1}{2}(\alpha _1+\alpha _3),\quad a_{\alpha }=\frac{1}{2}(\alpha _1-\alpha _2),\quad b_{\alpha }=\frac{1}{2}(\alpha _2-\alpha _3), \end{aligned}$$

(A15)

$$\begin{aligned} {\bar{\beta }}&=\frac{1}{2}(\beta _1+\beta _3),\quad a_{\beta }=\frac{1}{2}(\beta _1-\beta _2),\quad b_{\beta }=\frac{1}{2}(\beta _2-\beta _3), \end{aligned}$$

(A16)

and the Jacobian determinant becomes

$$\begin{aligned} J(u,v)=\left| \frac{D(\alpha ,\beta )}{D(u,v)}\right| =a_\alpha b_\beta -b_\alpha a_\beta . \end{aligned}$$

(A17)