The problem of the minimum induced drag of wings having a given lift and a given span is extended to include cases in which the bending moment to be supported by the wing is also given. The theory is limited to lifting surfaces traveling at subsonic speeds. It is found that the required shape of the downwash distribution can be obtained in an elementary way which is applicable to a variety of such problems. Expressions for the minimum drag and the corresponding spanwise load distributions are also given for the case in which the lift and the bending moment about the wing root are fixed while the span is allowed to vary. The results show a 15-percent reduction of the induced drag with a 15-percent increase in span as compared with results for an elliptically loaded wing having the same total lift and bending moment

Jones, R. T.

English

NASA Technical Reports Server

TECHNICAL NOTE 2249 
THE SPANWISE DISTRIBUTION OF LIFT FOR MINIMUM 
INDUCED DRAG OF WINGS HAVING A GIVEN LIFT AND 
A GIVEN BENDING MOMENT 
Robert T. Jones 
Ames Aeronautical Laboratory 
December 1950 
539 
https://ntrs.nasa.gov/search.jsp?R=19760012005 2020-03-22T16:53:46+00:00Z

NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 
TECHNICAL NOTE 2249 
THE SPANWISE DISTRIBUTION OF LIFT FOR MINIMUM INDUCED 
DRAG OF WINGS HAVING A GIVEN LIFT AND A 
GIVEN BENDING MOMENT 
By Robert T. Jones 
SUMMARY 
The problem of the minimup induced drag of wings having a given l i f t  
and a given span is extended t o  include cases i n  which the bending moment 
t o  be supported by the wing is a l s o  given. A s  i n  the  c lass icd l  problem 
of induced drag, the theory is  limited t o  l i f t i n g  surfaces traveling a t  
subsonic speeds. 
t r ibu t ion  can be obtained i n  an elementary way which is  applicable t o  a 
var ie ty  of such problems. 
responding spanwise load d is t r ibu t ions  are a l so  given fo r  the case i n  
which the  l i f t  and the  bending moment about t he  wing root are fixed while 
the span is  allowed t o  vary. 
the  induced drag with a 15-percent increase i n  span as compared with 
r e su l t s  fo r  an e l l i p t i c a l l y  loaded wing having the s a m e  t o t a l  l i f t  and 
bending moment. 
It is  found t h a t  the required shape of the downwash dis- 
Expressions fo r  the minimum drag and the cor- 
The r e su l t s  show a 15-percent reduction of 
INTRODUCTION 
I n  the problem of minimum induced drag as or ig ina l ly  t reated by Munk 
(references 1 and 2) the span of the wing and the  t o t a l  l i f t  w e r e  supposed 
t o  be given and the d is t r ibu t ion  of l i f t  over the span resul t ing i n  a min- 
innun of drag was  sought. The solut ion of t h i s  problem thus provided a 
convenient lower bound f o r  the induced drag of a wing of given dimensions. 
In  the prac t ica l  design of wings the requirements for  low induced 
drag and the requirements for  s t ruc tu ra l  strength are opposed. 
bending moment developed by the l i f t  becomes an important considera- 
t ion - more important i n  many cases than the ac tua l  spanwise dimension 
of the wing. 
minimum drag with l imi ta t ions  imposed on the bending moment as w e l l  as 
on the t o t a l  l i f t .  It is the  purpose of the present paper t o  show how 
the methods of t he  earlier analysis can be extended i n  a very simple way 
t o  the solution of problems involving the bending moment of the load 
dis t r ibut ion.  
H e r e  the  
Such considerations lead t o  the problem of determining the 
541 
NACA TN 2249 
A complete list of symbols employed i n  t h e  ana lys i s  w i l l  be found i n  
t h e  appendix. 
GENERAL FORMULAS FOR LIFT, DRAG, AND BENDING MOMENT 
Reference may be  made t o  t h e  o r i g i n a l  papers of Prandt l  and Munk 
(references 1 and 2 ) ,  o r  t o  any of t h e  standard t e x t  books on aerodynamics, 
f o r  t h e  fundamental developments of wing theory which form t h e  b a s i s  f o r  
