1、,.-,/TECHNICAL NOTE 2699CALCULATION OF LIFT AND PITC IZING MOMENTS DUE TO ANGLE OFATTACK AND STEADY PITCHING VELOCITY AT SUPERSONIC SPEEDSFOR THIN SWEPTBACK TAPERED WINGS WITH STREAMWii3E TIPSAND SUPERSONIC LEADING AND TRAILING EDGESBy John C. Martin, Kenneth Margolis, and Isabella JeffreysLangley A
2、eronautical LaboratoryLangley Field, Va._= .- .,. - -. - .- . . . . .Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,- TECH LIBRARY K#WB, NM “Ii lllMllMlllllllfllunll,ID club5752 , NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS -.I!fEHNICAL NOTE 2699CALC
3、ULATION OF LIFT AND PITCHING MOMENTS DUE TO ANGLE OFATTACK AND STEADY :ITCHING VELOCITY AT SUPERSONIC SPEEDSFOR TEUN SWEPIBACKTAPERED WINGS WITH STRMMWISE TIPS,AND SUPERSONICLEADING AND TRAILING EDGESBy John C. Martin, Kenneth Margolis, and Isabella Jeffreysr suMMARYIOn the basis of linearized super
4、sonic-flowtheory the stability1 derivatives C% and Cmq (moment coefficients due to ane of attackand steady pitching velocty, respectively) and CLq (lift coefficientdue to steady pitching velocity) were derived for a series of thin swept-back tapered wings with stresawisetips supersonicleading andtra
5、iling edges. The results are valid for a range of Mach number for.- which the Mach lines from the-leading edge of the center section cut thetrailing edges. An additional limitation is that the foremost Mach ltie,.from either tip may not intersect the remote half of the wing.The results of the snalys
6、is are presented as a series of designcharts. Some illustrativevariations of the derivatives and of the chord-wise center-of-pressurelocation with the various wing design parametersare also included. To facilitate the transformation of the calculated results to arbi-trary moment-reference locations,
7、 the required data for ma have beenselected or computed from the charts and equations in NACA IN2114 andsre also presented in the form of design charts.INTRODUCTION The development of the linearized supersonic-flowtheory has enabledthe evaluation of stability derivatitiesfor a variety of wing config
8、ura-.tions at supersonic speeds. Fairly complete information is now availablefor the theoretical stabilityderivatives of rectangular, triangular, and-arrowhead plan forms (references1 to 6). For the sweptback tapered wingwith stresmwise tips, some of the available stability derivatives are theI. -+
9、- .- - -. - - - .- . . . . -Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-I . .2 NACA TN 2699lift-curve slope cczcL.E.free-stream Mach numberslope of leading edge (cotA)mr. =BcotAAI? local pressure difference between upper and lower surfacesof airf
10、oil; positive in sense of liftq steady pitching velocity- . . . .- . -.- . - - -.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,- -.- .4 NACA TN 2699s wing areaS3S4 sreas of integrationu incremental flight velocity along x stability aXisIx, Y, z forc
11、es parel to x, y, and z stability axes,”respectivelyX2 Y) z Cartesian coordinates (see fig. 2(a)l)xl Y1 coordinates of a source point in xy-planea) Ya Caxtesian coordinatesmeasured from leading edge of tip section(*a =x- )b on right half-wingm ; Ya Y da =d-Ldistance f?mm wing apex to center of press
12、ure due to angle of()c%attack -Fz 1Idistance from w3ng apex to center of pressure due to steady pitchingdistance from wing apex to assumed center-of-gravityposition(note that Z - d when expressed as a function of F isdefined as static msrgin) -angle of attackleading-edge This procedure is applicable
13、 formany steadymotions, as can be seen from the following arguments. Onlythe potential will be considered;hover, since the pressure is directlym?omortional to the x-derivative of the potential, the conclusions will.L.also apply to the pressure. From referehce 17, the.potential at wpoint (xjy) in reg
14、ion IV can be expressed asfj(x,y)= -:-JS4$. dxl dyl (2)d(x - X1)2 - B*(Y - y)2is indicated in figure 4. Similsrly, theThe area of integration S4potential in region III is givenby$(x,y)= - ; JS3The area of integration S3L dX1 dyl (3)ix - 1)2 - 2(Y - Y1)2is indicated in figure 5.Figure 6 indicates the
15、 effect on the area of integration S4 whenthe point (x,Y) is moved from region IV to region III. Note that thewing area inside the effective forward Mach cone from the point (x,Y) is .the same as the area S3. For-a point (x,y) in region III, the rightside of equation (2) can be written as -_ _-_._._
16、. _ . . . _ . . . _ .= _ _ . _ _ _. . _ . . JProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN 2699 9d-xldyl =The area S4 -Mach cone fromwill be purely- al x -X1)2 -B2(Y -yl)2l-rJS4-S3s is the$. dxlq (4)ti x- X1)2 - B2(y - yl)2portion of.S4 whi
17、ch is outside the forwardthe point (x,y). The integral over the area S4 - S3imaginary because the radicd- of the integrand is alwaysimaginary. Only the integral over the area s will contribute to thereal part of equation (4), and the integrsl over 53 is, by equation (3),the potential in region III.
