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NAVIER-STOKES SIMULATIONS OF !_, ' _ '_"

WECS AIRFOIL FLOWFIELDS ,',_ 'G ''_*__Gregory E Homicz

Sandia National LaboratoriesComputational Fluid Dynamics Department/Mail Stop 0827

P. O. Box 5800Albuquerque, NM 87185-0827 USA

ABSTRACT

Sandia National Laboratories has initiated an effort to apply Computational Fluid Dynamics (CFD) to thestudy of WECS aerodynamics. Preliminary calculations are presented for the flow past a SAND 0018150airfoil. The flow solver used is F3D, an implicit, finite-difference code which solves the Thin-Layer Navier-Stokes equations. 2D steady-state calculations are presentedat various angles of attack, ix. Sectional lift anddrag coefficients, as well as surface pressure distributions, are compared with wind tunnel data, and exhibitreasonable agreement at low to moderate angles of attack. At high ix, where the airfoil is stalled, aconvergedsolution to the steady-state equations could not be obtained. The flowfield continued to change with

- successive iterations, which is consistent with the fact that the actual flow is inherently transient, andrequires the solution of the full unsteady form of the equations.

NOMENCLATURE

a** freestream speed of sounde internal energyL characteristic length scaleM** freestream Math number, U**/a**p static pressure Thbwork wa._supenrtedby Ihe UnitedReL Reynoldsnumber,p.Uj-4t. St_es Departmentof EnergyunderT static temperature Coetr_'t DE-ACn4-94ALRSOO0.U** freestream velocity magnitude, (u_ + w_);/zu, v, w velocity components along x, Z z respectivelyx, y, z cartesian coordinates in physical space(x angle of attack7 specific heat ratio, c/c vIx viscosityp density_, rl, _ body-fitted coordinate system in computational space

INTRODUCTION

The aerodynamics of Wind Energy Conversion Systems (WECS) is particularly challenging because itinvolves three-dimensionality, viscous effects, and in many circumstances, significant unsteadyphenomena. Present theoretical models can predict with confidence the aerodynamic performance of onlythose machines that represent incremental departures from earlier, tested designs. The computer modelscommonly used for such predictions (multiple-streamtube, local circulation, vortex, etc.[ 1]) all rely to someextent on tabulated airfoil characteristics (typically obtained from a wind tunnel test program), and variousempiricisms to represent those effects that cannot be simulated directly. Radically different designs typicallyrequire extensive experimental testing, followed by code modifications and adjustment of the empiricismsto the new data set. Moreover, blade fatigue life predictions demand even more detailed knowledge of theloads than is presently available from any of the current aerodynamic codes. Computational Fluid Dynamics

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DISCLAIMER

This reporg was prepared as an account of work sponsored by an agency of theUnited States Government. Neither the United States Government nor any agencythereof, nor any of their employees, makes any warranty, express or implied, orassumes any lesal liability or responsibility for the accuracy, completeness, or use.fulness of any information, apparatus, product, or process disclosed, or representsthat its use would not infringe privately owned rights. Reference herein to any spe-cific commercial product, process, or service by trade name, trademark, manufac-,turer, or otherwise does not necessarily constitute or imply its endorsement, recom-mend_,tion, or favoring by the United States Government or any agency thereof.The views an_ opinions of authors expressed herein do not necessarily state orreflect those of the United States Government or any agency thereof.

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(CFD) offers an alternative means of investigating WECS aerodynamics that addresses some of theseconcerns. CFD distinguishes itself from file semi-analytical theories by its attempt to calculate the relevantflow physics from "first principles", with as little recourse to empiricism as possible. The past few decadeshave witnessed gains in computer speed and algorithm efficiency that now make it feasible to view CFD asan attractive complement to experimental testing. Though such calculations may still be too time-consuming and expensive for use as production design tools by the WECS community, they can often beperformed at considerably less expense than a comprehensive test program. The insight thus gained canultimately lead to improved models for incorporation into day-to-day design codes.

