Renewable Energy 194 (2022) 875e884
Contents lists available at ScienceDirect
Renewable Energy
journal homepage: www.elsevier.com/locate/renene
Hydrodynamics and load shedding behavior of a variable-geometry
oscillating surge wave energy converter (OSWEC)
Michael Choiniere a, Jacob Davis b, Nhu Nguyen b, Nathan Tom c, Matthew Fowler d,
Krish Thiagarajan b, *
a
Department of Mechanical Engineering, University of Maine, Orono, ME, USA
Department of Mechanical and Industrial Engineering, University of Massachusetts Amherst, Amherst, MA, USA
National Renewable Energy Laboratory, Golden, CO, USA
d
Advanced Structures and Composites Center, University of Maine, Orono, ME, USA
b
c
a r t i c l e i n f o
a b s t r a c t
Article history:
Received 21 July 2021
Received in revised form
16 May 2022
Accepted 31 May 2022
Available online 3 June 2022
In order to improve their long-term viability, wave energy converters (WECs) need to be able to shed
loads when a threshold wave condition is exceeded. As shown by Tom et al. (2016) [1], provision of
adjustable flaps within the body of an oscillating surge wave energy converter (OSWEC) allows wave
energy to pass through the device. A control system may then be able to open and close the flaps when
waves approaching the device exceed preset thresholds. The variable-geometry OSWEC (VG-OSWEC)
concept studied in this paper is a bottom-hinged, rectangular wave paddle with five flaps of elliptical
cross-section embedded into the face of the paddle. System ID tests were conducted on this VG-OSWEC
device at a 1:14 scale in a wave basin. Free decay tests showed that the damping was distinctly nonlinear
when the flaps were fully open, and the natural frequency increased by 40% when compared with the
flaps in a fully closed configuration.
Tests with regular wave conditions were used to develop the response amplitude operator for the
rotational motion about the hinge. These response amplitude operator results when compared with
numerical simulations run using WEC-Sim/WAMIT and ANSYS AQWA, show strong agreement with the
flap open and closed conditions. The regular-wave condition measurements also show that the wave
excitation moment about the hinge was reduced by up to 60% when the flaps were fully open. The
experiments serve to demonstrate the potential of the variable geometry design to shed loads and
survive harsh ocean environments.
© 2022 Elsevier Ltd. All rights reserved.
Keywords:
Wave energy conversion
Oscillating surge wave energy converter
Wave hydrodynamics
Model tests
Response amplitude operator
Load shedding
1. Introduction
A hinged paddle is a terminator wave energy converter (WEC)
concept that is the subject of this paper. The intriguing hydrodynamics of a hinged paddle placed normal to oncoming ocean waves
has been studied from the mid-20th century. More recently the
hinged paddle has been incorporated into WEC concepts like the
Oyster [2,3], Waveroller [4], and Resolute Marine Surge WEC [5].
Long-term survival of WECs in the ocean can be challenged by two
common issues: fatigue damage resulting from the cyclic nature of
waves [6], and extreme loading from storms [7e9]. Because a
* Corresponding author. Department of Mechanical and Industrial Engineering,
University of Massachusetts Amherst, Amherst, MA, USA.
E-mail address: kthiagarajan@umass.edu (K. Thiagarajan).
https://doi.org/10.1016/j.renene.2022.05.169
0960-1481/© 2022 Elsevier Ltd. All rights reserved.
typical WEC works off a relative displacement of one structural
component or the other, there is an inevitable penalty on the fatigue life of components.
With a view toward long-term survivability, the National
Renewable Energy Laboratory (NREL) has been developing a novel
WEC concept that combines an oscillating surge wave energy
converter (OSWEC) with controllable surfaces [1]. The body of the
OSWEC consists of five horizontal control surfaces (elliptical flaps)
embedded longitudinally within the bottom-hinged solid rectangular paddle structure. The variable-geometry OSWEC (VGOSWEC) from NREL is more similar to a pitching device with a
single large flap [10]; however, increasing the number of variable
surfaces allows for greater refinement of the hydrodynamic properties [11]. A variation on the original OSWEC design incorporating
different positioning of the flaps was studied experimentally by
Choiniere et al. [12] and is shown in Fig. 1. The main body is a paddle
M. Choiniere, J. Davis, N. Nguyen et al.
Renewable Energy 194 (2022) 875e884
addressed in Ref. [13]. Models sized to occupy a portion of the tank
width remain susceptible to the interference of reflected waves due
to relatively limited tank length, [19,20]. The analysis by Choiniere
et al. [12] reported the first harmonic from a Fourier analysis for
both the wave elevation input and OSWEC rotational response.