18、Thus, the real part of the expression forthe potential of region IV will yield the expression for the potentialin region III as the point (x,y) is moved from region IV to region III.Analogous reasoning can be presented for regions I and II. The integra-tions were performed for the various regions an
19、d the resulting potentialsare presented in table IV. Application of equation (1) yielded thecorrespondingformulas for the pressures; these are presented in table V.Some illustrative chordwise and spanwisepressuredistributionsarepresented in figure 7 for the angle-of-attack case and in figure 8 forth
20、e steady-pitchingcase. Integration of the chordwise,pressuredistri-bution yields The spanwise loading (also obtainable by use of the pre-viously derived potentials evaluated at the trailing edge - see, forexample, reference 10,.equation(4). The circulation along the span Ijwhich is directly related
21、to the spanwise loading, is presented, forillustrativepurposes, in figures 9 and 10.thetheDerivation of Formulas for C CLq, and CmqThe derivati=s C, CLq, and Cm sre obtained by integratingqlifting pressure or the first moment of the lifting pressure overwing mea. These derivatives can be expressed a
22、s follows:. . . . . . . . . - -. .- - .- - - - -Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,- - .10c%=. J xACpdxdys4x*where AC = ;PNACA 2699(5)(6)(7)As indicated previously, the real pert of the expression for thepressure-differencecoefficient in
23、regionIV is also the expression forthe pressure-differencecoefficient for all other regions. Equations (5),(6), andwhere theretained,(7) c-, therefore, be written asCmq= -RI?. and (12) arepresented in the appendix. The formulas for these derivatives are alsogiven for the case where the leading edge
24、is sonic.The results of computations for cm sre presented in figures 11to 1, for CL in figures 16 to 20, and for Cm in figures 21 to 25.q The data are shown for a range of taper ratios,from O to 1.0 and for arange of the aspect-ratioparameter (A* =AB) from 3 t,o20. (CurvesforA1 = 2 are included for
25、the X = 1.0 cases.) The range of leading-edge-sweepbackangles is included in values of cot-% from 0 to 45.The dashed portions of the curves do not represent actual calculations,since these regions correspond to the condition where the Mach line fromthe wing apex intersectsthe tip. However, calculati
26、onswere madefor the sonic-leating-edgecondition and the dashed extensions of thecurves to these calculated end points thus, if a is a positivenumber,/( )2ta .aAlso, note thatI J?ormulasfor %“Sweptback leading edge, mt = 1 (sonic leading edge).-. . . - - - -. - Provided by IHSNot for ResaleNo reprodu
27、ction or networking permitted without license from IHS-,-,- . . .16For arbitrary taper ratio,aJ2#(32(1 - 4%)c% =k2 - 12Jk(k - 1) + J2(k - 1)2 J3(1 - k2)2,+32k3(l+ 21#)Lti-l J(l - k) + 2k a71JSti-1 + +J3(1 - #)2i14 2k -k2t4k + J - Jk60 2(k - 1) l-4(4k - 1)-J k - J15(1 - k)2(k + 1)5/%I2 7(2J-t4k)2 4(2