This paper describes recent efforts at Sandia to apply a suite of previously developed computational toolsto the prediction of WF_,CSairfoil flowfields. The next section presents the viscous flow equations and theirtransformation to computational space. Then the flow solver, F3D, and its capabilities are briefly described.Steady-state solutions are presented for the 2D flow past the SAND 0018/50, an airfoil specifically designedfor Vertical-Axis Wind Turbine (VAWT) applications, at various angles of attack. These are believed to bethe first Navier-Stokes solutions obtained for this airfoil. The sectional lift and drag coefficients, as well assurface pressure distributions, are validated by comparison with available wind tunnel data.

GOVERNING EQUATIONSThe flow of a compressible, viscous, Newtonian fluid such as air is governed by equations representing theconservation of mass, momentum, _ad energy, collectively referred to as the Navier-Stokes equations. Wewill work with a system of normalized variables: all lengths are nondimensionalized by some characteristiclength L, velocities by a**,time by L/a**,density by p**,pressure by p**a**2 , temperature by T**,energy bya**2 , and viscosity by Ix**.In a fixed frame of reference, using cartesian coordinates (x, y, z), the governingequations can be written in strong conservation form as (see, e.g., Hoffmann[2]):

_Q _E _F+_G aEv aFv _Gv

The boldfacequantityQ isthecolumnvectorofunknown variables,andE, F,andG areknown astheinviscidfluxvectors:

p pu pv pwpu pu2+p pvu pwu

Q = pv E = puv F = pv2+p G = pwv (2)pw puw pvw pw 2 + p

pe t (pe t+p) (pe t+p) (pe t+p) w

Here et is the total energylunit mass, et = e+ (1/2) u +v +w = cvT+ (1/2) u +v +w 2 .E_,Fv, and G v represent the viscous flux vectors:

E = F = G =P t' V

I "0 0 0Xxx "r'yx "r'zx (3)T,yy %z),

T'xY ' i%xz T'yz %zz

'_'r.xx+ V'rxy + w'rxz - q_ i_ z _-"ry), + W'_yz - q_ '_'_zx+ vxzy + WXzz - q_

-2

$ 4 t

[x] and_ represent the symmetricstress tensor and heat fluxvector, whose elements are given respectivelyby, e.g.,

Xxy = xyx = Re---L Ix +-_ qx R%Pr(r-I)M2. axBy considering the 1st ... 5th elements in the above column vectors, Eq. (1) is seen to express theconservation equations of mass,x-momentum, y-momentum, z-momentum, and energy, respectively.

For flows bounded by curvilinear surfaces, which do not lie conveniently along x, y, or z = constant, theimposition of boundary conditions on Eq. (1) is very difficult in cartesian coordinates. To get around this,the equations are transformed fromphysical space (x, y,z, 0 to a boundary-conforming computational space(_ rl, _, x). In general, the transformation must be performed numerically, rather than analytically, andserves two purposes. First, the originally curvilinear flow domain is mappedinto one which is rectangular(planar in 2D, or a "box" in 3D), and thus ideally suited for the use of an equally-spaced cartesian grid in(_, _, _). Secondly,surfaces on which boundary conditions are to be imposed will, by definition, lie along asurface of either _,,rl, or _ = constant. This will become clearer when the airfoil and grid are discussedbelow.

When the above equations are transformed to computational space, after much manipulation they can stillbe written in strong-conservation form similar to Eq. (1):

The transformedvectorofunknownsisgivenby [0.]= [Q]/J ,whereJ istheJacobianofthetransformation.TheexpressionsforJ,thetransformedinviscidfluxvectors,P_...,andtheviscousfluxvectors,/_v"',aresomewhatlengthy,andastheyarereportedelsewhere[2],willnotberepeatedhere.

