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RESEARCH ARTICLE
|
FEBRUARY 02 2022
Phase-averaged dynamics of a periodically surging wind
turbine
Nathaniel J. W
ei
;
John O. Dabiri
J. Renewable Sustainable Energy
14, 013305 (2022)
https://doi.org/10.1063/5.0076029
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Phase-averaged dynamics of a periodically surging
wind turbine
Cite as: J. Renewable Sustainable Energy
14
, 013305 (2022);
doi: 10.1063/5.0076029
Submitted: 20 October 2021
.
Accepted: 17 December 2021
.
Published Online: 2 February 2022
Nathaniel J.
Wei
1
and John O.
Dabiri
1,2,a)
AFFILIATIONS
1
Graduate Aerospace Laboratories, California Institute of Technology, Pasadena, CA 91125, USA
2
Department of Mechanical and Civil Engineering, California Institute of Technology, Pasadena, CA 91125, USA
a)
Author to whom correspondence should be addressed:
jodabiri@caltech.edu
ABSTRACT
The unsteady power generation of a wind turbine translating in the streamwise direction is relevant to floating offshore wind turbines, kite-
mounted airborne wind turbines, and other nontraditional wind-energy systems. To study this problem experimentally, measurements of
torque, rotor speed, and power were acquired for a horizontal-axis wind turbine actuated in periodic surge motions in a fan-array wind tun-
nel at the Caltech Center for Autonomous Systems and Technologies. Experiments were conducted at a diameter-based Reynolds number of
Re
D
¼
6
:
1
10
5
and at tip-speed ratios between 5.2 and 8.8. Sinusoidal and trapezoidal surge-velocity waveforms with maximum surge
velocities up to 23% of the free-stream velocity were tested. A model in the form of a linear ordinary differential equation (first-order in
time) was derived to capture the time-resolved dynamics of the surging turbine. Its coefficients were obtained using torque measurements
from a stationary turbine, without the need for unsteady calibrations. Its predictions compared favorably with the measured amplitude- and
phase-response data. Furthermore, increases in the period-averaged power of up to 6.4% above the steady reference case were observed in
the experiments at high tip-speed ratios and surge velocities, potentially due to unsteady or nonlinear aerodynamic effects. Conversely,
decreases in mean power with increased surge velocity at low tip-speed ratios were likely a result of the onset of stall on the turbine blades.
These results inform the development of strategies to optimize and control the unsteady power generation of periodically surging wind
turbines, and motivate further investigations into the unsteady aerodynamics of wind-energy systems.
Published under an exclusive license by AIP Publishing.
https://doi.org/10.1063/5.0076029
I. INTRODUCTION
New innovations in wind energy technology motivate the study
of wind-turbine performance in previously unexplored operational
regimes. In particular, while traditional land-based wind turbines are
fixed in place, wind-energy systems such as floating offshore wind tur-
bines (FOWTs) and airborne wind turbines undergo streamwise oscil-
lations that may potentially complicate the aerodynamics of these
systems. The periodic motions of the turbine rotor in these situations
introduce additional dynamics that can affect the power generation of
the wind turbines and the fatigue loading on their blades, thereby
impacting their contribution to global energy demands. Therefore, this
study investigates the dynamics of a periodically surging wind turbine
through analytical modeling and laboratory-scale experiments.
A. Current progress in surging-turbine aerodynamics
Previous studies have generally considered surge motions typical
of wave-driven FOWTs,
1
which are of increasing relevance as the
emerging offshore-wind sector continues to expand. The majority of
attention regarding turbine aerodynamics has been focused on time-
averaged quantities. Using a model FOWT in a wind tunnel and wave
tank, Farrugia
et al.
2
found that the time-averaged coefficient of power,
C
p
, increased above the steady case by 1% when oscillations in the tur-
bine were present. A similar increase in
C
p
by 1% was observed in
wind tunnel experiments by Khosravi
et al.
3
and free-vortex wake sim-
ulations by Shen
et al.
4
Farrugia
et al.
5
showed using free-vortex wake
simulations that
C
p
increased with surge frequency at tip-speed ratios
above the rated value by up to 13.7%, but decreased with surge fre-
quency at tip-speed ratios below the rated value. Independent simula-
tions by Wen
et al.
6
yielded similar results. Johlas
et al.
7
suggested that
the increases in average power with surge velocity can be described by
a simple quasi-steady model, where the term “quasi-steady” refers to
effects for which successive instances in time can be considered as
being in independent states of local equilibrium. Since the model is
derived from the cubic dependence of power on the incident inflow
velocity at the rotor, the relative power gains over the steady value of
J. Renewable Sustainable Energy
14
, 013305 (2022); doi: 10.1063/5.0076029
14
, 013305-1
Published under an exclusive license by AIP Publishing
Journal of Renewable
and Sustainable Energy
ARTICLE
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06 October 2023 17:49:11
C
p
from surge motions in the upwind direction outweigh the relative
losses from surge motions in the downwind direction. Their model
agrees well with the time-averaged power results from their numerical
simulations of a surging turbine, but the validity of the model has not
yet been evaluated over a wide range of surge velocities and operating
conditions. A fully characterized, quantitatively accurate explanation
for the observed increases in
C
p
thus remains elusive.
The time-resolved dynamics of a turbine in surge have been
explored as well. These unsteady dynamics determine the unsteady
loads on the turbine and its support structures, and therefore inform
the design of FOWT control systems. The presence of fluctuations in
turbine thrust, torque, and power at the same frequency as the
imposed surge motion is well-documented in the literature.
2,4–8
These
fluctuations increase in amplitude as the surge frequency is increased.
5
Mancini
et al.
,
9
however, showed in wind tunnel experiments with a
surging turbine that the relationship between torque amplitude and
surge frequency increases above the prediction of their linear quasi-
steady model at high frequencies. They attributed this to mechanical
resonance, and not to a breakdown of their linearization of power as a
function of the surge velocity or the influence of unsteady aerodynam-
ics. By contrast, torque amplitudes measured in wind tunnel experi-
ments by Sant
et al.
10
were much lower than those computed by
quasi-steady and dynamic-inflow codes.
The phase response of the turbine similarly lacks a single consis-
tent characterization in the literature. The model developed by Johlas
et al.
7
predicts that the instantaneous power from the turbine will be
in phase with the surge velocity. The linear quasi-steady model of
Mancini
et al.
9
supports the same prediction. Some computa-
tional
5,6,8,11
and experimental
9
results, however, have shown phase
differences in excess of 90
, while others displayed close to zero phase
offset.
