NASA Contractor Report 3357 

Induced Drag Ideal Efficiency 

Factor of Arbitrary 

Lateral-Vertical Wing Forms 

John DeYoung 

CONTRACT NASl-16000 

DECEMBER 1980 
NASA Contractor Report 3357 

Induced Drag Ideal Efficiency 

Factor of Arbitrary 

Lateral-Vertical Wing Forms 

John DeYoung 

Kentron Intermtiorzal, Inc. 

Hampton, Virginia 

Prepared for 

Langley Research Center 

under Contract NAS l- 16000 

National Aeronautics 

and Space Administration 

Scientific and Technical 

Information Branch 

1980 
SUMMARY 

A relatively simple equation is presented for estimating the induced drag 

ideal efficiency factor e for arbitrary cross-sectional wing forms. This 

equation is based on eight basic but varied wing configurations which have 

exact solutions. The e function which relates the basic wings is developed 

statistically and is a continuous function of configuration geometry. The 

basic wing configurations include boxwings shaped as a rectangle, ellipse, and 

diamond; the V-wing; end-plate wing; 90 degree cruciform; circle dumbbell; and 

biplane. Example applications of the e equations are made to many wing forms 

such as wings with struts which form partial span rectangle, diamond, triangle, 

ellipse, and semi-ellipse wings; the ellipse and rectangle dumbbell wings; bowtie, cruciform, winglet, and fan wings; and multi-wings. Derivations are 

presented in the appendices of exact closed form solutions found of e for the 

V-wing and 90 degree cruciform wing and for an asymptotic solution for multiwings. 

INTRODUCTION 

With the objective of better economics through improved aerodynamic and 

structural shapes, many unconventional aircraft configurations are being 

investigated. Particularly desirable are wing forms for which both drag and 

structural stresses decrease. A measure of aerodynamics efficiency is the 

induced drag ideal efficiency factor denoted by e. 

Historically, exact solutions for induced drag have been found for several 

nonplanar wing configurations. These include the rectangle boxwing, end-plate 

wing, and biplane reported in reference 1; the ellipse boxwing and circle 

dumbbell wing from the methods of reference 2; and the diamond boxwing and 

V-wing in reference 3. These solutions form the basis for the present study. 

The objective of this study is to develop a generalized equation for estimating the ideal e for arbitrary yz-plane wing forms. For a given wing form 

the ideal e is the maximum that e can become, and at ideal e all the wing form 

surfaces have ideal loading distributions such that the total induced drag is 

minimized. A generalized equation for estimating e provides the designer with 
a tool for evaluating the aerodynamic efficiency for the configuration under 

consideration. 

b 

Di 

E 

e 

ee 

e V 

f 

GO 

Gl 

H 

Ha 

hm 

K 

k 

L 

N 

P 

P 

pe 

9 

2 

SYMBOLS 

lateral span along y-axis of wing configuration 

induced drag, equation (1) 

complete elliptic integral of the second-kind, with modulus k 

induced drag ideal efficiency factor, equation (1) 

e for end-plate winq, equation (16) 

e for V-wing, equation (15) 

function of parameter P, equation (20) 

function of nlo H, equation (18) 

function of ev, ee, naf, equation (19) 

dimensionless maximum height, h,/b 

average dimensionless height, equation (25) 

maximum height of wing configuration 

complete elliptic integral of the first kind, with modulus k 

modulus for K and E, k = sina where ~1 is the modular angle 

total lift of wing system, equation (1) 

total number of wings 

parameter, equation (21) 

perimeter of wing system, equation (29) 

perimeter of ellipse, equation (22) 

free stream dynamic pressure, 1 2 pv2 
Y 

r 

Y 

rl 

na 

n10 

naf 

no' n1 

nn 

P 

u 

'e 

aM 

lateral coordinate, zero at midspan, positive to right 

gamma function, equation (14) 

dihedral angle 

dimensionless lateral coordinate, 2y/b 

averaged n of outboard face of wing, equations (28) and (76) 

span station of difference, equation (27) 

rla for fin wings, equations (30), (31), (32) 

span stations, equation (27) 

span station of multi-wings, equation (76) 

air density 

perimeter parameter, equations (29) and (77) 

perimeter parameter of ellipse, equations (22) through (24) 

perimeter parameter of multi-wing, equation (78) 

IDEAL EFFICIENCY FOR BASIC CONFIGURATIONS 

The present analysis is based on a group of varied wing configurations 

which have exact solutions for predicting e. These include the wings described 

as rectangle boxwing, ellipse boxwing, diamond boxwing, V-wing, end-plate wing, 

90 degree cruciform wing, circle dumbbell wing, and biplane. The induced drag 

ideal efficiency factor listed throughout the paper relates the total induced 

drag to the total lift of the wing system by the equation: 

L2 Di = - 

nqb e 

(1) 

where b is the maximum span in the lateral plane, q is the dynamic pressure, and 

L is the sum of all the lifts, including wing, winglets, end-plates, struts, 

or tail. For ideal e, the lifts and loadings will have ideal distributions. 

3 
Rectangle and Ellipse Boxwings 

From reference 1, p. 205, for the rectangle boxwing, the equations for 

the dimensionless maximum height for H and for e can be developed. 

H=hmBE-(l -- - k2)K 

b E' - k2K' 

1 

e = 

- k2 

(E' - k2K1F -b/2 0 b/2 

where K and E are complete elliptic integrals of the first and second kinds, 

respectively, with modulus k, while K' and E' are the complementary integrals. 

That is, K' is K with modulus m, and E' is E with modulus m. The 

modulus k is related to the modular angle c1 by k = sincl. Equations (2) and 

(3) do not permit a direct solution of e for a specified value of H. A value 

of k is specified and e from equation (3) is determined, then the corresponding 

H from equation (2) is determined using the same k value. Geometric characteristics of the rectangle include the nondimensional perimeter given by 

u=& 

2b + 2h 

2b 

"=l+H (4) 

The rectangle averaged cross-sectional area per wing span squared is 

hmb 

Ha=P= b2 H 

(5) 

The e for the ellipse boxwing can be 

determined from the work given in reference 

2. The results are 

f(-> 

-b/2 b/2 

e=l+H (6) 

The perimeter parameter of the ellipse is an elliptic integral of the second 

kind given by 

E,k=m,H<l' 

HE, k = m, H .> 1 

4 

(7) 
The ellipse averaged cross-sectional area per wing span squared is 

Ha 

vhmb 

=-=,H 4 b2 

(8) 

Relating Rectangle and Ellipse Boxwings 

Examination of equations (2) and (3) shows that at H = 0, e = 1, and at 

H + ~0, e -+ ~H/IT. Then for the rectangle boxwing, e can be expressed as 

e = (1 + % H)(l + f) (9) 

where f is a function of H to be determined. Equation (9) can apply to both 

rectangle and ellipse boxwings by letting H = Ha, and f be a function of perimeter difference. Then 

e = (1 + + H,)(l + f) 

f = 1.4512 P'841 (1 - fi)1'07g5 1 + .585 P3 (P - .37)3] ('0) 

P=cl-u, 

where oe is given 

(4). The f funct 

in equation (7), and CI for the rectangle is given in equation 

ion is statistically developed from the e and H data of equations (2) and (3). Equation (10) applies to either the rectangle or ellipse 

boxwing and gives an estimate of e for any shape between these two. Example 

values of e are listed in table 1. 

