Spanwise Distribution of Lift for Min Induced Drag

7/24/2019 Spanwise Distribution of Lift for Min Induced Drag

1/16

TECHNICAL NOTE

2249

THE SPANWISE DISTRIBUTION OF LIFT FOR MINIMUM

INDUCED DRAG OF

WINGS

HAVING

A

GIVEN LIFT AND

A

GIVEN BENDINGMOMENT

Robert T. Jones

Ames Aeronautical Laboratory


2/16


3/16

NATIONAL

ADVISORY COMMITTEE FOR AERONAUTICS

TECHNICAL NOTE

49

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 th e minimup induced dra g of w ings having a given l i f t

and a given span i s extended to inc lude

cases

i n which t h e ben ding moment

t o be supported by t he wing i s also g iven . A s i n t he c l a ss i cd l p roblem

of induced drag, th e theory i s l i m it e d t o l i f t i n g s u r fa c es t r a ve l in g a t

subsonic speeds.

t r ib ut io n can be obtained i n an elementary way which i s app l i cab le t o a

v a ri e ty of such problems.

responding spanwise load distribut ions are a l so g iven fo r t he case in

which th e l i f t and t h e bending moment about th e wing roo t are f ixed w hi le

the span i s al lowed t o vary.

the induced drag with a 15-percent incr ease i n span as compared with

re su l t s fo r an e l l i p t i c a l ly l oaded wing hav ing the

s a m e

t o t a l l i f t and

bending moment.

It

i s found th a t t he r equ ired shape of th e downwash di s-

Expre ssions f o r th e minimum drag and t h e cor-

The results show

a

15-percent redu ction of

INTRODUCTION

I n t h e problem of minimum induced dr ag

as

or ig in al ly t r ea te d by Munk

(references 1 and 2) t h e s pan of t h e wing and t h e t o t a l l i f t w e r e supposed

t o be g iv en and t h e d i s t r i b u t i o n of l i f t o ve r t h e sp an r e s u l t i n g i n

a

min-

innun of

drag

wa s

sought . The so lu t io n of t h i s problem thus provided

a

convenient lower bound for the induced drag of a wing of g iven dimensions.

In t h e p r a c t i c a l d e s i g n of wings the requirements for low induced

drag and the requi remen ts fo r s t ru c t u r a l s t reng th are opposed.

bending moment developed by th e l i f t becomes an imp ortant conside ra-

H e r e

t h e


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NACA TN

2249

A

complete

l i s t

of symbols employed i n t he a na ly s i s

w i l l

be found in

the appendix.

GENERAL FORMULAS FOR LIFT,

DRAG, AND BENDING MOMENT

Reference may b e made t o t h e o r i g i n a l papers of P ra n dt l and Munk

( r e f e r ences

1

and

2 ) ,

o r t o any of th e s tandard te 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 c a l c u la t io ns of i nduced d rag .

i s

given by

I n t h e s e d ev elo pm en ts t h e o v e r -a l l l i f t

S

r d y

1)

J S

and th e drag

i s

given by

S

wi

r dy

In the se formulas th e wing span

i s

supposed t o ex tend a long th e y ax i s

between - s and + s ,

r

i s

t h e l o c a l c i r c u l a t i o n o r v o r t e x ' s t re n g t h , and

V

i s th e consta nt ve lo c i ty of f l ig h t . The induced downwash ve lo c i ty

w i

i s

var iable a long the span and

i s

c on ne ct ed w i t h t h e v o r t e x d i s t r i b u t i o n

l

(y)

t hro ug h t h e r e l a t i o n

With t h i s v a l u e f o r

W i

a

d ou bl e i n t e g r a l i n vo lv in g t h e s pan wise d i s t r i b u t i o n of l i f t

as

repre-

s e n te d by t h e c i r c u l a t i o n s t r e n g t h

th e express ion fo r t h e drag may be conver ted

t o

I

This i n te g ra l may be reduced t o a more symmetric fo rm i f

i t

i s i n t e g r a t e d

by p a r t s on t h e s u p p o s it i o n t h a t

I'

f a l l s t o zero

a t

t h e wing t i p s .

