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Abstract and Figures

Free-stream turbulence characteristics play an important role in the mechanisms of power harvesting for wind turbines. Acquisitions of power and thrust from a model wind turbine of diameter 0:18m have been carried out in a wind tunnel for a wide range of turbulent base flows, with varying free-stream turbulence intensity in the range between 3% and 16% and integral time scale spanning from 0:1 to 10 times the turbine rotation period. The results demonstrate that power is significantly affected both by the inflow turbulence scales and its intensity, while thrust is scarcely affcted by free-stream turbulence. Fluctuations in the generated torques are also measured, with their behaviour dominated by the free-stream turbulence scale, and only moderately affected by turbulence intensity. The frequency response of thrust fluctuations has been measured for a selected subset of operating conditions, demonstrating that the turbine thrust is unaffected by high-frequency components in the in flow. Conclusions are drawn on the necessity to match both turbulence intensity and base flow frequency content in wind-tunnel studies if realistic results are to be obtained from small-scale studies.
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The effects of free-stream turbulence on the performance1
of a model wind turbine2
Stefano Gambuzzaa) and Bharathram Ganapathisubramani3
Aerodynamics and Flight Mechanics Group, Faculty of Engineering and Physical Sciences,4
University of Southampton, Southampton, UK, SO17 1BJ5
(Dated: 10 March 2021)6
Free-stream turbulence characteristics play an important role in the mechanisms of7
power harvesting for wind turbines. Acquisitions of power and thrust from a model8
wind turbine of diameter 0.18 m have been carried out in a wind tunnel for a wide9
range of turbulent base flows, with varying free-stream turbulence intensity in the10
range between 3 % and 16 % and integral time scale spanning from 0.1 to 10 times11
the turbine rotation period. The results demonstrate that power is significantly af-12
fected both by the inflow turbulence scales and its intensity, while thrust is scarcely13
affected by free-stream turbulence. Fluctuations in the generated torques are also14
measured, with their behaviour dominated by the free-stream turbulence scale,15
and only moderately affected by turbulence intensity. The frequency response of16
thrust fluctuations has been measured for a selected subset of operating conditions,17
demonstrating that the turbine thrust is unaffected by high-frequency components18
in the inflow. Conclusions are drawn on the necessity to match both turbulence19
intensity and base flow frequency content in wind-tunnel studies if realistic results20
are to be obtained from small-scale studies.21
Keywords: Free-stream turbulence, Model wind turbine, Power harvesting, Bluff-22
body drag23
I. INTRODUCTION24
In recent years, electrical power harvested from wind energy sources has been constantly25
growing, with peaks of 30 % of the United Kingdom power demand being provided from26
wind and tidal turbines. Wind turbines generate power from winds naturally present in27
the atmospheric boundary layer, which are characterized by complex turbulent structures,28
variable both in time and over the rotor swept area. These structures span a large range29
of scales, from larger meso-scale eddies that are generated by the interaction of the atmo-30
spheric boundary layer with the surface to frequent smaller-scale gusts1, and cause large31
fluctuations in power generated by a wind turbine2. The interaction of wind turbines with32
these structures is not straightforward to characterize.33
To study how wind turbine performance figures (namely, the thrust generated or the34
power harvested by the turbine) are affected by different turbulent flows, a wide number35
of experimental studies have been carried out. These range from experiments with porous36
discs3,4, to model-scale turbines in wind tunnels or water channels5–7, to observations on37
real-scale turbines generating usable electrical power8–10; the last are obviously scarcer due38
to inherent difficulties in undertaking rigorous measurements in natural environments.39
The effect of the oncoming flow on drag experienced by bluff bodies has received con-40
siderable attention. Previous studies11 identify the free-stream turbulence intensity as the41
main parameter driving drag variations. For model wind turbines at Reynolds numbers42
in the order of 105and turbulence intensities under 20 %, conflicting results are reported43
in literature: some studies observe consistent reductions in the thrust generated by model44
turbines with increasing turbulence intensity5,7, while other studies report an increase in45
thrust for more turbulent flows6or comparable values of velocity deficit in the near wake46
a)Electronic mail: s.gambuzza@soton.ac.uk
2
(thus, comparable thrust) for different turbulence intensities12. It must be noted that none47
of these studies (and neither does this one) employ a common model wind turbine design,48
and thus different results can, in principle, be expected; however, such large discrepancies49
in the general trends cannot be explained by different designs alone, and some other mech-50
anisms previously overlooked could affect these trends. The nature of these mechanisms is51
not evident from the works currently present in literature: for this reason, we want to high-52
light the necessity for a thorough parametric study on the effects of different turbulence53
parameters on a wind turbine thrust to be undertaken, to ascertain whether previously54
overlooked quantities can explain these conflicting results.55
More consistent results are obtained from the investigation of porous and thin bodies in56
turbulent flows; drawing parallels between these bodies and model-scale turbines is justified57
by the good match between their wakes, especially at low Reynolds numbers and moderate58
turbulence intensities13,14. In particular, ref. 3 obtains, for discs of different porosity, an59
increasing drag with free-stream turbulence intensity, with a smaller effect of free-stream60
integral time scale on the results also present. Similar trends are observed for thin solid61
bodies perpendicular to the free-stream15,16.62
Similarly, there is a strong interest in modelling the effects of free-stream turbulence on63
the power harvested by a wind turbine. For this aspect, some analytical models have been64
proposed in literature; the perhaps simplest of these is presented by ref. 8, where power is65
assumed to increase with the square of free-stream turbulence intensity:66
P(U,Iσ) = 1
2ρ U 3
(1 + 3I2
σ)πR2CP(1)67
for a wind turbine of rotor radius R, operating in a free-stream with speed Uand turbu-68
lence intensity Iσ. This model relies on the basic assumption that the turbine power coeffi-69
cient CPis independent of turbulence, instead being only function of the turbine operating70
regime (tip-speed ratio and Reynolds number). While this might seem a strong assumption,71
with the authors mentioning this is an upper bound for extracted power, some field stud-72
ies on real-scale turbines have elaborated on this model obtaining accurate predictions9,10,73
suggesting this assumption is reasonable for large-scale wind turbines operating in the at-74
mospheric boundary layer.75
Some studies have however highlighted that free-stream frequency content also affects76
power harvesting mechanisms: for instance, ref. 17 reports simulations of harmonic gusts77
on a wind turbine, observing that power harvested from the flow is maximum for gust78
frequencies between 0.05 Hz and 0.7 Hz, and decreases rapidly for higher frequency gusts,79
while lower frequencies do not provide any benefit in terms of generated power; ref. 18 also80
reports that the inflow frequency content modulates a model tidal turbine power output.81
