BASIC DESIGN OF FLYING WING MODELS
Basic Design of Flying Wing Models Preface A survey of the available literature on the topic of flying wing and tailless model airplanes shows, that in most cases, the airfoil selection is mentioned, but no reliable data for the selection are available. Due to the dedicated work of various modelers, a huge collection of airfoil coordinates is available today [UIUC UIUC, , [!ender"#iechers $ a comparable set of aerodynamic data is more difficult to find, though. %his situation and, the numerous &uestions of modelers, concerning the performance and the selection of airfoils for flying wing models, initiated this paper. 'n this web page, you will find a way to perform the necessary calculations for stable flying wing models. %he accompanying collection of airfoil coordinates and polars has been published in [() (). .
An indoor flying wing model by R. Eppler in 1942.
Airfoils for Flying Wings %ailless planes and flying wings can be e&uipped with almost any airfoil, if sweep and twist distribution are chosen accordingly. %hus, the one and only *flying wing airfoil* does not e+ist. owever, if we want to design a tailless plane with a wide operating range, the wing should have a small amount of twist only, or none at all, to keep the induced drag at reasonable levels throughout the whole flight envelope. Under these conditions, the wing must not create a large variation in moment coefficient, when the angle of attack is varied. %his makes it necessary, to use airfoils with a low moment coefficient. In the case of an unswept wing -*plank*, even an airfoil with a positive moment coefficient is necessary, to avoid upward deflected flaps under trimmed flight conditions. /uch airfoils usually have a refle+ed camber line.
A modern radio controlled F3B flying wing model of 1994.
Longitdinal Sta!ility 0ike its full si1ed cousins, each model airplane should have a minimum amount of stability, i.e. it should be able to return to its trimmed flight condition after a disturbance by a gust or a control input. ow much stability is re&uired, depends on the pilots personal taste2 contest pilots prefer a small stability margin, beginners like to fly with a large margin. ere, only a brief introduction into the topic will be given, which will make it possible to find a first guess for the center of gravity and a reasonable combination of sweep and camber for a flying wing.
"# $ns%e&t Wings 'Plan() #hile the hori1ontal tailplane provides the necessary amount of longitudinal stability on a conventional plane, it is the wing, which stabili1es an unswept wing. In most cases, airfoils with reflexed -s$shaped -s$shaped mean lines are used on flying wing models to achieve a longitudinally stable model. So*e i*&ortant Aerodyna*ic and Mec+anical Mec+anical Facts
%o understand, why a refle+ed airfoil is able to provide longitudinal stability to a wing, two things are important2 Total Force and oment! c"4 point %he pressure forces, which act on the surface of each wing section, can be replaced by a single total force and a single total moment. !oth act at the #$arter%c&ord point of of the airfoil. #hen the angle of attack changes -e.g. due to a gust, the moment stays nearly constant, but the total force changes. Increasing the angle of attack increases the force. 'enter of (ra)ity %ranslations and rotations of *free floating* bodies are performed relative to their center of gravity. #hen the angle of attack of a plane changes, the plane rotates -pitches around its center of gravity -c.g. -c.g.. . E,ili!ri*
0et3s have a look at a trimmed flight condition, where all forces and moments are in e&uilibrium and let3s compare a conventional, cambered airfoil with an airfoil with a refle+ed camber line. %he moments and forces for this trimmed state are denoted with an asterisk -4. %he forces are the weight of the model m, multiplied with the gravity acceleration g acceleration g -9.*1m"+ -9.*1m"+ and the aerodynamic lift , lift ,,, which have to cancel out -sum of forces in vertical direction e&uals 1ero. %he drag forces are neglected here. %he sum of the moments around c.g. -caused c.g. -caused by the airfoil moment moment and and the lift force , force ,,, acting at a distance from c.g. c.g. must also be 1ero.
con-entional airfoil %it+ ca*!er
airfoil %it+ refle.ed *ean line
E,ili!ri* State
%his airfoil has a nose heavy moment. As stated above, the center of gravity is also the center of rotation of the wing. #hen it is shifted behind the c"4 point, the air force ,- in front of the c.g. counteracts the nose heavy moment 54 to achieve e&uilibrium. %he distance between c.g. and c"4 point is depending on the amount of -. A symmetrical airfoil has -/, which means we have to place the c.g. at the c"4 point.
%he refle+ed camber line makes the moment coefficient positive, which means, that the moment around the c"4 point is working in the tail heavy direction. %herefore the center of gravity has to be located in front of the c"4 point to balance the moment - by the lift force ,-. %he larger the moment -$coefficient of the airfoil, the larger the distance between c"4 and the c.g. for e&uilibrium.
Distr!ed State
#hen the angle of attack is increased -e.g. by a gust, the lift force 0 increases. 6ow ,0,- and the tail heavy moment due to the lift is larger than the moment around c"4, which still is -. %hus the wing will pitch up, increasing the angle of attack further. %his behavior is instable and a tailplane is needed to stabili1e the system.
ere, we have the air force acting behind the c.g., which results in an additional nose heavy moment, when the lift increases. #ith ,0,-, the wing will pitch down, reducing the angle of attack, until the e&uilibrium state is reached again. %he system is stable.