28、J - 4)2(1 - k) + 1(3J+4k+kJ) -k2J2 2r 130J2ti(k + 1)5/2.2(8-J)-(kJ-J-4k) r(J+4k - 32(4k - Jk + J)1-2kJ(k + 1) L 8Jfi(k + 1)3/2iii)(+ J+4k)2 m 1 and1At-l AY+l6%nkA (1+1.) (m -1)”-For arbitrary taperc%. -flJ=l2(k- 1)2-12kmJ(k-1)+ 48k2m17(A2)km(3k2-1) co-l +3(k2- 1)2 mlkm- J(k-l2fi (1-4mk2+ 2mk-3k)32 6
29、mfi(k-1)2(1+ m!c)lfiJ(3k+2mk2 +2m%+l)i=i 11-. . . . ._ _ - .- . _ - - .- -Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-. - .181For taper ratio of 1,41 2mr2(-3m + 10m2 - 4) o-l 1 ,4 *,2c% =- .A B 3fi(m2- 1)2ml ml 2 2+=+fi(mt - 1)Am(2m + 1) +(m+ l)m
30、2 -1Unswept leading edge (m = ).- .For arbitrary taper ratio,2J- 1)4(1- x)For taper ratio of 1,4-6Ac%= !B(A4)(A5),.M) a71 .,1.I . . I. ._ - -=. _ . . _ .Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-. . Formulas for C%. _ .-Sweptback leadng edge, m
31、t = 1 (sonic leadhg edge)h-%-For arbitrary taper ra%io,!$II c=%jI,3*4J2 + fL*;a+4k,+(k + U2;:l+)(,J.4)2- 2(U - 42J + 4k) - 4Jk; 42 -(Equation continued on next page)GIProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-. - 20 . NACATN 269996k-l J(l-k)+2k
32、1)ti-l - 32(2 - 5k2)sin +J3 l-i 2k - k k3J36c-(3kJ+7J +k8k)(4k+ J-W) 4k+J-kJm- l2J2J(5 - 3k)(k + 1) + kk(9k + 5) -J - 4k + 3H a(hk - kJ+J)-16?Z2J M(kJ+J+4k)2 ti-14k+J-3kJ2k2J2 ( 4k+J+kJ “+;For taper ratio of 1,16(A+ 7)(A+ 1)2os-lA 2 1 16cL =q 3TCAB 32 A+1 -E+(81A2+ 190A + 105)I(A7)(A8)I -_-lProvided
33、 by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-I1IIIIiIi,NACATN 2699knkSweptback leading edge, m 1 and A (1+ A)(m - 1)”-,For arbitrary taper ratio,iiCL = BITJ(A2+ X + l)k%(m2 - 1)3/2.2Jm2(-k4m4+mzkl- 2 -mlkz)(m2k2-1)(1 .-l+ , -.,k%12(m121# -l)(210n2+ m%3) +
34、2k5m6(k2 + 3) + m?k3(k4 - 10k2 - 15) +12m2)4+2+6) - 4k(3 - k2) Cos-1 -3(m2k2 - 1)2(1 - k2)2(#k2 .1)2 -1-12k%nt3(-4. (J(l-/12mYk - 7m2k + %3k) + J(-6 - hm + 8m3k +L,2 - 8m,k- 6k + 7m2# + 4m3#-, /(A9)3W(l+mk)Z /.,LProvided by IHSNot for ResaleNo reproduction or networking permitted without license fro
35、m IHS-,-,-. -I NACAForCL!l.TN 2699taper ratio of 1,a/.12 -21m16 + 8Z4 -.92rnt2 + 28 +Afi(nIt2- 1)3 9(m2 - 1)IIIIIIw(4km - k - 1)co2m+3)(m + 1)=1(A15) 2VzGz1ll)m126(m!2 - l)ii=l JAt(lz - 4 - 74) (m + 1) - mt (Jlm13 + 32m2 + 32mt)+90 -t90A(m + 1)2Am(13&ui3 + 2 - 257m - 183)(m + 1) + (m + I) - m(m + I)
36、($M!2 - 16) - I&?+$Q(m + 1)2A1 .,. . .rn12(-964m15 - )U9&n4+ E?71n3+ 97&n2 - 353u11- 1.72) stif(mfz- 1)2 . c_-_ . . . .Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-iTN 2699Unswept leading edge, m = CO.,-29I!Forc%arbitrary taper ratioj ,rIJ(J - 4)1
37、-Li_AB(A2+ A+ 1)2 J - 4(1 - L)3(1 -A)(J - 4)12=-(1 - X)2(J - 4)2 H(1 - A)3(J - 4)3 +- J-4 +4&4(l-k3 a -4(1-X)(J - 4)4 1+4- 4(1 - X14+ 4 -3(J - 4)2 (J - 4)3J4 J4- l- -1-E-4( -zl4 4 (,-4(,J J+ 8(1-X) +- xy+8(1-k) - - 4(1-xg2 +1)2(1 - x)2!(l - h) - y p-4(1 -A3 +2(1 - x)J- 4(1 - x) -1-.(A17)1,_ -. . . . .Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
