REG NACA-TR-865-1947 Method for calculating wing characteristics by lifting-line theory using nonlinear section lift data.pdf

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1、FOR AERONAUTICS REPORT No. 865 _. :- : I METHODFORCA$ULiiTING$VICHAiACTiSTi , - I G !z . . L. . , D . DO iaoumc sm5oLs -, _ 1. FUNDAMENTAL AND Dl!WD UNITS . Meti Y-. ,jbgliak - symbo! : _ unit / ._ YEiT - foot (or mile)-,: _ .- _ eeoond (or. hour)-,-y, sea (or hr) -Fore _-_ weight of 1 kilogram-,- k

2、g weight of l-pound- lb , - - -. sayer _ , P - homepower (met or 0.002378 lb-ftA sd .- Momen! of .mertia-mk. Indicate axis oft - Specific weight of “stqlard” air, I,2255 /ma -or_ 9.07651 Ib/cu ft radius of gyration-k by propersubtiript.) / Coefhcient of -viscosity , : -. _ - ; .,-I- , . ., .a; - _ S

3、p&;n -r ,. -. , , , 2,., ., , METHOD FOR CALCULATING WING CHARACTERISTICS 3 equal to four times the corresponding coefficient in reference 5. The induced angle of attack (in degrees) at a point y1 on the lifting line is - 180 b S b/Z d ffi=- - u 87r -b/2 dy dy (2) Y1-Y This integral (in different no

4、menclature) was given by Prandtl in reference 6. If equation (1) is substituted into equation (2) and the variable is changed from y to 0, the induced angle of attack at the general point 0 becomes, according to reference 5, CY= and -yml;, respectively, for r=20. Similar tables for go AWLI; and goym

5、n- are given in TABLE I.-INDUCED-ANGLE-OF-ATT.4CK MULTIPLIERS pmk FOR ASYMMETRICAL LIFT DISTRIBUTIOKS %=,z,( y),“?“k 2.4 IF -0.9877 -n. 9511 -0.8910 -0. 8090 -0. 70il -0.5878 -Il. 4540 -0.3OJo -0.1564 0 211 _- -_- -_ -. _- _- IL I; m 19 18 17 16 15 14 13 12 11 10 -0.9877 19 915 651 -166.985 0 -7.019

6、 0 -1.401 0 -0.486 0 -0.230 1 O.%i7 _-_-_-_ _ _- -.- - -. 9511 18 -329.859 463.533 -122.749 0 -7.438 0 -1.792 0 -. 701 0 2 .9511 _ -_- _- -_- -. 8910 17 0 -180.336 315.512 -96. i3i 0 -i. Oi3 0 -1.920 0 -_ 819 3 .a910 - _ -_-_ _- -. 8090 16 -26.374 0 -125.246 243.694 -81.067 0 -6.680 0 -1.97i 0 4 .a0

7、90 -_-_- _- _- -_- _- -. 7071 15 0 -17.020 0 -9;. 524 202.571 -71.139 0 -6.391 0 -2. n2G _- _-_- _ -_ 5878 14 -i. 246 0 -12.604 0 -81.392 lii. 054 -64. i35 0 -G. 228 0 - _- _ -_ -_ _- -_ 4540 13 0 -5.166 0 -10.126 0 -il. 296 160.761 -fiO. i25 0 -6.192 7 4540 - -_- - - _ - - _- -. 3090 12 -2.956 0 -4

8、.022 0 -x. 396 0 -fvl. Bli 150.611 -58.514 0 8 .309G _-_ -_- -_-_ _-_ -_ 1564 11 0 -2. 241 0 -3.322 0 -i. 604 0 -60. iG8 145.025 -ST. 812 9 15R4 _ _- - -_ -_-_-_-_ -_- 0 10 -1.4m 0 -1.804 0 -2.865 0 -6.950 0 -58.533 143.239 10 0 _- -_- _-_ _-_- _-_-_-_-_-_ .1564 9 0 -1.153 0 -1.518 0 -2.554 0 -6.530

9、 0 -5i. 812 11 -. 1564 _- - -_- -_ _- _- .3090 8 -. 810 0 -.946 0 -1.319 0 -2.340 0 -6.288 0 12 -. 3090 _-_- - - - _ -_- .4540 i 0 -. G4G 0 -.son 0 -1. Ii6 0 -2.192 0 -6.192 13 -_ 4540 _- - -_- -_ _ _- .5878 6 -.46i 0 -. 530 0 -. 691 0 -1.068 0 -2.092 0 14 -. 5Si8 - -_- _- - -_-_ _ -_- ion 5 0 -_ 36

