Proceedings of the Tenth (2000) International Offshore and Polar Engineering Conference
Seattle, USA, May 28-June 2, 2000
Copyright©2000 by The International Society of Offshore and Polar Engineers
ISBN 1-880653-46-X (Set); ISBN 1-880653-47-8 (Vol. 1); ISSN 1098-6189 (Set)
Maximum Wave-Energy Absorption by Oscillating Systems Consisting of Bodies and Water Columns with Restricted or Unrestricted Amplitudes
J. Falnes
Norwegian University of Science and Technology
Trondheim, Norway
ABSTRACT
The theoretically maximum power absorbed from ocean waves by a system of several interacting oscillating bodies and water columns is studied for the case of unrestricted oscillation, as well as for the case when certain restrictions are applicable for the oscillation amplitudes. For the case when the radiation-damping matrix is singular, it is shown that the unconstrained maximum absorbed power is unambiguous even if the optimum oscillation is not then unique. Two different kinds of constraints are investigated for an axisymmetical system of an OWC in a floating body.
KEY WORDS: optimum oscillation, singular radiation-damping matrix
INTRODUCTION
Maximising the power absorbed from ocean waves by a system of several interacting oscillating bodies was studied during the late 1970s by Budal[1], Evans[2] and Falnes[3]. A particularly simple proof of the optimum condition, for the case of unrestricted amplitudes, was presented by Evans, when it was assumed that the radiation-resistance matrix is non-singular, i.e. invertible.
If amplitude constraints are applicable, numerical analysis has to be used except in certain simple problems which may be studied analytically. One such example was presented in 1981 by Evans[4] when one common global constraint was applied to all body amplitudes. Also in this analysis a non-singular radiation-resistance matrix was assumed.
In the present paper this analysis and the above-mentioned simple proof is generalised to the case where the radiation-resistance matrix is singular. Moreover, the derivation is also generalised to a situation where the radiation-resistance matrix, which is symmetric and real, is replaced by a hermitian complex radiation-damping matrix.
The motivation for the latter generalisation is the following.
It appears that if the system of oscillating bodies is extended include also a group of oscillating pressure distributions (or oscillating water columns (OWCs) with pneumatic power take-off), then the system's radiation-damping matrix is complex and hermitian[5].
SYSTEM OF OWC'S AND OSCILLATING BODIES
Wave interaction with a group of several oscillating bodies, immersed in water, was studied in the late 1970s by Evans[2] and independently by Falnes[3]. Subsequently, the interaction with a group of pressure distributions (OWCs with pneumatic power take-off) was analysed by Evans[6]. A few years later the wave interaction with several oscillators of both kinds, bodies as well as pressure distributions (or OWCs), were studied independently by Fernandes[7] and by Falnes and Mclver[5].
Consider an oscillating system consisting of
N=Nu + Np (1)
independent oscillators, where Nu is the number of body oscillators and Np the number of OWC oscillators. If each body is free to oscillate in all its six degrees of freedom (surge, sway, heave, roll, pitch and yaw), then Nu is equal to six times the number of oscillating bodies. Some of the OWCs may be contained in a fixed structure, while the remaining ones are contained in a floating structure which belongs to the set of oscillating bodies.
Each OWC structure is hollow, whereby a certain amount of air is contained in a chamber closed, at the lower end, by the internal water surface (the top of the OWC). Above the water the contained air may communicate with the outer atmosphere through a pneumatic power take-off system (air turbine, valves, etc.). In the lower end the OWC, at its mouth, communicates with the waves through the water external to the OWC structure.
The oscillation of the immersed bodies and of the OWCs may be utilised for converting power by take-off devices, which may be e.g. mechanical, hydraulic or pneumatic, but which are not discussed any further in the present paper. We shall, however, assume that by means of such devices any desired oscillation for each oscillator can be established, possibly within restrictions to be specified later.
