Finite Wing Theory.doc

8/10/2019 Finite Wing Theory.doc

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Finite Wing Theory

Wing Models

One may apply the results of 3-D potential theory in several ways. We first consider thetheory of finite wings.

We might start out by saying that each section of a finite wing behaves as described byour 2-D analysis. If this were true then we would still find that the lift curve slope was 2per radian that the drag was ! and the distribution of lift would vary as the distribution ofchord. "nfortunately! things do not wor# this way. $here are several reasons for this%

One e&planation is that the high pressure on the lower surface of the wing and the lowpressure on the upper surface causes the air to lea# around the tips! causing a reduction inthe pressure difference in the tip regions. In fact! the lift must go to 'ero at the tipsbecause of this effect. We will ne&t see how and why we must model the 3-D wingdifferently from 2-D.

If we were to ta#e the naive view that the 2-D modelwould wor# in 3-D! we might have the picture shown onthe right. If each section had the distribution of vorticityalong its chord that it had in 2-D! the lift would beproportional to the chord! and would not drop off at thetips as we #now it must.

$his sort of model does not conform to our physicalpicture of what happens at the wing tips. (nd indeed! it does not satisfy the e)uations of3-D fluid flow. $he reason that this does not wor# is that in this case the streamlines arenot confined to a plane. $hey move in 3-D and the flow pattern is )uite different.

We could go bac# to the governing e)uations and start simply with the linear *aplacee)uation. +y superimposing #nown solutions we could obtain a simple model of a 3-Dwing. We might start by superimposing vortices on the wing itself%


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+ut this is no more than the strip theory model that did not wor#. $he reason that thismodel ,which seems ust to be a superposition of #nown solutions is not ade)uate is thatit violates the governing e)uations in certain regions. $he model does not satisfy the/elmholt' laws since vorticity ends in the flow near the tips. 0ome additionalre)uirements must be imposed on the model. $he re)uirements for such a model are ust

the /elmholt' vorte& theorems! discussed previously.

Our simple 3-D model above may be modified as shown below to satisfy the first of the/elmholt' theorems.

In fact! as can be seen from the picture here! this vorte& model is not too far from reality.

$he downwash field and the e&istence of trailing vortices are not ust some strangemathematical result. $hey are necessary for the conservation of mass in a 3-D flow.

(ir is pushed downward behind the wing! but this downward velocity does not persist farfrom the wing. Instead it must move outward. $he outward-moving air is then s)uee'edupward outboard of the wing and the flow pattern shown above develops.

$he trailing vorte& is visuali'ed by 1(0( engineers by flying an agricultural airplanethrough a sheet of smo#e.


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$he main effect of this vorte& wa#e is to produce a downwash field on the wing.

$his downwash field has several very significant effects%

It changes the effective angle of attac# of the airfoil section. $his changes the liftcurve slope and has many implications.

Induced drag% *ift acts normal to flow in 2D. $his accounts for about of thefuel used in a commercial airplane! and as much as 4 of the drag in the criticalclimb segments.

It produces interference effects that are important in the analysis of stability andcontrol.

$he magnitude of the downwash can be estimated using the +iot-0avart law! discussedpreviously.

When applied to our simple model with two discrete trailing vortices! the e)uationpredicts infinite downwash at the wing tips! a result that is clearly wrong. In fact! theinduced downwash is not even very large.

$he failure of this simple model led 5randtl to develop a slightly more sophisticated onein 6764. 8ather than representing the wing with ust one horseshoe-shaped vorte&! thewing is represented by several of them%


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In this way the circulation on the wing can vary from the root to the tip. $he strength ofthe trailing vorte& filaments is related to the circulation on the wing then by%9wa#e: ;9wing( vorte& is shed from the wing whenever the circulation changes.

In the limit as the number of horseshoe vortices goes to infinity! the trailing wa#e is asheet of vorticity.

$he trailing vorte& strength per unit length in the y direction ,vorticity is the derivativeof the total circulation on the wing at that station. dyand since the wing circulation changes most )uic#lynear the tips! the trailing vorticity is strongest in thisregion. $his is why we see tip vortices! and not acomplete vorte& sheet! as in this 1(0( photo of an


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Basic Theory

We could try using 2-D flow results for each section! but correct them for the influence ofthe trailing vorte& wa#e and its downwash. $his is the idea of lifting line theory.

We use the 2-D result that%

together with the relation%

to obtain%

+ut the angle of attac# used here is reduced through the effects of downwash so that theeffective angle of attac# is the true angle? minus the downwash angle%

Where the induced downwash! Wind! is given by the +iot-0avart *aw%

@ombining the e&pression for gamma%

with the e&pression for the downwash angle%

provides an integral e)uation for the circulation distribution along the wing.

Aust as in thin airfoil theory! the integral e)uation can be solved by assuming a


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(fter integrating we have%

$he solution of this e)uation for all values of y is not )uite so easy as in the case of thinairfoil theory where we could get closed form e&pressions for the (nBs. $his is generallydone numerically. /owever! several interesting and simple results appear from thisanalysis without ever actually computing the (nBs from the distribution of local angle ofattac#. 0ome of these are discussed in the ne&t section.

Elliptic Wing Results

If! for e&ample! we represent the lift distribution with only a single term in the 2.

In this domain%

0ince the downwash distribution is constant the @ ldistribution is ust%

If the angle of attac# is also constant along the wing ,no twist then the @lis constant andsince%


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$hen in this case the section @lis e)ual to the wing @*and%

or%

8ecall that this holds for unswept elliptical wings.

General Lift Distributions

If we are given the lift distribution we can compute the An's as we would with any Fourierexpansion. And once we know the Fourier coefficients, we may compute the downwashdistribution and the induced drag:

ubstitution and evaluation of the definite integral!! leads to:

"his formula gives the downwash in the plane of the wing for arbitrary load distributions. For thesimple elliptical case, closed form solutions for the downwash and sidewash at the start of the

wake sheet exist. "he simple relation for the velocity induced by an elliptic wing tailing vortexsheet is:

#ere, the variable $ is the complex coordinate y % i& and wo is the downwash at the wing root: y & (.

"his formula permits computation of induced velocities behind a wing as they effect downstreamsurfaces such as hori&ontal tails.


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)ote that the downwash is only constant in the plane of the wing and behind the wing. As wemove outboard of the wing or out of the plane of the wake, the downwash varies considerably

and there is a rather large upwash beyond the wing tips.

"his downwash field produces several important effects. It changes the lift of surfaces in othersurfaces wakes. "his is important in the analysis of airplane stability and the effectiveness ofhori&ontal tails. As can be seen from the downwash plot, the interference of a canard wake with awing is extreme: the wing lift is reduced behind the canard and the part of the wing outboard ofthe canard has increased lift.

"he downwash also produces induced drag as discussed in the next section.

Induced Drag and the Trefftz lane

Funda!entals

"he *+ paradox that surfaces in inviscid flow produce no drag no longer applies in -+. "hedownwash created by the trailing wake changes the direction of the force generated by eachsection:

In three dimensions the force per unit length acting on a vortex filament is

#ere, the local velocity includes the component from the freestream and a component from theinduced downwash. "his latter component produces a component of force in the direction of thefreestream: the induced drag.