Diamond Boxwing 

From reference 3, the added mass M is given by 

M = 2rpcr 

where (here y replaces B of ref. 3) 

a = 1 (” - y) TT 2 , Y = tan -' H 

5 
Then the dimensionless added mass is 

2 

M-'-T= 

2Y 4' - yy-1 

2 

= 

m a2 I!( r ; + Z) r(1 - ;,I' 

(121 

C( r + + f tan -' H) r(1 - i tan -' H)P ~(1 - p tan-' H) 

where a2 is derived in reference 3, and H = h,/b. In reference 3, Di and L are 

related by 

L2 

Di=-= 

L2 . 

2MV2 rqb 2 4a2 ( M 

(13) 

- - 

b2 2) 

va 

where from the sketch, 4a2/b2 = 1 + H2, then 

e=(l+H2)(%)= 

~(1 + H2) (1 - G tan-' H) 

(14) 

p-3 r(i + i tan -' H) r( 1 - 5 tan-' H)12 

where r is the gamma function. For the diamond boxwing, values of e for 

various values of H are presented in table 1. Tables of the gamma function 

are given in reference 4. 

TABLE 1. - IDEAL e VALUES FOR RECTANGLE, ELLIPSE, 

AND DIAMOND BOXWINGS 

H 

0 

.05 

10 

:15 

.20 

.25 

:43 

.5 

.6 

- 

rectangle 

eq. (10) 

ellipse 

eq. (6) 

; 15178 

1:26814 

; 05 

1:10 

1.37327 1.15 

1.47189 1.20 

1.56610 1.25 

1.65709 1.3 

1.83204 1.4 

2.00026 1.5 

2.16362 1.6 

- 

diamond 

eq. (14) 

1 

1.01270 

1.02622 

1.04056 

1.05573 

1.07171 

1.08850 

1.12442 

I.16330 

1.20490 

H 

i80 

1:2 

1.4 

1.6 

1.8 

2 

3 

4 

co 

XI -- 

rectangle 

w (10) 

I__-- 

2.47997 

2.78642 

3.08606 

3.38041 

3.67066 

3.96086 

4.24189 

5.63505 

6.99909 

+4H/n 

. .-I_-----. -- 

ellipse 

eq. (6) 

.---_-- 

1.8 

2.0 

22:: 

22:: 

4" 

5 

-k! 

_. 

___-~ 

diamond 

eq. (14) 

--I_- 

1.29518 

1.39320 

1.49715 

1.60562 

1.71751 

1.83206 

1.94866 

2.55051 

3.16840 

+2H/7r 

-1-1_ 

6 
V-Wing 

An exact closed function solution was found for 

the e of a V-wing. The derivation is given in 

appendix A. From equation (A15) 

e V 

Y = tan -' 2H 

-b/2 0 b/2 

(15) 

where y is the dihedral angle in radians. Values of e for various H values 

are listed in table 2. 

End-Plate Wing 

An accurate but simplified e function for 

end plate wings is developed in appendix B. 

From equation (B14) d 

2-$ 

e = 

e ( 

1 + .732 H - 

-0612 H(.8 - H) 

2 H 1 

') 

bj2 

16) 

1 + .3688 HL 

Values of e for the end-plate wing computed from equation (16) are presented 

in table 2. 

90" Cruciform Wing and Circle Dumbbell Wing 

The 90" cruciform wing has H = 1, and as 

shown in appendix C, e = 2 for the unbanked wing. 

x0 ' o 

Th 

0 

Also shown in appendix C is the variation of e 

IL m 

b/2 

with bank angle, and that when induced drag is taken as relative to wing length 

(and not lateral span) the e is unity at all bank angles. 

The e for the circle dumbbell wing can be developed from induced drag 

methods presented in reference 2. 

7 
GENERALIZED EQUATIONS FOR ESTIMATING e 

The objective is to develop an e function, with continuity, which becomes 

the proper e function of each of the basic configurations when that configuration geometry is inserted. Because there are many basic configurations and 

the e function is continuous, the application to general configurations should 

be within an acceptable order of accuracy. The general e function is statistically developed so that each basic configuration is taken into account. 

H 

1-(l-$na$)(l-naFi- Ha) Go] (1 + f) + 

7r 

[1 - (&)1'7]G1 

a 

Go = 

(1 - .19 nlo H)(l + .4826 nTo Hz) 

(1 + 1.81 I-I,~ H)(l + -255 nlo5" H5") 

G, = 

(6 - 4fl be - 1) (ev - 1) nzf 

---- -- 

1 Jee-i---1$7-2eJe,-r)Jli,f]2 

Oaf" 'laf 1/Z 

= e 

e 

-1 = e,-1 

where e v and e, are given in equations (15) and (16) and in tab 

f = 1.4512 f-r P IPI .841 (1 - (P/1'2)1S07g5 [l + .5851Pi3((P 

Ha 

P= 

t 

90 Fl- - 

1 + 3(1 - rl10 j+(l - 2 rllD +2 I 

3'2 ( - cre) 

u 

07) 

(18) 

(1% 

le 2. 

I - .37)3] (20) 

(21) 

8 
- 

u =P erimeter of full or of partial span symmetric ellipse boxwing 

e twice the wing span 

Also functionally 

Y &) = H oetFi-4 

1 n1 

Numerical values of oe/n, are presented in table 3. 

An approximation for oe is 

‘e -zz 

n1 

- 

1 

- 

+ 

2 

I 

1.11374 

(H+l.11374 

n1 __ 

1 

1 + (;)1.11374 

- 

l+ 

q20 - ;7)2 

32 (1 + K-)6 

n1 

For a semi-ellipse in a n,-flo interval on the wing semi-span 

/ 

- 

ue = 0’0 + (11, - noI E, k;JiH = cross-section area of boxwing 

a b2 

(23) 

(24) 

(25) 

h 

H = $ , where hm is the maximum height of wing configuration. (26) 

9 
nlo = n1 - oo, where Q,~ and no are the spanwise stations of the 

outboard and inboard sides of the boxwing at the wing. (27) 

1 

na 

= nd(k), averaged n of outboard face of wing. (28) 

0 m 

o = &-, where p is perimeter of full or of partial span boxwing 

in which h is constant or decreases with n. (2% 

For bow-tie and dumbbell type wing configurations, the effective perimeter 

is determined from the shape with the maximum dumbbell height translated onehalf dumbbell width towards the inboard side but limited to the inboard side 

of the dumbbell which is no. Some examples are 

where--- -----is the translated shape needed to determine p and o for this type 

of wing. The value of cre of the dumbbell wing with symmetrical ellipses is 

given in equation (24). This ae and the dumbbell type wing value of G are 

used in equation (21) for evaluating P. 

Qaf = Qa’ for full or for partial span boxwings in which h 

is constant or decreases with n. (30) 

1 + .05 &' -I me I 

'af = 

1 + .05 vie 

na, for bow-tie, dumbbell type, and 

vertical fin wing configurations, 

where, 

(31) 

10 
I 

0 

s 

S 8 

i Hl 

c-::]! 

I 

: 

\ " 

qO 

'me rt, H2 

For fin wings I 

f 'me is mean lateral span station 

for S, = S2. For a vertical 

fin wing, qme = no = q1 = qa. 