Similarly, ref. 19 models the effects of turbulence on harvested power spectrally by means82
of a transfer function83
φP(f) = G(f)φu(f) (2)84
where φP(f) and φu(f) are the power spectral densities of harvested power and incoming85
velocity respectively, with G(f)'1 at low frequencies and G(f)f2in the inertial86
subrange. As such, this transfer function is heavily skewed towards low-frequency contri-87
butions and their study suggests that a wind turbine will harvest more power from a flow88
where more energy is present as slower fluctuations, or spatially as larger eddies. A similar89
behaviour is observed in ref. 20 for wind turbines of radically different sizes, ranging from90
model-scale to full-scale, with diameters in the order of 101m to 102m, and an analyti-91
cal approach is presented to explain the presence of the 2 slope in the transfer function;92
ref. 21 also reports a similar behaviour with a transfer function favouring low-frequency93
fluctuations, noting how this effect is mediated by the slope of the blade’s lift curve near94
stall.95
Lastly, some papers have shown that intermittency in the base flow velocity can also96
be used to predict torque fluctuations, and thus in principle power fluctuations, of a wind97
turbine subject to a turbulent inflow; ref. 22 reports that the probability distribution of98
3
torque increments can be directly related to base flow intermittency, then showing that some99
synthetic gusts models paired with blade-element method codes can successfully replicate100
these torque fluctuations. This is even more important when considering that the wake101
generated by a wind turbine is also highly intermittent23, and can affect the power harvesting102
mechanisms of downstream turbines in wind farms.103
In the current study, the performances of a model wind turbine will be characterized, in104
the mean sense by observing the thrust and power generated by the turbine, statistically105
by observing the magnitude of power and torque fluctuations, and in Fourier-space by106
presenting the spectra of thrust fluctuations, when subject to different turbulent base flows.107
As the reviewed literature has outlined, turbulence intensity alone might not be the single108
parameter driving changes in the turbine performance figures; nonetheless, this is often109
the sole turbulence parameter presented in many of the studies reviewed. To ascertain110
whether this is a valid assumption, it is necessary to separate the effects of different free-111
stream turbulence characteristics on the turbine performance by operating a turbine in flows112
with similar values of free-stream turbulence intensity and different timescales along with,113
conversely, flows with comparable timescales coupled with different intensities; this is the114
main novelty of this paper, which has been made possible by the usage of an active grid to115
generate bespoke flows. Comparison with similar studies undertaken on aerofoils at similar116
and larger Reynolds numbers will be carried out to determine whether these observations117
will hold at larger Reynolds numbers, and thus for real-scale wind turbines, where research118
on these aspects is scarce.119
The paper is structured as follows: section II includes a description of the wind tunnel,120
the active grid and the model wind turbine used during this study, along with a detailed121
description of the measurement techniques and their calibration procedures, as well as a122
detailed presentation of the data reduction techniques that have been used to obtain mean-123
ingful data from the experimental measurements; section III reports the results obtained124
during the measurements, with the effects of turbulence intensity and scale separately an-125
alyzed on the power and thrust generated by the model turbine, in conjunction with a126
discussion on the validity of these results when compared to previous published works in127
literature; section IV summarises the main findings of this current work.128
II. METHODOLOGY129
A. Facility130
The experiments were carried out in an open-return suction wind tunnel in the University131
of Southampton. This has a rectangular test section with cross-section of 0.9 m ×0.6 m and132
a fetch of 4.5 m.133
An active grid similar in design to the one outlined in ref. 24 has been placed at the inlet134
of the test section to generate turbulent flows. This is composed of an 11 ×7 grid of rods,135
with a mesh spacing of M= 81 mm; a picture of the active grid installed in the wind tunnel136
test section is reported in fig. 1. A Pitot probe has been placed 2.5Mupstream of the137
active grid to measure and set the value of Uin the wind tunnel; the bulk free-stream138
velocity has been kept constant to 8m/s by a PID controller implemented in Matlab139
for all test cases presented in this work. Two sets of diamond wings similar to the ones140
employed in ref. 25 have been installed on the grid; the first set was pierced to reduce the141
maximum grid blockage to 70 %, while the second set was completely solid, for a maximum142
grid blockage of 95%. The active grid has been operated in double-random asynchronous143
mode as described in ref. 26, with both the angular velocity of the rods and the time between144
changes in direction being chosen randomly in a predetermined interval.145
Previous research undertaken in this facility using the same active grid have found tur-146
bulence intensity to decay with a power law having exponent of 1.25 with distance from the147
grid27, for different wing geometries and grid routines, in line with studies carried out in148
similar facilities.28,29
149
4
FIG. 1. The active grid in the wind tunnel, seen from downstream, with the low-blockage wings
installed
B. Model wind turbine150
The model wind turbine used during these test is a speed-controlled, fixed-pitch three151
bladed turbine, whose rotor has been connected to a permanent-magnet brushed DC gen-152
erator. The generator has not been connected to external power sources, with the only153
source of motion being the aerodynamic torque generated by the turbine rotor. The model154
turbine rotor has been placed at a streamwise distance of 36 Mdownstream of the active155
grid, with the rotor hub at the center of the test section. The distance between the rotor156
plane and the active grid has been chosen as a balance between concerns on the maximum157
turbulence intensity the model wind turbine would be subject to, decreasing with increasing158
distance from the active grid26,29, and the necessity of having enough fetch of the test sec-159
tion downstream of the model to allow full wake development not to invalidate the results;160
this location results in roughly 1.6 m available for the wake to develop, corresponding to 9161
rotor diameters.162
The blades installed on the model wind turbine have been 3D-printed in-house with163
a stereolitography printer; these have a tip-radius of 90 mm, for a rotor diameter of164
2R= 0.18 m, resulting in a blockage ratio of 4.7 % in the facility used, computed as the ratio165
between the rotor swept area and the facility cross-sectional area. A NACA 63-418 aerofoil166
has been used along the whole blade, with the chord and twist distributions along the blade167
span reported in fig. 2. The diameter-based Reynolds number of the experiments, based168
on a mean free-stream velocity of 8 m/s is ReD= 9.6×104; the chord-based Reynolds169
number, based on the tip chord and the angular velocity at maximum power generating170
tip-speed, is Rec= 2.6×104.171
The turbine speed is measured by means of a Broadcom AS22 incremental optical rotary172
encoder installed on the turbine shaft, whose output is processed by an Arduino Uno board.173
This consists of a codewheel having a resolution of 360 counts per shaft revolution, acting as174
a reflective optical switch; the encoder outputs two square wave trains having 360 periods175
