Analytical study on hydrodynamic performance of a raft-type wave power device

Abstract

In this paper, an analytical model is developed for the motion response and wave attenuation of a raft-type wave power device. The analytical solution of diffraction and radiation problem of multiple two-dimensional rectangular bodies floating on a layer of water of finite depth is obtained using a linearized potential flow theory. Wave excitation forces, added masses and wave damping coefficients for these bodies are calculated from incident, diffracted and radiated potentials. Upon solving the motion equation, response, power absorption and wave attenuation of a raft-type wave power device are obtained. The model is validated by comparison of the present results with the existing ones, and energy conservation is checked. The validated model is then utilized to examine the effect of power take-off damping coefficient, raft draft, spacing between two rafts, water depth, and raft numbers on power absorption and wave transmission coefficient of raft-type wave power device. The influence of structure length ratio is also discussed. It is found that the same wave transmission coefficient can be obtained by any certain raft-type wave power device, regardless of wave propagation direction.

Appendix: Proofs of the same transmission coefficient for raft devices with inverse a 1/a 2

For convenience, the comparison between the raft devices with inverse length ratio a ₁/a ₂ and a ₂/a ₁ can be transformed into the hydrodynamic problems of a raft device with a ₁/a ₂ suffering from the waves with incoming angle equal to 0 and 180 degree, respectively.

The wave transmission coefficient when for the opposite coming waves corresponding to Eq. 27 is given by

$$\hat T_{\text w}^\prime = \hat T_{{\text w},0}^\prime + \frac{{{\omega ^2}\cosh \left( {kh} \right)}}{{Ag}}\boldsymbol{A}_R^{\prime T}{\boldsymbol{A}^ - },$$

(31)

where $\hat T_{{\text w},0}^\prime = 1 - \frac{{\omega \cosh \left( {kh} \right)}}{{{\text{i}}Ag}}A_{1,1}^{\prime D}{{\text{e}}^{{\text{i}}k{x_{l,1}}}}$ is the complex transmission coefficient of the fixed jointed structures for the waves coming from the opposite direction; $A_{1,1}^{\prime {\text{D}}}$ is the coefficient of the diffraction spatial velocity potential at Subdomain 1 for the waves coming from the opposite direction.

It is believed that for two-dimensional wave diffraction problem of fixed arbitrary shape (not limited to rectangular section) floaters and/or submerged bodies, the complex transmission coefficient of the structures suffering from waves with incoming angle = 0 is equal to that suffering from waves propagating in the opposite direction [20, 21], leading to

$${\hat T_{{\text w},0}} = \hat T_{{\text w},0}^\prime ,$$

(32)

Apart from computing the integral of the incident wave potential and the diffracted wave potential on the wetted surface as shown in Eq. 20, the wave excitation vectors for waves coming in x direction F _e and in the opposite direction F _e′ can also be expressed, respectively, as follows using Haskind relation:

$${{\varvec{F}}_{\text{e}}} = - {\text{i}}\rho gD\left( {kh} \right){{\varvec{A}}^ - };{{\varvec{F'}}_{\text{e}}} = - {\text{i}}\rho gD\left( {kh} \right){{\varvec{A}}^ + },$$

(33)

where $D\left( {kh} \right) = \left[ {1 + \frac{{2kh}}{{\sinh (2kh)}}} \right]\sinh (kh)$,

Use of Eqs. (23) and (33) gives

$$\begin{aligned} {\mathbf{A}}_{{\text{R}}}^{T} {\mathbf{A}}^{ + } & = \left[ {\begin{array}{*{20}c} {\left( {{\mathbf{F}}_{{\text{e}}} } \right)^{T} } & {{\mathbf{0}}} \\ \end{array} } \right]\left( {{\mathbf{S}}^{{ - 1}} } \right)^{T} \left\{ {\begin{array}{*{20}c} {{\mathbf{A}}^{ + } } \\ {\mathbf{0}} \\ \end{array} } \right\} \\ & = - {\text{i}}\rho gD\left( {kh} \right)\left[ {\begin{array}{*{20}c} {\left( {{\mathbf{A}}^{ - } } \right)^{T} } & {{\mathbf{0}}} \\ \end{array} } \right]\left( {{\mathbf{S}}^{{ - 1}} } \right)^{T} \left\{ {\begin{array}{*{20}c} {{\mathbf{A}}^{ + } } \\ {\mathbf{0}} \\ \end{array} } \right\}, \\ \end{aligned}$$

(34)

$$\begin{aligned} {\mathbf{A}}_{{\text{R}}}^{{\prime T}} {\mathbf{A}}^{ - } & = \left[ {\begin{array}{*{20}c} {\left( {{\mathbf{F}}^{\prime } _{{\text{e}}} } \right)^{T} } & {{\mathbf{0}}} \\ \end{array} } \right]\left( {{\mathbf{S}}^{{ - 1}} } \right)^{T} \left\{ {\begin{array}{*{20}c} {{\mathbf{A}}^{ - } } \\ {\mathbf{0}} \\ \end{array} } \right\} \\ & = - {\text{i}}\rho gD\left( {kh} \right)\left[ {\begin{array}{*{20}c} {\left( {{\mathbf{A}}^{ + } } \right)^{T} } & {{\mathbf{0}}} \\ \end{array} } \right]\left( {{\mathbf{S}}^{{ - 1}} } \right)^{T} \left\{ {\begin{array}{*{20}c} {{\mathbf{A}}^{ - } } \\ {\mathbf{0}} \\ \end{array} } \right\}, \\ \end{aligned}$$

(35)

The authors observe that because both $\boldsymbol{A}_{{\text{R}}}^{T} \boldsymbol{A}^{ + }$ and ${{\varvec{A}}_{\text{R}}}^{\prime T}{{\varvec{A}}^ - }$ are scalars, it follows that they equal their own transpose. Because of the symmetry of the matrix ${\mathbf{S}}$, ${\mathbf{S}}^{{ - 1}}$ is also a symmetric matrix $({\mathbf{S}}^{{ - 1}} )^{T} = {\mathbf{S}}^{{ - 1}} .$ By transposing Eqs. 34 or 35, the following equation is obtained:

$$\boldsymbol{A}_{{\text{R}}}^{T} \boldsymbol{A}^{ + } = \boldsymbol{A}_{{\text{R}}} ^{{\prime T}} \boldsymbol{A}^{ - } .$$

(36)

According to Eqs. 27, 31 and 36, the following equation is obtained:

$${\hat T_{\text w}} = {\hat T'_{\text w}},$$

(37)

which means that a raft device suffering from waves in opposite directions leads to the same transmission coefficient.