10、8 0 -_ 441 0 -.GO4 0 -.981 0 -2.02G 15 -_ iOi1 _-_-_-_ _- _- _ 1 2 3 4 _- _-_-_-_-_.-_ .98i7 .9511 .a910 .809G .iOll ) 6.S8i8 1 7.4540) .iOLa) .1564 ?fFT, 1 Vslues of I: at top to hr used with values of m at left side; valurs of I: at bottom to be used with values 01 m at right side. Provided by IHS

11、Not for ResaleNo reproduction or networking permitted without license from IHS-,-,- m Im nm Cm Cm.3 ! -_ IL- _ -0.9877 -_ 9511 :i -. 8910 li -.8090 -. 7071 : -.6878 14 -. 4540 13 -. 3090 12 -. 1564 0 :i : :ii 8” .4540 .5878 i .7071 5 .809a .8910 i .9511 2 .98ii 1 where u -22a, ma- Values of rlrnr vr

12、ns, urn, and uma are given in table III for r=20. 0.07854 .I5515 .I4939 .139+6 .12708 :Ei .07131 .04854 .02457 -0.00607 -. 01154 -. 01589 -. 01867 -. 01964 -. 0186i -. 01589 -.01154 -. c?!ml7 0 .00607 .01154 .01589 .01867 .01964 .01867 .01589 .01154 .00607 0 .01214 .02308 .03177 .03735 .03927 .03735

13、 .0317i .02308 .01214 - Wing lift coefficient.-The wing lift coefficient is obtained by means of a spanwise integration of the lift distribution, For asymmetrical lift distributions (15a) For symmetrical lift distributions (15b) Induced-drag coefficient.-The section induced-drag co- efficient is equ

14、al to the product of the section lift coefficient and the induced angle of attack in radians, The wing induced-drag coefficient is obtained by means of a spanwise integration of the section induced-drag coefficient multiplied by the local chord, S b/2 cDi=i -b/2 180 =lccv dy A CT ;l!?f!=!d() S Provi

15、ded by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,- .:$ETHOD FOR CALCULATING WtVING CHARACTERISTICS .J 7 For asymmetrical lift distributions For asymmetrical lift distributions For symmetrical lift distributions (16b) ProAle:drag coefficient.-The section pro

16、file-drag coefh- cient can be obtained from section data for the appropriate airfoil section and local Reynolds number. For each span- wise station the profile-drag coefficient is read at the section lift coefficient previously determined. :. The wing profile- drag coefficient is then obtained by me

17、ans of a span -b,2 Cd,0 dy For asymmetrical lift distributions : (174 For symmetrical lift distributions cDQ=mzl (Cd, ;) qrns m , (17b) Pitching-moment coefficient.-The section pitching-moment coefficient about its quarter-chord point can be obtained from section data for the appropriate airfdil sec

18、tion 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 -Icl sin (C%-cr()-c+, cos () may usually be neglected. The wing pitching-moment

19、coefficient is obtained by the spanwise integration L .,756-494 . ” .(. ,_ /.,. -:“.,y- ., . _ . j; ; ., _. : : :. .C- For symmetrical lift distributions Wb) Rolling-moment coefficient.-The rolling-moment coeffi- cient is obtained by means of a spanwise integration (204 For an antisymmetrical lift d

20、istribution Induced-yawing-moment coefficient.-The induced- yawing-moment coefficient is due to the moment of the induced-drag distribution, A S CIc rfft 3, d 2y - =4 -1 b 180 b 0 7 (21) The induced-yawing-moment coefficient for an antisymmet- rical lift distribution is equal to zero and has little

21、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 mome

22、nt of the profile-drag distribution, .C -A S b2 $ dy nQ sb-b/z _. : .- =!$lT 7 d ) _, (22) APLICATIONOFMETHODUSINNONLINEARSECTIONJFT 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 symmetri

23、cal lift distributions are considered hereinafter inasmuch as these are believed to b-e sufllcient for illustrating . ( (%8) and Y (S,) The y (LlQ 0 distribution is the additional lift distribu- tion corresponding to a wing lift coeflicient C, ) determined s in table IX through the use of the multip

