Renewable Energy 41 (2012) 105e114
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Renewable Energy
journal homepage: www.elsevier.com/locate/renene
Hydrodynamics of the IPS buoy wave energy converter including the effect
of non-uniform acceleration tube cross section
António F.O. Falcão
a,
*
, José J. Cândido
a
, Paulo A.P. Justino
b
, João C.C. Henriques
a
a
b
IDMEC, Instituto Superior Técnico, Technical University of Lisbon, 1049-001 Lisbon, Portugal
Laboratório Nacional de Energia e Geologia, Estrada Paço do Lumiar, 1649-038 Lisbon, Portugal
a r t i c l e i n f o
Article history:
Received 15 June 2011
Accepted 6 October 2011
Available online 6 November 2011
Keywords:
Wave energy
Wave power
IPS buoy
Oscillating body
Hydrodynamics
a b s t r a c t
An important class of
floating
wave energy converters (that includes the IPS buoy, the Wavebob and the
PowerBuoy) comprehends devices in which the energy is converted from the relative (essentially
heaving) motion between two bodies oscillating differently. The paper considers the case of the IPS buoy,
consisting of a
floater
rigidly connected to a fully submerged vertical (acceleration) tube open at both
ends. The tube contains a piston whose motion relative to the
floater-tube
system (motion originated by
wave action on the
floater
and by the inertia of the water enclosed in the tube) drives a power take-off
mechanism (PTO) (assumed to be a linear damper). To solve the problem of the end-stops, the central
part of the tube, along which the piston slides, bells out at both ends to limit the stroke of the piston. The
use of a hydraulic turbine inside the tube is examined as an alternative to the piston. A frequency domain
analysis of the device in regular waves is developed, combined with a one-dimensional unsteady
flow
model inside the tube (whose cross section is in general non-uniform). Numerical results in regular and
irregular waves are presented for a cylindrical buoy with a conical bottom, including the optimization of
the acceleration tube geometry and PTO damping coefficient for several wave periods.
Ó
2011 Elsevier Ltd. All rights reserved.
1. Introduction
The concept of the point absorber for wave energy utilization
was developed in the late 1970s and early 1980s, mostly in Scan-
dinavia
[1].
This is in general a wave energy converter of oscillating
body type whose horizontal dimensions are small compared to the
representative wavelength. In its simplest version, the body reacts
against the bottom. In deep water (say 50 m or more), this may
raise difficulties due to the distance between the
floating
body and
the sea bottom. Multi-body systems may then be used instead, in
which the energy is converted from the relative motion between
two bodies oscillating differently. This is the case of several devices
presently under development, like the Pelamis, the Wavebob and
the PowerBuoy.
Sometimes the relevant relative motion results from heaving
oscillations. This paper considers the special situation when
a
floater
reacts against the inertia of the water contained in a long
vertical tube open at both ends, located underneath. This is the case
of the spar-buoy OWC, possibly the simplest concept for a
floating
oscillating water column (OWC) device equipped with an air
turbine, in which the upper end of the tube extends through the
*
Corresponding author.
E-mail addresses:
falcao@hidrol.ist.utl.pt, antonio.falcao@ist.utl.pt
(A.F.O. Falcão).
0960-1481/$
e
see front matter
Ó
2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.renene.2011.10.005
buoy above the sea water level. Masuda, in Japan, developed
a navigation buoy based on the OWC spar-buoy concept
[2,3].
The
spar-buoy OWC will not be analysed in this paper.
The IPS buoy is another type of spar-buoy and will be analysed
here in detail. It was invented by Noren
[4]
and initially developed
in Sweden by the company Interproject Service (IPS). The device
consists of a buoy rigidly connected to a fully submerged vertical
tube (the so-called acceleration tube) open at both ends (Fig.
1).
The
tube contains a piston whose motion relative to the
floater-tube
system (motion originated by wave action on the
floater
and by
the inertia of the water enclosed in the tube) drives a power take-
off (PTO) mechanism. The same inventor later introduced an
improvement that significantly contributes to solve the problem of
the end-stops: the central part of the tube, along which the piston
slides, bells out at both ends to limit the stroke of the piston
[5].