per physical revolution of the shaft, at a phase angle of 90 electrical degrees from each other.176
The Arduino board determines the shaft angular velocity by counting the number of rising177
5
FIG. 2. Chord and twist distributions along the blade span
edges in one of these two signals over a predefined time interval of 250 ms; as the angular178
position of the shaft is discretised to 1/360 of a revolution, the turbine angular velocity179
is measured as multiples of 0.67 rev/min. Moreover, as the resulting angular velocity is180
obtained as a summation of a 250 ms long signal, fluctuations having a frequency higher181
than 4 Hz are filtered out.182
The rotor angular velocity is set by means of an H-bridge circuit to which the DC machine183
is connected, controlled by a 1.6kHz pulse-width modulation signal generated by the same184
Arduino board; a PID controller implemented by the Arduino actively sets the turbine185
angular velocity by controlling the H-bridge MOSFET switches.186
The mechanical torque generated by the turbine has been estimated from the current187
in the DC machine winding; this is an often-used approach for lab-scale wind turbine188
experiments30,31, where the installation of a rotating torque transducer is challenging due189
to the small dimensions of the turbine. The torque exerted on the DC machine shaft is190
directly proportional to the winding current32; the torque–current relationship has then191
been modelled as192
Qsh =KTI+Qf(ω) (3)193
where Qsh is the mechanical torque to the machine shaft, Iis the current in the generator194
winding, Qfis the torque lost to friction, in this case assumed only function of the rotor195
angular velocity ω, and KTis a proportionality constant. To measure KT, one can take196
advantage of a property of permanent magnet DC machines for which, under negligible197
loading,198
V=KEωR0I(4)199
where Vis the voltage drop across the machine poles, R0is the machine internal resistance,200
and201
KE=K1
T(5)202
when expressed in appropriate units. Measurement of KEhas therefore been carried out203
by operating the DC machine as a motor for different values of V, measuring the machine204
angular velocity and current draw. Similarly, Qf(ω) has been estimated by logging the205
current draw of the machine, disconnected from the turbine rotor, at constant ω. The206
current in the generator winding has been measured by measuring the voltage drop across a207
0.1 Ω shunt resistor, sampled by a 12 bit analog-to-digital converter at a frequency of 4Hz.208
The aerodynamic forces generated by the turbine have been measured with a Mini40 load209
cell, manufactured by ATI, sensitive to forces and torques along and around all axes. The210
sensitivity of the instrument to streamwise loads has been estimated by repeatedly loading211
6
and unloading the load cell with a calibration weight having mass of 0.1 kg, generating a212
force comparable to the one exerted by the turbine on the load cell; from this procedure, a213
sensitivity of 0.5 mN has been found. Calibration of this load cell has been carried out by214
the manufacturer, and no additional calibrations of this device have been undertaken prior215
to the experiments. Drift of the load cell with respect to time has been corrected for by216
taking zero-force readings before and after an acquisition, then assuming a linear drift of the217
balance output with time. The load cell output has been sampled with a dedicated National218
Instruments USB-6212 data acquisition card at a frequency of 10 kHz. The force timeseries219
have been filtered with a digital Butterworth low-pass filter having a cutoff frequency of220
1 kHz; additionally, a sixth-order Butterworth band-stop filter between the frequencies of221
18 Hz and 27 Hz has also been applied to filter out the contributions given by structural222
vibrations of the turbine mast. Defining the Strouhal number St as the reduced frequency223
St =f2R
U
(6)224
where R= 0.09 m is the turbine radius and Uis the free-stream speed, this corresponds225
to a band-stop filter removing spectral contributions between St of 0.4 and 0.6. Observing226
the inflow spectra reported in fig. 6, these frequencies fall in the inertial subrange for all227
test cases here reported; as the angular velocities at which the wind turbine operates are228
always higher than half the filtered frequencies, it can be expected from the data reported229
in refs. 19 and 20 that this band-stop filter will act in the power-law range of the power230
spectra.231
FIG. 3. Frequency response of the ATI Mini40 load cell to an impulse force in the mast direction;
the vertical dashed line denotes the cutoff frequency of the Butterworth low-pass filter
The force signals recorded by the load cell will be used to compute the spectra of thrust232
fluctuations: to determine whether the load cell can adequately measure high-frequency233
force components, the output of the load cell to an impulse in the vertical direction (parallel234
to the turbine mast) has been recorded; this impulse has been applied to the load cell by235
means of a small aluminium rod. The frequency content of the force signal recorded by the236
load cell is plotted in fig. 3 versus both frequency fand reduced frequency St. The load237
cell response is flat for frequencies under 130 Hz, or St = 3, after which it exhibits a more238
irregular response settling to a value approximately 1.5dB lower. The frequency response239
then falls considerably for St larger than 15, or frequencies of 660 Hz and higher, lower than240
the filter cutoff frequency. The frequency content of the thrust measured by the load cell241
has been presented in fig. 23: no meaningful data is observed for Strouhal numbers larger242
than 1 and, as the load cell response is mostly flat in this frequency range, no correction to243
the acquired force signals has been applied.244
Prior to the experiments, the power generated by the wind turbine has been estimated245
with the blade-element method implemented in QBlade33. The aerofoil Cland Cdvalues246
have been estimated with the panel method implemented in Xfoil34, specifying a free-247
stream Reynolds number of 2.4×104and a free-transition coefficient Ncrit of 5; these have248
been reported in fig. 4a. The power generated by the model turbine is presented as the249
7
FIG. 4. (left )Cland Cdfor the NACA 63-418 aerofoil estimated by Xfoil; (right) estimated and
actual power coefficient CPversus tip-speed ratio λfrom BEM simulations and measurements in
laminar free-stream
250
adimensional power coefficient CP, defined as251
CP=Qsh ω
1
2ρ U 3
πR2(7)252
where ωis the turbine angular velocity, Ris the turbine rotor radius, Uis the free-stream253
velocity, ρis the fluid density, Qsh is the mechanical torque generated as estimated from254
eq. (3), and the mechanical power harvested by the turbine is estimated as the product of255
shaft torque and angular velocity; the turbine speed is presented as the tip-speed ratio λ,256
defined as257
λ=ω R
U
(8)258
The comparison between the power coefficient predicted by the BEM simulation and the259
wind tunnel results is reported in fig. 4b. The experimental curve has been obtained in260
the wind tunnel described in section II A, with the active grid removed, in which a baseline261
turbulence intensity of 0.3% is attained. Important differences between the predicted and262
measured turbine CPare visible, with the BEM code overestimating the peak power gen-263
erated by the turbine, and underestimating the CPat higher λ. Nonetheless, this method264
can still provide a reasonable first approximation for the flow around the turbine blades in265
laminar free-stream conditions.266
To visualize the nature of the flow around the turbine blades, fig. 5a reports the starting267
position of the separation bubble on the turbine blade suction side, as a function of the268
tip-speed ratio λand the adimensional position along the blade span r/R, with Rbeing the269
