Stevens, V. I.

English

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PRl7DICTION OF TEIE mODYNAl4 IC  CHARACTERISTICS 
OF WINGS OF mfTRARY PLAN FCJI~M 
By Victor I. Ste-recs 
Aaes Aeronautical Labora5oq~ 
I n  our present effor t  t o  fly i n  and through the transonic-npeed 
range we have resorted to  widely diversif ied plan f o r m .  ' Ranges of 
sweep, espect r a t io ,  and taper rp t io  a re  being considered which 
extend f e r  'oeyond those considered prac t ica l  several years ago, and 
the e f fec t  of wide variations i n  those pai-te~s on the ~ubsonic  
aerodynamic character is t ics  of wings is  a0 yet  largely unknown. 
The mult ipl ic i ty  of posaible plan form preclLide9 an inveeti&ation 
of each experimentally. A s  a result,considereblc e f f o r t  has been 
directed towslds developirq a theoret ical  method of predicting the 
loading over the wine since, once t h i s  loading has been deterained, 
not only can' s t ruc tura l  loads be estimated but values of the various 
aercx5ymmi.c-chamcteristics such a s  l i f t -curve slope, aeroa-mamic- 
center location, m d  induced drag can a l so  be found, The invest i -  
gation of several theo:%tical methods f o r  the grediction of loading 
and the applicatzon of one of these methods t o  8 wide range of plan 
forme i s  the subJect of t h i s  paper. 
The theoret ical  methods studied were those developed by Falkner, 
b;r Mutterperl, and by Weissinger. I n  each of these meth0d.s the wing 
is replaced by a diatr ibut ion of vortices.  The strength d is t r ibu-  
t i o n  of these vortices i s  fixed by the bounbry condition which 
requires that the induced veloci t ies  of these vortices produce no 
flow through the plane of the wing. The difference among the method6 
lie's i n  differences i n  physical location of the vortices,  i n  the 
disposit ion of the control points where the boundary condition 1s- 
applied, and i n  the reathematical manipulation. F i w e  1 compares 
the layout of vortices end control points f o r  the msthods. Falkner 
replaced the wlng with a dis t r ibut ion of f i n i t e  horseshoe vortices,  
both spanwise and chordnlse. He likewise dis t r ibuted the control 
points both spanwise and chordwlse, hence h i s  method is c lass i f ied  
as a lift ing-eurface method. A s  a resu l t  of his work Falkner has 
recommsnded a particular vortex and ccmtrol point diatr ibut ion which 
was followed i n  our studieo. I n  contrast t o  the Falkner l i f t i n g -  
surface method, both the Weissinger and the Nutterperl methods a re  
l i f t i ng - l ine  methods; that is, the loading 13 concentrated on the 
q~mrter-chord l ine ,  and the control points a re  distri'outea along the 
three-quarter-chord l ine ,  The Weissinger and Mutterperl methods 
d i f f e r  i n  the spanwise location of the control points and i n  the 
mathematical development. 
https://ntrs.nasa.gov/search.jsp?R=19790079931 2020-03-21T20:14:53+00:00Z
Van Dorn and DeYoung have examined each of these method8 fo r  
accuracy and ease of a ~ p l i c a t i o n  (reference 1 ) .  The accuracy was 
evaluated by comparing the predicted and the experimentally deter-  
mined aerodynamic charactor is t ics  of f ive winge having sweep angles 
ranging from -45O t o  45O. By keeping an account, of the time required, 
each method was a l so  evaluated with r e e r O  t o  ee.se of application. 
The resu l t s  of t h i s  study are  shown i n  f igure  1. From a comparison 
of the predicted and experimental values of ep~nwise loading, l i f t -  
cume slope, ' a d  spmwi se center of pressure, the Falkner method 
wns judged t o  be veqy trccurate and the Weissinger method only 
s l ight ly  lees  accurate. The Mutterperl method, whl l e  predicting 
with moderate accuracy the character is t ics  of the sweptback wings, 
did not give accuracy conrparnble to-other methods i n  the case of 
the m e p t f o m r d  wings. The time required per  solution was  l ea s t  
f o r  the Weissinger method (3  hr). I n  contrast 28 hours wes required 
per solution f o r  the Mutterperlmethod and 30 hours f o r  the F a l h e r  
method. 
