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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 U∞and 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)∝f−2in 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 10−1m 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 U∞in 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=KEω−R0I(4)199

where Vis the voltage drop across the machine poles, R0is the machine internal resistance,200

and201

KE=K−1

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 U∞is 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, U∞is 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)dτ (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=U∞Trev

2RT0(12)331

332

L0

ctip

=T0U∞

2R=U∞Trev

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, U∞is 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∞

ΩM(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 CP–λspace 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(λ−λonset) (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 U∞is the mean ﬂow velocity during the tests. Normal-674

isation of the vertical axis is performed as675

GT=ΓT

¯

T /U∞2(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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