A Comparative Study of Numerical Schemes and Turbulence Models for Wind Turbine Aerodynamics Modelling

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1 WIND ENGINEERING VOLUME 8, NO., 4 PP A Comparatve Study of Numercal Schemes and Turbulence Models for Wnd Turbne Aerodynamcs Modellng Catherne A. Baxevanou and Ncolas S. Vlachos Department of Mechancal and Industral Engneerng Unversty of Thessaly - Athens Avenue, 84 Volos, Greece emal <cbaxe@me.uth.gr> Correspondng author, emal <vlachos@me.uth.gr>, tel , fax ABSTRACT Ths paper s a comparatve study of combnng turbulence models and nterpolaton schemes to calculate turbulent flow around a NACA arfol before and after separaton. The calculatons were carred out usng the code CAFFA of Perc, whch was approprately modfed to nclude more numercal schemes and turbulence models. Ths code solves the Naver-Stoes equatons for D ncompressble flow, usng fnte volumes and structured, collocated, curvlnear, body ftted grds. Seven dfferencng schemes were nvestgated: central, upwnd, hybrd, QUICK, Harten-Yee upwnd TVD wth fve lmters, Roe-Sweby upwnd TVD wth three lmters, and Davs-Yee symmetrc TVD wth three lmters. Turbulence effects were ncorporated usng four turbulence models: standard -ε, -ω hgh Re wth wall functons, -ω hgh Re wth ntegraton up to the wall, and the -ω low Re model. A parametrc study showed that best results are obtaned: a) for the -ε model, when usng the Harten-Yee upwnd TVD scheme for the veloctes and the upwnd nterpolaton for the turbulence propertes and ε, and b) for the -ω models, when usng the Harten-Yee upwnd TVD scheme wth dfferent lmters for the veloctes and the turbulence quanttes and ω. The turbulence models that ntegrate up to the wall are more accurate when separaton appears, whle those usng wall functons converge much faster. Keywords: CFD, turbulence models, numercal schemes, arfol flow, separaton, wnd turbne aerodynamcs, stall. INTRODUCTION The long-term obectve of ths wor s to develop an approprate numercal model for unsteady turbulent flows around wnd turbne blades at stall condtons. Usually the flow around wnd turbne blades s turbulent, wth the whole or part of the blade very often operatng under stall condtons. Ths s, due ether to sudden changes of the local wnd or to desgn to control the wnd turbne power output. Under such condtons, the aerodynamc behavour of the blade s of partcular mportance because of the appearance of dynamc stall flutter, for whch numercal predcton becomes dffcult because of non-lnear characterstcs. A number of researchers have made both numercal and expermental studes of arfol stall and ts mplcatons for the dynamcs of wnd turbne blades. For example, Eaternars and Platzer [] have revewed extensvely the most common numercal schemes and the turbulence and transton models, as used for the predcton of arfol dynamc stall. They compared the man computatonal results wth avalable expermental measurements. The effects of the aerodynamc loadng on blade dynamcs have been dscussed by, among