On the basis  of these resu l t s  it was concluded that f o r  a 
detailed stukv of ,a glqren plan form, where a high degree of ~ccurncy  
was desired ~ n d  whe~e ease of a.pplication assuaed l e s se r  jmportance, 
the Fallney method was best .  However, f o r  a general study of a 
variety of plan forms the Weissingel method ~ppeared  best  sui ted 
since good accuracy could be had a t  a 90 percent awing  i n  conguting 
time. 
kccordingly we have u t i l i zed  the Weissingar method t o  invest1 - 
gate the loading and a s a o c i ~ t e d  aerodynainic character is t ics  of a wide 
ranae of g3.m forns. Figure 2 pictures the & w e  covered but does 
not indicate the number of plen f o m  considered. Actually the 
characteristics of about 200 w-lngs were calculated, The general 
range of variables included sweep from 45O f o m r d  t o  600 back, 
aspect r a t io s  from 1.5 t o  8.0,and taper -:atios from 0 t o  1.,5, The 
s t ruc tura l  f eas ib i l i t y  of the varioue shapes m s  used a s  a rough 
guide i n  selecting the l imlting values of the geometric parameters. 
The resu l t s  of t h i s  investigetion are  found i n  reference 2 .  
Charts presented i n  t h i s  reference allow e. rapid rtnd sirnple Ceter- 
mination of the nost i lqortant  ae rod~nar i c  c h a ~ c t s r i s t i c s  of any 
wing haying a plan form fal l ing within the rsxlge of t h i s  study, 
Aerodynamic character is t ics  which c8n 7?e i-ead d i rec t ly  from these 
charts include the span-loading coeff ic ients ,  s~anwise center of 
pressure, l i f t -curve slope, and a .erodyne~c  center. These ps.rameters 
a re  given a s  a function of sweep f o r  families of sspect rcttio and 
f o r  'various taper ra t ios .  Sufficient values of aspect r a t i o  and 
tapor r a t i o  were chosen t o  allow rapid and accurate interpelation. 
Sample charbe taken from reference 2 a re  shown i n  figure 3. For 
the sake of c l a r i t y  i n  t h i s  presentation, the data sho~m have been 
limited t o  one taper r a t io ,  t o  a few a s ~ e c t  r a t io s ,  and,in the case 
of spanwise loading, t o  one spenwiee s ta t ion .  In the reference paper, of 
course, d a t a  a r e  given f o r  a complefe r a n ~ e  of these geometric 
parameters. The simplicity of obtaining the desired charrrcteristics 
is obvioua. The chart shming the charecter is t ics  is entered a t  the 
proper value of sweep of the quarter-cho;;"d l ine  end the desired 
value of the character is t ics  obtained d i rec t ly .  In this Tanner it 
is possible t o  obtain the win,ploading coefficient a t  four spanwise 
s tat ions,  the l if t-curve slope, the spanwise center of pressure, snd 
the eerodyhamlc center. AerodpLmic charac ter i s t ics  obtained from 
these charts have been correlated with experimental results,and i n  
general the agreement i s  good. S t r i c t l y  epealcing, the method appl ies  
only a t  ze?o l i f t .  However since the aerodynernic charec ter i s t ics  
axe i n  general l i nea r  up t o  angles of a t t ~ c l c  where s e p a ~ t i o n  
occurs, the theoretically predicted c b r a c t e i i s t i c s  can be used with 
good accuracy up t o  t h i s  point. Specific correlations of lift- 
curve slope and aerodynamic center mensured a t  ze-?o l i f t  w i l l  'l?e 
discussed l a t e r .  
- 
Most of the qual i ta t ive effecte  of weep and tape? r a t i o  on - 
span loading a re  not new and hence V i l l  not be discussed i n  t h i s  
paper. However, m e  of the most interest ing i ~ s u l t s  of t h i s  invest i -  
gation showed t h a t ,  f o r  each angle of sweep there i a  a taper r a t i o  
f o r  which aspect r a t i o  has l i t t l e  effect on the s p m  loading and 
f o r  which the span loading i s  pract ical ly  elliptic3.1. This relat ion-  
ship is shown i n  figure 4, A s  the wine is swept forward more inverse 
' , taper  i s  required, and a s  the wing is swept back more of the usual 
type of taper i a  requiredo Because of the e l l i p t i c  loading, minimum 
induced dmg and maximum l i f t -curve slope 8.m obtained for wings .on 
t h i s  l ine.  For plan forme f a l l i h g  on t h i s  l i ne ,  aspect r a t i o  had no 
e f fec t  on the loading. For pl8.n forme above the l ine ,  loading moves 
outboard with inc~eas ing  aspect rrrtio,and, convei-ssly, f o r  plen 
forme below the line,loading moves inboard with increasing aspect 
r a t io .  