'af 

= 

1 

1 + ; (2 Ila2 - Qo,) yg=J (1 - 

sin y2 

sin y, 1 

- (Tl 

l/2 

- qol) JO11 - %I 1 + 

1 .77 + .9H -05 - + 2.1527 g + 1 .670- (93 qol 1 

(91 + q12 - 2q01 

'al 

+ 

.77 + .gH 

1 + .05 2 1+.67H 

r-IO1 

(1 - 

sin yzn 

sin Y2n-,) [.05 + tq1y2n-1 -2:;;;/10,n-ii3'2j) qa,2n-1 (32) 

where the variables are defined through the following fin wing semispan 

sketch: 

,----~~~ -~. ' 

0 7713 

= 1, Y3 = Y4 = 0 

H3 = H4 = 0 

11 
TABLE 2. - IDEAL e OF V-WING AND END-PLATE WING 

I 

e 

V 

H V-wing 

eq. (15) 

0 1 

.05 1.00189 

.lO 1.00750 

-15 1.01665 

.20 1.02909 

-25 1.04452 

1; 1.06257 1.10524 

.5 1.15470 

.6 1.20903 

I 

H 

T- 1 

0 1 .oooooo 

.Ol 1.000275 

.02 1.000960 

.03 1.001977 

-04 1.003286 

.05 1.004856 

.06 1.006667 

.07 1.008700 

.08 1.010940 

.09 1.013375 

-10 1.015994 

.ll 1.018786 

.12 1.021742 

.13 1.024856 

.14 1.028119 

15 

:16 

1.031525 

1.035068 

.17 1.038742 

.18 1.042543 

.19 1.046464 

12 

I 

e e I 

end-plate H 

eq. (16) 

.---_ - - _ 

1:19523 ; 09878 1.2:; 

1.28888 1.4 

1.38192 1.6 

1.47248 1.8 

1.56140 2 

1.73481 1.90313 i 

2.06720 03 

e 

V-wYng 

eq. (15) 

.--". - _ -__ 

1.32715 

1.45265 

1.58206 

1.71362 

1.84636 

1.97974 

2.11345 

2.78235 

3.44833 

+ZH/r 

- ----.--- 

2.38527 

2.69328 

2.99408 

3.28952 

3.58082 

3.86878 

l-----e 4.15394 

5 -5489 

6.9085 

-+4H/r 

TABLE 3. - PERIMETER PARAMETER OF THE ELLIPSE BOXWING 

H 

T 

‘e 

T 

.20 1.050502 

.21 1.054653 

.22 1.058912 

.23 1.063276 

.24 1.067740 

.25 1.072303 

.26 1.076960 

.27 1.081709 

.28 1.086547 

.29 1.091470 

30 

:31 

1.096478 

1.101567 

.32 1.106734 

.33 1.111977 

.34 1.119741 

.35 1.122685 

.36 1.128146 

.37 1.133675 

.38 1.139271 

.39 1.144932 

H 

F 

‘e 

5- 

H 

9 

r---- ‘e 

T 

.40 1.150656 .60 1.276350 

.41 1.156442 .61 1.283110 

.42 1.162288 .62 1.289908 

.43 1.168192 .63 1.296744 

.44 1.174155 .64 1.303617 

.45 1.180724 .65 1.310526 

.46 1.186245 .66 1.317471 

.47 1.192371 .67 1.324450 

.48 1.198549 .68 1.331464 

.49 1.204778 .69 1.338512 

.50 1.211056 .70 1.'345592 

-51 1.217383 .71 1.352705 

-52 1.223758 -72 1.359850 

.53 1.230179 .73 1.367025 

.54 1.236658 .74 1.374232 

.55 1.243157 .75 1.381468 

.56 1.249712 .76 1.388734 

.57 1.256310 .77 1.396029 

.58 1.262949 .78 1.403353 

.59 1.269630 .79 1.410704 

H 'e H 'e 

c Q1 T < 

- 

.80 1.418083 1.0 1.570796 

.81 1.425490 1.2 1.73145 

.82 1.432922 1.4 1.89808 

.83 1.440381 1.5 1.98318 

-84 1.447866 1.6 2.06931 

.85 1.455376 1.8 2.24423 

.86 1.462910 2.0 2.42211 

.87 1.470470 2.5 2.54309 

.88 1.478053 3.0 3.34118 

.89 1.485660 3.5 3.81272 

.90 1.493290 4 4.28921 

.91 1.500943 5 5.25251 

.92 1.508619 6 6.22502 

.93 1.516317 8 8.18624 

.94 1.524037 10 10.15994 

.95 1.531778 20 20.09713 

.96 1.539541 30 30.07148 

.97 1.547324 40 40.05720 

.98 1.555128 50 50;0;"/"," 

.99 1.562952 m 1 
APPLICATION TO PARTIAL SPAN BOXWINGS 

When a strut is designed so that it contributes to the lift of the vehicle 

and since the strut is attached to the wing, the configuration is the same as 

that of the boxwing, in particularly, partial span boxwings. One objective of 

this section is to demonstrate usage of the generalized equation ,(17) for 

estimating e for this type of wing. 

For the rectangle boxwing, Ha = H, no = 0, n, = n,. = n, = 1, u = 1 + H, 

and equations (17) and (21) reduce to 

e = (1 + 1 H)(l + f) 

P=u-u 

e 

(33) 

Equation (33) is the same as equation (10) which is a solution for the 

rectangle boxwing. 

For the ellipse boxwing, Ha = ITH/~, na=~H/4, u = ue, then equation (17) 

becomes e = 1 + H, which is equal to equation (6). 

For the diamond boxwing, ~~ = l/2, Ha = H/2, no = 0, n, = 1, ,-,,. = 1, 

and u = m, then equations (17) with (18) and (21) reduce to 

e = 

c 

1 + i H (1 - .44604 i 1 i'i;H ' + '4826H2 )](I 

1 + .255H5" 

+ f) (34) 

P = .35355 (m - ue) (35) 

where with the P of equation (35), f is determined from equation (20). These 

f values are negative since P is negative. Values of e computed from equation 

(34) compare within one-tenth of one percent accuracy with those of the diamond boxwing listed in table 1. 

13 
Reflexed Ellipse Boxwing 

For this wing, ~-,a = naf = 1 - IT/~, 

Ha/H = 1 - IT/~, n,O = 1, CI = a,; then 

P = 0, f = 0, and equation (17) becomes 

e'= 1 + (4- 

7-r 1)H [1 - (5 - f - :)Go] 

= 1 + .27324H (1 - .881725 Go) (36) 

where Go is given in equation (18) with n,. = 1. Some typical values of e are 

. 

H 0 .l .2 .5 1 2 

e 1 1.00721 1.02008 1.07519 1.19110 1.46881 

Partial Span Ellipse Boxwing 

For this wing, na = naf = @/4)~,, Ha/H = @/4)ri1, 

y. = r-q 3 0'1 is a specified value, 0 2 n, 5 1, 

cl=0 e, then P = 0, f = 0, and equation (17) becomes 

e = 1 + nlH 1 - (1 - n:)(l - $ n:)Go] (37) 

When H = n,, the central wing is circular. For n, = 0.5, the equation and some 

values of e are 

e = 1 + .5H (1 - .63434 G > 0 

where Go is given in equation (18) with n,. = 0.5. 

