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FILE COPY NO. I-W CASE FILE
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NATIONAL ADVISORY COMMITTEE
FOR AERONAUTICS
TECHNICAL NOTE
No. 1269
METHOD FOR CALCULATING WING CHARACTERISTICS
BY LIFTING-LINE THEORY USING NONLINEAR
SECTION LIFT DATA
By James C. Sivells and Robert H. Neely
Langley Memorial Aeronautical Laboratory Langley Field, Va.
•O^ FILE COPY Washington „ To be returned to
** A* of thi Htuhad April 1947 Advisory Cmmftm
Ö.C,
^
NATIONAL ADVISORY COMMITTEE FOE AERONAUTICS
TECHNICAL NOTE NO. 12Ö9
METHOD FOR CALCULATING WING CHARACTERISTICS
BY LIFTING-LINE THEORY USING NONLINEAR
SECTION LIFT DATA
By James C. Si veils and Robert H. Neely
SUMMARY
A method is presented for calculating wing characteristics "by lifting—line theory using nonlinear section lift data. Material from various sources is combined with some original work into the single complete method described. Multhopp'e eystems of multipliers are employed to obtain the induced angle of attack directly from the spanwise lift distribution. Equations are developed for obtaining these multipliers for any even number of spanwise stations, and values are tabulated for ten stations along the semispan for asymmetrical, symmetrical, and antisymmetrical lift distributions. In order to minimize the computing time and to illustrate the procedures involved, simplified computing forms containing detailed examples are given for symmetrical lift distributions. Similar forms for asymmetrical and antisymmetrical lift distributions, although not shown, can be readily constructed in the same manner as those given. The adaptation of the method for use with linear section lift data is also illustrated. This adaptation has been found to require less computing time than most existing methods.
The wing characteristics calculated from general nonlinear section lift data have been found to agree much closer with experimental data In the region of maximum lift coefficient than those calculated on the assumption of linear section lift curves. The calculations are subject to the limitations of lifting-line theory and should not be expected to give accurate results for wings of low aspect ratio and large amounts of sweep.
INTRODUCTION
The lifting-line theory is the best known and most readily applied theory for obtaining the spanwise lift distribution of a wing and the subsequent determination of the aerodynamic character- istics of the wing from two-dimensional airfoil data. The character- istics so determined are in fairly close agreement with experimental results for wings with small amounts of sweep and with moderate
2 NACA TN No. I269
to high values of aspect ratio; for this reason, this theory has served aa the "bads for a large part of pre cent aeronautical knowledge.
The hypothesis upon which tho theory is "based is that a lifting wing can "be replaced "by a lifting line and t'uit the incremental vortices ahed along the span trr.il "behind the wing In straight lines in the direction of the free-stream velocity. The strength of these trailing vortices is proportional to the rate of change of the lift along the span. The trailing vertices induce a velocity normal to the direction of the free—stream velocity and to the lifting line. The effective angle of attack of each section of the wing is therefore different frcm the geometric angle of attack by the amount of the engle (called the induced angle of attack) whose tangent is the ratio of the value of the induced velocity at the lifting line to the value of the free—stream velocity. The effective angle of attack is thus related to the lift distribution through the Induced angle of attack". In addition, the effective angle of attack is related to the section lift coefficient according to two-dimensional data for the airfoil sections incorporated in the wing. Both relationships must "be simultaneously satisfied in the calculation of the lift distribution of the wing.
If the section lift curves are linear, these relationships may "be expreused "by a single equation which can "be solved analytically. In general, however, the section lift curves axe not linear, particularly at high angles of attack, and analytical solutions are not feasible. The method of calculating the sponwise lift distribution using nonlinear section lift data thus becomes one of making successive approximations of the lift distribution until one is found that simultaneously satisfiea the aforementioned relationships.
Such a method has been used by Wieselsberger (reference 1) for the region of maximum lift coefficient and by Boshar (reference 2) for high—subsonic speeds. Both of these writers used Tani's system of multipliers for obtaining the induced angle of attack at five stations along the semisprn of the wing (reference 3)« Tani, however, considered only the case of wings with symmetrical lift distributions. Multhopp (reference h), using a somewhat different mathematical treatment from that which Tani used, derived systems of multipliers for symmetrical, antioy'-maetrical, and asymmetrical lift distributions for four, eight, and sixteen stations along the semispan. Multhopp'0 derivation, in slightly different form and nomenclatures is presented herein and tables are given for the multipliers for ten stationo along the semispan (the "usual number of stations considered in many reports in the United States).
ITACA TN No. 1269
For symmetrical distributions of wing chord and angle of attack, the multipliers for symmetrical lift distributions may be used with nonlinear or linear section lift curves. For asymmetrical distributions of angle of attack, the multipliers for asymmetrical lift distributions must be used if nonlinear section lift curves are used. If an asymmetrical distribution of angle of attack can be broken up into a symmetrical and an antisymmetrical distribution, the antisymmetrical part may be treated separately if the Bection lift curves can be assumed to be linear.
