JOURNAL OF AIRCRAFT Vol.37,No.4, July –August 2000
Modern Adaptation of Prandtl’s Classic Lifting-Line Theory
† W.F.Phillips ¤ andD. O.Snyder UtahState University, Logan, Utah 84322-4130
The classicalsolution to Prandtl’ s well-knownlifting-line theory applies only to a singlelifting surface withno sweep andno dihedral. However ,Prandtl’s originalmodel of anitelifting surface hasmuch broader applicability. Ageneralnumerical lifting-line method based on Prandtl’ s modelis presented. Whereas classicallifting-line theory isbasedon applying the two-dimensionalKutta –Joukowskilaw to a three-dimensional ow,the present method isbasedon a fullythree-dimensional vortex lifting law. The methodcan be usedfor systems of lifting surfaces witharbitrary camber ,sweep, anddihedral.The accuracyrealized fromthis method is shownto be comparableto thatobtained from numerical panel methods and inviscid computational uiddynamics solutions, but at asmall fractionof the computationalcost.
Nomenclature V =magnitudeof V 1 1 Ar =globalreference area V =local uidvelocity b V =velocityof theuniform owor freestream =twicethe lifting surface semispan 1 C`i =sectionlift coef cient for wing section i =uidvolume vi j =dimensionlessvelocity induced at control point j by C`a =airfoilsection lift slope vortex i,havinga unitstrength C`a i =sectionlift slope for wing section i C` =sectionlift coef cient for an innite wing v =unitvector in directionof freestream 1 1 i Cmi =sectionmoment coef cient for wing section i a i =localangle of attackfor wing section c =localsection chord length a L0i =localzero-lift angle of attack with no apde ection c¯ =overallaerodynamic mean chord length C =vortexstrength in the direction of r0 i c¯i =aerodynamicmean chord length for wing section i C i =strengthof horseshoevortex dAi =differentialplanform area at controlpoint i D G =dimensionlessstrength correction vector dF =differentialaerodynamic force vector d Ai =planformarea of wing section i d` =directeddifferential vortex length vector d i =apdeection for wing section i F =netforce exerted by uidon the surroundings d `i =spatialvector along the bound segment i d Mi =quarter-chordmoment for wingsection i fdi =localsection induced drag per unitspan f` =localsection lift per unit span e i =apeffectivenessfor wing section i G =dimensionlessvortex strength vector ³i =dimensionlessspanwise length vector Gi =dimensionlessvortex strength for section i h = angle from r1 to r2 [J] = N by N matrixof partial derivatives q =uiddensity L r =globalreference length X =relaxationfactor M =netmoment about the c.g. exerted by the uid ! =local uidvorticity N =totalnumber of horseshoevortices n =numberof horseshoevortices per semispan R =residualvector Introduction 1,2 Ra =aspectratio HEdevelopmentof Prandtl’s lifting-linetheory, providedthe ri =vectorfrom c.g. to controlpoint i T rstanalytical method for accurately predicting lift and induced ri1 j =vectorfrom node i1 to control j dragon anitelifting surface. In this theory, Prandtl hypothesized
ri2 j =vectorfrom node i2 to control j thateach spanwise section of anitewing has a sectionlift equiva- r1, r2 =magnitudesof r1 and r2 lentto thatacting on asimilarsection of aninnite wing having the Downloaded by UNIVERSITY OF ILLINOIS on September 24, 2019 | http://arc.aiaa.org DOI: 10.2514/2.2649 r0 =vectorfrom beginning to endof vortexsegment samesection circulation. With this hypothesis, the two-dimensional 3 4 r1 =vectorfrom beginning of vortexsegment to arbitrary vortexlifting law of Kutta andJoukowski wasapplied at each sec- pointin space tionof the three-dimensional wing, to relate the localaerodynamic r2 =vectorfrom end of vortexsegment to arbitrarypoint in forceto the local circulation. However, to xthedirection of the space aerodynamicforce vector, the undisturbed freestream velocity in the s =spanwisecoordinate Kutta–Joukowskilaw was, intuitively and without proof, replaced uai =chordwiseunit vector at controlpoint i withthe vector sum of the freestream velocity and the velocity in- uni =normalunit vector at control point i ducedby thetrailing vortex sheet. This theory gives good agreement usi =spanwiseunit vector at control point i withexperimental data for straightwings of aspect ratio greater than u =unitvector in directionof thefreestream aboutfour. Prandtl’ s lifting-linetheory has had a profoundimpact 1 onthe development of modernaerodynamics and hydrodynamics andis still widely used today. However, conventional lifting-line Received 8October1999; presented as Paper00-0653 at theAIAA 38th theoryapplies only to asinglelifting surface with no sweep and no Aerospace Sciences Meetingand Exhibit (Oct),Reno,NV ,12 –14 January dihedral. 2000;revision received 2February2000; accepted forpublication 10 Febru- Inmost modern textbooks (e.g.,Bertin and Smith 5 ),Prandtl’s hy- ary2000. Copyright c 2000by the American Instituteof Aeronauticsand Astronautics,Inc. All ° rights reserved. pothesisis justi ed basedon the provision that owin the spanwise directionis small. The failure of conventional lifting-line theory to ¤ Professor,Mechanical andAerospace EngineeringDepartment. Member AIAA. accuratelypredict the aerodynamic forces acting on a sweptwing †Graduate Student,Mechanical andAerospace EngineeringDepartment. isusually blamed on the violation of this provision. However, it Member AIAA. canbe shownfrom the three-dimensional vortex lifting law that the 662 PHILLIPS AND SNYDER 663