H 0 .l .2 .5 1 2 

e 1 1.02116 1.04710 1.14372 1.33842 1.78399 

14 
Partial Span Semi-Ellipse Boxwing 

For this wing, T-I~ = naf = fn/4)~,, Ha/H = I , 

("/439' 010 = n'1, then equations (17) and (21) -' 1 

become 

e = 

[ 1 - (1 - rl:)(l - 6 n:)GO 0 + f) 

P= 

2 

t T 

i 

3/2 

1 + 3(1 - ; n$(l - ; q2 

b - ae) 

(38) 

(39) 

The semi-ellipse perimeter parameter is 

and a, is given in equation (24). With r~, = 0.5, equations (38) through (40) 

reduce to 

e = 1 + : (1 - .63434 Go)] (1 + f) 

P = .033507 (0 - be) 

(41) 

15 
Using equation (41) with equations (18) and (22), for n, = 0.5, some typical 

values for e are 

H 0 .l .2 .5 1 2 

e 1 1.02324 1.05142 1.15340 1.35489 1.81216 

For the full-span semi-ellipse boxwing, 0, = 1, then equations (38) 

through (40) reduce to 

e = (1 + H)(l + f) 

P = .523103 (0 - ue) 

a=--$- 

: : 

Using equation 

H 0 

I I 

.- 

E, k=/'m,H<; 1 

2HE, k = 41 - (2H)-2, H > 

- 

421, for 0, = 1, some typical values of e are 

.l I .2 I .5 I 1 I 2 

(42) 

e 1 1.11669 1.23959 1.61191 2.22938 3.45462 

Partial Span Rectangle Boxwing 

For this wing, na = naf = Q,, H,/H = n,, 

u=q + H, y. = nlv 

(21) iecome 

then equations (17) and 

1 4n1 e= 1+- 3-r L 

l-(l- $ $1 - n:, Go (1 + f> 

312 

P= (17, + H - ue) (44) 

16 
With T-I, = 0.5, equations (43) and (44) reduce to 

e = 1 + $ (1 - .44604 Go)] (1 + f) 

P = .064 (H + .5 - ue) I 

where ue is determined from equation (22) with n, = 0.5. For n, = 0.5, 

some typical values of e are 

Partial Span Diamond Boxwing 

For this wing, n, = naf = (1/2)n,, 

Ha/H = (1/2)rl,, u =&v, TI,~ = I-I,, 

then equations (17) and (21) become 

- 

I 12 

cr n, 

P= L I 

1 + 3(1 - ; $)(l - $2 

- 

With 0, = 0.5 

(45) 

(1 - ; +Go (1 + f) 

3/2 

(Ap-w - u,) 

e = C 1 + + (1 - .84251 Go)] (1 + f) 

P = .011339 (4.25 + Hz - ue) 

For ql = 0.5, some typical values of e are 

(46) 

(47) 

(48) 

(49) 

H 0 .l .2 .5 1 2 

e 1 1.00645 1.01685 1.06521 1.17650 1.44816 

17 
Partial Span Right Triangle Boxwing 

For this wing, qa = qaf = o/2)0,, 

Ha/H = (l/2) nl ,' u = (l/2) r-q + ~igYp7F, 

n,o = 01, then equations (17) and 1 21) 

become 

"1 

{ 

2 rl,H 

e= l+- 

t 1 IT - (1 - $ $t’ - ; $Go] 

> 

(1 + f) 

- 

- 3/2 

P= 

; IIf 

1 + 3(1 - ; $)(l - $2 ' (7 Ql ’ +Jv-ue, 

- 

For qi = 0.5, typical values of e are 

(50) 

(51) 

For ql = 1, typical va lues of e are 

H 0 .l .2 .5 1 2 

e 1 1.03178 1.07181 1.23909 1.60153 2.37390 

Ellipse Dumbbell Boxwing 

In this wing 

'a 

= 

Oaf= 2 ' h, + rlo) + g (11, - 11,) 

H a-n --- 

H 4 ( Ql - 

Qo) 3 Q'O = Q' - T-lo (52) 

and u = u ,soP=f=O 

where ae is fro: equation (24). 

18 
With these values and using equations (17) through (19), e can be 

determined for arbitrary ellipse and position in the wing semispan. For the 

special case of circular boxwings with n, = 1, the wing appears as shown below. 

0--D nO 00 

For the circle dumbbell boxwing with n, = 1, the height parameter is 

H=(l- no)/2, and the wing terms simplify to 

'a = ; (1 t oo) + ; (1 - r10) 

H 

n10 =1-no 

y. H = ; (1 - noI 

(53) 

With the values of equation (53) and equations (17) through (19) typical 

values of e are determined at no = .8, .7, .5, and 0, which correspond to H 

values of .l, .15, .25, and .50, respectively. 

These circular dumbbell boxwing values compare well with the theory results 

from the application of the methods of reference 2. 

19 
Rectangle Dumbbell Boxwing 

In this wing, 

na 

= 

'af = '1 

H 

n10 = n1 - Q. =$ 

u=q +H 1 

7 

3 

(54) 

With the terms of equation (54), e is determined by using equations (17) through 

(21) and (24). For the special case of a square dumbbell boxwing with n, = 1, 

then H = (1 - no)/2 and the equations simplify to 

e= 

1 

$ H)(l - 2H)G0] (1 + f) + 

Go = (1 - .38 H2)(l + 1.9304 H4) 

(1 + 3.62 H2)(l + 1.4425 H5) 

(55) 

u=l+H,u 

e 

= 1 + .422112H 

P= 8(.577888)H4 

L 1 + 3(1 - 4 H2)(l - 8 H2)2]3'2 J 

where H = (1 - no)/2. With equation (55), typical values of e of the square 

dumbbell boxwing are determined at no = .8, .7, .5, and 0, which correspond to 

H values of .l, -15, .25, and .50, respectively. 

H 0 .lO .15 .25 .50 

e 1 1.20451 1.29238 1.50320 2.00026 

20 
For this wing, 

Bow-Tie Boxwing 

Ho + &H - Ho)2 + (1 + rlo)L ] I (56) 

With the terms of equation (56), e can be determined by using equations (17) 

through (21) and (24). For the special 

case of no = 0, and HO/H = (r2 - 8)/8 = 

-28373 the equations for determining e 

simplify to 

e = (1 + $ H)(l + f) + .560149 (ee - 1) 

(57) 

; + .61685 H + ; /l + .5 8722 H2 - E, k = ./x 

[HE, k = +'!-I} 

- 

Typical values of e for this bow-tie boxwing computed from equation (57) are 

i++Gii*j~ 

APPLICATION TO FIN-WINGS 

Fin-wing configurations include end-plate wings, winglets, V-wings, crosswings, and fan wings. Since in this class of wings Ha = 0, equation (17) 

simplifies to 

e=l+G, (58) 

21 
where G, as a function of naf is given in equation (19), and naf, in equation 

(32). The prediction of'e for fin-wings can be demonstrated by application of 

the method for estimation of naf. 

End-Plate Wings and V-Wings 

For the end-plate wing 

' I--+; 

na, = na2 = 0" = y)' = 7-73 = , 

-1 1 

Y2 = -Y, ¶ 

then equation (32) reduces to 

H2 1 + .05 j-j-- 

1 

'af = 1.05 , at H2 = H,, oaf = 1 

At 'af = 1, G, = ee - 1, so by equation (58), e = ee, (eq. 16). 

For the V-wing 

'al 

= 

'a2 = 

.5, rl,' = 1, ilo, = ill3 = 03 Y2 = Y’) 

then equation (32) reduces to naf = l/2. At this value of naf, G, = ev - 1, 

so by equation (58), e = ev, (eq. 15). 

