Actuator forces in CFD: RANS and LES modeling in OpenFOAM

Transcription

1 Home Search Collectons Journals About Contact us My IOPscence Actuator forces n CFD: RANS and LES modelng n OpenFOAM Ths content has been downloaded from IOPscence. Please scroll down to see the full text J. Phys.: Conf. Ser ( Vew the table of contents for ths ssue, or go to the journal homepage for more Download detals: IP Address: Ths content was downloaded on 30/09/2015 at 22:28 Please note that terms and condtons apply.

2 Actuator forces n CFD: RANS and LES modelng n OpenFOAM P. Schto, A. Zasso Department of MechancalEngneerng Poltecnco d Mlano, va La Masa 1, Mlano, Italy E-mal: [email protected] Abstract. Wnd turbne wakes are a very challengng topc for scentfc computatons, but modern CFD frameworks and latest HPC centers allow settng up numercal computatons on the wake nduced by the wnd turbne. The man ssues s that the correct modelng of the wake s related to the correct modelng of the nteracton between the blade and the ncomng flow. The am of the proposed work s to estmate the aerodynamc forces actng on the blades n order to correctly generate the rotor wake applyng equvalent aerodynamc force source on the flow. The defnton of a blade forces s done developng a model able to correctly estmate ths aerodynamc forces as a functon of the local flow seen by the blade durng ts revoluton. Introducton The numercal method s mplemented as an actuator-lne approach n the open-source framework OpenFOAM [1]. The key pont of the problem s the correct defnton of the wnd ncdence angle and wnd speed that s responsble for the load on the blade. Standard approach reles on Blade Element Momentum theory to defne the actual wnd speed and ncdence angle that produces a certan load. Ths approach s useful for structural calculatons, but CFD provdes very detaled flow felds that are useful for a more physcal descrpton of the local wnd. Current AL models calculate the aerodynamc forces generated by a secton of the blade, usng as reference the wnd evaluated n a sngle pont, at some dstance upstream of the rotor plane [2] or located n the regon close to the blade secton [3] [4]. These methods are qute dffused, however the results n terms of wnd turbne power predcton can be mproved; n general the thrust actng on the rotor s reproduced wth a good accuracy. The goal of ths paper s the defnton of a model able to defne the relatonshp between flow characterstcs around a wnd turbne blade secton and the forces actng on the arfol: ths model wll be called Effectve Velocty Model, snce t ams to evaluate from local nflow velocty felds the equvalent undsturbed velocty ncdent to the arfol and then use ths value to correctly nterpolate the arfol tabulated polar curve and defne the forces actng on the blade secton. Effectve Velocty Model As sad n IEA Task 29 fnal report the prncpal varable n ALM mplementaton consst the angle of the angle of attack (AOA) defned as the geometrcal angle between the undsturbed flow drecton and the chord lne and the effectve velocty on the rotor plane, these because the normal and tangental force dstrbuton on the blades are drectly related to ths two varable and transtvely turbne torque, thrust and wake depend on force dstrbuton. Content from ths work may be used under the terms of the Creatve Commons Attrbuton 3.0 lcence. Any further dstrbuton of ths work must mantan attrbuton to the author(s) and the ttle of the work, journal ctaton and DOI. Publshed under lcence by Ltd 1

3 The concept of the Effectve Velocty Model (EVM) s to effectvely use the bg amount of nformaton provded by CFD about the flow around an arfol: calculatng numercally the flow t s possble to evaluate the velocty on any pont located around the wnd turbne blade n order to get the nformaton about the ncomng wnd and therefore t wll be possble to nsert the correct aerodynamc forces n the flow feld usng a method that uses ths nformaton. The prncpal hypothess n EVM development s that t s more effectve to evaluate AOA and effectve velocty on the rotor plane not samplng a sngle pont but through the average of velocty sampled on a segment, and thus on multple ponts, postoned perpendcular to the mean relatve velocty drecton The man ssue about EVM s the defnton of sample segment length and poston and the evaluaton of relatonshp between the generated aerodynamc forces and flow sampled on ths segment. Fgure 1. Identfcaton of the arfol on the rotatng blade and generc arfol on secton A-A The synthess of the EVM s done lookng at a generc 2D plane perpendcular to the rotatng blade secton A-A n fgure 1, where α s the local angle of attack dfferent to AOA for the effect of the nduced velocty by the bound vortex generated by the arfol [5]. In ths plane the actuator lne (AL) collapses n a punctual applcaton of force becomng an ActuatorForce () model of blade secton. Referrng to ths generc secton and choosng as blade profle a generc NACA0012 symmetrc arfol, a seres of numercal smulatons have been conducted for dfferent angles of attack. The am of these smulatons was to set up the EVM n order to make t able to nsert lft and drag forces as close as possble to the ones generated by the arfol profle for every possble blade operatonal angle of attack. Flud dynamc equvalence of NACA profle and ActuatorForce The bass of the AL model s the equvalence of flow velocty feld resultant from physcal nteracton wth blade profle and from the nserton of an equvalent force source n ncompressble statonary Naver-Stokes momentum equatons, reported n equatons 1-2 neglectng turbulence modellng. A NACA0012 symmetrc profle, wth a chord length c=1m was taken for ths analyss. Two smulatons have been carred out, the frst wth a mesh modelled around the NACA 0012 profle (and hereafter named NACA0012) and the second wth a regular and coarse mesh wth force source (that s called ) where the grd s completely Cartesan, wth a mesh sze equal to 1m, that s the length of the profle chord. The total doman extenson for both calculatons s 50x50m, and the arfol s placed n the center of the doman. The smulatons are set-up by performng a 2D set of smulaton on dfferent 2