2 76 A COMPARATIVE STUDY OF NUMERICAL SCHEMES AND TURBULENCE MODELS others, Chavaropoulos [], who studed the flap lead-lag aeroelastc stablty of wnd turbne blades. Voutsnas and Rzots [] have developed a vscous-nvscd nteracton model for the smulaton of dynamc stall on arfols. The performance of varous numercal schemes and turbulence models have been studed by, for example, Kral [4], who evaluated turbulence models for compressble flows n arcraft applcatons, such as the Baldwn-Lomax and Thomas algebrac models, the Baldwn-Barth and Spalart-Almaras one-equaton models, fve low Re models (JL, CH, SP, HC and SO) and the Menter SST blended -ε/ω model. He used a zonal, upwnd, mplct-factored algorthm to solve both the mean flow and turbulence equatons. The equatons were dscretzed usng a cell-centred, fnte-volume representaton and a generalzed form of Roe s upwnd fluxdfference splttng technque wth an optonal TVD operator. Zngg et al [5] evaluated several dscretzaton schemes for subsonc and transonc flows over arfols n order to compare ther accuracy. The dscretzaton for the nvscd fluxes ncluded: a) second-order-accurate centred dfferences wth thrd-order matrx numercal dsspaton, b) second-order convectve upstream splt scheme (CUSP), c) thrd-order upwnd-based dfferencng wth Roe s fluxdfference splttng, and d) fourth-order centred dfferences wth thrd-order matrx numercal dsspaton. The effects of turbulence were modelled usng the Baldwn-Lomax model, and a far-feld crculaton correcton at the outer boundary. In the present study, varous turbulence models and nterpolaton schemes are combned and compared for ther ablty to calculate steady-state turbulent flow around a NACA arfol, before and after flow separaton. The code CAFFA, developed by Perc [6] and avalable on the Internet, was modfed approprately and developed further to nclude more numercal schemes and turbulence models [7]. These are compared, based on ther accuracy and rate of convergence. The numercal results are also compared wth the experments of Bragg & Khodadoust [8].. FLOW GOVERNING EQUATIONS AND TURBULENCE MODEL. Flow governng equatons The governng Naver-Stoes equatons of the present flow, whch s consdered ncompressble, may be expressed as follows: Contnuty equaton U () Momentum equaton U ρ ρ U P U µ µ t t ( ) U () where, the turbulent vscosty, µ t, depends on the flow and s calculated from the specfc turbulence model.. Turbulence models The effects of turbulence on the flow were mplemented va the followng turbulence models: a) Hgh Re -ε model (standard) It s the most wdely used model but t cannot predct flow separaton accurately because t nether ntegrates up to the wall nor does t account for modfcaton of turbulence dsspaton due to an adverse pressure gradent [6].

3 WIND ENGINEERING VOLUME 8, NO., 77 U ρ ρ U ( µ µ t ) τ ρε t () ρ ε ε ε ρ µ µ U ( t ) t ε ρc ε C ε ε P (4) where, the turbulent vscosty s: µ t ρc µ ε (5) and C µ.9, σ κ, C ε.44, C ε.96, σ ε. are model constants. b) Hgh Re -ω model (wth wall functons) As wth the standard model, ths model does not ntegrate up to the wall. However, t accounts for modfcaton of turbulence dsspaton due to adverse pressure gradent [9] and t s, therefore, more sutable for arfol flow predctons. ε ω (6) β * ( ) ρ ρ U µ µ σ * U * t τ β ρω t x x (7) ρ ω ρu t ω ω ( µ µ σ α ω t ) τ x U βρω (8) where, µ t ρ ω (9) and, α 5/9, β /4, β* 9/, σ / and σ*/ c) Hgh Re -ω model (wth wall condtons) Ths model uses the same equatons as the prevous model. However, t does ntegrate up to the wall provdng a specfc value of ω on the boundary [9]. It s, therefore, more sutable for flow separaton calculatons. τ ωw u S R ν () where, ( ) 5, S R R, R R R < 5 5 () u R R τ ν where, R wall roughness () d) Low Re -ω model Ths model ntegrates up to the wall and adusts the constants n the regon close to the wall [9]. The ntegraton requres a fner grd than for the other models, but ths s less demandng than for the -ε low Re model.

4 78 A COMPARATIVE STUDY OF NUMERICAL SCHEMES AND TURBULENCE MODELS µ t α * ρ ω () * U * ρ ρ U ( µ µ tσ ) τ β ρω t x x (4) ρ ω ρu t ω ω ( µ µ σ α ω t ) τ x U βρω (5) Re α τ R α* Re τ R (6) α Re α τ 5 Rω ( α ) * 9 Re τ R ω (7) β* 4 5 Reτ 9 8 Rβ 4 Re τ Rβ (8) where, β β, σ* σ, α*, α R β 8, R 6,, 4 7 Rω τ, Re ων. NUMERICAL DETAILS. The numercal code To carry out the present comparatve study, the CFD code CAFFA (whch was developed by Perc [6] and s avalable n the Internet) was modfed approprately and developed further to nclude more numercal schemes and turbulence models [7]. Ths code solves the Naver- Stoes equatons for D ncompressble flow, wth or wthout separaton, usng fnte volumes and structured, collocated, curvlnear, body ftted grds. It uses the algorthm SIMPLE and solves the resultng systems of algebrac equatons wth the SIP method. The effects of turbulence were mplemented usng four models: a) standard - ε hgh Re, b) -ω hgh Re wth wall functons, c) -ω hgh Re wth ntegraton up to the wall, and d) -ω low Re model. The last three models were mplemented as proposed by Wlcox [9]. Central dfferencng was used for the dffuson terms, whle, for the convecton terms, seven dfferent schemes were consdered: a) upwnd nterpolaton, b) central dfference, c) hybrd scheme, d) QUICK [-], e) Harten-Yee upwnd TVD wth fve dfferent lmters, f) Roe-Sweby upwnd TVD wth three dfferent lmters, and g) Davs-Yee symmetrc TVD wth three dfferent lmters. The frst four schemes wll not be dscussed here as they have been well documented prevously.