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"he induced drag is related to the lift by:

From the results of lifting line theory for lift and downwash in terms of the Fourier coefficients ofthe lift distribution:

so we have:

"he induced drag is often written as:

"his may be written in coefficient form as:

with the same definition of e. )ote that e simply depends on the shape of the lift distribution. It iscalled the span efficiency factor or /swald's efficiency factor. )ote also that the induced dragforce depends principally on the lift per unit span, 01b.

2e can determine 3uickly, from the expression for induced drag above that drag is a minimum fora given lift and span when all of the Fourier coefficients except the A4term 5which produces lift6are &ero. "his corresponds to the elliptic loading case mention previously. In this case thedownwash is constant and e 4.

Far Field "nalysis and the Trefftz lane

"he analysis above works 3uite well for analy&ing the drag of wings given the distribution of lift. Itwas invented by 7randtl and 8et& around 49*(. It does involve a bit of hand+waving, though, andre3uires some very approximate ideali&ations of the flow. For example: when we talk about thedownwash induced by the wake on the wing, ust where on the wing do we mean; 2e assume asingle bound vortex line and compute velocities there, but a real wing does not have a singlebound vortex and the velocity induced by the wake varies along the chord. Fortunately, theanswers from this model are more general than the model itself appears. "he induced dragformulas can also be derived from very fundamental momentum considerations.

If the box is made large, contributions from certain sides vanish. In the limit as the box sides go toinfinity we obtain the following expressions for lift and drag:


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#ere, u, v, and w are the perturbation velocities induced by the wing and its wake. )ote that thedrag only depends on the velocities induced in the


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$he result that the drag of a lifting system depends only on the distribution of circulationshed into the wa#e leads to some very useful results in classical aerodynamics.

5erhaps the most useful of these is called Eun#Bs stagger theorem. It states that%The total induced drag of a system of lifting surfaces is not changed when the elements

are moved in the streamwise direction.

$he theorem applies when the distribution of circulation on the surfaces is held constantby adusting the surface incidences as the longitudinal position is varied.

$his implies that the drag of an elliptically-loaded swept wing is the same as that of anunswept wing. It also is very useful in the study of canard airplanes for which the canarddownwash on the wing is )uite complicated. Eoving the canard very far behind the wingdoes not change the drag! but ma#es its computation much easier. One may use thestagger theorem to prove several other useful results. One of these is the mutual induceddrag theorem which states that% $he interference drag caused by the downwash of one

wing on another is e)ual to that produced by the second wing on the first! when thesurfaces are unstaggered ,at the same streamwise location.

$hese results are especially useful in analy'ing multiple lifting surfaces.

Trefftz lane Lift Deri'ation

We have discussed the calculation of drag based on the velocities induced in the $refft'plane! but can lift be calculated in a similar wayF

$he answer is not so easy. We start with the e&pression for force based on the momentume)uation.

*etBs assume ,naively! for now that the contribution of each of these integrals goes to'ero on each side of the bo&! e&cept for the bac# side! as the dimensions of the bo& are

increased. $his leaves the contribution in the $refft' plane due to the wa#e.

In the $refft' plane! if we assume that all of the induced velocities are normal to theplane! the lift becomes%


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$he evaluation of this integral seems straightforward.+ut! it is not.

@onsider the integral

when the wa#e is modeled simply by a pair of vortices.

$he induced velocity! w! is given by%

$hus! the inner part of the above integral becomes%

0o! the integral for lift is%

$his loo#s e&actly rightG however! letBs consider the same integral when the order ofintegration is reversed%

$he inner part of the above integral now becomes%


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$his integrand is antisymmetric as shown in the plot below. 0o! the integral for lift is% * :.

1ow we have a parado&. We get two values for the same integral.(ctually! this is not a parado&G it is rather a function that is not *ebesgue integrable.

In order to evaluate an integral unambiguously! the function must satisfy two conditions%6. It must be continuous! e&cept at a countable number of points.2. $he integral of its absolute value must be finite.

$o avoid this problem! the integral for lift can be first evaluated over finite limits. $a#ing' from -( to ( and y from -+ to + we find%

* : ,2 H " 9 > ,+Js atan,(>,+Js - ,+-s atan,(>,+-sJ (>2 lnK ,(2J,+Js2 > ,(2J ,+-s2L M

$he limit as ( and + get large depends on the ratio of ( to +.When (NN+ the value goes to 2 H " 9 s! when +NN( the value is and when( : + the value is H " 9 s. $hus the integral remains ambiguous when evaluated over aninfinite domain.

(s noted by *arry Wigton of +oeing! this dilemma is resolved by using a different modelof the flow field. When the vertical velocity associated with thisvorte& system is


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integrated over the $refft' plane! no ambiguities arise. +ut! the results are surprising.

$he result is that the contributions from the finite length trailing vortices goes to 'ero.$he contribution from the bound vorte& is found to be independent of the length of thetrailing vortices and is%

$he starting vorte& contribution is similarly independent of the trailing vorte& length andis e)ual to the bound vorte& contribution. $hus! this lift is due to momentum flu&! but notfrom the trailing vortices.

We finally need to loo# at how the pressure term on the upper and lower sides of thecontrol volume is involved.

(s might be e&pected! integrals once again are not unambiguous. $hey depend on the

relative si'es of the bo& sides! even though everything is infinite.( careful analysis leads to the following basic results%

6. If the wa#e length is small compared with the bo& width and height then the lift isassociated with the momentum term of the starting and bound vortices.

2. If the wa#e length is large! and the bo& height is large compared with the width! thenthe lift is associated with the momentum term of the trailing vortices.

3. If the wa#e is long and the width is large compared with the height! then the lift isassociated with the pressure terms on the top and bottom.

#onplanar Wings

ven after the issues with infinite-domain integrals have been resolved! we must worryabout the assumed wa#e position. (lthough we could argue that streamwise wa#es canusually be used for far-field drag computataions! streamwise wa#es can still support liftforces. When the wing or wa#e is substantially nonplanar! these effects can be significant.


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In fact! the vorte& lift generated by highly swept wings can be estimated by far-fieldmethods only when the roll-up of the wa#e sheet is accurately computed. $he alternativesin such cases are to use near field methods or to compute the wa#e shape.

(o!putational Models

anel Methods

>any computational models and analysis methods are based on linear three+dimensionalpotential flow theory. "hese are discussed in the overview of panel methodsin an earlier chapter.

In this section we take a look at the simplest panel method in more detail.

Weissinger Method

Weissinger theory or e&tended lifting line theory differs from lifting line theory in severalrespects. It is really a simple panel method ,a vorte& lattice method with only onechordwise panel! not a corrected strip theory method as is lifting line theory. $his modelwor#s for wings with sweep and converges to the correct solution in both the high andlow aspect ratio limits.

$he version of this model used in the Wing Design program is actually a variant ofWeissingerBs method% it uses discrete s#ewed horseshoe vortices as shown.

ach horseshoe vorte& consists of a bound vorte& leg and two trailing vortices. $hisarrangement automatically satisfies the /elmholt' re)uirement that no vorte& line ends in

the flow. ,$he trailing vortices e&tend to infinity behind the wing.