4 meshes: the NACA0012 mesh has cells, wth 0 grd ponts along the arfol chord and cell sze ncreases wth the dstance from the arfol as can be nferred n fgure 2, whle the mesh has unform cell sze and a total of 2500 cells as can be seen n fgure 3. Steady-state RANS equatons were solved usng a k-ω-sst turbulence model. In the numercal smulatons the boundary velocty magntude V =m/s s fxed, whle the boundary velocty angle α = that s the AOA fxed as boundary condton. On NACA0012 smulatons the force on the arfol was measured by ntegratng the pressure over the arfol surface, see fgure 4, whle n calculatons the force measured n the NACA0012 s drectly ntroduced n the doman. In fgure.5 the force nserted on the cell where the arfol s located, s spltted n ts component F x and F y along x and y axes. ρuu μ U = p U arfol wall = 0 (1) ρuu μ U + f = p f = arfol aerodynamc force (2) Fgure 2. NACA0012 mesh. Fgure 3. mesh. F y F x Fgure 4. mesh around NACA0012 arfol Fgure 5. mesh around nserton force cell, the poston yelds by NACA0012 arfol s reported The frst step n EVM valdaton work s to confrm the effectve equvalence of velocty feld from NACA0012 modellng, V NACA, and analyss, V. The NACA0012 smulaton has been evaluated and taken as benchmark for result, from NACA smulaton the polar curve for ths blade profle s extracted, ths nformaton s then used n smulaton, fxed the angle of attack by boundary condton, whle the correct aerodynamc force s nserted nterpolatng the aerodynamc coeffcents and the correspondence between V NACA, and V s analysed. 3

5 Dstrbuton of force As reported n several prevous works [2] [4] [6] [7] the aerodynamc forces need to be dstrbuted smoothly on several mesh ponts n order to avod sngular behavour. In practce the aerodynamc blade forces are spatally dstrbuted wth a 2-dmensonal normal dstrbuton f x = 1 1 2π 2 ϵ 4 e 2 d2 ϵ 2 (3) Beng d the dstance between cell-centered grd ponts and the ponts where the actuator pont les, the ϵ parameter defnes the shape of normal dstrbuton functon. In ths work the value of ϵ s studed to fnd the narrowest kernel functon that prevents the occurrence of a sngular numercal behavour. Fgure 5. ε value study. Boundary condtons: V = m/s; α =. The wake generated by for ϵ equal to 0.5,1 or 2 tmes the length of computatonal mesh sze s sampled on the two control lnes placed 2 and 5 chord length, equal to 1m, behnd NACA0012 profle and pont, those lne are shown n fgure 5. In Fgure 7 and Fgure 8 the velocty magntude and angle of smulaton wth dfferent ϵ value, V and V are plotted together wth velocty magntude and angle of NACA0012 smulaton V NACA and V NACA.versus lne extenson from upper pont. For smaller ϵ and consequently narrower spatal dstrbuton of force the wake reproduces better the local shape of NACA0012 wake, however t s vsble a spatal oscllaton n the velocty and the angle for ϵ equal to 0.5: the solver s steady state and the smulaton reached convergence, therefore these oscllatons have no physcal motvaton and have consequently only numercal explanaton. To avod ths numercal oscllaton, ϵ = 1 was chosen and ths value s used for all the followng smulatons. 4

6 .5 V [m/s] 9.5 NACA0012 =0.5 =1 = [m] 9 V [deg] [m] Fgure 7. Wnd velocty (V) and angle ( V) sampled on lne A for ϵ=0.5, 1 and 2.1 V [m/s] NACA =0.5 =1 = [m] 9.5 V [deg] [m] Fgure 8. Wnd velocty (V) and angle ( V) sampled on lne B for ϵ=0.5, 1 and 2. Defned the regularzaton kernel extenson a pont to pont comparson between the flow feld calculated usng the geometrcal reproducton of the NACA profle V NACA and the flow feld calculated usng the Actuator Force approach V.s proposed: the result of NACA profle smulaton has been nterpolated for the cells of smulaton, then the error between the two was calculated. The relatve error v; V,computed for every smulaton cell for both magntude and phase, equatons 3-4, s globally small and ncreases only n the regon very close to the arfol as can be observed n Fgure 9 and Fgure. 5