The purpose of the present paper is to combine the contributions of Multhopp and several other writers, together with some original work, into a single complete method of calculating the lift distributions and force and moment characteristics of wings, using nonlinear section lift data. Simplified computing forms are given for the calculation of symmetrical lift distributions and their use is illustrated by a detailed example. The adaptation of the method for use with linear section lift data is also illustrated. No forms are given for asymmetrical or antisymmetrical lift distributions inasmuch as such forms would be very similar to those given.
SYMBOLS
S wing area
b wing span
c chord at any section
cß root chord
c.(. tip chord
c" mean geometric chord (s/b)
/2 |VD/2 ?
c1 mean aerodynamic chord [— I c dy
A aspect ratio (b2/s)
x coordinate parallel to root chord
y coordinate perpendicular to plane of symmetry
z coordinate perpendicular to root chord and parallel to plane of symmetry
k KACA TH No. 1269
q free-etreem dynamic pressure ( -xP-v )
/pVc PVC8\ E Reynolds number ( =• or -— )
p mass density
V free-stream velocity
\i coefficient of viscosity
CL wing lift coefficient (l/qS)
C2 section lift coefficient (l/qc)
L wing lift
I section lift
Cj) wing drag coefficient (D/qS)
CD0 wing profile-drag coefficient
Cj). wing induced-drag coefficient
c^ section profile-drag coefficient
c^. section induced-drag coefficient
D wing drag
Cm wing pitching-moment coefficient (M/qSc')
CJJJ ,. section pitching-moment coefficient about section quarter— ' chord point
M wing pitching moment
Cj wing rolling-moment coefficient (iZ/qS)
L* wing rolling moment
Cn. wing induced—yawing-moment coefficient
Cno wing prcfile-yawing-moment coefficieot
a angln of' attack of any section along the Bpan referred to its chord l:'no
NACA TN No. 1269 *
a-, angle of attack of root section referred to it3 chord lins
aa angio of attack of root section referred to its zero 3 lift line
oj_ section induced an^le of attack
CLQ effective angle of attack of any section
aQ section angle of attack for two-dimensional airfoils
a7 angle of aero lift of any section 'o
a7 angle of zero lift of root section
<x„, v wins en^le of attack for sero lift -(L-0/
c geometric angle of twist of any section along the span (negative if washout)
e ' aercayrrmic a-gle of twist of any section along the spen (negative if washout)
e geometric angle of twist of tip section
e^* aerodynamic angle of twist of tip section
a wing lift—curve slope, per degree
aQ section lift-curve slope, per degree. / Two-dimensional lift—curve slopo\
s. Edge—velocity factor /
cos 0 coordinate (2j/b)
An coefficients in trigonometric series
ßafc multiplier for induced angle of attack (asymmetrical distributions)
X^ multiplier for induced angle of attack (symmetrical distributions)
y^, multiplier for induced nngle of attack (entisymmetrlcal distributions)
NACA TTT No. 12Ö9
Tj multiplier for lift, drag, and pitching-moment coefficients m (asymmetrical distributions)
T) multiplier for lift, drag, and pitching—moment coefficients ms (symmetrical distributions)
a multiplier for rolling— and yawing-monent coefficients (asymmetrical di otrituti ons)
0 multiplier for rolling-moment coefficient (antisymmetrical distrlbutions)
E edge-velocity factor />emir»rijggter\ \ span /
Subscripts
max maximum va]ue
al ralue for additional lift (C^ = l)
b value for basic lift (CL = 0)
/a„ "\ value fcr constant value of a„ V a"J as (^M value for fdven value of e^'
THEORETICAL DEVELOrMFJuT OF METHOD
Lift Distribution
The methods of Tani (reference 3) and Multhopp (reference if-) for determining tho induced an^le of attack are fundamentally the same, differing only in the mathematical treatment. The method presented herein is essentially the came as that given by Multhopp, In the following derivation the spanwise lift distribution is expressed as the trigonometric series
C f*
as in reference 5, vhere 9 is defined by the relation cos 9 = 2Z ,
It may be noted that each coefficient An, as used herein, is equal
KACA TN No. 12Ö9
to four times the corr©spending coefficient in reference 5. The induced angle of attack (in degrees) at a point y-, on the lifting line is
* <m dy (2)
This integral (in different nomenclature) wa,3 civan by Prandtl in reference C. If equation (l) is substituted into equation (2) and the variable is changed fron y to 6, the induced anrjle of attack at the general point ß becomes, according to reference 5*
a. = ——— > nfi, sin nO (3) hn sin 0-'—- ^
The problem of obtaining the induced angle of attack is thus reduced to cne of c'stemming the coefficients of the trigonometric series.