4,12,13
Thediscrepanciesintheliteratureregardingtheamplitude
and phase of the torque and power output of surging turbines moti-
vates the current study.
The lack of consensus with respect to mean quantities and their
amplitude and phase stems in large part from unanswered questions
regarding the relative importance of quasi-steady and unsteady effects.
The models of Johlas
et al.
7
and Mancini
et al.
9
can be classified as
purely quasi-steady models that neglect unsteady effects. Other models
have incorporated unsteady effects directly. For example, de Vaal
et al.
14
compared the results of different dynamic-inflow models,
which include corrections for time-varying inflow velocity and acceler-
ation, that were paired with blade-element momentum (BEM) simula-
tions of a surging turbine. They concluded that these engineering
models were capable of capturing global forces on FOWTs in typical
offshore conditions. In a different approach, Fontanella
et al.
15
derived
a state-space model that maps linearized turbine aerodynamics and
wave dynamics to the time derivatives of the kinematic parameters of
the turbine. The model was shown to perform well both in simulations
and as the basis for control systems.
16
In addition to these models,
others have suggested various unsteady flow phenomena that could
influence the turbine dynamics. For instance, several of the aforemen-
tioned studies have considered the effects of airfoil stall, particularly at
the blade root, on time-averaged and fluctuating quantities.
5,6,8,11
In
addition to blade stall, Sebastian and Lackner
17
postulated the forma-
tion of unsteady recirculation regions in or downstream of the rotor
plane during turbine surge, as a result of slip-stream violations.
Furthermore, Wen
et al.
11
attributed the phase differences observed in
their simulations to added-mass effects, blade-wake interactions, and
unsteady aerodynamics. These unsteady flow phenomena may affect
the structure, dynamics, and recovery of the wake of a surging tur-
bine.
8,18–22
However, it still remains to be seen which (if any) of these
unsteady effects must be accounted for in a model to capture the tor-
que and power production of real surging turbines, or whether existing
quasi-steady models are sufficient for this purpose.
Finally, since nearly all existing work on surging-turbine aerody-
namics has been conducted in the context of surge oscillations typical
of FOWTs under normal operating conditions, the dynamics of wind
turbines surging through larger amplitudes or higher frequencies
remain relatively unexplored. Larger surge oscillations would be rele-
vant not only to FOWTs in more extreme conditions, but also to air-
borne wind turbines mounted to aircraft or crosswind kites.
23
Crosswind kites generally fly through large periodic orbits with length
scales much greater than the size of the aircraft itself.
24
Turbines
mounted to these kites would therefore undergo surge motions at
amplitudes far larger than those experienced by FOWTs. In addition,
Dabiri
25
recently suggested that streamwise unsteadiness could be
leveraged to increase the efficiency of wind-energy systems above the
theoretical steady limit. Since increases in time-averaged power have
already been observed at the relatively low levels of unsteady motion
typical of FOWTs, an investigation of higher surge amplitudes and fre-
quencies could provide insights toward the practical realization of
these theoretical efficiency gains.
B. Research objectives
This study aims to address several open questions regarding the
time-resolved dynamics of a wind turbine in the surge. The amplitude
and phase of torque and power relative to the surge motions are inves-
tigated in wind tunnel experiments. Trends in the data are parameter-
ized by a model that accounts for quasi-steady aerodynamic torques
and unsteady generator torques. The model is first-order in time and
linear in the turbine surge velocity and rotor speed; thus, it shall hence-
forth be referred to as a first-order linear model. An important feature
of the model is that its coefficients can be computed from measure-
ments obtained under steady conditions; no data from actual surge
tests are required to obtain time-resolved torque and power predic-
tions. The experiments span higher levels of unsteadiness than previ-
ous studies in the literature, with scaled amplitudes up to
A
¼
A
=
D
¼
0
:
51 and nondimensional surge velocities up to
u
¼
fA
=
u
1
¼
0
:
23, where
A
is the surge amplitude,
f
is the surge frequency in
radians per second,
u
1
is the free-stream velocity, and
D
is the turbine
diameter. By contrast, the highest values reported in the literature are
A
¼
0
:
13 (Tran and Kim
8
)and
u
¼
0
:
42 (Wen
et al.
11
)insimula-
tions, and
A
¼
0
:
15 (Sant
et al.
10
)and
u
¼
0
:
079 (Mancini
et al.
9
)
in experiments. The findings in this study may thus be generalized to
FOWTs operating under extreme conditions, as well as novel airborne
wind-energy systems and other emergent technologies. The combined
analytical and experimental results presented in this work provide a
foundation upon which questions regarding the influence of unsteadi-
ness and nonlinearity, including the dependence of the mean torque
and power on surge kinematics, may be more comprehensively inves-
tigated in future work.
The paper is structured as follows: In Sec.
II
,afirst-orderlinear
model is derived that enables a disambiguation between aerodynamic
and generator torques. Its amplitude and phase characteristics are also
Journal of Renewable
and Sustainable Energy
ARTICLE
scitation.org/journal/rse
J. Renewable Sustainable Energy
14
, 013305 (2022); doi: 10.1063/5.0076029
14
, 013305-2
Published under an exclusive license by AIP Publishing
06 October 2023 17:49:11
analyzed. In Sec.
III
, the experimental apparatus is described, and
methods for computing the coefficients of the analytical model from
measurements in steady conditions are given. Phase-averaged results
from experiments with sinusoidal and trapezoidal surge-velocity wave-
forms are presented in Sec.
IV
, and the results are compared with
model predictions. Finally, a discussion regarding model capabilities
and limitations, nonlinear and unsteady effects, and application to
full-scale wind turbines is provided in Sec.
V
.
II. ANALYTICAL MODEL
In this section, we derive a model for the torque generated by a
surging horizontal-axis turbine from an ordinary differential equation
that is first-order in time and linear in the turbine surge velocity and
rotor speed (or rotation rate). We linearize the aerodynamic torque
with respect to the inflow velocity and rotor speed and combine it
with a model for the generator torque to obtain a differential equation
fortherotorspeedoftheturbine.Wethenderivetransferfunctionsin
the frequency domain to characterize the amplitude and phase relative
to the surge-velocity waveform of the aerodynamic and generator tor-
que. A notable advantage of this model is that the model coefficients
can be extracted directly from torque and rotation-rate measurements
of the turbine in steady conditions (i.e., without surge motions); these
methods will be described in Sec.
III D
for the turbine used in these
experiments.