Netral Point and Sta!ility
As we learned above, an unswept wing with a refle+ed airfoil is able to stabili1e itself. Its c.g. must be located in front of the c"4 point, which is also called ne$tral point -n.p.. %he distance between the neutral point -&uarter chord point for an unswept wing and the center of gravity is defining the amount of stability $ if the c.g. is close to the n.p., the straightening moment is small and the wing returns -too
slowly into its e&uilibrium condition. If the distance c.g. % n.p. is large, the c.g. is far ahead of the c"4 point and the wing returns &uickly to the e&uilibrium angle. 7ou will re&uire larger flap deflections to control the model, though. If the distance is too large, the wing may become over$stabili1ed, overshooting its trimmed flight attitude and oscillating more and more until the plane crashes. A measure for stability is the distance between c.g. and n.p., divided by the mean chord of the wing. %ypical values for this number for a flying wing are between /./2 and /./, which means a +tability coefficient sigma of 2 to percent. #e can e+press the e&uilibrium of moments around c.g. for our design lift coefficient
by
, which can be transformed to find the moment coefficient needed to satisfy a certain stability coefficient2 . E.a*&le #e want to use an unswept flying wing -a plank for ridge soaring and decide to use a
target lift coefficient of /.. #e want to have a stability coefficient of and are looking for a matching airfoil. #e calculate the necessary moment coefficient 'm /. - /./ /./2. /earching through a publication about 8ppler airfoils [(9, we find that the airfoils 8 :9; and 8 (<= could be used for our model.
/# S%e&t Wings /#" Netral Point and Sta!ility #e have already learned, that the center of gravity must be located in front of the neutral point. #hile the n.p. of an unswept, rectangular wing is appro+imately at the c"4 point, the n.p. of a swept, tapered wing must be calculated. %he following procedure can be used for a simple, tapered and, swept wing. >irst, we calculate the mean aerodynamic chord length sweep angle2
of a tapered wing, which is independent from the
with the root chord l r , the tip chord l t and the taper ratio #e can also calculate the spanwise location of the mean chord
.
, using the span b,
. %he n.p. of our swept wing can be found by drawing a line, parallel to the fuselage center line, at the spanwise station y. %he chord at this station should be e&ual to . %he n.p. is appro+imately located at the c"4 point of this chord line -see the sketch below.
(eometric parameter+ of a tapered! +wept wing.
Instead of using the graphical approach, the location of the neutral point can also be calculated by using one of the following formulas, depending on the taper ratio2
, if taper ratio ? =.<@
, if taper ratio B =.<@. %he c.g. must be placed in front of this point, and the wing may need some twist -washout to get a sufficiently stable wing.
/#/ 0%ist %he selection of the location of the c.g. to be infront of the n.p. is not a guarantee for e&uilibrium $ it is only a re&uirement for longitudinal stability. Additionally, as e+plained above for unswept wings, the sum of all aerodynamic moments around the c.g. must be 1ero. !ecause we have selected the position of the c.g. already to satisfy the stability criterion -c.g. in front of n.p., we can achieve the e&uilibrium of the moments only by airfoil selection and by adusting the twist of the wing. 'n conventional airplanes with a hori1ontal stabili1er it is usually possible to adust the difference between the angles of incidence of wing and tailplane during the first flight tests. 'n the other hand, flying wings have the difference built into the wing
-twist, which cannot be altered easily. %hus it is very important to get the combination of planform, airfoils and twist right -or at least close before the wing is built. Again, the calculation of these parameters is &uite comple+ and shall not be presented here the relations are shown in great detail in [(@. ere I will present a simple, appro+imate approach, which is based on two graphs, and can be used for swept, tapered wings with a linear airfoil variation from root to tip. #e start with the same geometric parameters, which we have used for the calculation of the n.p. above. Additionally, we calculate the aspect ratio - AR b"5 , where 5 is the wing area of the wing. %he selection of the airfoil sections also defines the operating range of the model. Airfoils with a small amount of camber are not well suited for slow, thermaling flight, but good for >
Finding t+e 1e,ired 0%ist 2 re,
Using graph :, we enter the graph with the aspect ratio AR on the hori1ontal a+is, and draw a vertical line upwards, until we intersect the curve, corresponding to the sweep angle of the c"4 line. Continuing to the a+is on the left border, we find the standard value β4re& for the re&uired twist angle. %his standard value is valid for a wing, which2
•
is trimmed at 1./ and, has a +tability coefficient of σ- 1/ -see above, and
•
uses airfoils with a moment coefficient of 1ero.
•
>rom the standard value we calculate the true, re&uired twist angle, using the formula inset into the graph. %herefore, we calculate the ratio of our target lift coefficient to the standard lift coefficient -' , "
and the ratio of our desired
stability coefficient to the standard . #e see, that a reduction of the lift coefficient to ' ,/. also reduces the re&uired twist by =E. Also, if we use a smaller stability margin σ, we need a smaller amount of twist.