7 error v = (V error V = ( V V NACA ) 3 V NACA ) V NACA V NACA Where V and V s velocty feld magntude and angle computed on the th cell for smulaton, V NACA and V NACA s velocty feld magntude and angle computed by NACA smulaton averaged on the th cell of smulaton. (4) Fgure 9. Magntude absolute relatve error Fgure. Phase absolute relatve error EVM defnton The tunng of EVM s based on the equalty of the flow felds between the smulaton of the full blade arfol and the one related to the. The goal s to defne the sample lne through whch the ncomng velocty s evaluated and then the forces are calculated. The frst step nvolves the defnton of the samplng lne length and ts poston n the computatonal doman relatng to the pont. Through the comparson between the full blade secton case and the smulaton, the analyss focuses on the velocty feld n the regon upwnd of the profle. Fgure 11. Poston of the dfferent samplng lnes n the computatonal doman. The poston of the pont s marked by the crcle 6

8 In Fgure 11 dfferent lnes perpendcular to the undsturbed ncdent wnd, placed a dfferent dstance from the cell-centre of the cell whch the pont belongs and crossng the entre doman, are used to collect data and compare the dfferences on the velocty feld n terms of magntude and angle, plotted n Fgure 12 versus s, lne extenson from upper pont..5.4 NACA NACA V [m/s].1 V [deg] s [m] s [m] Fgure 12. Comparson between the velocty calculated on NACA0012 and smulatons n terms of magntude (left) and angle (rght), detected on the lne placed 1.5 m n front of the pont. In order to establsh the best poston for the samplng lne of the EVM the relatve errors n magntude error v (s, d) and phase error V (s, d) between V NACA (s) and V (s), sampled on lne at dfferent dstance form pont, have been calculated. error v (s, d) = (V s V NACA (s)) V NACA (s) on samplng lne dstance =d error V (s, d) = ( V s V NACA (s)) V NACA (s) on sam plng lne dstance =d The mean error on the lne s then evaluated especally takng nto account the phase error, the tested case shows that a good compromse between the error values and the dstance of the lne, that should be as close as possble to better represent the nstantaneous velocty value nvestng the pont, s reached when the samplng lne s postoned at a dstance equal 1.5chord lengths, as can be deduced from the plot of Fgure (6) 7

9 1 0,1 0,01 0,001 0,0001 0, , , ,5 1 1,5 2 2,5 3 3,5 d mean phase error mean magntude error Fgure 13. Comparson between the NACA0012 and the smulatons on U (magntude and angle) functon of the coordnate s, detected on the lne placed 1.5 m n front of the pont. The second step deals agan wth a comparson between the arfol NACA and the, but the purpose now s the defnton of the length of the EVM samplng lne. Focusng on the chosen lne placed at a dstance of 1.5 m upstream the pont and set a new coordnate q (representatve of the porton of the lne that s consdered), a new examnaton of the two smulatons s done. For each value of q, between 0 and the total length of the lne, the varaton n the ncdence angle Δα s calculated, where α = mean α q α (6) For both the NACA0012 and the smulatons. The relatve dfference between the curves decreases ncreasng the porton of samplng lne consdered coordnate, because a long segment takes nto account also the nformaton gven by the cells that are far from the arfol poston, where the ar s very close to the undsturbed flow. Agan a compromse s needed and through the analyss of the data reported n Fgure 14, t s chosen a wdth of 5 tmes the mesh dmenson for the EVM samplng segment. As can be seen by the mentoned plot for a segment of 5 m the descendant trend of the relatve dfference between NACA0012 and smulatons s clearly vsble. In Fgure 15 the chosen poston of the EVM samplng lne n the computatonal doman respect to the pont s represented. 8

10 V[deg] NACA q [m] V NACA- [deg] V absolute dfference q [m] Fgure 14. Dfference between Δα and absolute dfference between NACA0012 and data. Fgure 15. EVM lne and poston n the computatonal doman. The am of the thrd phase of the EVM constructon s the defnton of the emprcal formula relatng the angle detected by the EVM close to the profle and the undsturbed angle used to calculate and then to apply the aerodynamc forces delvered by the arfol. The dea for ths model s that the mentoned formula s gven n terms of a correcton that must be appled to the angle of α provded by the EVM; consderng that the generc aerodynamc force (lft or drag) s defned as: lft = 1 2 ρu2 C l α c (7) drag = 1 2 ρu2 C d α c (8) 9

