Stochastic modeling of lift and drag dynamics under turbulent conditions

Transcription

1 Stchastic mdeling f lift and drag dynamics under turbulent cnditins Muhammad Ramzan Luhur Matthias Wächter Jachim Peinke FrWind - Center fr Wind Energy Research University f Oldenburg, Germany Abstract In this cntributin we present the develpment f a stchastic mdel f the lift and drag dynamics f an airfil under turbulent inflw cnditins. The measurements have been perfrmed fr an airfil FX 79-W-151A in a clsed wind tunnel f University f Oldenburg. The turbulent inflws were generated using different grids including a fractal grid. The frces were measured using tw strain gauge frce sensrs installed at tw end pints f an airfil in the span-wise directin. The analysis f the measured data is dne fllwing the classical averaging prcedure as well as a nvel stchastic apprach. The stchastic mdeling is dne using measurements with fractal grid having clser characteristics t natural wind. The mdel is based n measurement time series f lift and drag cefficients, using a first-rder stchastic differential equatin called the Langevin equatin. The basic mdel btained thrugh this new apprach is etended t accunt fr the scillatin effects cntained in lift and drag dynamics. The stchastic analysis brings further insight int the high frequency lift and drag dynamics. The results are ptimized using a χ 2 -test n the prbability density functins. The mdeled time series, their prbability density functins and cnditinal prbability density functins shw gd agreement with the actual measurement. The mdel is being develped with the aim t integrate it int a general wind energy cnverter mdel. in every secnd, which impses different risks. The dynamical nature f the wind has a significant impact n the peratin f WECs especially in terms f highly dynamic lads. As ne cnsequence, the behavir f the lift and drag frces acting n rtr blades changes fundamentally cmpared t static, laminar flw. The well-knwn dynamic stall effect can lead t a substantial enlargement f the lift dynamics when fast changes f the angle f attack ccur, cmpare [1]. The aim f this wrk is deeper understanding f lift and drag dynamics especially at high frequencies and the mdeling f these dynamics. Fr this purpse a stchastic lift and drag mdel is being develped with the gal t be integrated int aerdynamic mdels based n BEM thery which then culd be cmbined with wind energy cnverter mdels like FAST [2] r FLEX5 [3]. 2 Eperiments 2.1 Measurements The measurement set up is shwn in fllwing Figure 1. The lift and drag frce measurements Keywrds: Wind tunnel measurements, fractal grid, turbulent inflws, stchastic mdeling, lift dynamics, drag dynamics. 1 Intrductin Wind energy cnverters (WECs) are permanently epsed t turbulent atmspheric flws changing muhammad.ramzan@uni-ldenburg.de matthias.waechter@frwind.de jachim.peinke@frwind.de Figure 1: View f the test sectin. Black arrws shw the psitin f frce sensrs installed n the end pints f vertically munted airfil in the test sectin.

2 have been taken in the wind tunnel f the University f Oldenburg n an airfil FX 79-W-151A having chrd length f 0.2 m. The wind tunnel has a clsed test sectin f 1m wide, 0.8m high and 3m lng. The frces n the airfil were measured at sampling frequency f 1 KHz using tw strain gauge frce sensrs munted n tw end pints f the airfil in span-wise directin. The mean wind velcity was 50 m/s leading t a Reynlds number f Re=700,000. The turbulent inflws were generated using different types f grids including a fractal grid, which lead t a turbulence intensity f 4.6%. Further details f the measurement can be fund in [4]. 2.2 Calculatin f lift and drag cefficient The frce sensrs which were installed at tw end pints f the airfil in span-wise directin measure lift and drag frces directly. The frces perpendicular and parallel t the wind inflw represent the lift and drag frce, respectively. Frm measured frces, the lift and drag cefficients are calculated as [4, 5]: = F L qa = F D qa (1) (2) Where F L is the lift frce, F D the drag frce, q the inflw dynamic pressure and A the area f the airfil. 