A. Aerodynamic-torque model
A first-order linear model for the aerodynamic torque can be
derived using a local linearization with respect to the inflow velocity
and rotor speed,
s
aero
s
0
þ
@
s
@
u
u
¼
u
1
;
x
¼
x
ð
u
u
1
Þþ
@
s
@
x
u
¼
u
1
;
x
¼
x
ð
x
x
Þ
;
(1)
where
u
¼
u
1
þ
U
ð
t
Þ
is the instantaneous inflow velocity relative to
the turbine,
U
(
t
) is the turbine surge velocity in a stationary frame of
reference,
x
is the rotor speed, and
s
0
is the steady aerodynamic tor-
que, i.e., the mean torque measured on a stationary turbine at a wind
speed of
u
1
. In this work, bars denote time averages over a single
streamwise motion oscillation period for time-dependent variables in
the case of unsteady streamwise motion, while the subscript 0 denotes
the value of a variable in the reference case corresponding to a steady
inflow at speed
u
1
. We then define the performance coefficients,
K
‘
¼
@
s
@
u
u
¼
u
1
;
x
¼
x
(2)
and
K
d
¼
1
R
@
s
@
x
u
¼
u
1
;
x
¼
x
;
(3)
where
R
is the radius of the turbine. These coefficients qualitatively
correspond to lift and drag terms in a blade-element expression for
aerodynamic torque. Values for these constants can be obtained
empirically from measurements of the turbine torque taken with a sta-
tionary turbine over a range of wind speeds and loading conditions
(cf. Sec.
III D
). Simplifying the above expression yields the following
model:
s
aero
K
‘
U
K
d
R
ð
x
x
Þþ
s
0
:
(4)
The accuracy of this aerodynamic model depends on whether
s
is suffi-
ciently linear in
u
and
x
in the neighborhood of the steady operating
condition (
u
¼
u
1
and
x
¼
x
). Since the model is inherently quasi-
steady, its accuracy will also depend on whether any unsteady effects
such as dynamic stall on the turbine blades are present.
B. Generator-torque model
The torque applied by the generator (
s
gen
) in opposition to the
aerodynamic torque (
s
aero
) represents the torque that is converted to
usable power at each instant in time. It thus also represents the
mechanically measurable torque on the turbine shaft (
s
meas
). It is
importanttonotethatthegeneratortorqueisnotnecessarilyequalto
the aerodynamic torque in the case of unsteady rotation, with any dif-
ference between the two inducing a change in the angular velocity of
the rotor.
The equations of motion for a permanent-magnet generator are
identical in principle to those for a permanent-magnet motor, which is
frequently modeled as a first-order ordinary differential equation in
time,
26
s
gen
¼
s
meas
¼
K
2
d
x
dt
þ
K
1
x
þ
K
0
;
(5)
where
K
2
is the moment of inertia of the generator about its rotational
axis,
K
1
is the generator constant, and
K
0
is an empirical zero-speed
offset. Since the generator torque is proportional to the current
through the generator coils,
K
1
and
K
0
scale inversely with the resistive
load applied to the generator.
26
Hence, a higher resistive load applied
to the generator corresponds to a lower physical load on the turbine.
This formulation assumes that the generator is directly driven by the
turbine; a gear-ratio scaling could be incorporated to map the rotor
speed to the generator rotation rate for turbines with gearboxes.
It should also be noted that the generator model in its current
form does not include any effects of control, such as tip-speed ratio
control systems that are typically present in utility-scale wind turbines.
For the purposes of this study, the use of a direct-drive generator with
fixed resistive loading and no speed control simplify the modeling of
the turbine dynamics and subsequent model validation against experi-
mental data. However, the linear form of the model means that linear
or linearized tip-speed ratio controllers can readily be incorporated
using classical analytical techniques.
C. Governing equation and its transfer functions
The dynamics of a turbine under the influence of competing
aerodynamic and generator torques are given by the swing equation,
27
J
d
x
dt
¼
s
aero
s
gen
;
(6)
where
J
is the moment of inertia of the turbine, its shaft assembly, and
the generator about the axis of rotation, and is thus in practice much
larger than
K
2
. Deviations of the instantaneous aerodynamic torque
away from equilibrium, if not immediately matched by the generator
torque, will lead to a change in the rotor speed until the generator
torque overcomes inertia and restores the torque balance. This implies
that the torque measured by a torque transducer, and consequently
Journal of Renewable
and Sustainable Energy
ARTICLE
scitation.org/journal/rse
J. Renewable Sustainable Energy
14
, 013305 (2022); doi: 10.1063/5.0076029
14
, 013305-3
Published under an exclusive license by AIP Publishing
06 October 2023 17:49:11
the power measured from the generator, will lag behind the aerody-
namic torque.
Substituting Eqs.
(4)
and
(5)
into the above relation, we arrive at
the equation of motion,
J
d
x
dt
¼
K
‘
U
K
d
R
ð
x
x
Þþ
s
0
ðÞ
K
2
d
x
dt
þ
K
1
x
þ
K
0
:
(7)
In the limit of equilibrium, in which all time-derivatives are zero, the
steady aerodynamic torque
s
0
must equal the generator torque, i.e.,
s
0
¼
K
1
x
þ
K
0
. We can therefore simplify the model into a more
informative form,
J
d
x
dt
¼
K
‘
U
K
d
R
ð
x
x
Þ
K
2
d
x
dt
K
1
ð
x
x
Þ
:
(8)
The resulting model is conceptually similar to that of Fontanella
et al.
,
15
though in this work the linearization for the aerodynamic tor-
que is obtained differently and the generator torque is an output,
rather than an input, to the system. An additional benefit of the model
in Eq.
(8)
is that it requires no data from unsteady surge experiments
to make time-resolved predictions, since all of its coefficients can be
computed either from measurements in steady flow or from geometric
properties of the turbine and generator. The model has the form of a
linear time-invariant (LTI) system, which allows transfer functions of
the aerodynamic and measured torques to be computed in order to
quantify the phase and amplitude behavior of the system. Taking the
Laplace transform of Eq.