relationshipbetween section lift and section circulation is not af- Glauert.7 Themost popular method, based on Gaussian quadrature, fectedby owparallel to the bound vorticity (see Saffman6). The wasoriginally presented by Multhopp. 8 Mostrecently, Rasmussen three-dimensionalvortex lifting law requires that, for any volume and Smith9 havepresented a morerigorous and more rapidly con- enclosedby astreamsurface in aninviscid,incompressible, steady vergingmethod, based on aFourierseries expansion similar to that ow,aforcemust be exerted on the surroundings equal to the product rstused by Lotz 10 andKaramcheti. 11 ofthe uiddensity and the cross product of the local uidvelocity Purelynumerical methods for solving the lifting-line equation withthe local uidvorticity, integrated over the volume. Saffman 6 havealso been proposed. McCormick 12 haspresented a numerical presentsa proofof thisvortex lifting law . methodthat can be used for a singlelifting surface having a straight Onlyin thelimiting case of two-dimensional potential owcan liftingline. This method is based on applyingthe two-dimensional thelocal uidvelocity vector in the vortex lifting law be replaced Kutta–Joukowskilaw to thethree-dimensional owand neglects the withthe freestream velocity. For this special case, the general vor- downwashgenerated by thebound vorticity. Results obtained from texlifting law reduces to the two-dimensional Kutta –Joukowski thismethod are essentially identical to thoseobtained from the series law.Strictlyspeaking, the two-dimensional Kutta –Joukowskilaw solution.A numericallifting-line method has also been developed cannotbe used to relate section lift to section circulation in athree- byAndersonet al. 13 thatrelaxes the assumption of linearitybetween dimensionalpotential ow.However,the three-dimensional vortex sectionlift and section angle of attack.For a singlestraight lifting liftinglaw ,appliedto Prandtl’s modelof the nitewing, requires a surface,this method gives good agreement with experimental data localsection lift equal to thecross product of thelocal uidveloc- atanglesof attack both below and above stall. However, the method ityvector with the local circulation vector, multiplied by the uid stillassumes a straightlifting line and ignores the downwash pro- density, q (V C ).Thiscross product results in a sectionlift that is ducedby thebound vorticity. Thus, as isthecase with all methods independentof £ thecomponent of uidvelocity that is parallel to the usedto obtain solutions to the classical lifting-line equation, this boundvorticity. numericalmethod applies only to a singlelifting surface with no Ifthe uid owcomponent parallel to the bound vorticity has sweepand no dihedral. noeffecton thesection lift, why does conventional lifting-line the- Here,a numericallifting-line method is presented that can be oryfail to predict the performance of sweptwings? The answer is usedto obtainthe forces and moments acting on a systemof lift- alsoprovided by the general vortex lifting law .Whencomputing ingsurfaces with arbitrary position and orientation. This method, sectionlift from the general vortex lifting law, the local velocity basedon Prandtl’s originalmodel of a nitewing, accurately pre- inducedon eachvortex segment must include the velocity induced dictsthe effects of bothsweep and dihedral as wellas theeffects of bythe remainder of that same vortex as well as that induced by aspectratio, camber, and planform shape. Results obtained from this allother potentials included within the oweld. This means that, methodare compared with experimental data and with results from whencomputing the lift on any wing section, we must include the othernumerical methods. The accuracy realized from the present velocityinduced by all other vortex segments, free or bound,that methodis shown to becomparableto thatobtained from numerical arecontained within the oweld. In thedevelopment of classical panelmethods and inviscid computational uiddynamics (CFD) lifting-linetheory, Prandtl intuitively added the velocity induced solutions,but at asmallfraction of the computational cost. In ad- bythetrailing vortex sheet to the undisturbed freestream velocity ditionto the obvious applications to aeronautics, this method has specied by thetwo-dimensional Kutta –Joukowskilaw. However, broadapplication to the eldof hydrodynamics,including hydro- hedid not suggest including the velocity induced by one bound foils,marine propellers, and control surfaces. Unlike the classical vortexsegment on another. lifting-linesolution, the presentmethod is notbasedon alinearre- For owover a straightlifting surface, the bound vortex laments lationshipbetween section lift and section angle of attack. Thus, areall reasonably parallel. For any two parallel vortex laments,the themethod could conceivably be applied,with caution, to account forceresulting from the velocity induced on the rst lamentby the approximatelyfor the effects of stall. secondis equal,opposite, and collinear with the force resulting from thevelocity induced on thesecond lamentby the rst.Thus, for Formulation straightlifting surfaces, the parallel nature of the bound vorticity Inwhat is commonly referred to as the numerical lifting-line makesit reasonable to neglectthe interaction between bound vortex method (e.g.,Katz and Plotkin 14 ),anitewing is synthesizedusing lamentsand compute the section lift based only on the velocity acompositeof horseshoeshaped vortices. The continuous distribu- inducedby the trailing vortex sheet and the undisturbed uniform tionof bound vorticity over the surface of thewing, as well as the ow. continuousdistribution of freevorticity in the trailing vortex sheet, For owover a sweptwing, the bound vortex lamentson each isapproximatedby anitenumber of discretehorseshoe vortices, sideof thewing are roughly parallel to each other and to thelocal asshownin Fig. 1. wingquarter chord. However, the bound vortex lamentson one Thebound portion of eachhorseshoe vortex is placed coincident sideof thewing are not parallel to the bound vortex lamentson withthe wing quarter-chord line and is,thus, aligned with the local Downloaded by UNIVERSITY OF ILLINOIS on September 24, 2019 | http://arc.aiaa.org DOI: 10.2514/2.2649 theother side. Thus, for a liftingswept wing, the bound vorticity sweepand dihedral. The trailing portion of eachhorseshoe vortex generatedon oneside of the wing produces downwash on theother isalignedwith the trailing vortex sheet. The left-hand corner of one sideof thewing. This downwash reduces the net lift and increases the totalinduced drag for the wing. The downwash resulting from the boundvorticity is greatest near the center of the wing, whereas thedownwash resulting