CODE DESCRIPTION

ThesystemofpartialdifferentialequationsrepresentedbyF-xl.(5)isfartoocomplextosolveanalyticallyexceptforthemosttrivialofcases.ThecomputercodeusedtogeneratetheresultsreportedhereisF3D,aprogramdevelopedbyProfessorJosephStegerandseveralcolleaguesandstudents[3]-[7].ForhighReflows,theeffectsofviscosityareoftenconfinedtorelativelythinshearlayersadjacenttothesurface.Assumingthat_isthecoordinatenormaltothesurfaceincomputationalspace,itcanbeargueduSattheviscoustermsin_andI]willbemuchsmallerthanthosein_,andhencecanbeneglected.F3D solvestheresultingsystem,termedtheThin-LayerNavier-Stokes(TLNS)equations.Ofcourse,caremustbeexercisedinapplyingsuchasystemtoflowsinvolvingmassiveseparationand/orrecirculation,wheretheunderlyingassumptionsmay nothold.F3D hasbeensuccessfullyusedtopredicttheaerodynamicsofsubsonic,transonicandsupersonicflowfields[5]-[7],includingviscousflowsatlargeincidence[5].SolutionAl_orithm

An evenly-spaced,cartesiangridofdiscretepointsisoverlaidontheflowin(_,TI,_)space,andthepartialderivativesintheequationsareapproximatedbyfinitedifferences.Foraddednumericalstability,a flux-splitupwinddifferencingschemeisappliedtotheconvectivetermsin_,whilecentraldifferencesareretainedinrland_.Theresultingsystemofalgebraicequationsissecond-orderaccurateinspace,andfirst-orsecond-orderaccurateintime.Itcanbeinvertedusingatwo-factor,implicit,approximate-factorizationalgorithmwhichisbothstableandrelativelyefficient[5].An explicitalgorithmisavailableasanoption.Tofacilitateapplicationtocomplexgeometries,multi-blockgrids,bothoverlappingandembedded,canbeaccommodaled.F3D canbeusedtostudyflowswhichareeithertwo-orthree-dimensional,steadyorunslcady,inviscid(Euler)orviscous,andlaminarorturbulent.

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Turbulence & Transition

Strictly speaking, the equations discussed above apply only to laminar flows. At high Re, the flow can beexpected to transition to turbulence. This introduces additional diffusive terms into the equations termedturbulence shear stresses, or more usually, Reynolds stresses. The Reynolds stresses represent additionalunknowns, and the system of equations becomes underdetermined.

At this point recourse is made to empiricism, in the form of a turbulence model. Turbulence models providethe additional relations required to close the system of equations. The results reported here were generatedusing the Baldwin-Lomax algebraic turbulence model[8]. At the risk of over-simplifying, it postulates thatthe effects of turbulence may be modeled by solving the laminar flow equations (Eq. (5) above) with themolecular viscosity, ix, replaced by an effective viscosity, Reff = ix+ ixt ixtis the so-called eddy viscosityarising from turbulence. Algebraic relations are used to relate ixtto the mean flow variab[es in [Q] ;typically ixt >_ix' as turbulent mixing is much more efficient than viscous (i.e., laminar) diffusion. Detailscan be found in Ref. [8].Transition from laminar to turbulent flow is still very much an active area of research. For practicalcalculations it is necessary to resort to empiricism or experimental knowledge of its location. F3D wasmodified so that transition locations could be independently input on both the upper and lower airfoilsurfaces. Upstream of these points the flow was treated as fully laminar (ixt= 0), and downstream it wasassumed to be fully turbulent. Limited experimental data are available on the transition locations for theSAND 0018150 from the work of Reda[9]. These locations were used as input to F3D; specific values willbe mentioned in connection with the numericai results below.