(8)
with respect to an arbitrary surge velocity
U
(input) and the resulting rotor speed
x
(output) yields the transfer
function,
x
ð
s
Þ
U
ð
s
Þ
¼
K
‘
ð
J
þ
K
2
Þ
s
þ
K
1
þ
K
d
R
:
(9)
This transfer function has the form of a first-order lowpass filter
with critical frequency
f
c
¼
K
1
þ
K
d
R
J
þ
K
2
. Using this transfer function, we
can also derive transfer functions for the aerodynamic and generator
torques,
s
aero
ð
s
Þ
U
ð
s
Þ
¼
K
‘
K
d
R
x
ð
s
Þ
U
ð
s
Þ
¼
K
‘
ð
J
þ
K
2
Þ
s
þ
K
1
ð
J
þ
K
2
Þ
s
þ
K
1
þ
K
d
R
(10)
and
s
gen
ð
s
Þ
U
ð
s
Þ
¼ð
K
2
s
þ
K
1
Þ
x
ð
s
Þ
U
ð
s
Þ
¼
K
‘
K
2
s
þ
K
1
ð
J
þ
K
2
Þ
s
þ
K
1
þ
K
d
R
;
(11)
which share the same critical frequency
f
c
. The frequency response can
be computed from these transfer functions using the imaginary part of
the Laplace variable
s
, i.e., Im
ð
s
Þ¼
f
. Phase and amplitude predictions
from the model can thus be obtained analytically, and the mean torque
is given by the steady-state value
s
0
.Powercanthenbecomputedas
P¼
sx
. The linear form of the model dictates that, for periodic surge
motions with zero net displacement, the period-averaged mean torque
and power are not functions of the surge motions. According to this
model, then, unsteady surge motions will not affect the period-
averaged power generation of the turbine. The model thus forgoes the
ability to predict time-averaged quantities in favor of an analytical for-
mulation of the time-resolved turbine dynamics. The consequences of
this trade-off will be discussed in Sec.
VB
.
The transfer functions suggest that the relevant nondimensional
parameters for the surge dynamics are the nondimensional surge
velocity,
u
¼
fA
=
u
1
, and the normalized surge frequency,
f
¼
f
=
f
c
.
The analysis suggests that the amplitude of the unsteady torque oscilla-
tions scales directly with
u
, with a frequency dependence dictated by
f
. Either the reduced frequency
k
¼
fD
=
u
1
or the nondimensional
surge amplitude
A
¼
A
=
D
would complete the nondimensional
parameterization by including the length scale of the turbine, but in
contrast to suggestions in the literature,
6,19
these parameters do not
appear to follow directly from the transfer-function formulation of the
model.
III. EXPERIMENTAL METHODS
In this section, the experimental apparatus used to study the tor-
que and power production of a wind turbine in periodic surge motions
is described. First, the wind tunnel and turbine apparatus are
described. Then, the parameter space explored in these experiments is
presented, and the experimental procedure is outlined. Finally, meth-
ods for empirically determining values for the scaling coefficients of
the analytical model derived in Sec.
II
and an overview of sources of
uncertainty are provided.
A. Experimental apparatus
Experiments were conducted in a large open-circuit fan-array
wind tunnel at the Caltech Center for Autonomous Systems and
Technologies (CAST). The fan array was composed of 2592 computer
fans arranged in two counter-rotating layers within a 2
:
88
2
:
88 m
2
frame, mounted 0.61 m above the floor of the facility (cf.
Fig. 1
). The
open-air test section downstream of the fan array vented directly to
the atmosphere, while the other three sides and ceiling of the arena
were enclosed with walls or awnings to mitigate the effects of atmo-
spheric disturbances. The experiments in this study were carried out at
a free-stream velocity of
u
1
¼
8
:
06
6
0
:
16ms
1
, corresponding to a
diameter-based Reynolds number of approximately
Re
D
¼
6
:
13
10
5
.
The relevance of this study to Reynolds numbers typical of utility-scale
wind turbines is discussed in Sec.
VC
. The turbulence intensity in the
tunnel, represented by the standard deviation of the velocity fluctua-
tions normalized by the average streamwise velocity, was measured to
be 2.20
6
0.17%. These measurements were obtained with an ultra-
sonic anemometer (Campbell Scientific CSAT3B) placed at the hub
height and streamwise zero position of the wind turbine. Because the
facility was exposed to the atmosphere and experiments were con-
ducted over a range of atmospheric conditions and times of day, tem-
perature and relative-humidity readings were recorded with
measurement precisions of
6
1
Cand
6
5% from a portable weather
station (Taylor Precision Products model 1731) so that the air density
could be calculated accordingly.
The turbine apparatus was constructed on an aluminum frame
(80/20 1515T-slotted profile) that was bolted to the floor and secured
with sandbags. The frame was 2.00 m long, 0.69 m wide, and 0.87 m
tall. Two 2 m long rails with two ball-bearing carriages each (NSK
NH252000AN2PCZ) were mounted on top of the frame, parallel to
the streamwise direction and spaced 0.65 m apart in the cross-stream
direction. A traverse was mounted on the ball-bearing carriages, which
supported a 0.99 m tall, 0.038 m wide turbine tower. The wind-turbine
shaft assembly was placed on top of this tower at a hub height of
1.97 m above the floor. The origin of the surge motions of the turbine
Journal of Renewable
and Sustainable Energy
ARTICLE
scitation.org/journal/rse
J. Renewable Sustainable Energy
14
, 013305 (2022); doi: 10.1063/5.0076029
14
, 013305-4
Published under an exclusive license by AIP Publishing
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was located 3.09 m downstream of the fan array, and the rails afforded
a maximum surge stroke of 1.52 m upstream of the origin. A sche-
matic of the apparatus and its position relative to the wind tunnel is
given in
Fig. 1
.
A three-bladed horizontal-axis wind turbine (Primus Wind
Power AIR Silent X) with a rotor diameter of
D
¼
1.17 m and hub
diameter of 0.127 m was attached to a 25.4 mm diameter steel shaft
supported by two axially mounted shaft bearings (Sealmaster NP-
16T). The blade chord ranged from 100 at the root to 32 mm at the
tip. The blades were constructed from a laminated carbon-fiber com-
posite. A rotary encoder (US Digital EM2-2-4096-I) with a resolution
of 4096 counts per revolution was attached between the shaft bearings.
A rotary torque transducer (FUTEK TRS300, 20 Nm torque rating)
was connected to the turbine shaft on one end and to the turbine gen-
erator (Primus Wind Power AIR 30, 48 V) by means of a floating
19.0 mm diameter steel shaft on the other. The shaft assembly was
surrounded by a 4.76-mm thick acrylic nacelle (0
:
610
0
:
152
0
:
152 m
3
) with a slanted rear section intended to reduce bluff-body
separation effects in the turbine wake. The estimated blockage of the
swept area of the turbine and all support structures, relative to the
fan-array surface area, was 14%.