turbine radius. The local angle of attack is reported in fig. 5b, with the angles of 15°and270
25°being highlighted in the plot: these correspond, in the aerofoil polar reported in fig. 4a,271
to the first value of αfor which the Cl(α) curve is no longer linear and the value of αthat272
maximises Clrespectively. The flow around the turbine blades is expected to be separated273
for values of λ < 3, and separation is expected to occur uniformly along the blade.274
C. Hot-wire anemometry275
Turbulent characteristics of the base flows generated with the active grid have been mea-276
sured by means of hot-wire anemometry. Two single-wire probes have been manufactured277
8
FIG. 5. (left ) Position of the tripping point on the suction side of the blades and (right ) local angle
of attack as a function of tip-speed ratio λand spanwise position on the blade r/R
in-house from a 5 µm platinum wire with a sensing length of 1 mm, isolated via copper plat-278
ing of the non-sensing segment of the wire. The probes have been installed on a traverse279
and moved on a plane 36 Mdownstream of the active grid, at the same streamwise location280
as the turbine rotor; the measurements have been carried out in an otherwise empty test281
section.282
The properties of the low turbulence base flows obtained with low blockage grid wings283
have been sampled on a 5 ×2 equally spaced grid on a stream-normal plane, spanning284
vertically from z/2R=0.55 to 0.55, and horizontally from y/2R=0.28 to 0.28; the285
higher-turbulence base flows generated by the higher blockage wings have been measured286
instead on a 3 ×4 grid ranging from z/2R=0.55 to 0.55 and y/2R=0.42 to 0.42.287
The probes have been connected to a constant-temperature anemometer operating at an288
overheat ratio of 0.8; the signal has been conditioned with an analog low-pass filter having289
a cutoff frequency of 9.6 kHz, then sampled at a frequency of 20 kHz with a 16 bit analog-290
to-digital converter integrated in a National Instruments USB-6212 DAQ board. For each291
position, acquisitions have been carried out for 3 min to 6 min, to observe convergence of292
the velocity standard deviation to ±0.5 %. This confidence interval has been computed, for293
each velocity signal acquired, with a bootstrap algorithm as delineated in ref. 35: 100 boot-294
strap replications of the original velocity signal, each consisting of 5000 samples, have been295
extracted, and the uncertainty on the obtained velocity standard deviation is obtained from296
the probability distribution function of the standard deviations of all bootstrap replications.297
The probes have been calibrated against a Pitot tube located at the same streamwise298
distance from the grid. While calibrating, the grid has not been removed from the test299
section, and its wings have been rotated parallel to the incoming flow to result in the smallest300
blockage possible; a turbulence intensity of 3% was measured for this configuration, and its301
effect on the Pitot probe readings has been neglected. To compensate for the hot-wire drift302
over time, pre- and post-calibrations have been carried out, with the probes drift assumed303
linear with time. Additionally, the temperature correction outlined in ref. 36 has been304
applied to the acquired hot-wire signals.305
D. Data reduction306
The effects of turbulence on the turbine performance characteristics have been investi-307
gated by means of a reduced set of parameters, namely the flow turbulence intensity308
Iσ=u0
U(9)309
9
where u0=qu0(t)2is the standard deviation of the fluctuating (i.e. zero-mean) streamwise310
component of velocity u0(t), and Uis the mean streamwise velocity; and the integral time311
scale312
T0=Zτ0
0
ρuu(τ)(10)313
where ρuu(τ) is the autocorrelation factor of u0(t) and τ0is the first zero of ρuu(τ). To314
better estimate the integral time scale for a given velocity timeseries, an ensemble-averaging315
procedure similar to the one described in ref. 37 has been carried out: initially, the full zero-316
mean velocity signal u0(t) has been used to compute a first approximation of T0according to317
eq. (10); the original velocity signal is then divided in Nwwindows, each having a duration of318
200 times the initial T0approximation, with a different autocorrelation factor ρuu(τ) being319
computed for each window; these Nwdistinct autocorrelation factors are then averaged320
to obtain an ensemble-averaged ρuu(τ) which is then used in eq. (10) to obtain the final321
estimate of T0.322
To relate the flow integral time scale to the turbine properties, this last quantity is323
normalised by the rotation period of the wind turbine at peak power harvesting regime324
Trev = 20 ms325
T0=T0
Trev
(11)326
Physically, this is a measure of how many rotations the turbine completes while being327
traversed by a single large scale structure. It is possible to relate this quantity to the more328
commonly used ratios between integral length scale L0and turbine diameter 2Ror turbine329
tip-chord ctip by means of Taylor’s frozen turbulence hypothesis330
L0
2R=T0U
2R=UTrev
2RT0(12)331
332
L0
ctip
=T0U
2R=UTrev
ctip T0(13)333
where U= 8 m/s, R= 90 mm and ctip = 15 mm for this study.334
Both Iσand T0are computed from the instantaneous velocity signals obtained via hot-335
wire anemometry, as described in section II C: these are single-wire probes aligned with the336
horizontal direction, and thus measure the instantaneous velocity in the streamwise-vertical337
plane, perpendicular to the wire, including the zero-mean vertical velocity component. Us-338
ing the fluctuating component of this signal to estimate Iσand T0instead of u0(t) in eqs. (9)339
and (10) does in principle lead to an error in the estimation of those quantities; however,340
given the mean flow is oriented in the direction of u0, we have neglected the effect of this341
on the computed turbulence parameters.342
The flows generated by the active grid are not characterized by a uniform distribution343
of turbulence intensity and integral time scale on the turbine swept area; for this reason,344
the values of these parameters as presented in section III are obtained by a disc-averaging345
process conceptually identical to the definition of rotor-equivalent velocity carried out in346
ref. 9. For no test case, the maximum deviation of turbulence intensity from the mean has347
exceeded 3.5 % of the mean, while integral time scale has never deviated more than 10%348
from the rotor-averaged values.349
In addition to the power coefficient CP, defined in eq. (7), the model turbine performance350
will be analyzed in terms of its torque and thrust coefficients (respectively CQand CT),351
considered functions of the tip-speed ratio λdefined in eq. (8). These are defined as352
CT=T
1
2ρ U 2
πR2(14)
CQ=Qsh
1
2ρ U 2
πR3=CP
λ(15)
10
where ωis the turbine angular velocity, Ris the turbine rotor radius, Uis the average353
free-stream velocity during the tests, ρis the fluid density, Tis the thrust generated by the354
turbine, and Qsh is the mechanical torque generated as estimated from eq. (3). The thrust355
figures measured by the load cell contain both the contributions of the turbine and the drag356
generated by the supporting structure (nacelle and mast). To estimate the fraction of drag357
generated by the turbine rotating parts, the forces acting on the support structure have358
been removed from the acquired forces with the turbine rotor in place. These last ones have359
been measured by mounting a dummy turbine hub without blades on the motor shaft, with360
measurements on the forces generated in this configuration measured for all base flows the361
turbine has been subject to.362
In addition, to quantify the unsteady mechanical load to the generator, we define, given363
a torque timeseries Qsh(t), the unsteady torque coefficient364
C0
Q=Q0
1
2ρ U 2
πR3(16)365
where Q0=qQ0
