Maximum wave-power absorption under motion
constraints
D. V. E V A N S
School of Mathematics, University of Bristol, Bristol BS8 1 714/,, UK
An expression is derived for the maximum mean power that can be absorbed by a system of
oscillating bodies in waves under a global constraint on their motions. The particular case of a
single half-immersed sphere is used to show how the 'point absorber' result predicting capture
widths in excess of unity must be modified. The theory is also applied to the submerged cylinder
wave-energy device and curves are presented which show how the maximum efficiency is affected
by restricting the motion of the device.
INTRODUCTION
Results for the maximum power which a single body in
two or three dimensions can absorb when oscillating in
resonance with an incident sinusoidal wave train have
been given by Evans ~, Mci 2 and Newman 3. More recently
these results have been generalized to an arbitrary
number of bodies by Evans 4 and Falnes 5. In each case it
was assumed that the body was free to oscillate with
whatever amplitude was necessary to achieve the optimal
conditions for maximum power. An exception to the
papers mentioned is that due to Newman 6 who imposed
motion constraints in considering the power absorption
capability of a slender elongated body.
In practice most wave-energy devices will have physical
limitations placed upon their excursions due to restraints
such as mooring lines or pump stroke. Whereas there
exists a significant band of wave frequencies and lengths
for which the device does not reach these limits, in long
waves a relatively small device would, ideally, need to
make large excursions to achieve optimum power
absorption. Again larger amplitude waves require larger
device excursions for optimum absorption.
Here, simple expressions are presented for the
maximum efficiency of power absorption of a body when
its velocity and hence amplitude is constrained such that
its magnitude never exceeds a given multiple of the
incident wave amplitude and also for a system of
absorbing bodies under global velocity constraints.
Examples of a floating hemisphere in three dimensions
and a submerged circular cylinder in two dimensions are
presented.
damping matrix. Let X be the complex N-vector denoting
the amplitude and phase of the exciting force on each
body in its direction of subsequent motion, when all
bodies are held fixed.
Then it can be shown 4'5 that:
P{U)=¼(x*u + u*x)-½u*Bu
: 1X*B-
I X - ~ ( U - ½ n - 1X ) * B ( U
- ½B - 1X)
(1)
where * denotes conjugate transpose, and the second
expression follows from the first by elementary
manipulation provided B 1 exists. It is clear that the
maximum value of P when U is unconstrained is given by
the first term in equation (11 when U =½B-~X.
We shall assume that
IIuII ~- u * u ~</~2
(2)
and we shall optimize equation (1) subject to equation (2).
It is convenient to define Y=½B 1X so that equation (1)
becomes:
P(U) = ½ Y ' B Y - ~ U - Y)*B(U - V )
{3)
Clearly if Y*Y ~<f12 then Uop, = Y giving Pmax(U)=½Y'BY
(=~X*B 1X). IfY*Y>~fl z, Uop~is given from geometrical
considerations. Thus Pmax(U)is the largest value of C for
which the closed surfaces P(U) = C touch the surface U*U
=/32, in the complex N-dimensional U-space. This is most
easily determined by introducing a Lagrange multiplier #.
Consider the scalar:
Q(U) = P( U ) - ½#{U*U - f12)
THEORY
We consider N independently oscillating bodies making
simple harmonic motions of radian frequency o~ in one
degree of freedom in response to a long-crested incident
wave-train also of frequency ~. Each body can absorb
energy from the incident wave field. Let U be the complex
N-vector denoting the amplitude and phase of the velocity
of each body, and B the N x N real, symmetric radiation
0141 1187/81/040200-0452.00
©1981 C M L P u b l i c a t i o n s
200
Applied Ocean Research, 1981. Vol. 3, No. 4
A necessary condition for maximizing equation (3) subject
to U*U = fi2 is that V~o(U)= 0 where V denotes the 2Ndimensional vector gradient operator involving the
partial derivatives of both real and imaginary parts of Ui
for each i.
It follows that:
-B(U- Y)-pU =0
Maximum wave-power absorption under motion constraints: D. V. Evans
Thus
or
U = Uopt = (B + #I)- 1BY
Pmax
(4)
Substitution of this value back into U*U=/~ 2 gives a
single scalar equation to determine/a:
1
2[1-H(1-f)(1-6)2]/B
=~lX,I
where
=
Y*B(B +/al)- 2By ~- ¼X*(B + #I)- 2X =/32
----II¥*FR( B +LV
p l ) t- 2--/ B
~ 2- ] X s ~ , - s
for
/~ ~<½lIB- 1X, l[- IIYII
and
P ~ , ( U ) --~..~
1 Y ' B - 1X~ for /~>~{IIB-1X~lI
B/(B +
2~BIXI-
Ia) =
(10)
'
(5)
and H(x) is the Heaviside step function.