The load on the turbine was controlled electrically with
resistors. The three-phase alternating current produced by the
generator was converted to DC by a bridge rectifier (Comchip
Technology SC50VB80-G) and passed to a bank of resistors.
Different combinations of fixed 10
X
resistors (TE Connectivity
TE1000B10RJ) in series or parallel and an 8
X
rheostat (Ohmite
RRS8R0E) were used to achieve a range of loading conditions that
spanned the operational envelope of the turbine. An emergency
short-circuit switch built from a solid-state relay (Crydom D1D12)
and a 12-A fuse (Schurter 4435.0368) were included in the circuit
for safety purposes.
The turbine traverse was driven in the streamwise direction by a
piston-type magnetic linear actuator (LinMot PS10–70x320U-BL-QJ).
Its sliding cylinder was attached to the traverse at rail height and its
stator was mounted to the downstream end of the frame. The motions
of the traverse were controlled by a servo driver (LinMot E1450-LU-
QN-0S) with an external position sensor (LinMot MS01–1/D)
mounted along one of the two rails for increased repeatability. Power
for the system was provided by a step-up transformer (Maddox MIT-
DRY-188). Motion profiles were loaded onto the driver, and a TTL
pulse was used to start each successive motion period. In these experi-
ments, the maximum surge velocity was
U
¼
2.40 ms
1
and the maxi-
mum surge acceleration was
dU
dt
¼
23
:
7ms
2
, while the average
absolute position error was 0.68 mm and the average absolute velocity
error was 8
:
33
10
3
ms
1
.
FIG. 1.
Schematic of the experimental apparatus, including fan-array wind tunnel (left) and surging turbine (center-right). The turbine is illustrated at
its maximum upstream
position relative to the origin. The estimated blockage of the swept area of the turbine and all support structures, relative to the fan-array surface
area, is 14%.
Journal of Renewable
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Data were collected from the rotary encoder and torque trans-
ducer using a data acquisition card (National Instruments USB-6221),
as well as an amplifier (FUTEK IAA100) for the raw torque voltage
signals. Data collection occurred at a sampling rate of 1 kHz. A
LabVIEW control program coordinated data collection and triggering
for the linear actuator. It was also used to adjust the resistance of the
rheostat remotely via a stepper motor (Sparkfun ROB-13656). After
each experiment, the program converted the raw voltage signals from
the amplifier to dimensional values by interpolating between calibrated
points, and it performed a fourth-order central-difference scheme on
the angular measurements from the encoder to obtain a rotation-rate
signal. Furthermore, the measured torque signals were filtered using a
sixth-order Butterworth filter with a cutoff frequency of 100 Hz to
reduce the influence of electrical noise.
B. Experimental parameters
The apparatus described in Sec.
III A
was used to investigate the
unsteady torque and power production of a wind turbine actuated in
surge motions. Surge amplitudes between
A
¼
0.150 and 0.600 m
(
A
¼
A
=
D
¼
0
:
128 and 0.514) were tested in combination with
motion periods between
T
¼
0.5 and 12 s. These combinations resulted
in reduced frequencies
k
¼
fD
=
u
1
between 0.079 and 1.821 and non-
dimensional surge velocities
u
¼
fA
=
u
1
between 0.039 and 0.234.
The specific combinations of
A
and
T
explored in this study, and their
respective values of
u
, are given in
Table I
. Sinusoidal and trapezoidal
surge-velocity waveforms served as motion profiles for these experi-
ments. The trapezoidal waveforms consisted of alternating segments
of constant acceleration and constant velocity. The relative duration of
the constant-acceleration phases was parameterized by
n
, defined as
the total time of nonzero acceleration in a single cycle divided by the
cycle period. Hence,
n
¼
0 corresponded to a square wave, while
n
¼
1 corresponded to a triangle wave. The types of waveforms used in
these experiments are shown in
Fig. 2
.
The wind turbine was tested with resistive loads of 7.39, 7.48,
9.80, 10, 20, and 40
X
. The first three loads were attained using the
rheostat and fixed resistors, and the second three were achieved using
fixed 10
X
resistors in series. These corresponded to steady tip-speed
ratios
k
0
¼
R
x
0
=
u
1
between 5.21
6
0.22 and 8.77
6
0.35, and coeffi-
cients of power
C
p
;
0
¼P
0
=
p
2
q
R
2
u
3
1
between 0.176
6
0.010 and
0.288
6
0.013. A summary of the loading conditions and their corre-
sponding steady nondimensional parameters is given in
Table II
.
Additionally, a power curve consisting of a collection of measurements
at different wind speeds and loading conditions is shown in
Fig. 3
.
Each of these steady measurements was conducted 3.09 m down-
stream of the fan array and over a duration of at least 2 min.
C. Experimental procedure
Experiments were conducted in the CAST fan-array wind tunnel
between May and July 2021. A zero-offset reading was taken for the
torque sensor at the start of every day of measurements. To bring the
turbine from rest to its prescribed operating condition, a higher wind
speed was initially applied for at least 2 min to mitigate the effects of
startup hysteresis in the shaft assembly. Each unsteady experiment
was paired with a corresponding steady reference case, conducted
within 1 h of the unsteady case to minimize the influence of changing
atmospheric conditions. The steady measurements were taken 3.09 m
downstream of the fan array, defined as
x
¼
0(where
x
is positive in
the upstream direction). In the unsteady experiments, the turbine
moved between
x
¼
0and
x
¼
2
A
at a prescribed frequency
f
¼
2
p
=
T
. Each unsteady test began with five to ten startup periods to
allow the system to reach cycle-to-cycle equilibrium. After these initial
cycles, torque and rotation-rate measurements were recorded over 100
successive motion periods. For the shortest motion periods (
T
¼
0.5 s),
200 motion periods were measured. The torque data were filtered and
numerical derivatives of the angular-encoder readings were obtained,
and these data were used to compute temporal means and time-
resolved, phase-averaged profiles. The amplitudes and phases of these
phase-averaged profiles were calculated by means of a fast Fourier
transform. Finally, the aerodynamic torque was inferred via Eq.
(6)
,
where
s
gen
was supplied by the phase-averaged measured torque and
d
x
dt
was computed using a second-order central differencing scheme
and was filtered using a sixth-order Butterworth filter with a cutoff
frequency of 100 Hz to attenuate numerical-differentiation errors.
D. Computing model constants
To compare the experimental results with the model derived in
Sec.
