1、cooI1.r,j/1NATIONAL (./ADVISORY COMMITTEEFOR AERONAUTICSTECHNICAL NOTE 2335A PLAN-FORM PARAMETER FOR CORRELATING CERTAINAERODYNAMIC CHARACTERISTICS OF SWEPT WINGSBy Franklin W. DiederichLangley Aeronautical LaboratoryLangley Field, Va.WashingtonApril 1951REPRODUCEDBYNATIONAL TECHNICALINFORMATION SER
2、VICEU. S. DEPARTMENTOFCOMMERCESPRINGFIELD,VA. 22161Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,- i_ _!_ ,_ ,Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NATIONAL ADVISORY CONNITTEE FOR AERONAUTI
3、CSTECHNICAL NOTE 2335A PLAN-FORM PARAMETER FOR CORRELATING CERTAINAERODYNAMIC CHARACTERISTICS OF SWEPT WINGSBy Franklin W. DiederichSUNNARYOn the basis of approximate expressions for the lift-curve slopeand the coefficient of damping in roll of swept wings at subsonicspeeds, the finite-span effects
4、on these aerodynamic characteristicsare shown to be functions primarily of a plan-form parameter, which isthe aspect ratio divided by the cosine of the sweep angle and by theratio of the section lift-curve slope to 2n. The use of this parameterin presenting concisely and in correlating certain aerod
5、ynamic charac-teristics and the limitations attendant upon such use are discussed.INTRODUCTIONThe conventionally defined geometric aspect ratio has long beenrecognized as a very convenient means for correlating, interpreting,and analyzing certain aerodynamic parameters of unswept wings.Specifically,
6、 aerodynamic parameters which depend primarily on theover-all level of pressures on the wing surface rather than on thedistribution of these pressures depend more on the geometric aspectratio than on any other geometric parameter. For swept wings the signif-icance of the aspect ratio is not obvious.
7、 In fact, neither the geometricaspect ratio nor any other known parameter associated with the geometryof the plan form serves to correlate aerodynamic parameters for sweptwings as readily as does the geometric aspect ratio for unswept wings.In this paper approximate expressions are derived for the l
8、ift-curve slope and for the coefficient of damping in roll of swept wingsin compressible subsonic flow. On the basis of these expressions, aplan-form parameter is defined which is a function of the aspect ratio,the sweep, and the section lift-curve slope. As is shown in this paper,this parameter aid
9、s in the correlation and interpretation of certainaerodynamic properties for swept wings in a manner similar to that ofthe aspect ratio for unswept wings.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-2 NACATN2335AbCD ic_CLCLclC_pFMSASYMBOLSaspectsp
10、an (wing tip to wing tip)wing induced-drag coefficientlift-curve slope of section perpendicular to leading edge orquarter-chord line at a Mach number equal to M cos Aper radianwing lift coefficientwing lift-curve slope, per radiancoefficient of damping in rollplan-form parameter (_ cosAA)free-stream
11、 Nach numberwing areasection-liftefficiencyfactorLangle of sweeptaper ratio ( Tip chord 1Root chord/ANALYSISLift-Curve SlopeIncompressible flow.- According to lifting-line theory, the lift-curve slope of an elliptic unswept win_ is given exactly and that ofmost other unswept wings, approximately by
12、the relationProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACATN 2335 3ACLn - A + 2N cga (i)where the section-lift efficiency factor _ is defined byczaT= (2)2_The lift on a section perpendicular to the leading edge of aninfinite swept wing, accordi
13、ng to the effective-velocity-componentconcept (reference i), is the same as that of a section of an infiniteunswept wing which has the same chord and section as those perpendicularto the leading edge of the swept wing, which is exposed to a free-streamvelocity equal to the component of the free-stre
14、am velocity perpendicularto the leading edge of the swept wing, and which is at an angle of attackequal to that of the swept wing relative to this component. As a resultof this concept, the lift-curve slope of a section of an infinite sweptwing parallel to the free-stream velocity is(c_alSwept = c_
15、cos A (3)so thatA_.)m = c cos ACL_ _where c_a is the lift-curve slope of the section perpendicular tosome swept reference line, such as the quarter-chord line, and whereis measured in planes parallel to the plane of symmetry.On the basis of reasoning concerning induction effects of sweptwings of fin
16、ite span which takes into account the results of lifting-line theory for unswept wings and the results of the effective-velocity-component concept for infinite swept wings, an approximate expressionfor the lift-curve slope of swept wings of finite span has been _ivenin reference 2:ACLa = A + 20 cos
17、A cza cos A (5)Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACATN 2335This expression indicates that the lift-curve slope of a swept wingdepends separately on the aspect ratio and the sweep. However,equation (5) may be rewritten in the formCLa FC
18、_a cos A F + 2(6)orCLa F + 2 Za Swept (7)where the plan-form parameter F is defined asF = A (8)cos AAs indicated by equation (7) the use of the plan-form factor serves toreduce equation (5) to an expression which depends only on the plan-form factor, rather than on the aspect ratio and sweep separat
19、ely,although the sweep is also contained implicitly in the section lift-curve slope of the swept wing.By comparing equations (i) and (7) the plan-form parameter is seento determine the lift-curve slope of swept wings in the same way as doesthe aspect ratio in the case of unswept wings. For this reas
20、on, andbecause the plan-form parameter reduces to the aspect ratio in the caseof unswept wings with a theoretical (incompressible-flow, thin-airfoil)section lift-curve slope of 2_, the plan-form parameter may convenientlybe regarded as an equivalent aspect ratio for certain purposes.Since equation (
21、i) and hence equations (5), (6), and (7) are basedon lifting-line theory, they are valid only for wings of moderate andhigh aspect ratios. At low aspect ratios they yield results that aretoo high. A modification to equation (i) based on lifting-surfacetheory for elliptic wings has been introduced in
22、 reference 3. With aProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACATN 233_generalized interpretation of the results of reference 3 for the caseof arbitary section lift-curve slope, the wing lift-curve slope of anunswept wing may be written asAI
23、C_ (9)CLaA 2In reference h this relation has been modified for swept wings.The expression for the lift-curve slope obtained in this manner givesgood results at low-subsonic speeds but does not yield the correctlimit for wings of very low aspect ratio as given in reference 5, thatis_lim CL_ : _ A (I0
24、)A-_O 2or, in terms of the plan-form parameter,lim CLa = ! FA-CO c_ cos A hHowever, if in equation (9) the value of the section lift-curve slopethat appears both as such and in the factor _ is corrected for sweepas in equation (3), equation (9) becomes= ACLa Cga cos A (Ii)A _/1 + hD2 + 2N cos AVwhic
25、h does go to the correct limits given by equations (h) and (i0).With the plan-form parameter defined by equation (8), equation (ii)may be rewritten either as= _ _ SweptCL_ F + 2Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-6 NACATN 2335analogous to
26、 equation (9) or asCL_ Fcos A ,_ Lc_a F + - + 2F2(13)analogous to equation (6). As in the case of equations (6) and (7),the expressions for the lift-curve slope given by equations (12)and (13) depend on the plan-form parameter F rather than the aspectratio and the sweep separately. The fact that the
27、y serve to reduce thelift-curve slopes of wings with widely varying aspect ratios and anglesof sweep to a single function is illustrated by figure l, which showsCLcthe lift-curve slope ratio of a wide variety of plan formsc_a cos Acalculated by the method of reference 6 as a function of the plan-for
28、mparameter F. Also shown in figure I are a few ponbs which correspondto lift-curve slopes measured in some of the wind-tunnel tests mentionedin references h and 7; these points are in good agreement with the linedefined by equation (13).Compressible subsonicl flow.- The lift-curve _slopes _iven by_e
29、quations (12) and (13) can be corrected for subsonic compressibilityeffects by means of the three-dimensional Glauert-Prandtl rule. Thisrule states that the pressures and forces on a thin wing at a low angleof attack in compressible subsonic flow may be calculated by multi-plying by i/_i- M2 the inc
30、ompressible-flow pressures and forces ona fictitious wing which is obtained from the actual wing by stretchingall its coordinates parallel to the free stream by the factoriI_ - N 2 and which is set at the same angle of attack as the actualwing. According to this rule, equation (12) becomesA_I - M2i
31、T cos A e- c _ cos A eCLc _f_ - 112 A _i N2 / hrl2CS2Ae- _i+ +2D cos A e A2(1 - 1!2 )where c_ is the incompressible-flow lift-curve slope of the sectionof the fictitious wing perpendicular to its leading edge orProvided by IHSNot for ResaleNo reproduction or networking permitted without license from
32、 IHS-,-,-NACA TN 2335 7quarter-chord line, where _ is based on this value of c_, and whereA e is the angle of sweepback of the fictitious swept wing, so thatitan Ae = tan A- M2and hence_I- M2cosA - cos A (15)e_l - M2cos2ABy combining the effective-velocity-component concept and the two-dimensional G
33、lauert-Prandtl rule the value(cza)Compre ssibleVI - M2cos2 A(c_a)Incompressible (16)may be obtained for the lift-curve slope of the section of a swept wingperpendicular to the leading edge or the quarter-chord line. (The sameresult may be obtained by an application of the three-dimensionalGlauert-Pr