(rap& 16 Finding t&e re#$ired twi+t. 3ariation of 4ero lift angle
If we use different airfoils at root and tip, they may have different 1ero lift directions, which influences the e&uilibrium state. %he geometric twist has to be reduced by the difference of the 1ero lift directions α / of tip and root sections2 . Using the same airfoil for both sections, we can set βα= to 1ero. Inflence of t+e Airfoil Mo*ent coefficients
%he moment coefficient of the airfoils contributes to the e&uilibrium, and has to be taken into account for the calculation of the twist. Fraph ( can be used to find the e&uivalent twist due to the contribution of 'm, which has to be subtracted from the re&uired twist. If we use airfoils with positive moment coefficients, the contribution will be positive, which results in a reduction of the amount twist, highly cambered airfoils yield negative values β 'm, which force us to build more twist into the wing. /imilar to the previous graph, we enter with the aspect ratio, intersect with the sweep curve and read the value for β∗ 'm from the lefthand a+is.
(rap& 26 Finding t&e additional twi+t d$e to t&e airfoil+ moment coefficient. Again, the graph has been plotted for a certain standard condition, which is a moment coefficient of cm- /./ -note2 positive value. #e apply the ratio of the moment coefficients -cm "cm- to find the contribution β 'm of the moment coefficient to the geometric twist. %his contribution has to be subtracted from the re&uired twist angle, too. Using the usual, cambered airfoils with negative moment coefficients will change the sign of the ratio cm "cm-, which results in negative β ∗ 'm values. %his means, that the subtraction from β re# will actually be an addition, increasing the geometric twist angle. If we have different airfoils at root and tip, we can use the mean moment coefficient 7cm!tip cm!root 8"2 to calculate the ratio cm "cm-. >inally, we can calculate the geometric twist angle β geo, which has to be built into the wing2 . E.a*&le As you have noticed, the graphs contain an e+ample, which is used here. #e consider a flying wing model with the following data2
wing span
b 2.3 m
chord length at root
l r /.2/ m
chord length at tip
l t /.1:/ m
sweep angle at c"G line
ϕ /.2 2/;
design lift coefficient
' , /.
root section 8 :9(
cm!r /./1 and α /!r %/.3;
tip section 8 :9G
cm!t /./3 and α /!t /.;
desired stability coefficient +igma /./ #e calculate the wing area /2 5 7l
11.* - 7/."1./8 - 7/./"/.18 2.9; . %his means that our model would need a twist angle of (.) -wash out from root to tip, if we would use a symmetrical airfoil section. %he difference of the 1ero lift angle of tip and root section is . 6ow we read the twist contribution of the moment coefficient from graph (, which is β4Cm H .9, which has to be corrected for our smaller mean moment coefficient2
.* - 7/./2"/./8 2.32; . >inally, we calculate the geometric twist from
2.9; % /.*; % 2.32; %/.1:; . %he negative value means, that we could use a small amount of wash$inJ %his is because we have already enough stability due to the selection of airfoils with refle+ed camber lines. /ince the calculated amount is very small, we can use the same angle of incidence for the root and tip ribs. /ince the presented method is not perfect, we can assume an accuracy to : degree, which is also a reasonable assumption for the average building skills.
Bi!liogra&+y 7es, lots of Ferman references, but there are at least +ome papers in 8nglish below... :. epperle, 5artin2 =e$e >rofile f?r =$rfl?gelmodelle! >5%$Kolleg 9, Lerlag fMr %echnik und andwerk, !aden$!aden, Fermany, :)99. (. Unverfehrt, ans$NMrgen2 Fa+@ination =$rfl?gel , Lerlag fMr %echnik und andwerk, !aden$ !aden, Fermany, :))=. <. epperle, 5artin2 =e$e >rofile f?r Eletro%>ylonmodelle! >5%$Kolleg :=, Lerlag fMr %echnik und andwerk, !aden$!aden, Fermany, :)):. G. epperle, 5artin2 =e$e >rofile f?r >ylon $nd 5peed , >5% ;, Lerlag fMr %echnik und andwerk, !aden$!aden, :)9;. . Nakob, 'tto2 Eppler 22/ $nd Eppler 221, >5% (, Lerlag fMr %echnik und andwerk, !aden$ !aden, :)9. ;. epperle, 5artin2 =e$e >rofile f?r +c&nelle Fl$gmodelle, >5%$Kolleg :@, Lerlag fMr %echnik und andwerk, !aden$!aden, :))G. @. epperle, 5artin2 5egelfl$gmodell i@@ard mit >rofil C 42, >5% (, Lerlag fMr %echnik und andwerk, !aden$!aden, :)99. 9. Anderson, D. A.2 Dntrod$ction to Flig&t , ropeller+, AIAA$9<$=:)=. Also published in the Nournal of Propulsion and Power, Lol. :=, 6o. , /eptember$'ctober :))G. ::. 0arrabee, 8. 8.2 >ractical He+ign of inim$m Dnd$ced ,o++ >ropeller+, /A8 paper @)=9, !usiness Aircraft 5eeting and 8+position, #ichita, April :)@). :(. ansen, .2 indanalme++$ngen im Reynold+@a&lbereic& )on odellfl$g@e$gen, >5%$ Kolleg ;, Lerlag fMr %echnik und andwerk, !aden$!aden, Fermany, :)9). :<. 