the  ca l cu la t ions  of induced drag. 
i s  given by 
I n  these  developments t h e  over-all  l i f t  
+S 
r d y  (1) 
J - S  and t h e  drag i s  given by 
+S 
wi r dy 
I n  these  formulas t h e  wing span is  supposed t o  extend along t h e  y a x i s  
between -s  and +s ,  r is the  l o c a l  c i r c u l a t i o n  o r  vor tex ' s t rength ,  and 
V i s  t h e  constant ve loc i ty  of f l i g h t .  The induced downwash ve loc i ty  w i  
i s  va r i ab le  along the  span and i s  connected with t h e  vor tex  d i s t r i b u t i o n  
l' (y) through t h e  r e l a t i o n  
With t h i s  va lue  f o r  W i  
a double i n t e g r a l  involving the  spanwise d i s t r i b u t i o n  of l i f t  as repre- 
sented by t h e  c i r c u l a t i o n  s t r eng th  
t h e  expression f o r  t h e  drag may be converted t o  
I' 
This i n t e g r a l  may be  reduced t o  a more symmetric form i f  i t  is  in t eg ra t ed  
by p a r t s  on the  supposit ion t h a t  I' f a l l s  t o  zero a t  t h e  wing t i p s .  Thus1 
lThe v a l i d i t y  of equations ( 3 ) ,  ( 4 ) ,  and ( 5 )  can be demonstrated by 
r e f e r r i n g  t o  t h e  l i m i t i n g  values of complex i n t e g r a l s  taken along a pa th  
a shor t  d i s t ance  above t h e  s ingular  po in t  on t h e  real axis. I n  t h e  case 
of equations ( 3 )  and ( 4 )  t h i s  process y i e l d s  t h e  Cauchy p r inc ipa l  value. 
542 
NACA TN 2249 
I n  mathematical terms t h e  problem is  t o  minimize t h e  double i n t e g r a l ,  
equation (5), while holding f ixed  values of 
L = pV lS+' I' dy 
and 
where B is t h e  bending moment spec i f i ed  about t he  poin t  so. For the  
t i m e  being 
although later another example w i l l  appear. 
so w i l l  be taken as t h e  o r ig in ,  o r  wing roo t  (so = 0), 
Although t h e  de r iva t ion  of t h e  formulas f o r  induced drag lfiakes use of 
the  concept of the  l i f t i n g  l i n e ,  i t  is important t o  no te  that t h e  r e s u l t s  
are not a c t u a l l y  r e s t r i c t e d  t o  t h i s  approximation. 
well-known stagger theorem the  induced drag of a l i f t i n g  sur face  w i l l  be  
equal t o  t h a t  of a l i f t i n g  l i n e  i f  t he  spanwise load d i s t r i b u t i o n s  are 
t h e  same. 
According t o  Munk's 
It should be noted f u r t h e r  t h a t  t he  induced drag of a wing having a 
given l i f t  and a given spanwise load d i s t r i b u t i o n  is not  a f f ec t ed  by t h e  
compressibil i ty of t h e  air  a t  subsonic speeds. 
add i t iona l  drag assoc ia ted  with t h e  formation of waves arises and t h e  
induced drag, which i s  assoc ia ted  with t h e  vor tex  wake, becomes only a 
pa r t  of t h e  t o t a l  p ressure  drag. 
A t  supersonic speeds an 
THE DISTRIBUTION OF DOWNWASH FOR MINIMUM DRAG 
I n  general ,  i f  t h e  drag is t o  be a minimum, a s m a l l  v a r i a t i o n  i n  t h e  
shape of t h e  curve of spanwise loading w i l l  produce no f i r s t -o rde r  change 
i n  t h e  drag. 
t o  t h e  o r i g i n a l  loading; i t  is  then necessary t o  f ind  conditions under 
which t h e  drag added by a s m a l l  add i t iona l  loading is  zero. 
The v a r i a t i o n  i n  shape may take  t h e  form of a small addition 
The so lu t ion  of t h i s  lat ter problem is  rendered e spec ia l ly  simple 
by t h e  mutual drag theorem (reference l), which arises from t h e  evident 
symmetry of t h e  i n t e g r a l  t o  be  minimized (equation (5)) .  
states t h a t  i f  t h e  l i f t  d i s t r i b u t i o n  (represented by I' (y)) is t h e  sum 
of two d i s t r i b u t i o n s  r l ,  and I'2, t h e  drag of I'l a r i s i n g  from t h e  
downwash f i e l d  of r2 is  exac t ly  equal t o  t h e  drag of r2 a r i s i n g  from 
the  downwash of 
The theorem 
543 
NACA TN 2249 
Consider now an i n i t i a l  d i s t r ibu t ion  designed t o  achieve minimum 
drag. (See f ig .  1.) The drag added by a s m a l l  addi t ional  loading w i l l  
be composed of three par ts ,  namely: 
1. 
2. 
3. 
The drag of the additional l i f t  act ing alone 
The drag of the or ig ina l  loading a r i s ing  from the downwash 
'J$e drag of the additional loading induced by the downwash f i e ld  
f i e l d  of the addi t ional  loading 
of the or ig ina l  loading 
I t e m  1 is  of second order i n  terms of the  magnitude of the  added l i f t  f o r  
smooth dis t r ibut ions,  t ha t  is, so-called "weak variations." (The f ac t  
tha t  t h i s  second-order t e r m  is invariably posi t ive insures tha t  the  drag 
w i l l  be a minimum and not a maximum.) 
mutual drag theorem. The f i rs t -order  var ia t ion i n  drag can then be com- 
puted by considering only the drag of the  small addi t ional  l i f t  act ing in  
the induced downwash f i e l d  wi(y) of the or ig ina l  l i f t .  
Items 2 and 3 are equal by the 
The conditions of fixed bending moment and fixed t o t a l ' l i f t  are m e t  
by allowing only those curves of l i f t  var ia t ion tha t  produce no change i n  
these quant i t ies ,  t ha t  is, curves having zero area and zero moment. It 
can be seen tha t  such curves of var ia t ion  must have a t  least three ele- 
ments t o  m e e t  the conditions of zero area and zero moment. Furthermore, 
any curve meeting these conditions can be subdivided in to  groups of three 
elements so tha t  the individual groups a l so  s a t i s f y  the conditions. 
Hence, as the representative of such r e s t r i c t ed  curves of var ia t ion w e  
may adopt three small elements having areas 21, 22, and Z3 ( f ig .  1 ) .  
These elements, together with t h e i r  posit ions y l ,  y2, and y3 and the 
loca l  values of the  downwash w -  etc., due t o  the or ig ina l  loading 
must s a t i s f y  the following three equations: I1 , 
> fo r  6L = 0, Z l  + 22 + Z3 = 0 
f o r  
for  
6B = 0, 
6 D i  = 0, 
Z l Y l  + Z2Y2 + Z3Y3 = 0 
Zlwil + Z2wi2 + Z3wi3 = 0 
It can be seen tha t  these equations w i l l  be consistent if 
w - a+by2 and wi3 - a+by3, where a and b are constants t o  be 
determined from the given conditions. 
f ied  f o r  a l l  posit ions y ~ ,  y2, etc. ,  i t  is concluded tha t ,  i n  general, 
w i l  - a+byl, 
52 
Since such equations must be satis- 
w - a + b y  (9) i 
NACA TN 2249 
Hence, f o r  a minimum induced drag with a given t o t a l  l i f t  and a given 
bending moment the  dowzlwash must show a l inea r  var ia t ion  along the span.2 
(See f ig .  2.) 
The foregoing method may be r'eadily extended t o  a more general class 
of problems involving bending moments o r  ro l l i ng  moments. Suppose, f o r  
example, a braced wing is considered, as i n  the dotted out l ine  of f ig-  
ure  3. I n  t h i s  case the bending moment developed by tha t  portion of the 
l i f t  act ing inboard of the  point of bracing attachment may be of no con- 
cern, but it may be desired t o  l i m i t  the bending moment developed by tha t  
portion of the  spanwise load curve extending between t h i s  point and the 
t ip .  I n  t h i s  case so w i l l  not be zero. A t  least three elements are 
required t o  preserve s ta t ionary values of the l i f t  and bending moment, 
and it i s  evident that at least two of the  elements must l i e  t o  the r igh t  
of the point sQ. The three  simultaneous equations are (see f ig .  2): 
Here y2 and y3 are to  the r igh t  of the point so and y1 lies t o  the 
l e f t  of t h i s  point. For these equations3 t o  be consistent wi must have 
the form 
yil - a; wi - a+b(y2-s0); wi3 - a+b(yg-s0) 
2 
Hence, i n  general, the  downwash w i l l  be a constant over the portion of the 
span f o r  which the moment i s  not specified,  as i l l u s t r a t ed  i n  f igure  3. 
I f  no r e s t r i c t ion  whatever is placed on the moment there  is obtained the 
solution of Munk's or ig ina l  problem, namely, t ha t  the  downwash should be 
constant over the e n t i r e  span. 
21t may be noticed a t  t h i s  point tha t ,  whereas the discussion has empha- 
sized the idea of minimizing the  drag, the analysis  actual ly  makes no 
d is t inc t ion  between the l i f t ,  bending moment, or  drag, i n  tha t  station- 
ary values of a l l  three are demanded. Thus equation (9) may be consid- 
ered a necessary condition f o r  the  solution of the following problems: 
(1) given the  t o t a l  l i f t  and the induced drag t o  f ind the d is t r ibu t ion  
of l i f t  over the span tha t  w i l l  r e su l t  i n  a minimum bending moment, 
and (2) given the  bending moment and the induced drag t o  f ind the dis- 
t r ibut ion resu l t ing  i n  the maximum t o t a l  l i f t .  
3See reference 3 fo r  a discussion of solutions of such equations. 
545 
NACA TN 2249 
Determination of Span Loading and Induced 
Drag From t h e  Downwash Dis t r ibu t ion  
The case of b i l a t e r a l  symmetry with moment spec i f i ed  about t h e  
roo t  s ec t ion  w i l l  serve as an example of t h e  ca l cu la t ion  of t he  a c t u a l  
span loading and induced drag. 
t h a t  t h e  downwash d i s t r i b u t i o n  w i l l  cons i s t  of two s t r a i g h t - l i n e  
segments w i th  a reversal of s lope  a t  t h e  plane of symmetry. 
necessary t o  compute t h e  spanwise v a r i a t i o n  of T corresponding t o  
such a curve of downwash. 
It w i l l  be evident from t h e  foregoing 
It is then 
To perform t h i s  c a l c u l a t i o n  by standard methods of a i r f o i l  theory, 
use is made of t h e  idea  t h a t  a t  a g r e a t  d i s t ance  behind t h e  wing t h e  
vor tex  shee t  forms a two-dimensional f i e l d  of motion, with t h e  discon- 
t i n u i t y  i n  t h e  lateral v e l o c i t y  across  t h e  shee t  given by and 
t h e  downwash w given by t w i c e  t h e  va lue  of t h e  induced downwash wi 
a t  t h e  wing. Hence, t h e  quan t i ty  1 / 2  (dI'/dy) - 2iwi can be evaluated 
by means of t he  f ami l i a r  complex ve loc i ty  func t ion  v - i w  of t h e  two- 
dimensional p o t e n t i a l  theory using f o r  
vor tex  sheet.  
is given along t h e  l i n e  represent ing  t h e  trace of t h e  span, then t h e  
ve loc i ty  vec tor  a t  any o ther  po in t  i n  t h e  f i e l d  
obtained from the  r e l a t i o n  (reference 4) 
dr/dy, 
v i t s  value j u s t  abbve t h e  
I n  t h i s  theory i f  t h e  v e r t i c a l  component of ve loc i ty  w 
5 = y + i z  may be  
A s  noted above, 
- -  dr - v(y + o i l  - v(y - o i )  = 2vfy + o i l  
dY 
Introducing w = a + b y  f o r  y > 0 and w = a - by f o r  y < 0 i n t o  
equation (11) y ie lds ,  a f t e r  i n t eg ra t ion ,  
and hence 
r = 2 (a + $) ,/-+ y2 cosh-' - S 
IYI 
lr 
The spanwise loading thus  conta ins  t h e  e l l i p t i c a l  d i s t r i b u t i o n  as one 
component. 
546 
NACA TN 2249 
Equation (13) f o r  the  spanwise d is t r ibu t ion  of c i rculat ion enables 
the determination of the  over-all l i f t ,  bending moment, and drag i n  t e r m s  
of the unassigned constants a and b. The use of equations (2), ( 6 ) ,  
and (7), together with the  wing semispan s, yields  the following values: 
B = 0 V s 3  f 6 a  + I b s )  \ 
a b Di = - 2 V L + V B  
It is  convenient t o  specify the bending moment of the l i f t  i n  t e r m s  of 
the lateral posit ion of the centroid, or  center of pressure, of the load 
curve. The lateral centroid as a f rac t ion  of the  semispan s may be 
denoted by y' (i .e. ,  y'  = 2B/Ls). Then, solving fo r  a and b, 
The expression for  induced drag in  terms of the l i f t  and the lateral 
center of pressure becomes 
This equation y ie lds  the minimum drag for  the given posit ion of 
the lateral center of pressure is specified so as t o  coincide with tha t  
for  an e l l i p t i c a l  loading (i.e., b = 0; y' = 4/33), then the above 
formula reduces t o  
y'.  I f  
The optimum dis t r ibu t ion  of loading for  a given posit ion of the centroid 
y9  
The re su l t  is  
may be obtained from equation (13) with the  a id  of equations (15). 
547 
NACA TN 2249 
Drag f o r  a Given Bending Moment with Unrestricted Span 
The foregoing calculations show, as w a s  t o  be expected, t ha t  the 
However, i f  the  r e s t r i c t ion  on the span is removed, 
e l l i p t i c  loading yields  a smaller drag than any of the others  within a 
r e s t r i c t ed  span. 
s t i l l  lower values of the  induced drag can be obtained without any 
increase i n  the bending moment at  the wing root. 
obtained by permitting the span t o  increase and a t  the  s a m e  t i m e  adopt- 
ing a more tapered form of the  loading curve. 
The lower values are 
Equation (16) which contains the three var iables  l i f t ,  span, and 
center of pressure can be eas i ly  rearranged t o  show the var ia t ion of 
drag with span when the bending moment and the l i f t  are held a t  fixed 
values. I n  t h i s  case, the lateral posit ion of the  center o f  pressure 
y's w i l l  be fixed, while the form and extent s of the load curve w i l l  
vary. I n  order t o  provide a convenient basis  f o r  comparison the span 
and shape of the load curves w i l l  be re la ted t o  the e l l i p t i c  loading. 
I f  S/Se denotes the r a & h  of the  semispan of the  wing t o  that of an 
e l l i p t i c a l l y  loaded wing having the s a m e  t o t a l  l i f t  and bending moment, 
then equation (16) can be rewritten: 
The quantity i n  the bracket is the r a t i o  of the  induced drag t o  that of 
the corresponding e l l i p t i c a l l y  loaded wing. 
f igure 4 t o  show the  decrease of drag possible by increase of the span. 
The forms of load curve required fo r  the minimum drag a t  various values 
of s/se are shown i n  f igure  5 .  
This r a t i o  is plot ted i n  
It w i l l  be noted t h a t  a 15-percent reduction of the induced drag 
below tha t  fo r  e l l i p t i c  loading can be achieved with a 15-percent 
increase i n  span. 
50 percent (s = 1.15 t o  1.50) yield no s ignif icant  reductions, however. 
A t  still la rger  values of s 
a t  an i n f i n i t e  value of s. For extreme values of s/se the curves begin 
t o  show negative loadings a t  the t i p s  and eventually the bending moment 
at  ce r t a in  points along the  span w i l l  exceed t h a t  a t  the wing root. 
Further increases of span between 15 percent and 
the  drag becomes lower, and approaches zero 
Ames Aeronautical Laboratory 
National Advisory Committee for  Aeronautics, 
Moffett Field, Calif . ,  Sept. 25, 1950. 
548 
NACA TN 2249 
L 
2 
Di 
B 
P 
r 
wi 
W 
V 
V 
Y,rl 
60 
S 
Y’ 
a,b 
APPENDIX 
DEFINITIONS OF SYMBOLS 
t o t a l  l i f t  
element of l i f t  
induced drag 
bending moment 
air density 
circulation 
induced downwash velocity a t  wing 
downwash velocity, a t  i n f in i ty  
lateral velocity 
velocity of f l i gh t  
distances along wing semispan 
point of orggin fo r  bending moment 
length of wing semispan 
lateral position of load centroid as a fraction of 
constants 
( w - 2wi) 
s 
549 
NACA TN 2249 
REFERENCES 
1. Munk, Max M . :  The Minimum Induced Drag of Aerofoi ls .  NACA Rep. 121, 
1921. 
2. Prandtl, L.: Applications of  Modern Hydrodynamics to Aeronautics. 
NACA Rep. 116, 1921. 
3. Bocher, Maxime: Introduction to  Higher Algebra. The MacMillan eo . ,  
N.  Y .  1907, p. 49. 
4 .  Munk, Max M . :  Elements of  the Wing Section Theory and o f  the Wing 
Theory. NACA Rep. 191, 1924. 
550 
NACA TN 2249 
FiGURE 1.- SPANWISE LOAD 
ELEMENTS OF VARIATION 
-=T$$&7 
CURVE WITH THREE 
a +  by 
551 
NACA TN 2249 
1. zii 
M
 (3
 
L
rd
a 
Lr, 
0
 
a
 
n
 
0
 
a
 
W
 
I- W 
I
 
I- 
5 52 
NACA TN 2249 
553 
5 54 


The spanwise distribution of lift for minimum induced drag of wings having a given lift and a given bending moment

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