Bischoff, O.

Hofsäss, M.

Kühn, M.

Mann, Jakob

Mikkelsen, Torben

Rettenmeier, A.

Schlipf, D.

Trabucchi, D.

Trujillo, J.

Online Research Database In Technology

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Downloaded from orbit.dtu.dk on: Dec 17, 2017Testing of Frozen Turbulence Hypothesis for Wind Turbine Applications with aScanning LIDAR SystemSchlipf, D.; Trabucchi, D.; Bischoff, O.; Hofsäss, M.; Mann, Jakob; Mikkelsen, Torben Krogh;Rettenmeier, A.; Trujillo, J.; Kühn, M.Published in:Detaled ProgramPublication date:2010Document VersionPublisher's PDF, also known as Version of recordLink back to DTU OrbitCitation (APA):Schlipf, D., Trabucchi, D., Bischoff, O., Hofsäss, M., Mann, J., Mikkelsen, T., ... Kühn, M. (2010). Testing ofFrozen Turbulence Hypothesis for Wind Turbine Applications with a Scanning LIDAR System. In DetaledProgram ISARS.Testing of Frozen Turbulence Hypothesis for Wind Turbine Applicationswith a Scanning LIDAR SystemD. Schlipf1, D. Trabucchi2, O. Bischoff1, M. Hofsäß1, J. Mann 3,T. Mikkelsen3, A. Rettenmeier1, J.J. Trujillo1, M. Kühn11 Endowed Chair of Wind Energy at Universität Stuttgart, Germany2 Politecnico di Milano, Italy3 Risø, Technical University of Denmark, DenmarkABSTRACTTaylor’s frozen turbulence hypothesis is tested in itsapplicability for wind turbine applications. In this re-search full field measurements are performed at a testsite for multi-megawatt wind turbines by means of apulsed LIDAR with a scanning device. The systemis installed at the top of the nacelle of a 5 MW windturbine. It provides simultaneous wind speed, with amaximum sampling rate of 5 Hz, at different stationsparallel to the mean wind. Measurements in a rangebetween 0.4 and 1.6 rotor diameter are performed fol-lowing several two and three dimensional trajectories.The spectral characteristics of measurements takensimultaneously at different separation distances arestudied. The scanning strategy which maximizes thewavenumber region where results are consistent withTaylor’s hypothesis is assessed. The best results areachieved by a horizontal sliding trajectory with validwavenumbers up to 0.125 rad/m.1 INTRODUCTIONThe study of atmospheric turbulence covers a widerange of topics which arises interest in several fieldsof both science and engineering. This research dealswith the investigation of Taylor’s hypothesis over anopen field in the inflow of a 5 MW wind turbine. The as-sumption states that, under certain conditions, the tur-bulent structures move as frozen entities transportedby the mean wind. This allows deriving the spatial pat-tern of turbulence from its temporal description. Thehypothesis of frozen turbulence is relevant for wind en-ergy because it provides a simple and robust model tosimulate the flow of a wind field. It simplifies the use ofremote wind measurements for predictive control andthe study of the wind turbine wake.The upstream flow in front of a wind turbine is notfully conform with the assumptions required by Tay-lor’s hypothesis. However in regards to the intendedapplication, the validity of the frozen turbulence hy-pothesis has to be assessed. This research is in-tended to detect the eddies’ scales and the correlatedfrequency and wavenumber range where Taylor’s hy-pothesis best fits the real turbulence behavior. In or-der to achieve this purpose, a system composed by apulsed LIDAR with an integrated scanning device [1]was set on the nacelle of a 5MW wind turbine andused to check the existence of certain scales of turbu-lent structures which could be considered as frozen.A pulsed LIDAR system gives the advantage of mea-suring simultaneously the line of sight wind speed atdifferent distances and facilitates the study of Taylor’shypothesis by means of a cross-spectral analysis.2 TAYLOR’S HYPOTHESISIn a turbulent air flow with a mean speed ui, the turbu-lent oscillations u′i can be defined as:u′i = ui−ui i = 1,2,3 (1)where ui is the air velocity and i= 1, 2, 3 indicatesrespectively the longitudinal, lateral and vertical di-rection. Most commonly the flow is assumed to beisotropic. In such cases the longitudinal componentis enough to represent the whole field of turbulence.As the white light can be imagined like the superposi-tion of waves characterized by all the different wave-lengths, it is possible to figure turbulence like the su-perposition of several structures, each one character-ized by a wavelength vector λ which describes its spa-tial distribution. In the case of turbulence, the largerstructures are known as eddies, and, according to Tay-lor’s hypothesis [2] they do not evolve and are con-vected by the average wind flow. A consequence ofthis can be