5ueller, %. et al.2 ,ow Reynold+ =$mber ind T$nnel ea+$rement+6 T&e Dmportance of being Earne+t , Conference on Aerodynamics at 0ow Qeynolds 6umbers, 0ondon, :)9;. :G. 8ppler, Q. and /omers, D.2 A 'omp$ter >rogram for t&e He+ign and Analy+i+ of ,ow%5peed Airfoil+, 6A/A %5$9=(:=, :)9=. :. 8ppler, Q.2 >rati+c&e Berec&n$ng laminarer $nd t$rb$lenter Ab+a$ge%(ren@+c&ic&ten, Ingenieur$Archiv. :;. 8ppler, Q.2 T$rb$lent Airfoil+ for (eneral A)iation, Nournal of Aircraft, Lol. :, 6o. (, :)@9. :@. Fuglielmo, N. and /elig, 5.2 5panwi+e Jariation+ in >rofile Hrag for Airfoil+ at ,ow Reynold+ =$mber+, Nournal of Aircraft, Lol. <<, 6o. G, :));. :9. >elske, U. und /eubert, Q.2 >rofilme++$ng bei leinen Reynold+@a&len im odellwindanal , /tudienarbeit, Institut fMr Aero$ und Fasdynamik, UniversitRt /tuttgart, :)9<. :). Anderson, N. D.2 F$ndamental+ of Aerodynamic+, (nd 8dition, 5cFraw$ill, :)):, I/!6 =$ =@$:==@;@$). (=. Althaus, D.2 >rofilpolaren f?r den odellfl$g , 6eckar$Lerlag, Lillingen$/chwenningen. (:. /elig, 5. /., Donovan, N. >., >raser, D. !.2 Airfoil+ at ,ow 5peed+, /oartech 9, :)9). ((. Krause, !.2 odellmotorentec&ni , transpress L8!, !erlin, :)9;. (<. Nennings, F.2 Two%5troe T$nerK+ Candboo , .P. !ooks-S :)@<. (G. !Tnsch, . #.2 Her +c&nella$fende Lweitatmotor , 5otorbuch Lerlag, /tuttgart, :)9(.
(. 0yon, C., /elig, 5. /., !roeren, A.2 Bo$ndary ,ayer Trip+ on Airfoil+ at ,ow Reynold+ =$mber+, AIAA )@$=::, :))@. (;. Kat1, N., Plotkin, A.2 ,ow%5peed Aerodynamic+, 5cFraw$ill, :)):, I/!6 =$=@$:==9@;$G. (@. 6ickel, K., #ohlfahrt, 5.2 5c&wan@lo+e Fl$g@e$ge, !irkhRuser Lerlag, :))=, I/!6 <$@;G<$ (=($, also available in english as2 Taille++ Aircraft in T&eory and >ractice, AIAA 8ducation /eries, I/!6 :$;<$G@=)G$(. (9. epperle, 5.2 Eppler%>rofile, 5%! :"(, Lerlag fMr %echnik und andwerk, !aden$!aden, :)9;, I/!6 <$99:9=$:==$;. (). epperle, 5artin2 >rofile f?r =$rfl?gelmodelle! >5%$Kolleg (=, (:, and ((, Lerlag fMr %echnik und andwerk, !aden$!aden, Fermany, :))@. <=. /elig, 5., Fuglielmo, N., !roeren, A., FiguVre, P.2 5$mmary of ,ow%5peed Airfoil Hata, Lolume :, /oar%ech Publications, Lirginia !eac h, :)), I/!6 =$);G;@G@$:$9 <:. /elig, 5., 0yon, C., FiguVre, P., 6inham, C., Fuglielmo, N.2 5$mmary of ,ow%5peed Airfoil Hata, Lolume (, /oar%ech Publications, Lirginia !each, :));, I/!6 =$);G;@G@$($; <(. /abbagh, K.2 21+t 'ent$ry Met % T&e aing of t&e Boeing ::: , Pan !ooks, :));, I/!6 =$ <<=$<(9)=$. <<. 0ennon, A.2 R"' odel Aircraft He+ign, AirAge Publishing -5A6, :==;, I/!6 =$)::()$ G=$( lows, /pringer:))), I/!6 <$G=$;;:9:$;
Airfoils for 0ailless Air&lanes5 Design and Selection Remar6 %he following paper is a summary of a presentation, given on the : |
Air&lane 0y&es and Mo*ent Coefficient
5oment Coefficient and Airfoil /hape Qefle+ and 5oment Coefficient Qefle+ and 0ift W Drag 0ocation of Camber and 5oment Coefficient Lelocity Distribution and !oundary 0ayer Dangers everywhere Airfoil Design for 0ight %ailless Airplanes Conclusions
Air&lane 0y&es and Mo*ent Coefficient !esides lift and drag coefficients, the moment coefficient cm is of importance for the behaviour of an airplane $ it has a big impact on the longitudinal stability. #hile a conventional airplane can compensate the moment of the wing with its hori1ontal tail, a tailless plane obviously can3t. It is possible to divide tailless airplanes in three groups, depending on how they achieve longitudinal stability. %he re&uirements for the moment coefficient of the airfoil is a direct result of the stabili1ing mechanism. %ing %it+ot s%ee& '&lan() 0ongitudinal stability is created solely by the airfoil. A plank re&uires an airfoil with a po+iti)e moment coefficient. • s%e&t %ing It is possible to use any airfoil, because longitudinal stability can always be achieved by selecting a suitable combination of sweep and twist. >or best allround performance, airfoils with low moment coefficients -around 1ero are better suited although. %hey need smaller amounts of twist, which results in a broader speed range without paying too much penalties off the design point. • %ing %it+ a lo% &osition of t+e center of gra-ity '&arafoil) %he moment coefficient is less important and it is possible to use traditional airfoils with negative moment coefficients. %he position of the c.g. can be chosen to guarantee stability, but usually airfoils with medium moment coefficients are chosen to achieve higher penetration speeds and a wider speed range. •
'la++e+ of taille++ airplane+ and t&eir typical moment coefficient+. Remar6 %o achieve static longitudinal stability, the center of gravity -c.g. must be located in front of the neutral point, which makes the momentum derivative dCm"dAlfa -rel. c.g. negative. Placing the c.g. in the neutral point results in dCm"dAlfa H =, making the plane indifferent, i.e. it will not stabili1e itself after a disturbance. %hese re&uirements lead to the moment coefficients presented above. %o achieve dynamic stability oscillations must be damped out, which can be difficult, when the c.g. is located too far