3 Stchastic mdel 3.1 Basic mdel Since the wind is highly dynamic by nature, the resulting lift and drag frces are dynamic in nature t. T understand this phenmenn n a deeper level a nvel stchastic apprach is applied n the high-frequency dynamics f the system. This new apprach is based n the Langevin prcess used t mdel the lift and drag cefficient. The Langevin equatin [6 8] is a first rder stchastic differential equatin based n drift and diffusin functins cupled with a nise term. It is applied n the time series f bth lift and drag cefficients: dx(t) dt = D (1) (X) + D (2) (X)Γ(t), (3) where Γ(t) is a Gaussian white nise termed as Langevin frce [6] with mean value f Γ(t) = 0 and variance Γ 2 (t) = 2. The D (1) (X) and D (2) (X) are the drift and diffusin functins, als knwn as first and secnd rder Kramers-Myal cefficients fr X(t) which can be estimated frm actual measurement data using the relatin [6, 7]: D (n) (X; α) (n=1,2) = lim τ 0 1 n!τ (X(t + τ) X(t))n X(t)=X;α, (4) where α is the fied angle f attack. In general, results f the direct estimatin frm equatin (4) may suffer frm different surces f errrs, such as finite sampling and additinal measurement nise [6]. Therefre an ptimizatin apprach was applied in rder t btain the drift and diffusin functin estimatin which define the best pssible mdel f the system under investigatin. The apprach is based n χ 2 -test applied in terms f prbability density functins (PDFs) f bth mdel and measurement. T calculate the PDFs f the mdel, an analytical slutin t the Langevin equatin, i.e. the statinary slutin t the crrespnding Fkker Planck equatin [8] is used here. As the Langevin equatin invlves nise term and is randm by nature, it is timecnsuming t btain stable values by numerical simulatin. The statinary slutin t Fkker Plank equatin used t btain the mdel PDF is p (Xmdel ) = N D (2) (X mdel ) ep [ ] Xmdel D (1) (y) D (2) (y) dy (5) where N is a nrmalizatin factr [6, 8]. The equatin epresses that the mdel signal is cmpletely defined by the behavir f drift and diffusin functins. T find ut the best drift and diffusin functin values in an autmatic way, the χ 2 -test was cupled with Inverse Parablic Interplatin algrithm [9] which reads = b 1 2 (b a) 2 [f f(c)] (b c) 2 [f f], (6) (b a)[f f(c)] (b c)[f f] where is the estimated abscissa value f new pint, the a, b and c are the abscissa values f the three randmly selected pints and f, f and f(c) are respective rdinate values f three pints. The algrithm wrks in a way that it discards ne pint after each iteratin and decides fr new set f three pints fr net iteratin like the pint with minimum rdinate value is always in the mid f three pints. Hence, it is the minimum rdinate value which decides fr which furth pint is t be discarded. The algrithm cntinues fr several iteratins until the best value is btained.

3 T check the quality f mdel results, finally the χ 2 value based n PDFs f mdel and measurement is cmpared with the intrinsic standard errr estimated as [10] Ni S Errr =, (7) N T tal where N i is the number f cunts in the i th bin and N T tal the size f the sample. Fr the best quality f the mdel, the χ 2 value is t be in rder r less than the standard errr. 3.2 Etensin t basic mdel The basic mdel reprduces gd matching time series as well as the statinary PDFs f lift and drag cefficients with the actual measurements. Hwever t gain deeper insight int the underlying prcess, the cnditinal PDF p(x(t + τ) X(t)) has t be cnsidered, as it cntains the cmplete infrmatin n the prcess described by equatin (3). In ur case the basic mdel des nt reprduce the cnditinal PDFs satisfactrily fr all time lags τ. T vercme this limitatin, the mdel was etended t a tw-dimensinal embedding f the prcess, hwever that has als cntributed the same results as basic ne-dimensinal mdel. The main challenge was t incrprate the ut f phase (in case f lift and drag) and variable scillatin int the lift and drag mdel. Mrever an amplitude mdulatin f the scillatin (breathing) is als fund alng lift and drag cefficient time series, cmpare Figure 4. Finally, satisfactry results are btained by the fllwing etensin ( ) [ ( k ) ] 2πk S X mdel = X langevin +A sin ep, T k (8) where X langevin is the result btained frm equatin (3), A the cnstant t fi the scillatin amplitude f lift and drag cefficient, k the discrete time variable and T the mst prbable scillatin perid fr lift and drag cefficients crrespnding t the central peak in the pwer spectral density (PSD). The epnential functin in the