19th Australasian Fluid Mechanics ConferenceMelbourne, Australia8-11 December 2014

Characterization of Turbulence and Deterministic Stresses through Phase-DependentMeasurements in the Near Wake of a Wind Turbine in an Infinite Turbine Array

N.M. Hamilton1, M. Tutkun2 and R.B. Cal1

1Department of Mechanical and Materials EngineeringPortland State University, Portland,Oregon, 97201, USA

2Institute for Energy Technology, Kjeller, Norway

Abstract

A wind tunnel experiment was conducted to investigate the evo-lution of the stress field in the wake of a wind turbine in thefully developed turbine canopy layer. Phase-locked stereo parti-cle image velocimetry measurements were taken in planes par-allel to the turbine rotor and progressing throughout the nearwake. Stresses vary significantly as a function of phase an-gle of the turbine rotor blades. The resupply of kinetic en-ergy to the momentum-deficit area of the wake is accomplishedlargely through the flux term in the mean kinetic energy equa-tion. Phase-dependent contributions to the total flux into thewake indicate that turbulent structures impart periodic increasesin entrainment of high-momentum fluid. Deviations of phase-averaged velocities from total mean values are used to formulatedeterministic stresses, which make smaller overall contributionsto the flux of kinetic energy into the wake.

Introduction

The turbulent wakes of wind turbines has been discussed insome detail for individual devices (e.g. [11, 13]) but only re-cently has the interaction between wakes in large arrays beeninvestigated. Complex flow generated by atmospheric forcingand interaction with rotating blades is still the subject of inter-est in making wind energy more productive and efficient to helpmeet the increasing energy demands worldwide.

Numerical simulations [3, 12] have been a common means ofresearching full wakes and wake interaction in infinite windturbine arrays. The idea of the ‘infinite’ array of wind tur-bines arose from the observation that a periodicity of turbu-lence statistics exists in regularly arranged wind farms beyondthe fourth row of devices [3, 4]. The periodicity in the stream-wise spatial coordinate allows the convective and pressure gra-dient terms to be effectively omitted from the energy balancefor wind turbines.

Phase-related flow effects have been investigated for wind tur-bines by Lignarolo [9] and Hu [7] illuminating the developmentof tip vortices evolving from the blades under various operat-ing conditions. The significance of phase or standing wavesin turbulent flows was formulated through a triple decomposi-tion [10] of velocity measurements. Deterministic stresses havebeen shown to be significant in turbomachinery by Adamczyk[1].

The current work undertakes to decompose velocity measure-ments in planes spanning a wind turbine wake into phase-averaged and deterministic components. The flux of kinetic en-ergy, responsible for much of the resupply of high-momentumflow into the wake [2, 6, 5], it formulated with the phase-averaged and deterministic stresses.

Theory

The mean kinetic energy equation in a wind turbine boundary

layer can be described through a slightly modified set of turbu-lent boundary layer equations.

U j∂ 12U

2i

∂x j=−Ui

ρ∂P∂xi

+uiu j∂Ui∂x j−

∂uiu jUi∂x j

−Fxi (1)

Above, the forcing term Fxi represents the thrust force added tothe flow as power is extracted by the wind turbine device. Inmany studies of wind turbine wakes [2, 6, 5, 8] the unsteadyterm is omitted from the mean kinetic energy budget throughensemble averaging of a large set of random samples in the pe-riod flow of wake. With the infinite array assumption equation(1) reduces to a balance between the flux of kinetic energy, theproduction of turbulence, and the thrust force from the windturbine.

The Reynolds stress tensor is significant to the overall momen-tum balance in the wind turbine boundary layer. It contributesdirectly to the flux of kinetic energy and production of turbu-lence terms (third and fourth terms in the right hand side ofequation (1)) and can account for a relatively large portion ofthe total kinetic energy in the turbine canopy.

The conventional description of the Reynolds stress tensor po-sitions normal stresses along the diagonal of the tensor andsymmetrically distributes the shear terms in off-diagonal posi-tions. In polar-cylindrical coordinates the turbulent stress tensoris written,

uiu j = (ũi−Ui)(ũ j−U j) (2)

=

uxux −uxur −uxuθ−urux urur −uruθ−uθux −uθur uθuθ

where the x, r, and θ subscripts refer to the axial, radial, andazimuthal coordinates, respectfully. In the following discussionand figures, the phase angle and azimuthal coordinate are dis-tinguished as φ and θ, respectively.

The total Reynolds stress denoted in equation (2) is ensembleaveraged over all measurements, regardless of phase angle ofthe turbine rotor. Throughout the following, a capital letter or anoverbar indicates that a quantity has been averaged over time,rendering the quantity independent of phase. Angle bracketsindicate that phase-dependent quantities have been averaged.Making an average of phase-dependent values over all phaseangles yields an approximation of the time-averaged quantity.