A half-scale prototype of the IPS buoy was tested in sea trials in
Sweden, in the early 1980s
[6].
The AquaBuOY is a wave energy
converter, developed in the 2000s, that combines the IPS buoy
concept with a pair of hose pumps to produce a
flow
of water at
high pressure that drives a Pelton turbine
[7,8].
A prototype of the
AquaBuOY was deployed and tested in 2007 in the Pacific Ocean off
the coast of Oregon.
A variant of the initial IPS buoy concept, due to Stephen Salter, is
the sloped IPS buoy: the natural frequency of the converter may be
106
A.F.O. Falcão et al. / Renewable Energy 41 (2012) 105e114
analysis using tools like the commercially available codes (e.g.
WAMIT, AQUADYN) based on the boundary element method for the
computation of the hydrodynamic coefficients, including the inter-
ference between the buoy and the acceleration tube, as done in
[12].
2.1. One-dimensional
flow
inside the tube
We consider now the
flow
inside the acceleration tube, whose
total length is
L
(Fig.
2).
The position of the tube sections are defined
by a longitudinal coordinate
x
(with
x
¼
0 at the lower end of the
tube). The piston is allowed to move, relative to the tube, inside
a central part 3e4 (working part),
b
2
þ
b
3
x
b
1
þ
b
2
þ
b
3
, of
length
b
1
and cross sectional area
A
1
, as shown in
Fig. 2.
The
working part is continued downwards and upwards by tube parts
1e2 and 5e6, of lengths
b
3
and
b
4
, respectively, both of cross
sectional area
A
2
¼
a
2
A
1
(
a
!
1). The transitions are provided by
conical connections 2e3 and 4e5, of cross sectional areas
A(
x
). If
a
>
1, there may be a significant axial force on the tube resulting
from the pressure distribution on the inner conical walls.
The added mass of the oscillating water column contained in
a semi-infinite open tube of radius
r
and of negligible wall thickness
in an unbounded perfect
fluid
is
r p
l r
2
, where
l
¼
0.6133
r
is an
added length (see
[13,14]).
We assume this result to apply to our
case (small tube thickness close to the tube ends), with
p
r
2
¼
A
2
.
The introduction of the added mass (or the added length
l)
in
a theoretical model that assumes the
flow
inside the tube to be one-
dimensional allows the following two effects to be accounted for:
(i) the inner
flow
close to the tube end is non-uniform; (ii) the outer
flow field
in the vicinity of the tube end is affected by the motion of
the water column inside the tube. These effects were analysed in
two- or three-dimensional potential
flow
in
[13,14].
In a
fixed
frame of reference, the buoy-tube pair moves along its
own vertical axis with velocity
W(t)
(positive for upward motion),
where
t
is time. We note that the water
flow
inside the tube is
unsteady in any referential. In our analysis, we adopt a non-inertial
piston
Fig. 1.
Schematic representation of the IPS buoy.
reduced, and in this way the capture width enlarged, if the buoy-
tube set is made to oscillate at an angle intermediate between
the heave and the surge directions. The sloped IPS buoy has been
studied since the mid-1990s at the University of Edinburgh, by
model testing and numerical modelling
[9e11].
2. Theoretical modelling
The IPS buoy consists basically of a buoy rigidly connected to
a fully submerged tube (the acceleration tube), oscillating in heave,
by the action of the waves, with respect to a piston that can slide
along the tube. The wave energy is absorbed by means of the
relative motion between the piston and the buoy-tube set. The
concept is represented in
Fig. 1.
We note that most of the inertia against which the buoy-tube set
moves is that of the water contained inside the acceleration tube
(obviously in addition to the mass of the piston itself). In the
simplified mathematical modelling adopted in this paper, we
assume that the buoy-tube set is constrained to oscillate in heave,
an assumption that seems reasonable taking into account the axial
extent of the device.