Tiie lift distribution (equation (l)) mR.y he approximated by a finite tr'.gciioretric series of r -- 1 terns where> for subsequent
c7c usage, r is assumed to he even. The values of —- at the equally
b spaced points <9 = —-- in the range 0 < 9 < JT are expressed as
(?£) = 5" ^ sin n 52 (k)
CiO where m = 1, 2, 3,. . ., r — 1. Conversely, if the values of —£
b are known at each point the coefficients An of the finite series may be found by hamonic analysis as
0 X-l /c7c\ A = -t\ LL.I 0in nM ^ n rm=l Wm
r
8 NACA TW No. 1269
If equation (5) is substituted in equation (3), a double summation is obtained for the induced angle of attack as
Oi(9) = - I80 'r-1
kn ein 9\ n sin n6
r J~ \ b An. r
E=l
r—1 r—1
to sin 0 ^— V b A -^—- m=l n=i
coa
If the induced angle of attack is to be determined at the same points 6 et -fchich the load distribution is known, that is, at the
points 0 = —£, then
°li
lQ. r-1 , . r~l
. —S2-T- V (C4C-) Y n to sin ^ ---r \ b /m •—T
(k - m)ir (k + m)it cos n — cos n
" / c 7 c \
m=l
(6)
vhero
ßmk = 180
r-1
4ifr ein JSS. <-— •-» n=.- cos
r — cos n (k + m)jt (7)
It can be shown that, if cos f> •/ 1,
/ n cos n$ = n=l
r cos (r - 1)0 - (r - 1) cos r$ - 1
2(1 - cos 0)
NACA TW No. 1269
If $ = 0, a numerical series is obtained
f1 n . rir^li fei 2
By use of these relationships in equation (7) it is found that, when kirn is odd
ßmk = 3&£L W sin ^L r
1
1 - cos (k + m)n T
1 - COS Xk^jüit (8a)
when k = m
ßmk l80r
8n sin &L (8b)
and when k i m is even and k / m
ßmk = 0 (8c)
For a symmetrical lift distribution
v b /m \ b /r-m
and
^k = «ir-k
so that the summation for a^, needs to be made only from 1 to r/2
10 KACA TN No . 1269
r/2
<Hk
where, when k ± m is odd
( hi) * A m \mk
m=l
Ink = ßmk + ßr-m,k (far m t r/2)
(9)
l8o
2nr sin ^
cot 1*±»>JL cot (k - m>"~ .£ C sin jCk_+-ffiL>L 8in (k ,- m)n
r r
(10a)
*mk = ßmk (for m = r/2)
l8o - / 2kir -v itrfcoß + 1) \ r
(10t>)
vhen k = m
\te = ^mk
l80r
8it sin ^ r
(10c)
and when k ± m is even and k ^ m
»Bfc- ° (lOd)
For an antisymmetrical lift distribution
(2il) m-(22±)
NACA TN No. 12Ö9 11
and
«lk * "" °lr-k
In this ca8e the summation for oc^ needs to be made only from 1
to (§ - l) since (^) - 0; then
Olfc-
I-1 m. ?mk (11)
where, when k ± m is odd
?mfc ~ nak ßr-m,k
- ^2 ' 1 1
2itr sin2 (k + m)ir r sin^ (k - m)n
r
when k - m.j
?mk = ßmk
_ 3.80r
8it el Ln £2. r
(12a)
(12b)
and when k i m is even and k ^ m
?mk = 0 (12c)
Multipliers can thus be calculated so that the induced angle C T C
may be readily obtained by multiplying the known values of -*—
by the appropriate multipliers and adding the resulting products.
12 WACA TN No . 1269
The multipliers are independent of the aspect ratio and taper ratio of the wing. Tables I and II present values of ßu^, and X^
and 7-sfa} respectively, for r = 20. Similar tables for YFn^rnk
and Tn^yßiir are given in references 7 and 8, respectively, tut
no derivation is given therein. Tables for jjrrßak» ng^ndc' and ißZ7^
are given in reference k for values of r = 8, lb, and 32. An inspection of tables I and II shows that positive values occur on3y on the diagonal from upper left to lower right and that almost half of the values are equal to zero, The multipliers ß^ and X^ may be used with either nonlinear or linear section lift data whereas the multipliers for 7^ may. be used only with linear section lift data.
The method of determining the lift distribution becomes one of successive approximations. For a given geometric angle of attack, a distribution of c-j is assumed from which the load distri—
bution — -— is obtained. The induced angle of attack is then b
determined by equation (6), (9), or (11) through the use of the appropriate raultip?.iers and subtracted from the geometric angle of attack to give the effective angle of attack at each spanvise station. From section data for the appropriate airfoil section and local Eeynolds number, values of Cj are road which correspond to the effective angle of attack of each section. If these values of c^ do not ag^ee with those originally assumed, a second assumption is am de for c^ and the process is repeated. Further assumptions are made until the assumed values of c^ are in agreement with those obtained from the soction data.
Wing Characteristics
Once the lift distribution of a wing has been determined, the main part of tho problem of calculating the wing characterietics is completed. The induced-drag and induced-yawing-moment coefficients are entirely dependent upon the lift distribution and it is assumed that the section profile-drag and pitching-moment coefficients are the same functions of the lift coefficient at each section of the wing as those determined in two-dimensional tests.
The calculation of each of the wing coefficients involves a span-wise integration of the distribution of a particular .
function f(T?)' This integration can be performed numerically
through the use of additional sets of multipliers which are found in the following manner.