equatin cntrls the increase and decrease in the scillatin amplitude alng the lift and drag time series, where k is half the length f mst frequent breathing perid ccurring alng the time series, k = (k md k ) and S is described as fllws: { +1, fr (2n)k < k (2n + 1)k S = (9) 1, fr (2n + 1)k < k 2(n + 1)k, where n = 0, 1, 2, Results 4.1 Cmparisn f classical averaging and Langevin apprach The cmparisn f said appraches is presented in fllwing Figure 2. In the classical averaging v ^ < > σ CL D (1) (,AOA)<0 D (1) (,AOA)=0 D (1) (,AOA)> AOA [ ] v ^ < > σ CD D (1) (,AOA)<0 D (1) (,AOA)=0 D (1) (,AOA)> AOA [ ] Figure 2: Static lift and drag cefficient curves with drift functin. The slid line represents the mean lift curve whereas the dashed lines the standard deviatin frm the mean lift curve. The red and green arrws demnstrate the psitive and negative drift functin trying t mve twards black crsses which represent the stable fied pints where the drift functin is zer. Same eplanatin like applies fr drag cefficient curve. prcedure, the static lift and drag curves f an air-

4 fil are defined and determined by taking mean and standard deviatin f lift and drag cefficients ver each angle f attack. The lift and drag cefficient curves are btained by using equatins (1) and (2) fr frce measurements using a fractal grid. The standard deviatin demnstrates the fluctuatin f lift and drag cefficient frm the mean. The drift functin is btained thrugh Langevin apprach using equatin (4) which prvides deeper understanding f the lcal dynamics f the lift and drag cefficient fr each angle f attack. The red arrws indicate a deterministic increase f lift and drag cefficients ver the time, similarly green arrws demnstrate the case ppsite t this. The black crsses represent the stable fied pints matching the usual mean lift and drag curves, where the drift functin is zer. 4.2 Stchastic lift and drag mdel Depending n the dynamic behavir f the lift and drag, the stchastic apprach as eplained in sectin 3 is applied. At an initial stage, the mdel is develped fr measurements with fractal grid. The results presented here are fr the angle f attack AOA = 28 and similar results are btained fr ther AOAs als. The mdel is based n measurement time series measure (t) and measure (t). Frm the measurement time D (1) (C ) 0.10 L D (1) (C ) D 0.00 D (1) (,initial) D (1) (,ptimised) D (1) (,initial) D (1) (,ptimised) (c) D (2) (C ) 0.00 L D (2) ( ) D (2) (,initial) D (2) (,ptimised) D (2) (,initial) D (2) (,ptimised) (d) Figure 3: Drift and diffusin functin fr and fr AOA = 28. Red clr represents the initial estimated values whereas black the ptimized values. Drift functin, Diffusin functin, (c) Drift functin and (d) Diffusin functin. series, the drift D (1) (X, AOA) and diffusin D (2) (X, AOA) functins can be estimated using equatin (4). The drift and diffusin functins presented here in Figure 3 are ptimized using Inverse Parablic Interplatin algrithm equatin (6), which is based n χ 2 -test [9] applied in terms f PDFs f bth mdel and measurement. Fr ptimizatin purpse, the statinary slutins f the Fkker Plank equatin (5) have been used t btain the mdel PDFs in rder t get stable χ 2 value, as the Langevin apprach is randm by nature. This ptimizatin cnsists in a quadratic mdificatin f D (2) (X, AOA), as its estimatin fllwing equatin (4) suffers thrugh finite-sampling deviatins [6]. The ptimized diffusin functin D (2) (X, AOA) fund here, is cnstant in X rather than quadratic. Basing n ptimized drift and diffusin functins, when slved in time, the Langevin equatin (3) can mdel measurement time series measure (t) and measure (t). Up t this basic stage, mdel can satisfactrily reprduce the time series as well as statinary PDFs but fails t reprduce the cnditinal PDFs p(x(t + τ) X(t)) fr all time lags τ as discussed in sectin 3.2. As the etended mdel wrks fr all required cnditins, therefre, we apply and present here all the results frm etended mdel using equatin (8). Figure 4 shws the mdel and measurement time series fr AOA = 28. Bth the mdeled time series fr lift and drag cefficient shw gd agreement (t) (t) measure mdel t[ms] measure mdel t[ms] Figure 4: Ecerpt f lift and drag cefficient time series fr AOA = 28. measure(t) in red and mdel(t) in black. Similarly measure(t) in red and mdel(t) in black.