II
, empirical methods were developed to compute the coefficients
of the model. Both the generator constants
K
1
and
K
0
and the aerody-
namic model coefficients
K
‘
and
K
d
were computed from steady
TABLE I.
Combinations of nondimensional amplitude
A
¼
A
=
D
and motion period
T
tested in this study, tabulated with their respective nondimensional surge velocities
u
.
u
(s)
A
¼
0
:
128
A
¼
0
:
257
A
¼
0
:
385
A
¼
0
:
514
T
¼
12
–
–
–
0.039
T
¼
6
–
–
–
0.078
T
¼
3
–
–
–
0.156
T
¼
2
–
0.117
0.175
0.234
T
¼
1
–
0.234
–
–
T
¼
0.5
0.234
–
–
–
FIG. 2.
Surge-velocity waveforms used in these experiments. Sinusoidal profiles
and four types of trapezoidal profiles (parameterized by
n
) were tested.
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torque measurements, without requiring any information from
unsteady tests.
The generator constants
K
1
and
K
0
were obtained by applying a
constant resistive load to the turbine and measuring torque over a
range of wind speeds. From these data, linear fits for the generator tor-
que as a function of rotor speed could be extracted for each loading
condition. The fit coefficients corresponded directly with the generator
constants
K
1
and
K
0
from Eq.
(5)
. The calculated values are reported
in
Table II
above, and, as expected for this type of generator, they
scaled inversely with resistance. The accuracy of the generator-torque
model depends primarily on the linearity of the generator within the
typical operating conditions of the turbine. As evidenced by the data
and linear fits shown in
Fig. 4
, the generator used in this study fulfilled
this condition well (
R
2
>
0
:
999 in all cases).
The same steady torque data were used to compute the aerody-
namic model coefficients
K
‘
and
K
d
. The steady torque data were
plotted as a three-dimensional set of points with respect to the wind
speed and rotor speed, and a second-order polynomial surface fit was
applied to the points. The fit equation gave an analytical expression for
the partial derivatives with respect to
u
1
and
x
0
at any point, from
which the coefficients
K
‘
and
K
d
were calculated by means of the defi-
nitions given in Eqs.
(2)
and
(3)
. The accuracy of the method as a
quasi-steady aerodynamic representation is limited by the topology of
the torque surface and the fidelity of the applied surface fit. For the tur-
bine used in this study, the second-order polynomial surface (shown
in
Fig. 5
) was only weakly quadratic and fit the data with
R
2
>
0
:
999,
so the model was expected to perform well over the range of condi-
tions tested in these experiments.
The moment of inertia of the generator was estimated from man-
ufacturer specifications of the mass and radius of its rotor to be
K
2
¼
6
:
96
10
4
kg m
2
. The moment of inertia of the turbine and
shaft assembly (including the generator),
J
, could be estimated in this
TABLE II.
Performance characteristics and model constants for the six loading conditions investigated in this study. The values of
K
1
and
K
0
were not measured directly for the
7.39
X
case; the values from the 7.48
X
case were used instead.
Resistive load (
X
)
7.39
7.48
9.80
10
20
40
Case identifier
k
0
, sinusoidal cases
5.21
6
0.22
5.34
6
0.22
6.11
6
0.25
6.21
6
0.25
7.72
6
0.31
8.64
6
0.35
k
0
, trapezoidal cases
–
–
6.11
6
0.25
6.27
6
0.26
7.67
6
0.31
8.77
6
0.35
C
p
;
0
, sinusoidal cases
0.261
6
0.013
0.270
6
0.012
0.264
6
0.011
0.288
6
0.013
0.250
6
0.012
0.181
6
0.009
C
p
;
0
, trapezoidal cases
–
–
0.264
6
0.011
0.286
6
0.014
0.249
6
0.012
0.176
6
0.010
K
‘
kg
m
s
, sinusoidal cases
0.458
0.456
0.445
0.444
0.422
0.409
K
‘
kg
m
s
, trapezoidal cases
–
–
0.445
0.443
0.423
0.407
K
d
kg
m
s
, sinusoidal cases
0.026 2
0.026 4
0.027 6
0.027 8
0.030 2
0.031 7
K
d
kg
m
s
, trapezoidal cases
–
–
0.027 6
0.027 9
0.030 1
0.031 9
K
1
ð
kg
m
2
s
Þ
–
0.014 1
0.011 1
0.011 2
0.006 49
0.003 76
K
0
ð
Nm
Þ
–
0.153
0.125
0.119
0.085 0
0.067 6
FIG. 3.
Steady power curve for the turbine used in this study, measured over a
range of resistive loads (3.5–100
X
) and wind speeds (6.05–12.09 ms
1
).
FIG. 4.
Measured torque from steady experiments with different resistive loads,
plotted against rotor speed. The two coefficients from the linear fits (shown as
dashed lines,
R
2
>
0
:
999) correspond to the generator constants
K
1
and
K
0
.
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manner as well. However, given the number of parts and nontrivial
geometries in the assembly, a more empirical approach was taken. A
3D-printed spool with a diameter of 6 cm was attached to the turbine
shaft, and the generator circuit was disconnected. A string was wrapped
around the spool and connected to a weight, suspended from the tur-
bine tower by means of a pulley. The weight was dropped ten times,
and using the average measured torque and a fit based on the measured
rotation-rate signal and the equation of motion of the system, an aver-
age moment of inertia was found to be
J
¼
0
:
026 6
6
0
:
000 8 kg m
2
.
This value aligned with the results of geometric estimates.
Thus, the only information required to obtain first-order predic-
tions of the dynamics of a surging turbine were two moments of iner-
tia and a series of steady torque measurements over a range of rotor
speeds and wind speeds. Using the constants obtained from these pre-
requisites, the amplitude and phase of the aerodynamic and measured
torques could be calculated analytically using the transfer functions
given in Eqs.
(10)
and
(11)
. To predict the time-resolved dynamics of
the turbine for general surge-velocity waveforms, a numerical integra-
tion of the model as an initial-value problem was required. This was
carried out using a fourth-order Runge–Kutta integration scheme with
a time step of 10
3
s over ten motion periods, with the rotor speed ini-
tialized at
x
j
t
¼
0
¼
x
0
. The final period served as a representation of
the steady-state model prediction.
E. Sources of uncertainty
Before the results of the experiments can be discussed in detail,
the nature of the experimental facility requires a consideration of the
sources of uncertainty in the measurements. These sources can be
divided into two types: those that occurred over short time scales rela-
tive to a single test case, and those that occurred over longer time
scales and were thus not captured in the error estimates computed
from each data set.
Sources of uncertainty that occurred over short time scales
contributed to the error bounds that will be shown in the figures in
Sec.
IV
. The standard deviation of the wind speed in the tunnel, mea-
sured over 5 min, was 2.20
6
0.17% of the average wind speed. This
variability includes fluctuations due to turbulence and short-period
fluctuations in the bulk flow velocity due to atmospheric disturbances.
Though external winds were generally stronger during the afternoon,
no corresponding increase in torque or rotation-rate uncertainty was
evident for measurements conducted under these conditions.
Therefore, despite the exposure of the facility to local atmospheric con-
ditions, gusts and pressure fluctuations had a small influence on the
results compared to other factors. Electrical noise from the torque sen-
sor also contributed to measurement uncertainties, though these
effects were reduced by the 100 Hz lowpass filter applied to the raw
torque signal. Finally, a slight misalignment in the turbine shaft was
responsible for small variations in the measured torque within every
full rotation of the turbine. These intracycle variations accounted for
much of the uncertainty on the mean-torque measurements, and were
occasionally visible in the phase-averaged torque and rotation-rate
profiles. However, since their time scales were dictated by the rotor
speed and were thus one to two orders of magnitude faster than the
time scales of the surge oscillations, they did not directly affect the
surgedynamicsthatwerethefocusofthisstudy.
The open-air nature of the facility, however, meant that changing
conditions throughout the day and across multiple days introduced
additional sources of uncertainty that was not explicitly captured in
the error estimates computed directly from each data set.
Measurements of the wind speed at a single location over temperatures
between 24.7
Cand31
:
9
C showed a mild dependence of wind speed
on temperature (
R
2
¼
0
:
535), resulting in a 3.0% overall difference in
wind speed. Given that temperatures in the facility ranged from
16.8
Cto32
:
6
C across all experiments, a linear model would predict
an uncertainty in the wind speed of
6
3
:
2%. However, since no wind-
speed measurements were taken at temperatures below 24
C, the
fidelity of a linear model across the entire range of temperatures could
not be directly confirmed. Because of these uncertainties, no tempera-
ture corrections were applied
a posteriori
to the wind-speed data. This
source of uncertainty complicated comparisons of mean torque and
power between test cases, though this was to some extent ameliorated
by normalizing the mean data from unsteady tests by a temporally
proximal steady case. By contrast, the amplitude and phase depended
primarily on the surge kinematics, which were very repeatable on
account of the precision of the linear actuator, and were thus less
affected by relatively small differences in wind speed.
In addition, the zero offset of the torque sensor exhibited a hys-
teretic dependence on temperature. To test this, the torque sensor was
left in the facility for a period of 28 h, and a voltage measurement was
recorded every 10 min. After the measurements were completed, dur-
ing which the temperature in the facility ranged from 19.5
Cto
33
:
3
C, the difference between the initial and final voltage measure-
ments corresponded to a torque difference of 0.014 Nm. Compared to
the average torque measurements reported in this study, this repre-
sented a total shift of 1.2%–2.9%. Since the torque sensor was zeroed
at the start of every day of experiments, this served as an upper bound
on the uncertainty introduced by the zero-offset drift. This additional
uncertainty again primarily obfuscated the mean measured torque,
without affecting the amplitude and phase.
Finally, the intracycle torque variations described previously
increased in magnitude in the tests conducted during June and July
FIG. 5.
Measured torque from steady experiments with different resistive loads,
plotted against both wind speed and rotor speed. A second-order polynomial sur-
face (
R
2
>
0
:
999) was fitted to the points to facilitate the computation of the linear-
ized sensitivities
K
‘
and
K
d
.
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2021, increasing the reported experimental error of the longest-period
tests (
T
¼
6 and 12 s). Higher temperatures and direct sunlight on the
apparatus during these experiments may have caused the shaft assem-
bly materials to expand, thereby exacerbating the rotational asymme-
tries in shaft friction responsible for the torque variations. This
temperature dependence of friction could have influenced both the
mean torque and power as well as their amplitude.
To demonstrate the cumulative effect of these sources of uncer-
tainty, a series of 13 test cases, composed of six sinusoidal motion pro-
files at two different loading conditions, and a single sinusoidal
motion profile at a third loading condition, was tested twice at differ-
ent times of day. All of these data were included in the results pre-
sented in Sec.
IV
for visual comparison. These thus serve as qualitative
indicators of the influence of environmental conditions and system
hysteresis on the results and analysis. As suggested previously, the
effects of these factors will be most evident in the mean torque and
power measurements, and to a lesser extent in the measured torque
amplitudes.
IV. EXPERIMENTAL RESULTS
Experiments were conducted with 32 different sinusoidal wave-
forms over 6 loading conditions and 42 different trapezoidal wave-
forms over 4 loading conditions and 4 values of the waveform
parameter
n
. As mentioned previously, 13 of the sinusoidal cases were
repeated at different times of day to convey the additional uncertainty
due to changing conditions in the facility: 6 at
k
0
¼
6
:
11, 6 at
k
0
¼
8
:
64, and 1 at
k
0
¼
7
:
72. The data from these experiments are
presented in this section, in terms of torque amplitude, torque phase,
and mean power. A selection of phase-averaged power measurements
and their associated model predictions are provided as
Figs. 12–16
in
Appendix
.
A. Torque amplitude response
Aerodynamic-torque amplitudes, referenced to their respective
surge-velocity amplitudes
U
¼
fA
, are shown for sinusoidal and
trapezoidal waveforms in
Fig. 6
, while similar plots for the generator
(measured) torque are given in
Fig. 7
. The transfer-function magni-
tudes were normalized by the steady torque for each case
s
0
and the
wind speed
u
1
. The aerodynamic torque was predicted by the model to
increase in amplitude and asymptotically approach a maximum value
above
f
1. The generator torque predictions showed behavior
more characteristic of a lowpass filter, where the amplitude decreased
with increased frequency. The data for both the aerodynamic torque
and generator torque showed good agreement with the model predic-
tions for low values of
f
, except for the two lowest tip-speed ratios in
the sinusoidal-waveform cases. Excluding these tip-speed ratios and
the highest three values of
f
, the average relative error between the
model predictions and measured data were 2.79% and 2.13% for
the aerodynamic- and generator-torque amplitudes, respectively. At
the two lowest tip-speed ratios, the turbine was close to stall, a flow
regime not accounted for in the model. Decreasing the resistive load
by 0.1
X
in the steady case caused the turbine to stall completely, slow-
ing to a rotor speed of less than 10 rads
1
. Unsteady tests undertaken
at these two loading conditions with higher amplitudes and frequen-
cies than those shown also caused the turbine to stall completely.
Therefore, it was likely that the downstream surge motions caused the
turbine to experience stall due to a reduction in the incident wind
velocity relative to the turbine, thus decreasing the amplitude of the
torque oscillations. A less severe decrease in amplitude could be seen
at steady tip-speed ratios close to
k
0
6
:
2, which represented the
optimal operating condition for the turbine (cf.
Fig. 3
). At normalized
frequencies approaching
f
¼
1, the amplitude began to drop below the
model prediction. Given the evidence of stall at lower tip-speed ratios, it
is likely that this deviation was an effect of stall onset along a portion of
the turbine blades as well. This conjecture is also in agreement with the
simulations of Tran and Kim,
8
who suggested that large surge motions
at low to intermediate tip-speed ratios can cause stall to occur at the
roots of the turbine blades and propagate radially outwards.
In contrast to the decreases in amplitude observed at low tip-
speed ratios, the amplitudes at higher normalized frequencies (
f
>
1)
FIG. 6.
Aerodynamic-torque amplitude data (markers), compared with model predictions (dashed lines), for (a) sinusoidal and (b) trapezoidal surge wavefo
rms. Markers repre-
sent
n
¼
0.01 (
), 0.25 (
), 0.5 (
) and 1 (
ä
).
Journal of Renewable
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and tip-speed ratios above the optimal value increased above the
model predictions. Increases in amplitude were also correlated with
increasing
n
(i.e., increasing proportions of streamwise acceleration
within each motion cycle) in the trapezoidal-waveform experiments.
The higher tip-speed ratios implied that the local angle of attack along
the turbine blades was lower in these cases, reducing the likelihood
that these increases in amplitude were due to stall phenomena. Since
the model appeared to accurately capture the torque amplitudes at
lower frequencies, it could be hypothesized that these trends were evi-
dence of additional dynamics that became more salient at higher levels
of unsteadiness. The specific nature of these dynamics cannot be iden-
tified definitively in the absence of flow measurements or a higher-
order model for validation, but the present measurements do permit
speculation as to the source of the observed discrepancies with the
model. These considerations will be discussed in Sec.
VB
.
B. Torque phase response
The aerodynamic and generator phase-response data for the
same experimental cases shown in
Figs. 6
and
7
are given in
Figs. 8
and
9
and are compared with model predictions. The generator-
torque phase lagged behind the forcing signal
U
(
t
) across all tested fre-
quencies, while the aerodynamic-torque phase led the forcing signal
for all cases except those at the lowest two tip-speed ratios. Excluding
the lowest two tip-speed ratios, the average difference between the
model predictions and measured data were 3
:
59
and 3
:
21
for the
aerodynamic- and generator-torque phase, respectively. The phase in
the anomalous cases showed an approximately 20
lag relative to the
predictions of the model, suggesting again that the turbine blades were
experiencing the effects of stall under these conditions. The tests con-
ducted at higher tip-speed ratios followed the qualitative trends
FIG. 7.
Generator-torque amplitude data (markers), compared with model predictions (dashed lines), for (a) sinusoidal and (b) trapezoidal surge waveform
s. Markers represent
n
¼
0.01 (
), 0.25 (
), 0.5 (
) and 1 (
ä
).
FIG. 8.
Aerodynamic-torque phase data (markers), compared with model predictions (dashed lines), for (a) sinusoidal and (b) trapezoidal surge waveforms.
Markers represent
n
¼
0.01 (
), 0.25 (
), 0.5 (
) and 1 (
ä
).
Journal of Renewable
and Sustainable Energy
ARTICLE
scitation.org/journal/rse
J. Renewable Sustainable Energy
14
, 013305 (2022); doi: 10.1063/5.0076029
14
, 013305-10
Published under an exclusive license by AIP Publishing
06 October 2023 17:49:11
predicted by the model, though the model slightly overpredicted the
aerodynamic- and generator-torque phase by around 6
at low nor-
malized frequencies. Unlike the torque amplitude, the torque phase
was relatively insensitive to changes in the trapezoidal-waveform
parameter
n
.
C. Mean power
While the amplitude and phase responses of the aerodynamic
and generator torque were nontrivial, the first-order linear model pre-
dicted zero change in the mean torque and power for all surge
motions. The period-averaged measured torque and power, however,
diverged from these predictions as the surge motions became more
pronounced. When plotted against the nondimensional surge velocity
u
and a nondimensional surge acceleration defined as
a
¼
f
2
A
=
u
2
1
D
(cf.
Fig. 10
), the normalized period-averaged measured power
P
=
P
0
diverged from unity at higher levels of unsteadiness. For tip-speed
ratios at or below the optimal operating point, the average power
decreased with increasing surge velocity and acceleration. This was
especially the case for the lowest two tip-speed ratios tested; at higher
surge velocities than those shown, the turbine stalled completely and
the normalized average power dropped nearly to zero. It is thus rea-
sonable to infer that the occurrence of stall along portions of the tur-
bine blades was responsible for the decrease in power observed at
lower tip-speed ratios and higher surge velocities.
For higher tip-speed ratios, the average power increased with
increasing surge velocity and acceleration, with a maximum mea-
sured value of
P
=
P
0
¼
1
:
064
6
0
:
045. The increases appeared
FIG. 9.
Generator-torque phase data (markers), compared with model predictions (dashed lines), for (a) sinusoidal and (b) trapezoidal surge waveforms. Ma
rkers represent
n
¼
0.01 (
), 0.25 (
), 0.5 (
) and 1 (
ä
).
FIG. 10.
Period-averaged measured power
P
, normalized by the reference steady power
P
0
, plotted for all cases against (a) the nondimensional surge velocity
u
and (b) the
nondimensional surge acceleration
a
. Circular markers (
) represent sinusoidal-waveform cases; non-circular markers represent trapezoidal-waveform cases with
n
¼
0
:
01
(
), 0.25 (
), 0.5 (
), and 1 (
ä
). For the sake of clarity, error bars are only plotted for every sixth point.
Journal of Renewable
and Sustainable Energy
ARTICLE
scitation.org/journal/rse
J. Renewable Sustainable Energy
14
, 013305 (2022); doi: 10.1063/5.0076029
14
, 013305-11
Published under an exclusive license by AIP Publishing
06 October 2023 17:49:11