Thus1


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NACA

TN 49

I n mathemat ica l

terms the problem

i s

t o m in im iz e t h e d o ub le i n t e g r a l ,

equa t ion (5), whi le ho ld ing f ixe d va lue s o f

L = pV

lS+ dy

and

where

B i s

th e bending moment sp ec i f ie d about th e po int s o . For the

t i m e being

although

l a t e r

another example w i l l appear.

s o

w i l l

be taken as t h e o r i g i n , o r wing r o o t s o =

0),

Although t h e de r i va t i on of t h e formulas f o r induced drag lfiakes use

of

t h e c on ce pt o f t h e l i f t i n g l i n e , i t is i m po rt an t t o n o t e that t h e r e s u l t s

are n ot a c t u a l ly r e s t r i c t e d t o

t h i s appr ox ima tion .

well-known sta gg er theorem th e induced drag of

a

l i f t i n g s ur fa ce

w i l l

b e

e q ua l t o t h a t of

a

l i f t i n g l i n e i f t h e spanw ise l oa d 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 fur ther tha t the induced drag of

a

wing having

a

g iv en l i f t a nd

a

given spanwise load d i s t r i bu t i on i s n o t a f f e c t e d by t h e

compr ess ib i l i t y o f t he

a i r

a t

subsonic speeds.

a d d i t i o n a l d r a g a s s o c i a t e d w i t h t h e f or m at io n o f waves

arises

and the

induced drag, which

i s

as so ci at ed with th e vo r t ex wake, becomes only

a

p a r t o f t h e t o t a l p r e s s u r e d ra g.

A t

supersonic speeds an

THE

DISTRIBUTION

OF DOWNWASH FOR MINIMUM DRAG

I n g e ne r al , i f t h e dr a g

i s

t o b e

a

minimum,

a

s m a l l

v a r i a t i o n i n t h e

shape of th e curve of spanwise loading

w i l l

produce no f i r s t -o rd er change

i n t h e d ra g.

t o t h e o r i g i n a l l o ad in g ; i t i s t hen necessa ry t o f i nd cond i t ions unde r

which t h e drag added by

a

s m a l l a d d i t i o n a l l o a d i n g i s zero.

The var ia t i on i n shape may t ak e t h e form of

a small

add i t ion


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NACA TN 2249

Consider now an i n i t i a l di st r i bu ti on designed t o achieve minimum

drag. (See f i g . 1.)

The

drag added by

a

s m a l l

addi t iona l loading

w i l l

be composed of three parts,

namely:

1.

2.

3.

The drag of the addi t iona l l i f t ac t in g a lone

The drag of th e o ri gi na l loading ar i si n g from th e downwash

'J e

drag of the additional loading induced by the downwash field

f i e l d o f t he add i t i ona l loadi ng

of the or i g in a l load ing

I t e m

1

s of second ord er in terms of th e magnitude of t he added l i f t f or

smooth dis t r ib ut i ons , th at i s , so-called weak va ria tio ns. (The fa c t

that this second-order

t e r m is

invar iab ly pos i t i ve insures th a t the drag

w i l l be a minimum and not a maximum.)

mutual drag theorem.

The

f ir s t- or de r va ria t io n i n drag can then be com-

puted by consid ering only the drag of th e small a d di ti on a l l i f t a c ti n g

i n

the induced downwash f i e l d wi(y) of the or ig in al l i f t .

Items 2 and 3 are equal by th e

The condit ions of f ixed bending moment and f ixed t o t a l ' l i f t

are

m e t

by al lowing only those curves of l i f t var i at i on th at produce no change i n

t hese quan t i t i e s , t ha t

i s ,

curves having zero area and zero moment. It

can be seen th at such curves of va ri at io n must have a t l e a s t t h r ee ele-

ments to m e e t the condi t ions of zero area and ze ro moment. Furthermore ,

any curve meeting t hes e condit io ns can be subdivided in to groups of th re e

elements

s o

tha t the ind iv idua l g roups a l so sa t i s f y the condi tions.

Hence,

as

the r epresen ta t ive of such re s t r i c te d curves of v ar ia t io n

w e

may adopt three small elements having areas

21,

22, and

Z3

( f i g . 1 ) .

These

elements, toge ther with t h e ir po sit i ons yl , y2, and y3 and th e

l o c a l val ue s of t h e downwash

w -

etc.,

due to the or ig ina l load ing

must sa t i s f y the fol lowing thre e equations :

1

,

o r

6L

=

0,

l

+ +

Z3 = 0

f o r

f o r

6B

= 0,

6 D i =

0,

Z l Y l Z2Y2 Z3Y3 =

Zlwil

+

Z2wi2

+

Z 3w i 3

=

0

I t can be seen that these equat ions

w i l l

be cons i s ten t

if

w

a+by2 and wi3 a+by3, where

a

and b are cons tan t s t o be

w i l a+byl,

52


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NACA

TN

2249

Hence, for a minimum induced drag with a giv en t o t a l l i f t and

a

given

bending moment t h e dowzlwash must show

a

l in ea r var ia t ion a long the span .2

(See f ig . 2.)

The for egoin g method may be r'eadily extended t o a more general class

of problems inv olvin g bending moments or ro ll in g

moments.

Suppose, for

example,

a

braced wing

i s

considered,

as

i n t he do tt ed ou t l i n e of f i g -

u r e

3.

I n t h i s

case

th e bending moment developed by t h a t po rt io n of th e

l i f t a c t i n g i nb oa rd

of

the point of bracing attachment may be of no con-

cern, but

it

may be des ire d t o l i m i t the bending moment developed by that

porti on of th e spanwise load curve extending between t h i s point and the

t i p. 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 st at io na ry values of the l i f t and bending moment,

and

it

i s

evident

that

a t least two

of t h e elemen ts must

l i e

t o t h e ri g h t

of the point

sQ.

The th re e s imul taneous equat ions

are

(see f i g . 2):

Here y2 and y3 are t o t h e r i g h t of the po in t

so

and y1 l i e s t o t h e

l e f t o f th i s po in t. For these equa t ions3 t o be cons i s ten t wi must have

the form

yil

a;

wi

a+b(y2-s0);

wi3

a+b(yg-s0)

2

Hence, i n gen er al , t h e downwash w i l l be

a

constant over the portio n of t he

span f o r which t he moment

i s

not spec i f i ed ,

as

i l l u s t r a t e d i n f ig u r e

3.

I f no res t r ic t i on whatever i s plac ed on th e moment th e r e i s obtained the

so lu ti on of Munk's o r i g i n a l problem, namely, t h a t th e downwash should be

cons tant over the en t i re span.

21t

may be noticed a t t h i s poi nt th at , whereas th e discussion has empha-

size d th e id ea of minimizing t he drag, the a na ly si s ac tu al ly makes no

di s t i nc t io n between the l i f t , bending moment, or drag, i n th at s ta t io n-

ary values of a l l t h r ee a re demanded.

Thus

equation

(9)

may be consid-

ered a necessary condit ion f o r th e solu tio n of th e following problems:

1)

g iven t he t o t a l l i f t and t he induced d rag t o f i nd t he d i s t r i bu t i on


8/16

NACA TN 2249

Determination of Span Loading and Induced

Drag From t h e Downwash D i s t r i b u t i o n

The case of b i l a t e r a l symmetry with moment sp ec if ie d about t he

r o o t s e c t i o n w i l l serve as an example of t h e c a l c u l a t i o n o f t h e a c t u a l

span load ing and induced drag.

th a t th e downwash d i s t r ib u t io n w i l l c o n s i s t of t w o s t r a i g h t - l i n e

segments w it h

a

reversal of s lope a t t h e pl ane of symmetry.

necessary t o compute t he spanwise var ia t i on of

T

corresponding t o

such a curve of downwash.

It w i l l be evident f rom the foregoing

It

i s then

o

per fo rm t h i s ca lc u l a t io n by s tandard methods o f a i r f o i l theory ,

use i s made of th e idea th a t

a t a

gre a t d i s ta nce beh ind the wing t he

vor tex sheet 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 l a t e r a l ve lo c i t y ac ros s th e shee t g iven by and

t h e downwash

w

given by

t w i c e

t h e va lu e of t h e induced downwash

wi

a t

th e wing. Hence, t h e qu an ti ty 1 / 2 (dI'/dy)

-

2iwi can be eval uate d

by means of th e fam il ia r complex ve lo ci ty fun ct io n

v

- i w of t h e two-

d imens iona l po te n t i a l theory using f o r

vor tex shee t .

i s

g iv en a lo ng t h e l i n e r e p r e s e n t i n g t h e

t race

of th e span, then the

v e l o c i t y v e c t o r a t any o t he r p o i nt i n t h e f i e l d

obta ined f rom th e re la t i on ( reference 4)

dr/dy,

v

i t s va lue ju s t abbve th e

I n t h i s t h eo r y i f t h e v e r t i c a l c omponent of v e l o c i t y

w

=

y

+

i z may be

A s noted above,

- -r -

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 fo r y

< 0

i n t o

equat ion 11) y i e l d s, a f t e r i n t e g ra t i o n ,

and hence


9/16

NACA

TN 2249

Equation

(13)

fo r th e spanwise d i s t r ib u t io n of c i r cu la t ion enables

th e determination of t he over-all l i f t , bending moment, and drag i n

t e r m s

of t he unassigned co nstan ts

a

and

b.

The us e of equation s (2),

6 ) ,

and (7),

tog eth er wit h t h e wing semispan

s,

yields the fol lowing values :

B

=

0 V s 3 f 6 a + I b s )

\

a

b

Di

=

V L + V B

It

i s

convenient to sp ecify th e bending moment of th e l i f t i n t e r m s of

the l a t e r a l posi t ion of the centroid, or cente r of pressure, of the load

curve. The

l a t e r a l

cent ro id

as

a fr ac ti on of th e semispan

s

may be

denoted by

y

( i . e . , y ' = 2B/Ls). Then, sol vin g f o r

a

and b,

The expression for induced drag

i n terms

of t he l i f t and t he

l a t e r a l

center of pressure becomes

This equation y ie ld s th e minimum drag fo r th e given po sit i on of

the l a t e ra l cen t e r of pressure i s spec i f i ed so as t o co inc ide wi th tha t

f o r an e l l i p t i ca l l oad i ng

( i .e . ,

b

=

;

'

=

4/33), then the above

formula reduces to

y ' .

I f


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NACA

TN 2249

Drag for

a

Given Bending Moment with Unrestricted Span

The foregoing calculations show,

as w a s

to be expected , th a t the

However,

i f t h e r e s t r i c t i o n on

the

span

i s

removed,

e l l i p t i c l oa din g y i e ld s a

smaller

drag than any of th e oth er s within a

r es t r i c ted span .

s t i l l lower valu es of th e induced drag can be obtain ed without any

in cr ea se i n th e bending moment a t th e wing roo t.

obtained by permitt ing the span to increase and

a t

t h e

s a m e t i m e

adopt-

ing a more tapere d form of t h e load ing curve.

The

lower values

are

Equation (16) which contains the th ree var ia ble s l i f t , span, and

cente r of pressure can be e as i l y rearranged t o show the var i at io n of

drag with span when th e bending moment and t h e l i f t

are

held a t f ixed

val ues. I n t h i s

case, t he

l a t e r a l

pos i t ion of t he cen ter

o f

pressure

y ' s w i l l be f ixe d, while t he form and exte nt s of t he load curve w i l l

vary. In order t o provide

a

convenient basis for comparison the span

and shape of th e load curv es w i l l be r e l a t e d t o t h e e l l i p t i c lo ad in g.

I f

S/Se denotes the

r a & h

of th e semispan of th e wing t o

that

of

an

e l l ip t i c a l l y loaded wing having the s am e t o t a l l i f t and bending moment,

then equation (16) can be rewrit ten:

The quant i t y i n the bracket

i s

the ra t i o of th e induced drag to that of

the corresponding el l i p t i c a l l y loaded wing.

f i gu r e 4 t o show th e decrease of drag po ssib le by in crea se of the span.

The forms of load curv e requ ir ed f o r t h e minimum dra g a t var ious values

of

s/se

are shown i n f ig ur e

5 .

T hi s r a t i o i s p l o tt e d i n

It w i l l

be noted t ha t

a

15-percent redu ction of t he induced drag

below tha t fo r e l l i p t i c load ing can be achieved wi th a 15-percent

incre ase i n span.

50 percent

s =

1.15 t o 1.50) yi el d no sig ni f ic an t reductions, however.

A t s t i l l

l a rg er va lues of

s

a t

an i n f i n i t e va l ue o f s. For extreme values of

s /se

the curves begin

to show negative loadings a t the tips and eventually the bending moment

a t ce r ta in po in t s a long th e span

w i l l

exceed that

a t

th e wing ro ot.

Further increases of span between

15

percent and

th e drag becomes lower, and approaches ze ro


11/16

NACA

TN

2249

L

2

i

B

P

r

wi

W

V

V

Y,rl

6

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

a i r

dens i ty

c i r c u la t io n

induced downwash velocity a t wing

downwash velocity,

a t

i n f i n i t y

l a t e ra l v e l o c i t y

v e lo c i ty of f l i g h t

distances along wing semispan

point

of

orggin

f o r

bending moment

length of wing semispan

l a t e ra l pos i t ion

of

load centroid

as

a f r a c t i o n of

constants

w -

2wi)

s


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REFERENCES

1. Munk,

Max M :

The Minimum Induced Drag

of

Aerofoi l s .

NACA

Rep.

121,

1921.

2. Prandtl,

L. :

Applications of Modern Hydrodynamics to Aeronautics.

NACA

Rep. 116, 1921.

3.

Bocher,

M a x im e :

Introduc tion to Higher Algebra.

The MacMillan

e o . ,

N . Y . 1 9 0 7 p. 49.

4 . Munk, Max M : Elements o f the Wing S ec ti on Theory and o f the Wing

Theory. NACA Rep. 191, 1924.


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FiGURE 1

SPANWISE LOAD

ELEMENTS OF VARIATION

-=T &7

CURVE WITH

THREE

a

by


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N

A

C

A

N

1

z

M

3

d

L

n

W

I

W

I

5


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N

T

5

5

3


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Documents

Spanwise Distribution of Lift for Min Induced Drag