$he basic concept is to compute the strengths of each of the PboundP vortices re)uired to#eep the flow tangent to the wing surface at a set of control points.
http://www.desktop.aero/appliedaero/solnmethods/panelmethods.htmlhttp://www.desktop.aero/appliedaero/solnmethods/panelmethods.html


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If the vorte& of unit strength at station produces a downwash velocity of (I@ iat stationi! then the linear system of e)uations representing the boundary conditions may bewritten%

where alphaM represents the angle of incidence of the sections along the span ,assumed a

flat plates. If the section has camber! the local angle of attac# is ta#en as the angle fromthe 'ero lift line of the section.

$he linear system of e)uations to be solved may also be written in terms of the angle ofattac# at the wing root and the twist amplitude. 2.

RM is the total twist ,washout in the wing from root to tip

$hus! the wing circulation distribution can be written as the sum of two distributions%

0ince the section lift ,lift per unit length along the span is related to the circulation by%

$he lift distribution can be e&pressed as%

where l6 and l2 are independent of the incidence angles and depend only on the planformshape of the wing.

0ince the lift coefficient of the wing! @*! is linearly related to the angle of attac# we canalso write the lift distribution in the following form%


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$he first term is #nown as the additional lift distribution and the second term is called thebasic lift distribution. $hey scale linearly with the wing lift coefficient and the twist anglerespectively. (dditional information on basic and additional lift distributions is availablein the section on wing design.

Wing "nalysis rogra!$his Aava application computes the lift and @ ldistribution over a wing with sweep andtwist. $o increase the angle of attac#! clic# near the upper part of the plotG to reducealpha! clic# in the lower area.

Details:

$he analysis is a discrete vorte& Weissinger computation. 5itching moment is based onthe mean geometric chord and is measured about the root )uarter chord point. $he twist isassumed linear and is ta#en to be positive for washout ,tip incidence less than rootincidence.

&i!ple &)eep Theory

*ifting line theory wor#s only for unswept wings.

Weissinger theory provides a means for computing the distribution of lift on swept wings!but not the chordwise distribution of pressures.

Sorte& lattice models! panel methods! and nonlinear @


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would remain unchanged.

We have ust created an infinite! obli)uely-swept wing that is moving with respect to the

air at a speed%

We can use this fact to design a wing which can fly at a high speed with a pressuredistribution associated with a lower speed.

$he main idea behind sweeping the wing is to reduce the effects of compressibility. $hecomponent of the flow parallel to the wing is not effected by the presence of the wingG thenormal component is decoupled from the tangential component.

$his is true not only according to linear flow theory! but also in the case of nonlinearcompressible flow with shoc# waves. It is an interesting e&ercise to show how the fullpotential e)uations decompose into a normal term and a tangential term when one assertsthat nothing changes in the tangential direction. $his idea is called simple sweep theory.We can consider sections normal to the wing edges as operating in a flow with lower

Each number and dynamic pressure. $he effective normal Each number is then%

but because of the reduced normal dynamic pressure! the section lift coefficient based onthis component of the freestream velocity must be increased if the total lift is fi&ed%


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airfoils were designed to operate at a Each number of .U ,normal Each number withthe wing swept V. (lthough the airplane lift coefficient was not meant to e&ceed .2T!the airfoils had a design @lof 6. because of the reduced normal dynamic pressure fromsimple sweep theory.

$he reason for designing a supersonic wing with sweep and subsonic airfoil sections canbe seen in the results of thin airfoil theory which predicted no drag for subsonic sections!but did indicate that supersonic airfoil sections would produce drag due to thic#ness!camber! and lift.

0ince the effect varies with cosine of the sweep angle! we e&pect that either forward or aftswept wings would reali'e similar benefits. $his is basically true! although! as discussedin the section on wing design and forward swept wing! there are some importantdifferences. 0imilarly! wings with obli)ue sweep have been designed and tested. research aircraft! the X-27was built by Yrumman for 1(0(! D(85(! and the (ir


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0everal aerodynamic advantages of the forward swept wing have been suggested. One ofthe more interesting of these is illustrated below. $he claim is that the lower surface of aswept forward wing contributes a larger share of the total lift than the lower surface of anaft-swept wing.

(lthough e&aggerated in this figure! this effect is predicted and observed. It is due in partto perturbation velocities induced by the 3-D thic#ness distribution and in part to thevelocities induced by streamwise vorticity.

0ome of the advantages and disadvantages of forward sweep%

"d'antages

+etter off-design span loading ,but with less taper% @ladvantage! weight penalty (eroelastically enhanced maneuverability

0maller basic lift distribution


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8educed leading edge sweep for given structural sweep

Increased trailing edge sweep for given structural sweep - lower @Dc

"nobstructed cabin

asy gear placement

Yood for turboprop placement

*aminar flow advantagesF

Disad'antages

(eroelastic divergence or penalty to avoid it *ower C@lZC ,effective dihedral

*ower @nZ,yaw stability

+ad for winglets

0tall location ,more difficult

*arge @mwith flaps

8educed pitch stability due to additional lift and fuse interference

0maller tail lengthFFF

+bli,ue Wings

"hese results suggested to ?.". @ones that obli3uely+swept wings would be the ideal shape forsupersonic aircraft wings. #e first proposed the concept in the 49('s and flew flying wing modelsat the first IBA meeting in >adrid in 49CD. A great deal of work has been done since on obli3uewing aircraft including design work by 8oeing, =eneral ynamics, and 0ockheed, wind tunneltesting and analysis by )AA, and flight testing of models and piloted aircraft.

"he picture below shows the A+4 low speed obli3ue wing demonstrator.


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Although one of the principal advantages involves reduced supersonic wave drag, the concepthas other merits.

2hen compared to an aircraft with symmetric variable sweep 5such as the 8+4 or F+4446, theobli3ue wing has little change in the aerodynamic center position. "his keeps the stability atreasonable levels and avoids large trim changes or complex fuel+pumping schemes.


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In addition, several structural advantages have been suggested, especially for variable+sweepaircraft: bending loads on the pivot are avoided, only a single pivot is re3uired, the actuator loadsare reduced, and the straight carry+through structure reduces weight in other portions of theaircraft.

(n all-wing version of the obli)ue wing was first proposed by Y. /. *ee in 67V2. $heidea has been revived with the advent of active control systems and a recent artist conceptof the obli)ue flying wing is shown below.

Despite several )uestions about stability and control! this concept can be made to fly .

+bli,ue Wing Flight


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$his model weighed about 4 lb ,3V #g with a 2 ft ,V m wing span andwas powered by two ducted fan engines. $he aircraft is unstable andcontrolled actively with a custom-built flight control computer. $he proectwas underta#en at 0tanford "niversity in 6776-677! principally by 0teveEorris and +en $igner.

"erodyna!ics of &lender Bodies

0ometimes we are not interested in generating lift! ust reducing drag! and when we haveto reduce the drag of a given volume! the best shape is often a slender body -- and oftennearly a body of revolution. $he aerodynamics of such shapes is )uite different fromairfoils and wings! but follows some of the the same basic principles.

+odies including fuselages are important because they produce drag! lift and moment.

$hey also produce important interference effects with wings and substantially change thestability of an airplane.