We introduce the following assumptions. (i) The tube is suffi-
ciently far away underneath the buoy for the hydrodynamic inter-
action between both to be negligible. (ii) The interaction between
the wave
fields
induced by the two ends of the tube may be
neglected. (iii) The distance from the free surface to the tube upper
end is large enough for the excitation and radiation forces on the
flow
about the two tube ends to be neglected. (We note however
that the added mass at the two ends of the tube will be accounted
for.) (iv) Finally, the
flow
inside the tube is modelled as one-
dimensional. Admittedly, some of these simplifications may be
rather drastic. This is specially the case of assumptions (i) and (iii) if
the distances from the acceleration tube to the buoy and to the free
surface are not large enough. In spite of this, the present paper is
expected to provide useful insights into the relationships between
device geometry, PTO parameters and wave energy converter
performance. Naturally, in cases of special practical interest, this
simplified approach should be complemented by a more rigorous
6
b
4
5
4
L
A
2
A
(
ξ
)
b
2
b
1
A
1
A
(
ξ
)
3
2
b
2
b
3
A
2
ξ
1
Fig. 2.
IPS buoy with acceleration tube.
A.F.O. Falcão et al. / Renewable Energy 41 (2012) 105e114
107
frame of reference
fixed
to the buoy-tube pair. In this referential,
the piston velocity is
V(t)
and the one-dimensional
flow
velocity at
a section
x
of area
A(
x
) is
The force on the piston is
f
p
(t)
¼
A
1
(p(z
À
,t)
À
p(z
þ
,t)), where
z
À
and
z
þ
are the coordinates of the lower and upper surfaces of the
piston. We
find
vð
x
;
tÞ
¼
A
1
VðtÞ:
Að
x
Þ
2
b
3
À
x
ð
a
À
1Þ
þ
a
b
2
(1)
f
p
ðtÞ ¼ ÀM
W
where
dW
dV
À
M
V
;
dt
dt
(8)
For a conical transition, it is
Að
x
Þ
¼
A
1
ðb
3
x
b
2
þ
b
3
Þ;
M
W
¼
r
A
1
ðL þ
2lÞ;
(2)
M
V
¼
r
A
1
b
1
þ
a
À2
ðb
3
þ
b
4
þ
2lÞ
þ
2b
2
a
À1
:
(9)
(10)
2
x
À
b
1
À
b
2
À
b
3
Að
x
Þ
¼ ð
a
À
1Þ
þ
1
b
2
A
1
ðb
1
þ
b
2
þ
b
3
(3)
x
b
1
þ
2b
2
þ
b
3
Þ:
The total axial force
f
t
(t) on the internal surface of the two
conical parts of the acceleration tube is
b
Z
þb
3
2
In what follows, we assume that the piston is of negligible
length and mass (this is equivalent to assuming that its length is
non-zero and its mean density is equal to water density). Since the
tube is totally submerged, the net force on the piston is not affected
by gravity and so we simply ignore the acceleration of gravity for its
calculation and denote by
p
out
the uniform pressure of the
supposedly unbounded water far away from the tube ends.
Let
x
¼
z
be the instantaneous position of the piston (assumed of
negligible length). Applying Bernoulli’s equation for unsteady one-
dimensional
flow
(see e.g.
[15]),
we
find,
for the pressure at
a section
x
<
z
below the piston,
f
t
ðtÞ ¼
b
3
dAð
x
Þ
pð
x
;
tÞ
d
x
À
d
x
LÀb
4
Z
pð
x
;
tÞ
LÀb
4
Àb
2
dAð
x
Þ
d
x
:
d
x
(11)
It may be written as
f
t
ðtÞ ¼ Àm
W
where
dW
dV
À
m
V
;
dt
dt
2b
3
þ
a
À
1
ðb
3
þ
b
4
þ
2lÞ
;
2
(12)
m
W
¼
r
A
1
a
þ
a
À
2
2
2
(13)
"
pð
x
;
tÞ
¼
p
out
þ
r a
À4
À
Z
x
Àl
A
2
1
Að
x
Þ
2
#
V
2
dW
À
r
ð
x
þ
lÞ
dt
2
(4)
m
V
h
i
À1
À2
b
2
þ
1
À
a
ðb
3
þ
b
4
þ
2lÞ
:
¼
r
A
1
2 1
À
a
(14)
À
r
vvð
x
;
tÞ
d
x
:
vt
The second term on the right-hand-side of Eq.