Cruciform or Cross-Wing 

For the cross-wing 

nal = na2 = .5, nil = rl12 = 1, 

l-lo1 = VI3 = 0, 

then equation (32) reduces to 

T 

Hl 

+ 

H2 

4 

22 
rl af 1' - (60) 

When y2 = y,, then naf = l/2, the configuration becomes the V-wing, and as 

before, e = ev. When y2 .= -y, and ti, = H,, the configuration becomes a 

symmetrical cross-wing, and equation (60) leads to naf = .757267. With this 

value of naf, and equation (19), equation (58) gives e for the symmetrical 

cross-wing as 

e=l+ 

1.84922 (ev - 1) 

J- 

(61) 

-1 

( .397872 + ,"' _ , )2 

e 

where ev and ee are given in equations (15) and (16), and in table 2. Some 

typical values of e are 

H 0 .l .2 .5 1 2 

e 1 1.03997 1.11847 1.43415 2.00000 3.09219 

where the H = 1 configuration is a 90 degree symmetrical cross-wing which as 

shown in appendix D has a value of e = 2. 

Winglet Wings 

For this wing, 

Tl = 1, 7-7 3 = no1 

23 
then with equation (32) 

(1 - 11 - ' + .05 + yy) 0’01 v2 no1 

2.1527 

20 + .05 

2 qo,) 

(62) 

(63) 

At H = 1, equation (58), with (19) and the ee and ev values at H = 1, results in 

e=l+ 

2.14961 nEf 

(1 + .12672qf)2 

(64) 

Typical values of e for various locations of a single winglet for H = 1, are 

no1 0 -25 .50 .75 

naf .50000 .62305 .74074 .85106 

e 1.45265 1.68961 1.95892 2.24811 

where the no, = 0 value is the same as for the V-wing. 

For vertically symmetric winglets, 

y2 = -Y,, n,2 = 1, H2 = H,, then 

equation (62) becomes 

(1 - 

naf 

= [ 1.05 + 

y&l - 110J’2 

2.1527 - I 2(1 + 05 q2 ) 

’ + no1 

. 01 

(65) 

24 
Then using equation (64), typical values of e for various t, values of a wing 

with vertically symmetric winglets, for H = 1, are 

1.00 1.05 

t-: 

1.00 1.01498 

2.69328 2.74146 

where the no, = 0 value is the same as for the symmetric cross-wing of H = 1, 

and the no1 = 1 value is the same as the end-plate wing of H = 1. 

For the case of y2 = 0, n,2 = 1, 

then H2 = 0, and equation (62) becomes 

I I 

"01 

-I + qol 

2 2(1 + .05 llo,) 

Pm 

Then using equation (64), typical values of e for various no, values of a wing 

with upper surface winglets, for H = H, = 1, are 

.50 .75 1.00 

-68880 .77623 .86222 .95238 

2.04810 2.27930 2.54421 

25 
Fan Wings 

To analyze a 2N - 1 fan wing where 

then from equation (32) 

= 1.05 + 

(1 - ,bl)3’2 ' + nO1 

naf 2.1527 2 + 

2 + .I iI'01 

2 

.77 + .9H ' - n0 

1 + .67H 1.15 N 

For equally spaced fins along H 

H2n-l _ n-2 

H1 N-l 

2 

H2n-l _ 

n=3 H1 

; (N - 2) 

(67) 

(68) 

For a three-fin fan wing, N = 2, then from the first part of equation (67), 

the oaf values for various no1 span stations and typical e values at'H = 1 

from the equation (64) are 

nO1 0 .25 .50 .75 1 

naf -86838 .92008 .94059 .95486 1 

e 2.29667 2.44668 2.50828 2.55182 2.69328 

26 
For other values of H, e can be determined from equation (58) with G, given in 

equation (19), where qaf are those listed above. 

For a five-fin fan wing, N = 3, and with equations (67) and (68), the naf 

values and typical e values at H = 1 are 

no1 0 .25 .50 .75 1 

-. 

naf .94295 .98166 .98352 -97640 1 

e 2.51544 2.63513 2.64098 2.61862 2.69328 

For an infinite-fin fan wing, N = m, then with H = 1 

"01 O .25 .50 .75 1 

naf 1.0921 1.1048 1.0694 1.0195 1 

e 2.9992 3.0433 2.9216 2.7561 2.6933 

The no, = 0 value of e is the same as the bow-tie wing with "o = Ho = 0, 

and H = 1. All the above %, = 1 values of e correlate with the e of the 

symmetric end-plate wing with H = 1. 

For the infinite-fin wing with no, = 0, equation (67) simplifies to 

.77 + .9H 

naf = .75727 + .3348 , + .67H (6% 

Then with various H values, typical values of e from equations (58) and (19), 

are 

.l .2 .5 1 2 

1.02712 1.03775 1.06324 1.09207 1.12498 

1.2525 1.4575 2.06.77 2.9992 .4.7419 

27 
These values of e for the infinite-fin wing with nO, = 0, are close to those 

listed for the bow-tie boxwing where HO/H = .2337, no = 0. It is apparent 

the Ho = 'b = 0 bow-tie and infinite-fin cross-wing have the same e. 

An example usage of the fin wing 

equations for other shapes is the case H1 

= H5 = ; 

where 

nll = fl12 = 1, y2 = -yl 3 H2 = H 

y6 = -y5 = -~/r/2, H5 = H6 = H,, 

- 

'la5 - na6 = 

1 

then equation (32) reduces to 

naf = 

77 + .9H -75727 + .3344 j + .67H (70) 

Equation (70) is the same as equation (69), hence e is equal to that listed 

under equation (69). 

APPLICATION TO MULTI-WINGS 

Multi-Wings with Equal Spans 

A vertical stack of wings with equal spans is pictured as 

-b/2 

h h2 

b(2 

\ wing 1 

+ 

h3 ,, 

' wing 2 

h4 :: lwing 3 

h 

A 

IN " .wing 4 

--c 

wing N 

28 
then Ha = H, na = 1, and equations (17) and (21) become 

e = (1 + 1 H)(l + f) 

P = UM - ae 

(71) 

(72) 

Comparing equations (71) and (10) shows, for small H so that P and f are small, 

that the e for multi-wings of equal spans approaches that of the rectangle 

boxwing. This is in accord with multi-plane thoery. The perimeter p for 

multi-wings requires a redefinition here since there are no vertical panels. 

In effect the h distances in p are subdued so that for equal span wings 

= 1 + H 

OF1 

1 + 3.06 w H1'6 

(73) 

where 0 M denotes CT of a multi-wing aircraft and N is the number of wings and 

N - 1 is the number of spaces between wings. For equally vertically spaced 

wings, h2 = h3 = . . . hN, then 

(&’ = (N _ l)(!+’ = N-l H2 

1 

2 1 

H H (N _ 1)2 I-I/(N = N-l 

thus for wings of equal spans which are vertically equally spaced, the effective 

perimeter parameter is 

OF1 

= 1 + 

1 + 3.0; H1/6 

N 

(74) 

The e is determined by inserting aM of equation (73) or (74) into equation (72) 

and using equations (20) and (71). Equation (71) is accurate in the range 

O<H<l, - but somewhat higher as N becomes large. As mathematically proven in 

appendix D, in the limit for H very large, the e of the multi-wing and the 

lifts of wings of equal spans are simply 

29 
L 

e = N, and"= 1 

Ll 

(75) i 

which is independent of Hn distribution. As the number N of wings becomes 

large, then as shown in equation (74), u = 1 + H, which is also the value for 

the rectangle boxwing. Thus the e of the multi-wing with an infinite nutier 

of equal span wings is identical to the rectangle boxwing of the same H. 