$he flow over more general fuselages and bodies can be predicted in much the same wayas the flow over airfoils and wings. 0uperposition of sources and doublets in the form ofpanel methods or simpler forms ,ala thin airfoil theory are often used. 1avier 0to#ese)uations are used when flow separation is suspected.


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In the first part of this section! we e&amine the results of such calculations rather than themethods themselves! while the second part briefly introduces the concept of slender bodytheory.

Flo) o'er Bodies

0ome closed-form solutions for the potential flow over bodies of revolution are availableand are useful as reference results.

We noted that in 2-D the ma&imum velocity on an ellipse was given by%uma&>" : 6 J t>c

In 3-D the surface velocity over an ellipsoid of revolution is given by%

& is the distance from the midpoint in units so that the length is 2. and r is the radius ,orin these units! the ratio of diameter to length.

$he ma&imum velocity is given by%

$he figure below ,from 0chlichting illustrates the pressure distribution on bodies ofrevolution. D>* : .6


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1ote that perturbation velocities are much smaller than in 2-D. $hese velocities may beestimated by superimposing point sources. In this case for an ellipsoid%

1ote that the ma&imum velocity is sensitive to the actual shape! with a paraboloid havingabout T larger perturbations. $he results from such a distribution of sources on thea&is! slightly underpredicts the velocity perturbations.

$he figure below shows the pressure distribution on a typical fuselage shape with D>* :.7 computed by a source distribution on the &-a&is. 1ote how the pressure falls near thecenter of the cylindrical portion of the fuselage.


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(s indicated below! fuselages in inviscid flow produce a nose-up pitching moment whenthe angle of attac# is increased. $his effect is destabili'ing and is an importantconsideration in the si'ing of the hori'ontal tail.

(lthough the inviscid picture suggests that no lift is produced! the viscous flow actuallyseparates at the bac# of the fuselage! ma#ing the moment somewhat smaller and the liftlarger than predicted by inviscid theory. $his lift produces induced drag and the fuselagebehaves as a low aspect ratio wing.

$he figure below shows the effect of angle of attac# on fuselage lift! drag! and momentbased on e&perimental data. (lso shown is the estimated moment based on inviscidtheory.

&lender Body Theory

( simple theory can be used to estimate the aerodynamic characteristics of bodies thatvary slowly in X-direction% wings of very low aspect ratio and high sweep or slender


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fuselages.

In such cases! the rate of change of all )uantities in the & direction is small and thegoverning e)uation becomes%

1ote that the ,6-E2 du>d& term has been dropped because we assume that everythingvaries slowly in the X direction. $he remaining e)uation is an e)uation for 2 dimensionalincompressible flow in the cross-plane! and since the Each dependence has dropped out!it is valid for both E6 and EN6.

$he 2-D cross flow can be computed using a conformal mapping method or even a 2-Dpanel method. 1ote that in the cross flow plane! the flow is unsteady as the span of theswept wing or the diameter of the fuselage changes as it cuts through the plane. 0o whilethe solution to *aplaceBs e)uation still provides the correct velocity distribution! thepressures must be computed using the unsteady +ernoulli e)uation.

One particularly simple and useful case is that of a highly-swept! low aspect ratio wing.

"sing the slender body concept we can solve for the lift of a low aspect ratio wing.

0uch low aspect ratio wings tend to produce constant downwash and thus nearly ellipticloading until the angle of attac# gets large.

$he rate of change of momentum of the air in the cross-flow plane is e)ual to the force

per unit length on the wing%

If we consider a certain area in the cross plane

given a velocity w , then the force becomes%It can be shown from 2-D unsteady theory ,not discussed here that the effective area ! (


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is given by a circle around the plate of diameter e)ual to the local span! [.

$hus!

$he total force on the wing is then given by%

$his force acts normal to the wing plane. 0o for small angles of attac#%

$his shows that in the low aspect ratio limit! the lift curve slope differs from what lifting

line theory predicted by a factor of 2. (n e&pression for lift curve slope derived fromsecond order corrections to lifting line theory is given by Aones%

where p is the ratio of wing semi-perimeter to wing span. b : 6 J 6>(8 so @*: 2(8 > ,(8J3.$his e&pression is in close agreement with e&periment over a wide range of aspect ratios.

( similar analysis can be done for slender bodies of revolution.

$his leads to the result that the lift produced by a body with a cut-off base is given by%

1ote again that these results are independent of Each number. $he 5randtl-Ylauertcorrection still applies! but the reduction in forces due to effective stretching in the &-direction ust cancels the increase in the velocities ,6>6-E2 that would have been appliedin 2-D.

Wing Design


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"here are essentially two approaches to wing design. In the direct approach, one finds theplanform and twist that minimi&e some combination of structural weight, drag, andB0maxconstraints. "he other approach involves selecting a desirable lift distribution and thencomputing the twist, taper, and thickness distributions that are re3uired to achieve thisdistribution. "he latter approach is generally used to obtain analytic solutions and insight into theimportant aspects of the design problem, but is is difficult to incorporate certain constraints andoff+design considerations in this approach. "he direct method, often combined with numericaloptimi&ation is often used in the latter stages of wing design, with the starting point establishedfrom simple 5even analytic6 results.

"his chapter deals with some of the considerations involved in wing design, including theselection of basic si&ing parameters and more detailed design. "he chapter begins with a generaldiscussion of the goals and trade+offs associated with wing design and the initial si&ing problem,illustrating the complexities associated with the selection of several basic parameters. Eachparameter affects drag and structural weight as well as stalling characteristics, fuel volume, off+design performance, and many other important characteristics.

2ing lift distributions play a key role in wing design. "he lift distribution is directly related to thewing geometry and determines such wing performance characteristics as induced drag, structuralweight, and stalling characteristics. "he determination of a reasonable lift and B ldistribution,combined with a way of relating the wing twist to this distribution provides a good starting point fora wing design. ubse3uent analysis of this baseline design will 3uickly show what might bechanged in the original design to avoid problems such as high induced drag or large variations in

Blat off+design conditions.

A description of more detailed methods for modern wing design with examples is followed by abrief discussion of nonplanar wings and winglets.

Wing Design ara!eters


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&pan

electing the wing span is one of the most basic decisions to made in the design of a wing. "hespan is sometimes constrained by contest rules, hangar si&e, or ground facilities but when it is notwe might decide to use the largest span consistent with structural dynamic constraints 5flutter6."his would reduce the induced drag directly.

#owever, as the span is increased, the wing structural weight also increases and at some pointthe weight increase offsets the induced drag savings. "his point is rarely reached, though, forseveral reasons.

4. "he optimum is 3uite flat and one must stretch the span a great deal to reach the actualoptimum.

*. Boncerns about wing bending as it affects stability and flutter mount as span is increased.

-. "he cost of the wing itself increases as the structural weight increases. "his must beincluded so that we do not spend 4( more on the wing in order to save .((4 in fuelconsumption.

. "he volume of the wing in which fuel can be stored is reduced.

C. It is more difficult to locate the main landing gear at the root of the wing.

G. "he ?eynolds number of wing sections is reduced, increasing parasite drag and reducingmaximum lift capability.


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/n the other hand, span sometimes has a much greater benefit than one might predict based onan analysis of cruise drag. 2hen an aircraft is constrained by a second segment climbre3uirement, extra span may help a great deal as the induced drag can be H(+D( of the totaldrag.