AIRFOIL & GRID

The SAND 0018150 was designed to have character- ('I _-----..--._istics favoring its application to the Darrieus Verti- _ --1"'-cal-Axis Wind Turbine (VAWT)[10], and is currentlyin use on the Sandia 34-m Test Bed in Bushland, Tex- FIGURE 1. SAND 0018/50 AIRFOILas[ll]. Its profile is shown in Fig. 1. It is a symmetric, 18% thickness-to-chord ratio section designed tomaintain laminar flow over - 50% of chord at zero angle of attack. The resulting lower drag improves itsperformance at low-to-moderate windspeeds. It also has an abrupt-stall characteristic; this is advantageousin high winds, where passive stall regulation is used to limit output power.

The grid used in this study was generated using the Eagle grid generation code developed at MississippiState University[12]. Figure 2 shows the grid transformation in schematic form. The airfoil lies in the (x, z)plane in physical space (Fig. 2a), where it forms the inner boundary of the flow domain; the outer boundaryis chosen to be a circle. The region between them is mapped to a rectangle in the (_, _) plane in

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A E F G A'(a) Physical Space (b) Computational Space

FIGURE 2. SCHEMATIC OF 2D .,XlI,_r:(_.lLMAPPING (not Io scale)

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ecomputational space (Fig. 2b). The points labelled A...G silow that the airfoil is mapped to the bottom ofthe rectangle, and the outer boundary to the top.

Figure 3a shows an overall view of the actual grid, which is an "O-type" structured grid that wrapscircumferentially around the airfoil. JMAX = 201 grid lines were used in the circumferential (J or _)direction, and LMAX = 110 were used in the radial (L or _) direction. Uniform freestream conditions areimposed on the outer boundary, which has a radius of - 100c (c being one chord length). This large radiuswas chosen in an attempt to simulate as nearly as possible "unconfined" flow conditions. At this scale theairfoil itself, at the center of the grid, is not visible.

(a)OverallGrid (b)Magnified(200x)GridFIGURE 3.FINITE DIFFERENCE GRID FOR SAND 0018/50

InFig.3b theviewismagnified~ 200timesfora betterlookatthegridinthevicinityoftheairfoil.Gridpoints are packed much more tightly in the vicinity of the airfoil, so as to resolve the very thin boundarylayers which occur at high Re. In units of chord length, the thickness of the first row of grid cells adjacentto the airfoil, bsu'c, is on the order of 10.7 With 201 points circumferentially, the average grid spacing inthe streamwise direction, z_s/c, is about 10"2. However, as can be seen in Fig. 3b, the points were alsoclustered near the leading and trailing edges, where &s/c ~ 10-4 and 10"3, respectively. These values werechosen based on the results of the grid-refinement studies of Zingg[13]. On the airfoil surface, "no-slip"boundary conditions were imposed, Le., u = w = O.

RESULTS & DISCUSSION

Preliminary calculations were run for the SAND 0018/50 at three different angles of attack: 4, 10, and20 , chosen to see how the code performed in various flow regimes. At 4* one would expect the flow toremain attached, with the airfoil on the linear portion of its lift characteristic; o_= 10 is just below the angleat which the airfoil stalls; and at 20 the airfoil is deeply stalled. Experimental wind tunnel data for thesectional lift and drag coefficients, ct, c o - (L, O)/(l/2pV2,c), as well as the surface pressure coefficient,5, = (P-P-)/( l/2p,,u2-)' were available for comparison._= 4The first case considered had freestream conditions of M** = 0.26, Re c = 1.5 106, and o_= 4. Theseconditions correspond to those for Run No. 4193 in Ref. [14]. Based on Reda's flow visualization studies[9],the transition locations were set at x/c = 0.65 and 0. 70 on the upper(suction) and lower(pressure) surfaces,respectively. The convergence of CL and CD with respect to iteration number is shown in Fig. 4a below.Note that the values of Co have all been multiplied by a factor of 10 to allow visualization on the same scalewith CL. After 3000 ilcrt!I!, ,:_:the computed CL had settled down to a value of 0.510, vs. the experimentallymeasured 0.40, a differcn,. :, ,f +27%. The predicted CD was 0.0118, also +27% higher than the measuredvalue of 0.0093. At this i,,x_ angle _,1attack, the contributions to CD from pressure and skin friction arecomparable. This undersct,res the need to use very tightly clustered grids near the airfoil surface, to allowbetter resolution of the ;,: ,l_:_rslress.