Measurements made at particular phase angles of the rotorblades yield a set of phase-averaged deviations from time av-eraged quantities. These deviations can be multiplied and en-semble averaged to acquire deterministic stresses [1, 10].

u′′i u′′j = (Ui−〈ui〉)

(U j−〈u j〉

)(3)

The total flux of kinetic energy can be then decomposed into tur-bulent, phase-dependent, and deterministic contributions. Con-sidering the velocities in a cylindrical coordinate system, the

component of the flux tensor bring high-momentum flow intothe wake can be written as,

Fxr = uxurUx =(〈uxur〉φ +u′′x u′′r

)Ux (4)

Experiment

Measurements of a wind turbine wake in the fully developedcanopy layer were made in the wind tunnel at Portland StateUniversity. The wind tunnel was furnished with a passive grid atthe entrance of the test section to introduce turbulence, remov-able vertical strakes shaped to precondition the boundary layerto in the wind tunnel to more closely match observed the atmo-spheric boundary layer, and semi-porous surface roughness viasmall-diameter chains. Characteristic profiles of the approachflow to the model array are presented in Statistical developmentof wind turbine wakes for the fully developed wind turbine arrayboundary layer characterized via wall-normal-spanwise planesby Hamilton and Cal.

x/D=3

x/D=2.5

x/D=2

x/D=1.5

x/D=1

x/D=0.5

x

θ

r

Figure 1: Schematic of a model wind turbine with and mea-surement planes across the wake. Flow is from left to right.The turbine shown is in the fully developed region of the windturbine array.

Figure 1 shows an example turbine from the current experiment.In the figure measurement locations accessed through 2D−3Cstereographic particle image velocimetry (SPIV) are shown aswell as the polar-cylindrical coordinate system used in the fol-lowing analysis. Measurements were made behind a wind tur-bine in the fully developed region, the fourth row for arraysarranged in a Cartesian grid. Rows of turbines were spaced sixrotor diameters apart (72 cm) and columns separated by 3 rotordiameters (36 cm) hub to hub.

The wind turbine models were fabricated in-house and consistof a hollow steel mast and rotor blades cut from 0.5 mm sheetsteel. The blades of the turbine were given pitch and twist viaa press to ensure uniformity. Each blade was pitched approxi-mately γroot = 22◦ out of the rotor plane at the root of the bladeand had a 7◦ twist from root to tip, resulting in a pitch of γtip15◦at the tip of each blade. The nacelle of each turbine was com-posed of a DC electric motor, loaded with resistive elements toslow the rotation of the turbine blades, allowing each row of tur-bines to be matched to its peak power coefficient. Model windturbines were mounted in steel plates (0.75 cm thick) spanningthe wind tunnel. Figure 2 shows details of the model wind tur-bines.

The position of the rotor blade was located with a Monarch re-mote optical sensor placed outside the wind tunnel and ableto see a small square of reflective tape affixed to one bladeof the rotor. With each pass of the tape the sensor initiateda square wave signal that in turn triggered the SPIV measure-ments. Four positions of the rotor were considered in the exper-iment, φ ∈ [0◦,30◦,60◦,90◦], where 0◦ was one blade oriented

γtip = 15◦

γroot = 22◦

1 cm

0◦

30◦

60◦

90◦

Hubheigh

t=

12cm

0.75 cm

120◦

120◦

D = 12 cm

Figure 2: Schematic of a fully assembled wind turbine modelincluding the mast and a section of the mounting plate. Thephase angles indicated are measured from the position of a sin-gle blade at top-dead-center.

vertically upward, rotating clockwise facing the front of the tur-bine.

Results

Th momentum deficit of the wake of the wind turbine is visi-ble in the contours of the mean axial (streamwise) velocity Uxshown in figure 3. The minimum value of Ux = 0.5 ms−1 oc-curs directly following the nacelle of the model wind turbineat x/D = 0.5. In the conditions of this experiment is it not un-common to see recirculation in this location in an instantaneoussense especially in leading row wind turbines. However, overa large set of samples any negative axial velocities are washedout.

Figure 3: Time averaged streamwise velocity, Ux. The contoursabove show the mean velocity averaged over all measurementsand are independent of phase.

The choice of polar-cylindrical coordinate system is intuitivegiven the geometry and operation of the wind turbines. Whenviewed in this sense, the dominant Reynolds stresses contribut-ing to the energy budget are the normal stress u2x and the shearstress −uxur (figures 4 and 5, respectively). The second orderquantities show the growth of the wake more clearly than themean velocities.

Figure 4: Time averaged Reynolds normal stress composed offluctuations of axial velocity, u2x . The contours above show themean velocity averaged over all measurements and are indepen-dent of phase.

Peak values of u2x occur approximately x/D = 1.5 downstreamfrom the wind turbine rotor similar to previous studies of windturbine wakes [5, 6]. The axial normal stress is on the order of1 m2s−2, more than twice the magnitude of the radial and az-imuthal normal stresses (not shown for brevity). Contributionsto the overall flux of kinetic energy by u2x are small due to gra-dients that are small in the streamwise direction compared tothose along the radial coordinate.