"he selection of optimum wing span thus re3uires an analysis of much more than ust cruise dragand structural weight. /nce a reasonable choice has been made on the basis of all of theseconsiderations, however, the sensitivities to changes in span can be assessed.

"rea

"he wing area, like the span, is chosen based on a wide variety of considerations including:

4. Bruise drag

*. talling speed 1 field length re3uirements

-. 2ing structural weight

. Fuel volume

"hese considerations often lead to a wing with the smallest area allowed by the constraints. 8utthis is not always true sometimes the wing area must be increased to obtain a reasonable B0atthe selected cruise conditions.

electing cruise conditions is also an integral part of the wing design process. It should not bedictated a priori because the wing design parameters will be strongly affected by the selection,and an appropriate selection cannot be made without knowing some of these parameters. 8ut the

wing designer does not have complete freedom to choose these, either. Bruise altitude affects thefuselage structural design and the engine performance as well as the aircraft aerodynamics. "hebest B0for the wing is not the best for the aircraft as a whole. An example of this is seen byconsidering a fixed B0, fixed >ach design. If we fly higher, the wing area must be increased bythe wing drag is nearly constant. "he fuselage drag decreases, though so we can minimi&e dragby flying very high with very large wings. "his is not feasible because of considerations such asengine performance.

&)eep


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2ing sweep is chosen almost exclusively for its desirable effect on transonic wave drag.5ometimes for other reasons such as a c.g. problem or to move winglets back for greaterdirectional stability.6

4. It permits higher cruise >ach number, or greater thickness or B0at a given >ach numberwithout drag divergence.

*. It increases the additional loading at the tip and causes spanwise boundary layer flow,exacerbating the problem of tip stall and either reducing B0maxor increasing the re3uiredtaper ratio for good stall.

-. It increases the structural weight + both because of the increased tip loading, andbecause of the increased structural span.

. It stabili&es the wing aeroelastically but is destabili&ing to the airplane.

C. "oo much sweep makes it difficult to accommodate the main gear in the wing.

ee section 9.*.C for more detail on simple sweep theory and the effect of sweep.

>uch of the effect of sweep varies as the cosine of the sweep angle, making forward and aft+swept wings similar. "here are important differences, though as discussed further in the sectionon forward swept wings.

Thic$ness

"he distribution of thickness from wing root to tip is selected as follows:

4. 2e would like to make the t1c as large as possible to reduce wing weight 5therebypermitting larger span, for example6.

*. =reater t1c tends to increase B0maxup to a point, depending on the high lift system, butgains above about 4* are small if there at all.

-. =reater t1c increases fuel volume and wing stiffness.

. Increasing t1c increases drag slightly by increasing the velocities and the adversity of thepressure gradients.

C. "he main trouble with thick airfoils at high speeds is the transonic drag rise which limitsthe speed and B0at which the airplane may fly efficiently.


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Taper

"he wing taper ratio 5or in general, the planform shape6 is determined from the followingconsiderations:

4. "he planform shape should not give rise to an additional lift distribution that is so far from

elliptical that the re3uired twist for low cruise drag results in large off+design penalties.

*. "he chord distribution should be such that with the cruise lift distribution, the distributionof lift coefficient is compatible with the section performance. Avoid high B l's which maylead to buffet or drag rise or separation.

-. "he chord distribution should produce an additional load distribution which is compatiblewith the high lift system and desired stalling characteristics.

. 0ower taper ratios lead to lower wing weight.

C. 0ower taper ratios result in increased fuel volume.

G. "he tip chord should not be too small as ?eynolds number effects cause reduced Blcapability.

H. 0arger root chords more easily accommodate landing gear.

#ere, again, a diverse set of considerations are important.

"he maor design goal is to keep the taper ratio as small as possible 5to keep the wing weightdown6 without excessive Blvariation or unacceptable stalling characteristics.

ince the lift distribution is nearly elliptical, the chord distribution should be nearly elliptical for

uniform Bl's. ?educed lift or t1c outboard would permit lower taper ratios.

Evaluating the stalling characteristics is not so easy. In the low speed configuration we must knowsomething about the high lift system: the flap type, span, and deflections. "he flaps+ retractedstalling characteristics are also important, however 5B+4(6.

T)ist

$he wing twist distribution is perhaps the least controversial design parameter to beselected. $he twist must be chosen so that the cruise drag is not e&cessive. &tra washouthelps the stalling characteristics and improves the induced drag at higher @*Bs for wingswith additional load distributions too highly weighted at the tips.

$wist also changes the structural weight by modifying the moment distribution over thewing.

$wist on swept-bac# wings also produces a positive pitching moment which has a smalleffect on trimmed drag. $he selection of wing twist is therefore accomplished bye&amining the trades between cruise drag! drag in second segment climb! and the wing


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structural weight. $he selected washout is then ust a bit higher to improve stall.

Wing Lift Distributions

As in the design of airfoil sections, it is easier to relate the wing geometry to its performancethrough the intermediary of the lift distribution. 2ing design often proceeds by selecting adesirable wing lift distribution and then finding the geometry that achieves this distribution.

In this section, we describe the lift and lift coefficient distributions, and relate these to the winggeometry and performance.

"bout Lift and (l Distributions

$he distribution of lift on the wing affects the wing performance in many ways. $helift per unit length l,y may be plotted from the wing root to the tip as shown below.

In this case the distribution is roughly elliptical. In general! the lift goes to 'ero at thewing tip. $he area under the curve is the total lift.

$he section lift coefficient is related to the section lift by%

0o that if we #now the lift distribution and the planform shape! we can find the @ldistribution.

$he lift and lift coefficient distributions are directly related by the chord distribution./ere are some e&amples%


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$he lift and @ldistributions can be divided into so-called basic and additional liftdistributions. $his division allows one to e&amine the lift distributions at a couple of

angles of attac# and to infer the lift distribution at all other angles. $his is especiallyuseful in the process of wing design.


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We can use the data at these two angles of attac# to learn a great deal about the wing.


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+ut because the section @lis the lift divided by the local chord! taper has a verydifferent effect on the @ldistribution.

@hanging the wing twist changes the lift and @ ldistributions as well. Increasing the tipincidence with respect to the root is called wash-in. Wings often have less incidenceat the tip than the root ,wash-out to reduce structural weight and improve stalling

characteristics.

0ince changing the wing twist does not affect the chord distribution! the effect on liftand @lis similar.

Wing sweep produces a less intuitive change in the lift distribution of a wing.+ecause the downwash velocity induced by the wing wa#e depends on the sweep! thelift distribution is affected. $he result is an increase in the lift near the tip of a swept-bac# wing and a decrease near the root ,as compared with an unswept wing.

$his effect can be )uite large and causes problems for swept-bac# wings.

$he greater tip lift increases structural loads and can lead to stalling problems.

$he effect of increasing wing aspect ratio is to increase the lift at a given angle ofattac# as we saw from the discussion of lifting line theory. +ut it also changes theshape of the wing lift distribution by magnifying the effects of all other parameters.