Multi-Wings with Arbitrary Spans 

A vertical stack of wings with .various spans is pictured as 

A 

hm‘ 

V 

‘----I 

h2 1 

h3 $ -l 

h4 

, n2 

n3 

I 

hN 

n4 

, 

% 

The effective integrated value for na 

is the sum of the rectangle areas formed 

by each of the multiwings, then 

I I 

I J ’ 

Ha =A= Hn 

na H nn H' where H = Hn (76) 

where H,/H = (hn/b)/(hm/b), and b is the span of the widest wing. The 

effective perimeter of the multiwing -'I 

1 

is approx imated by perimeter-_ _._..- 

-at 

u= 

%’ + ‘N 

2 + H2 + 

nn -2nn+1)2 + H,2+l (77) 

30 
The multiwing perimeter parameter is 

J b - n2)2 + (1 - ,2)2B2 - (1 - n2)B 

UM = Q2 + - l+B 

where 

B = 3.06 !k$ Hli6 

( 78) 

(79) 

As N approaches infinity the multi-wings outline a boxwing and equations (76) 

and (77) become the values of the boxwings. Then since at N = m, B = 0, the 

prediction of e for the multi-wing of infinite wings is identical to that of 

the boxwing formed by the outline of the infinite wings. 

The nlo parameter is n2 which represents the span extent of the wing 

with second largest span, then 

n10 = n2 030) 

The elliptic ae parameter is for a partial span to the q2 or qlo span station. 

The values for ue are determined from using ue/q10 instead of ae/nl in 

equation (22) or in table 2. Thus with na and Ha/H from equation (76), and 

uM from equation (78), then P is evaluated from equation (21) which becomes 

P= 

c 

nlOna 

I 

3/z ( ‘e 

1 + 3(1 - lllpa)(l - 2nlona)2 'M - n10 nlo) (81) 

With P, f is found from equation (20);then e computed from equations (17) and 

0% 

Accuracy is ensured to H values such that e/(1 + f) is the same as the 

H= 03 value of e. The H = 03 value is derived in appendix D where from equation 

(D9), for maximum e at H = 03 

I2 L 

e =I+' 

n=2 

$, "=$ 

Ll 

(82) 

31 
For finite height parameter H, e and Ln are less than these values. 

The biplane solution has N = 2, then 

H2 = H, na = Ha/H = nlo = nN = q2 and Jy-yy-1 

equations (77), (78), and (79) become -n2 n2 

u = n2 + H, B = 1.53(2 - '12) 

2 H1/6 

OM = 9 + 

/n - (1 - n2)B 

l+B 

With these values P is determined from equation (81 

(18), and (20). 

(83) 

and e from equations (17), 

Diamond Shaped Multi-Wings 

A diamond shaped multi-wing can be 

represented by letting wing gaps and 

wing spans be I I I 

H, = $-T 

! 

where N is an odd integer. Then from equation (76 

H H 

a-2 z-s n4 

'a H 

-- 

n=2 % H N2 -1 

4 N+l N-f3 

N2 -1 ( 2 --l)(7) -7 1 (!!+l)(!y 

11 

' Q2 

1 n3 

'.5(N+l) 

(84) 

N+3 (7- n) = 

t1)tl =& 

I (85) 

32 
or for all odd-integer N, na = Ha/H = l/2. -we Q'10 value is 

n10 = n2 

JI-1 

N+l 

The perimeter is 

U= nN + 1 + 2H2 + &2 - nN + 1)2 + (H - 2H2)2 

2 

I__ 

2 

2 2H =-t----+ 

N+l r4-i 

(86) 

( 87) 

The multi-wing perimeter parameter oM is determined from equation (78) in which 

n2 is given by equation (86). B from equation (79) summed for this diamond 

shape is 

B = 6.12 H"6 1 '(7/m 

N 1 -t-N 038) 

With oM evaluated, P is computed from equation (81) in which na = l/2, nlo is 

from equation (86), and ae is with the ellipse extending to nlo. 

When N becomes infinite, these parameters become rla = Ha/H = l/2 (for any 

N), qlo = q2 = 1, B = 0, UM = o = m. These values are the same as those 

listed in equations (34) and (35) for the diamond boxwing, hence e is the same 

as those listed in table 1 for the diamond. 

Numerical Values of e for Multi-Wings 

For equal span wings and vertically equally spaced wings, for the biplane 

N = 2, and for the triplane N = 3, equation (74) becomes 

uM 

= 1 t 

1 + l.;3 H1/6 (bip1ane) 

=I t 1 + 1 coy Hl/6 hiplane) (8% 

33 
With equation (89), P is evaluated from equation (72) with ue of the ellipse 

extending to nl = 1. With P, f is evaluated from equation (ZO), then e from 

equation (71). These e's are listed in table 4 for various H values with the 

H= m values from equation (75). 

For a partial span wing on the biplane, N = 2, H2 = H, na = Ha/H = nlo = 

n2' and aM is given in equation (83), where n2 denotes the spanwise extent of 

the lower wing. Equation (81) becomes 

2 

P = F n2 312 

11 + 3(1 n$(l 2n;)2 3 CUM - ue) - - 

. 

(90) 

where ue is with the ellipse extending to the nlo = n2 span station. With P, 

f is determined from equation (20). Equation (17) becomes 

+ + n2H [l - (1 - 2 $)(l - $)Go (1 + f> 

Tr 

(91) 

where Go is given in equation (18) in which nlo = n2. The e for the biplane 

with n2 = 0.6 computed from equation (91) is listed in table 4 for various 

H values. The H = ~0 value of e is from equation (82). 

TABLE 4. - IDEAL e OF VARIOUS MULTI-WINGS 

r equal span 

H biplane 

equal span n2 = 0.6 span 

triplane biplane 

w (8% eq. (89) eq. (91) 

0 1 1 

:2 1 1.3529201 1:3834 ; 2176 1.087645 

.3 1.4810 1.5315 1.1301 

.4 1.5842 1.6654 1.1769 

.5 1.6407 1.7839 1.2154 

03 2 3 1.36 

34 
CONCLUSIONS 

It has been demonstrated that the application of the generalized equations 

(17) through (32) and equations (76) through (82) provides a direct method for 

estimates of ideal e for arbitrary lateral-vertical wing forms. Example predictions of e are made for a wide range of wing configurations, from multi-wings 

to fan and dumbbell wings. Several check points were verified with other 

methods including the cruciform at H = 1 (eq. 61), the circle dumbbell (eq. 53), 

and biplane and triplane (table 4 with ref. 1). Also, multi-wing solutions 

converge to those boxwings as the number of wings increase, and fan wing solutions converge to those of bow-tie wings as the number of fins increase. For 

the V-wing, an exact closed function was developed. An asymtotic unique maximization solution for multi-wings was found which resulted in exact expressions 

for ideal e and ideal lift of each wing. 

Hampton Technical Center, Kentron International, Inc. 