from the trailing vorticity is greatest near the wingtips. Thus, for a sweptwing, the lift is reduced both near the centerof thewing and near the tips. Prandtl’s classicallifting-line theory is based on alinearrelation- shipbetween section lift and section angle of attack.With this linear assumption,and with the assumption of astraightlifting line, the theoryprovides an analyticalsolution for the spanwise distribution ofliftand induceddrag acting on anitelifting surface. The solution isintheform of anin nite sine series for the circulation distribu- tion.Historically, the coef cients in this sine series have usually beenevaluated from collocation methods. T ypically,the series is truncatedto a niteseries, and the coef cients in the niteseries areevaluated by requiring the lifting-line equation to be satis ed ata numberof spanwise locations equal to the number of terms Fig.1 Horseshoe vortices distributedalong the quarterchord of a intheseries. A verystraightforward method was rstpresented by nitewing with sweep anddihedral. 664 PHILLIPS AND SNYDER
inducedat an arbitrary point in space, by a completehorseshoe vortex, is
C u r2 (r1 + r2)(r1 r2) u r1 V = 1 £ + £ 1 £ 4p r2(r2 u r2) r1r2(r1r2 + r1 r2) ¡ r1(r1 u r1) ¡ 1 ¢ ¢ ¡ 1 ¢ (3)
Asis thecase with panel methods, the user must specify the orien- tationof thetrailing vortex sheet. In obtaining the classical lifting- linesolution for a singlelifting surface with no sweepor dihedral, Prandtlassumed the trailing vortex sheet to be alignedwith the wing Fig.2 Positionvectors describingthe geometryfor a horseshoevortex. chord.This was done to facilitateobtaining an analyticsolution. In obtaininga numericalsolution, there is littleadvantage in aligning thetrailing vortex sheet with a vehicleaxis such as the chord line. horseshoeand the right-hand corner of thenext are placed on the Morecorrectly, the trailing vortex sheet should be alignedwith the samenodal point. Thus, except at thewing tips, each trailing vortex freestream.This is done easily in the numerical solution by setting u equalto the unit vector in thedirection of the freestream. Al- segmentis coincident with another trailing segment from the adja- 1 centvortex. If two adjacent vortices have exactly the same strength, thoughit is intuitively more appealing to align the trailing vortex thenthe two coincident trailing segments exactly cancel because one sheetwith the freestream, in reality, this makes very little difference hasclockwiserotation and the otherhas counterclockwise rotation. inthe nalresult. For typical wings, aligning the trailing vortex Thenet vorticity that is shed from the wing at any internal node is sheetwith the chord line rather than the freestream produces errors simplythe difference in the vorticity of the two adjacent vortices intheresulting forces and moments of less than 1%. Still, because thatshare that node. themethod allows the trailing vortex sheet to be alignedeasily with Eachhorseshoe vortex is composed of three straight vortex seg- thefreestream, this should always be done. ments.From the Biot –Savartlaw and the nomenclature de ned in Whena systemof liftingsurfaces is synthesizedusing N horse- Fig.2, thevelocity vector induced at anarbitrary point in space,by shoevortices, in amannersimilar to that shown in Fig. 1, Eq. (3) can anystraight vortex segment, is readily found to be, for example, see beusedto determine the resultant velocity induced at any point in Bertinand Smith 5 orKatzand Plotkin, 15 space,if the strength of eachhorseshoe vortex is known.However, thesestrengths are not known a priori.T ocomputethe strengths of the N vortices,we must have a systemof N equationsrelating these C r r r r N strengthsto someknown properties of thewing. In what has com- V = 1 £ 2 r 1 2 (1) 2 0 14 4p r1 r2 ¢ r2 ¡ r2 monlybeen referred to asthenumericallifting-line method, these j £ j N equationsare provided by forcinga Neumanncondition, which species zero normal velocity at thethree-quarter chord of thewing AlthoughEq. (1) isintheform commonly found in modern text- sectionmidway between the trailing legs of eachhorseshoe vortex. books,it isnot themost useful form for numerical calculations. The Thismethod works remarkably well for planar swept wings with no inducedvelocity computed from Eq. (1) isindeterminate whenever camber.However, not surprisingly, this method does not work well r1 and r2 arecollinear, even for points that lie outside the vortex forcambered wings or for wings including de ected apsand/ or segment.T oeliminatethis division by zero for points that are not on controlsurfaces. thevortex segment, we can makeuse ofthetrigonometric relations Inreality, this numerical lifting-line method is simply the vortex latticemethod 16,17 appliedusing only a singlelattice element, in the chordwisedirection, for each spanwise subdivision of the wing. r = r r , r r = r r cosh , r r = r r sinh 0 1 ¡ 2 1¢ 2 1 2 j 1 £ 2j 1 2 Applyingthe Neumann condition at only one pointin the chordwise directionis clearly not adequate for wing sections with camber or Usingthese relations we have apdeection. This method gives a resultthat depends only on the positionand slope of the camber line at thethree-quarter chord. The r r r r r r r predictedperformance is completely independent of camber line 0 1 2 = 1 ¡ 2 1 2 shapeat any other chordwise location. This is clearlynot realistic. r r 2 ¢ r ¡ r r r 2 ¢ r ¡ r j 1 £ 2j 1 2 j 1 £ 2j 1 2 Fora morepragmatic approach, we turn to the general three- dimensionalvortex lifting law , 6 1 r1 r2 r1 r2 Downloaded by UNIVERSITY OF ILLINOIS on September 24, 2019 | http://arc.aiaa.org DOI: 10.2514/2.2649 = r1 + r2 ¢ ¢ r 2r 2 sin2 h ¡ r ¡ r 1 2 2 1 F = q (V !) d £ (r + r )(1 cosh ) r + r = 1 2 ¡ = 1 2 r 2r 2(1 cos2 h ) r 2r 2(1 + cosh ) 1 2 ¡ 1 2 UsingPrandtl’ s hypothesis,we assume that each spanwise wing sectionhas a sectionlift equivalent to that acting on a similarsection r1 + r2 = ofaninnite wing with the same local angle of attack.Thus, applying r r (r r + r r ) 1 2 1 2 1 ¢ 2 thevortex lifting law to a differentialsegment of theliftingline, we have and Eq. (1) canbe moreconveniently written as dF = q C V d` (4) £ C (r1 + r2)(r1 r2) V = £ (2) Ifowover a nitelifting surface is synthesizedfrom a uniform ow 4p r1r2(r1r2 + r1 r2) ¢ combinedwith horseshoe vortices placed along the quarter-chord line,from Eq. (3),thelocal velocity induced at a controlpoint placed Noticethat, unlike the result from Eq. (1),theinduced velocity anywherealong the bound segment of horseshoe vortex j is computedfrom Eq. (2) isnot singularwhen the angle from r1 to r2 iszero.It is, however, still singular when this angle is p . N § C i vi j Whenwe use Eq. (2) forthe nitebound segment and the two V j = V + (5) 1 c¯i semi-innite trailing segments shown in Fig.2, thevelocity vector i = 1 PHILLIPS AND SNYDER 665
