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Performance of a distensible-tube wave attenuator in a slender focusing channel

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Abstract

A wave energy device utilizing a water-filled distensible tube aligned head to waves is analyzed. The wave-excited pressure bulges in the tube activate a stationary forward-bent circular oscillating water column (OWC) at its stern. This system has been extensively tested in wave-tank, at different scales. However, further improvements to its efficiency are possible. The present research assesses its performance at 1:20 scale, while operating in a slender focusing channel. Complementarily, two other situations are considered: one at the nearshore, where the system is assembled to a bottom-standing jacket structure, and an onshore design where the tube operates in front of a sea wall in constant depth or over a sloping bottom. Deep and intermediate regular waves of small and finite amplitudes are generated in the wave-tank, representing the wave climate in the Occidental Group of the Azores islands. A non-linear power take-off impedance is imposed by a set of calibrated orifice plates and its characteristic is then obtained assuming compressible airflow. Measurements of air pressure in the pneumatic chamber, OWC free-surface displacement, and incident wave-field provide estimates of energy capture-width, which are then compared with analytical predictions and benchmark test results. Photographs further help to understand the physics underlying the tube’s working modes. Moreover, the performance of the system is obtained with and without the tube and then compared with standard values for OWCs. It is demonstrated that the addition of a distensible tube and a suitable focusing channel can significantly improve the capacity factor of a conventional OWC system.

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Abbreviations

\(A_{0}\) :

Orifice area

\(a\) :

OWC mouth/tube stern submergence

\(b\) :

Wave-tank width

\(b^{\prime}\) :

Reynolds number correction factor

\(C_{{\text{d}}}\) :

Orifice discharge coefficient

\(C_{0}\) :

Impedance coefficient

\(C_{{\text{W}}}\) :

Power capture-width

\(C_{\infty }\) :

Discharge coefficient for infinite Reynolds number

\(c\) :

Incident wave celerity

\(D\) :

Tube distensibility

\(d\) :

Pressurized rubber tube diameter

\(d_{0}\) :

Orifice diameter

\(E\) :

Young’s modulus

\(e\) :

Gap between channel mouth and tank sidewalls

\(H\) :

Incident wave height

\(h\) :

Water depth in the tank

\(h_{0}\) :

Pressure head in the shaft

\(h_{1}\) :

Height of pneumatic chamber

\(k\) :

Fundamental wave number

\(L\) :

Rubber tube length

\(l\) :

OWC length

\(\dot{m}\) :

Mass flow-rate across orifice

\(P\) :

Extracted power at the PTO

\(P_{i}\) :

Incident wave power

\(p\) :

Air pressure in the pneumatic chamber

\(p_{0}\) :

Atmospheric pressure

\(p^{*}\) :

Air pressure upstream of the orifice

\(Q\) :

Volume flow-rate of air across orifice

\(q\) :

Maximum instantaneous air volume flow

\({\text{Re}}\) :

Reynolds number

\({\text{Re}}_{{\text{u}}}\) :

Reynolds number upstream of the orifice

\(r\) :

Internal radius of pneumatic chamber

\(T\) :

Incident wave period

\(T_{0}\) :

OWC’s natural period

\(T_{1}\) :

Tube’s tuning period

\(T_{{\text{S}}}\) :

Full-scale significant wave period

\(t\) :

Time

\(U\) :

Bulge wave speed

\({\text{Ur}}\) :

Ursell number

\(w\) :

Tube wall thickness

\(X\) :

Longitudinal length of the focusing channel

\(X_{{\text{S}}}\) :

Horizontal length of the sloped bottom

\(Z\) :

Normalized impedance

\(\overline{Z}_{{{\text{PTO}}}}\) :

Mean PTO impedance

\(Z_{{{\text{tube}}}}\) :

Distensible tube impedance

\(\dot{z}\) :

Mean velocity of the OWC free surface

\(\beta \) :

Orifice diameter to upstream diameter ratio

\(\gamma \) :

Specific heat ratio of air

\(\Delta p\) :

Differential pressure

\(\varepsilon \) :

Expansibility factor

\(\zeta \) :

Amplification factor

\(\eta \) :

Water surface displacement in shaft

\(\theta \) :

Flare angle of the focusing channel

\(\lambda \) :

Incident wavelength

\(\mu \) :

Absolute viscosity of air

\(\mu_{1}\) :

Absolute viscosity of water

\(\rho\) :

Mass density of air passing through orifice

\(\rho_{1}\) :

Mass density of water

\(\rho^{*}\) :

Mass density of air upstream of orifice

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Acknowledgements

The present study follows the work done under FCT Project PTDC/EME-MFE/111763, developed from 2012 to 2016 within the framework of the COMPETE Programme, partially funded by the Portuguese Foundation for Science and Technology and the European Union. The research was carried out in the experimental facilities of the Laboratory of Fluid Mechanics and Turbomachinery of Universidade da Beira Interior, in Portugal, with the collaboration of the University of Southampton in the UK. The authors are grateful to FCT and UBI for their support of the research activities. Furthermore, the valuable contribution of Mr. Morgado and of the Electromechanical Engineering Msc students that have helped to develop and assemble the experimental apparatus is acknowledged. The first author also wishes to acknowledge the valuable teachings of the late Professor Francis Farley, whose enthusiasm will always be remembered.

Funding

The work was partially funded by the Portuguese Foundation for Science and Technology (FCT) and the European Union, within the framework of Programme COMPETE.

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Authors and Affiliations

  1. Laboratory of Fluid Mechanics and Turbomachinery, Electromechanical Engineering Department, Universidade da Beira Interior, 6201-001, Covilha, Portugal

    Antonio C. Mendes & Francisco P. Braga

  2. Energy & Climate Change Research Group, Faculty of Engineering and Physical Sciences, University of Southampton, Southampton, UK

    John R. Chaplin

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All authors contributed to the study conception and design. Material preparation, data collection, and analysis were performed by FPB. The first draft of the manuscript was written by ACM and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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Correspondence to Antonio C. Mendes.

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Mendes, A.C., Braga, F.P. & Chaplin, J.R. Performance of a distensible-tube wave attenuator in a slender focusing channel. J. Ocean Eng. Mar. Energy 9, 745–766 (2023). https://doi.org/10.1007/s40722-023-00297-8

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