1、NATIONAL. ADVISORY COMMITTEE FOR AERONAUTICS REPORT No. 921 THEORETICAL SYMMETRIC SPAN LOADING AT SUBSONIC SPEEDS FOR WINGS HAVING ARBITRARY PLAN FORM By JOHN DeYOUNG and CHARLES W. HARPER 1948 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-AEXONAUT
2、IC SYMBOLS W Q m .l cc S SW G 6 G. A V P L D . 0. DC D. 0 1. FUNDAMENTAL AND DERIVED UNITS Metric English Symbol unit Abbrevia- unit Abbrevis- tion ,tion - Length- 1 Time _-_ I Force _ F Power -_ P Speed- V meter _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ second- _ _- _ weight of 1 kilogrrtm- _ - m foot (o
3、r mile) _-._ ft (or mi) second (or hour)-,- see (or hr) or 0.00237s lb-ftw4 sec2 Momen! of inertia=mk?. (Indicate axis of SpecXc weight of %tandard” air, 1.2255 kg/m* or 0.07651 lb/cu ft radius of gyration k by proper subscript.) Coefficient of viscosity 3. AERODYNAMIC SYMBOLS Area .4rea of wing Gap
4、 Span Chord Aspect ratio, g . . True air speed . Dynarmc pressure, 3 5 ,. Lift, absolute coefficient C or for an airfoil of 1.0 m chord, 100 mps, the corresponding Reynolds number is 6,865,OOO) Angle of attack Angle of downwash Angle of attack, infinite aspect ratio Angle of attack, induced Angle of
5、 attack, absolute (measured f: om zero- lift position) Flight-path angle Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-REPORT No. 921 THEORETICAL SYMMETRIC SPAN LOADING AT SUBSONIC SPEEDS FOR WINGS HAVING ARBITRARY PLAN FORM By JOHN DeYOUNG and CHA
6、RLES W. HARPER Ames Aeronautical Laboratory Moffett Field, Calif. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-National Advisory Committee for Aeronautics Headpuarter8,I724 P Street NW., Washinqton F ( “(or angle of attack of the plate 3 CU,) indu
7、ced by the total downwash at each control point Y, (2) the load coefficient ( “9 a,=% of the lifting line at each span- wise point n on the quarter-chord line, and (3) influence coefficients A., which relate the influence of the circulation at any point n to the downwash at any point v and are a fun
8、ction of wing geometry only. The method shows that for an arbitrary loading the equations have the following form: av=AvnGn, v=1,2, . . . m 7Z=l 2 It is important for the render to realize here that B choice has been mode between B number of possible procedures. These possibilities arise from the fa
9、ct that the exact location, on a tapered wing, of constant-percent-chord lines depends upon the orientation of the reference line along which the chord is measured. The orientation of the reference line is usually chosen such that an airfoil section so defined will have aerodynamic characteristics c
10、losely resembling those found two-dimensionally for the same section. This then enables an esti- mation of the effect of changes in section characteristics on over-all wing characteristics. For unswept wings, there is little reason to consider my orientation of the reference line other than parallel
11、 to the free-stream direction. However. when a wing is swept. the question of orientstfon of the reference chord cannot be so easily answered. Insufficient experimental data exists to determine the most sntisfnctory orientation. and strong nrgumenti can be presented for at least two orientations, na
12、mely. pnrallel to the free streamand perpendicular to some swept referenca line. In the present analysis the reference chord was chosen as being parallel to the free stream since it greatly slmpll5es the mathematical procedure and since wnsideretion of the differences expected from use of the altern
13、ate choice indicates they will be small. The render should note that the boundary condition is given by UJ,= V, sin (I., from which (G), is seen equal to sin m The substitutio6 of P, for sin (I. has the effect of increasing the value of loading on the wing above that necessary to satisfy the boundar
14、y condition. However, the boundary condition was fixed assuming that the shed vortices moved downstream in the extended chord plane. A more realistic picture is obtain if the vortices ore assumed to move downstream ln a horizontal plane from the wing trailing edge. It am be readily seen that, if thi
15、s occurs, the normnl component of velocity induced by the trails at the three-quarter-chord line is reduced and, if the boundary condition is to continue to be saCsBed, the strength of the bound vortex m.ust increase. It follows that substitution of PY for sin LII then has the effect of accounting f
16、or t,he bend up of the trailing vortices. It is not known how exact the correction is, but calculations and experimental veri5ution show it of to be the correct order. Each equation gives the downwash angle at the control point v, the spanwise location of which is defined by t=cos y where the downwa
17、sh results from the circulation at m points n on the wing the spanwise locat,ions of which are defined also by vj=cos 7 In the case of symmetrically loaded wings, each panel pro- duces an identical equation for the corresponding semispan point. Since only one of these identical equations is of value
