NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS WARTIME REPORT ORIGINALLY ISSUED October 19k2 as Advance Restricted Report THEORETICAL DISTRIBUTION OF LOAD OVER A SWEPT-BACK WING By Doris Cohen Langley Memorial Aeronautical Laboratory Langley Field, Va. NACA WASHINGTON 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. L - 221 t pill*. 310 Znsn NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS ADVANCE RESTRICTED REPORT ., i THEORETICAL DISTRIBUTION OF LOAD i OVER A 5WEPT-BACK WING By Doris Cohen SUMMARY Tho load over an elliptical wing with 30° sweepback has been calculated by 3 method, based on vortex theory, which takes account of the chordwise distribution of lifti ing area. The theory indicates a 14-percent loss in to- tal lift due to the introduction of sweepback, with the greatest loss taking place at the center of the span. An increase in concentration of load at the tips is also in- dicated. The results are compared with res\ilts previous- ly obtained by somewhat simpler calculations based on the assumption of a sinelo lifting vortex. I INTRODUCTION Until recently, theoretical treatments of the effect of sweepback on the aerodynamic characteristics of a wing have failed to consider any deviation of the span loading from that of the corresponding straight wing. In a recent report (reference l), the load over a swept-back wing is determined by cor.sidering the effect of a lifting line at the quarter-chord line of the wing on the flow at the three-quarter-chcrd line. A method of load determination has since been developed (reference 2) that takes into account the continuous chordwise distribution of lift. Application of this more accurate method to the case of a swept-back wing indicated (see reference 2) that tho in- troduction of sweepback causes the lift at 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 thoreforo 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 reference 1. METHOD 0? OBTAINING- THE LIFT DISTRIBUTION The method of calculation of the lift di st ri "but i on , described in detail in reference 2, consisted in replac- ing the wing and 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 that, in order to satisfy the boundary con- ditions, the induced velocities must be proportional to the slope of the surface at these points and, in particu- lar, for a flat surface they must all be equal, The vor- tices coincide with the contour lines of the circulation function T, 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 quarter-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 p. discontinuity at the center, where the downwash was infinite. A second approximation, designed to elimi- nate the peak in the downwash at the center, proved to be too far in the other direction. A third approximation gave again a linear variation of downwash, but with slight- ly lower values at the center than at the tip. Values for the quarter-chord points obtained for this same lift dis- tribution fell along a 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 assumed 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 AND DISCUSSION Figure 1 shews the complete configuration of vortices determined for a flat swept-back wing without thickness. °^ The vortex lines were derived from the lift distribution »-} in such a way that adjacent lines enclose a fixed amount cf lift; the concentration of lift in pny region is there- fore proportional to the density of the lines. The entire pattern is independent of nnglc of attack, except is the basic theory breaks down at large angles of attack. In figure 2 is shown the span loading derived "'or the elliptical wing with 30 sweepback. The calculated load is compared with the elliptical load, which has been shown (reference 2) by the same method to bo a reasonablt accu- rate assumption for ar elliptical wing with no swespback. At the sane angle of attack of the two wings, measured in accordance with the thin-wing-section theory by the slope s*t the three-quarter-chord line, the area under the curve for sweepback is 85 percent of that under the ellipse, indicating a loss, due to the introduction of sweepback, of 14 percent of tho total lift. This rosult is twice that obtained by Mutter perl for an airfoil of constant chord (reference l), using rectilinear vortices concen- trated on the quarter-chord line. The effect of sweepback on the spanwise variation of the lift, indicated by the curves drawn for the same to- tal lift, is in general the same as is given by hut t erper 1 ' s simplified treatment, except for the pronounced falling off of lift at the center. Because Muttterperl chose his downwash points at 50 percent of tho semi span and beyond, no comparison of the results at the center is possible. The present method is, however, considered to be particu- larly valid in that region. The present calculations are made for elliptical v/ings. In Muttarperl ' s work and in the experiments avail- able for comparison, wings with constant chord distribu- tion or straight taper wore considered. In some tests (references 3, 4, and 5) the swoepback was effected ~oi~ ro- tating the wing about an axis in the plane of symmetry, thus changing the section profile in the direction of the air stream as well as tho chord distribution. Thus, no real check of tho theory is available. The following table is a summary of pertinent test da' on the Ions in total lift due to the Introduction c sweepback. Tho values tabulator! give the total lift en the swe p t -hack wings, expressed as fractions of the lift on the corresponding straight wings. Theoretical values for the total lift for other than 30 sweepback were ob- tained by interpolation, en the assumption that the lift varies as the cosine of the angle of sweepback. Angle of sweep- back (deg) Theoretical values Experimental data Theory of reference 1 (rectangular chord distribution) Theory of reference 2 (elliptical chord distribution) Value Reraar :.s Reference 20 23 27i 30 0.97 J . J~-- .39 .86 O.So • Si • 96 \.ei Slightly rounded tips Aspect ra.tio, So3 No tip fairings; 2:1 taper i'z tip fairings Corrected for aspect ratio 3 7 3 and 9 U 5 Unless otherwise noted, tho wings were of constant chord and aspect ratio 5. The reason for the discrepan- cies among the test results is net understood, but it is possible that differences in plan form introduce first- order effects not predictable by potential-flow theory. Pressure-distribution tests have been made by Knight and Moycs (references 3 and 6) on rectangular wings with 20 sweepback. Ho measurements were made over the central 35 percent of the span, however, where the chief effect of the sweepback is to be expected. The incompleteness of the ©xperiments, combined with tho distortion of the chord distribution and of the section profiles introduced with the sweepback, makes the data unsatisfactory for checking the present results. Erperi mental verification of the dropping off of the lift in the center is therefore still needed . CONCLUDING EEMAHXS -i The effects of sweepback shown "by the prosent analy- sis arc similar to those shown in reference 1: Sweep- s' back promotes higher concentration of load at the wing tips and reduces the total lift for a given angle of at- tack. The theoretical reduction of lift in the case of an elliptical chord distribution, aspect ratio 5, and 30° 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 presented. The accuracy of the theory should be checked by further tests, especially pressure-distribution measure- ments to detormin -ether or not the large loss of lift near the center actually occurs. Langley Memorial Aeronautical Laboratory, National Advisory Committee for Aeronautics, Langley Field, Va . REFERENCES 1. i'iutt erperl , William: The Calculation of Span Load Distributions on Swept-Back Wings. T.N. No. 834, NACA, 1941. 2. Gohen, Deris: A Method for Determining the Camber and Twist of a Surface to Support a Given Distri- bution of Lift. T.N. No. 855, NACA, 1942. 3. Knight, Montgomery, and Noyes, Richard W. : Span-Load Distribution as a Factor in Stability in Roll. Rep. No. 3 93, NACA, 1931. 4. Williams, D, H., and Halliday, A. S.: Experiments on Swept-back and Swept-f orward Aerofoils. R. & M. No. 1491, British A. R.C. , 1953. 5. Rossell, H. E., and Brand, C. L.: Swept Back Wings. Part VIIJ, Reports on Wind Tunnel Experiments in Aerodynamics. Smithsonian Misc. Coll., vol. 62, no. 4, 1916, pp. 55-75. 6. Knight, Montgomery, and Noyes,, Richard W.: Span Load Distribution on Two Monoplane Wing Models as Af- fected by Twist and Sweepoack. T.N. No. 346, NACA, 193C. 7. Wie so? sber ger , C.: Measurements on Wings with Sweep- back and Warping. (Trans, frcm Results of Aero- dynamic Test Plant at G-ottingen, vol. II, 1923.) Memo. rep. No. 123, War Dept . Air Service, McCook Pi eld, Ohio, 19 24. 8. Anderson, Raymond P.: Determination of the Character- istics cf Tapered Wing?. Rop. No. 572, NACA, 1936. 9. Anderson, Raymond P.: The Experimental and Calculated Characteristics of 22 Tapered Wings. Rep. No. 627, NACA, 193 8. IIACA Fig. 1 M o CD ft a tu 3 £ r. 10 o - £t ID £H II • •P <$ •H 00 +3 • cd rH II £ fn O u CO o -p c3 o •H • ■d (-1 d •H c! o CO .H o rH O -P -P o •H •p O r d O 3 d o •H -P a o m •H o o d >j-P O -P •H .H M -P CO O d a •h o d fs i-H CO o •H O Q to > = 'C I CO u •r) P=l f-. O o Digitized by the Internet Archive in 2011 with funding from University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundation http://www.archive.org/details/theoreticaldistrOOIang ttACA Fi, °o /o T-o - J 'uoiq.T?XTi.oj;TO Z^%°Ii jffSiffiK9n.pA 1 3 12 62 mwmf i SbstoSceubrary