Although these harmonic results were sufficient to prove the VGOSWEC in concept, additional validation is needed to build confidence in the numerical models. This spurred a new experimental
testing campaign at the University of Maine's Harold Alfond W2
Ocean Engineering Laboratory.
In this work, we present results from this new set of experiments conducted at the University of Maine, and their agreement
with those obtained from numerical simulations. An important
difference in geometrical variation from previous work is that the
hinge is sufficiently separated from the tank bottom. Furthermore,
the width of the basin allows the flow around the paddle to be
three-dimensional. The variable geometry modules, which
comprise of the five elliptical flaps that rotate in unison (Fig. 1), are
considered in the following two configurations:
1. “0-degree” with flaps fully closed (aligned vertically with
respect to the seafloor, allowing no flow passage)
2. “90-degrees” e with flaps fully open (horizontally, allowing the
flow to pass through).
For these two configurations, we present results from system ID
tests (free decay), and then examine wave induced moments about
the hinge, as well as the response to the waves. These results are
compared with those obtained from simulations conducted with
the open-source software WEC-Sim [21]. The VG-OSWEC wave
tank model in practice could have set the elliptical flaps at intermediate orientations between 0- and 90-degrees to generate
experimental data that could be used to compare against numerical
simulations; however, the emphasis of this test, given the limited
access to wave tank time, focused on generating a data set to understand the two possible extremes of the VGOSWEC in terms of
load shedding and device oscillation. One can imagine that any
intermediate elliptical flap angles will produce wave-excitation
loads and pitch oscillation amplitudes that fall within the two
bounding cases. The reader is directed to Husain et al. [22] to view
WEC-Sim results of a modified VG-OSWEC, relative to this work,
that considered intermediate elliptical flap orientations.
Fig. 1. Definition of the variable geometry OSWEC used in this paper. For c ¼ 0; the
geometry is identical to that described by Choiniere et al. (2017).
with some thickness, hinged at the base and extending up to the
mean water level. When subject to the incident wave field normal
to its surface, the paddle response is an oscillatory rotation (4)
about its hinge. In still-water conditions the paddle surface does
not pierce the free surface, which contrasts with other similar
studies e.g., Refs. [13e15]. Other studies, e.g. Ref. [16] examined the
effect of paddle height from the water surface on the wave energy
conversion performance. The control surfaces were shown to be
effective in tuning the natural frequency of the device to the
dominant wave frequency, thus allowing for optimal power
extraction in operating sea states. At higher sea states, the design
incorporates sufficient flexibility to reduce loads on the foundation.
The flaps, located in the lower part of the plate in principle, can be
adjusted to any angle about their axes of rotation, and together,
serve as a load-shedding mechanism. Limited two-dimensional
experiments by Ref. [12] on the OSWEC with the hinge located at
the tank bottom (c ¼ 0 in Fig. 1) showed that the pitch motion with
flaps fully open and closed agreed with numerical simulations. The
model OSWEC spanned the width of the wave tank in an attempt to
emulate the two-dimensional flow problem. However, this
configuration led to a non-sinusoidal, fluctuating wave profile
resulting from the superposition of the incident and reflected
waves from the OSWEC and the wave maker, which prevented
meaningful time-domain comparisons between numerical simulations and experimental data. Similar observations were made by
Brito et al. [17] and explored further by the same authors in
Ref. [18], including the effects of overtopping and flow rotation.
Nonlinear aspects of the flap motion are more comprehensively
2. Theoretical and numerical simulation details
The motion of the OSWEC device under consideration is limited
to 1 -of- freedom (DOF) in pitch ð4Þ due to the fixed shaft, (Fig. 1).
The general 1-DOF equation of motion of the paddle can be written
as:
_ þ K4 ¼ M *
ðJ þ A55 Þ€
4 þ B4_ þ CD 4_ j4j
(1)
Here, J and A55 represent the paddle's mass moment of inertia
and frequency-dependent added moment of inertia about an axis
passing through the hinge, respectively. The damping force is
composed of linear and quadratic components that are formulated
to ensure the force direction is maintained with respect to the
_ The damping coefficients, B and CD , are also
paddle velocity ð4Þ.
frequency-dependent and can have contributions from various
sources, including wave radiation and mechanical and viscous
sources. This could be further augmented by power-take-off (PTO)
damping if a PTO mechanism were present. M * is the excitation
moment from incident waves. Assuming small amplitude oscillations, the coefficient of stiffness K can be represented as:
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M. Choiniere, J. Davis, N. Nguyen et al.