5 with the actual measurement time series f lift and drag cefficients. Figure 5 shws the results in terms f statinary PDFs fr same AOA = 28. Bth mdel and measurement PDFs shw gd agreement with each ther. PDF[a.u.] PDF[a.u.] measure mdel measure mdel Figure 5: Statinary PDFs f lift and drag cefficient fr AOA = 28. measure in red and mdel in black. Similarly measure in red and mdel in black. A Gaussian PDF is added as a slid line. In additin t statinary PDFs, the cnditinal PDFs p(x(t + τ) X(t)) are crucial t check as they cntain the full infrmatin epressed by equatin (3). The mdel in the eisting etended state reprduces the cnditinal PDFs well fr all time lags τ as the limitatin discussed in sectin 3.2 (i.e. the lift and drag ut f phase variable sinusidal scillatin with breathing behavir alng the lift and drag time series) has been cvered in the frm f equatin (8). Since t is used fr the cntinuus time, whereas ur etended mdel is based n discrete time series, thus it is replaced by k in equatin (8). The results in terms f cnditinal PDFs fr mdel and measurement fr time lag τ = 5, AOA = 28 are shwn in Figure 6 and Figure 7. The mdel cnditinal PDFs fr different time lags fr bth lift and drag cefficient mdel shw gd agreement with the actual measurements. measure (k + 5) mdel (k + 5) measure (k) mdel (k) Figure 6: cnditinal PDFs fr time lag τ = 5, AOA = 28. fr measurement in red, fr mdel in black. PDFs are shwn as cntur lines

6 measure (k + 5) mdel (k + 5) measure (k) mdel (k) Figure 7: cnditinal PDFs fr time lag τ = 5, AOA = 28. fr measurement in red, fr mdel in black. PDFs are shwn as cntur lines. The necessity fr the scillating term in equatin (8) is believed t stem frm the ut f phase scillatin f vrte shedding due t flw separatin ver the suctin side f airfil. T incrprate these additinal effects, the ptimized value f diffusin functin D (2) (X, AOA) fr Langevin equatin (3) is reduced up t abut 67% in case f lift and 77% in case f drag. The scillatin perids fr lift and drag cefficients fr equatin (8) are taken as 32 sample steps and 25.5 sample steps, respectively. These are the mst prbable scillatin perids at the central peak f the PSD. Additinal scillatin perids at higher frequency and lwer energy, d nt have a significant influence n the system. The breathing length is average breathing length ccurring alng the lift and drag cefficient time series. The values used here as breathing length are 510 sampling steps fr lift and 420 sampling steps fr drag cefficient. In this way fr lift k = 255 sampling steps and fr drag k = 210 sampling steps, i.e. half f the breathing length apply fr equatin (8). The value f A is used t fi the apprpriate scillatin amplitude fr mdel time series and is decided n the basis f minimum χ 2 value. In rder t cmpare the behavir f mdel and measurement, the phase-space trajectry f the and fr AOA = 28 has been shwn in fllwing Figure 8. The mdel trajectry seems Measurement Mdel Figure 8: Mdel and measurement phase-space trajectry fr and fr AOA = 28. Measurement in red and mdel in black t have gd agreement with the actual measurements. Obviusly, it is impssible t get eactly the same trajectry like measurements, as the apprach used fr mdeling purpse is stchastic (i.e. randm in nature). In every simulatin, the mdel trajectry adpts different shape due t stchasticity but lks similar in behavir t actual measurements. T check the quality f results, the final btained value f χ 2 is cmpared with intrinsic standard errr (statistical errr) btained thrugh equatin (7). The quantitative cmparisn f χ 2 and statistical errr based n mdel and measurement statinary PDFs gives χ 2 larger than the statistical errr. The current btained values in case f are χ 2 = 0 ± 5 and S Errr = 762 ± whereas fr are χ 2 = 0±6 and S Errr = 796±