where vi j isthe dimensionless induced velocity where
u ri2 j V d`i C i 1 £ v 1 , ³i c¯i , Gi 1 ri2 j ri2 j u ri2 j ´ V ´ dAi ´ c¯i V ¡ 1 ¢ 1 1
c¯ ri1 j + ri2 j ri1 j ri2 j and,from Eqs. (5) and (9),thelocal angle of attackwritten in terms i + £ , i = j 4p r r r r + r r 6 ofthedimensionless variables is given by i1 j i2 j i1 j i2 j i1 j ¢ i2 j u ri1 j 1 N vi j £ v + v ji G j uni ¡ 1 j = 1 ´ ri1 j ri1 j u ri1 j 1 ¢ ¡ 1 ¢ a i = tan¡ (12) N u ri2 j v + v ji G j uai 1 £ 1 j = 1 ¢ c¯ ri2 j ri2 j u ri2 j i ¡ 1 ¢ , i = j (6) 4p u ri1 j Equation (11) canbe writtenfor N differentcontrol points, one 1 £ ¡ associatedwith each of the N horseshoevortices used to synthesize ri1 j ri1 j u ri1 j ¡ 1 ¢ thelifting surface or system of liftingsurfaces. This provides a sys- At thispoint, c¯i couldbe any characteristic length associated with tem of N nonlinearequations relating the N unknowndimensionless thewing section aligned with horseshoe vortex i.Thischaracteristic vortexstrengths Gi toknown properties of thewing. At anglesof lengthis simply used to nondimensionalize Eq. (6) andhasnoeffect attackbelow stall, this system of nonlinear equations surrenders ontheinduced velocity. The choice of characteristiclength will be quicklyto Newton’ s method. addressedlater. The bound vortex segment is excluded from Eq. (6), ToapplyNewton’ s method,the system of equationsis written in when i = j,becausea straightvortex segment induces no downwash thevector form: alongits own length. From Eqs. (4)and (5),theaerodynamic force acting on aspanwise (G) = R (13) differentialsection of thelifting surface located at controlpoint i is given by where N N C j i (G) = 2 v + v ji G j ³i Gi C`i (a i , d i ) (14) dFi = q C i V + v ji d`i (7) 1 £ ¡ 1 c¯ j £ j = 1 j = 1 Whenwe allow for the possibility of apde ection, the local section Wewishto ndthe vector of dimensionless vortex strengths G that liftcoef cient for the airfoil section located at control point i is a makesall components of theresidual vector R gotozero.Thus, we functionof localangle of attackand local apde ection, wantthe change in theresidual vector to be R.Westart with an initialestimate for the G vectorand iteratively ¡ re ne the estimate C`i = C`i (a i , d i ) (8) byapplyingthe Newton corrector equation
Thelocal angle of attackat controlpoint i is [J]D G = R (15) ¡ 1 Vi uni N N a = tan¡ ¢ (9) where [J] is the by matrixof partial derivatives i V u i ¢ ai 2wi (v j i ³i ) where uai and uni are,respectively, the unit vectors in the chordwise ¢ £ Gi wi directionand the direction normal to thechord, both in the plane of j j , j = i thelocal airfoil section as shown in Fig.3. If therelation implied @C v (v u ) v (v u ) 6 `i ai ji ¢ ni ¡ ni ji ¢ ai by Eq. (8) isknown at eachsection of the wing, the magnitude of 2 2 ¡ @a i vai + vni theaerodynamic force acting on aspanwisedifferential section of @ i Ji j = = thewing located at control point i canbe writtenas @G j 2wi (v ji ³i ) 2 wi + ¢ £ Gi 1 2 ( ) j j wi dFi = 2 q V C`i (a i , d i ) dAi 10 j j 1 j j , j = i v v Settingthe magnitude of theforce from Eq. (7) equalto that obtained @C`i ai (v ji uni ) ni (v j i uai ) ¢ 2 ¡ 2 ¢ from Eq. (10) andrearranging, we can write ¡ @a i vai + vni Downloaded by UNIVERSITY OF ILLINOIS on September 24, 2019 | http://arc.aiaa.org DOI: 10.2514/2.2649 N (16)
2 v + v ji G j ³i G i C`i (a i , d i ) = 0 (11) N 1 £ ¡ j = 1 wi v + v ji G j ³i (17) ´ 1 £ j = 1
N
vni v + v j i G j uni (18) ´ 1 ¢ j = 1
and
N
vai v + v ji G j uai (19) ´ 1 ¢ j = 1
Using Eq. (16) in Eq. (15),wecompute the correction vector D G. Thiscorrection vector is usedto obtain an improved estimate for thedimensionless vortex strength vector G accordingto Fig.3 Unitvectors describingthe orientationof the localairfoil sec- tion. G = G + X D G (20) 666 PHILLIPS AND SNYDER
Thisprocess is repeated until the magnitude of thelargest residual Theaerodynamic moment generated about the center of gravity islessthan some convergence criteria. For angles of attackbelow is stall,this method converges very rapidly using almost any initial estimatefor G anda relaxationfactor X ofunity.At anglesof attack N N C i C j beyondstall, the method must be highlyunder relaxed and is very M = q ri C i V + v j i d `i + d Mi (27) £ 1 c¯ j £ sensitiveto the initial estimate for G. i = 1 j = 1 Forthe fastest possible convergence of Newton’ s method,we requirean accurate initial estimate for the dimensionless vortex Ifweassume a constantsection moment coef cient Cmi over each strengthvector. For this purpose, a linearizedversion of Eq. (11) is spanwiseincrement, then useful.For a straightlifting surface of in nite aspect ratio at small
anglesof attack,the downwash is zero, the section lift is a linear s2 1 2 2 functionof angleof attack,and all nonlinear terms in Eq. (11)vanish. d Mi = q V Cmi c ds usi (28) ¡ 2 1 Fora liftingsurface of high aspect ratio with no sweepor dihedral, s = s1 atsmallangles of attack, we can still ignore the nonlinear terms and
computean approximate dimensionless vortex strength vector from where usi isthelocal spanwise unit vector shown in Fig.3, thislinear system. At smallangles of attack, the local section lift coef cient can be usi = uai uni (29) approximatedas £