(4)
accounts for the
difference in kinetic energy at cross sections with different areas,
and is zero where the cross sectional area is
A
2
. The third term on
the right-hand-side of Eq.
(4),
proportional to dW/dt, is due to the
fictitious
body force per unit mass
ÀdW/dt
associated with the non-
inertial frame of reference. The last term results from the
unsteadiness of the velocity
v(
x
,t) and may be written as
À
r
Z
x
Àl
vvð
x
;
tÞ
dV
l
þ
b
3
b
2
ð
x
À
b
3
Þ
þ
d
x
¼
r
a
ð
a
b
2
À
b
3
þ
a
b
3
þ
x
À
ax
Þ
vt
dt
a
2
ðb
3
The preceding equations show that the forces on the piston,
f
p
,
and on the tube,
f
t
, depend on the sum
b
3
þ
b
4
, not on the lengths
b
3
or
b
4
separately, a result that is not unexpected. They also show
that the expressions of those forces are linear in the accelerations
dW/dt and dV/dt, with no dependence on velocities. From the
viewpoint of the axial force
f
t
on the tube,
m
W
may be regarded as
an inertial mass associated with the tube (and
floater)
acceleration
dW/dt; the same applies to
m
V
in connection with the acceleration
dV/dt of the piston in the frame of reference
fixed
to the buoy-tube
pair. If the whole tube is of uniform inner cross section, i.e. if
a
¼
1,
it is simply
M
V
¼
M
W
and
f
t
¼
m
V
¼
m
W
¼
0, a situation that was
studied in
[16].
2.2. Piston versus hydraulic turbine
The original IPS buoy was conceived with a piston, sliding along
the acceleration tube, whose relative motion activates a secondary
hydraulic ram (or linear pump) that supplies high pressure liquid
(water or oil) to a hydraulic circuit
[4].
If energy is to be absorbed
from large amplitude waves, the excursion of the primary piston is
also relatively large which requires a long rod (possibly longer than
20 m), which, when subjected to compression forces, can cause
serious buckling problems. An alternative to the piston pump is
a pair of hose pumps as in the AquaBuOY
[7,8],
which avoids
compression loads but whose hydraulic circuit working pressure is
much lower than what is attainable by piston pumps.
A self-rectifying hydraulic turbine located in the narrower part
of the tube may be used instead of a piston, although this seems not
to have been proposed before. This avoids the problem of limiting
the piston excursion. In order to avoid cavitation, the turbine
should be deeply submerged. Naturally, the
flow
through the
turbine is far more complex than the (assumed one-dimensional)
flow
in the tube. However, for the purpose of accounting for the
inertia of the
flow
through the turbine, we may define an
x
b
2
þ
b
3
Þ;
(5)
À
r
Z
x
Àl
vvð
x
;tÞ
dV
lþb
3
b
2
þ þ
x
Àb
2
Àb
3
ðb
2
þb
3
x
zÞ:
d
x
¼
r
a
vt
dt
a
2
(6)
We note that Eq.
(4)
gives
p(Àl,t)
¼
p
out
, as if the tube were
extended downwards by a length equal to the added length
l.
Above the piston,
x
>
z,
we have
pð
x
;
tÞ
¼
p
out
þ
r a
À4
À
þ
r
Lþl
Z
A
2
1
Að
x
Þ
2
!
V
2
dW
þ
r
ðL À
x
þ
lÞ
dt
2
(7)
x
vvð
x
;
tÞ
d
x
:
vt
Expressions similar to
(5) and (6)
can be derived for the last term of
Eq.
(7).