•NACA TN No . 1269 13
If
then
?(&) = f(cos 0) =/ An sin n£
f'1 f / f/Ez\ JZz) = (y A sin no ] ein 0 dO
• = a A 2 1
Since the values of f( -v- ) are determined at the points 9 = —,
A-, can he found hy harnonic analysis as in equation (5)
r-1
Al = I X' f^£) sin ~ 1 r / \ b /m r
Therefore
! r-1
, if) <?; - f 1 <?1 - ? m=l
r-1
-2lf(f7mV '13a)
m=l
where
TU. = JL sin M ,m 2r r
Ik NACA TN No. 1269
If the distribution is symmetrical., f( &•) = f( §£) and V b /m \ 0 /r-m
where
%s = 2% (m ^ 2 )
^ms = ^m (-S) The moment of the distribution f(^) can bo found in a similar manner. ,b/
P1 nrt
j f(?) (¥) if) - I (I An »"> *9) 3ln * - » «
fA0 4 2
-i\ f(?z) 8in22HI ?r /__ \b/m r
m=l
r-1
mh )-*$)„ ** (l4a) Dl=l
NACA TN No . 1269 15
where
„ = JL ein &£ m 8r
If the distribution is antisymmetrir.al, ?(~) - ~ f(~-) \b /m \b /r-ffi
5-1
I ff^Y^ d/2yV. ^ f(gyA (14b)
vhere
iaa = P yam
Values of r\ , r\ , c , and 0 are given in table III for r = 20. 'm' 'ins' mJ
Wing lift coefficient.- The ving lift coefficient is obtained
by means of a spanwiae integration of the lift distribution,
nb/2 1 D
czc dy
-0/2
A f1 « 2<L * 12 a(£) b \b/
If the lift distribution is asymmetrical
CL - A >m (15a)
16 NACA TN No. 1269
If the lift distribution is symmetrical
r/ CL = A
2
m=l
Induce«?-dra^ coefficient.— The section induced-drag coefficient
is equal to tlie product of the section lift coefficient and the induced angle cf attack in radians,
Ttc7a. 1 180
The wing induced-drag coefficient is obtained by means of a sparwise integration of the section induced-drag coefficient multiplied by the local chord;
rtci c % " i I "W dy
(UA
A . c,o b 130 Vb /
ii-1
For asymmetrical lift distributions
r-1 «A "v /c7c \ ,,s .
= T£ ; i-f- °dL nm (16a) m=l
For symmetrical lift dictributions
r/2
m-1
KACA TN No. 1269 17
Profile^drag coefficient,— The section profile-drag coefficient
can lie obtained from section data for the appropriate six-foil section and locol "Reynolds number. For each spanvise station the profile- drag coefficient is read at the section lift coefficient previously determined. The vin£ profile—drag coefficient Is then obtained by me e no of a sp.-iirwif.e integration of the section profile-drag coefficient multiplied ly the local chord:
CT Dr
nVa
ii-i
cd0 c dy
-b/2
i 2
-1
f)
For asymmetrical lift distributions
r~l
>° -> (c*4 ^ m-1
(17a)
or for syiuiretrical lift distributions
r/2 V—
'o 3 ^
^ ~ / [ ^o~ ) \a (17b)
Pitching-moment coefficient.— The section pitchmg-moment
coefficient about its quarter-chord point can be obtained from section data for the appropriate airfoil section and local Reynolds number. For each spanwise station the pitching-moment coefficient is read at the section lift coefficient previously determined and then transferred to the wing reference point by the equation
18 NACA TN No. 1269
cm = cmc/^ ~ J cos (as - aj.) + cä ein (aB - a^)
cl aln (as ~ al) ~ Gd0 G0S (ae * ai) (18)
vhere x and z are Treasured frou the wing reference point to the quarter—chord point of the section under consideration and upward and backward forces and dietex>cüs are taken a3 positive. The section pitcliingnncment coefficient about its aerodynamic center may be used instead of cm n , in which caße x and z arc- measured to the section aerodynamic center. The term c^ Bin (^3 - o^) may usually bo neglected. The wing pitching-fiioment coefficient is obtained by the epanwiee integration
Cm Sc7
J-b/2 cm c2 dy
\fmm For asjTmaetrical lift distributions
z—r \ C C * /r m=l m. 'in (19a)
For symmetrical lift dibtrlbutions
ra X /CTT,C
^AcV m=l Tme (19b)
Rolling-moment coefficient.- The rclling-moment coefficient
is obtained by means of a spanwise integration
MACA TN No . 1269 19
|V2
C, = 1_
Sb Cjcy dy
-"b/2
_ A k
I •b 1) \b/
(20a)
For an antisymmetrical lift distribution
I-1
H-*L(? m=l (20b)
Induced—yawing-moment coefficient.— The induced-yawing-
mcment coefficient is due to the moment of the induced-drag distribution
0 b/2
c„. --L
L JtC l£2t
ni Sb J 180 •b/2
y d-y
A 4 b ISO b Q\b/
r^l nA \ ~/£ic
I80 Z—V b m=l
/czc \ LA b °y» a- (21)
20 KACA TN Nc . 1269
The induced-yawing-moment coefficient for an antieymmetrical lift distribution is equal to zero and has little meaning inasmuch as the lift coefficient is also zero. The induced-yawing-moment coefficient is a function of the lift and rolling-moment coefficients and must be found for asymmetrical lift distributions.