C (a , d ) = C (a a + e d ) (21) Whenwe use Eq. (28) in Eq. (27) andnondimensionalize `i i i » `a i i ¡ L0i i i Usingthe small angle approximation for both the geometric angle M N N ( ) = 2ri Gi v + G i G j v ji ³i ofattackand theinduced angle of attack,after applying Eq. 12 to 1 q V 2 A L £ 1 £ 2 r r i = 1 j = 1 Eq. (21), we have 1
N C s2 d A mi c2 s i ( ) C`i = C`a i v uni + v j i uni G j a L0i + e i d i (22) d usi 30 » 1 ¢ ¢ ¡ ¡ d Ai Ar Lr j = 1 s = s1
ApplyingEq. (22) to Eq. (11) andignoring second-order terms, we Tothispoint, the local characteristic length c¯i hasnot been de- obtainthe linear system ned.It couldbe any characteristic length associated with the span- wiseincrement of the lifting surface that is associatedwith horseshoe N vortex i. From Eq. (30),wesee that the natural choice for this lo- calcharacteristic length is the integral of the chord length squared, 2 v ³i Gi C`a i v ji uni G j = C`a i (v uni a L0i + e i d i ) j 1 £ j ¡ ¢ 1 ¢ ¡ j = 1 withrespect to the spanwise coordinate, divided by theincremental (23) area.If we also assume a linearvariation in chord length over each spanwiseincrement, we have Equation (23) givesgood results, at small angles of attack, for a singlelifting surface of high aspect ratio with no sweep or dihedral. s2 c + c d A = c ds = i1 i2 s s (31) Forlarger angles of attack, highly swept wings, or for interacting i 2 i2 ¡ i1 systemsof lifting surfaces, the nonlinear system given by Eq. (11) s = s1 shouldbe used. However, Eq. (23) providesa reasonableinitial estimatefor the dimensionless vortex strength vector, to be used and withNewton’ s methodfor obtaining a solutionto this nonlinear s2 c2 + c c + c2 system. 1 2 2 i1 i1 i2 i2 c¯i c ds = (32) ´ d Ai 3 ci + ci AerodynamicForces and Moments s = s1 1 2 Oncethe vortex strengths have been determined, the total aero- Whenwe use these de nitions in Eq. (30),thedimensionless aero- dynamicforce vector can be determinedfrom Eq. (7).Ifthe lifting dynamicmoment about the c.g. is surfaceor surfaces are synthesized from a largenumber of horseshoe vortices,each covering a smallspanwise increment of one lifting N N M surface,we can approximate the aerodynamic force as beingcon- = 2ri Gi v + G i G j v ji ³
Downloaded by UNIVERSITY OF ILLINOIS on September 24, 2019 | http://arc.aiaa.org DOI: 10.2514/2.2649 i 1 q V 2 A L £ 1 £ stantover each spanwise increment. Then, from Eq. (7), the total 2 r r = 1 = 1 1 i j aerodynamicforce is given by
N N d Ai C i C j Cmi c¯i usi (33) F = q C i V + v ji d `i (24) ¡ Ar L r 1 c¯ j £ i = 1 j = 1 Once the N dimensionlessvortex strengths Gi areknown, Eqs. (25) where d `i isthe spatial vector along the bound segment of horseshoe and (33) areused to evaluate the aerodynamic forces and moments. vortex i fromnode 1 tonode2, inthe direction of segmentvorticity. Likepanel methods, lifting-line theory provides only a potential Whenwe nondimensionalize Eq. (24),thetotal nondimensional owsolution. Thus, the forces and moments computed from this aerodynamicforce is methoddo not include viscous effects, so that the parasitic drag is unobtainable.In addition to thisrestriction, that also applies to panel N N F d Ai methods,lifting-line theory imposes an additional restriction that = 2 Gi v + Gi G j v ji ³ (25) 1 2 i doesnot applyto panel methods. For lifting surfaces with low aspect q V A 1 £ Ar 2 r i = 1 j = 1 1 ratio,Prandtl’ s hypothesisbreaks down, and the usual relationship betweenlocal section lift and local section angle of attack no longer where d Ai istheplanform area of segment i, applies.It has long been established that lifting-line theory gives
s2 goodagreement with experimental data for lifting surfaces of aspect 2 d Ai = c ds (26) ratiogreater than about four. Forlifting surfaces of loweraspect s = s1 ratio,panel methods or CFD solutionsshould be used. PHILLIPS AND SNYDER 667
Fig.4 Lifting-linegrid with cosine clustering and 20 elements per semispan.
GridGeneration and Control Points Eachlifting surface must, of course, be divided into spanwise elements,in a mannersimilar to that shown symbolically in Fig. 1. InFig.1, thewing is dividedinto elements of equalspanwise in- crement.However, this is not the most ef cient way in which to grida liftingsurface. Because vorticity is shedfrom a liftingsur- facemore rapidly in the region near the tips, the nodal points should beclusteredmore tightly in this region for best computational ef - Fig.5 Sectionlift distribution for three ellipticwings. ciency.The authors have found conventional cosine clustering to be quiteef cient. For straight lifting surfaces, clustering is only needed nearthe tips and the cosine distribution can be appliedacross the entirespan. However, for a liftingsurface with sweep and/ ordihe- dral,there is astepchange in theslope of the quarter chord at the root.This step change causes the downwash to changevery rapidly inthe region near the root. Thus, in general, the authors recom- mendapplying cosine clustering independently over each semispan ofeachlifting surface, as shown in Fig.4. Thisclusters the nodes moretightly at boththe tip and theroot. This clustering is based on thechange of variables, s/ b = [1 cos(u )]/ 4 (34) ¡ Overeach semispan, u variesfrom zero to p as s variesfrom zero to b/2.Distributingthe nodes uniformly in u willprovide the desired clusteringin s.Ifthe total number of horseshoeelements desired on eachsemispan is n,thespanwise nodal coordinates are computed from, s /b = 1 [1 cos(ip / n)], 0 i n (35) Fig.6 Sectioninduced drag distribution for three ellipticwings. i 4 ¡ · · i wherethe bound segment of horseshoevortex extendsfrom node Thissame solution gives the local section induced drag i to node i 1ona leftsemispan and from node i 1 to node i onarightsemispan. ¡ The authors have found that using ¡ this nodal fdi 4Ra 2 1 = [1 (2s/ b) ] 2 (38) distributionwith about 40 horseshoe elements per semispan gives 1 2 2 2 ¡ 2 q V cC¯ ` p R + C thebest compromise between speed and accuracy. Figure 4 shows 1 1 a `a asystemof liftingsurfaces overlaid with a gridof thistype using 20elementsper semispan. Asa rst-ordertest, the numerical method presented here should Becausesingularities occur at the junctures of adjacent bound- agreewith Eqs. (37) and (38),atsmallangles of attack, for a straight vortexsegments, for maximum accuracy and computational ef - ellipticwing with no geometric or aerodynamic twist. Figure 5 ciency,the location of controlpoints must be criticallyassessed. At showsa comparisonbetween the section lift distribution predicted rstthought, it wouldseem most reasonable to placecontrol points by Eq. (37) andthat predicted by the present numerical method onthebound segment of each vortex, midway between the two trail- forthree straight elliptic wings of different aspect ratio. Figure 6 comparesthe induced drag distribution predicted by Eq. (38) to