108
A.F.O. Falcão et al. / Renewable Energy 41 (2012) 105e114
equivalent tube element of length
b
1
and an equivalent cross
sectional area
A
1
, as shown by the shaded area in
Fig. 3.
Since the
diameter of the turbine is expected to be much smaller than the
diameter of the main part of the acceleration tube, the area ratio
a
2
¼
A
2
/A
1
should be much larger than unity. Obviously, the
expressions derived above for the piston in tube may be applied to
this case, the volume
flow
rate through the turbine being
Q(t)
¼
A
1
V(t);
the turbine pressure head is
D
p
¼
f
p
(t)/A
1
, where
f
p
is given
by Eq.
(8).
2.3. Hydrodynamic analysis in regular waves
We consider now the IPS buoy, and denote by
x(t)
the coordinate
for the heaving motions of the
floater-tube
set, with
x
¼
0 in the
absence of waves and
x
increasing upwards. Let
y(t)
be the oscil-
lations in piston position relative to the buoy-tube set. We note that
_
_
it is
x
¼
W
and
y
¼
V,
where
W
is the tube velocity and
V
is the
relative velocity of the piston as defined in Section
2.1.
We consider
a linear PTO such that a relationship
_
ðM
1a
þ
M
1b
Þ
€
þ
Bx
þ
r
gSx
¼
f
e
ðtÞ þ
f
t
ðtÞ þ
f
p
ðtÞ:
x
(16)
_
f
p
¼
Ky
þ
C y
Here
M
1a
¼
m
1a
þ
m
1a
and
M
1b
¼
m
1b
þ
m
1b
are the mass plus added
mass of bodies 1a and 1b, respectively,
r
is water density,
g
is
acceleration of gravity,
B(
u
) is radiation damping coefficient of the
buoy (body 1a), and
S
is the cross sectional area of the
floater
defined by the undisturbed free surface. We note that
m
1a
is
_
a function of
u
and recall that it is
x
¼
W
where
W
is the tube
velocity defined in Section
2.1.
On the right-hand-side of Eq.
(16),
f
e
is the hydrodynamic excitation force on the
floater
due to the
incoming waves,
f
t
is the force on the inner surface of the tube given
by Eq.
(12)
and
f
p
is the vertical force on the piston (given by Eq.
(8))
that is transmitted to the buoy by the piston rod or by the pair of
hose pumps. We recall that negligible hydrodynamic interference is
assumed. We note also that
m
1b
is supposed not to be a function of
frequency
u
(as a consequence from the assumption of deep
submergence of body 1b).
Since we have a linear system acted upon by a simple-time-
harmonic excitation force of frequency
u
, we may write, after the
transients related to the initial conditions have died out,
(15)
fx;
y; V; f
e
g ¼ fX;
Y; V
0
;
F
e
ge
i
u
t
:
(17)
holds between the force on the piston
f
p
and the relative
_
displacement
y
and velocity
y
of the piston. The constant
C
is the
PTO damping coefficient and the constant
K
may be regarded as
a spring stiffness. The instantaneous power absorbed by the PTO is
_
P
¼
f
p
y.
The following hydrodynamic analysis is based on linear water
wave theory, which, as is well known, requires the wave amplitude
and the amplitude of body oscillations to be small (compared with
wavelength). The equation of motion can be found in
[17].
We
consider the body to consist of a
floater
(subscript 1a) and a tube
(subscript 1b), and denote by
m
1a
,
m
1b
and
m
1a
,
m
1b
the corre-
sponding masses and added masses. The added mass
m
1b
of the
tube is supposed to be independent from the wave frequency (as in
an unbounded medium).