11 t s possble to suppose that the angle correcton Δα relaton s dependent on the parameters drectly nfluencng the forces: α = α EVM α = α(c l, C d, c M ) (9) Where α EVM s the calculated angle of attack from EVM, α s the boundary one equal to AOA, C l and C d are the aerodynamc coeffcents of the consdered arfol and c s the chord length, M the mesh sze, so c/m s a non-dmensonal value. To quantfy the Δα correcton a large test campagn was made consderng a large varaton for all the comparng parameters: dfferent test cycles were made to study the nfluence of varyng C l (wthn the cycle), whle C d and c/m were fxed for the consdered cycle tself. Snce the flow s statonary, the valdaton of the EVM just requests to compare the detected α EVM wth the undsturbed angle α, that s known a pror. So here are presented the dfferent tests made: C l varyng n the range from 0 to 2, step 0.2, wth C d fxed once equal to 0, once equal to 0.25, once equal to 0.5; fxed C l = 1, C d = 0, a test cycle was made varyng the profle chord from 0.1 to 1, step 0.1. These tests gve an dea of the nfluence of the parameter c/m, even f t s qute obvous the lnear nfluence of ths parameter, because the forces depend lnearly by c, when C l, C d and U are fxed. Fgure 16. Influence of C l and C d on Δα value. For all tests M was equal to 1 m and the ntal condton for V s horzontally drected, magntude of m/s. Relatng to Fgure 16 the followng evdences are detected and used to defntvely estmate the EVM correcton: Δα(C l ) s a lnear functon; the correcton depends prmarly by C l, but the nfluence of C d s clearly vsble, as an alteraton of the slope of Δα (C l ) lne; the non-dmensonal rato c/m mpact lnearly to the Δα value. Hence ths s the proposed correcton: α = c M C d C l () An mportant observaton s that no correcton s needed for the absolute value of V, snce the presence of profle nduces only a deflecton on the ncomng flow. Studes are made also n ths drecton and confrmed the hypothess. Of course, must be remnded that the Δα correcton prevously presented s nextrcably assocated to the chosen poston for the EVM samplng lne. The fnal step of the tunng of the EVM s a frst valdaton of the ntroduced model. A new test campagn s set-up, consderng a statonary case, V =m/s s keep fxed, whle the boundary velocty

12 angle α sweep from -16 to 16. The tests are made to show the robustness of the model for dfferent angles of attack, consderng also the stall condton. The EVM effectveness s proven by the good agreements between forces generated by dfferent NACA profles and the ones calculated by. Fgure 17 and Fgure 18 show that the model works correctly wth every type of arfol, known polar curves smply, because the Δα correcton s a functon of non-dmensonal coeffcents NACA NACA63.4 Lft [N] 0 Lft [N] [deg] [deg] Drag [N] NACA0012 Drag [N] NACA [deg] [deg] Fgure 17. NACA0012 vs forces wth EVM. Fgure 18. NACA63.4 vs forces wth EVM. Concluson The defnton of an nteracton model for ntroducng the aerodynamc forces usng an actuator force approach has been studed on a two dmensonal model, by comparng the results on a geometrcally reproduced arfol and usng the actuator force approach. The flow feld upwnd and downstream of the profle show a good agreement only f a sutable regularzaton kernel s appled: a small kernel suffers of numercal nstabltes, whle a too large kernel dstrbutes wdely the forces and cannot reproduce the near wake of the profle. An effectve velocty model tuned on lft and drag correcton allows to defne correctly the forces generated on the arfol, and therefore allows to reproduce not only the forces actng on the profle, but s able to generate a wake smlar to the one of a geometrcally reproduced arfol. Acknowledgments The authors would lke to thank Matteo Caccalanza and Luca Bernn for the help n conductng ths research. The author thank also PRACE platform for fundng the numercal analyss wth Project INCOME4WINDFARMS Innovatve Computatonal Methods for Wnd Farms. References [1] The OpenFOAM Foundaton, [2] Wu Y, Porté-Agel F. Large-eddy smulaton of wnd-turbne wakes: Evaluaton of turbne parametrsatons. Boundary - layer meteorology. 2011;138(3): [3] Martnez L, Leonard S, Churchfeld M, Morarty P. A comparson of actuator dsk and actuator lne wnd turbne models and best practces for ther use. In: Amercan Insttute of Aeronautcs and Astronautcs; / [4] Troldborg N. Actuator lne modelng of wnd turbne wakes. [PhD Thess]. Techncal Unversty of Denmark, Lyngby, Denmark; [5] Wen Zhong Shen, Martn O.L, Hansen and Jens Nørkær Sørensen. Determnaton of the Angle of Attack on Rotor Blades..02/we

13 [6] Sorensen JN, Shen. Numercal modelng of wnd turbne wakes.journal of fluds engneerng. 2002;124(2): [7] Ivanell. Numercal computatons of wnd turbne wakes. Trta-MEK