Renewable Energy 194 (2022) 875e884
K ¼ rgczb Wzg
(2)
M * ðtÞ ¼
Here, r is the density of water, g is the acceleration due to
gravity, c is the submerged volume of the paddle, and W is the
paddle's weight. rb and rg are the centers of buoyancy and gravity,
respectively. If the far-field incident wave and the paddle motion
response are assumed to be sinusoidal with frequency, u, and with
amplitudes A and 40 , respectively, then a nondimensional response
amplitude operator (RAO) is defined as:
RAO ¼
40
kA
Ke5 ðtÞ ¼
(4)
Here, Xk and nk are the exciting force and the normal vector,
respectively, in the k (k ¼ 1; 2; … 6Þ direction. Sb is the body surface of the OSWEC. Alternatively, the Haskind relation, which correlates the radiation and the scattered potentials, can also be
employed to solve for wave loads [24,25]. This technique provides
the solutions without the need to obtain the diffraction potential
explicitly.
(5)
Here, 4R;k refers to the radiation potential in the k direction. The
second method is employed in WAMIT to obtain the wave load for
this study. For this problem specifically, the exciting wave torque in
the pitch is computed as:
v4I
dS
X5 ðuÞ ¼ iur∬ 4I n5 4R;5
vn
Sb
1
2p
∞
ð
Me5 ðuÞeiut du
(9)
∞
Here, Ke5 is called the pitch wave-excitation torque kernel and h
is the wave elevation.
The added inertia, linear radiation damping, and excitation coefficients were calculated for an angular frequency range of
0.1e11 rad/s at 0.1-rad/s increments. The mesh imported to WAMIT
consisted of nearly 1:1 104 panels with the water plane area also
meshed to remove irregular frequencies. The pitch coefficients
were calculated with the simulation coordinates placed at the device hinge to reflect the coefficients used in the general, linearized,
1-DOF equation of motion shown in Eq. (1).
The pitch hydrodynamic coefficients for both 0- and 90-degree
configurations, being primary inputs to the numerical simulations,
are presented in dimensionless form in Fig. 2. The linearized,
combined pitch-hydrostatic and gravitational-restoring coefficients remained nearly unchanged; experimentally derived
restoring coefficients for the 0- and 90-degree configurations were
27.14 and 27.31 kg-m2s2, respectively. The hydrodynamic coefficients decrease significantly as the flaps were opened from 0 to
90 ; a reduction of over 50% is maintained between the pitchadded moments of inertia up to a wave angular frequency of
4 rad/s, and 46% and 50% reductions in maximum pitch radiation
damping and wave excitation torque magnitude, respectively, were
achieved. The peak values of all curves shifted to a slightly higher
frequency with the 90-degree configuration. For instance, the 0degree configuration's added moment of inertia declined dramatically from its peak value at 4 rad/s through to its minimum near
7 rad/s, while a similar trend was observed for the 90-degree
configuration's added inertia from 5 to 8 rad/s. The wave radiation damping also decreased in this frequency range but not by the
same factor as the former (90% compared to a 50% decrease), with
the 0-degree configuration remaining larger than the 90-degree
geometry. The effect of opening the flaps can create an
appendage effect and can increase viscous drag effects. This aspect
is explored further in the results section of this paper.
Radiation coefficients are key for estimating the maximum power and effective absorption bandwidth of a WEC [27,28]. The WEC
hydrodynamic efficiency for transferring a wave's kinetic energy to
mechanical power is maximized when the resistive component of
the WEC impedance dominates the inertial. In the case of the
VGOSWEC, because the mass moment of inertia and linearized
spring coefficient do not change with the geometric configuration
one should expect the 0-degree geometry to have a larger resistiveto-inertial-impedance ratio than the 90-degree case. Therefore, in
the high-frequency range, after the peak in radiation coefficients,
the 0-degree configuration will have a higher theoretical capture
efficiency; however, other factors, such as forces in the foundation,
matching the required PTO torque, and other conversion efficiency
characteristics of the PTO may make the 90-degree case preferred
in certain sea states [29].
Sb
v4I
dS
Xk ðuÞ ¼ iur∬ 4I nk 4R;k
vn
Sb
(8)
where:
Here, k is the wave number corresponding to the linear
dispersion relation of progressive waves and can be computed from
the wave frequency and water depth.
