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NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS
WAItllTIME RI EPO RT
October 1942 as
Advance Restricted Report
THEORETICAL DISTRIBUTI(I OF LOAD
OVER A SWEPT-BACK WING
By Doris Cohen
Langley Memorial Aeronautical Laboratory
Langley Field, Va.
N ; A .r C ._ r
NACA WARTIME REPORTS are reprints of papers originally issued to provide rapid distribution of
advance research results to an authorized group requiring them for the war effort. They were pre-
viously held under a security status but are now unclassified. Some of these reports were not tech-
nically edited. All have been reproduced without change in order to expedite general distribution.
UTATIONAL ADVISORY COMMITTEE FOR AERONAUTICS
ADVANCE RESTRICTED REPORT
ThEORETICAL PISTRIBUTION OF LOAD
OVER A SWEPT-BACK WIlNG
By Doris Cohen
Tho load over an elliptical wing with 300 sweepback
has been calculated by a method, based on vortex theory,
-.-hich takes account of the chordwise distribution of lift'-
ing area. ?]hc t'.eory indicates a 14-nercent lose in to-
tal lift duc to the introduction of sweepback, with the
greatest loss taking place at tno center of the span. An
increase in concentration of load at the ti-s is also in-
c.icat,-d.. The r-sults n.re compared with results previous-
ly obtained ly soumewhnt simpler c-lculations based on the
assumption of a ringl- lifting vortex.
Until recently, theoretical treatments of the effect
of sweepbick on the aerodynamic characteristics of a wing
have failed to consider any deviation of the span loading
from that of tne corresponding straight wing. In a recent
report (reference i), the load over P swept-back wing is
determined by considering the effect of a lifting line at
the quarter-chord line of the wing on the flow at the
three-quarter-chord line. A method of load determination
has since been developed (reference 2& thnt takus into
reccount the continuou-s chordwise distribution of lift.
Application of this more accurate method to the case of a
swept-back wing indicated (seea reference 21 that the in-
troduction of swcepb.ack causes the lift qt the center to
fall considerably below that of the corresponding wing
without sweepback. The calculations of reference 1 did
not show this effect. Further calculations were therefore
undertaken to determine the correct load distribution,
with special attention given to the load at the middle of
the span. The calculations were made for the case of a
wing of aspect ratio 6 with an elliptical distribution of
chord, the center line of which is swept back 30.
The results obtained -re compared with the data of
METHOD OF OBTAIITING THE LIFT DISTRIBUTI01:
The method of calculation of the lift distribution,
described in dit'il in reference 2, consisted in replac-
ing the wing nnd its wake by a continuous distribution of
vortices and computing the induced vertical velocities
caused by this vortex system at several points on. the wing.
It is evident thrt, in order to satisfy the boundary con-
ditions, the induced velocities must be proportional to
the slope of the surface .t these points qnd, in particu-
lDr, for flat surface they must all be equal, The vor-
tices coincide with the contour lines of the circulation
function F, which, in turn, is obtained by integrating
the lift back along the chord from the leading edge.
Points for which the downwash was calculated were
taken along the auarter-chord line and the three-quarter-
chord line, at the center section and at 30, 60, and 86.7
percent of the semispan. The lift distribution derived
from two-dimensional theories resulted in a linear varia-
tion of downwash along the three-quarter-chord line ex-
cept for a discontinuity at the center, where the downwash
was infinite. A second approximation, designed to elimi-
nate the Tpeak in the downwrsh at the center, proved to be
too fir ir. the other direction. A third approximation
gave again a linear variation of downwash, but with slight-
ly lower values qt the center than at the tip. Values for
the nuarter-chord points obtained for this same lift dis-
tribution fell along P line parallel to that for the
three-quarter chord and approximately 8 percent below it.
This result indicates a small amount of camber, about
equal to the average camber of the straight elliptical
wing used for comparison, but in any case negligible. It
was assui.cd that interpolation between the third load dis-
tribution and the first (two-dimensional) approximation,
at the same angle of attack, would be a fairly accurate
solution to the problem, especially since the third ap-
proximation was already a close one. The curves presented
are the result of this interpolation.
RESULTS .'.I2D DISCUSSION
Figure 1 shcws theo cc-lte ccnf igurnt io- of vortices
determined for a flpt siept-i-ck vilig -.withcut thic-.:nes?.
The vortex lines were derived from the lift distribution
in such n way thnt ndjicent lines enclose i fixed amount
of lift; the concentration of lift In -ny region is there-
fore proportiunal to the den-ity of the lines. The entire
pattern is independe,,t of nngle of rttnck, except -s the
bnsic theory bre.hs down at Inrge angles of att-ck.
In figure 2 is shown the spin loading derived "or the
elliptic l wing witl 3C) spwoJoptnck. The calculated lond
is conp-red with the cllipticil loid, wi-,ich hns been shown
(reference 2) by the s-:.,e method t*o be a rce sonblt accu-
r-te asur-ption for or lllptic.nl 'ing *.-ith no swo,.pbhck.
At the same angle of rtt-ck of the ti.,o wings, measured in
accordance with the: thin-wing-section theory by the slope
-t the throo-au.-rt. r-chord lino, the sron under the curve
for sweepback is 86 percent of th-t under the cllipso,
indicating loss, due to the intro-iction of sweapbnc]:,
of 14 percent of the total lift. This result is twice
thit obtained by 'uittcrmjrl for -r. nirfoil of constant
chord (reference 1i, using rectilino--r vortices concen-
trated on the quarter-chord line.