5 WIND ENGINEERING VOLUME 8, NO., 79. Interpolaton schemes In addton to beng less accurate, the frst-order nterpolaton schemes used for the convecton terms ntroduce false dffuson. The hybrd schemes may produce frst-order accuracy n cases of sgnfcant convecton and may lead to large numercal dffuson normal to the man flow drecton. Therefore, they should not be used for hghly convectve flows n regons of large angles to the grd and n separaton zones. The central dfference and QUICK schemes wor very well n regons wth sgnfcant dffuson, but fal to produce oscllatons n regons where convecton s domnant. In order to overcome these dffcultes,, a seres of TVD schemes (Total Varaton Dmnshng) were ntroduced nto the numercal code. These are, n general, second-order accurate but, n regons of numercal oscllatons, may become of frstorder, thus securng at least convergence, but wth reduced accuracy [-4]. In the TVD scheme, the value of the varable at the surface of the computatonal cell s calculated from: ( ) ϕ f ϕ f ϕ x e e P e E e φ e, Fe (9) where, X e coeffcent wth value X e f e for F e > and X e -f e F e < φ ( / ), e flux vector lmter (mass weghted) on the nterface between the and cells For the calculaton of the flux vector lmters (TVD lmters), three schemes are used as descrbed below (a to c). a) Harten-Yee upwnd TVD scheme Flux vector lmter: φ ψ α G ψ α () G ( ) β where, α egenvalue of the RANS system Jacoban X ϕ ` ϕ Frst row ( ) () where, ϕ value of the relevant scalar (ϕ u, v, ϕ) X RHS egenvalue of the RANS system Jacoban ψ( ) correcton determned by the entropy condton β ψα G G () The lmters proposed for the specfc scheme are the followng: () G f mn mod otherwse G (), and, G mn mod f otherwse G (4), ( ) ( ) where, ϕ ϕ and ϕ ϕ (5)

6 () f otherwse G (6a) and, f otherwse G (6b) () (7a) and, (7b) where, (v) f otherwse G (8a) and, f otherwse G (8b) f otherwse G (9a) G abs abs abs * max, mn, *, mn, * G mn mod,, G mn mod,, 7 5 ω G ω ω ω G ω ω ω G G 8 A COMPARATIVE STUDY OF NUMERICAL SCHEMES AND TURBULENCE MODELS (v)

7 and, f otherwse G (9b) b) Roe-Sweby upwnd TVD scheme () wth the followng lmters: () () where n the general case: f otherwse r () where, () () Van Leer lmter: (4) () Charavarthy and Osher lmter [5] (5) c) Davs-Yee Symmetrc TVD scheme (6) wth the followng lmters: () (7) G mn mod,,, φ ψ α G G r r ( ) [ ] ( ) [ ] 4 4 max., mn., max, mn., G r r r σ α α α Sgn abs r X X σ σ σ ϕ σ ϕ G r r [ ] ( ) [ ] mn mod, max, mn, φ α G ( ) G abs abs abs * max, mn, *, mn, * WIND ENGINEERING VOLUME 8, NO., 8