Profile-yawing-moment coefficient.- The profile-yawing— moment coefficient is due to the moment of the profile-drag distribution,
C -i- no Sb
b/2
cdQcy dy
tU/2
£dof £y c b if)
r-l K—
2L m=l. Y(CJr
\ u 'm ) °m W
APPLICATION OF METHOD USING NONLINEAR SECTION LIFT DATA
FOR SYMMETRICAL LIFT DISTRIBUTIONS
The method described is applied herein to a wing, the geometric characteristics of which are given in table IV. Only symmetrical lift distributions are considered hereinafter inasmuch as these are believed to be sufficient for illustrating the method of calculation. The lift, profile-drag, and pitching-moment coefficients for the various wing sections along the span were derived from unpublished airfoil data obtained in the Langley two-dimensional low—turbulence pressure tunnel. The original airfoil data were cross-plotted against Reynolds number and thickness ratio inasmuch as both varied along the span of the wing. Sample curves are given in figures 1 and 2. From these plots the section characteristics at the various spanwise stations were determined and plotted in the conventional manner. (See fig. 3.) The edge-velocity factor E, derived in reference 9 for an elliptic wing, has been applied to the section angle of attadc for each value of section lift coefficient as follows:
NACA TW Ho . 1269 21
ae = E(a0 -alQ)+ alo
Lift Distribution
Computation of the lift distribution at an angle of attack of 3° is 3liovn in table V. This table is designed to be used where the multiplication is done by means of a slide rule or simple calculating machine. Where calculating machines capable of performing accumulative multiplication are available, the spaces for the individual products in columns (6) to (15) may be omitted and tb'.i table made smaller. (See tables VII and VTII.) The mechanics of computing are explained in the table; however, the method for approximating the lift coefficient distribution requires some explanation. The initially assumed lift-coefficient distribution (column (3) of first division) can be taken ae the distribution given by the geometric angles of attack but it is best determined by some simple method which will give a close approximation to the actual distribution. The initial distribution given in table V was approximated by
Cl = A 1 + 21
A + 1.8 j_2 nc -iff 'Ha)
where cj(a) i"3 the lift coefficient read from the section curves for the geometric angles of attack. This equation weights the lift distribution according to the average of the chord distribution of the wing under consideration and that of an elliptical wing of the same aspect ratio and span. When the lift distributions at several angles of attack are to be computed and after they have been obtained for two angles, the initial assumed cj distribution for subsequent angles can be more accurately estimated in the following manner: Values of downwash angle are first estimated by extrapolating from values for the preceding wing angles, and then, for the resulting effective angles of attack, the lift coefficients are read from the section curves.
The lift coefficients in column (10) of table V, read from section lift curves for the effective angles of attaok, will usually not check the assumed values for the first approximation. In order to select assumed values for subsequent approximations, the following simple method has been found to yield satisfactory results. An incremental value of lift coefficient Ac7 is obtained
tin according to the relation (numbers in parenthesis are columns in table V) :
22 NACA TN No. 1269
"(18) - oflaHL + 3[(18) - (3)]m * [(18) - QlJan Acim =
K
where K has the following values at the spanwise stations
b K
0 to 0.8910 8 to 10
.9511 11 to 13
.987T Ik to l6
and (l8) - (3)Jm is the difference "between the check and assumed values for the mth spanwise station. The incremental values BO determined are added to the assumed values in order to obtain new assumed values to be used in the next approximation. This method has been found in practice to make the check and assumed values converge in about three approximations if the first approximation is not too much in error.
Wing Coefficients
Computations of the wing lift, profile-drag, induced-drag, and pitching-moment coefficients are shown in table VI. Since the lateral axis through the wing reference point contains the quarter- chord points of each section, the x and z distances in equation (18) are zero, and the pitching-moment coefficient of the wing is determined solely by the values of cm /• .
APPLICATION OF METHOD USING LINEAR SECTION LIFT DATA
FOR SYMMETRICAL LIFT DISTRIBUTIONS
Although the method described herein was developed particularly for use with nonlinear section lift data, it is readily adaptable for use with linear section lift data with a resulting reduction in computing time as compared with most existing methods. When the section lift curves can be assumed linear, it is usually convenient to divide any symmetrical lift distribution (as 5n reference 10)
NACA TN Wo. 1269 23
Into two parts — the additional lift distribution due to angle of attack changes and the basic lift distribution due to aerodynamic twist. The calculation of these lift distributions is illustrated in tablea VII to X for the wing, the gsroaetrJc characteristics of which were given in table IV,
It should be noted that tables VII and "VIII are essentially the same es table V but are designed primarily for use with calculating machines capable of performing accumulative multi- plication. If such machines are not available, these tables may be constructed similar to table V to allow spaces for writing the individual products.