. Tyo of the characteristic8 f o u r d d i r e c t l y  from the Weissinger 
method, l i f t -curve slope and aerodynamic center,  a re  of par t icu lar  
value because of t h e i r  irrrportcnce i n  longitudinal s t a b i l i t y  e m l y s i s  
and design. Since they are  so important the accuracy with which the 
Weissinger method predicts these the.-mcteristica end the e f f ec t s  of 
plan form on these character is t ics  as  predicted by the Weissinger 
method should be examined. 
I n  figw'e 3 the theoret ical  and expeillmental ia lues  of lift- 
curve slope are  c o r r e h t e h  fo r  z nimber or" random p l a n  formo . Where 
suff ic ient  c leannce  between points existed, the ving plan forms 
heave been superimposed. Included i n  th ie  col-rele.tIon ?re t r iangular  
wings, highly sweptbaclc wing ,  sweptfomrd wings, and wings wlth - 
inverse tcper.  Most of the eqerimental  b t a  rnre 'taken from refer- 
ence 3 e n d  the --mi nder from other American payers. Devlatlon from 
the 45O l ine  indicates the e c o r  of co:relaflon. On the aveyage, 
t h i s  deviation ici less thm j percant. Although not shown herein, 
we have a l so  corrrpared the l i f t -curve slopes of ~mswept wir,gs a s  
estimated by the Weissinger method rnd by the method employing the 
nodified Jones ' edge-velocity correction, and the P greement i s  
nearly perfect.  - 
A similar coiTela.:ion f o -  nerodynamlc cente-.- is  given i n  
figure 6. E-rperi~ent end theory .do not ahoy a s  goo,d a&-:eeiner,t f o r  
t h i s  parameter a s  f o r  the l i f t -curve slope. The;-e is no c lear  
sys t emt ic  variation i n  the c o r n l a t i o n  with plan fom,end mst of 
the discrepm~cies i n  c o r e l a t i o n  are of the order of the accup-cy 
with which the aerodywnic cente;. usually con 7?e determined Iy exyeri- 
merit. I n  any evont, f o r  75 percent of the plan foilcs the di~crepancy 
Is leas  than 2 >ercent of the M .A .C . ; which discrepency i s  sr i l l  
compared t o  the e f fec ts  of p h  fo-rm. It i s  OUT bel ief  that the 
Weissinger method glves both l i f t -curve slope and aerodyrarnic center 
with sufficient acctl;.lacy f o r  use i n  p~el iminary design studies.  
Sample charts of l i f t -curve slope a re  shown in  figure 7 .  L i f t -  
cume elope i s  given a s  a function of sweep fo r  taper  rp t ios  of 0, 
0.5, and 1.5 and f o r ~ . s p e c t  ra t io8 of 1.5, 3.5) e, ana =. Me~y of 
the curves b.ve again been omitted f o r  c l a r i ty .  The curve f o r  
in f ln f t e  aspect r a t i o  i n  each case is obteined from sirnple sweep 
theorg and. hence is a cosine c w i ~ e .  Note that i n  eech case the 
e f fec t  of aspect m t i o  falls off with incl-ease i n  sweep. Also note 
tha t  f o r  low aspect ra t ios ,  small angles of sweeg huve l i t t l e  e f fec t  
on l i f t - c u n e  slope. A s  the wing appToaches e rnoro pointed pb-n 
form,the l i f t -curve slope increases on swept'oac!~ wings and decreases 
on sweptfo;.wa.:rd wings,  hi l e  inverse taper  reduces l l f  t -cl~rve slope 
on m p t b a c k  wings and increases it on sweptfomrd wlngs. On highly 
swept wings th ie  effect i s  of such mgnitu6e th t  tape- m t i o  exerts  
aa great en influence on l i f t -curve slope cs does aspect m t i o .  Thus 
i n  a q  theoret ical  ~ p p ~ o a c h  t e inport~.nce of including the e f f ec t s  
of taper  m t i o  i o  obvious. 