ahead of the neutral point and the moments of inertia around a spanwise a+is are small -as with an unswept flying wing. Air&lane 0y&es and Mo*ent Coefficient
5oment Coefficient and Airfoil /hape
Airfoils for 0ailless Air&lanes5 Design and Selection Airplane %ypes and 5oment Coefficient Mo*ent Coefficient and Airfoil S+a&e
Qefle+ and 5oment Coefficient Qefle+ and 0ift W Drag 0ocation of Camber and 5oment Coefficient Lelocity Distribution and !oundary 0ayer Dangers everywhere Airfoil Design for 0ight %ailless Airplanes Conclusions
Mo*ent Coefficient and Airfoil S+a&e %he shape of an airfoil can always be thought of being composed of two parts2 a camber di+trib$tion -camber line and a t&icne++ di+trib$tion, which is distributed along the camber line. %he re&uirements for the three classes of tailless planes can be translated in corresponding airfoil shapes and wing twist distributions. %he only way to achieve a positive moment coefficient and the re&uired amount of lift is to use a 5%+&aped camber line, which is also called a reflexed camber line. •
•
•
%ing %it+ot s%ee& '&lan() A po+iti)e moment coefficient results in a reflexed camber line. A twist does not help for stability, but can improve the stall characteristics of the wing. s%e&t %ing 0ow moment coefficients and a small amount of twist can be achieved by airfoils with little camber and a neutral or slightly refle+ed camber line. %ing %it+ a lo% &osition of t+e center of gra-ity '&arafoil) %he moment coefficient poses no strong restriction on the airfoil shape.
'la++e+ of taille++ airplane+ and t&eir typical airfoil +&ape+. Airfoils with strongly refle+ed camber lines are usually not used on conventional airplanes $ they are uni&ue to tailless airplanes. >rom the aerodynamicists view, airfoils with Xrefle+Y are &uite difficult and very interesting to develop, because they are very sensitive with respect to changes in Qeynolds number . %hus the following sections will concentrate on airfoils with refle+ed camber lines, as they are used in tailless planes with no or little wing sweep. Airplane %ypes and 5oment Coefficient Mo*ent Coefficient and Airfoil S+a&e
Qefle+ and 5oment Coefficient
1efle. and Mo*ent Coefficient #e already know, that the moment coefficient ' m and the shape of the camber line are closely connected. If we e+amine airfoils with a refle+ed camber line more closely, we find, that the shape of the rear part of the camber line has a big influence on ' m. In fact, it is possible to adust the shape near the trailing edge to achieve nearly any desired ' m. %he figure below shows how c m can be controlled2 starting from a symmetrical airfoil, a flap is deflected smoothly upwards by and := $ the moment coefficient follows the deflection.
T&e plot of moment coefficient )+. angle of attac +&ow+! &ow cm depend+ on t&e amo$nt of reflex. Using this trick, the problem seems to be solved. #e simply bend the railing edge upward until we achieve the moment coefficient necessary to stabili1e our tailless plane and there we go... !ut we probably prefer an airplane, which not only flies safe and stable, but also performs with a low sink speed, a high penetration speed and a good ,"H ratio $ that3s where all the trouble begins.
1efle. and Lift 6 Drag %he shape and location of the lift and drag polar of the airfoil is the key to airplane performance. %he images below show, how the cd $cl polar changes, when the camber line gets refle+ed.
,ift )+. drag coefficient+ for different amo$nt+ of reflex. 6ow we have a problem2 while we add refle+ to the camber line, in order to shift the moment coefficient towards the positive values, we shift the lift vs. drag polar down. %his means, that we actually reduce the lift at a certain angle of attack and, what3s even worse, we also reduce the ma+imum lift coefficient. A reduced ma+imum lift coefficient leads to higher stall and landing speeds, which is not e+actly our aim. 'f course the aerodynamicist already knows a remedy against
low lift2 he increases the amount of the ma+imum camber. Indeed, this increases the lift, but also reduces the positive moment coefficient.
Location of Ca*!er and Mo*ent Coefficient 0uckily, there is still one parameter left to compensate for the destabili1ing effect of increased camber $ it is the location of the ma+imum camber. Its location has a small influence on the lift vs. drag polar, but a strong impact on the moment coefficient. !elow, the moment coefficient vs. the angle of attack for a family of airfoils is shown. %he airfoils differ in the location of the ma+imum camber xc "c.