34、andtl rule, except that _-1_llncompressible refers to thefictitious/_rather than the actual wing.) If the values of cos A eand of _)Compressible given by equations (15) and (16) are substi-tuted in equation (i_), equation (i_) reduces to equation (12). Conse-quently, equations (12) and (13) are vali
35、d for compressible subsonicflow, provided that the section lift-curve slope c_a , which entersinto equations (12) and (13) both directly and through the definitionsof _ and F, is that of the section of the actual wing perpendicularto its leading edge or quarter-chord line at a Mach number equal toM
36、cos A . If that value is unavailable, the quantity i_I- M2cos2Atimes the lift-curve slope of the same section in incompressible flowmay be used instead.In figure i several points represent lift-curve slopes measuredat Mach numbers in the vicinity of 0.7 in the tests mentioned inreferences _ and 7. T
37、hese points are close to the line defined byProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-8 NACA TN 2335equation (13) and follow the trend of the points representative of lift-curve slopes which were calculated theoretically or obtained in low-spee
38、d tests.Coefficient of Damping in RollThe coefficient of damping in roll may be obtained for swept wingsof moderate or high aspect ratio in incompressible flow from reference 2,modified for lifting-surface effects in a manner similar to that employedfor the lift-curve slope (see also reference h), a
39、nd corrected forcompressibility effects by means of the three-dimensional Glauert-Prandtlrule. The resulting expression may be written asorK2 F= - - _ (cZa)Swept+ 16 + hF F2(17)C _ K2p _ Fc_ cos a 8 ._ 16F +_+hVF2(18)where K is the ratio of the ordinate of the effective lateral centerof pressure in
40、roll to one-half the semispan and is equal to twice theYLterm P used in references 2 and h. The factor K varies betweenb/2about 0.92 and 1.09 as a function primarily of the taper ratio and, toa lesser extent, of the aspect ratio and the angle of sweep. Thevariation with aspect ratio can be expressed
41、 equally well as a variationwith the plan-form parameter F or the aspect ratio proper in view ofthe smallness of the aspect-ratio effect. As F approaches _ thefactor K approaches _/$ i + 3k for tapered wings or i for ellipticV3 i + kwings; as F approaches 0 the factor K approaches i for all planform
42、s.Equations (17) and (18) give values of which approach theC;pproper limit as F approaches 0 and _, that is,i I +3klim C - (cF-_ ;P 12 i + X _a/Swep tProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-2 NACA TN 2335 9andlim C gp AF-_O 32orF(See referenc
43、e 8.)As in the case of the wing lift-curve slope, the section lift-curve slope C_a that appears in equations (17) and (18) both directlyand through the term D in F is that of the section of the actualwing perpendicular to its leading edge or quarter-chord line at a Machnumber equal to M cos A . Also
44、, as in the case of the wing lift-curveC_pslope, the coefficient of damping in roll and the ratioC;a cos Aare functions only of the plan-form parameter F, except for the presenceof the term K2. The plan-form factor can therefore be used to plotthe coefficients of damping in roll of a wide variety of
45、 plan forms(with a given taper ratio) and a wide range of subsonic Mach numberson a single line.Induced DragThe drag of a wing is usually considered to consist of two parts,the profile drag and the induced drag. At low Mach numbers the profiledrag is largely independent of aspect ratio and sweep. Th
46、e induceddrag associated with a given lift distribution depends only on theaspect ratio of the wing; it may be expressed in the formCDi i + 6CL2 _A(19)where 6 is a positive number, usually small compared with I, whichdepends on the deviation of the spanwise lift distribution from anelliptical distri
47、bution.If the lift-curve slopes and coefficients of damping in roll fora variety of plan forms and Mach numbers are plotted in the form ofProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-t0 NACA TN 2335C_CL_ and Pc_a cos A cos A respectively, against
48、the plan-form param-c_eter F as suggested by equations (13) and (18), a parallel method ofpresenting information concerning the induced drag may be of interest.Such a method can be deduced by rearranging the terms of equation (19)to obtain the relationc ; cos Aa l+ nonetheless, adiscussion of the limitations of these equations serves to shed someiight on the applicability of the parameter F as well.The lift-curve slope given by equations (12) and (13) has beenderived from incompressible-flow lifting-line results with approximatelifting-surface correction
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