sh(t)2is the standard deviation of the fluctuating torque timeseries Q0
sh(t).366
Similarly, we define the unsteady power coefficient as367
C0
P=Q0
sh(t)ω0(t)
1
2ρ U 3
πR2(17)368
where ω0(t) is the fluctuating component of angular velocity for a given timeseries. Note369
that the prime symbol in C0
Pdoes not refer to the fluctuating (zero-mean) component of370
the CP(t) generated by the wind turbine, but rather to the component of time-averaged371
power coefficient CPwhich is generated by the coupled fluctuations in the turbine torque372
and angular velocity. As such, C0
Pis not a function of time, and its value is different from373
zero.374
For each base flow, the turbine has operated in a range of angular velocities ωbetween375
1000 rev/min and 4200 rev/min, corresponding to a range of reduced speeds λfrom 1 to376
5; multiple acquisitions of the performance coefficients have been carried out to reduce the377
uncertainty on the measurements.378
The plots presented in section III report the 95 % confidence intervals on the mean val-379
ues of all performance coefficients as a function of λ. The uncertainty εXon the generic380
measured coefficient CXis estimated as381
εX=s2std (x1,1. . . x1,N , x2,1. . . xM,N )
Nind 2
+εinstr2(18)382
where xj,k refers to the k-th sample in the j-th acquired timeseries of CX(λ), std() refers to383
the standard deviation of the set in parentheses, εinstr is the uncertainty introduced by the384
measuring instrument, and Nind refers to the number of statistically independent samples385
in the dataset analyzed.386
To estimate Nind, two instantaneous measurements xi,k and xj,k are considered to be387
statistically independent if these are separated in time by more than δt388
δt = max 2R
U
,2T0(19)389
with T0being the free-stream integral time scale of the particular base flow the turbine is390
subject to.391
For measurements of mechanical power and torque, four distinct time series of angular392
velocity and torque at each operating point of the turbine have been acquired in base flows393
having turbulence intensity lower than 10%, while six time series have been recorded in394
higher-turbulence conditions. The thrust measurements have instead been acquired once395
11
for the data reported in figs. 20 to 22, with an additional, longer acquisition being dedicated396
to the data presented in fig. 23. This was made necessary by the higher standard deviation397
of the torque signal, requiring a larger number of statistically independent samples to bring398
the confidence intervals down to an acceptable level, whereas the uncertainty on the mean399
thrust values is mostly dominated by εinstr as standard deviation of these is small.400
III. RESULTS AND DISCUSSION401
A. Base flows402
The model turbine performance has been measured in 11 base flows, with their main403
characteristics reported in table I. For convenience, the base flows are named with a number404
and a letter, respectively increasing with integral time scale and turbulence intensity; base405
flows with the same number thus have comparable values of T0, while test cases with the406
same letter have approximately the same Iσ. The case named 1A has been used as low-407
turbulence reference, and obtained with the same procedure described in section II C for408
hot-wire calibration. The active grid routine is represented by its nominal Rossby number409
Ro, defined as410
Ro =U
M(20)411
where Ω is the mean angular velocity of the rods, and Mis the grid mesh spacing.412
TABLE I. Base flow characteristics
Name Grid wings Ro Iσ(%) T0L0/2R L0/ctip Legend
1A - - 3.0 0.16 0.14 1.71
2B
Low blockage
2 7.5 1.05 0.93 11.20
2C 5 8.8 1.13 1.00 12.05
3D 10 9.6 1.56 1.39 16.64
4E 20 10.7 3.00 2.67 32.00 9
5F 40 11.3 7.05 6.27 75.20 /
6F 60 11.6 11.75 10.44 125.33 4
2F
High blockage
2 11.5 1.16 1.03 12.37 O
2G 5 13.2 1.14 1.01 12.16
3H 12 14.8 1.59 1.41 16.96
4I 25 16.2 3.19 2.84 34.03
To better appreciate the difference between these conditions, fig. 6 reports the spectra413
of two families of incoming flows: base flows at low turbulence scale (2B, 2C, 2F, and 2G)414
T0have been plotted in fig. 6a, while test cases at constant Iσ(2F, 5F, and 6F) have been415
reported in fig. 6b. From the first figure, it is evident that the flows generated by the active416
grid at low Ro exhibit, at least in the spectral domain, the same characteristics of canonical417
passive-grid generated turbulence, with a flat spectrum in the energy containing subrange418
and an approximately 5/3 slope in the inertial region; the main difference between these419
curves is a uniform shift towards more energetic eddies present at all frequencies, compatible420
with a different turbulence intensity. The flows generated by the active grid operating at421
Ro = 5 also present a small peak in their spectra for a range of frequencies close to the422
angular velocities of the active grid rods; this contribution is not present for test cases 2B423
and 2F, where the active grid has operated at a Rossby number of 2.424
The power spectral densities of the three flows sharing the same Iσ(2F, 5F, and 6F)425
are, on the contrary, quite different: a peak at low frequencies, corresponding to the range426
of frequencies at which the active grid has operated, is present for the two largest scale427
flows, and absent in test case 2F. This results in a different spectral distribution of power428
12
FIG. 6. Spectra of constant T0, increasing Iσflows (left) and constant Iσ, increasing T0flows
(right) with reference 5/3 slope (dashed line)
between these cases, with flows at high scale carrying significantly more energy as low-429
frequency fluctuations, while being markedly less energetic in the inertial subrange. As430
their turbulence intensity is similar, these three flows are extremely well suited to verify431
the validity of the analyses presented by previous literature, such as the models derived432
from ref. 8, where Iσis assumed to be the only parameter driving mean power variations433
regardless of the turbulence frequency content, or the low-pass turbine behaviour delineated434
in refs. 19–21.435
FIG. 7. Autocorrelation coefficient ρuu (τ) for the constant Iσflows 2F, 5F and 6F as a function
of lag τ
The autocorrelation coefficients for these last three flows are also reported in fig. 7: flow 2F436
is seen to behave like a classical grid-generated turbulent flow, rapidly losing correlation with437
itself; in comparison, flows 5F and 6F have a more delayed zero-crossing and afterwards show438
negative values of ρuu(τ). The negative value of the autocorrelation coefficient shows that,439
for small lags, the fluctuating velocity signal u0(t) is similar to its shifted negative u0(t+τ);440
this suggests that these two flows can be seen as quasi-periodic gusts superimposed to a441
turbulent base flow.442
13
B. Mechanical power and torque443
The power generated by the wind turbine has been measured for all base flows reported in444
table I, and for λranging from 1 to 5. The power curves on the CPλspace are self-similar445
between different base flows, as reported in fig. 8 with the parameters used to normalize446
these curves being the maximum power coefficient attained by the model turbine and the447
tip-speed ratio at which highest mechanical torque is generated λpeak; this last parameter448
was chosen as opposed to the value of λthat maximises CPbecause the torque curves are449
characterized by a clearer peak, as it will be seen in fig. 11, as opposed to the large range of450
λfor which a constant CPis attained. Being these curves self-similar, their difference can451
be analyzed by observing how peak CPvaries between base flows.452
FIG. 8. Self-similarity of power curves CP(λ)
FIG. 9. Dependence of maximum power generated on turbulence characteristics, and (dashed line)
parabolic increase in power from ref. 8; note the logarithmic scale of the colour bar. Markers
indicate the base flow as reported in the relevant column of table I.