Provided that the PTO is linear (as assumed above) and after the
transients related to the initial conditions have died out, we may
write, for the motion of the body in the presence of incoming
sinusoidal waves of frequency
u
,
Here
X, Y, V
0
¼
i
u
Y
and
F
e
are complex amplitudes. We may write
F
e
¼
A
w
G
(
u
), where
A
w
is the incident wave (real) amplitude, and
G
is the (in general complex) excitation force coefficient. The absolute
value of
G
(
u
) may be related to
B(
u
) by the Haskind relation (valid
for an axisymmetric body oscillating in heave in deep water, see
[17])
j
G
ð
u
Þj ¼
1=2
3
2g
r
Bð
u
Þ
u
3
:
(18)
By using the complex amplitude representation, we easily
obtain, from Eqs
(8), (12), (16) and (17),
À
u
2
ðM
1a
þ
M
1b
þ
m
W
þ
M
W
ÞX þ
i
u
BX
þ
r
gSX
À
u
2
ðm
V
þ
M
V
ÞY ¼
F
e
;
(19)
u
2
M
W
X
þ
u
2
M
V
Y
¼ ðK þ
i
u
CÞY:
buoy
(20)
For given wave frequency
u
and excitation force amplitude
F
e
, the
pair of linear algebraic equations
(19) and (20)
yield the complex
amplitudes
X
and
Y.
The time-averaged value of the power absorbed
by the PTO (piston or turbine) is
P
¼
u
2
CjYj
2
=2.
If the whole tube is of uniform inner cross section, i.e. if
a
¼
1, it
is simply
A
1
¼
A
2
,
f
t
¼
m
V
¼
m
W
¼
0 and
M
V
¼
M
W
¼
M
2
(say), where
M
2
is the mass plus added mass of the water contained in the tube.
In this case, Eqs
(19) and (20)
reduce to
À
u
2
ðM
1a
þ
M
1b
þ
M
2
ÞX þ
i
u
BX
þ
r
gSX
À
u
2
M
2
Y
¼
F
e
;
turbine
(21)
(22)
u
2
M
2
X
þ
u
2
M
2
Y
¼ ðK þ
i
u
CÞY:
Fig. 3.
IPS buoy with the piston replaced by a hydraulic turbine (in the shaded space).
Since we only consider heave oscillations, the equations of
motion are not affected by how the mass
m
1
¼
m
1a
þ
m
1b
is
distributed between bodies 1a and 1b. For convenience of presen-
tation of numerical results, we assume that
m
1a
is the mass of water
of volume equal to the submerged part of the buoy in the absence of
waves.
The mass
m
1b
and the added mass
m
1b
of body 1b appear
together in the equations as
M
1b
¼
m
1b
þ
m
1b
. For this reason, the
numerical results in Sections
3 and 4
are given for
M
1b
and not for
m
1b
and
m
1b
separately.
A.F.O. Falcão et al. / Renewable Energy 41 (2012) 105e114
109
3. Numerical results in regular waves
Numerical results were obtained for a cylindrical buoy of radius
a,
with a conical bottom (semi-angle of the cone equal to
p
/3). In
calm water, the cylindrical part of the buoy is submerged to a depth
equal to the radius
a.
The added mass
m
1a
and the radiation
damping coefficient
B
were computed with the software WAMIT for
a set of values of the frequency
u
and deep water. A dimensionless
plot of
m
*
¼
m
1a
/(
rp
a
3
) and
B
*
¼
B/(
rp
a
3
u
) versus
T
*
¼
T(g/a)
1/2
1a
(T
¼
wave period) is shown in
Fig. 4.
We define the dimensionless
value of the mass of body 1 (including the added mass of body 1b)
as
M
*
¼
(m
1a
þ
M
1b
)/m
1a
, where, for the geometry considered here,
1
it is
m
1a
¼
rp
a
3
(1
þ
3
À1
tan
p
/6)
¼
3.7462
r
a
3
. We note that the
minimum value of
M
*
is equal to unity, since
M
1b
¼
m
1b
þ
m
1b
1
cannot be negative. In the special case when
a
¼
1, i.e. an acceler-
ation tube of uniform inner cross section (A
1
¼
A
2
), we also define
the dimensionless mass plus added mass of body 2 (water con-
tained in the tube) as
M
*
¼
M
2
/m
1a
, where
M
2
¼
r
A
1
(L
þ
2l).