18、, the total series reduces to the equations correspond- ing to the wing midpoint and one panel,. For the seven- point solution, equation (1) is therefore written av= 5 a, G, v=1,2,3,4 n=1 where a”, represents the influence coefficients for the sym- metric seven-point solution. The set of four simult
19、aneous equations so formed can be easily solved to obtain the dis- tribution of total load (in terms of G,) on any wing for which the angle of attack at each spanwise station, sweep, and chord distribution are specified. The .distribution of load is specified at only four spanwise stations, namely 7
20、=0.924, 0.707, 0.383, and 0 (n=l, 2, 3, and 4, respectively). Values of loading at additional spanwise stations can be found by means of the interpolation function given in the appendix (equation (A52). The simplicity of the procedure depends to a largeextent on the fact that the solution can be fou
21、nd in terms of the coefficients avn. Even where these must be computed for each wing plan form the method offers computational ad- vantages over other equally accurate methods. However, because these avn coefficients are a function of geometry alone, it is possible to relate them in a simple manner
22、,such that a limited amount of comput(ation will give the a”, coefficients for all plan forms to which the method is applicable. Details of this procedure and the results of applying it are discussed in a later sect.ion of the report. The method assumes that the flow follows the wing surface and mak
23、es some allowance for the trailing sheet aft of the trailing edge becoming horizontal.3 Hence, the method should apply to higher angles of attack with considerable accuracJi, provided the flow remains along the wing surface. The method assumes incompressible flow but it will be shown how the effects
24、 of compressibility can be included within the limits of applicability of the Prandtl-Glauert rule. The method assumes the. theoretical section lift-curve slope of 2s (or with account taken of compressibility, 27r/p) but apro- cedure will be shown which accounts for the variation in section lift-cur
25、ve slope from the theoretical value. It is clear from the foregoing outline of the theory that the method can account for variations in those geometrical char- J see footnote 5. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-4 REPORT NO. 92 l-NATION
26、AL ADVISORY COMMITTEE FOR AERONAUTICS ncteristics of wings, namely plan form and twist, which have the greatest influence on the spanwise distribution of lift. With the exception of variations in cb the method cannot directly account for any effects dependent upon the geometry of the airfoil section
27、 even though these may affect the span loading. The substitution of uniformly cambered sections for uncambered sections across a wing span is assumed to change only the wing angle for zero lift and this change is assumed equal to that shown by the section. The substi- tution of variable camber is as
28、sumed equivalent to twisting uncambered sections. PROCEDURE FOR DETERMINATION OF AERODYNAMIC CHARACTERISTICS FROM SPAN LOADING The foregoing section has shown a method by which the symmetric span load distribution of any wing having a straight quarter-chord line over the semispan can be deter- mined
29、 from a knowledge of wing geometry only. With such loading determined it becomes possible to quickly find other characteristics. It should be noted, however, that these characteristics are derived directly from the wing load distribution and that no further aerodynamic theory is in- volved. It is po
30、ssible to find the gross load distribution and re- sultant characteristics directly for a wing at any angle of attack and having any plan form and twist. Past experience, however, has shown that gross characteristics can better be studied if the basic and additional type loadings are handled separat
31、ely. Since the two types of loading are additive, this procedure is permissible. Basic loading is that existing with zero net lift on the wing and is, therefore, due to twist or effective twist 4 (e. g., span- wise change in camber) of the wing-chord plane. The basic loading and characteristics depe
32、ndent on it are unchanged by the addition of load due to uniform spanwise wing angle- of-attack change and are equal at all angles of attack to that found for zero net lift on the wing. Additional loading is that due to equal geometric angle-of- attack change at each section of the wing. The distrib
33、ution of additional load is a function only of wing plan form and is thus independent of any basic load due to twist existing on the wing. The magnitude of the additional load is a function only of angle of attack of the wing and thus each equal increment of angle of attack will give the same increa