8 8 A COMPARATIVE STUDY OF NUMERICAL SCHEMES AND TURBULENCE MODELS f otherwse G wth and max () G (8) mn mod,, f otherwse G wth and () G (9) mn mod, mn mod, f otherwse G wth and. Computatonal grds The D measurements of Bragg and Khodadoust [8], for the NACA arfol n arflow at Re.5 6 and 8 angle of attac, and ther D measurements at the wng centrelne, are taen as reference cases. They provde dstrbutons of Cp at the centrelne (D measurements) and the values of lft and drag coeffcents, C L and C D for angles of attac from to 8 (D and D). Two dfferent computatonal grds were used, the frst to account for turbulence models that use wall functons and the second for those who ntegrate up to the wall. The grds were C type wth boundares at cords from the arfol [6], constructed carefully n order to handle the extreme cases of mnmum or maxmum frcton coeffcent. Wall boundary condtons were appled on the arfol surface. On the C outer part of the grd, nput boundary condtons were appled (specfc value on the boundary cells) whle the normal outer part was assumed to be an outlet boundary (no pressure gradent normal to the boundary). Prelmnary calculatons showed that ths coeffcent s of order.5 at the leadng and.5 at the tralng edge. Based on these values, the grd requrements related to the wall dstance (y ) were selected as follows: Grd - : A computatonal grd of sze 4 6 ( 464 cells) wth 6 computatonal cells on the arfol was used for the turbulence models wth wall functons: a) standard hgh Re -ω model, and b) hgh Re -ω model wth wall functons. The requrement s for the frst grd node to be n the regon < y < 4 [7]. The grd, therefore, had at the leadng edge: y 9 < 4 and at the tralng edge: y 8 < 4 Grd - : A computatonal grd of sze 5 9 ( 75 cells) wth 9 computatonal cells on the arfol was used for the -ω models that ntegrate up to the wall: a) hgh Re -ω model wth wall condtons, and b) low Re -ω model. The requrement s for the frst grd node to be n the regon y <.5 [8]. The grd, thus, had at the leadng edge: y.5 and at the tralng edge: y.58 < RESULTS AND DISCUSSION max max The four turbulence models and the numercal schemes (5 combnatons of schemes and lmters) are compared, based on ther accuracy and rate of convergence wth the experments of Bragg and Khodadoust [8]. For each turbulence model, 5 smulatons were performed n order to account for all possble combnatons of nterpolaton schemes for the veloctes and the turbulence quanttes, as follows: a) Hgh Re -ω model (standard) We performed 9 successful smulatons, out of the 5 possble. It should be notced that, n relaton to convergence, unsuccessful combnatons were, () those for whch the

9 WIND ENGINEERING VOLUME 8, NO., 8 dscretzaton of the turbulence transport equatons used hgher order schemes (central scheme or QUICK), and () the TVD schemes that produced the most accurate calculatons when used for the momentum equatons. Wth respect to the predcton of Cp dstrbuton and the C L and C D values, the results closest-to-experment were obtaned for the combnatons of schemes that use, for velocty dscretzaton, central dfferencng, the Roe-Sweby upwnd TVD wth the Charavarthy & Osher lmter, and the Harten-Yee Upwnd TVD scheme wth the lmter (v). The next most accurate combnatons were those usng, for the velocty, the QUICK scheme and the rest of the TVD lmters of the Roe-Sweby and Harten-Yee schemes, wth the excepton of the Harten- Yee Upwnd TVD wth the lmter () that led n most of the cases to dvergence. Fnally, the worst accuracy was produced by the upwnd and hybrd schemes, and by the Davs-Yee Symmetrc TVD scheme. Wth relaton to the rate of convergence, the slowest s the QUICK scheme, followed by central dfferencng and those of Roe-Sweby Upwnd TVD wth the Charavarthy and Osher lmter, and of Harten-Yee Upwnd TVD wth the lmter (v). Thus, the most accurate schemes were the slowest, whle the less accurate schemes were faster. In Fg., the dstrbuton of the pressure coeffcent Cp s shown for: a) upwnd nterpolaton for the veloctes and the propertes of and ε, b) the Harten-Yee Upwnd TVD scheme wth the lmter (v) for the velocty equatons and the upwnd scheme for and ε, and c) the measured dstrbuton of Cp at the centerlne [8]. We observe that the value of C p appears to be overestmated by all the nterpolaton schemes. Ths s because the present model does not tae nto account the D flow effects. Ths also occurs for all the results of all turbulence models, as presented below. b) Hgh Re -ω model wth wall functons We performed 5 successful smulatons; out of the 5, wth smlar convergence rates as prevously. The best combnatons were those usng central dfferencng for velocty dscretzaton. Such a case however may become dangerous for larger angles of attac where separaton appears, because of the nown oscllatory behavour of the central dfference -e HR,,4,6,8, Cp Cp(upwnd-upwnd) Cp(TVD-upwnd) Cp(exp)-D-CL 4 Fgure : Dstrbuton of Cp for Hgh Re -ε model and varous schemes (NACA, α 8, Re.5 6 ) x/c