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1.6 : , -2 _ ; , , ._- _ ."_ - , , ,'_'' - I'. i

t !t._ - ox%I ai!i"o,,- Q,. _

OR-I- _M _... Om O.t ii! \ .o..o--o--o-J- _: -"s-"_'...... ---v---o.41- _._ ,,-_- _ ,. ., ...-:'7--:[ |!!" _o'*.z_.c,.: .4o1'_

.... ,,..., .... _ .... , .... t .... ' 2-"--' "i-'_ ' ; " " ' ; ' ' ' ; " " ' i ' ' " '' """ ' ' ' '"0.4-,- _ , , , ' - ' ' i.0o 5oo tooo tsoo eooo esoo sooo -o,_ o.o o_ 0.4 0.6 o.8

Iteration Number x/c

(a) CL,CDConvergenceHistory Co)Surface Pressure Coefficient, cp

FIGURE 4. COMPARISON OF CL, CD/uND cp WITH MEASURED DATAFOR THE SAND 0018150 AT M**= 0.26, Re c = 1.5 x 106, o_= 4

Figure 4b compares the predicted surface pressure coefficients against the measured data. Note that thevertical axis has been inverted so that the upper portion of the curve corresl_nds to the upper (suction)surface of the airfoil, and the lower portion to the lower (pressure) surface. The agreement is reasonablygood, with the biggest difference occurring near the leading edge on the upper surface. Over most of theairfoil, the pressure is overpredicted on the lower side, and underpredicted on the upper side, which explainswhy the calculated CL was too high.

The computedstreamline pattern is shown in Hg. 5a. The airfoil chord is hofizontai, with the flow enteringat a 4 angle from the lower left. As expected, the flow remains attached to the airfoil with no evidence ofseparation. Figure 5b displays the contours of static pressure in the surrounding flowfield. This is a gray-scale reproduction of the full-color original, with each color corresponding to a different level of p/p,,,, asindicated by the color bar on the fight. The stagnationpointlies just below the nose of the airfoil. The muchlower pressures on the upper surface, as compared with those on the lower side, are of course responsiblefor the lift.

1.025J- ,, 1.022

T - -- ..... 1.0101.0141.011.0061.0020.9900.11940.990.91160.9620.0700.974

......,.

(a) Streamline Pattern (19)Pressure ContoursFIGURE5. COMPUTED FLOWFIELD FOR THE SAND ()018/5(1 AT

M_ - 0.26, Rec - 1.5 x 106, _ = 4

i...............

,

e ,

0_=10

The next case runwas for Moo = 0.21, Rec = 1.19 x 106, and (x= 10, correspondingto Run No. 731 inRef. [14]. Tramitionwas specifiedatthe leadingedge on the uppersurface,andatx/c = 0.79 on the lowersurface[9]. The convergencehistoryof CL andCOis shown in Fig. 6a. The Lz-norm, essentiallythe r.m.s.

4 -8 '_. ; . ,'_''--. i .7. d I i "-

S CL2

-4- Ohiot

-z toxcD o-

-4 ..... ''''''''I'''', , I'" z --',"--"=-:-":-'", ,''",'""o looo am sooo 4oeo .-o_. o.o o_. 0.4 o_ o.8 t.o t_.