*ow aspect ratio wings have nearly elliptic distributions of lift for a wide range oftaper ratios and sweep angles. It ta#es a great deal of twist to change the distribution.


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Sery high aspect ratio wings are )uite sensitive! however and it is )uite easy to departfrom elliptic loading by pic#ing a twist or taper ratio that is not )uite right.

1ote that many of these effects are similar and by combining the right twist and taperand sweep! we can achieve desirable distributions of lift and lift coefficient.


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$he effect of a high lift constraint on optimal wing designs. Wing sweep! area! span! andtwist! chord! and t>c distributions were optimi'ed for minimum drag with a structuralweight constraint.

Wing Design in More Detail

$he determination of a reasonable lift and @ l distribution! combined with away of relating the wing twist to this distribution provides a good startingpoint for a wing design. 0ubse)uent analysis of this baseline design will

)uic#ly show what might be changed in the original design to avoid problemssuch as high induced drag or large variations in @lat off-design conditions.

Once the basic wing design parameters have been selected! more detaileddesign is underta#en. $his may involve some of the following%

@omputation or selection of a desired span load distribution! theninverse computation of re)uired twist.

0election of desired section @pdistribution at several stations along thespan and inverse design of camber and>or thic#ness distribution.

(ll-at-once multivariable optimi'ation of the wing for desiredperformance.

0ome e&amples of these approaches are illustrated below.


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$his figure illustrates inverse wing design using the DI0@ ,direct iterativesurface curvature method. $he starting pressures are shown ,top! followedby the target ,middle! and design ,bottomG light yellow : low pressure and

green : high pressure. $his is an inverse techni)ue that has been used verysuccessfully with 1avier-0to#es computations to design wings in transonic!viscous flows.+elow is an e&ample of wing design based on Pfi&ingP a span loaddistribution. When the U3U was re-engined with high bypass ratio turbofans! adrag penalty was avoided by changing the effective wing twist distribution.


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$he details of the pressure distribution can then be used to modify the cambershape or wing thic#ness for best performance. $his sounds straightforward!but it is often very difficult to accomplish this! especially when it ta#es hoursor days to e&amine the effect of the proposed change. $his is why simplemethods with fast turnaround times are still used in the wing design process.

(s computers become faster! it becomes more feasible to do full 3-Doptimi'ation. One of the early efforts in applying optimi'ation and nonlinear@D.

(lthough this was an inviscid code! the design variables were limited! and the


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obective function simplistic! current research has included more realisticobectives! more design degrees of freedom! and better analysis codes.

--but we are still a long way from having Pwings designed by computer.P

#onplanar Wings and Winglets

/ne often begins the wing design problem by specifying a target Bpdistribution and1or

span loading and then modifying the wing geometry 5either manually, by direct inverse, orby nonlinear optimi&ation6. In the case of planar wings, the elliptic loading results providea useful benchmark in the creation of target loadings. 5For high aspect ratio wings, *airfoil results may provide a useful reference for the chordwise loading.6

>ore complex methods for creating target Bp's are beyond the scope of this discussion,but we have little guidance at all when the wing is nonplanar.

"his section deals with the problem of optimal loading for nonplanar lifting surfaces. It iseasily generali&ed to multiple surfaces.

2hen the wing is not planar, many of the previous simple results are no longer valid.Elliptic loading does not lead to minimum drag and the span efficiency can be greaterthan 4.(.

#ere we will describe a method for computing the minimum induced drag for planar andnonplanar wings. First, consider the distribution of downwash for minimum drag. "his canbe obtained by using the method of restricted variations as follows.


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2e consider an arbitrary variation in the circulation distribution represented by JK4andJK*which do not change the lift:J0 L M JK4% L M JK* (.

"his implies: JK4 + JK*

If the drag was minimi&ed by the initial distribution:J L1* w4JK4% L1* w*JK* (.

o, w4 w*

"hat is, the downwash is constant behind a planar wing with minimum drag.

In the general case, with multiple surfaces or nonplanar wings, the same approach maybe used. In this case, the condition for constant lift is:J0 L M JK4cos N4% L M JK*cos N* (.

where N is the local dihedral angle of the lifting surface.

For minimum drag:J L1* n4JK4% L1* n*JK* (.

where nis the induced velocity in the "refft& plane in a direction normal to the wakesheet 5the normalwash6.

In this case, JK4cos N4 + JK*cos N*

so, n k cos N.

"he normalwash is proportional to the local dihedral angle. "hus, the sidewash onoptimally+loaded winglets is (, for example.

2e may then solve for the distribution of circulation that produces this distribution ofnormalwash.


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Alternatively, one may use a more direct optimi&ation approach. 2ith the circulationdistribution represented as the row vector, OKP and the wake modeled as a collection ofline vortices of strength OKwP, we may write the wake vorticity in terms of the surfacecirculation, based on a discrete vortex model as shown below.

"he drag is then given by: L1* OnP Q OKPwhere nis the normal wash in the "refft& plane computed using the 8iot+avart law.

OnP is related to the circulation strengths by:OnP RIBS OKPwhere RIBS is a function of the geometry.

o, L1* RIBS OKP Q OKP

"he lift is also a function of the circulations:0 L M OKP Q Ocos NPwith N the local dihedral angle.

Finally, the optimal values of OKP are given by settingd 5 % T50+0ref6 6 1 dKi ( where T is a 0agrange multiplier.

"his problem is sometimes done as homework, but some results are summari&ed below:

Q 2hen the wing1winglet combination is optimi&ed for minimum drag at fixed span, itachieves about the same drag as a planar wing with a span increased by about C ofthe winglet height.

Q "he wing lift distribution is as shown below with increased lift outboard compared withthe no winglet case.

"his increased tip loading along with the extra bending moment of the winglet leads toincreased structural weight. 2hen a bending moment constraint replaces the spanconstraint, wings with winglets are seen to have about the same minimum drag as thestretched+span planar wings. "his is shown below.


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Induced drag of wings with winglets and planar wings all with the same integratedbending moment 5related to structural weight6. )ote that solutions to the left of the spanratio 4.( line are not meaningful.

"he same approach may be taken for general nonplanar wake shapes. "he figure belowsummari&es some of these results, showing the maximum span efficiency for nonplanarwings of various shapes with a height to span ration of (.*.

everal points should be made about the preceding results.

4. "he result that the sidewash on the winglet 5in the "refft& plane6 is &ero for minimuminduced drag means that the self+induced drag of the winglet ust cancels the wingletthrust associated with wing sidewash. /ptimally+loaded winglets thus reduce induceddrag by lowering the average downwash on the wing, not by providing a thrustcomponent.

*. "he results shown here deal with the inviscid flow over nonplanar wings. "here is aslight difference in optimal loading in the viscous case due to lift+dependent viscous drag.>oreover, for planar wings, the ideal chord distribution is achieved with each section at


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its maximum Bl1Bdand the inviscid optimal lift distribution. For nonplanar wings this is nolonger the case and the optimal chord and load distribution for minimum drag is a bitmore complex.

-. /ther considerations of primary importance include:tability and controltructures/ther pragmatic issues

E-a!ples. Multilanes Eultiplanes include biplanes such as the Wright 672 glider shown below.