Dnfl$ence of t&e location of t&e camber 7xc "c8 on t&e moment coefficient. As we can see, moving the location of the ma+imum camber backwards, also shifts the moment coefficient down towards negative values. %hus it might be advisable, to concentrate the camber in the first &uarter of the chord length, if we want to compensate the lift loss -introduced by the refle+ed camber line by an increased amount of camber.
Dangers e-ery%+ere #e have already learned, that several parameters, linked closely together, have an influence on the design of a low moment, high lift airfoil. #e have not yet talked about the additional problems, introduced by the behavior of the boundary layer.
>roblem+ facing t&e de+igner of an airfoil for taille++ airplane+. %he >igure above and the enumeration below show the most important parameters and how they are linked together. •
•
•
5oving the ma+imum camber towards the leading edge Z more stable moment coefficient -more positive. $ puts more stress on the boundary layer near the leading edge -suction peaks. Increasing the amount of camber Z better lift -cl vs. cd shifts towards positive cl values. Z cl%max may be increased. $ less stable moment coefficient -more negative. $ puts more stress on the boundary layer -more pressure drop towards the trailing edge. Increasing the amount of refle+ Z more stable moment coefficient -more positive. $ less lift -cl vs. cd shifts towards negative cl values. $ cl%max decreases.
3elocity Distri!tion and Bondary Layer %o shed more light on the boundary layer problems, we will have a look at velocity distributions of airfoils with refle+ed camber lines.
Cow camber and reflex c&ange t&e )elocity di+trib$tion. %he image above presents velocity distributions of four different airfoils with different combinations of refle+ and camber. %he moment coefficient of all airfoils is similar. In general, the velocity distributions of the upper surfaces show high velocities in the first third of the chord length, steadily decreasing as the flow reaches the trailing edge. Depending on the Qeynolds number, the steepness of the velocity drop -which, according to !ernoulli3s e&uation, represents a pressure rise is limited. #hen the pressure rise is too strong, the flow separates, causing loss of lift and increasing drag. %he velocity distribution on the lower surface pose less problems, with a danger for boundary layer separation near the trailing edge. %his will have a destabili1ing effect and will cause an increase in drag. A uni&ue feature of airfoils with a refle+ed camber line is the crossover of the velocity distributions in the second half of the chord length. Increasing the amount of refle+, speeds up the flow on the lower surface, while it slows down the flow on the upper side. %he enclosed area of negative lift near the trailing edge increases, driving the moment coefficient towards positive -more stable values. Increasing the camber will increase the velocity on the upper surface and decrease the speed on the lower surface. %he enclosed area of positive lift in the front half of the airfoil also increases, contributing to the lift. #hen the camber is increased too much, the ma+imum lift may decrease, because the camber has to be compensated by a larger amount of refle+, putting more stress on the boundary layer. %ypically, the ma+imum lift can be increased to a certain amount, by increasing camber and refle+, but at the cost of a harder stall, which might be dangerous during takeoff and landing.
Airfoil Design for Lig+t 0ailless Air&lanes /maller tailless airplanes, like ultra light and foot launched gliders should be designed with a larger safety margin regarding stall speed and stall behavior. Due to their low flight speed, the are responding lively to guests during the takeoff and landing phase, which may cause large angle of attack variations. Additionally they operate in a Qeynolds number range between (==3=== and : million, where the boundary layer has a strong impact on airfoil performance and behavior.
>irst we will have a look at two e+isting airfoil sections, which have been used in man carrying tailless airplanes. %he first one is a modern airfoil -> =$$:(; with a refle+ed camber line, which was not designed for amateur built tailless planes, but for helicopter blades. 6evertheless it has been used in some amateur proects, by people seeking for ma+imum performance. %he second airfoil -orten II has been designed for light tailless planes, but is much older, having been designed in the :)<=s. F7 89:;:"/<
%his airfoil has been designed by >. . #ortmann during a stay at the !ell helicopter company in the U/A.
Jelocity di+trib$tion of t&e FN /%C%12 airfoil at 3 different angle+ of attac. T&e de+ign point corre+pond+ approximately to t&e middle of t&e t&ree c$r)e+. %he velocity distribution shows a typical low speed laminar flow airfoil. %he velocity on the upper surface increases steadily to =E chord length, where a rather steep, concave velocity drop starts. %he velocity on the lower surface is nearly constant, slightly increasing towards the trailing edge. /uch a distribution results in laminar flow over the first half of the airfoils upper surface and along the total length of the lower surface. %he drag polar, shown below, indicates the benefits and the problems of such a velocity distribution2 very low drag and good performance, but only in a narrow range of lift coefficients -laminar b$cet . 'utside of the bucket, the drag increases rapidly and the danger of flow separation at low Qeynolds numbers is &uite high. %he airfoil has a hard stall behavior and suffers from flow separation at Qeynolds numbers below ( millions. >or an amateur built, light tailless glider, it is probably not a good choice -but well suited for the blade of a helicopter rotor.
Hrag polar+ of t&e FN /%C%12 airfoil.