These peaks have been reported in fig. 9 as functions of turbulence intensity Iσand scale453
T0. From this data, a trend of increasing generated power with turbulence is evident, with454
both Iσand T0proving beneficial from the point of view of harvested power. However,455
the expected parabolic trend with turbulence intensity presented in ref. 8, which is usually456
assumed to hold true for real scale wind turbines, is not present. In fact, for high values of457
14
turbulence, the increase in power observed from the data presented is almost double what458
would be predicted by the model — compare, for instance, the 16 % increase in power yield459
at highest turbulence intensity versus the 8 % predicted by the analytical model. Moreover,460
the power extracted by the turbine from different base flows at highest Iσis constant, while461
a large difference could be expected if the parabolic trend were to hold true.462
FIG. 10. Power curves for constant T0'1.1 (left) and constant Iσ'11.5 % (right)
The effect of turbulence intensity alone can be isolated by analysing the power trends for463
base flows 2B, 2C, 2F and 2G, which share a close value of T0'1.1 and a large range of Iσ
464
from 7.5 % to 13.2 %; the relative power curves have been reported in fig. 10a. The effect465
of turbulence intensity alone is that of a clear increase in the turbine power yield, with466
the curves obtained for flows 2B, 2F and 2G distinctively staggered along the vertical axis,467
for all values of λ; some overlap between the two intermediate flows 2C and 2F is however468
present, despite the important difference in turbulence intensity between these base flows.469
Likewise, the effect of turbulence scale can be observed from the power curves obtained470
with flows 2F, 5F and 6F, sharing a close value of Iσ, reported in fig. 10b . Despite the471
larger uncertainty in the measurements for base flows 5F and 6F, a trend of increasing power472
can be clearly observed connected with larger flow scales, with flow 2F clearly resulting in473
less power extracted by the turbine despite the same turbulence intensity. Unlike the power474
curves obtained in different Iσ, negligible differences in power are observed for low values475
of λ, with the turbulence scale affecting the power curves only at high values of tip-speed476
ratio.477
To more easily tackle the mechanisms that lead to power variations, it is advantageous to478
independently study the effects of turbulence on the individual parameters on which power479
depends. Given a time window with a timeseries of torque and angular velocity, the mean480
power coefficient is481
CP=Q(t)ω(t)
1
2ρ U 3
πR2=(Q+Q0)(ω+ω0)
1
2ρ U 3
πR2=Q ω
1
2ρ U 3
πR2+C0
P(21)482
where C0
Prefers to the fraction of power generated by the combined fluctuations in torque483
and angular velocity, as defined in eq. (17). As such, under constant U, the mean wind484
turbine power output is increased either with an increase in the torque generated at the485
same angular velocity, or with greater angular velocity at which the same torque is attained.486
In addition, a less evident component of power is connected to the coupled fluctuations of487
torque and angular velocity, with this last contribution not necessarily positive, depending488
on the combined fluctuations signs.489
The torque curve obtained for flow 4E is reported in fig. 11 as representative of torque490
curves for all base flows. The curve exhibits a clear peak and is otherwise linear, with the491
peak closely related to the stall angle of attack of the blades, being the tip-speed ratio492
15
FIG. 11. Torque curve CQ(λ) for base flow 4E (Iσ= 10.7 %, T0= 3.00), and (red) estimation of
peak torque position; 95 % confidence intervals on power omitted where smaller than 1% of the
measurement
loosely inversely proportional to the blades’ average angle of attack. The flow around the493
blades is attached for λ > λpeak and otherwise separated, with the slope of the CQcurve494
at high λthen connected to the Cl,α of the aerofoils. This is compatible with the BEM495
simulation results reported in fig. 5, which predicted the turbine blades to stall at a λof 3.496
FIG. 12. Dependence of λpeak on the turbulence characteristics
The position of λpeak as a function of the base flow properties is reported in fig. 12,497
from which it is evident that this aspect can only account for a fraction of power variations498
previously observed, since the increase in λpeak is always smaller than 4%. Note that to499
improve the estimation of λpeak, the torque curves have been interpolated with a second-500
order polynomial with a five-point stencil around the curve maximum, as reported in fig. 11.501
Due to the small magnitudes of these changes, finding a definite trend of this parameter502
with free-stream turbulence properties is not straightforward: the data suggests a simple503
dependence of λpeak on Iσ, increasing linearly until a constant value is reached at turbulence504
intensities greater than 9 %, with flow 2B (Iσ= 11.5 %, T0= 1.16) being an outlier for this505
trend. Alternatively, this trend can be interpreted mostly as an increasing function of T0,506
with constant λpeak attained for T0>3, which would explain the lower value observed507
for flow 2F. While this might be tentatively connected to an earlier onset of stall, this508
16
is not found in literature, with stall angle of attack mostly unaffected, if not delayed, by509
free-stream turbulence38,39.510
FIG. 13. Torque generated by base flows 1A and 3D, normalised to their maximum; 95 % confidence
intervals omitted where smaller than 1 % of the measurement
From the point of view of power production, this leads to the result of an increased CP
511
due to the higher value of λat which this peak is attained. To better visualize this aspect,512
fig. 13 reports the torque curves obtained for the reference base flow and a moderately513
turbulent test case; to isolate the tip-speed ratio shift of torque, the curves are normalised514
to their maximum. It appears that the shift in λpeak is representative of a uniform shift of515
the torque curve towards higher values of λ, which results in a higher power harvested by516
the turbine due to the increased angular velocity at which the shaft torque is exerted. While517
this aspect can explain the initial increase of power for low values of Iσ, its magnitude is518
limited, with the increase in λpeak being always lower than 4% of the baseline value, and519
thus cannot account for the totality of CPvariations observed in fig. 9. Furthermore, with520
λpeak being constant for high turbulence intensities, this phenomenon cannot result in the521
variations in power observed for Iσ>10 %.522
FIG. 14. Dependence of maximum torque on the base flow properties
The maximum torque attained by the turbine also clearly affects the power harvested by523
the model turbine, with fig. 14 reporting the trend of maximum turbine torque as a function524
of the turbulence characteristics. Similar to what was observed for maximum power, the525
17
torque clearly increases with turbulence intensity and the asymptotic trend observed for526
λpeak is not present, thus explaining the differences in power obtained for large values of527
Iσ. Contrarily to what observed for power, peak torque does not appear to be affected by528
T0, with flows 2F, 5F and 6F resulting in similar maximum CQ.529
FIG. 15. Torque coefficient attained for base flows 2F, 5F and 6F (Iσ'11.5 %)
To explain the effect of T0on power, the peak torque might thus not be a parameter530
representative of the full torque curve, and the full curves have to be compared. From the531
curves reported in fig. 15 it can be appreciated that the main difference caused by the base532
flows at constant Iσresides in the torque generated close to the blade stall, with the higher533
T0flows resulting in a less steep torque curve and a more gentle stall, with high torques534
being kept for higher λ, thus leading to an increase in power akin to the one highlighted by535
the data in fig. 13.536
FIG. 16. Dependence of power due to torque and angular velocity fluctuations on the turbulence
characteristics
The last contribution to power comes from the coupled fluctuations in torque and angular537
velocity contained in the term C0
P. This parameter is predominantly affected by free-stream538
turbulence scale, although a smaller, linear dependency on Iσis also found for base flows539
18
with T0<2; for no test cases this component of power amounts to more than 1 % of the540
maximum power generated by the model turbine, thus suggesting this contribution can be541
neglected with respect to the total model turbine power output.542
FIG. 17. Power from fluctuating torque and velocity for base flows 2F, 5F and 6F; all confidence
intervals included as none is smaller than 1 % of the measurements
As reported in fig. 17, this last component of power is highly affected by the turbine543
regime. Low values of λ, corresponding to stalled flow around the blades, result in small544
fluctuations in power, which rapidly increase once the flow around the blades is attached.545
It can however be noticed that the contribution of C0
Pis always positive, corresponding to546
Q0and ω0being of the same sign, as confirmed by field data19.547
From these curves, it can be understood that the main difference between the power548
curves obtained for constant Iσand varying T0is on the different stall behaviour of the549
turbine blades, as reported in fig. 15, with the variations in λpeak and C0
Pbeing negligible550