The effect of sweepbnck on the s-.snn'ise variation of
the lift, indicated by the curves drawn f-r t le same to-
tal lift, is in general the same as is given b:' Eutterperl's
simplified treatment, except for the pronounced fnllinr.
off of lift at the center. Because ut-tcrperl chose his
downwash points nt 50 percent of tnro scmispan and beyond,
no corp'nrison of the results At the center is possible.
The present method is, however, considered to be pnrticu-
larly,' valid in that region.
The present cnlculntions "re mind for elliptical
wings. In Muttarperl's work and in the epzorimants qvail-
able for comparison, wings with constant chord distribu-
tion or straight taper wore con-iderod. In somu tests
referencess 3, 4, and 5) the sweopb-ck wis affected by ro-
tating the wing about ann axis in the planc of symmetry,
thus changing the section profile in the direction of the
air stream as woll qs the chord distribution. Thus, no
real chock! of the t.hcory is av-,ilnblo.
The following table is a s-umnary of pertinent test
dat-i on the lore in total lift due tno the intrnducticn of
sweopbqk. The values tab 'viatce give the total lift on
th. s-.'c -back -.inoga, exprcss.d as frac i.zns of the lift
en the corresnor..ing straight ,-rings. TneLretieal values
for the total 1-lft for o r-er than J 0 sweepback were ob-
tain d b"o interpolation, .-n tic ass-.:mptizn that the lift
varies as the cosine of the angle of sweenbaca.
T.-orotit,.l values Ex erimental. data
TjiPcry of The ory f I
reference 1 reference 2
(rcctang ilar (ell ir ':-cal ViT7L'-v -crr':s Aefercnce
d istriluti oa) disLrijr.t n Ion)
C0.97 0.5 I Sli-ntl. ro-idud 3
.9 .*1 Atr.ct r;tlo, E,3 7
-9 .%:6 1o til f:.irlngs; S La'. 9
F.7 1[ D tip iriri-s 4
3 Corrected for 5
r npcct ratin
______ __ i ___ L _
Unless c'therwi se r.cte:, the uing1s were of constant
chord and aspect ratio 6. The reason fcr the discrepan-
cies a wrong the test res' lt, is not anc.:rstood but it is
possible th.-t differences in pl-i. form intrnduco first-
order 3ffacts not predictable by rot ntial-flow theory.
Prcssi.rc-distribut ion csts i vc; been *:_ndio b Knight
rind IToyc s .references 2 and 6) on roctar:gulrr wings with
200 sweepback. 2o moa sura.T.nts were ;.ipce over the central
35 parent cf the npan, ho- .wcr, ,hero the c-.ief effect
of the swa-pback is to bto xp-ctcd. The incorplet ness of
the oxperimcnts, combined with the dist ortic n of the chord
distribution arnd of tnc section profiles Introduced with
the sucopback, mnkcs the dat unsatisfactory for chocking
the present results. Epoarim-ontni verification of the
dropping off cf tno lift in the center is therefore still
n c c do d.
The effects of swcepback shown by the present analy-
sis arc similar to those shourn in reference 1: Sweep-
bnck promotes higher concentration of load at the wing
tips and reduces the total lift for a given 9ngle of at-
t-ck. The thcoratical reduction of lift in the case of an
elliptical chord distribution, aspect ratio 6, amd 300
sweepback amounts to 14 percent of the load for the straight
wing. This loss, which is about twice as large as would be
expected from reference 1, results from a pronounced reduc-
tion of the load carried at the center of the wing, a fac-
tor which was not covered by the calculations of the refer-
ence. Available experimental data on sweepback are not
considered to provide a conclusive check of the results
prcsonted. The accurqc,- of the theory. should be checked
by further tests, especially pressuro-distribution measure-
monts to dotormin- -*.hethor or not the large loss of lift
near the center actually- occurs.
Langley Memorial Aeron.auticsl Laboratory,
Urtional Advisor.- Committee for Aeronqutics,
L-.ngle;- Field, Va.
1. -utterperl, William: The Calculation of Span Load
Distributions on Swept-Back Wings. T.I. INo. 834,
2. Cohen, Dcris: A Method for Determining the Camber
andi Twist of P Surface to Support n Given Distri-
bution of Lift. T.lj. io. 855, hACA, 1942.
3. Knight, Montgomery, nnd lToyes, Richard W.: Span-Load
Dirtrioution as a actor in Stability in Roll.
Rep. Ho. 393, IACA-, 1931.
4. Willi-ams, I1. H., and H-1lid.y, A. S.: Experimants on
Swe t-brck ind S rept-forw!rd Aerofoils. R. & M.
Uo. 1491, British A.R.. 1933.
5. Rossell, H. E., and Brand, C. L.: Swept Pack Wings.
Part V'IT, ReT'crts on Wind funnel Experirents in
Aercdvinr.ic3. Szithsonimn Ifisc. Coll., vol. 62,
no. 4, 1916, pp. 5E-73.
6. Knight, Hontgomcry, and .To.es., Richard W.: Span Load
Distribution on Two Monoplane Wing Models as Af-
f;cted by Twi.L t and Sueepbnck. T.Ir. loe. 346, ITACA,
7. Wieso1sborfer, C.: Measurerents on Wings with Sweep-
back anCd Warping. (Trnns. frcm Results of Acro-
lynapmic Test Plant ;t CG6ttina-cn, vol. II, 1923.)
AnI.o. rep. THo. 123, War Lept. Air Service, McCook
Field, Ohio, 1924.
8. Anderson, Raymond F.: Determination of the Character-
isrics of Tap.ered Wings. Rop. Ho. 572, IIACA, 193G.
9. Anderson, RPymond F.: The Experimental and Calculated
CLar-cteristics of 22 T-pored Wings. Rep. ITo. 627,
NACA Fig. I
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