Downloaded by UNIVERSITY OF ILLINOIS on September 24, 2019 | http://arc.aiaa.org DOI: 10.2514/2.2649 inglegs. However, the authors have found that this does not givethe bestnumerical accuracy. A signicant improvement in accuracy, for thatpredicted from the present method for the same three elliptic agivennumber of elements,can be achieved by placingthe control wings.A similarcomparison for tapered wings shows very similar pointsmidway in u ratherthan midway in s.Thus,the spanwise results.The numerical results shown in Figs. 5 and6 wereall gen- controlpoint coordinates are computed from eratedusing 40 horseshoeelements per semispan for wings having aNACA 2412airfoil section. Similar comparisons were made for 1 ( ) si / b = 4 {1 cos[(ip / n) (p /2n)]}, 1 i n 36 otherstraight wings having varying amounts of camber,taper, and ¡ ¡ · · washout.Camber was varied from 0 to8%, taper ratio was varied Thisdistribution places control points very near the spatial midpoint from0.1 to 1.0, and washout was varied from 0 to5deg.In all cases, ofeachbound vortex segment, over most of thewing. However, near thenumerical solution using 40 horseshoeelements per semispan theroot and the tip, these control points are signi cantly offset from agreedwith the classical lifting-line solution to within two-tenths thespatial midpoint. of1%forthe induced drag and towithin ve-hundredthsof 1% for thelift. Thus, for all practical purposes, results obtained from this Results numericallifting-line method are identical to those obtained from Forstraight elliptic wings with no geometric twist and no aero- theclassical lifting-line solution for a singlelifting surface with dynamictwist, Prandtl’ s classicallifting-line theory gives a very nosweepor dihedral.Because, for straight wings of aspect ratios simpleand well-known closed-form solution. From this solution, greaterthan about four, the classical solution is known to adequately thesection lift distribution in thespanwise direction is givenby predictthe effects of aspectratio, camber, and planform shape, the samecan be saidfor the present numerical solution. f` 4Ra 2 1 = [1 (2s/ b) ] 2 (37) Unlikethe analytical solution to Prandtl’ s classicallifting-line 1 2 p R + C ¡ 2 q V cC¯ ` a `a theory,the present numerical method can be applied to wingswith 1 1 668 PHILLIPS AND SNYDER
sweepand/ ordihedral. T oexaminehow well the present method predictsthe effects of sweepand dihedral, results obtained from this methodwere compared with results obtained from a numericalpanel methodand from an inviscidCFD solution.The results predicted for theeffects of sweepwere also compared with limited experimental data. Forthe numerical panel method comparison, the commercial code PMARC18,19 wasselected. This code was developed at NASA Ames ResearchCenter and is one of the most ef cient numerical panel codesavailable. PMARC uses atquadrilateral panels with uni- formsource and doublet distributions on eachpanel. This code also accountsfor the effects of wake rollup, using an unsteady wake developmentapproach. Thecommercial code WIND, 20 developedby the NP ARC Al- liance,was chosen for the inviscid CFD comparison.This code uses anode-centered nitevolume approach to solve the Euler equa- tionson a structuredgrid. The available fth-orderupwind-biased discretizationscheme was used to reduce the effects of numerical Fig.8 Comparisonbetween the induceddrag coef cient predicted by viscosity. the numericallifting-line method, PMARC, and WIND withthat ob- Forall three numerical methods the grid size was repeatedly re- tainedfrom experimental data, for a straightwing and a wingwith 45degof sweep. neduntil the solution was no longersigni cantly affected by further gridre nement. By this procedure, an optimumgrid for each of the threemethods was selected as that which gave the best combina- Fromthe results shown in Fig.7, we see that the lift coef cient tionof accuracy and computationalspeed. With this method of grid predictedby all three methods is in good agreement with experi- resolution,the numerical lifting-line method required 40 sectionel- mentalobservations for both wings. From Fig. 8 wesee that, for ementsper semispan. Grid-resolved solutions for PMARC required thestraight wing, the induced drag predicted by both the numeri- 4500panels per semispan.The selected grid was partitioned to give callifting-line method and by PMARC isin goodagreement with 45segmentsalong the semispan and 90 aroundthe circumference, experimentaldata, whereas WIND givesan induced drag that is withpanels clustered near the leading edge, the trailing edge, and the somewhathigher. The predictions for induced drag on the swept wingtips.For WIND, atwo-blockH –Htypemesh with 1 .25 106 £ wingare not nearly as good.For this wing, the induced drag pre- gridpoints per semispan gave the best compromise between accu- dictedby thepanel code is about40% less than that observed exper- racyand computational ef ciency. The computational domain for imentally.However, both the numerical lifting-line method and the theWIND solutionsextended approximately 30 chords from the CFD solutiongive induced drag values that are about 25% above wingin all directions, with grid points clustered near the wing and theexperimental values. Still, the values of induceddrag predicted thetrailing-tip vortices. When these resolved grids were used for bythenumerical lifting-line method are as goodas or betterthan