The added inertia, radiation damping, and excitation coefficients of the OSWEC when subjected to waves (Eq. (1)) were
calculated from WAMIT version 7.3 [23]. These coefficients are used
as inputs for WEC-Sim [21], an open-source, mid-fidelity simulation tool, developed as part of a collaboration between NREL and
Sandia National Laboratories, that runs on the MATLAB/SIMULINK
platform. The six rigid-body equations of motion are solved in the
time domain through a Cummins convolution approach.
In WAMIT, wave loads can be estimated using the diffraction
theory and by direct integration. Let 4D , 4S , and 4I be the total
diffraction, the scattered, and the incident wave potentials of the
surrounding flow, respectively. The wave force in the k direction
can be approximated as:
Sb
Ke5 ðt tÞhðtÞdt
∞
(3)
Xk ðuÞ ¼ iur∬ 4D nk dS ¼ iur∬ 4I þ 4S nk dS
∞
ð
(6)
If Across were to denote the OSWEC's body's cross-sectional
surface area, then dS ¼ dAcross ðx; yÞdz. The moment due to wave
action, Me55 ðuÞ, on the OSWEC about the y-axis at the hinge is
estimated using the wave load in the surge direction and the distance from the connecting point:
v4I
ðz þ h cÞdAcross ðx; yÞdz
Me5 ðuÞ ¼ iur∬ 4I n1 4R;1
vn
Sb
(7)
3. Experimental program and analysis methodology
Employing the frequency-dependent results from WAMIT, the
time-domain, exciting wave moment is then calculated in WECSIM as [21,26]:
The model used in the experiments was the same one used in
Ref. [31] and is briefly described in this section, with the principal
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M. Choiniere, J. Davis, N. Nguyen et al.
Renewable Energy 194 (2022) 875e884
Fig. 2. Dimensionless pitch hydrodynamic coefficients,
calculated about the hinge point: added inertia and radiation damping (left); and excitation torque magnitude and phase
(right). Here, A*55 ¼ A55 ðI 55 Þ1 , B*55 ¼ B55 ðuI 55 Þ1 , X *5 ¼ jX 5 jðrgh2 uÞ1 , 4*5 ¼ 45 ðpÞ1 and u* ¼ uðh=gÞ1=2 .
dimensions captured in Fig. 3 and Table 1. The main paddle of the
OSWEC was constructed from sheets of high-density polyethylene,
heat welded at the joints to create watertight buoyant chambers
above and below the flap window and connected by solid side
plates. The five variable-pitch elliptical flaps were 3D-printed in
sections of poly-lactic acid plastic with 40% infill. The individual
sections of each flap were joined with two continuous wooden
dowels through the interior of each section spanning the width of
the flaps, and reinforced with fiberglass strips on the outer surface.
A final coat of epoxy was applied to protect each flap from leaks.
The paddle was mounted to a steel plate at its bottom surface to
connect corrosion-resistant bearings with integrated shaft clamps.
Experiments were performed at the Harold Alfond W2 Ocean
Engineering Laboratory (Fig. 4) at the University of Maine. The facility is 30 m long, and 9 m wide, with a working water depth of
4.5 m. It is equipped with a wave maker that has 16 independently
controlled paddles capable of producing regular or random waves
of varying frequencies with heights up to 0.8 m. At the other end, an
elliptical beach was installed to reduce wave reflection. To perform
this test campaign, the test article's hinge was mounted to a rigid
support structure on the tow carriage and located in the center of
the wave basin (Fig. 4).
The paddle was instrumented with Qualisys markers for the 6degrees-of-freedom motion tracking system. Data were collected
using the W2 data acquisition system, which is built on the National
Instruments PXIe platform. Wave environments were calibrated
through point measurements from a wave elevation probe.
Without the model in the water, the calibration probe was installed
Table 1
Dimensions and mass properties.
Parameter
Value
Unit
Depth (h)
Hinge to seabed (c)
Hinge depth (hH)
Paddle height (Hp )
Paddle width
Paddle thickness
Mass (m)
Moment of inertia (about center of gravity) I55
4.5
3.85
0.65
0.61
0.94
0.095
25.7
1.37
m
m
m
m
m
m
kg
kg-m2
at the model location and the desired wave environments were run.
Data from this probe were analyzed, and wave corrections were
applied to the wave maker. This process was iterated until the
desired environments were realized. The probe was then removed
and the model was installed for testing using the calibrated
environments.