10 84 A COMPARATIVE STUDY OF NUMERICAL SCHEMES AND TURBULENCE MODELS -o HR WF,,4,6,8, Cp Cp(upwnd-upwnd) Cp(CD-TVD) Cp(TVD-TVD) Cp(exp)-D-CL 4 x/c Fgure : Dstrbuton of Cp for Hgh Re -ω model wth wall functons and varous schemes (NACA, α 8, Re.5 6 ) scheme. We prefer, therefore, the next best scheme,.e. the Harten-Yee Upwnd TVD wth the lmter (v) for the veloctes combned wth the lmter (v) for and ε. Fg. shows the dstrbuton of Cp for: a) upwnd nterpolaton for the veloctes and the and ε equatons, b) the Harten-Yee Upwnd TVD scheme wth the lmter (v) for the veloctes and the lmter (v) for the and ε equatons, c) central dfference for the momentum equatons and the Roe-Sewby Upwnd TVD scheme wth the mnmod lmter for and ε, d) the dstrbuton of Cp measured at the centerlne. c) Hgh Re -ω model wth ntegraton to the wall We performed 4 successful smulatons, out of the 5, wth smlar observatons as above. The best results were obtaned wth central dfferencng for the veloctes, whch for the reasons dscussed above should be reected at much larger angles of attac. Agan, the next best combnaton s the Harten-Yee Upwnd TVD wth the lmter (v) for the veloctes and the Harten-Yee Upwnd TVD wth the lmter (v) for and ε. Fg. shows the dstrbuton of Cp for: a) upwnd scheme for the veloctes and the and ε equatons, b) the Harten-Yee Upwnd TVD wth the lmter (v) for the veloctes and the lmter (v) for and ε, c) central dfference for the veloctes and the Roe-Sewby Upwnd TVD wth the mnmod lmter for and ε, d) the measured dstrbuton of Cp at the centrelne. d) Low Re -ω model We performed 8 successful smulatons, out of 5, wth smlar observatons as prevously. Fg. 4 shows the dstrbuton of Cp for: a) upwnd scheme for the veloctes and the and ε equatons,

11 WIND ENGINEERING VOLUME 8, NO., 85,5 -o HR WC,5,5,,4,6,8, Cp,5,5,5 Cp(upwnd-upwnd) Cp(CD-TVD) Cp(TVD-TVD) Cp(exp)-D-CL 4,5 Fgure : Dstrbuton of Cp for Hgh Re -ω model wth wall condtons and varous schemes (NACA, α 8, Re.5 6 ) x/c b) the Harten-Yee Upwnd TVD scheme wth the lmter (v) for the veloctes and the lmter (v) for and ε, c) central dfference for the veloctes and the Roe-Sewby Upwnd TVD scheme wth the mnmod lmter for and ε, d) the dstrbuton of measured Cp at the centrelne. In all models, smlar results were obtaned for the combnatons that use, for veloctes, central dfferencng and the TVD schemes of Roe-Sweby Upwnd TVD wth the Charavarthy and Osher lmter, and of Harten-Yee Upwnd TVD wth the lmter (v).,5 -c LR,5,5,,4,6,8, Cp,5,5,5 Cp(upwnd-upwnd) Cp(CD-TVD) Cp(TVD-TVD) Cp(exp)-D-CL 4,5 Fgure 4: Dstrbuton of Cp for Hgh Re -ω model and varous schemes (NACA, α 8, Re.5 6 ) x/c

12 86 A COMPARATIVE STUDY OF NUMERICAL SCHEMES AND TURBULENCE MODELS,,4,6,8, Cp Cp(e-hr) Cp(o-hr-wf) Cp(o-hr-wc) Cp(o-lr) Cp(exp)-D-CL 4 Fgure 5: Dstrbuton of Cp for (4) turbulence models and best combnaton of schemes (NACA, α 8, Re.5 6 ) x/c As mentoned above, the central dfference scheme was reected because of ts oscllatory behavour at separaton. The TVD schemes of Roe-Sweby Upwnd TVD wth the lmter of Charavarthy and Osher, and of Harten-Yee Upwnd TVD wth the lmter (v) gve smlar results n terms of accuracy and requred teratons. In partcular, the Roe-Sweby Upwnd TVD scheme wth the Charavarthy and Osher lmter produces slghtly better estmate of the drag coeffcent, whle the Harten-Yee Upwnd TVD scheme wth the lmter (v) gves better fttng of the coeffcents C p and C L, and for ths reason the latter s preferred. In terms of teraton tme and number of teratons, the two schemes have smlar behavour. After the best combnaton was determned for each turbulence model, comparson was made between them. From Fg. 5, where the dstrbuton of Cp s shown, and from Table, where the values of the aerodynamc coeffcents C L and C D are gven at 8 o angle of attac, t appears that the -ε hgh Reynolds model gves sgnfcantly better results for drag coeffcent and lft coeffcent, whle the other models behave about the same. However, detaled examnaton of the Cp dstrbutons at the tralng edge (Fg. 6) shows that the models that ntegrate up to the wall produce smoother behavour n ths regon. The dfferences n the rate and tme of convergence are also notceable; the models usng wall functons converge n Table. Values of C D and C L for NACA arfol at 8 angle of attac (Re.5 6 ) Model C D C L Number of teratons (convergence at 4 ) -ε HR ω HR WF ω HR WC ω LR D experment D exp. centerlne..6 -