seen considering the following exampletaken from [3] and represented in Figure 1.An eddy with a diameter of 100m has a temperaturedifference of 5oC between the front and back side. Thesame eddy, 10 seconds later, is blown downwind at awind speed of 10 m/s. The relative equation which de-scribes this behavior is:dTdt=∂T∂ t + u∂T∂x = 0 ⇒∂T∂ t = −u∂T∂x , (2)where T is the temperature, t the time, x the wind direc-tion and u the mean wind speed. The same conceptcan be extended in all directions to other variables thatcharacterize the flow:∂ξ∂ t =−(u∂ξ∂x + v∂ξ∂y +w∂ξ∂ z), (3)1Figure 1. Taylor’s hypothesis example [3]where ξ stands for a general variable of the flow andu, v, w for the average wind components in x, y, z di-rection, respectively. A simple test of Taylor’s hypothe-sis validity is the comparison of two data series of thelongitudinal turbulence u′A and u′B, taken at two fixedpoints A and B along the wind flow. In such a case,the two measures are expected to be equal and to berelatively shifted by the time required for the wind tocover the separation distance between A and B. Sinceit evaluates the similarity of two series in function oftheir relative lag τ, the correlation coefficient is a suit-able function for the intended purpose. It is the nor-malization of the correlation function R(τ) by means ofthe covariance σ2AB between u′A and u′B, that varies inthe interval [-1,1] approaching 1 as much as two seriesare correlated and reaches its maximum at the lag thatsynchronizes best the two series and therefore corre-sponds to their relative shift.ρAB(τ) = R(τ)σ2AB=E[u′A(t)u′B(t + τ)]E[u′B(t)u′A(t)](4)A deeper characterization can be achieved by meansof the spectral analysis, in particular with the coher-ence γ2AB( f ) and the phase φ ( f ) of the cross energyspectra density SAB( f ):SAB( f ) =∫ +∞−∞RAB(τ)e−i∗2π f τdt= |SAB( f )|e−i∗φAB( f )γ2AB( f ) = |SAB( f )|2SAA( f )SBB( f )φ ( f ) =  SAB( f ).(5)To compute the described functions, it was chosen toperform the Welch periodogram, in order to minimizethe statistical errors. According to Taylor’s hypothe-sis, γ( f )2 should be one for all frequencies, while φ ( f )should be proportional to the delay between uA and uB:dφ ( f )d f= 2πτ0 = 2π xABu(6)3 NACELLE LIDAR MEASUREMENTSThe experimental tests analyzed in this research in-volved a pulsed LIDAR enhanced with a sophisticatedscanning device developed by the Endowed Chair ofWind Energy (SWE). The whole system is mountedat a height of about 102 m, on the nacelle of a 5 MWwind turbine whose rotor diameter D measures 116 m.The turbulence oscillations were first recorded alonga fixed direction (staring mode) at five different Fo-cus Distances (FD) in the range of 0.4-1.2 D. In thisway, the wind speed in a single point might not pro-vide an accurate description of the whole flow whichpasses through the turbine rotor. Because of this rea-son, measurements were performed not only in star-ing mode, but also following different two and threedimensional trajectories centered on the turbine axis(see Figure 2), with reference dimensions comparablewith the rotor diameter D, varying the distances andtheir separation (see Table 1).Figure 2. Trajectories sketchid Trajectory Points Proj. at 1D [D] Time Range [D]h. x v width x height [s] From Step ToA Staring 1x1 0.00x0.00 0.20 0.40 0.20 1.40B Sliding 5x1 0.75x0.00 1.60 0.50 0.25 1.50C Sliding 9x1 0.75x0.00 3.20 0.60 0.20 1.40D Lissajous2Grid 4x3 0.80x0.15 2.08 0.40 0.20 1.20E Lissajous2Grid 6x5 0.50x0.50 5.40 0.40 0.20 1.20F Lissajous2Grid 7x7 0.75x0.75 8.39 0.50 0.25 1.50G Lissajous2Grid 7x7 0.75x0.75 8.29 0.40 0.20 1.20Table 1. Description of the measuring campaigns.4 DATA ANALISIS AND RESULTSThe data series are first separated from low qual-ity records, which are mainly the consequence oflack of particles in the atmosphere or impact of thelaser beam with the rotating blades. Those data canbe filtered with the help of the Carrier-to-Noise-Ratio(CNR). For exemple an inpact with the rotating bladesusually results in a Gaussian distribution below -17dBCNR [dB]Points[-]removed data 27.6%considered data 72.4%-17dB-30 -25 -20 -15 -10 -50200400600800Figure 3. Typical 10min-histogram of CNR values.t [s]yH [m]z H[m]2702752802852902953006 m/s7 m/s8 m/s9 m/s10 m/s11 m/s12 m/s13 m/s14 m/s-50050-40-200204060Figure 4. Example of 3D interpolation of filtered mea-surements. The dots mark the focus points.γ2Coherencek [rad/m]φ[rad]Phase10−3 10−2 10−1 10010−3 10−2 10−1 100-2-1012300.20.40.60.81Figure 5. Coherence and phase between the two clos-est FD of scanning trajectory-C (black) and maximumvalid wavenumber kˆ (dark gray).