Figure 5: Time averaged Reynolds shear stress composed ofaxial and radial fluctuations of velocity, −uxur. The contoursabove show the mean velocity averaged over all measurementsand are independent of phase.

In many wind energy studies (e.g. [2, 6]) the Reynolds shearstress composed of the streamwise and wall-normal fluctuationsis discussed as its contribution to the flux and production termsof the kinetic energy budget is the most significant. The analogto this term in the polar-cylindrical coordinate system is −uxurwhich describes turbulence involving streamwise variations andthose radially inward or outward from the hub. Figure 5 showsthat the −uxur is positive for most of the near wake, with theexception being directly behind the mast of the wind turbine.The contours of −〈uxur〉30◦Ux and −〈uxur〉60◦Ux show regionsin the near wake where the flux is of greater magnitude thanthe average because the position of the rotor allows the flow tobe advected into the wake. For other phase angles, the bladesdisturb the direct advection into the wake.

The above statistics have been shown in a time-averaged sense.The flux of kinetic energy has been formulated in figure 6 ina phase-averaged sense to demonstrate that there are periodiccontributions to the entrainment of kinetic energy. Typicallythis quantity is discussed as bringing high-momentum flow intothe wake from above the turbine canopy. In the current formu-lation −〈uxur〉φUx shows the flux radially inward or outward.The general trend of the flux of kinetic energy is into the wakefrom outside. Below the nacelle of the wind turbine there are afew areas in which the flux of kinetic energy is away from thecenter of the wake as seen in −〈uxur〉30◦Ux and −〈uxur〉60◦Ux.

Figure 6: Phase-averaged contours of the flux of kinetic energy〈uxur〉φUx.

A further consideration for the flux of kinetic energy is the con-tribution by the deterministic stresses from equation 3, shownin figure 7. Although the turbulent stresses show significantdependence on the orientation of the rotors, the deterministicstresses are approximately two orders of magnitude smaller.The deterministic shear stress−u′′x u′′r demonstrates periodic val-ues in the azimuthal coordinate θ around the rotor diameter. Thecontours in figure 7 represent the wave-like contribution addedto the mean flow field. They are independent of orientation ofthe rotor blades and can be viewed as the underlying periodicityin the wake.

The flux composed of the deterministic stresses is shown infigure 8. The magnitude of this contribution, like the stressesabove, is approximately two orders of magnitude smaller thanthe phase-dependent or time averaged quantities. The magni-tudes of the flux composed with the deterministic stresses indi-cates that the periodic contributions to the total behavior can beneglected at the first order.

Conclusions

z /D

y/D

u ′ ′xu′ ′

x

−0.5 0 0.5

0.6

0.8

1

1.2

1.4

1.6

1

2

3

4

5

6

x 10−3

z /D

y/D

u ′ ′xu′ ′

r

−0.5 0 0.5

0.6

0.8

1

1.2

1.4

1.6

−3

−2

−1

0

1

x 10−3

Figure 7: Normal (top) and shear (bottom) deterministicstresses composed of ensemble averaged deviations betweenphase-averaged and total mean velocities at x/D = 0.5.

Figure 8: Contours representing the contribution of the totalflux by the deterministic stress −u′′x u′′r .

The current experiment made phase-locked 2D−3C stereo PIVmeasurements across the wake of a wind turbine in a fully-developed wind farm. The total mean values of axial velocityshow the momentum deficit area in the near wake quite clearlybut it is in the turbulent stresses that the growth of the wake ismost easily distinguished.

The flux of kinetic energy is composed with the phase-averagedshear stress and compared showing that entrainment of highmomentum flow is predominantly into the center of the wakeradially. Following the rotor blades in certain orientations, thereis flux of kinetic energy out of the wake below hub height.

Deterministic stresses representing the combined deviations ofphase-averaged velocities from the total mean quantities areapproximately two orders of magnitude smaller than turbulentstresses. This difference in magnitude arises from the natureof the flow forcing the passage of the rotors, as opposed toother turbomachinery where the flow is forced by the blades.Although the deterministic contribution to the flux of kineticenergy is smaller in magnitude, there is some theoretical in-terest arising from azimuthal periodicity at the rotor diameter.That the deterministic stresses make a small contribution to theoverall behavior suggests that they may merit further investiga-tion. Alternate decompositions directly quantifying deviationsof phase-locked stresses from time-averaged values may be ofmore interest in terms of the flux of kinetic energy.

An experiment detailing the evolution of turbulence statisticsthroughout the near and far wake for an identical experimen-tal arrangement can be found in Statistical development of windturbine wakes for the fully developed wind turbine array bound-ary layer characterized via wall-normal-spanwise planes byHamilton, Tutkun, and Cal, presented in the current conference.

References

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