thethree methods, the computational time required to obtain a solu- thosepredicted by theother two methods, even for this highly swept tionfrom PMARC wasabout 2 104 timesthat required using the £ wing.Furthermore, the induced drag on ahighlyswept wing, as pre- presentnumerical lifting-line method. The WIND solutionsrequired dictedby thepresent method, appears to besomewhat conservative, approximately2 106 timesas longas the lifting-line solutions. £ whereasthe panel code predicts an induced drag that is too low . Acomparisonbetween results obtained from these three numer- Thoughnone of the numerical methods tested seemed to do very icalmethods and previously published experimental data is shown wellat predicting the effects of sweepon induceddrag, all three of inFigs.7 and8. The solid lines and lledsymbols correspond to themethods appeared to do agoodjob of predicting the effects of astraightwing of aspect ratio 6.57, with experimental data ob- 21 sweepon lift. tainedfrom McAlister and T akahashi. Thedashed lines and open Toobtainthe numericallifting-line solutions shown in Figs.7 and symbolsare fora 45-degswept wing of aspectratio 5.0, having ex- 22 8,wehave used the linear relationship between a two-dimensional perimentaldata reported by Weberand Brebner. Bothwings have sectionlift coef cient and a sectionangle of attack that is predicted symmetricalairfoil sections with no geometrictwist and constant bythinairfoil theory. Because the downwash and induced angle of chordthroughout the span. The straight wing has a thicknessof 15% attackare obtained from the solutionto thenonlinearsystem given andthe swept wing has a thicknessof 12%. by Eq. (11),wehave no reason to expect that the lift coef cient predictedfor the complete wing should be alinearfunction of geo- metricangle of attack.However, the results in Fig. 7 showthat this Downloaded by UNIVERSITY OF ILLINOIS on September 24, 2019 | http://arc.aiaa.org DOI: 10.2514/2.2649 relationshipis, in fact, very nearly linear for both wings. The result predictedfor the straight wing is almost exactly linear, as is also predictedby theclassical lifting-line solution. Even for the highly sweptwing, the deviation from linearity is quite small. Both the panelcode and the experimental data con rm this result. Theoverprediction of induceddrag by theCFD codefor both the straightwing and theswept wing is likelydue tonumerical viscosity andis expected. However, the reason that the numerical lifting-line methodoverpredicts induced drag for theswept wing, but not forthe straightwing, may be lessobvious. The foundation of lifting-line theoryrequires the bound vorticity to followthe chordwise aerody- namiccenter of thewing. The numerical lifting-line solution shown inFig.8 wasobtained by assuming that the lifting-line follows the wingquarter chord, which is the theoretical aerodynamic center fromthin airfoil theory. With this assumption, the discontinuity in theslope of thequarter chord at the spanwise midpoint of ahighly sweptwing produces a ratherstrong singularity (see,for example, 23 Fig.7 Comparisonbetween the liftcoef cient predicted bythe numer- Chengand Meng ).Thisresults in theprediction of strongdown- icallifting-line method, PMARC, and WIND withthat obtained from washand large induced drag in theregion near this singularity. In experimentaldata, for a straightwing and awingwith 45 degof sweep. reality,the experimental results of W eberand Brebner 22 show that, PHILLIPS AND SNYDER 669
Fig.9 Deviationof the aerodynamiccenter fromthe quarterchord in the regionnear the spanwisemidpoint of ahighlyswept wing. Fig.11 Pressure forces andshed vorticity for a systemof liftingsur- faces aspredicted bythe numericallifting-line method.
arecombined and forced to satisfy Eq. (11) asasinglesystem of coupledequations, the interactions between lifting surfaces are di- rectlyaccounted for. An example of suchinteractions can be seenin Fig.11. The lifting surfaces shown in Fig. 11 all have symmetric air- foilsections and are oriented to givea zerogeometric angle of attack. Theonly direct production of lift in thiscon guration comes from adeection of theailerons. The de ected ailerons produce lift and vorticitythat in turn induce lift and vorticity on allother surfaces. In Fig.11 thearrows indicate the magnitude and direction of the local sectionforce, and the streamwise lines at thetrailing edge of each liftingsurface indicate the magnitude of thelocal shed vorticity. As canbe seen from Fig. 11, the lifting surface interactions predicted bythe present method are at least qualitatively correct. However, furtherstudy is needed to determinethe quantitative accuracy of the predictedlifting surface interactions. Fig.10 Comparison between the rollingmoment coef cient predicted Thepresent numerical method contains no inherentrequirement bythe numericallifting-line method, PMARC, and WIND forwings fora linearrelationship between section lift and section angle of at- with10 and20 deg of dihedral. tack.Thus, the method could conceivably be applied,with caution, toaccount approximately for the effects of stall. The lifting-line methodrequires a knownrelationship for the section lift coef cient nearthe spanwise midpoint of a highlyswept wing, the aerodynamic asafunctionof sectionangle of attack. Because such relationships centermoves considerably aft of thewing quarter chord, as shown mustbe obtainedexperimentally beyond stall, the present method inFig.9. Thisremoves the discontinuity in slope at thespanwise wouldpredict stall by usinga semi-empiricalcorrection to an other- midpointof thelifting line and signi cantly reduces the predicted wisepotential owsolution. For this reason, the method should be induceddrag. Unfortunately, there is currentlyno simplemeans to usedwith extreme caution for angles of attackbeyond stall. How- predictthe true chordwise aerodynamic center of a highlyswept ever,the method may be ableto predictthe onset of stall.Further wingin the region near the spanwise midpoint. Thus, the present studyis neededto determine to whatextent the method can be