Free-decay tests were conducted to evaluate the system parameters. To estimate the magnitude of linear and nonlinear
damping coefficients for the OSWEC, a nonlinear system ID method
was adopted [32]. Assuming the decayed oscillation is approximately sinusoidal over a half cycle, the quadratic velocity term is
linearized using a Fourier series expansion:
_ 4jy
_
4j
8
un 4k 4_
3p
Fig. 3. Dimensions of interest (mm) of the model-scale VG-OSWEC with variable flaps in 0 configuration.
878
(10)
M. Choiniere, J. Davis, N. Nguyen et al.
Renewable Energy 194 (2022) 875e884
Fig. 5. Quadratic system ID fit for a 0-degree-configuration, free-decay run. The
damping ratio is represented by the intercept, and the slope is proportional to the
quadratic drag coefficient.
experimental 0-degree configuration free decay run is presented in
Fig. 5.
For the regular-wave runs, a steepness value in the upper linear
wave range was selected: gTH2 ¼ 0:0008 where H is the wave height
and T is the wave period. For the range of periods spanning the
wave maker's capabilities, the wave heights were calculated
directly. The model-scale test matrix is shown in Table 2. The fullscale periods range from 2.25 to 12.72 s, and the wave heights
range from 0.04 to 1.27 m. These tests were run for 3 min each,
corresponding to about 11.2 min at full-scale. For each run, wave
elevation was probed at five locations and pitch angle response
were collected at 50 Hz.
Much like the wave data collected prior to device installation, in
response to incident waves, VG-OSWEC exhibited small multiple
harmonics in the angular displacement time history. To be consistent with the method used to characterize the incident wave
amplitude, a fast Fourier transform analysis was also performed on
the angular displacement time history collected in each run. The
amplitude of the first harmonic was considered to be the VGOSWEC's angular displacement amplitude and used to determine
the pitch RAO.
Wave-excitation moment experiments were run using the same
model scale test matrix as the regular wave response runs (Table 2).
The model was fixed via a rigid connection at the top of the flap,
through which a load cell was mounted to record the horizontal
reaction force. Moments about the model's hinge were calculated
post-experiment by multiplying this force measurement by the
moment arm (rlc ¼ 0:955 m). Tests were run for 3 min, at the model
scale, with wave elevation and force reaction sampled at 50 Hz.
Fig. 4. a) W2 facility with the OSWEC model installed on the carriage (Top). The wave
machine (not shown)is in the near end of the picture, and the beach is at the far end; b)
Computer-aided drawing of the OSWEC model mounted on the carriage (bottom).
where un is the natural frequency and 4k is the amplitude of the kth
oscillation cycle. If we consider peaks spaced two periods apart, the
following linear equation is obtained (see Ref. [32]):
1
4
4
CD
4
ln k1 ¼ z þ
2p 4kþ1
3p ð J þ A55 Þ k
(11)
Here z is the damping ratio comprising only the linear damping
terms:
z¼
B
2ð J þ A55 Þun
(12)
Eq. (11) is fit with the quantity on the left-hand side as the
dependent variable and 4k as the independent variable, such that
the intercept represents the damping ratio and the slope can be
related to the quadratic drag coefficient, CD . Hence, contributions
from the linear and quadratic damping terms are isolated. Using the
resulting damping ratio, the cumulative linear damping coefficient,
B, can be calculated from Eq. (12). By subtracting the known
damping coefficients, the pitch wave radiation damping and PTO
damping (here BPTO ¼ 0) from the cumulative coefficient, the residual damping coefficient can be estimated. This is a rudimentary
approximation of this coefficient because the pitch wave radiation
damping is obtained from WAMIT, and hence the resulting difference may also account for discrepancies in the magnitude of the
latter at the natural frequency. A representative fit to Eq. (12) for an
4. Results and discussion
System identification was based on four free-decay runs for each
flap-angle configuration. Two runs were set at a small initial pitch
angle ð 5 Þ, and another two runs at larger value ð 10 Þ. The
results of the quadratic system ID procedure, including the damping ratio, natural frequency, linear drag coefficient, and quadratic
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M. Choiniere, J. Davis, N. Nguyen et al.
Renewable Energy 194 (2022) 875e884
Table 2
Model scale regular wave test matrix.
Run No
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
T (s)
H (m)
1
0.008
1.2
0.011
1.4
0.015
1.6
0.02
1.8
0.025
2
0.031
2.2
0.038
2.4
0.045
2.6
0.053
2.8
0.062
3
0.071
3.2
0.08
3.4
0.091
3.6
0.102
3.8
0.113
damping coefficient, CD , from one configuration to the other, for
both runs 1 and 2; contributions to this coefficient result from the
viscous fluid drag on the outside frame of the OSWEC, which does
not change with flap angle orientation.