13 WIND ENGINEERING VOLUME 8, NO., 87,,5,,5, Cp,5,5,85,87,89,9,9,95,97,99,,, Cp(e-hr) Cp(o-hr-wf) Cp(o-hr-wc) Cp(o-lr) Cp(exp)-D-CL,5, Fgure 6: x/c Detals of Cp dstrbuton at the tralng edge for (4) models and best combnaton of schemes (NACA, α 8, Re.5 6 ) much fewer teratons than those ntegratng up to the wall, and they requre less computatonal cost per teraton, because they use coarser grds. Fnally, a seres of smulatons for angles of attac from to o were performed. Fg. 7 shows the comparson of the coeffcent C L for each turbulence model and the D and D measurements, whle Fg. 8 shows the coeffcent C D. It s mportant to note that n the lnear regon, the slope of C L for all turbulence models compares extremely well wth the expermental D results, whle the slope of the D experments s much less. Devatons start to appear n the regon of stall. It s clear that, as the angle of attac ncreases, the turbulence models that ntegrate up to the wall gve better results than those usng wall functons. On the,8 Cl,6,4, Cl,8,6,4, Cl(e-hr) Cl(o-hr-wf) Cl(o-hr-wc) Cl(o-lr) Cl(exp)-D Cl(exp)-D 5 5 Angle 5 Fgure 7: Dstrbuton of C L for (4) models and best combnaton of schemes (NACA, Re.5 6 )

14 88 A COMPARATIVE STUDY OF NUMERICAL SCHEMES AND TURBULENCE MODELS,8 Cd,6,4, Cl,8,6,4, Cl(e-hr) Cl(o-hr-wf) Cl(o-hr-wc) Cl(o-lr) Cl(exp)-D Cd(exp)-D,,,,4 Cd,5,6,7 Fgure 8: Dstrbuton of C D for (4) models and best combnaton of schemes (NACA, Re.5 6 ) other hand, there s constant overestmaton of the drag coeffcent, attrbutable to the lac of any transton model; the - ε hgh Reynolds model gves results closest to the D expermental results. Moreover, the former predct separaton at smaller angles of attac, as shown n Fg. 9. The comparson between the two models usng wall functons shows that the -ω model s better because t taes nto account the exstng pressure gradent. Fnally, t should be notced that all smulatons were carred out wth the same grds constructed for the 8 o angle of attac and, as a result, these grds requred many more teratons for angles of attac much larger or smaller., Separaton pont,8 x/c,6,4, Sp(e-hr) Sp(o-hr-wf) Sp(o-hr-wc) Sp(o-lr) Angle Fgure 9: Poston of separaton for each turbulence model (NACA, Re.5 6 )