Iteration Number x/c

(a)CL,CDConverg_coHistory Co)SurfacePressureC_cicnt, Cp

FIGURE6. COMPARISON OF CL, CD AND ce WITH MEASURED DATAFORTHESAND 0018/50 ATMoo= 0.21, Rec = 1.19 x I06, a = I0

value of [A0.] between successive iterations, is monitored as an overall measureof convergence. Byiteration 3000 or so, this had only dropped by a little more than an orderof magnitude from its peak.Generally,a dropof two tothreeordersof magnitudeis considerednecessaryforadequateconvergence.Forthis reason the calculationwas continued andafter4500 iterations it hadfallen more than two ordersofmagnitude. The predicted force coefficients were CL = 1.102 vs. the measured 0.858 (+28%), andCD =0.0467 vs. the measured0.0427 (+9%). The surfacepressuredistributionsare comparedin Fig. 6b. Overmost of the chord the agreementis reasonablygood. As was thecase at 4, the code appears to generallyoverestimatepressures on the lowersurface,andunderestimatethem on theupper,which is againconsistentwith the predicted CLbeing high. However, near the leading edge on the suction surface there are two"spikes" in the predictedpressures not presentin the data.More will be said aboutthese shortly.

(a)OverallView (b)MagnifiedView

FIGURE7. STREAMLINEPATI'ERNFORTHE SAND 0018/50 ATM_ = 0.21,ReC = 1.19 I0_, AN[.)(_ = lO

i

i . t

Figure 7a shows the streamline pattern for tllis flow. The stagnation poinl has moved I'urlher down on lhelower surface, relative to that in Fig. 5a. Though the flow remains largely attached, the lirsl slreamline abovethe upper surface does not follow the surface as closely as it did at 4. As there appears to be some"waviness" near the leading edge, this region was greatly magnified, resulting in the view shown in Fig. 7b.Several additional streamlines were interpolated adjacent to the surface to better illuminate the flow pattern.Two separation/reattachment bubbles are now evident on the suction surface. One would expect lowerpressures there, and indeed, these bubbles correspond in location to the negative "spikes" in the predictedsurface pressures in Fig. 6b. It was found that the bubbles' location and size, as well as the height of thespikes, varied somewhat during the calculation, though CL and CD were changing only slightly. Theseoscillations are taken as evidence of incipient separation from the vicinity of the leading edge. Whether suchbubbles occurred in the experiment is unknown, though the smooth pressure variation in Fig. 6b wouldseem to argue against it. On the other hand, the pressure instrumentation used to obtain the data in Fig. 6bdid not have a particularly fast response time[ 15], and so in some sense those data represent a time-averagedresponse which may have masked such behavior.

The predicted pressure contours are shown in t_p=Fig. 8. At this magnification, the negative pressure ,.0-

1.022spikes near the leading edge noted above are not ,.o,visible, The same color bar was used here as was ,.o,used in Fig. 5b, to facilitate comparing the two. The ,.o_1.006contours are qualitatively similar to those at 4, ex- ,.dO=cept that the pressures on the bottom are generally o.,,0.994higher, and the low pressure region on the suction o.,side extends out further into the flow. This is con- ..m0.g82sistent with the higher lift achieved at this angle of o.oTeattack, o.,7,

o_= 2.0FIGURE 8. PRESSURE CONTOURS FOR

The last case attempted was for M** = 0.2, Rec = S - "!1,Rec1.8 x 106, and o_--.20. These conditions unfortu-nately did not admit a converged solution. Cb Co,and the L2-norm never stopped oscillating, andeventually the calculation "blew up". The surfacepressure distribution, streamline pattern, and pres-sure field contours also exhibited significant varia-tion as the iterations proceeded. Figure 9 displays atypical streamline pattern; it is strongly suggestiveof the massive separation and vortex shedding phe-nomena associated with a stalled airfoil. The pat-terns at other stages in the calculations arequalitatively similar, though the location of the vor-tical elements changes. Though care should be tak-en not to read too mucil into unconverged solutions, FIGURE 9. STREAMLINE PATTERN FOR SANDas they do not satisfy the governing equations, this 0018/50 AT M,,, = 0.2, Rec = 1.8 10 6,strongly suggests that at 20 the flowfield is inher- o_= 20 (not converged)ently transient, and characterized by massive sepa-ration and vortex siledding. This is consistent with Ihe observation made elu'lier, thai ;_ttill. _ itlc ;ti_l(_ilisknown to be deeply stalled. Such flows require that all the unsteady terms in the govcml_, cqUztli_,_l_b.,."retained. Though F3D is capable of treating such cases, this greatly increases computing titl_'_, and llt_Stl,,:'!_.calculations have been altempled as yel.