Free-decay simulations for each experimental pitch's initial
conditions were performed with WEC-Sim, using the hydrodynamic coefficients produced by WAMIT. The linear and quadratic
pitch damping coefficients, estimated using the quadratic damping
system ID method and taken as the mean value of the results for
runs 1 and 2, were applied to the rotational constraint at the hinge
location of the OSWEC. The WEC-Sim no-wave convolution integral
wave class, noWaveCIC, was used for the simulation. Across all
initial conditions and for both configurations, the simulated time
histories compared very well with the experimental results.
Representative time histories are presented in Fig. 6. The representative 0-degree time history shown is that of run 1 listed in
Table 3 with an initial condition of 12.5 . The simulation captures
the first peak well, whereas the experimental and simulated natural periods begin to deviate past the first period. This is likely due
to a changing experimental hydrostatic and gravitational restoring
stiffness, which is not captured by the constant, linearized restoring
coefficient used in the simulation. The representative 90-degree
time history shown is run 4 with an initial condition of 6.0 . The
first five periods of the traces compare exceptionally well, whereas
the simulated response appears to decay slightly faster past this
point, indicating a slight overestimation of system damping in the
90-degree configuration that is especially prevalent at low angular
velocities. In contrast to the 0-degree configuration, the natural
period of the experimental results is matched by the simulation
throughout the entire duration of the recorded decay.
Experimental pitch response time histories for the 15 regularwave-condition runs (Table 2) were obtained for the both 0- and
90-degree configurations. Wave elevation time histories, probed
during the experiments, were input into WEC-Sim using the
etaImport wave class and used to simulate the model response for
each condition. Linear and quadratic drag coefficients were applied
to the models using the same method as described in the free decay
simulations. A time-history comparison of the experimental and
simulated results for a representative wave condition (run 11 with a
wave height and period of 0.062 m and 3.0 s, respectively) is shown
in Fig. 7. The WEC-Sim simulations capture the initial transients
well for both flap angle configurations, while deviations are
observed during the following transition to steady state. Simulated
results exceed the experimental response amplitude by as much as
30% in this range. The simulated and experimental responses
realign past the fifth period for the 0-degree configuration and past
the ninth period for the 90-degree configuration, remaining in
excellent agreement for the remainder of the recorded time history.
The amplitude and phase of the pitch response amplitude
operator were obtained from the measured and simulated pitch
and wave elevation time histories through a Fourier transform
approach. RAO values from Eq. (3) were calculated with the
wavenumber corresponding to the observed periods and compared
for 0- and 90-degree configurations in Fig. 8. Excellent agreement
between the experimental and simulated RAO for both flap configurations was obtained over the dimensionless angular frequency
range of 1.75e4.25 rad/s (dimensional angular frequency of about
Table 3
Quadratic system ID results for 0- and 90-degree configurations produced using the
experimental free-decay time histories.
0-degree
90-degree
Run no.
40
z
un
B
CD
1
2
3
4
1
2
3
4
12.5
12.3
6.6
6.5
9.6
9.8
5.4
6.0
0.034
0.035
0.037
0.046
0.032
0.036
0.059
0.060
1.04
1.04
1.04
1.04
1.45
1.45
1.45
1.46
5.68
5.78
6.25
7.68
3.62
4.03
6.74
6.86
90.59
87.54
82.79
63.04
94.48
80.81
61.42
47.12
damping coefficient for each initial condition, are presented in
Table 3.
A comparison of the high- and low-initial condition runs (runs
1e2 and 3e4, respectively, for both flap configurations shown in
Table 3) suggests a significant increase in linear damping with a
corresponding decrease in quadratic damping as the initial pitch
angle is decreased. Simulated results run with damping coefficients
predicted from the smaller set of initial conditions (runs 3e4) were
significantly overdamped compared to the corresponding experimental runs. In these low-initial-condition cases, it is likely there
are not enough peaks to accurately predict the damping coefficients
from the slope the quadratic system ID fit. As a result, damping
coefficients from the first two runs were averaged and used to
produce simulation results across all initial conditions for both flap
angle configurations.
A significant increase in natural frequency, from 1.04 to 1.45 rad/
s, is observed from the 0- to 90-degree-configuration results, as
estimated from the experimental, free-decay runs. Because the
linearized restoring coefficients are nearly the same (27.14 and
27.31 kg-m2s2) for the 0- and 90- degree cases, respectively) this
shift in natural frequency is indicative of a decrease in added inertia
from 0 to 90 . Using the standard definition of the natural frequency we find that the total inertia of the 90-degree configuration
is nearly 50% that of the 0-degree configuration, a result supported
by the boundary-element-method simulation results presented in
Fig. 2.