;orten II
As a reminiscence to the past, this airfoil is astonishingly well suited for our purpose. 'riginally this airfoil had been designed and used by the orten brothers, who built a number of high performance sailplanes during the :)<=s and :)G=s. %heir designs are legend, although some of their published performance data are a little bit too good. %here planes were well known for their good handling characteristics, which can be of greater importance that the actual performance figures of a sailplane.
Jelocity di+trib$tion of t&e Corten airfoil at 3 different angle+ of attac. %he velocity distribution of the orten II airfoil looks &uite different from the > airfoil. %he distribution on the upper surface, is much smoother. Due to the blunt leading edge, no suction peaks are visible for the selected angles of attack. %he pressure rise -velocity drop starts near (=E of the chord length, which leads to a less steep gradient. %he lift vs. drag characteristics shows smoother polars with no distinctly separated low drag region. %he airfoil does not reach the low drag of the > airfoil.
Hrag polar+ of t&e Corten DD airfoil. %he stall characteristics of the orten II airfoil are smoother than the > =$$:(; airfoil, but due to its e+cessive thickness, it also suffers from boundary layer separation at lower Qeynolds numbers. Its moment coefficient is not positive enough for an unswept flying wing. A thinned version of this airfoil with a slightly increased refle+, might be usable for a light, tailless glider. Instead of trying to modify an e+isting airfoil a new design has been performed to find a compromise between performance and handling.
M; =>
During the design of this airfoil, the emphasis has been to design an airfoil, suited for light, man carrying gliders, with moderate wing sweep. #hile performance was an issue, ma+imum lift capabilities and good handling &ualities were more important. Qegarding the usual construction techni&ues, it was decided to allow transition on the upper surface near :E of the chord length. %he blunt leading edge and the smooth movement of the trailing edge separation is responsible for the comfortable stall characteristics. >light Qeynolds numbers can start at G==3===, where the ma+imum lift coefficient is appro+imately :.G.
Jelocity di+trib$tion of t&e C :* at 3 different angle+ of attac.
,ift )+. drag polar+ of t&e C :*.
Conclsions Depending on the type of tailless airplane, stability re&uirements lead to different criteria for airfoil selection. >or most tailless planes, airfoils with low moment coefficients yield the best performance. 0ow moment coefficients and high lift coefficients can be achieved by using refle+ed camber lines, but the corresponding velocity distributions are sensitive to low Qeynolds numbers, and may result in problems with stall behavior. %he best compromise for light, tailless airplanes seems to be a moderately refle+ed camber line, combined with the ma+imum camber shifted towards the leading edge and a rather blunt nose. %ogether with a smooth velocity distribution, a soft stalling character can be achieved. Concerning the stall characteristics, the airfoil plays an important role, but only in conunction with the spanwise lift distribution, which is a result of the wing planform and the spanwise twist distribution. /wept and tapered wings tend to have a higher loading near the wing tip, which causes the tip to stall first, if no additional twist is used.
Alas c8n !a?a relacion de as&ecto Wings of lo% as&ect:ratio are known for having poor aerodynamic efficiency 0"D at low speeds, along with stability problems, both static and dynamic.
%he high angle of attack stall of the low aspect$ratio wings has been known since the early days of aerodynamics. #hy short wings, then S $ %here can only be practical limitations to use such wings. %he most interesting case is the wing$in$ground ship -AQ H : < there are also short wings in racing cars -typical aspect ratios AQ B < short wings are also part of control devices in competition sailing boats and micro air vehicles. 0ow aspect ratios -about ( < in fighter aircraft are necessary to maintain a high degree of manouvrability, though at the e+pense of some stability -see below. #e define a low aspect$ratio wing a wing having AQ B . /&uared wings and wings with even lower aspect ratios are still interesting from a theoretical point of view, but of uncertain application. %heoretical analysis was performed by !et1 in the :)(=s. 0ater a number of non$linear lifting line theories -5ulthopp, Fersten, %ruckenbrodt, and others provided simple means to predict the vorte+ lift created by the leading edge separation. >ig. : below shows a &ualitative e+ample of how strong an influence the leading edge vorte+ can have on the lifting characteristics of a short wing.
Figure 1: Lift characteristics at very low aspect-ratios
Wing Perfor*ance at Lo% S&eeds S+ort %ings for low speeds have been studied for a long time. %he high angle of attack stall of the low aspect$ratio wings has been known since the early days of aerodynamics. andley$Page -:):: found that a s&uared wing -AQH: stalled at angles above G= degrees, while a moderately slender wing -AQH;.( stalled at incidences as low as := degrees.