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Timin_ Statisti_

It is reasonable to ask what level of resources,in terms of both hardware andcomputer time, are requiredto perform sucl_calculations. The above cases were runeither on aCray Y-MP8/864, a Sun SPARC 10/MODEL30, or a SPARe10/MODI_L51 desktop workstation, using 64-bit arithmetic; this corresponds to single-precision on tile Cray and double-precision on the SPARe 10s. Table 1 shows the cpu time required to

ti i ' "perform 100 iterations, the cptJ me/(teration-grxd-pt.), and the total cPU time required to perform theapproximately 3500 iterations typically needed for a converged steady-statbsolution. (The latter figure isprobably conservative.) The timesquoted are only approximate,andmay varyby 10-15%depending on theparticular run. The last row also assumes a calculation begun with uniform freestream propertieseverywhere in the flow.Convergence is possible in considerably fewer iterations by starting the calculationfrom an earlier, converged, solution for condRionsclose to those desired.

TABLE 1. STEADY--STATEAIRFOIL CALCULATION CPU TIMES

Platform: Cray Y-MP Sun SPARC10/30 Sun SPARe10/51Pe'r 1_ iterations .... i5 rain ..... 115min ............ 80 rainPer iter./pergrid ptl............. 4.1 x 10.4 see 311x 10"3see 2.2 x ]0 "-__cConvergedsolu'tion......... 8.8 hrs 67 hrs 47 'hrs

'' '' .... / .......

SUMMARY & CONCLUSIONS

Numerical simulations based on the TLNS equations have been performed for the flow past the SAND0018/50 airfoil using the finite-difference code F3D. Steady-state solutions have been attempted at anglesof attack of 4, 100,and 20. Surface pressures agree reasonably well with the data over most oft.he chordat 4* and 100.Predicted valuesof CL and CDare high by about 25-30%. CLis largely the result of (inviseid)pressure forces, and hence should exhibit a smaller error; why it doesnot remains a matter of investigation.CD is more difficult to predict than CLbecause of the contribution from skin friction, particularly at lowerot's. More study is needed regarding the effects of grid resolution, the turbulence model used, and theassumed transition locations on Co.The steady-state calculations would not converge for ot= 20", where the 0018/50 is known to be deeplystalled. The computed streamline patterns were consistent with this, and suggest that such flows must besimulated with the unsteady terms included. Future plans call for trying to simulate the 0018/50's responsewhile undergoing sinusoidal oscillations in pitch, forwhich experimental data have also been obtained[ 14].These results indicate that while CFD holds much promise for improving our understanding of WECSflowfields, we are a long way from being able to predict their aerodynamics without any recourse toempiricism or wind tunnel data. The phenomena of turbulence and transition, in particular, produce effectswhich presently must be modeled rather than directly simulated. While the time required for suchcalculations may currently prevent their use in day-to-day design activities, the continual improvements indesktop computer power and algorithm efficiency can be expected to lead to progressively moresophisticated engineering codes.

ACKNOWLEDGMENTS

This work was supported by the WindEnergyTechnology Department at Sandia National Laboratories. Theauthor wishes to acknowledge numerous technical discussions with Dr. Dale Berg of that department. He isalso deeply indebted to Dr. Daniel Barnette, currentlyin the P_uallelComputational Sciences Department;without his early development efforts and guidance in tht. use of F3D, this work would not have beenpossible.