in comparison.551
FIG. 18. Sample trend of C0
Q/CQfor base flow 4E, and ramp fit (dashed line)
While their effect on mean power generation might be negligible, torque fluctuations552
severely decrease the mean time to failure of full-scale turbines by gearbox fatigue loading,553
one of the preeminent causes of failure in the field40,41. Figure 18 reports the magnitude554
of torque fluctuations as a percentage of the total torque generated for base flow 4E; for555
all base flows investigated the ratio C0
Q/CQcan be reasonably approximated by a ramp556
function,557
19
C0
Q
CQ
(λ) = C1+C2R(λλonset) (22)558
where C1,C2and λonset are fit coefficients, and R(x) is the ramp function, defined as559
R(x) = max(0, x) (23)560
FIG. 19. Dependence of the fit parameters (left to right )C1,C2and λonset on turbulence charac-
teristics
The effect of turbulence intensity and scale on the baseline torque fluctuations C1is561
reported in fig. 19a. While there is a limited effect of turbulence intensity on the value of this562
baseline, it is evident that the main parameter driving the value of C1is the turbulence scale,563
with the three flows at Iσ= 11.5 % resulting in different torque fluctuations, compatibly564
with the results reported in fig. 17 regarding power extracted from torque fluctuations. A565
similar trend is observed for the ramp slope C2reported in fig. 19b, for which a linear566
increase with turbulence intensity is observed for flows at low T0, and the test cases at high567
free-stream turbulence scale exhibiting consistently larger values of this parameter.568
The value of λthat onsets higher torque fluctuations has been reported in fig. 19c; the569
trend this parameter shows with free-stream turbulence is somewhat similar to what ob-570
served for λpeak in fig. 12, with a negligible effect of turbulence scale and a marked linear571
effect of turbulence intensity until a constant value is kept at Iσ>10.5 %.572
To summarize the effects of free-stream turbulence on the power extraction mechanisms,573
power is seen to be positively affected both by flows with large turbulence intensities and574
long integral time scales; these parameters have effects on the turbine power yield of com-575
parable magnitude, suggesting that neither of these aspects can be overlooked in a realistic576
forecasting model.577
In more detail, free-stream turbulence intensity has a general effect of increasing the578
turbine power mostly by increasing the torque the individual blades generate for a given579
regime; this effect is seen for all values of λ, being the power curves obtained for flows580
at different Iσvisibly staggered for all values of tip-speed ratio (see fig. 10a). The trend581
of power harvested with turbulence intensity obtained with this model turbine does not582
follow the customary parabolic behaviour used for large-scale turbines, with the magnitude583
of power increases being considerably larger than predicted, and an unexpectedly constant584
power output for Iσ>12 % observed. To explain this behaviour, it can be assumed that585
the 1 + 3I2
σterm correctly represents the increase in kinetic energy flux through the rotor586
20
swept area, and any additional power is due to an improvement in the blades performance,587
observed as an increase in generated torque in fig. 14. This aspect might not be dependent on588
the low Recof this present study, as it is also found in data published for aerofoils operating589
at higher Reynolds numbers38,39,42, which might suggest that the parabolic model might590
underestimate power increases even for full-scale wind turbines. Free-stream turbulence591
intensity also moderately affects the magnitude of torque fluctuations, albeit in a much592
smaller amount when compared to turbulence scale, as well as the onset tip-speed ratio of593
these fluctuations.594
Similarly, free-stream integral time scale also has an important effect on the power har-595
vesting mechanisms, with longer time scales translating to a higher power yield for the596
same turbulence intensity; this is in line with previous findings19,20 that have delineated597
a low-pass behaviour of the wind turbine with respect to incoming velocity fluctuations.598
The spectra previously reported in fig. 6b at constant Iσ'11.5 % are particularly suited599
to confirm this finding: despite the same total energy from fluctuations, base flows 5F and600
6F present low-frequency contributions, absent in flow 2F, that the wind turbine is able601
to harvest, thus resulting in higher power despite the same turbulence intensity (and thus,602
the same total energy in the incoming flow). This is reversed at higher frequencies, where603
flow 2F exhibits more energy than the other two; this does not result in a larger power604
yield as the turbine is not able to convert those fluctuations into usable power. From the605
data reported in this study and from the analytical modelling presented in ref. 19, it is606
unclear whether further increasing free-stream integral time scale will keep increasing the607
wind turbine power output; data from ref. 17 suggests this might not be the case, with608
increasing T0resulting in values of power closer to the analytical parabolic trend; however,609
the low-pass approach of ref. 19 conversely would suggest that increasingly larger values610
of T0would generate increasingly higher power until an upper bound, to the point where611
all the energy in the inflow is contained in large eddies, which the turbine can convert in612
mechanical power.613
It must be noted that the process of power extraction is slightly dependent on the turbine614
tip-speed ratio λ, being visibly different between low λ, where the blades are operating in615
stalled condition, and high regimes, where the flow around the blades is mostly attached.616
In particular, it appears that the mean power generated by the turbine is insensitive to the617
inflow timescales at low tip-speed ratios, with the power curves resulting from the family of618
flows at constant Iσmatching considerably for stalled flow around the blades. This turbine619
behaviour is unexpected, as previous research in the topic21 finds that the turbine low-pass620
behaviour is not a property of the operating conditions.621
This distinction between low- and high-λbehaviour of the wind turbine is also observed622
as an effect on the values of fractional torque unsteadiness C0
Q/CQ, whose values are mostly623
affected by the free-stream integral time scale. This suggests some independence of the624
blades aerodynamic properties on the turbulence scale in stalled conditions, possibly medi-625
ated by the presence of a large separation bubble around the aerofoils; direct measurements626
of the velocity and pressure field around the blades in stalled and attached flow conditions627
are necessary to investigate the causes of this behaviour.628
C. Thrust629
Turbine thrust has been recorded for λranging from 1 to 5, and for all base flows reported630
in table I.631
The thrust curves obtained in different base flows appear self-similar, as reported in632
fig. 20. Unlike the power and torque curves presented in section III B, thrust generated633
by the turbine does not have a clear peak in the CT-λspace and, as such, identifying the634
parameters to normalize λand CTby is more challenging. In this case, we introduce two635
normalisation coefficients, kλand kT, which minimize the least square difference between636
the reference thrust curve, obtained for base flow 1A, and the curve kTCT(kλλ) obtained637
in turbulence. The trends of these parameters as a function of the base flow properties can638
21
FIG. 20. Self-similarity of thrust curves CT(λ)
describe the behaviour of the thrust generated by the wind turbine in different operating639
conditions, which are reported in fig. 21; by definition, the value of both kλand kTfor test640
case 1A is 1.641
FIG. 21. Trends of kλ(left ) and kT(right ) with turbulence intensity Iσand scale T0
By far, the most evident effect of turbulence on the turbine thrust is the one on kλ,642
which is seen broadly and consistently increase with Iσ, while the effects of T0are smaller:643
conceptually, this results on a steeper drag curve for higher values of Iσand thus a larger644
drag for the same tip-speed ratio. As the effect of T0on kλis marginal, it would be possible645
to define a rotor-equivalent velocity that, similarly to what previously presented for power646
in ref. 8, is only a function of Iσ; however, this velocity would not increase as the square of647
Iσ, instead reaching an upper bound as the inflow turbulence intensity increases.648
Conversely, the effect of Iσon kTappears to be that of a minor increase, possibly con-649
centrated in the range of turbulence intensities between 3 % and 7.5 %, followed by a steady650
decrease, while its values are never exceeding a 2% difference from the low-turbulence651
baseline flow.652
A behaviour similar to the one observed for kTis also seen on the maximum thrust coeffi-653
cient reached by the turbine in different base flows, reported in fig. 22: once again, maximum654
CTdecreases steadily with turbulence intensity in the range of Iσhere presented, and the655
total variations are moderate in size, never reaching more than 2 % or less than 3 % of the656
22
thrust generated in the low-turbulence test case. However, given that the confidence inter-657
vals size is often larger than the difference between consecutive points in fig. 22, it is difficult658
to individuate a clear trend, especially for average values of turbulence intensities between659