Lift Characteristics
Two lift distributions are required for the determination of the additional and bae?c lift distributions. The first one is obtained in table VII for a constant angle of attack cc^ (c ' ^ 0) and the second one in table VIII for the angle of attack distribution
c c due to the aerodynamic twiat (aag = 0). The check values of —i—
(column (l8)) are obtained by multiplying the effective angle
of attack 0^ . by -,-C-. The final approximations are entered in table IZ as (923.) ° and i'-^)
(ccas) V /(Gt*)
(211) The (--*- ) distribution is the additional lift distribution
corresponding to a wing lift coefficient CL/.^ \ determined in
table IX through the use of the multipliers r^. It is usually CJ 1 c
convenient to use the additional lift distribution —-=-— corresponding
to a wing lift coefficient of unity. This distribution is found
by dividing the values of {——) by Or. ,. V "b /(aa3) (aag)
The ( —- ) . distribution is a combination of the basic lift \ b /(et-)
distribution and an additional lift distribution corresponding to a wing lift coefficient CL/e ,\ also determined in table IX. The
cl>c basic lift distribution —ep- is then determined by subtracting the
additional lift distribution —^— ^L(^. >\ from l'T~//e >\-
2k NACA TN No. 1?69
Inasmuch ao the wing lift curve is assumed to "be linear, it is defined sy its slo^e and angle of attack for zero lift which ar6 also found in table IX. The maximum wing lift coefficient is estimated according to the method of reference 10 which is illustrated in figure k. The maximum lift coefficient is considered to be the wing lift, coefficient at which some section of the wing "becomes the first to reach its maximum lift, that is, c7, + Or ci = c7 4
D ^ ta]_ 'max This value cf C^ is moat conveniently determined "by finding the
minimum value of —'Zax Ji along the span aB illustrated in table 33. Cial
Induced—Drag Coefficient
The section induced-drag coefficient is equal to the product of the pection lift coefficient and the induced angle of attach in radians. The lift distribution for any wing lift coefficient is
£i£ = _M_ cL + -^- (23) b b L b v
The corresponding induced rngle of attack distribution may be written as
ai = ai„i CL + ^.-h (2^
The values of aj, and aj, are determined in table X in the
same manner a3 —~~ and —k_ j.n table IX. The induced-drag b b
distribution is therefore
cd-,c c7c a-.
b b 57-3
or cdn c cäi -,-u0 cdn- c
Cd-!c ^al 9 alb ^ / -~ = —~- CL2 + — QL + —-£- (25) b b b b
N/.CA TN No . 1269 25
where
13 13 57.3
Cfl1n1, c C) ,c a., c7,,,
^ "b 57.3 * 57.3
and
b > I O (28)
The calculation of each of these iiidueed-drag distributions is illustrated 'n table X together with tho numerical integration of each distribution to obtain the wing induced-drag coefficient.
Profile-Drag and Pitching—MoMent Coefficients
Tho ^rofile-drag and pitching-ironent coefficients for the wing depend directly upon the section data and therefore their calculation is the saue whether linear or nonlinear section lift data are used. For tue linear case the section lift coefficient is
c l°cl&1°L+°h
for any wing coefficient C^. By use of this value for c, the
profllc-d^ag and pitching-inonent coefficients are found as in table VI.
DISCUSSION
The characteristics of three wings with symmetrical lift distributions have been calculated by use of both nonlinear and linear section lift data and are presented in figure 5 together with experimental results. Those data were taken from reference 11. The lift curves calculated by use of nonlinear section lift data are in clooe agreement with the experimental results over the entire range of lift coefficients whereas those calculated lay use of linear section lift date are in agreement only over the linear portions of the curves as would be expected.
26 IRC;. TN No . lc-69
It mr>Bt be remembered that the methods presented are subject to the limitations of lifting—line theory upon which the methods are based; therefore, the close agreement shown in figure 5 should not b3 exoected for wings of low aspect ratio or large uweep. The use of the edge—velocity factor more or lees ccmpenaateß for some of the effects o^ aspect ratio and, in fact, e.pueare to over compensate at the larger values of aspect ratio as shown in figure 5«
Additional comparisons of calculated and experimental data are given in reference 11 for wings with symmetrical lift distributions, but very little comparable data are available for wings with asymmetrical lift distributions. Such data are very desirable in order to determine the reliability with which calculated data may be used to predict experimental wing characteristics.
Langley Memorial Aeronautical Laboratory National Advisory Committee for Aeronautics
Langley Field, Va. December 20, 1946
NACA TN No. 1269 27
HEF3REUCE1
1. WieBelebergor, C: On the Distribution of Lift across the Span near und beyond the Stall. Jour. Aero. Sei.., vol. 4, no. y, July 1937, pp. 363-365.
2. Boshar, John: The Determination of Span Load Distribution at High Speeds by Jre of High-Speed Wind-Tunnel Section Data. NACA ACE No. 4B22, 19kk.
3. Tani, Itiro: A Simple Method of Calculating the Induced Velocity of a Monoplane Wing. Rep. No. Ill (vol. IX, 3), Aero. Res. Inet., Tokyo Imperial Univ., Aug. 1934.
4. Multhoop, II.: Die Berechnung der Auftriobsvertoilung von Tragflügeln. Laftfahrtforschunj Bd 15, Nr. 4, April 6, 193$» pp. 1^3-109.