For a vertically oscillating buoy with a vertical axis of
symmetry it can be shown that:
It follows from equations (4) and (3) that:
Pmax(U) = ½U*(B + 2/~I)U
(9)
(6a)
2rrlX[ 2 = 8B2P w
(6b)
where Pw = ½P9A2.0)/29 is the mean power per unit crestlength of the incident wave, and 2 is the incident wave
length. We can define a capture width I= P/Pw so that:
(7)
2
ema,/P~, ----/max= 2~{ 1 -- H(1 - 6)(1 -- 6) z }
Clearly Pma,(U) is continuous in ft.
where
5 = 20)AotB/IX[
= o~.
EXAMPLES
In practice each of the N bodies would have constraints
on their amplitudes and velocities which may be of the
form:
lUll ~<0)A0¢i=fli
(i = 1, 2 ..... N)
(8)
where A is the incident wave amplitude and ~i are
constants governing the permissible body amplitude
compared to the incident wave amplitude. In normal
operating conditions ~ = 0 ( 1 ) but in larger waves ~ may
be small.
It is not obvious how the N constraints (8) can be
handled analytically and in particular cases a numerical
optimisation procedure would have to be used. The global
constraint (2) goes some way towards the solution, in the
sense that with fl = min{fl~, i = 1, 2 ..... N} then if equation
(2) is satisfied so is equation (8) since:
IUilZ<~u*u<~3z<~3[
Pmax klXI 2 E1 p2/(B +/~)2]/B
=
for
-
/~ ~<½IB 'XI
where IX[ = 2/~(B + #) determines/~.
(11)
where K = 2rt/2 = 0)2/0 is the wave number, a the radius of
the buoy, and 2aa=B/M0) the damping coefficient in
heave, non-dimensionalized with respect to a typical
mass, m.
Figure 1 shows the variation of lmax/2a with Ka for
different c¢ for a half-immersed sphere with M=2rrpa 3
where values of).33(Ka) have been taken from Havelock 8.
It can be seen that if the amplitude of motion of the sphere
is not allowed to exceed the wave amplitude the capture
width never exceeds 70% of the sphere diameter in
[ II~
1.4
1.2
a=2
(i=1, 2 . . . . . N)
The global constraint applied here may be more
appropriate to devices such as the French 7 flexible bag
where a fixed air volume is driven through a turbine and
re-circulated due to the compression of segmented flexible
bags.
To apply the above theory to more than one oscillating
body requires knowledge of B and X for such
configurations and accurate values are not available even
for the simplest cases. Instead we shall concentrate on a
single body in two or three dimensions.
For a single body oscillating in one degree of freedom
all vectors and matrices become scalars and, from
equation (6b):
(2M/pa3)U2(Ka)3/2).13/32
I
1-0
o,8
06
0'4
0-2
[
0
0"4
15 10
I I
I
0"8
6
[
4
1
1'2
3
I
Ka
I
I
I
1"6
2"0
2"4
2
I
1-5
I
2 8
X/2a
Figure 1. Variation of maximum capture width/diameter
ratio with wavelength for half immersed energy-absorbin 9
sphere for different motion constraints. , unconstrained
sphere
Applied Ocean Research, 1981, Vol. 3, No. 4
201
Maximum ware-power absorption under motion constraints: I). li Et'an.~
where 14"[(14',.) is a wave-making coefficient being the
ratio of the wave amplitude far upstream {downstream)
caused by unit amplitude oscillatory motion of the body.
For instance, for a symmetric body for which 14 +]-]A l-
1.o
t/ ..... = ½ [ 1
where
~- 0'6
- H(1 - 5 ) ( 1
- ,~)2]
(16)
(17)
,5 = 2c~W,.
and ~i- is the wave-making coefficient.
For a symmetric section absorbing energy through
oscillations in two uncoupled degrees of freedom
0,4
2
0.2
J
t/max=
]--21
E
i
0'2
0
0"4
i
1
0"6
0"8
1!0
1.i2
ga
3020
t
i
12
8
6
4
3
I
I
I
I
i
"X/2a
Figure 2. Variation in loss of maximum e/[ficiency of an
energy-absorbing submerged circular cylinder with
wavelength for different motion constraints. Submergence c
.I a
contrast to the result from unconstrained motions where
/max( --= '~/2Yt) increases indefinitely as Ka--,O. For sphere
amplitudes up to twice the wave amplitude (~=2) /max
peaks at a value 8% greater than the sphere diameter
showing that energy is being absorbed from outside the
region occupied by the sphere. In steeper waves where
may have to be as small as 0.5 the capture width never
exceeds 40% of the sphere diameter.
Note that we are not concerned here with how the
power is absorbed. In particular the difficulty in
implementing the required spring rate for optimal
absorption in this case, as discussed by Evans ~, is not
considered.
Many wave-energy devices begin life as cylindrical
sections spanning a narrow wave-tank, so it is
appropriate to consider the form of equation (9) in this
case.
Now for a tank of width Lwe have
IXl
and
B =½PcolA +127 tL
where 6~=2~iW~ in an obvious rotation. Now a
submerged cylindrical section making a combination of
vertical and horizontal oscillations has been proposed as
a potentially attractive wave-energy device 9. In this case
W~,I = W,,2 in deep water and
t/m~,= 1 - H ( 1 - c~)(1_,5)2
(18)
with ,5 as in equation (17). Here we have made the not
unreasonable assumption that the constraints on vertical
and horizontal motions of the cylinder are the same so
that % =~2 =~.