used methodcannot easily be correctedto remove this error. atanglesof attacknear or beyondstall. Theaccuracy of the present numerical lifting-line method for predictingdihedral effects was also tested by comparingthis method withboth PMARC andWIND. Figure10 shows the variation in Conclusions rollingmoment coef cient with sideslip angle for two nonplanar Theinsight of Ludwig Prandtl (1875–1953) wasnothing short of rectangularwings having different dihedral, as predicted by allthree astonishing.This was never more dramatically demonstrated than in methods.Both wings have symmetric NACA 0015airfoil sections thedevelopment of hisclassicallifting-line theory, during the period withno sweep or twistand an aspectratio of 6.57.The solid line and from 1911–1918.The utility of this simple and elegant theory is so Downloaded by UNIVERSITY OF ILLINOIS on September 24, 2019 | http://arc.aiaa.org DOI: 10.2514/2.2649 lledsymbols correspond to a wingwith 20 degdihedral, whereas greatthat it isstill widely used today. Furthermore, with a fewminor thedashed line and open symbols are for a wingwith 10 degdihedral. alterationsand the use of amoderncomputer, the model proposed FromFig. 10, wesee that all three methods agree very closely in byPrandtlcan be usedto predict the inviscid forces and moments theirpredictions for the wing with 10 degdihedral. For this wing actingon liftingsurfaces of aspectratio greater than about four with theagreement between the lifting-line method and the panel code is anaccuracyas goodas thatobtained from modern panel codes or within2%, and the CFD solutionagrees with the lifting-line solution CFD, butat a smallfraction of thecomputational cost. towithin7%. The three methods do, however, diverge somewhat in theirpredictions for the wing with 20-deg dihedral. This divergence References becomesmore pronounced at very large sideslip angles. For the 1Prandtl,L., “Trag ugel¨ Theorie,” Nachrichtenvon der Gesellschaftder wingwith 20-deg dihedral at sideslip angles in therange of 12deg, Wisseschaftenzu G ottingen¨ , Geschaeftliche¨ Mitteilungen,Klasse, 1918, thepanel code predicts a rollingmoment coef cient that is about 9% pp. 451–477. abovethat predicted by the lifting-line method, whereas the CFD 2Prandtl,L., “Applicationsof Modern Hydrodynamics to Aeronautics,” NACA 116,June 1921. solutionpredicts a resultthat is nearly10% below the lifting-line 3 solution. Kutta,M. W.,“Auftriebskr afte¨ in Stromenden¨ Flussigkeiten,”¨ Illustrierte AeronautischeMitteilungen ,Vol.6, 1902,p. 133. Thepresent numerical method could be usedto predictthe aero- 4Joukowski,N. E.,“ Surles TourbillonsAdjionts,” Trarauxde laSection dynamicforces and moments acting on asystemof lifting surfaces Physiquede laSocieteImperiale des Amis des Sciences Naturales , Vol. 13, witharbitrary position and orientation. Each lifting surface would No. 2, 1906. besynthesized by distributing horseshoe vortices on a gridstruc- 5Bertin,J. J., and Smith, M. L.,“IncompressibleFlow About Wings of turedsimilarly to that shown in Fig. 4. Because all of the horseshoe FiniteSpan,” Aerodynamicsfor Engineers ,3rded., Prentice –Hall, Upper vorticesused to synthesizethe complete system of liftingsurfaces SaddleRiver, NJ, 1998,pp. 261 –336. 670 PHILLIPS AND SNYDER
6Saffman,P .G.,“ VortexForce and Bound V orticity,” Vortex Dynamics , Hill,New York,1991, pp. 291 –294. CambridgeUniv. Press, Cambridge,England, U.K., 1992, pp. 46 –48. 16Falkner,V .M.,“TheCalculation of Aerodynamic Loading on Sur- 7Glauert,H., TheElements ofAerofoil and Airscrew Theory , 2nd ed., faces ofAny Shape,” Reportsand Memoranda 1910 ,AeronauticalResearch CambridgeUniv. Press, Cambridge,England, U.K., 1959, pp. 142 –145. Council,London, Aug. 1943. 8Multhopp,H., “ Die Berechnungder Auftriebs V erteilungvon Trag- 17Multhopp,H., “ Methodfor Calculating the Lift Distribution of Wings ugeln,” Luftfahrtforschung ,Vol.15, No.14, 1938,pp. 153 –169. (SubsonicLifting Surface Theory ),” Reportsand Memoranda 2884 , Aero- 9Rasmussen, M.L.,andSmith, D. E.,“ Lifting-LineTheory for Arbitrary nauticalResearch Council,London, Jan. 1950. ShapedWings,” Journalof Aircraft ,Vol.36, No. 2, 1999,pp. 340 –348. 18Katz, J.,andMaskew, B., “ UnsteadyLow-Speed Aerodynamics Model 10Lotz,I., “Berechnungder Auftriebsverteilug beliebig geformter Flugel,” forComplete Aircraft Congurations,” Journalof Aircraft ,Vol.25, No. 4, Zeitschriftf ur¨ Flugtechnik und Motorluftschiffahrt ,Vol.22, No. 7, 1931, 1988,pp. 302 –310. pp. 189–195. 19Ashby,D. L.,Dudley, M. R.,and Iguchi, S. K.,“ Developmentand 11Karamcheti, K.,“ Elements ofFinite Wing Theory,” Ideal-FluidAero- Validationof anAdvanced Low-Order Panel Method,” NASA TN-101024, dynamics,Wiley,New York,1966, pp. 535 –567. Oct. 1988. 12McCormick,B. W.,“ TheLifting Line Model,” Aerodynamics,Aeronau- 20Bush,R. H.,Power, G. D.,andTowne, C. E.,“WIND: TheProduction tics andFlight Mechanics ,2nded., Wiley, New York,1995, pp. 112 –119. FlowSolver of theNPARC Alliance,” AIAA Paper98-0935, Jan. 1998. 13Anderson,J. D.,Jr.,Corda, S., andV anWie, D. M.,“Numerical Lifting- 21McAlister,K. W.,andT akahashi,R. K.,“ NACA 0015Wing Pressure LineTheory Applied to Drooped Leading-Edge Wings Below and Above andTrailing V ortexMeasurements,” NASA TP-3151,Nov. 1991. Stall,” Journalof Aircraft ,Vol.17, No.12, 1980,pp. 898 –904. 22Weber,J., and Brebner, G. G.,“Low-SpeedTests on45-degSwept-Back 14Katz, J.,and Plotkin, A., “ Lifting-LineSolution by Horseshoe Ele- Wings,Part I: Pressure Measurements onWingsof Aspect Ratio5,” Reports ments,” Low-SpeedAerodynamics, from Wing Theory to Panel Methods , andMemoranda 2882 ,AeronauticalResearch Council,London, May 1958. McGraw–Hill,New York,1991, pp. 379 –386. 23Cheng,H. K.,and Meng, S. Y.,“TheOblique Wing as aLifting-Line 15Katz, J.,and Plotkin, A., “ Constant-StrengthV ortexLine Segment,” Problemin Transonic Flow,” Journalof Fluid Mechanics ,Vol.97, No. 3, Low-SpeedAerodynamics, from Wing Theory to Panel Methods , McGraw– 1980,pp. 531 –556. Downloaded by UNIVERSITY OF ILLINOIS on September 24, 2019 | http://arc.aiaa.org DOI: 10.2514/2.2649