(lthough the Wright brothers e&ploited the structural advantages of biplanes!rather than the lower vorte& drag for fi&ed span and lift! their motivation waspartly aerodynamic. +ased on their own tests and those of Otto *ilienthal! it wasapparent that at very low 8eynolds numbers ,typical of test conditions used bythese pioneers highly cambered! thin sections performed much better than thic#ersections! ma#ing the cable-braced *ilienthal designs or the Wright biplaneconcepts especially attractive. +ecause of the low flight speeds re)uired for*ilienthalBs ta#e-offs and landings and for the power plants available to theWrights! the designs needed to be light and incorporate large wing areas. $hisre)uirement was satisfied well with the biplane configuration.

$he multiplane concept was ta#en to e&tremes by 5hillips in 67. $he aircraftshown below with 2 wings would have had a high span efficiency! but the verylow 8eynolds number of each wing would lead to poor performance. $he strutsand cables of early biplane designs also led to large parasite drag! so the effects ofimproved span efficiency were not obvious. 0everal modern proposals for


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cantilevered or semi-cantilevered biplanes have emphasi'ed the lower vorte& dragof such configurations at the e&pense of structural efficiency! 8eynolds number!and fuel volume.

$he induced drag of a multiplane may be lower than that of a monoplane of e)ualspan and total lift because the nonplanar system can influence a larger mass of air!

imparting to this air mass a lower average velocity change! and therefore lessenergy and drag. that of the single wing. $he inviscid drag of the system is then halfthat of the monoplane.

In addition to the well-#nown advantages in vorte& drag! the favorableinterference between two wings of a closely-coupled biplane can be used toimprove the section performance. $he lower-than-freestream velocity at thetrailing edge of the forward wing and the new boundary layer on the downstreamwing can be e&ploited and some of the difficulties with lower 8eynolds numbersfor the biplane as compared with a monoplane can be alleviated if not turned to

advantage. Yains in @*ma&! width of laminar drag buc#et! and drag divergenceEach number at fi&ed t>c are possible with good multiple element section design.(s an e&ample! a single fully-laminar section ,6 laminar flow on upper andlower surfaces can support a @* of about .. ( 2-element wing can be designedwith an overall @* of about .UT. $his may help to e&plain the preference forbiplanes in the low 8eynolds number world of insects.


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E-a!ples. (losed &yste!s

$he aerodynamics of nonplanar wing systems that form closed loops are veryinteresting. 0uch configurations include bo&-planes! ring wings! oined wings! andPspiroid-tipP devices. Wings that form closed loops! such as the ring-wingillustrated below! do not eliminate the Ptip vorticesP or trailing vorte& wa#es eventhough the wing has no tips. 0till! the vorte& drag of the circular ring wing is ustT that of a planar wing with the same span and total lift and the concept hasbeen studied at several organi'ations! including early aviation pioneers! a maoraircraft manufacturer! as well as several toy companies.

$he *oc#heed bo&-plane! shown below! achieves even greater drag reduction at agiven span and height than the circular ring wing ,in fact the theoretical minimum


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vorte& drag in a configuration with reasonable high-speed performance ,note thedesirable transonic area-ruling and some structural advantages.


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$he recently-patented Pspiroid wing tipP produces a reduction in induced drag!much li#e that of a winglet. /owever! its closed planform shape may ma#e itpossible to reduce local lift coefficients]often a problem for winglets.

(lthough a closed lifting system may eliminate the wing tips! it does not eliminatethe trailing vorte& wa#e. In fact! the lift produced by the system can be directlyrelated to the velocities in the wa#e that lead to induced drag. $hese systems are


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still interesting because one may add a constant circulation vorte& ring to thesystem without changing the wa#e. 0uch a constant strength vorte& distributiondoes not add any lift! but it may be used to produce moments without induceddrag penalties or to manipulate section lift coefficients in a desirable way.

E-a!ples. &trut*Braced Wings (ircraft concepts that employ au&iliary aerodynamic surfaces as struts to improve

both aerodynamic and structural efficiency have been studied e&tensively. In oined-wing designs ,below the hori'ontal tail sweeps forward and oins the

main wing! forming a strut. $he tail is then in compression! reducing wingbending moments. If the tail is large enough to be positively loaded! some induceddrag savings is achieved! while if it is carrying a down-load! the closed loopfeature of the system minimi'es trim drag.


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5fenningerBs laminar designs with lifting struts e&ploit the nonplanar strutgeometry primarily for structural weight and stiffness! although some induceddrag reduction may be achieved.

E-a!ples. Winglets and Tip De'ices $he most common contemporary nonplanar wing configuration is the wing with

winglets! as seen below on the EcDonnell-Douglas ED-66. $hese surfaces doreduce induced drag for a given span! as well as providing a means of )uic#lydistinguishing the airplane from a D@-6. $he ED-66 design includes smalldownward winglets! while the UU- employs a full-chord single winglet! andmany other variations are possible.


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( variant of the winglet concept! the @-wing is discussed later in this paper. Itinvolves adding a hori'ontal winglet e&tension ,a wingletletF and has interestingaerodynamic! structural! and control implications.

E-a!ples. #onplanar Wa$es


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$he induced drag of a nonplanar system can be lower than that of a planar system of thesame lift and span. $his is true even when the wing surfaces themselves are co-planar! buttheir vorte& wa#es are not. &amples of this phenomena include%

(mericaBs @up sailboat #eels. /ere the #eel and rudder ,or twin #eel surfaces arecoplanar! but due to the substantial leeway angle and longitudinal displacement of

the two surfaces! the wa#e downstream of the boat resembles that of a biplanesystem and the induced drag is reduced substantially. @rescent wings. $his phenomenon was postulated as the reason for the distinctive

planform shape of some bird wings and fish fins! although the effect is almostunmeasurable.

0plit-$ips. $his design was created to e&ploit the nonplanar wa#e geometry and isdiscussed in more detail in a subse)uent section of this paper.

Results. What is ossible/

ach of these configurations provides particular advantages and disadvantages! althougheach benefits from some reduction in induced drag compared with the conventionalmonoplane. $he reduction in vorte& drag is shown below for biplanes! bo&planes! andwinglets with varying ratios of height to span. $hese results were computed using anoptimi'ing vorte& lattice code! but agree with classical solutions from 5randtl! von\arman and +urgers! @one! and Aones. 1ote that the bo&plane achieves the lowest dragfor a given span and height! although winglets are )uite similar. @onsiderable savings ininduced drag are possible for a fi&ed span if large vertical e&tents are permitted.


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$he figure below illustrates the effect of nonplanar wing shape on span efficiency. achof the geometries! shown in front view below! is permitted a vertical e&tent of 2 of thewing span. ach design has the same proected span and total lift. $he results were

generated by specifying the geometry of the trailing vorte& wa#e and solving for thecirculation distribution with minimum drag. 0o! each of the designs is assumed to beoptimally twisted. $his was done by discreti'ing the vorte& wa#e and solving a linearsystem of e)uations for minimum drag with a constraint on overall lift. 0imilar results fora variety of shapes have been described by @one! Eun#! *etcher! Aones! and others.