2
Here, we consider regular waves of frequency
u
and assume that
the PTO consists solely of a linear damper (no spring, i.e.
K
¼
0). We
define a dimensionless damping coefficient
C
*
(
u
)
¼
C/B(
u
), where
B(
u
) is the radiation damping coefficient of body 1a. We also define
dimensionless values
X
*
¼ jXj/A
w
(A
w
¼
incident wave amplitude)
for the motion amplitude of body 1 (buoy-tube set) and
Y
*
¼ jY/Xj
for the amplitude of the piston motion relative to the buoy-tube
pair. Note that
Y
*
¼
0 means that the piston is rigidly connected
to the buoy. If the piston does not move (possibly because the
_
_
inertia of the water inside the tube is infinite) it is
x
¼ À
y
and
Y
*
¼
1.
The theoretical maximum limit for the time-averaged wave
power that can be absorbed from regular waves in deep water by
a heaving wave energy converter with a vertical axis of symmetry is
well known to be (see
[17])
equivalent to a two-body heaving wave energy converter in which
the mass plus added mass of bodies 1 and 2 are respectively
M
1a
þ
M
1b
and
M
2
, and the PTO is activated by the relative motion
between the bodies. This two-body case was theoretically analysed
in detail by Falnes
[18].
An optimization was performed that consisted in
finding
the
*
pair of dimensionless values
C
*
and
M
*
that maximizes
P
, for given
2
values of the dimensionless wave period
T
*
and of
M
*
. This two-
1
dimensional optimization was performed with the aid of the
FindMaximum subroutine of
Mathematica.
Results are shown in
Figs. 5e7
for
T
*
¼
10, 12 and 14. The following curves (dimension-
less values) are plotted (versus
M
*
): (i) amplitude
Y
*
¼ jY/Xj
of the
1
relative motion between bodies 1 and 2; (ii)
M
*
(mass plus added
2
mass of body 2); (iii) PTO damping coefficient
C
*
. For all plotted
*
points, it is
P
¼
P=P
max
¼
1, since maximum capture width
l
/2
p
(
l
¼
wavelength) is attained by the maximization process. We note
that the wave energy is absorbed solely from the motion of body 1a
(bodies 1b and 2 are assumed far away from the free surface). So
X
*
is the same as for a single body 1a optimally reacting against the
bottom; it depends only on
T
*
and is independent of the optimal
pair (M
*
,
M
*
). It is well known (see e.g.
[17])
that, for a single
1
2
heaving body, maximum absorbed power is attained for oscillation
amplitude
X
opt
¼ jF
e
j(2
u
B)
À1
, which, for axisymmetric
floating
body
1a, can be written in dimensionless form as
*
X
opt
¼ ð2
p
Þ
À7=2
B
*À1=2
T
*3
;
(24)
1.2
1
0.8
0.6
0.4
0.2
0
1
40
30
20
10
0
1
1.2
*
1.4
M
1
*
M
2
P
max
¼
g
3
r
A
2
w
:
4
u
3
(23)
Accordingly we define the dimensionless power
P
*
¼
P=P
max
1,
where
P
¼
u
2
CjYj
2
=2
is the time-averaged power absorbed from
the waves.
3.1. Tube of uniform cross section
a
¼
1
We consider
first
the case when
a
¼
1 (and
f
t
¼
0). Since the
inner cross section of the tube is uniform, the
flow
of water
(assumed one-dimensional) inside the acceleration tube is also
uniform (the water moves as a solid body) and the system is
Y
*
1.2
1.4
1.6
1.8
C
*
1.6
1.8
Fig. 4.
Dimensionless plot of the added mass
m
(solid line) and radiation damping
coefficient
B
*
(dotted line) versus wave period
T
*
, for the buoy (body 1a) in deep water.
*
1a
Fig. 5.
Dimensionless plots of
M
*
,
Y
*
and
C
*
versus
M
*
, for
a
¼
1 (tube of uniform inside
2
1
*
cross section) and wave period
T
*
¼
10. Maximum absorbed power
P
¼
1 is attained
for all plotted points.
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