24、liers vms. It is usually convenient to use the additional lift distribution C,j2 - - b corresponding to a wing lift. coe5cient of unity. This distribution is found by dividing the values of 2 ( 6 (aas) by c”(mas,. The * ( b (*r3 distribution is a combination of the basic lift distribution and an add

25、itional lift distribution corre- sponding to,a wing lift coe5cient CL (13 also determined in _-. -. table IX. The basic lift distribution CT is then determined ._._._ -. -_ . by subtracting the additional lift distribution y CL(tl,) Inasmuch as the wing lift curve is assumed to be linear, it is defi

26、ned by its slope and angle of attack for zero lift which are also found in table IX. The maximum wing lift coefficient is estimated according to the method of refer- ence 10 which is illustrated in figure 4. The maximum lift coe5cient is considered to be the wing lift coe5cient at which some section

27、 of the wing becomes the first to reach its maximum lift, that is, clb+CL c=c,. This value of CL is most conveniently determined by finding the minimum value of Czmaz-CzO . “al along the span as illustrated in table IX. -. I INDUCED-DRAG COEFFICIENT The section induced-drag coefficient is equal to t

28、he prod- uct of the section lift coefficient and the induced angle of attack in radians. The lift distribution for any wing lift coe5cient is _ *: (23) The corresponding induced angle of attack distribution may Il 1 /,Ti c 1.55 1.12 .91 .al 1 .62 .70 .99 I 1.37 I- TABLE VI.-CALCULATION OF WING COEFF

29、ICIENTS FOR EXAMPLE WING A=10.05; a.=3.00 .m - .224 - w CIC Multipliers 7 57.3+c (CT., 1 0 1 - 0 -65.803 .01138 .0981 - .01023 .OOOO I.C .1564- Fl .,o .3090- 2 a .4540- R e .5878- &; T-xiT- 03: - a .a : .8090 E. _- E .a910 / .105-i .0984 I .0893 I .0811 I .0722 - .0632 -/- .0534 (TiEiF/- I .0411 01

30、-1.491 1 0 1 -7.089 1 0 I-167.045 1 915.651 S+ Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-TABLE VIIT.-CALCULATION OF LIFT DISTRIBUTION FOR EXAMPLE WING FOR a.,=0 Multipliers A, Check - -72.472 0 -10.926 0 -.oG97 , 0 - - - -_- _ .8090 -2.138 -. 0

31、125 0 -2.880 0 -7.208 0 -81.434 243.694 -125.537 0 -26.635 - -p-pppp - .8910 -2.604 -.0124 -1.638 0 -2.371 0 -7.370 0 -96.962 315.512 -180.528 - _, .9511 -3.013 -1.456 -1.557 -67.157 I-.-9.916 0 _- -62.917 160.761 -72.472 0 -10.926 0 -5.812 0 -.095 -.784 -.0742 .01023 -.a77 -_-_.- _- -65.803 177.054

32、 -82.083 0 -13.134 0 -7.713 - -.a7 -1.028 -. 1019 .lwJ17 -.COQ4 - _ _- -71.743 202.571 -97.965 0 -17.388 0 -.331 -1.339 -.133 ar.,=lO.OO; y=-3.901 0 0.07854 0.1102 0.1323 -0.0029 -0.0105 0.0076 0.1429 0.926 0.053 1.421 1.477 - -_ -_- -_ _-_ _-_ .1564 .15515 .1057 .1269 -.a040 -. 0100 .0060 .I295 .9s

33、o ,046 1.418 I. 400 _-_- _ -_- _ _- - .3090 .14939 .0984 .1181 -.0057 -. 0093 .0036 .1161 1.015 .031 1.423 1.371 _-_ _ _- - _- .4540 .13996 .0899 .I079 -.0077 - 0085 .0008 .I040 1.038 INJS 1.432 1.372 - _- _- -_ _ _ - .5878 .12708 .0811 .0974 -. 0096 -.0077 -.a019 .0925 1.053 -.a21 1.441 1.388 _- -_

34、 .7071-.liiF .0722 .086i -. 0111 -.OOBS -.0043 .0823 1.053 -.a51 1.436 1.412 _- -_-_- .8090 .09233 .0632 oi59 -.0121 -.0060 -.0061 .a735 1.033 083 1.418 1.453 _- .8910 .07131 .0534 .0641 -. 0120 -. 0051 -. 0069 0665 964 104 1.404 1.564 _ _ _ _ _ -_-_ -_- - _- _- - I_ _- .9511 .04854 .a411 .0493 -.01