E.&eri*ental Data
A large body of data was published in the :)(=s as a result of a series of e+periments at Fottingen, Fermany. Data for different aspect$ratios can be correlated &uite well using the Prandtl$0anchester formula, originally derived for elliptic spanwise loading. aving reference data at aspect$ratio AQH, one can derive the polar characteristics of shorter wings with remarkable precision, at least in the linear range. Ma.i** Lift Coefficients
%he values of the ma+imum lift coefficient are largely independent from the aspect$ratios at aspect$ ratios above ( and a Qeynolds number : million. At aspect$ratios : B AQ B ( the most evident effect is the shift of the angle of attack at which ma+imum lift is reached. %his angle increases progressively and easily reaches <= degrees. /&uared wings show Clma+ as high as :.<. At even lower aspect ratios the wing is subect to strong vorte+ flows and C0 increases at a faster rate than that predicted with a linear theory. %his is due to the presence of the strong tip vortices that separate closer to the leading edge, according to a mechanism similar to that governing the delta wing. %he spanwise distribution of lift is another interesting aspect of these wings. %he lift is mostly concentrated in the inboard sections and reaches high values. Aft sweep moves the point of ma+imum lift outboard, which on turns may promote undesirable tip stall. Pressre C+aracteristics
8+perimental investigations showed that the root sections do not e+perience high 08 pressure peaks. In addition, the spanwise pressure gradients are such as to cause an outward drain of the boundary layer from the root sections. %he combined influence of these two effects is such as to make the root sections highly resistant to flow separation and therefore capable of developing local lift coefficients of such large magnitude as to more than compensate for the lift losses that occur when the tip sections of the wing stall. Sta!ility C+aracteristics
0ongitudinal and lateral stability of low aspect$ratio wings have been investigated over the years. 'ne important aspect is the behavior of the pitching moment with the lift coefficient, that is strongly dependent on the wing sweep and various technical devices -tip devices, fences, nacelles, etc.. 0ongitudinal stability depends mostly on the aspect$ratio and the sweep angle -/hortal$5aggin, :):. %he stability limit is appro+imated in the figure below for untapered wings -tapered wings have less conservative limits.
Figure 2: Longitudinal stability limits (empirical data). 0eading$edge separation due to induced camber in the three$dimensional wing may cause a bubble of large spanwise e+tent, so that airfoils that ordinarily stall from the trailing$edge actually stall from the leading$edge. %his phenomenon is dependent -at least on the sweep, on the leading edge radius and the twist. Inflection Lift
In the study of wing stability it is useful to define the terms inflection lift and $+able lift . %he inflection lift is the point at which there is a change in the pitching moment without conse&uences to the stability. Usable lift is the point at which the pitching moment breaks away and leads to a shift of the aerodynamic center with maor conse&uences for the longitudinal stability. >igs. < and G below show e+amples of pitching moment behavior at constant aspect$ratios and at constant sweep, respectively. %he data are &ualitative, but they are compiled from e+perimental works reported in Qef. :.
Figure 3: Pitching moment at constant sweep angle (45 deg)
Figure 4: Pitching moment at aspect-ratio (AR=4.) %he ratio between inflection lift and ma+imum lift is of the order =.$:.=, with the lowest values approached by low aspect$ratio wings, and the largest values proper of slender wings. Means for I*&ro-ing Perfor*ances
%ip stall is one of the problems encountered by short wings ->ig. (, especially at take$off and landing. 'ne very effective way to delay or even remove the stall is the use of fences -or vanes, which are vertical surfaces aligned with the flight direction.
%he fences provide a physical bound to the spanwise pressure gradients and constrain the boundary layer drain toward the tip, which is the prime cause of the tip stall.
Figure 5: Effect of fences on CL of swept back wing. 'ther methods include2 nacelles, stores, e+tensible leading$ edge flaps, droop nose, boundary layer control, chord e+tension, variable sweep, camber, and twist. Lertical" hori1onal tail and wing$body combinations make up for additional effects.
Perfor*ances at S&ersonic S&eeds #ing sweep and aspect ratio at supersonic speeds have a rather precise correlation, showing the stability limits of the wing. In this correlation the aspect$ratio decreases as the sweep increases. >or sweeps of G= to ;= degrees the aspect$ratios range is ( to G, although this limit can be occasionally e+ceeded by accurate design of the tails and other control surfaces. /ince high sweep is re&uired to fly with agility at supersonic speeds, the aspect$ratio is set as a conse&uence. %he %able below shows a summary of fighter aircraft wing aspect$ratio and ma+imum speeds -data compiled from Nane3s Information /ystem.
0a!le "5 As&ect:1atios of Fig+ter Wings Aircraft
A1
M
U/ >$: -5cDonnell$Douglas
<.=
(.
U/ >$:9 -5cDonnel$Douglas
<.
:.9
Dassault 5irage (===
(.=
(.(
Dassault Qafale 5'(
(.;
(.=
/ukhoi /u$(@
<.
(.<
5apo 5ig$()
<.G
(.<
%he wings of the aircraft on %able : are all swept back. Development of swept forward wings is still at the research stage, e+cept for one prototype fighter -$(). 1elated *aterial • • • • • •
'bli&ue >lying #ing >orward /wept #ing Delta #ing ypersonic #averiders 6on$planar #ings %ables of Aspect$Qatios
On t+e We!
These sites are not part of the aerodyn.org domain. There is no guarantee nor control over their content and availability. •
#ing$in$Fround /hips,
Selected 1eferences
>ink 5P, 0astinger N0, Aerodynamic '&aracteri+tic+ of ,ow%A+pect Ratio ing+ in 'lo+e >roximity to t&e (ro$nd , 6ACA %6 D$)(;, :);:. • /chlichting , %ruckenbrodt 8. Aerodyna*ics of t+e Air&lane , 5cFraw$ill, 6ew 7ork, :)@). •
•
@anes5 All t+e Worlds Aircraft ":/88/, edited by P. Nackson, :))) -published fully update every year J
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