7.5 % and 12 %, where a basically constant value of maximum thrust is attained. For values660
of turbulence intensity larger than 12%, however, the trend of decreasing maximum thrust661
is clear.662
FIG. 22. Dependence of maximum thrust generated on base flows
663
664
FIG. 23. Normalised transfer function GTbetween incoming flow and thrust fluctuations for flows
2B (top), 2F (midd le) and 5F (bottom), at tip speed-ratios of λ= 1.9 (left ), 3.8 (center ) and 4.7
(right); additionally, boundary St between energy containing and inertial subranges (red vertical
line) and turbine reduced angular velocity (black vertical line)
To visualize the effects of velocity fluctuations on thrust unsteadiness, fig. 23 reports the665
transfer function ΓTbetween incoming flow and unsteady thrust, defined in the frequency666
23
domain as667
ΓT(f) = φT(f)
φu(f)(24)668
where φT(f) is the power spectral density of the acquired thrust timeseries, as measured669
by the ATI Mini40 load cell, and φu(f) is the spectrum of incoming velocity measured via670
hot-wire anemometry as described previously in section II C.671
The frequency axis is presented as Strouhal number St, defined here as672
St =f2R
U
(25)673
where 2Ris the rotor diameter and Uis the mean flow velocity during the tests. Normal-674
isation of the vertical axis is performed as675
GT=ΓT
¯
T /U2(26)676
where ¯
Tis the time-average thrust for the given tip-speed ratio and base flow, so that GT
677
is adimensional.678
In addition to GT, the plots also report an estimate of the frequency at which the inertial679
subrange begins for a given base flow as a red vertical line, the turbine angular velocity as680
a black vertical line, and the values of GTin the range where the band-stop filter has been681
applied as dotted red; the best-fitting 1 slope line is also reported in dashed black to each682
subplot.683
The transfer functions are presented for flows 2B, 2F, and 5F, thus encompassing low684
and high values of both Iσand T0; the three tip-speed ratios at which the thrust timeseries685
has been measured represent a stalled-blade condition (λ= 1.9, low CTand CP), a peak686
power (λ= 3.8, high CP) and a high CToperating condition (λ= 4.7), thus representing687
synthetically the different regimes the model wind turbine has been subject to during the688
experimental campaign.689
Similarly to the power transfer functions presented by ref. 19, GTis characterized by three690
distinct regions: a first part of the curve at very low Strouhal numbers, where the turbine691
response to turbulence is mostly flat; a second region at intermediate frequencies, where692
the transfer function decreases rapidly following a power law, in this case with exponent693
1; and a third region for high St where the transfer function has visibly lower values,694
up to three orders of magnitude lower than at low frequencies. This overall shape of the695
thrust transfer function appears to be common between all inflow conditions. Additionally,696
peaks at the turbine angular velocity along with its higher order harmonics are present for697
all measurements reported: these can likely be attributed to vibrations generated by the698
turbine rotor during operation, arising from mechanical unbalance of the turbine, and thus699
might not be related to a fluid dynamics phenomenon.700
While the general shape of GTis comparable with previously published spectra of power701
frequency response, there are two main differences between power and thrust spectra,702
namely the slope of the transfer function in its second region, and the extent and bounds703
of these regions. With regards to the local slope, previously published power transfer func-704
tions report a slope of 2, while from the data here reported it is evident that the best705
approximating power law has an exponent of 1: as ref. 20 shows, the 2 slope can be706
derived from the angular kinetic energy conservation equation; as thrust is not related to707
this equation, the 2 slope need not apply here.708
The bounds of these regions are also different from the ones expected for the power fluc-709
tuations transfer function: literature reports the boundary between the first and second710
region to correspond with the boundary between the energy containing and inertial sub-711
ranges, which is reported as a red vertical line in fig. 23. From the data reported in this712
study, it appears that the actual border between region 1 and 2 lies at considerably lower713
values of St for the low-T0flows and at slightly lower values for flow 5F. A similar result is714
24
obtained for the critical frequency fcafter which the turbine becomes insensitive to the in-715
flow velocity fluctuations: ref. 18 reports this value to be approximately double the turbine716
angular velocity, while data reported in this study suggests that this is both unaffected by717
λand always lower than the expected 2 ω; no estimation of the actual value of fccan be718
done from the data here reported as it consistently falls in the range of frequencies removed719
by the band-stop filter.720
While these results are unexpected, they can explain the apparent insensitivity of the721
turbine thrust on the inflow conditions and simultaneously the high sensitivity of extracted722
power on the inflow: as the transfer function drops to low values for smaller frequencies, the723
wind turbine acts as a low-pass filter with a smaller ideal cutoff frequency and thus filters724
out more of the inflow spectral components. It can be assumed that there exists a range725
of frequencies for which the turbine cannot convert inflow fluctuations into thrust but can726
still be used to generate mechanical power.727
IV. CONCLUSIONS AND OUTLOOK728
The thrust, power output, and their unsteadiness generated by a model wind turbine have729
been measured in a wide range of turbulent base flows to identify the trends of these quan-730
tities with respect to turbulence parameters and to confirm whether simplified analytical731
models offer a sufficient degree of accuracy in predicting trends.732
Power generated by the model turbine is seen to increase with turbulence intensity, with733
the magnitude of this increase exceeding what was predicted by ref. 8: a maximum increase734
of up to 16 % from the low-turbulence baseline has been observed, whereas the simple735
quadratic model predicted a maximum power increase of 8 %. This has been justified as736
a dependence of the blades aerodynamic properties such as mean torque generated on the737
free-stream turbulence, thus invalidating the simplifying assumption on which the analytical738
model is based. From the data obtained in this study it is unclear whether this is due739
to the low chord-based Reynolds numbers at which the experiments have been carried740
out; literature however suggests that similar trends hold at higher Reynolds number and a741
similar trend of power with free-stream turbulence intensity might be observed for full-scale742
turbines. Free-stream turbulence scale is also positively affecting the turbine power yield;743
this is in line with previous studies modelling the turbine as a low-pass filter of incoming744
turbulence, better suited at harvesting power from low-frequency fluctuations.745
The ratio between unsteady and mean torque generated by the model turbine can be746
adequately modelled as a ramp function, constant for low tip-speed ratios and linearly747
increasing with λat high regimes. Free-stream turbulence scale has been individuated to748
be the main driving parameter for these fluctuations, with turbulence intensity playing a749
minor role.750
Thrust generated by the model wind turbine is seemingly unaffected by the inflow con-751
ditions, with the CTobserved in turbulence never differing from the one measured in low-752
turbulence conditions by more than 2 %. The spectra of thrust generated by the turbine753
have been related to the incoming flow by means of a transfer function, which highlights754
how high-frequency components in the free-stream are not translated into thrust by the755
turbine. This behaviour exhibits similarities with results previously published in literature756
regarding the power generated by turbines of radically different diameters, where the tur-757
bine is compared to a low-pass filter; for the case of thrust, the turbine response is seen to758
be considerably more skewed towards very-low frequencies than for power, which explains759
the simultaneous high sensitivity of the power harvested and small influence on thrust of760
the free-stream velocity fluctuations. This behaviour is markedly different from what was761
observed in literature for bluff bodies like solid plates or turbine simulators such as porous762
discs, where a clear effect of both turbulence intensity and scale is observed. Results there-763
fore hint at the possibility that these simplified models might not faithfully represent the764
actual phenomena of thrust-generation, and thus studies employing these simpler geometries765
might not obtain physically faithful results.766
25
ACKNOWLEDGMENTS767
This research did not receive any specific grant from funding agencies in the public,768
commercial or not-for-profit sectors. The PhD scholarship for author SG has been provided769
by the University of Southampton.770
DATA AVAILABILITY STATEMENT771
Data published in this article is available from the University of Southampton repository772
at https://doi.org/10.5258/SOTON/D1746773
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