The result (18) can be derived more directly by noting
that a combination of horizontal and vertical motions n/2
out of phase can give rise to a wave-train travelling
downstream only, of maximum amplitude 2~AW,,. The
incident wave is not reflected by the cylinder and the
resulting transmitted wave due to the superposition of the
incident wave with the downstream radiated wave has
minimum amplitude A(I - 2~W~). Energy considerations
now give equation (18). Similarly equation (14) can be
derived directly starting with Evans ~ equation (3.7).
It is clearer in this case, where the maximum efficiency
in the unconstrained motion is 1, to graph the maximum
efficiency loss At/ where
At/= 1 - t/m~,-=H(I -6)(1 --0) 2
Figures 2 and
3 show the variation of At/ with
1.0
(13)
where A +(A-) is the complex amplitude of the potential
far upstream, in the opposite direction to the incident
wave (downstream) due to the forced oscillatory motion
of the cylindrical section with unit velocity. Also
7={l+IA-/A+]2} -1. In this case we can define an
efficiency t/so that:
where
_@2
(I 2)
= pgAIA + IL
t/m~,-- Pm~/P,,,-/[1 - H ( 1 --b)(1
H(I-0,)(I
t
--5) 2]
0.8
c~
<3 0'6
0"4
(14)
0.5
0.2
5 = 2~(M/pa2L)1:2Ka(27)- 1.,2)" 31:2
3
J
and equations (12) and (13) have been used in equations (9)
and (10).
A simpler version of equation (14) in terms of A + is:
0
0"2
0"4
0.'8
0'6
1!o
, !2
Ka
3020
L I
12
I
8
I
6
I
4
I
3
I
kl2a
5=~KIA+]/7
=o~w)
202
/, ,
Applied Ocean Research, 1981, Vol. 3, No. 4
(15)
Figure 3. Variation in loss ¢?f maximum eJficiency of'an
energy-absorbing circular cylinder with wavelength for
d!fferent motion constraints. Submergence c = a
M a x i m u m wave-power absorption under motion constraints: D. V. Evans
w a v e n u m b e r for different values of~ and for two depths of
submergence. In Fig. 2 the submergence of the top of the
cylinder is c = a / 4 . F o r small waves, with ~ = 2 , say
corresponding to a m a x i m u m cylinder motion of twice the
wave amplitude a loss in m a x i m u m efficiency only
becomes apparent for wavelengths in excess of about 18
times the cylinder diameter. In bigger waves, a smaller
value of ~ is appropriate and then the loss in efficiency
occurs at much shorter wavelengths. F o r instance with
=0.25 there is a loss in efficiency at all incident
wavelengths reflecting the fact that the cylinder is
incapable of radiating a wave large enough to cancel the
incident wave. One would anticipate a reduced efficiency
for another reason, namely the tendency for bigger waves
to steepen and break up into higher harmonics past the
cylinder 9. Figure 3 shows similar results for a more deeply
submerged cylinder with clearance c = a. As expected the
loss in efficiency occurs more readily as the wave-making
capability of the cylinder is further reduced.
CONCLUSIONS
The results confirm that three-dimensional small point
absorber devices can only achieve capture widths in
excess of unity if amplified b o d y motions are allowed. F o r
long terminator devices the importance of the wavemaking capability of the device is illustrated by
considering a long energy absorbing cylinder submerged
to different depths.
Linear theory has been used t h r o u g h o u t so that the
constraints on the b o d y motion characterized by the
n u m b e r :t must assume to be applied by an increased
linear d a m p i n g force. In practice, however, such
constraints would invariably be non-linear and the above
theory would need modification.
REFERENCES
1 Evans, D. V. A theory for wave-power absorption by oscillating
bodies, J. Fluid Mech., 1976, 77, 1
2 Mei,C. C. Power extraction from water waves, J. Ship. Res., 1976,20,
63
3 Newman, J. N. The interaction of stationary vessels with regular
waves, Proc. l lth Symp. Naval Hydrodynamics, London, 1976, p. 491
4 Evans, D. V, Some theoretical aspects of three-dimensional waveenergy absorbers, Proc. 1st Symp. Ocean Wave Eneroy Utilization,
Gothenberg, 1979
5 Falnes, J. Radiation impedance matrix and optimum power
absorption for interacting oscillators in surface waves, Appl. Ocean
Res., 1980, 2, 75
6 Newman, J. N. Absorption of wave energy by elongated bodies, Appl.
Ocean Res., 1979, 1, 189
7 French, M. J. Hydrodynamic basis of wave energy converters of
channel form, J. Mech. Eng. Sci. (C) I. Mech. E., 1977. 19
8 Havelock, T. H. Waves due to a floating sphere making periodic
heaving oscillations, Proc. R. Soc., 1955, (A), 231, 1
9 Evans, D. V., Jeffrey, D. C., Salter, S. H. and Taylor, J. R. M.,
Submerged cylinder wave-energy device: theory and experiment,
Appl. Ocean Res., 1979, l, 3
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