5. Glaserb, H.: The Elements of Aerofoil and Airscrew Theory. Oiuoridge Univ. pre so, 1927«
6. Pi'DPv.tl, L.: Applications of Modern Hydrodynamica to Aeronautics. NACA Rep. KoVlIo, 1Q21.
7. Munkj Max M. : Calculation of Span Lift Distribution (Part 2). Aero. Digest, vol. 48, no. ?, Feb. 1, 1945, ?• 34.
8. Munk, Max M.: Calculation of Span Lift Distribution (Part 3). Aero. Digest, vol. 48, no. 5, March 1, 1945, p. 98.
9. Jonec, Robert T.: Correction of the Lifting—Line Theory for the Effect of the Chord. NACA TN No. 3l7, 1941.
10. Anderson, Raymond F.: Determination of the Characteristics of Tapered Wings. NACA Rep. Ho. 572, 1936.
11. Neely, Robert II., Bollech, Thomas V., Westrick, Gertrude C, and Graham, Robert R.: Experimental and Calculated Characteristics of Several NACA 44-Series Wings with Asnect Ratios 8, 10j and 12 and Taper Ratios 2.5 and 3» 5- NACA Tin No • 1270, 1947 •
NACA TN No. 1269 28
t— r-< o o r-t CO O o J- -3- o O co r-t o o r-t r— r-- rH r-t ON c— r-- -cf ON NO NO ON -d- r— r— ON r-i r-t ^- CO LfN ON O o CO LTN o LTN o LfN o t/N CO o o ON LfN co
tip / ON ON CO CO r— LTN ^1 rc\ rH rH rr\ -S LfN r- CO CO ON ON
o 1 1 1 ' i 1 1 1 1
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a ti t- o l-l ON -^ NO CO CO o t— r-t CO r- x^ LTN LTN r- -J LO NO r-t NO NO ,—t U) o f- NO CO KN C\J (ON J- CO J- CM r-! CT^ •a rr- • • • • • • • r-t
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r- r-t o o r-t CO o o J- -^ o o CO r-t o O ,-, t— c— f—< r-t ON r— c— -J ON NO NO CN
T* c^ c~- CN r-t r-t t— CG ITN ON o o CO LfN o LTN o LfN o CO o o ON LfN CO
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29 NACA TN No. 1269
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NACA TN No. 1269 30
TABLE III.- WING-COEFFICIENT MULTIPLIERS
¥ m ^ ^3 «m ^ma
-0.9877 19 0.01229 -O.OO607
-.9511 18 .021+27 -.01151+
-.8910 17 .03066 *- -- 0 jrtt* -.OI589
-.8090 16 .Ol+6l6 -.OI867
-.7071 15 .05551+ -.019614
-.5878 14 .O635I+ -.OI867
-.I+5U0 13 .O6998 -OI589
-.3090 12 .071+70 -OII5I+
-.156!+ 11 .07757 -.00607
0 10 .07851+ 0.0785I+ 0 0
.156]+ 9 .07757 .15515 .00607 0.01211+
.3090 8 .071+70 .11+959 .OII5I4 .02308
.1+51+0 7 .O6998 .13996 .OI589 .05177
.5878 6 .O635I+ .12708 .OI867 .03735
.7071 5 .O555I+ .11107 .OI96U .03927
.8090 h .Ol+6l6 .09233 .OI867 .03735
.8910 3 -^.03666 .07131 .OI589 .03177
.9511 2 / .021+27 .01+85)+ .OII5I+ .02308
.9877 1 .01229 .021+57 .00607 .01211+
,&]S~/£
NATIONAI COMMITTEE F
. ADVISORY OR AERONAUTICS
/täL
31 NACA TN No. 1269
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NACA TN No. 1269 32
TABLE __
First opproxlmotion
CALCULATION OF LIFT DISTRIBUTION FOR. . WING.
(1) (2) (3) (41 (5) (6) (7) I (8) 1(9) (10) I (II) I (12) I (13) | (14) | (15) (16) (17) (18)
¥ a
lac,' eS assumed
c b
ITobte 17) Ox (4)
X_k x column (5)
S(6)
«6
(21-06) (cl-«c») 0 .1564 1 .3090 .4540 .5878 .7071 .8090 8910 .9511 .9877
O It« 0.51 0.142« 0.071
143.239 -58.533 0 -6.950 0 -2865 0 -1.804 0 -1.468
1.88 1.12 O.libll 10. so -4.29 0 -.SI 0 -.21 0 - ___ 0 -.11
0.1564 2,76 .si7 .1295 .0670
-115.624 145.025 -67298 0 -10.158 0 -4840 0 -3 394 0
.«1.78 .511 -7.75 9.72 ___5J, ._
1506 M
g -67.15 7
-.68 0 -.52 0 -.23 0
3090 ?,li7 •^21 .1161, .06015
0 -64.802 0 -9916 0 -4.968 0 -3768
,90 w . _£ 0 -1.95 9.17 -4.09 0 -.60 0 -.10 0 -.2?