35、04 -. 0039 -.0065 .0613 ,804 -.106 1.419 1.897 _ _ _ _ _-_ -_-_ -_- - - - - -;- - .9877 .02457 .0232 .0279 -.0063 -.0022 -.OOIl .013i ,635 -. 094 cL (“5) =Az(xO)=o.sar, CL ,e,) =AZ(X) = -0.Oi9 a=cI.0?Lo,n,33 -cr. () = _- =o.gj aQ*( L-0) cl *a, CL,“, =Min. value in =1.37 ,(L-O, =q*+% (,-;! =-2.95 TAB

36、LE X.-CALCULATION OF ISDUCED-DRAG COEFFICIEXT FOR ESAlIPLE WISG A= 10.05; c+J =o.s33; CI.(.,) = -0.079 aid CLJlC 2Y Multipliers Ui(“Q i(,.) TJq,.) ai* 7 a nma (Table VII) (Table VIII) (XCq,/,) (Q-O) (Table IX) -_ ,_ 0.07854 / 2.060 0.1323 .15515 I 1.602 -.-12fi9-. .3090 .4540 .!a78 0.3273 _I o.Oi24

37、0.0031 .2442 I .0416 .0014 .1952 .0225 .0005 _-_ - _- 1559 .0032 n - _ -_- .7071 .11107 1.218 1. 4w -.331 -.I16 -.215 .086i -_ _ _- -_- - - .8090 .09233 1.492 1.792 -.550 -.142 -.408 -(- .n759 _ _-_-_- .8910 .07131 2.111 2.535 -.830 -.200 -.630 .0641 _- - -_ .9511 .04854 3.399 4.081 -1.351 -.322 -1.

38、029 -_I_ .n493 - _ -_ -.- .9877 .02457 4.840 5.812 -1.915 -.459 ) -1.456 j-=;-r-.0041- .0008 - -. 0019 -_ -.0042 _. -.0061 -_ - .0069 .-_ -.0065 - _-_ ,1359 -. 0121 .0002 -iii- -.0248 / .0009 _- -_- .I625 j -.05i9 .0013 .0067 .2012 ) -.0X3 .1622 ) -.0645 1 _ .0060 1 I I c”;=()cL2+()cL+* =o.n322cL*-n

39、.noo3cL+o.ooo3 - The values of atal and or+, are determined in table X in the where same manner as CT and 7 in table IX. cd i, cl,lc ia The induced-drag -=-b- g7x b (26) (Table IX) O.OOi6 .0060 -_. .0036 distribution is therefore cd .c czc at -L- - b - b 57.3 and CdibC C!,C ffi* -=- _ b b 57.3 The c

40、alculation of each of these induced-drag distributions is illustrated in table X together with the numerical inte- (25) gration of each distribution to obta.in the wing induced-drag coefficient. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-METHOD

41、FOR CALCULATING WING CHARACTERISTICS 17 FIGURE 4.-Estimation of CLmaz for esmqle wing. (CL,“. estimated to hc 1.37.) PROFILE-DRAG AND PITCHING-MOMENT COEFFICIENTS The profile-drag and pitching-moment coefficients for the wing depend directly upon the section data and therefore their calculation is t

42、he same whether linear or nonlinear section lift data are used. For the linear case the section lift coefficient is C1=CIa,L+Clb for any wing coefficient CL. By use of this value for c2 the profile-drag and pitching-moment coefficients are found as in table VI. DISCUSSION The characteristics of thre

43、e wings with symmetrical lift clistributions have been calculated by use of both nonlinear and linear section lift data and are presented in figure 5 together with experimental results. These data were taken from reference 11. The lift curves calculated. by use of non- linear section lift data are i

44、n close agreement with the experimental results over the entire range of lift coefficient, whereas those calculated by use of linear section lift data are in agreement only over. the linear portions of the curves as would be expected. It must be remembered that the methods presented are subject to t

45、he limitations of lifting-line theory upon which the methods are based; therefore, the close agreement shown in figure 5 should not be expected for wings of low aspect ratio or large sweep. The use of the edge-velocity factor more or less compensates for some of the effects of aspect ratio and, in f

46、act, appears to overcompensate 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 avail- able for wings with asymmetrical lift distributions. Such data arc very desirable in order to determine the reliability with which calculated data may be used to predict experi- mental wing characteristics. LANGLEY MEMORIAL AERONAUTICAL LABORA