15 WIND ENGINEERING VOLUME 8, NO., CONCLUSIONS The followng man conclusons can be drawn:. Successful convergence depends manly on the nterpolaton scheme used for the dscretzaton of the terms n the transport equatons of turbulence quanttes. The better the accuracy of the scheme used, the less s the possblty of achevng convergence.. The accuracy of the results depends on the nterpolaton scheme used for the dscretzaton of the terms n the momentum equatons. The best results were obtaned wth the central dfference scheme, and the TVD schemes of Roe-Sweby Upwnd TVD wth the Charavarthy and Osher lmter, and of Harten-Yee Upwnd TVD wth the lmter (v). The Harten-Yee Upwnd TVD scheme wth the lmter (v) s preferred because central dfferencng exhbts oscllatory behavour at separaton, whle the Roe-Sweby Upwnd TVD scheme wth the Charavarthy and Osher lmter s slghtly less effcent n the calculaton of C P and C L.. At small angles of attac, all models produce smlar results, whle those usng wall functons are computatonally more effcent. On the bass of accuracy, the best results are produced wth the - ε hgh Reynolds model. 4. At large angles of attac, the models that ntegrate up to the wall produce results that are more accurate and predct the appearance of separaton earler. 5. For all angles of attac, the -ε hgh Reynolds model gves better predcton of the drag coeffcent, but stll dfferent from the expermental results. Ths s attrbuted to the lac of a transton model. 6. Fnally, apart from the turbulence model lmtatons, the relatvely poor comparson wth experments s due to the lac of a transent model and to consderng the whole feld as turbulent. Addtonally, t s lely that the results could be mproved wth the use of more sutable grds. However, these suggested mprovements were beyond the scope of the present wor, whch amed at the comparson of numercal schemes and turbulence models n the predcton of turbulence flow around arfol for Reynolds numbers correspondng to wnd turbne operaton. ACKNOWLEDGEMENTS The leadng author thans the Gree Center of Renewable Energy Sources for a doctoral research scholarshp and Dr P. Chavaropoulos of CRES who provded useful nsght for the modellng of the flows. The outlne fndngs of ths paper were presented to the European Wnd Energy Assocaton s conference, whch presentaton was used extensvely n ths paper, however new wor s now ncluded. REFERENCES. Eaternars, J.A. and Platzer, M.F., Computatonal predcton of arfol dynamc stall, Progress n Aerospace Scence, 997,, Chavaropoulos, P., Flap lead-lag aeroelastc stablty of W/T blade sectons, J. Wnd Energy, 999, (), Voutsnas, S.G. and Rzots, V.A., A vscous-nvscd nteracton model for dynamc stall smulatons on arfols, AIAA paper Kral, L.D., Recent experence wth dfferent turbulence models appled to the calculaton of flow over arcraft components, Progress n Aerospace Scences, 998, 4,

16 9 A COMPARATIVE STUDY OF NUMERICAL SCHEMES AND TURBULENCE MODELS 5. Zngg, D.W., De Rango, S., Nemec, M. and Pullam, T.H., Comparson of several spatal dscretzatons for the Naver-Stoes equatons, Journal of Computatonal Physcs,, 6, Ferzger, J.H. and Perc, M., Computatonal Methods for Flud Dynamcs, Sprnger, Baxevanou, C., Development of a Naver-Stoes model for the aeroelastc analyss of wnd turbne blades, PhD thess, Unv. of Thessaly, n progress. 8. Bragg, M.B. and Khodadoust, A., Effect of smulated glaze ce on a rectangular wng. AIAA-89-75, 7 th Aerospace Scences Meetng, Reno Nevada, January 9-, Wlcox, D.C., Turbulence modelng for CFD, DCW Industres, Inc, Hayase, T., Humphrey, J.A.C. and Gref, R., A consstently formulated QUICK scheme for fast and stable convergence usng fnte volume teratve calculaton procedure, J. of Computatonal Physcs, 99, 98, Rahman, M., Mettnen, A. and Sonen, T., Modfed smple formulaton on a collocated grd wth an assessment of the smplfed QUICK scheme, Numercal Heat and Mass Transfer - Part B, 996,, Peyret, R., Handboo of Computatonal Flud Mechancs, Academc Press, London, Hrsch, C., Numercal computaton of nternal and external flows Vol. : Computatonal methods for nvscd and vscous flows, Wley, England, Hoffman, A.K. ND Chang, S.T., Computatonal Flud Dynamcs Volume II, Engneerng Educaton System, Wchta Kansas, Par, T.S. and Kwon, J.H., An mproved multstage tme steppng for second-order upwnd TVD schemes, Computers & Fluds, 996, 5(7), Weber. S. and Platzer, M.F., Computatonal smulaton of dynamc stall on the NLR 7 arfol, Journal of Fluds and Structures,, 4, Sanar, T., Sayer, P.G. and Fraser, S.M., Flow smulaton past axsymmetrc bodes usng four dfferent turbulence models, Appled Mathematcal Modellng, 997,, Wlcox, D.C., Comparson of two-equaton turbulence models for boundary layers wth pressure gradent, AIAA Journal, 99, (8), 44-4.