1、RESEARCH MEMORANDUM APPLICATION OF WING-BODY TIIEORY TO DRAG REDUCTION AT LOW SUPERSONIC SPEEDS By Barrett S. Baldwin, Jr., and Robert R. Dickey Ames Aeronautical Laboratory Moffett Field, Calif. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACARM
2、A-9 AEF!LICATION OF WING-BODY THEDRY TO IXRAG RELOWXCON AT mw SUPERSONIC SF= By Barrett S. Baldwin, Jr., and Robert R. Dickey J=-=Y 28, 1955 Page 10, lines ll and 14: Replace the expression f - . t NhCA-,-LangIey - S-4-65 - ZOO Provided by IHSNot for ResaleNo reproduction or networking permitted wit
3、hout license from IHS-,-,-c NACARM A54Jl.g i; -5 NATIONAL ADVISORY C however, ft was pointed out that, at higher supersonic Mach numbers, this modlffcation would sometfmes result in drags greater than that of the original config- uration. In reference 1 a method for contouring the fuselage of a wing
4、- b- Consider a wing-body cabination such as shown in sketch (a). Let x be the coordinate in the free-stream direction, y the spanwise coordi- nate, and z the remain- z ing Csztesisn coordinate HX in the thickss dfrec- tion, with the origin at the center of the body. AMachplane can be defined as a p
5、lane with its normal at 811 angle of tan-1(1/p) to the x axis. Let (xl,p,cp) denote the Mach plane which inter- sects the x axis at x and has the projection of its normal on the yz plane at an angle cp to the y axis. Let r* - S(x,fl,Q) be the area of the projection on the yz Sketch (a rt plane of th
6、e cross-eectfonal area intercepted on the configuration by the Mach plane (x,S,q). Then the drag of the configuration is the average with respect to cp of the drags of the equivalent bodies of revolution defined by the area distributfons S(x*,B,q). A method introduced in reference 4 is used in refer
7、ence 1 to evaluate the drag of each equivalent body of revolution. The variable 13 is defined by the relation X = k cos e 2 (1) where B is the length of the equivalent body. Then a set of quantities An(P,cp) are defined a8 the coefficients of sin n0 in a Fourier series expansion of Consequently the
8、An(l3,q) csn be determined from the relation An(P,cP) = $ s o as(x, ati(r -I ax* (2) Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-4 NACAHMA54Jlg Finally, the drag -of the configuration is given by (3) ” n=i Within the framework of the linear theor
9、y this result is valid only for equivalent bodies of revolution with no diecontinuities fn the gradienti of the area distributions. It should be noted that unless all parts of the configuration lie between the nose Mach cone and the forward Mach cone from the tafl, the equivalent body length, 2, wil
10、l be greater than the actual body length in scme cases. However, by consideration of streamwise body extensions of venishingly Bmall cross-sectional area, it can be seen that a constant value of 2 equal to or greater than the length of the longest equivalent body can be used In equation (1). Series-
11、expansion method.- In this section the Fourier series coef- ficfents defined in equation (2) till each be expanded in a finite series so that the drag formula can be expressed as a power series in powers of P* This manipulation leada convenient set of geometric plane method. to an expression of the
12、drag Fn terms of a parameters which were not apparent in theMach By the use of equation (l), equation (2) can be written as s z/2 as(x,p,ql) sinbe) at -z/2 ax sin 8 or after a partial integration * * i . + l .L .- - (4) provided that astxt, B,) ax* and S(x*,g,cp) are zero at the nose and t J;: c;i;:
13、 t(%Y)Yp,l ($Jm-p ax (13) ma0 Pro - P- a where yo( x) and y,(x) define the two edges of the configuration. The quantity in brackets izl equation (13) can be identified as the longitudtil disW.bution of the pth moment of area of the configuration. This Fndicates that the drag of the configura- tion c
14、an be expressed entirely In terms of moment dlstdbutions (lncludlng the area BlstrIbutim which correspcds to p E 0). The moment distrlbutlons can be defined as Mpk) = s Y,(X) t(x,Y)Yw (14) Y=-Y,(X) SubstitutFng thj.6 In equation (13) ylelds m AjW2 41“ Mp(x)($m-p ti BPcos% or lnferchengbg the or $ (;
15、)w2 4;: Mp(x)(+)-p dj =P as the desirea expansion of the An ,cp) Is In powers of f3. (15) (16) Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-8 NACARMA5kJVj - It can be seen in equation (14) that if the configuration has span- wise symmetry, the odd
16、 moment distributions will be identically zero, and the terms of equation (16) resulting from odd values of p will be zero. With streamwise symmetry of the moment distributions in addition, the odd values of n and m would be eliminated. In the process of substituting equation (16) into Jones drag eq
17、ua- tion, it is convenient to define several new symbols. . - c- _ .-.- Let -. dx (17) Y m=p so that n-2 An(Bt(P) = z - (-l)ppcosp bp P=o Then bb(p,cp I2 can be written as (18). * -L n-2 n-2 - .-. - An(P,cP) I” = cc i-11 p1+p2 hpl hp,cos .(pi+pj = Mob I - (+y32 (39) - *I n- ? , Provided by IHSNot fo
18、r ResaleNo reproduction or networking permitted without license from IHS-,-,-NACARMA5kJlg . - where MO(o) is the maximum value of the distribution. Then c and Mo( 0) L20 = 3 - 2 Mob 2 D = p ov2 b 1 15 (41) is obtained as the drag of the optknum configuration in the speed rsnge where p can be neglect
19、ed. Equations (39) snd (41) sre in agreement with the results of references 4 and 5. By substitution of equation (34) fnto equations (2l) and (!?7), it is found that I2 and I, will be zero for a configuration tith the opti- mum srea distribution and all other Iqs will be independent of the area dist
20、ribution. In that case equation (31) becomes $ c- -7 . 4.- -L where t(x,y) is the thickness distribution of the configuration includ- ing wing and b 9 npv2 M,y 1 2 2 + 3305 -ET xpv2 ya - Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-s NACA RM A+Jlg
21、 21 r bodies. of revolution mounted on the wing as shown in the second part of sketch (b). The arbitrarily chosen spsnwise locatfon of the auxilisry bodies determines their size in that small bodies at an outboard position can produce the same second-moment distribution as larger bodies a-tan Inboar
22、d position. It is etident that in order to prevent an ticrease ti the maximum value of the second moment of area the auxiliary bodies must be waisted in the ticinfty of the maxfmum thicluxess of the wing. The area distribution msy be made optimum by reshaping the body to satisfy the requirements of
23、the trsnsonic area rule after the auxilfary bodies have been added. *- In discussing the effects of modifications it is convenient to iso- late portions of the drag which will not be affected by the modifications under consideratfon. Considering pressure drag only, the quantity of pri- mary interest
24、 is the additional pressure drag caused by all additions to and alterations of the original body alone. The w5ng and auxiliary bodies are considered to be additions while the reshaping of the body is an alteration. Another reason for isolatfnF this additional pressure drag (4CD) is that the basic as
25、sumptions of the linear theory used to calculate 4CD for configurations with the transonic-srea-rule modification may not be tiolated, although the assumptions are violated at Mach numbers neer one for the body alone (see ref. 3). .F The additional pressure drag as just defined is obtained by cnm = zero otherwise Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