7 The values are fr 15 simulatins fr and separately, given in terms f mean and standard deviatin frm mean. Fr best quality χ 2 value shuld be in rder r less than the standard errr, therefre still the mdel needs sme minr crrectins especially in terms f breathing length, value f A and ptimized diffusin functin. 5 Cnclusin Lift and drag measurements were perfrmed in a wind tunnel n an airfil FX 79-W-151A. Static lift curves were determined using classical averaging methd and nvel stchastic apprach. The cmparisn between bth appraches shws that nvel stchastic apprach based n Langevin frmalism brings further insight n and dynamics in terms f drift functin displaying the lcal high-frequency behavir f the and dynamics. Furthermre a stchastic mdel f the and dynamics f an airfil has been develped with an etensin. The basic stchastic mdel reprduced well the statinary PDFs f the lift and drag cefficient, nly the cnditinal PDFs p(x(t+τ) X(t)) were nt satisfactry fr all time lags τ. T vercme the latter limitatin we prpse an etensin t basic stchastic mdel t incrprate the and ut f phase variable scillatin perid and breathing which is mst prbably stemming frm vrte shedding ver the suctin side f the airfil. The mdel in the eisting state with etensin, reprduces gd matching and time series, statinary PDFs as well as cnditinal PDFs p(x(t + τ) X(t)) fr all time lags τ. Hwever, fr further imprvement sme minr crrectins especially in terms f breathing length, amplitude f scillatin A and ptimized diffusin functin may be dne. Additinally, fr best visual matching f mdel time series with measurement, breathing length in equatin(8) culd be taken in randm way. The mdel is being develped with the aim t be integrated int aerdynamic mdels based n BEM thery t estimate the lads acting n wind turbine blades. Thereafter it culd be cmbined with wind energy cnverter mdels like FAST [2] r FLEX5 [3]. discussins. References [1] Wlken-Möhlmann, G.; Knebel, P.; Barth, S.; Peinke, J.: Dynamic lift measurements n a FX79W151A airfil via pressure distributin n the wind tunnel walls. Jurnal f Physics: Cnference Series75 (2007) [2] Jnkman, J.M. and Buhl Jr, M.L.: NWTesign Cde FAST. Natinal Renewable Energy Labratry, Technical Reprt NREL/EL August [3] Øye, S.: FLEX5 User Manual. Danske Techniske Hgskle, Technical Reprt [4] Schneemann, J.; Knebel, P.; Milan, P. and Peinke, J.: Lift measurements in unsteady flw cnditins. EWEC 2010, Warsaw Pland. [5] Luhur, M. R.; Schneemann, J.; Milan, P. and Peinke, J.: Stchastic mdeling f lift dynamics in turbulent inflws. EAWE 2010, NTNU Trndheim Nrway. [6] Gttschall, J. and Peinke, J.: On the definitin and handling f different drift and diffusin estimates. New Jurnal f Physics 10 (2008) (20pp). [7] Friedrich, R.; Peinke, J.;Sahimi, M. and Tabar, M. R. R.: Appraching cmpleityby stchastic methds: Frm bilgical systems t turbulence. Physics Reprts 506 (2011), [8] Risken, H.: The Fkker-Planck Equatin. Springer [9] Numerical Recipes in C. The art f scientific cmputing, secnd editin published by Cambridge University Press 1988, [10] Mrales, A.; Waechter, M. and Peinke, J.: Characterizatin f wind turbulence by higher rder statistics. Submitted t Wind Energy Jurnal, DOI: /we. Acknwledgements We wuld like t thank Jörge Schneemann fr using his measured data and Patrick Milan fr useful