Considering only runs 1 and 2, those that contain a sufficient
number of peaks to accurately apply the quadratic damping system
ID methodology, a decrease in the linear damping coefficient, B, of
approximately 33% is observed between the 0- and 90-degree
configurations. This is interesting, because simulated results from
WAMIT show near-zero damping at this frequency (Fig. 2). It is
conceivable that friction damping, which is linear with velocity
could contribute to the non-zero values. In addition, wave run-up
and other free-surface interaction effects could result in additional loss of kinetic energy, which could also manifest as damping.
When the flaps are open, higher viscous effects may be caused if the
open flaps are perceived as appendages to the paddle structure. On
the other hand, flow of water through the flaps could result in lower
damping. The values of CD for runs 1 and 2 do not show significant
difference between the two configurations. Other contributions
such as friction in bearings and instrumentation damping, are not
expected to change due to flap configuration. The system ID
method does not predict a significant change in the quadratic
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Fig. 6. Representative free-decay run results for 0-degree (left) and 90-degree (right) configurations. Responses are non-dimensionalized by the initial displacements, 12.5 and 6.0 ,
respectively.
with some spread in the data. This closeness to simulated value is
considered remarkable, considering local flow features that occur
when the paddle is nearly vertical, including the wave crest overtopping the edge of the paddle.
The wave excitation moments about the hinge in both configurations is shown in Fig. 10. To acquire this parameter experimentally, the OSWEC was rigidly fixed in its upright position and a
load cell was placed at the upper connection point to measure reaction force. Wave excitation moments about the hinge were then
calculated by multiplying the horizontal load cell readings by the
distance from the load cell axis to the hinge, with rlc ¼ 0:955m. The
measurements were recorded as time-history (Fig. 10), from which
the positive and negative peaks were extracted and combined to
obtain the final amplitudes and phases. The results are presented in
Fig. 11, with the wave excitation moments normalized by the hydrostatic pressure moment on one side of the device as:
M* ¼ 1
Fex rlc
2
6 rgwh A
M* ¼ 1
Fex rk
2
6 rgwh A
(14)
Similar configurations, in which rigid constraint was placed at
the VG-OSWEC hinge location, were modeled with WEC-Sim to
attain the numerical results for comparison. Fig. 11 shows that the
numerical data correlate very well with experimental measurements, especially in the range of u* from 1 to 3. At larger frequencies or smaller wave periods, the datasets for the flap open
configuration show large deviations up to 33% despite having
similar trends. Similar to the pitch motion, the moment phase angle
with respect to wave elevation hovers around p2 but only for smaller
Fig. 7. Representative pitch response time-histories for 0-degree (top) and 90-degree
(bottom) configurations for regular-wave-condition run 11 with a wave height and
period of 0.062 m and 3.0 s (u* ¼ 1.42), respectively. Non-dimensionalizations: t * ¼
t ðTÞ1 , 4* ¼ 4 ðkAÞ1 where t is time in seconds.
2.58e6.28 rad/s). Below this frequency range, results deviate; the
simulated RAO for the 0-degree configuration predicts a resonant
peak at a slightly lower frequency than the apparent trend of the
experiments, whereas the 90-degree simulated results predict an
early resonance. These trends appear to deviate from the natural
frequency values predicted by the free decay experiments, which
are 1.04 and 1.45 rad/s (with u* ¼ 0.70 and u* ¼ 0.98) for the 0- and
90-degree configurations, respectively. As the experimental natural
period of the model-scale VG- OSWEC fell outside of the wave
maker period capabilities for both configurations, these peaks were
not captured by the experiments. In terms of the phase, the simulations show that the angle is nearly uniform at p2. This is to be
expected if the mean position of the paddle coincides with the crest
of the wave. For example, Fig. 9 shows this phase difference between time histories of pitch and wave elevation for run 11. The
experimental measurements also show values that average 0:6p
values of u* . At larger frequencies, differences are significant with
reasonably similar trends.
A fundamental aspect of importance in this research is the
ability of the device to lower wave excitation loads on its foundation. In Fig. 11, it is shown that the configuration with the flaps open
(90-degree) has consistently lower wave excitation moments on
the device. The load shedding capabilities of the design accounts for
about 33%e55% of the load reduction across all the tested wave
frequencies. The relative decrease is observed for both numerical
and experimental cases despite the differences in the absolute
values between the two data sets.