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REFERENCES

1. Strickland, J.H., "A Review of Aerodynamic Analysis Methods for the Vertical-Axis Wind Turbine,"Proceedings of the 5th ASME Wind EnergySymp., New Orleans, LA, February 23-26, 1986, pp. 7-17.

2. Hoffmann, K.A., Computational Fluid Dynamics for Engineers, published by Engineering EducationSystem, Austin, TX, 1989.

i

3. Pulliam, T.H. and Steger, J.L., "Implicit Finite-Difference Simulations of Three-DimensionalCompressible Flow," AIAA Journal, Vol. 18, No. 2, February 1980, pp. 159-167.

4. Steger, J.L. and Warming, R.F., "Flux Vector Splitting of the lnviscid Gasdynamic Equations withApplication to Finite-Difference Methods," Journal of Computational Physics, Vol. 40, No. 2, April1981, pp. 263-293.

5. Ying, S.X., Steger, J.L., Sehiff, L.B., and Baganoff, D., Numerical Simulation of Unsteady, Viscous,High Angle of Attack Flows Using a Partially Flux-Split Algorithm, AIAA Paper 86-2179, presentedat AIAA 13th Atmospheric Flight Mechanics Conference,Williamsburg, VA, August 18-20, 1986.

6. Steger, J.L., Ying, S.X., and Schiff, L.B., "A Partially Flux-Split Algorithm for Numerical Simulationof Compressible lnviscid and Viscous Flow," Proceedings of the Workshop on Computational FluidDynamics, Institute of Nonlinear Sciences, University of California, Davis, CA, 1986.

7. Burring, P.G., Chiu, I.T., Obayashi, S., Rizk, Y.M., and Steger, J.L., "Numerical Simulation of theIntegrated Space Shuttle Vehicle in Ascent," AIAA Paper 88-4359, Proceedings of the AIAAAtmospheric Flight Mechanics Conf., Minneapolis, MN, August 15-17, 1988, pp. 265-283.

8. Baldwin, B.S. and Lomax, H., Thin Layer Approximation and Algebraic Model for Separated Turbul,entFlows, AIAA Paper 78-257, pre, ented at the AIAA 16th Aerospace Sciences Meeting, Huntsville, AL,January 16-18, 1978.

9. Reda, D.C., "Observations of Dynamic Stall Phenomena Using Liquid Crystal Coatings," AIAAJournal, Vol. 29, No. 2, February 1991, pp. 308-310.

10. Gregorek, G.M. and Klimas, P.C., "Tailored Airfoils for Wind Turbine Applications," Proceedings ofthe 6th ASME Wind Energy Symposium, Dallas, TX, February 15-18, 1987.

11. Berg, D.E. and Ashwiil, T.D., "An Update on the Structural Design of the Sandia 34-m Vertical-AxisWind Turbine," Proceedings of the 5th ASME Wind Energy Symposium, New Orleans, LA, February23-26, 1986.

12. Anonymous, Eagle User Manual, Vol. 1: Introduction and Grid Applications; Vol. 2: SurfaceGeneration Code; Vol. 3: Grid Generation Code, Mississippi State University, Mississippi State, MS,1988.

13. Zingg, D.W., "Grid Studies for Thin-Layer Navier-Stokes Computations of Airfoil Flowlields," AIAAJournal, Vol. 30, No. 10, October 1992, pp. 2561-2564.

14. Gregorek, G.M., Hoffmann, M.J. and Berchak, M.J., Steady-State and Oscillatory AerodynamicCharacteristics era Sandia 0018/50 Airfoil, Data Report, Aeronautical and Astronautical ResearchLaboratory, Ohio State University, Columbus, OH, August, 1989.

15. Hoffmann, M.J., Aeronautical and Astronautical Research Laboratory, Ohio State tiniversity,Columbus, OH, private communication.

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