Figure e presents sarrple charts of the aer-.mic center which 
is a lao  plotted a s  a function of sweeg f o r  taper r-atios of 0, 0.5, 
and 1.5 and f o r  aspect ~ t i o s  of 1.5, 3.5, and p.0, For the plan 
f o r m  investigated, the ne-odynamic-center locnticln wnged from a.s 
far  forward a s  15 percent of the M.A.C. t o  a s  f a r  back a s  40 percent 
of the M.A .C . I n  contraet , the aerodpamic center on unswept wings 
i s  eeldom more than 2 o r  3 percent of the M.A.C. from the 25 percent 
M.A .C . point. On highly tapered wings sweepback moves the aerody- 
namic center reamrard and sweepforward wves  it f 'omrd.  A s  taper  i s  
decreased, the trend is reversed so that, an wings havlng inverse 
taper,  sweepback moves the aerodynamic oenter forward while sweep- 
forward moves the eerodymmic center back. The magnitude of t h i s  
movement i n  each case is generally increased by increase i n  aspect 
r a t io ,  
As e ts ted  ea r l i e r ,  l i f t -curve slope and aerodynamic-center 
location a r e  important in longi tudiml  e t a b i l i t y  analyeis. That the 
Weissinger method can predlct  values of these chP.racteristics with 
euff ic ient  accuracy f o r  preliminary-design haa been shown. One other 
parameter must however be evaluated t o  complete the l.ongitudlna1 
e t a b i l i t y  a n a l p i s ,  and t h a t  pammeter is the downwash i n  the loca- 
t i o n  of the- tail. The Weissinger method can readily be extended t o  
compute downwash i n  the plane of the vortex sheet. We are  a t  present 
evaluating the accuracy of the  do\-sh resu l t s  obtained i n  t h i s  
m e r  by checking them with experimental data. Preliminary resu l te  
of such an evaluation are  @ven i n  figure 9. The maximum d m m e h  
angles as predicted and measured i n  a vertica1,plane approximately 
30 percent of wfng semispan out from the plan of symmstry a r e  s h m  
as a function of angle of a t tack  fo r  f ive swept wings tes ted  i n  the 
Amee 40- by &-foot tunnel, For these wings: t h e o r  predicts the 
var iat ion of downksh with angle of a t tack within 20 percent of the 
msasured value. Insuff icient  comparisons have been made t o  date,  
hatever, t o  warrant generalizations as t o  the accuracy of t h i s  method 
fo r  a wide range of plan f o m .  I n  our pi-esent investigation we 
plan t o  establieh t h i s  accuracy, improve the method where possible, 
and then extend the method so that  downwash mey be determined a t  
pointls above and below the vortex sheet. 
A l l  the resu l t s  obtained t h r o u ~ h  use of the Weissinger method 
apply only t o  incompressible flow. Bowever, t h r o y h  e n  application 
of the Pmdt l -Glauer t  rule, it i s  possible t o  account f o r  the e f fec ts  
of compreeaibilit~l on. span-loading character is t ice  f o r  speeds below 
the c r l t i c a l  speed. The method, which has been summarized i n  some 
d e t a i l  i n  reference 4, t r a r s l a t e s  the e f fec t  of compressibility in to  
an effect ive change i n  plan form i n  addition t o  the well-known 
increase i n  section pressures. 
It i s  apparent that, if such an approech serves t o  predict  
accurately the effectg-of: cmpreesibiUty,  i t . c a n  be used in  con- 
Junction with the subJect paper t o  give a rapid estimation of the 
character is t ics  of wings throughout the Mach number range below the 
c r i t i c a l  weed. A t  each Mach nlmbey the geometry of the wing would 
simply be Oistorted i n  the proper manner and rrsw c h e r a c t e ~ i s t i c s  
obtained from the charts. Only a few random e x p e ~ i m e n t ~ l  checks ,of 
t h i s  procedure have been made, but these compaidaons he.ve indicated 
model-ately good agreement between theory end eqeTiment. It is our 
plan t o  continue t h i s  study t b  es tabl ish the accurocy of the method, 
and, if necessaly, search f o r  means of improving the accu;-e.cy. 
1.. Van Dorn, Nicholas H., am3 DeYoung, John: A Comparison of Three 
Theo-etlcal Methods of Calculating Span Load Distribution on 
Swept Wings. NACA TN No. 14'6, 1947. 
- 
2 .  DeYoung , John: Theoreticbl Additional Span Loeding Characteristics 
of Wings with Arbitrary Sweep, Aspect Retio and Taper Ratio. 
NACA TN No.  1491, 1947. 
3 .  Shortal ,  Joseph A * ,  aW Mnggin, Bernard: Effect of Sweepback and 
. P.s$ect Eatio on Long1 tudim.1 S tab i l i t y  Charactey i s t i c s  of Wings 
a t  Low Speed. K4CA TN No. 1093 , 191~6. 
4. Dickson; R. :  Comparison of TWO Methods of Calculating Aerody- 
namic Loading on an Aerofoil with Large Sweeptack and Small 
Aspect Ratio. TN No. Aero 17W, B:-dtish R.A.E., June 1946. 
HORSESHOE VORTICES 
CONTROL POINTS ONTROL POINTS 
o TINUOUS SHEE 
a f TRAILING VORTICES 
FALKNER MUTTERPERL WEISSINGER 
ACCURACY VERY GOOD VARIABLE GOOD 
TIME 30 HR 28 HR 3 HR 
Ffgure 1.- Comparison of theoretical methods for determining loading. 
ASPECT RATIO = 3.5 v 
6.0 
1.5 v 
ASPECT RAT IO=  3.5 v 
ASPECT RATIO= 3.5 w 
V o a h  
TAPER RATIO * .5 
TAPER RATIO a 1.5 A 
- 
-30 0 30 45 60 
SWEEP ANGLE, DEGREES w 
Figure 2.- Range of plan forms investigated by Weissinger method. 
TAPER RATIO = .5 
SPAN 
LOAOLNG 
COEFF. 
SPANWISE 
c. P X 
SEMI- SPAN 
E N T E R  SECTION 
.5 
O.C. 
% 
MAC 
20 
30 
40 
-40 0 40 80 
SWEEP AMGLE,Aa,DEGREES 
Figui.e 3.- Sample charts of data obtained by Weissinger method. 
--• REGION WHERE LOADING 
MOVES OUTBOAR D W l T H  
INCREASING ASPECT RATIO 
ING MOVES IN- PLANFORMS WITH ELLIPTICAL 
BOARD WITH IN- LOADING INDEPENDENT OF 
ASPECT RATIO 
RATIO 
-45 -30 -15 0 15 3 0  45 60 75 90 
SWEEP ANGLE, As ,DEGREES 
4 
Figure 4. - Taper ratio - sweep angle relationship giving elliptic loadulg. 
1 3 3  { J )  
De 
D7 
a6 
THEORETICAL 
dCC/da .M 
.O 4 
1)3 
0 2  
EXPERIMENTAL d C q d a  
Figure 5.- Correlation of theoretical and experimental values of 
lift-curve slope. 
 THEORETICAL.^^ 
AEROOVNWIC 
CENTER 
LOCATION, 
X M.A.G. .25 
UNE OF PERFECT 
.I5 20 25 30 .35 .40 
EXPERIMENTAL AERODYNAMIC CENTER LOCAM.  X MAC. 
Figure 6.- Correlation of theoretical and experimental locations of the 
aerodynamic center. 
-
.- 
TAPER RATIO. o - TAPER RATIO- .5 
SWEEP ANGLE, 
- - - -  -. 
A%, DEGREES SWEEP ANGLE, A*, DEGREES 
TAPER RATIO 1.5 
SWEEP ANGLE, A s ,  DEGREES 
Figure 7.- Effect of sweep, taper ratio, and aspect ratio on lift- 
curve slope. 
TAPER RAT 10 0 
I I I 
I 
WEEP ANGLE, Aa/, , DEGREES 
20 
O.C. 
% 3 0 .  
M.A.C. 
40 T A P E R  RATIO = 0.5. 
I l l  
Figure 8.- Effect of sweep, taper ratio, and aspect ratio on aerodynamic- 
a 
,- A-1.5 
- 
\ 
\ 
MAXIMUM 
DOWNWASH 
ANGLE 
Figure 9. - Comparison of theoretical and experimental values 
of downwas h . 
 


Prediction of the aerodynamic characteristics of wings of arbitrary plan form

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