5. Conclusions
The performance of a fixed, bottom-hinged rectangular wave
paddle with five rotatable flaps was the subject of this paper. Tests
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Renewable Energy 194 (2022) 875e884
Fig. 8. Comparison of experimental and simulated response. Left: non-dimensional RAO for 0- and 90-degree configurations. Right: phase difference between the pitch response
and incident wave elevation. Angular frequency is non-dimensionalized as: u* ¼ uðh=gÞ1=2 .
Fig. 9. Comparison of nondimensional pitch response, 4* (left axis) and wave elevation, h* (right axis) time histories for 0-degree-configuration, regular-wave-condition
run 11 with a wave height and period of 0.062 m and 3.0 s (u* ¼ 1.42), respectively.
Non-dimensionalizations: t * ¼ t ðTÞ1 , 4* ¼ 4 ðkAÞ1 , h* ¼ h ðAÞ1 .
conducted on a 1:14 scale VG-OSWEC model in a wave basin
showed that the natural frequency was increased by 40% when the
flap configuration was changed from closed to open. On the other
hand experiments in regular wave conditions showed that the
wave excitation moment about the hinge was reduced by up to 60%
when the flaps were fully open. The experimental response
amplitude operator for the rotational motion about the hinge
showed good agreement with simulations for both flap conditions.
The WEC hydrodynamic efficiency for transferring a wave's kinetic energy to mechanical power is maximized when the resistive
component of the WEC impedance dominates the inertial. As
observed by Ref. [29] in the case of the VG-OSWEC, because the
mass moment of inertia and linearized spring coefficient do not
change with the geometric configuration, one should expect the 0degree geometry to have a larger ratio of resistive to inertial
impedance as compared to the 90-degree case. However, such high
frequencies result in small wavelengths, with accompanying small
wave heights in a practical sea state. If one were to consider a scaled
version of the experiment (e.g. at a water depth of 63 m and a wave
period typical of swell waves in large-storm conditions), then the
corresponding u* can range from 1 to 3. In this range, the 90-degree
case provides a higher reduction in base moment as compared to
the 0-degree case.
Practical considerations governing design and implementation
of variable geometry into a WEC system are important. Moveable
Fig. 10. Representative non-dimensional wave moment time-histories for 0-degree
(top) and 90-degree (bottom) configurations for regular wave condition run 3 with
wave height and period of 0.0136 m and
1.4 s (u* ¼3.05), respectively. Non1
dimensionalizations: M * ¼ M 16 rgwh2 A
, t * ¼ t ðTÞ1 .
components underwater pose challenges with inspection, maintenance and operations. Structural failures of most vulnerable
components become important considerations. A comprehensive
techno-economic feasibility study will be a future task of this
research team.
CRediT authorship contribution statement
Michael Choiniere: Formal analysis, Experimental program,
model instrumentation and preparation, preliminary analysis of
results. Jacob Davis: Formal analysis, Analysis of results, postprocessing, graphics, WEC-Sim data postprocessing. Nhu Nguyen:
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M. Choiniere, J. Davis, N. Nguyen et al.
Renewable Energy 194 (2022) 875e884
Fig. 11. Comparison of experimental and simulated wave excitation moments. Left:
wave moment magnitudes. Right: phase differences between the wave
non-dimensional
1
, u* ¼ uðh=gÞ1=2 .
excitation moment and incident wave elevation. Non-dimensionalizations: M * ¼ M 16 rgwh2 A
Formal analysis, data analysis. Nathan Tom: Conceptualization,
Idea conception, WEC-Sim model, simulated results development.
Matthew Fowler: Experimental facility preparation, instrumentation. Krish Thiagarajan: Conceptualization, Writing e original
draft, Writing e review editing, Idea conception, experimental
strategy, results review, manuscript development.
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Declaration of competing interest
The authors declare that they have no known competing
financial interests or personal relationships that could have
appeared to influence the work reported in this paper.
Acknowledgments
This work was authored in part by the National Renewable
Energy Laboratory, operated by Alliance for Sustainable Energy,
LLC, for the U.S. Department of Energy (DOE) under Contract No.
DE-AC36-08GO28308. Funding provided by the U.S. Department of
Energy Office of Energy Efficiency and Renewable Energy Water
Power Technologies Office. The views expressed in the article do
not necessarily represent the views of the DOE or the U.S. Government. The U.S. Government retains and the publisher, by
accepting the article for publication, acknowledges that the U.S.
Government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this
work, or allow others to do so, for U.S. Government purposes.
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