(see Figure 3). Then, the remaining data is inter-polated on a uniform time grid (see Figure 4). Forone/two dimensional measurements, also a space in-terpolation is required. In both cases, the grids arechosen in order to have the nodes as close as possi-ble to the measured points. The next steps first consistin the classification of the data according to the meanwind speed and the turbulence intensity, secondly incoupling the simultaneous data series in order to per-form the cross-analysis on all the possible pairs.Considering the frequency region limited by fi suchthat γ( f1)2 = 0.9, γ( f2)2 = 0.8 and γ( f3)2 = 0.7, it ispossible to transform the frequency to the wavenum-ber k = 1λ using the longitudinal average wind speed u:ki = 2π fu (7)Since the best result was achieved between the twoclosest FD of scanning trajectory-C (see Figure 5),this one was chosen for a first study of the phase.Using the relation displayed in (5) the theoretical de-lay τ0 was computed from the separation distance xABand the mean wind speed u. Then it was comparedwith the one evaluated from the phase φ . In order tok [rad/m]Delayerror[s]10−3 10−2 10−1 100-6-5-4-3-2-1012Figure 6. Error of the time delay estimation forthe phase between the two closest FD of scanningtrajectory-C(black), tolerance (light gray) and maxi-mum valid wavenumber kˆ (dark gray).k [rad/m]kSu 1u 1(k)10−3 10−2 10−1 10010−410−310−210−1Figure 7. Energy spectra for the closest FD of scan-ning trajectory-C(black), Kolmogorov cascade theory(light gray) and maximum valid wavenumber kˆ (darkgray).find a band of confidence of Taylor’s hypothesis alsofor the phase, the limit of the absolute difference be-tween the expected and the measured delay was setto 1 s . The error shown in Figure 6 is within the tol-erated range up to kˆ= 0.125 rad/m. It is interesting tocompare the results with the energy spectra kSxx. Ac-cording to Kolmogorov cascade theory [4], in turbulentflows the biggest eddies take energy from the mainstream and through an instability process transfer theenergy to smaller eddies until these turbulent struc-tures are no more unstable and vanish converted inheat. The instability process lies in the inertial range,which is the band where the slope of the energy spec-tra density Sxx is k−5/3. Because of the energy transfer,the eddies evolve and, therefore, in this range Taylor’shypothesis can not be considered applicable. In Fig-ure 7, kSxx is represented as well as the theoreticalslope, which in this case is k−5/3 × k = k−2/3. It canbe noticed that the inertial range begins exactly wherethe delay error overtakes the tolerance limit which isthe same wavenumber kˆ at which coherence is near tozero.If the measurements are used e.g. for look-ahead con-trol [5], the data has to be filtered to avoid incorrectcontrol action because of wrong prediction, dependingon the wavenumber kˆ and the mean wind u¯.In Figure 8 the considered signals was filtered (But-terworth, 2nd order) with the corresponding Cut-off-frequency f f = kˆu¯2π = 0.126 Hz .time [s]u i[m/s]u2u10 100 200 300 400 500 6005678Figure 8. Filtered Signals.5 CONCLUSIONThe experimental results of this research confirm Tay-lor’s frozen turbulence hypothesis concerning eddieswithin length scales that are relevant for multi-MWwind turbines. Based on the coherence, the hypothe-sis could be found valid with a 90% accuracy for eddylength scales in the order of two rotor diameters.Furthermore, considering the phase, the beginning ofthe inertial range can be considered as the limit of thefrozen turbulence model confidence band which wasfound at about 0.125 rad/m.ACKNOWLEDGEMENTThis work is part of the research project “Further de-velopment of LIDAR measurements for the offshoretest field” which is funded by the German EnvironmentMinistry under the code number 0327642.REFERENCES[1] Rettenmeier, A., Bischoff, O., Hofsäss, M., Schlipf,D., Trujillo, J. J., and Kühn, M., “Wind field anal-yses using a nacelle-based LIDAR system,” Proc.EWEC, 2010.[2] Taylor, G., “The spectrum of turbulence,” Proceed-ings of the Royal Society of London, 1938.[3] Stull, R., An introduction to boundary layer meteo-rology , Kluwer Academic Publishers, 1989.[4] Pope, S., Turbulent Flows, Cambridge UniversityPress, 2000.[5] Schlipf, D., Schuler, S., Grau, P., Allgöwer, F., andKühn, M., “Look-Ahead Cyclic Pitch Control Us-ing LIDAR,” Proc. Science of Making Torque fromWind , 2010.

Testing of Frozen Turbulence Hypothesis for Wind Turbine Applications with a Scanning LIDAR System

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Testing of Frozen Turbulence Hypothesis for Wind Turbine Applications with a Scanning LIDAR System

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