4540 ?.ll .51< .lolid .os4a
-12384 0 -62.917 160.76 1 -72.472 0 -10926 0 -5.812 __0
_Q -7.713
--77
.60
.65
1.36 • Sill -.67 0 -1.40 e.68 -1.91 0 -.59 0 -.11
5878 1-75] .501 •a?*: .0463
0 -8 320 0 -65.803 177.054 -82083 0 -13.134 0
J.15 .500 0 _,
-4.051 -_-___ 0 -3.05 8.20 -3.80 0 -.61 0 -.56
70 71 l.?f| .477 .082' _.o_9j
.031«
0 -7.372 0 -71.743 202.571 -97965 0 -17 388 0
.61 •k-u -•16 . 0
0 -2.880
__-..__9... 0
,--0- J -7.208
-.23
-2.82 7.96 -3.85 0 -.68 0
.8090 .BO •430 • 0735
0 -81.434 243.694 •i 25.537 0 -26.635 i _ I55J_-J25_._
4*4-10
.77|-87
•iiii
• 336
0 -•09 0 0 -2.57 7-70 -5-97 0 -.84 0
.8910
.9511
_.J2
-.10
.560
.281
.0665 l_ .025s -1.638 0
0 -1 06 2
-.02
-2 3 7 1
-.06 0
0 -7.370 0 :96 962
-2.J2 0
315.512 -180.528
-•OJi.H 0
0
0 -2 016
-.18 b
0 -7 599
L-5_,.. -122.880
- 2.11
- U.31 0
-329976
-5.68 .0613 .017?
463.533
7-97 0 -.05 0 -.13 0
.9877 -,19 .228
-0459 0 -0620 0
0
-1.491
-.01
0
0
• 65
-7.039
-.07
0
0
-167045
- i-_67_.
915651
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[ U94i-2.11 .165 .olid .010c 0 0 -.01
£ 1.88 • 98 .90 •77 .60 •55 •U2 •77
Second approx motion
0
01564
„1_0_
LJi
2 41
__5__
.524
.1),JQ .0712
143.239 -58533
145025
0
0 _ -67.298
-4.50
-6950
-.49
0
0
-2 865
-.20
0
0
-1.804
-.11
0 -1468
~ 0
0 -376 8
'-.23
1.07
.95
1.19.
1.69
1.52
.521
10.20 -115624
,1295 .
.1164
.0668
0 -10 158 0 -4 840 0 -3 394
-•?3 . 0
0
-5.812
-7.72 9.69 0 -67167
-.63 JJ -.32 . 0
0 -4.968
-5? ^ 3090
.0610
0
0
-6480«!
-3-95
150 611
9-19
0 -9916
_-i.iL
•517
-4.10 0 -.60 0
4540 2.11 .517 .1040 .0538
-12 384 0 -62.917 .160761
8.65
-72 472 0 -10.926 0 0
•74! 1-39 -.67 0 -1.18 -1.90 0 -.59 0 -31 0
5878 1.71 .500 .0925 .0461
0 -8 320 0 -65803 177.054 -82.083 0 -13.134 0 -7713
.60 1.11 .500 0 -•39 0 -1.05 ._!___> -I.80 0 -.61 0 -•36
7071 1 .?f .47e .0821 .0191
-4051 0 -7.372 0 -71.743 202.571 -97.965 0 -17.388 0
• s( •70 .480 -.16 0 -.29 0 -2.82 7.96 -3.es 0 -.68 0
8090 ,80 .0715 .0124
0 h2 880 0 -7 208 0 -81 434 243694 -125.537 0 -26635
.611 .19 .441 0 -.09 0 -.21 a -2.64 7.90 -4.07 0 -.86
8910 .1?
______
.182 .066S .0254
-1 638 0 -2 371 0 -7.370 0 -96.962 315512 -180528 0
•7< -.18 .586 -.04 0 -.06 0 -.19 0 -2.46 8.01 -4.59 0
.9511 -.10 .252 ,0611.
.0417
.0179
0 -1.062 0 -2.016 0 -7599 0 -122880 463533 -329.976
.8« -•99 .512 0 -.02 0 -.04 0 -.14 0 -2.20 8-30 -5.91
9877 -.19 .2iq .0096
-0.459 0 -0620 0 -1.491 0 -7.089 0 H67.045 915651
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33 NACA TN No. 1269
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Sivells, James Heoly, Robert H.
I DIVISION, Aerodynamic? (£) IHCTION: Wings end Airfoils (6) flcBOSS REFEBENCES. Hing» - Lift distribution (99170)
I
,70° 6073 Ö3IG AGENCY NUMQ
TM-1269 REVISION
AUTHORS) JL AMEB-TIUEi Method of calculating wing characteristics by lifting-line theory using non-
I lineer section lift data F03GN. TlTUi
OBK3INATING AGENCY, National Advisory Coimittee for Aeronautics, Washington, TOANSLATION,
D. C.
COUNTRY | P. S.
LANGUAGE tfOÄG'NjCLASa U. SjCLASS. I DATE I />AG£S I 1U.US. Eng. tfnclass. Apr'Ltf. 1+6 20
FEATURES tables, grephs
AOSTOACT Hateriel fron various sources is combinod with some original work into the method of
calculating wing characteristics by lifting-line theory. Kulthopp'B systems of multi- pliers sre employed to obtain the induced angle of attack directly from the spamriee lift distribution. Detailed examples sre given for symmetrical lift distribution. Winj characteristics cslculated from general nonlinear lift date have been fovnd to agree much cloeer with experimental deta in the region of maximum lift than thoee calculatod on the eesuoption of linesr section lift curves.
NOTEi Requeste for copies of this report must be eddressed toi N.A.C.A., Washington, D. C.
T-Z HO., AID MATERIEL COMMAND Aß VECHNICAI DNDEX WaiGHT FIELD. O IIO, USAAF >* or>o>» turn <a c:
