1、.-CORRECTIONTECHNICAL NOI E 3500OF ADDITIONAL SPAN LOADINGSWELSSINGER SEVEN-POINT METHOD FORTAPERED WINGS OF HIGH ASPECTCOMPUTED BY THEMODERATELYRATIOBy John DeYoung and Walter H. Barling, Jr.Ames Aeronautical LaboratoryMoffett Field, Calif.WashingtonJuly1955Provided by IHSNot for ResaleNo reproduct
2、ion or networking permitted without license from IHS-,-,-_ _TECIILIBRARYKAFB,M!I NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 111111111111-!lDbbb.?g.-.TECHNICAL NOTE 3500CORRECTION OF ADDITIONAL SPAN LOADINGSKISSINGER SEVEN-POINT METHOD FORTAPERED WINGS OF HIGH ASPECTCOMPUTED BY THEMODERATELYRATIOBy
3、John DeYoung and Walter H. Barling, Jr.suMMARYIt has been found that for wings combining high aspect ratio withlarge smounts of sweep, the Weissinger seven-point results are in error.A simple procedure is presented here which for a sizable range of plan. forms largely corrects these errors and resul
4、ts in more accurate spanloadings being read directly from the loading charts of NACA Report 921.This procedure consists of an alteration of the taper ratio used plus an4.additional correction applied at the wing root. In the above procedure,the lift-curve slope and the method of fairing the loading
5、are alsoimproved.The new results agree within *1 percent with theoretical resultsbelieved to be accurate; whereas maximum errors of the original Weissingerseven-point loading (for wings swept back 45) are approximately 2 and8 percent for aspect ratios equal to 3 and 10, respectively.INTRODUCTIONOf t
6、he several published methods for computing the span loading ofwings at subsonic speeds, the Weissinger “L” method with seven controlpoints across the span is one of the easier methods to use and, at onetime, appeared to afford the best compromise between labor and accacY.Solutions for many wings hav
7、e been plotted (ref. 1). The mathematicalcoefficients, avnj used in the four equations have also been presentedin graphical form for plan forms whose solutions are not plotted.Garner (ref. 2), Schneider (ref. 3), and others have indicated thatfor wings combininghigh aspect ratio with large sraountso
8、f sweepback,the seven-point loadings do not compare favorably with experimental resultsnor with theoretical results believed to be more accurate. The seven-point Weissinger loadings, C2C/CLCav, are generally too high outboard andtoo low inboard, as is illustrated in figure 1 (taken from ref. 3).Prov
9、ided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-2 NACA TN 3500The primary purpose of the present report is to find some simple,direct corrections to the solutions given in reference 1 for sweptbackwings of high aspect ratio.NOTATION!IRb=aspect ratio, coe
10、fficientsrelating the wing loading at station n to thedowmwash at station vwing span measured perpendicular to the plane of symmetry, ftlocal wing chord measured parallel to the plane of symmetry, f%average wing chord, ftlift coefficient,total liftClift-curve slope, per radian or degsection lift coe
11、fficientsection lift-curve slope, per radian or degclcspanwise loading coefficient or dimensionless circulation orrETconstants to be evaluated in theclc 2AGequal to CLca ,K=free-stresmMach numbernumber of terms used tnnumber of points acrossGuaspect-ratio-reductionfunctionsa numerical integration to
12、 obtain aVnthe wing span atdetermined and the boundary conditionscoefficient of q in the last term ofseriesfree-stresm dynamic pressure, lb/sq ftwhich the downwash issatisfied; also thethe trigonometric loading -.Reynolds number based on mean aerodynamic chordProvided by IHSNot for ResaleNo reproduc
13、tion or networking permitted without license from IHS-,-,-NACA TN 3500 3nv7wing area, sq ftfree-stresm velocity, ftsecwing angle of attack, radianscirculation, ft2/seca correction term to take into account errors in a four-termnumerical integration of span loadings, % to obtain liftdimensionless lat
14、eral coordinate Ysweep angle of the wing quarter-chord”linepositive for sweepbackdegsweep angle resulting from the Prandtl-Glauert transformation,% = an-=R%tip chordgeometricwing taper ratio, roo chordan effectivebeing reada correctiontaper ratio which results in more accurate answersfrom the loadin
15、g charts of reference 1term to increase lift in the vicinity of = Otrigonometric spanwise coordinate, cosz, radiansSubscriptsnumber pertaining to span station associated with the loadingVn = cos(nfi/m+ 1)number pertaining to span station associated with the downwashjVv = cos(vfi/m+ 1)values given by
16、 the procedure of referenceDEVELOPMENT OF CORRECTIONSOutline of Development1In the material to follow, the mathematical errors resulting fromseven-point solutionswill be approximately corrected by an effectivechange in wing taper ratio, the addition of a term to the loading at the., ,Provided by IHS
17、Not for ResaleNo reproduction or networking permitted without license from IHS-,-,-4wing root, and a correctiontobased on the easily determinedNACATN jQthe total lift. All correctionswill beerrors of the seven-pointmethod when.aspect ratio is infinite. Since for a given sweep angle the seven-pointso
18、lution gives correct values as A-O, a function which becomes mallas A diminishes is used to bring the corrections at A = cc down tofinite aspect ratios.The distribution of circulationparsmeter, czc/cav,will be correctedin two steps: (1) The taper ratio is altered so as to take into accountthe error
19、due to the M = 7 evaluation of the influence coefficients and(2) the loading at the root of a swept wing is multiplied bya factor totake into account an error that is largely due to the inability of anm = 7 trigonometric loading series to predict the loading at the root.Two corrective factors will b
20、e applied to the lift-curve slope,(1) a factor that takes into account the inability of a seven-pointnumerical integration to integrate the loading accurately, and (2) afactor that takes into account the added loading at the wing root due tothe above second correction. .With the distribution of the
21、circulationparameter ClC/Ca and thelift CL corrected,the loading coefficient /Czc CL.cav will be evaluated 4in terms of the above corrections and values read from the loading chartsaf reference 1. With the corrected loading at the four span stationsknown, the loading coefficient at other span statio
22、ns maybe found byusing the interpolationformula given as equation (A52) of reference 1.To overcome some shortcomings of this formula, simple additive correctionswill be developed. An equation for tke sPanwise center of pressure, cp,w5.11also be developed which will take into accountthe above correct
23、i.ans.The procedure by which the charts of reference 1 are used to obtainaccurate values is summarized in a section following the development. Nomarked increase of time or labor is involved in obtaining the more accurateresults over the ordinary usage of reference 1.madeCorrection of Distribution of
24、 CirculationCorrection at A = m.- If, with A = co,M in the Weissinger method isinfinitely large (i.e., the influence coefficients inteated exactly)while the method is developed primarily to apply to loading coeffi-cients given in charts in reference 1, it can also be applied to plan forms -eyond the
25、 scope of those charts by using values of the avn coefficientsin equation 1 of reference 1 correspondingto the wing with an alteredtaper ratio. .Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACATN 3500 5. then the loading due to angle of attack (“
26、additional”loading) is pre-dicted without error for 0. The result for M = w.CaCV/Ca = ( cos A)(C,/Ca) o0 a linear variation.? of loading related to the taper ratio.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-6 iTACATN 350064t Equation (1)2 0 Weis
27、singer 7-point valuex Equation (3)A=m, A=45; =0.45oo .2 .4 .6 .8 0 .It is seen that at q 0 the Weissinger loading values also have sm “almost linear distribution,but with less slope. The differencesbetweenthe Weissinger values at q 0 and the accurate values will be called theM error. It can be reduc
28、ed by using a taper ratio (in the Weissinger cal-culations)lower than the real geometric taper ratio. The ratio of tipvalue to root value of a line drawn equidistant from the Weissinger values(see sketch) divided intotaper ratio. This factortive taper ratio, he, fortics with regard to sweepcorrectio
29、n in simple formthe real taer ratio gives a factor fo reducingtimes.the wing taper ratio results in an effec-A = m. By limiting the plan-form characteris-and taper ratio, it is possible to present the(for A = W),Ie e 1 - 0.004(1 + ljAOA (4)forO.25 0 the loadingKn =is the distribtion at A = m. From e
30、quation (l),coefficient,Kn, (see list of symbols) reduces to()cP =.- +l+A) )7.1 -aV/n L+nfor straight-taperedwings. From equation (3), for q = O(c/ca14 2K4 =1+ (sinA)/2 = (1 + A)l + (sinA)/2(10a)(lObalso for straight-taperedwings. Such a loading distributionhas aninfinite loading gradient at the win
31、g tip and at the wing root; however,the aspect ratio would have to be extremely large to approach this con-dition. Here, in order not to demand too much from a seven-point integra-tion, the wing tip loading and the loading at the wing root will be roundedoff to approximate the behavfor of a wing of
32、A 20. At the wing tip,from = 0.8315 to 1.0000, the loading is assumed of the formK = alsinq + a3sin 3cp (11)where al and as are determinedby the requirement that the loading andloading slope equal that givenby equation (loaj at = 0.831.5. At thewing root, from = O to 0.1951, the loading is assumed o
33、f the formK =blsincp + b3sin + bsin (12where bl, b, and bs are determinedby the requirements that the loadingat q = O and 0.1951 and that the slope of the loading at ?l= 0.1951equal values given by equation (loo). The final distribution of loadingwith the rounding off at the tip and ?sootis shown in
34、 the followingsketch:.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACATN 3500 9.,.LoadingrEquations (10),= 0.1602 + 1.5806 I+A 70 .1951 .8315 LoSince the loading is known analytically, the area beneath the curvecan be determined exactly. The area
35、 is2.0028 + 1.8692A+ 0.8547 sin A+ 0.9346A sin A(1+X)(2+ sin A)It is convenient to raise the loading values so that the curve has unitarea by dividing by the above expression. The seven-point integrationformula of reference 1 can now be applied aud the errors of the formulacan be determined. The sev
36、en-point integration formula is (from ref. 1)(13)The ratio of the seven-point value to the exact value of c may bewritten(CLG)7(CIJ =% (0.7654Kl+ 1.4142K2+ 1.8474KS+ K.)exact(14)Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-10 NACATN 3500where the
37、Kis are the previously determined loading values with unitarea. Equation (14) does not yield unit area. The discrepancy will becalled the integration error, y, orcLa = (1 + 7)(CLJ7The quantity, 7, is virtually independent of taper ratio in the range0.25 A 0.75 and can be given as simplyWhile the abo
38、veerror at Am 20, iting the aspect-ratioMulthopp results forfollowing expressionsinAY=value is probably more representativeis convenient to take this value for.(16)of inteationA=x. Apy-function, 1 and setting k to yield the1+ (k/A)2 the Afor 7= 8 wing mentioned previously,at arbitrary aspect ratioss
39、in A201 ,(*yyields the(17)Correction for added root loading.- The product of czc/cav and agives the increase of root section loading due to the root correction.The increment of CLUFor the range of planventional taper), (C+e (21)4 of reference 1, y is givenbyequation (17, and o is given by equation (
40、9).Corrected Loading Coefficient and Interpolation.Corrected loading coefficient.-The loading coefficient,7 K = (czC/cLcs,v)n determined, say, from the charts of reference 1 byusing the effective taper ratio (eq. (8) and the root correction(eq. (9) will need a final correction since the total lift c
41、oefficient,CLa, has changed (eq. (21). The Knts become(Kn7)heKn = 1+7+ (0/5)1 (22)(1 + m+eK4 =1+ y+ (cT/5) JInterpolation.- The seven-point interpolation, equation (A51) ofreference 1, of loading at span stationsbetween the four known loadingvalues, gives the loading distribution which the seven-poi
42、nt numericalintegrationformula integrates when obtaining total lift. Thus the errorin interpolating is related to the error when integrating. The values ofKn for the preceding rounded-off loading assumed for A = w can beinterpolatedby equation (A52) of”reference 1. The differences of theinterpolated
43、from the accurate loading were found to be nearly directlyproportional to 7m (given in eq. (16). The effect of taper ratio is. nil and is omitted. Thus the increments of loading to be added to inter-polated results at q = 0.1951, 0.5556, and 0.8315, were found to be.Provided by IHSNot for ResaleNo r
44、eproduction or networking permitted without license from IHS-,-,-12 NACATN 3500%/2 = 5.157m4/2 = -1”557W1(23)%/2 = l*03ymrespectively. No reliable value of AK2is taken as zero.If the interpolation error is assumedin the same manner as the lift integrationaspect ratioswhich gives the correction7/2 =
45、5.157%5/2 = -1.557K3/2 = 1.03yAK2* Ocould be found and AK2to decrease with aspect ratioerror, 7,then at finiteIIto be added to interpolated loadings atv = 0.1951, 0.5556, 0.8315, and 0.9808, respectively.Spanwise Center of PressureThe spanwise center of pressure is givenpl. o(24)(25) (ctc/cav)dqq=oL
46、etting = cos q and multiplying the numerator and the denominatorbY cav/2b givesJd2G(cp)sin(pcosq) dcpncp = oJd.!?G(q)sinq oProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-13.NACATN 3500By use of theintegrals caninterpolatedthe spanwisevpwhere the Klo
47、ading seriesbe evaluated.given by equation (A17, reference 1, theTaking m = 15, and using the seven-pointvalues of loading correctedby-equation (24, we obtain forcenter of pressure0.352K1+0.503K2 + 0.344K + 0.041K4 + 0,364Y=(27)0.383K1+ 0.07K2 + 0.924K3 + 0.500K4 + 2.155Yvalues are for unit area.SUM
48、MARY OF PROCEDUREFor wings with taper ratios, X, between Q.25 and 0.75 and whose-quarter-chord lines are swept back between 30” and 80U,= the followingparameters are evaluatedFrom eitherof equation 10.004(1+ A)AO 1-1+ (7/A)2sinA201+ (11.7/A)2(1.75 - Ael3)sin A1 + (2h/A)2(2 + sin A)the loading charts of reference 1 or simultaneous(1) of reference 1, the loading coefficients K%solutionare obtainedfor the wing parameters A, A, and e. The Kn7 values are thenby correction factors a