Hampton, Virginia 23666 

June 20, 1980 

35 
APPENDIX A 

V-WING e FACTOR 

From reference 3, the added mass M is given by 

1-a M= TFP~ 

a 'rnX 

where (here y notation replaces 6 of ref. 3) -b/2 0 b:2 

The dimensionless added mass is given by 

M 1 - a -= 

2 2 

PT a aa 

where a is determined from the definite integral 

: 

1 

a= 

<dg 

(1 - &"(c + <)" 

0 

where 

(AZ) 

(A3) 

(A4) 

This integral was evaluated numerically in reference 3 for various values of y, 

then dimensionless added mass was computed and tabulated. However, as will be 

shown here, the integration in equation (A3) can be obtained by expanding the 

second parenthesis term into a binomial series. Thus 

5 =L +E52.L 

(c -I- [)" cc1 C 2!c2 

(c1+1) c2,. . .+ 

(-l)"a 

n!c" 

(a + l)(a + 2) . . . (a + n - l)gntl t . . . 1 (A5) 

36 
The definite integrals of the following type can be evaluated: 

1 

/ 

2?!9-& E0 -&+ E1 a at y at 2 

o (1-d 

1 

n=2 

= 3! 

CY, a+l)(a+ 2)(a+ 3) 

. 

1 ow 

n=n 

= (n + l)! 

a(a + l)(a + 2) .,. (a+ n + 1) 

Inserting equation (A5) into equation (A3) and applying equation (A6) leads to 

a=$ [ &- ( 2 t 3 

a+1 )(a •!- 2)c (a -I- NCX + 3)c2 

-...t 

1. 

(-U”(n + 1) . 

(a + n>(a + n + 1)~" 

+ - . 1 (A7) 

With c given in equation (A4), equation (A7) can be written as 

uw 

For the range of 0 s a 5 l/2, this summation is identically unity. Thus the 

exact solution for a is simply 

a l-a a=(+) 

With equation (A2) the dimensionless added mass is 

M - -- 

2 ( 

pra 

Z-)1+ 

1 

(A% 

(AlO) 

37 
From reference 3, the induced drag is given by 

L2 Di = - 

2MV2 

which equated to equation (1) results in 

e=$(L.-) 

pra 

From the V-wing sketch the geometrical a and y are 

2a _ 

-, b - & = (1 + 4 H2)1'2 , y = tan-' 2H 

(All) 

(A121 

(A13) 

With equation (Al), the dimensionless added mass and factor e, in terms of the 

y angle in radians, in closed functions, are 

3 

-- 2Y 71 

M 

1 

-= 

2 

pna I s ! lt-+ 

3 

2Y n 

e = ___ 

1 '-71 

I I 

cos2y 1 t 3 7r 

In terms of the height parameter, H = h,/b 

L tan -' 2H 

M 1 - i-tan -' 2H ' 

-= 

pna 

2 I i 2" 1 + ; tan -' 2H 

-' 2H 

e = (1 + 4 Hz) 

(A14) 

(A15) 

u-w 

(A17) 

Equations (A15) and (A17) give e in terms of elementary functions of either the 

dihedral angle or the height parameters of the V-wing. 

38 
APPENDIX B 

END-PLATE WING e FACTOR 

From page 211 of reference 1, 

KY = Bk sn n, $ = (7 5 l/2 

then 

sn n = 1 (K - E 112 

k K ) = sin9 

From the definition of snn related to sin9, n is 

I 

4 

n= F(W) = 

d+ 

(1 - sin26in 4) 2 l/2 

0 

(Bl) 

032) 

(B3) 

that is, the incomplete elliptic integral of the first kind. In addition, 

from reference 1, 

- (1 - ‘$I], 

a2 

b+, 

2 2E 

e = !$ (r _ k12) 

(B4) 

where k = sina, k' = (1 - k2)1'2 = COS a. With equations (Bl), (B2), (B3), the 

hm of equation (B4) becomes 

H =a IT E(+\a) -k F( +\a)] (B5) 

w 

where F(+\a) and E(+\a) are incomplete elliptic integrals of the first and 

second kinds with amplitude 4 and modular angle a, and K and E are complete 

elliptic integrals of first and second kind with modular angle a. The term 

39 
within the brackets of equation (B5) is the Jacobian Zeta Function. Then 

H = 1 KZ($\a) (B7) 

where the function KZ($\a) is tabulated in reference 4 for various values of 

4 and a, and has a maximum value at the $I given in equation (B6). From' 

equation (B4) 

e=!-K2(?$- 

IT2 

cos 2a) w 

As can be seen from equations (B5) through (B8), the factor e is not a 

direct function of H but is related to H only through the parameter a. That 

is, a value for ais specified, then 4 is determined from equation (B6), then 

H from either equation (B5) or (B7), and e from equation (B8). Some typical 

results for a range of ~1 values are shown as follows: 

dig H e r d:g H e r 

30 .07194 1.14138 .6996 80 .89782 2.53705 .7312 

45 .17351 1.33322 .7082 82 1.0165 2.71842 7296 

60 34824 

: 49550 

1.64573 7200 

17268 

84 1.1718 2.95205 17263 

68 1.89566 85 1.2713 3.10002 .7233 

75 .68710 2.20712 .7318 88 1.7821 3.84313 .7048 

This list includes the parameter r which is evaluated from 

r= 

1+2H -e 

(e - 1 - +H)H 

(Bg) 

Equations (85) through (B8) are quite complicated functions and do not 

permit e predictions for a specified value of H. It has been found that a 

relatively simple function exists between e and H. In the limit as a + 0 

equations (B7) and (B8) become 

H +1.k2 

4 ' 

e+ltlk2 

2 

40 
where k = sina + 0. Eliminating k2 results in 

H+O 

e = lt2H @lo) 

For a + a/2, equations (B7) and (B8) become 

2 4 H+-anT, e-t IT k 

where k' = (1 - k ) 2 112 = (--sa 

. Eliminating the log term, e becomes 

H+= 

e = ;H (BW 

A combination of equations (BlO) and (Bll) indicates a function of the following type 

e=lt(:t 

2 4 

1 i;H) H U-312) 

where r is a quasi-constant. The solution of equation (B12) for r is equation 

(Bg). As can be seen from the listed H and r values, r is nearly constant 

with H, but remains a weak function of H. An approximation for r which duplicates those in the list is given by 

r = -732 _ G2 C-8 - HI 

2 

1 + .3688 H2 

(Bl3) 

Then equation (812) gives e with a four to five digit accuracy for the wing with 

end plates as 

2 -- 4 Tl H 

1 + .732 H - .0612 H(.8 - H)2 

1 + .3688 H2 

(B14) 

41 
APPENDIX C 

90" CRUCIFORM WING 

As discussed in reference 5, page 39, in a 90" cruciform wing there is no 

induced velocity at one vorticity sheet 

due to the other. In this case each 

wing acts independently of the other, 

thus for minimum induced drag the 

spanwise loading distribution is elliptical along bw. With Lnl and Ln2 * 

representing the lift normal to wings 

1 and 2, with equation (1) 

L2 L2 

Did++=- L2 

rr9 b W nq bw nq b2 

W 

since Lzl + Lz2 = L2. Thus for this case e = 1. In reference 5 when the 

‘L’ ‘D.’ and A are reduced to dimensional quantities with bw, the solution is 

also e'= 1. 

For a 90" cruciform wing banked 

at angle $, the loading is elliptical 

along wings 1 and 2, and, similarly 

to equation (Cl), the induced drag is 

since again L2 + L2 = L2, nl n2 and hence e = 1. Thus when the induced drag of a 

90" cruciform is in terms of the wing width, bw, the efficiency factor is 

unity at any bank angle. 