34、se and . distribution of additional load irrespective of the gross load on the wing. The wing characteristics due to additional load of any given plan form are thus a function of the lift coefficient or angle of attack of the wing. In the following sections the procedure for determining - basic-type
35、 span load distribution and the characteristics associated with it (denoted by a subscript 0) is first presented and then the procedure for finding additional-type span load distribution and the associated characteristics (denoted by a subscript a). Finally, it is shown how characteristics due to gr
36、oss load distribution (denoted by absence of a subscript) can be found. Hereafter reference will be made to twist only. The reader will understand that effective twist will be handled in an identical manner. PROCEDURE FOR DETERMINATION OF AERODYNAMIC CHARACTERISTICS DUE TO BASIC LOADING Span load di
37、stribution and angle of zero lift for arbitrary twist.-It is not possible to obtain the distribution of basic load on a twisted wing directly from equation (2) since the values of (T” for each station are not generally known for the condition of zero net lift and thus, as written, eight unknown valu
38、es appear. However, only one ;mknown has actually been added to the four unknowns (the individual loads of equation (2) since, while four values of (Ye appear, they are related to one another through knowlege of the twist distri- bution. To form the fifth equation required for the solution in additi
39、on to the four represented by equation (2), use is made of equation (A46) of the appendix which gives the total lift in terms of the four individual loads. Thus Equating this to zero (basic loading) and with equa.tion (2), a set of five simultaneous equations is formed, the solution of which will gi
40、ve the values of loading at the spanwise points and the angle of zero lift of the reference station. Thus O= G4+2 i at the four spanwise stations q=O.924, 0.707, 0.383, and 0. Substitutions of the values of % in the expression for lift coefficient gives the wing lift coefficient for one radian chang
41、e in angle of attack, or in effect, lift-curve slope. Thus, with equation (3) =CL=$ (+=+1.848 +=+1.414 %+0.765 % (12) The dimensionless circulation per radian 2 can be es- pressed in the more usual loading-coefficient form through the relations Induced drag.-The induced drag due to additional load-
42、ing can be computed at any lift coefficient exactly as was Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-6 REPORT NO. 921-NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS the induced drag due to basic loading. Values of loading at the four spanwise poin
43、ts are found for the particular value of CL and substituted in the appropriate form of equa- tion (9). If the more usual expression fo; induced drag as a function of CL is desired, then values of 2 are substituted in the following expression which differs only algebraically from equation (9): c -(.y
44、 G4a -; ,( 0.056 G$+0.789 % 0.733 2-tO.845 $ I (13) Spanwise center of pressure,-Substitution of the values for individual loads in equation (Al7) of the appendix gives an expression for the continuous distribution of additional load. Integration of the increments of bending moment about an axis lyi
45、ng in the plane of symmetry will give total root bending moment. Then, with knowledge of the total load, an expression giving the spanwise location of the Center of load (on the quarter-chord line) can be found. In terms of the values of the dimensionless circulation, this exp,ression is (from refer
46、ence 5) 0.352 2+0.503 $+0.344 %+0.041 :E Ilc. v. ( pv2c From these two expressions it can be .found that cr=4a(;) (k) It can be seen that the term F indicates a change in the direction of flow with respect to the free-stream direction, which change decreases with distance from the lifting line. If t
47、he lifting line is assumed to be replacing a plate insofar as lift is concerned, then simplified lifting-surface theory requires that, at some distance from the lifting line, the direction of flow must be parallel to the plate. In. effect, then, the induced downwash angle becomes equal to the angle
48、of attack of the plate. It remains to determine how far aft of the lifting line the downwash angle must be meas- ured in order to properly relate the increase in circulation to the change in angle of attack of the plate. From the foregoing expressions and assuming small angles, the follow- ing relat
49、ion can be written: =(- can be regarded as an increase in the distance between the lifting vortex and the control point on the undistorted wing. From the previous section it will be recalled this corresponds to an increase in section lift-curve slope-in this case exactly in the ratio of L* B Thus, the theoretical section lift-curve slope, where compressible effects are inc
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