$he results illustrate the variability in span efficiency among these designs. 1ote therelatively small gain for the diamond-shaped device and the wing with dihedral! while the@-wing shape achieves essentially the same drag as the bo&plane.


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#onplanar Wa$es. The &plit Tip

wingletcombination. $he difficulty here is that we must twist the wing or create a planformshape that achieves the optimal load distribution that corresponds to this geometry.Eoreover! for reasonable wing planforms! the amount of out-of-plane wa#e deformationis very limited.


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$o e&aggerate this effect! a wing with the geometry shown below was created. $he idea

here was to generate a shape whose potential span efficiency gain for a given amount ofout-of-plane deformation was large. +ased on the previous figure! a split tip geometry forthe wa#e was selected as a shape that could be generated by wa#e deflection and the wingplanform shown below was investigated. $he figure shows the planform shape and theshape of the wa#e trace when the wing is at 7 degrees incidence. +ased on this wa#eshape! an induced drag savings of about T is possible when the wing is optimallyloaded! and more as the angle of attac# is increased.


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Of course! the wa#e does not trail from the wing in the streamwise direction and carefulcomputation of rolled-up wa#e geometry and inviscid drag shows that the effect of wa#e-rollup is to roughly double the gain e&pected for a streamwise wa#e. $he 66 incrementin span efficiency was significant and the concept was studied in more detail boththeoretically and e&perimentally. $he figure below shows the computed wa#e geometry

and wing paneling used to compute vorte& drag with the high-order panel code (T2.

$wo wings were constructed and tested at 1(0(^s (mes 8esearch @enter. $he first wasan untwisted planform with an elliptical chord distribution! unswept )uarter chord line!and an 1(@( 62 airfoil section. $he second wing of the same area and span! alsountwisted with a 62 airfoil section! incorporated the split tip geometry. +oth modelswere designed to incorporate a sensitive internal balance so as to minimi'e supportinterference. $he figure below shows the ratio of lift to drag for each of these wingsconfirming the predicted lower drag of the split tip geometry.


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$o further confirm the theoretical predictions! estimates of vorte& drag and wa#e shapewere compared from calculations! balance data! and a detailed wa#e survey.


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The C-Wing: A Novel Nonplanar Wing ConfigurationFrom the survey of nonplanar wing geometries discussed previously, one is struck by the fact thatone need not produce a closed system such as the box plane to achieve essentially all of theinduced drag savings that this configuration offers. In particular, extending the upper part of thebox only 4( of the span inward from the tip achieves a span efficiency within about 4 of thecomplete box. "hus, one could achieve the drag savings of the box plane without the ?eynoldsnumber and fuel volume penalties of the two+wing design. Furthermore, the small hori&ontal tipextensions have some interesting implications for airplane design. "his section addresses the B+wing concept in a bit more detail and focusses on the application to a large commercial transportdesign.

(*Wing "erodyna!ics $he optimal loading of this lifting system is shown in the figure below. $he

circulation of the main wing is carried onto the winglet so that the winglet isloaded inward. When the hori'ontal e&tension is added to the winglet! forming theP@P shape! the circulation is e&tended from the winglet as well! producing asurface that is loaded downward for minimum induced drag at fi&ed total lift. It isonly when the lifting surface is e&tended to the centerline to form a bo& plane thatthe upper wing can efficiently carry an upload. $his is because! as mentionedpreviously in connection with closed systems! we can superimpose a constantcirculation ring on the closed system to redistribute the lift without changing thewa#e.

$his download on the @-wing hori'ontal surfaces affects structural weight andtrim and the implications for aircraft configuration concepts was intriguing.


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$his configuration was independently PdiscoveredP by a genetic algorithm thatwas as#ed to find a wing of fi&ed lift! span! and height with minimum drag. $hesystem was allowed to build wings of many individual elements with arbitrarydihedral and optimal twist distributions. $he figure below depicts front views ofthe population of candidate designs as the system evolves. On the right! the bestindividual from a given generation is shown.


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C-Wing Summary

$he advantages of the @-wing configuration for a large capacity subsonic transport arelisted below. $hey include those directly associated with the nonplanar wing geometryand those that arise indirectly from the overall configuration shown on previous pages.

)onplanar 2ing:

4. ?educed span or reduced vortex drag at fixed span

*. Efficient trim with short fuselage

-. Improved lateral handling 5lower effective dihedral, reduced adverse yaw6

. 7otential for aeroelastic control: prevent aileron reversal, active flutter control

C. ?educed tendency for pitch+up, control at high alpha

G. ?educed vertical tail height

H. 7ossible reduction in wake vortex strength

Bonfiguration:

4. Improved aero1structural performance through span loading, potential forreduced wetted area

*. Effective use of redistributed wetted area reduces high lift system cost or "/thrust 1 noise, potential for laminari&ation.

-. ome advantages of all+wing design with reduced risks: egress, windows,growth, structure, acceptability

. * wing+mounted engines reduce obstacle problem with outer engine 1 engine outyaw

C. ingle deck in wing facilitates loading, emergency egress

isadvantages:

4. etails of emergency egress remain uncertain

*. Aerodynamics of thick inboard sections still an issue

-. Aeroelastics may be controllable but may need to be controlled


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(*Wing (onfiguration De'elop!ent

"sing the @-Wing configuration! the span of an otherwise conventional large aircraft canbe reduced. +ecause the fuselage tends to be rather short on double dec# configurations!the hori'ontal tail location is not much farther aft than the wing tips ma#ing it possible toconsider using the @-wing as the primary pitch control surfaces. ,$he hori'ontal @-wingsurfaces provide more stability for a given area as they are not affected by the aft fuselageflow field and are less affected by wing downwash. Eoreover! they provide a positivetrimming moment when optimally loaded. $he removal of the hori'ontal tail ma#es theuse of aft-fuselage-mounted engines a possibility! eliminating some of the severeproblems with the original outboard engine location. Despite some attractive features!however! the performance advantages for this configuration are not substantial! and

probably not worth the ris# associated with the unconventional design.

(s the number of passengers reaches V-4! the possibility of including somepassenger cabin area inside the wing appears more attractive.


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$his design comprises a three-surface configuration providing a large allowable c.g.range! with a relatively lightly loaded wing to simplify high-lift system re)uirements andaccommodate passenger cabins in the wing. $he vertical and hori'ontal tip e&tensionsprovide an efficient means of satisfying stability and control constraints.

(s the design evolved to the tri-et shown below! the wing span was increased! butremained substantially lower than the conventional design. Eore efficient use was madeof the e&isting UUU fuselage area and the thic# inner wing section was modified based onan investigation of high-speed thic# sections.

$he basic idea in this conceptual design study was not to obtain the highest performanacefor this large aircraft! but rather to provide a feasible solution to the large aircraftproblem. $he design addresses many of the problems that arise from the simple scaling-up of the conventional design.


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(*Wing +pti!ization and Initial Results


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0tudies at 0tanford and +oeing were underta#en as part of 1(0(^s (dvanced @oncepts5rogram in 677T. @oncurrently! initial si'ing and optimi'ation of the original conceptwere pursued at 1(0(^s *angley 8esearch @enter.

ach of these studies involved analysis and numerical optimi'ation of the basic concept.

(t *angley the

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Finite Wing Theory.doc

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