Generally the induced drag of a lifting configuration is in terms of the 

lateral span of the configuration, that is, b. Then from equation (l), the 

42 
induced drag of the 90" cruciform wing is 

Di = 

L2 

2 nqb e 

Equating equations (C3) and (C2) results in 

b 

e= ($)2= 

(c3) 

(c4) 

where 4 is the angle of bank of the 90" cruciform wing. When 4 = 0, +~/8, 

f~r/4, then e = 2, 1.176, and 1, respectively. In summary for the 90" 

cruciform wing 

e = 1, for all bank angles when Di is defined by equation (Cl), 

e = equation (C4), for all bank angles when Di is defined by 

equation (C3). 
APPENDIX D 

ASYMPTOTIC SOLUTION FOR OPTIMUM eAND LIFTS 

FOR MULTI-WINGS WITH LARGE GAPS 

If the gap height hn of a multi-wing is large enough such that the 

influence of one wing on another is negligible, then the ideal loading for 

each wing will be elliptic and induced drag is given by Din = Lz/qbE. The 

sum of these induced drags equals that of the multi-wing having the sum of all 

the lifts. Thus for a multi-wing with N planar wings 

L2 Di= 2 = 

rqb e 1 

Ln n=l 

where b 1 is the wing with maximum span. In dimensionless coordinates, 

then equation (Dl) becomes 

N L; 

=1+-ye n=2 nn 

solving for e 

(1 

e=- 

1 

(Dl) 

(D3) 

Equation (D3) is valid for any lift variation among the wings, however, there 

is some lift variation for which e is maximum. For equal loading e can be 

less than unity when 1 f ni2 + qi2 + . . . ni2 > N2. The maximization technique of reference 5 can be applied to equation (D3). A partial derivative 

44 
of e with respect to each of the loading ratios gives 

2(1 + En) 2 

ae - n=2 

-- 

at. 

In (1+ 

f2 

4,2 [ 2 

1+ g- (1 +& Rn) ?] 

n=2 0: 

n= % 

The conditions for maximum e are that these derivatives be zero for each of 

the loading ratios. Then 

L2 

l+-$ -J--(1+ 

n=2 qz 

c 

n=2 

!Ln) f$ _= 0 

Yll 

(D4) 

These are non-linear simultaneous equations in an with n = 2, 3, . . . N 

unknowns, and m = 2, 3, . . . N equations. For the biplane, N = 2, then 

equation (D4) with (D3) leads to 

2 2 

1 +2- g2 (1 + g2)7= R2 0, or 22 = 772 

e=l+ 2 

Q2 Q2 Q2 I 

(D5) 

For the triplane, N = 3, then the solution of two equations of (D4) leads to 

2 

+ 

2 

2 

2 

n; 1 + 

2 2 

n2 n3 3 - Q2 - 2 q2 - 

R3 + R3 + 

h3 

4 2 2 '13 - (1 + q2,, = 0 

n3 Q3 Q3 

which ion involves the solution of a cubic equation. As can be seen, the solut 

of equation (D4) becomes complicated when N t 3. What is needed is to have 

Rn as some function of T-I,, so that equation (D4) is satisfied. Such a funct 

has been discovered by the hint given in equation (D5). Let 

ion 
R 2 = 

n nn (D7) 

then equation (D4) becomes 

which completely satisfies the maximum conditions, and hence equation (D7) 

gives the individual wing lifts for maximum e. With equation (D7) inserted 

into equation (D3) the maximum or ideal e is 

2 R = 2 

n On' e=e 

wax 

=1+ nn 2 

n=2 

For equal span wings, rln = 1, so 

R 

n 

=l,e=N 

For biplanes 

2 2 g2 = 9’ e=l+q 2 

For triplanes 

2 2 2 2 ,t2 = q2, R3 = q3' e = 1 + n2 + q3 

(Dg) 

@lo) 

(Dll) 

(D72) 

and so forth. 

46 
 

REFERENCES 

1. von Karman, Theodore; and Burgers, J. M.: General Aerodynamic Theory - 

Perfect Fluids, Vol. II of Aerodynamic Theory, div. E, William F. 

Durand, editor, First Dover edition, Dover Publications, New York, 

1963, pp. 201-222. 

2. Cone, Clarence, D., Jr.: The Theory of Induced Lift and Minimum Induced 

Drag of Nonplanar Lifting Systems. NASA TR R-139, 1962. 

3. Letcher, John S., Jr.: V-Wings and Diamond Ring-Wings of Minimum Induced 

Drag. Journal of Aircraft, Vol. 9, No. 8, August 1972, pp. 605-607. 

4. Milne-Thomson, L. M.: Jacobian Elliptic Functions and Theta Functions, 

of Handbook and Mathematical Functions, M. Abramowitz and I. A. Stegun, 

editors, National Bureau of Standards AMS 55, March 1965, pp. 567-626. 

5. DeYoung, John: Minimization Theory of Induced Drag Subject to Constraint 

Conditions. NASA CR-3140, June 1979. 

47 
1. Report No. 2. Government Accerrion No. 

NASA CR-33 57 I 

4. Title and Subtitle 

Induced Drag Ideal Efficiency Factor of Arbitrary 

Lateral-Vertical Wing Forms 

3. Recipient’s Catalog No. 

I 

7. Author(s) 

I 

8. Performing Organization Report No. 

John DeYoung 

10. Work Unit No. 

9. Performing Organization Name and Address 

Kentron International, Inc. Hampton Technical Center 

3211 North Armistead Avenue 

Hampton, VA 23666 

11. Contract or Grant No. 

NASl-16000 

13. Type of Report and Period Covered 

12. Sponsoring Agency Name and Address 

National Aeronautics and Space Administration Contractor Report 

Washington, DC 20546 ' 

15. Supplementary Notes 

14. Sponsoring Agency Code 

530-04-13-01 

Langley Technical Monitor: Harry H. Heyson 

Topical Report 

16. Abstract 

A relatively simple equation is presented for estimating the induced drag ideal 

efficiency factor e for arbitrary cross-sectional wing forms. This equation is based 

on eight basic but varied wing configurations which have exact solutions. The e 

function which relates the basic wings is developed statistically and is a continuous 

function of configuration geometry. The basic wing configurations include boxwings 

shaped as a rectangle, ellipse, and diamond; the V-wing; end-plate wing; 90 degree 

cruciform; circle dumbbell; and biplane. Example applications of the e equations 

are made to many wing forms such as wings with struts which form partial span rectangle dumbbell wings; bowtie, cruciform, winglet, and fan wings; and multi-wings. 

Derivations are presented in the appendices of exact closed form solutions found of 

e for the V-wing and 90 degree cruciform wing and for an asymptotic solution for 

multi-wings. 

1 7. Key Words (Suggested by Author(s)1 

Induced Drag Endplates 

Winglets 

Biplanes 

Boxwings 

Multi-wings 

18. Oistribution Statement 

I Unclassified - Unlimited 

Subject Category 02 

1 9. Security Classif. (of this report) 

Unclassified 

I 

20. Security Classif. (of this page) 

Unclassified 

21. No. of Pages 22. Price 

47 A03 

I I I 

For sale by the National Technical Information Service, Spinefield. Vlrglnla 22161 

NASA-Langley, 1980