1、NASA TECHNICAL NOTE NASA e./ DESIGN CHARTS OF STATIC AND ROTARY STABILITY DERIVATIVES FOR CROPPED DOUBLE-DELTA WINGS IN SUBSONIC COMPRESSIBLE FLOW by John E. Lumur Lungley Reseurch Center Lungley Stution, Humpton, Vu. TN - D-5661 NATIONAL AERONAUTICS AND SPACE ADMINISTRATION WASHINGTON, D. C. 9 FEBR
2、UARY 1970 /iiI i Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-1. Report No. 2. Government Accession No. NASA TN D-5661 I 4. Title and Subtitle DESIGN CHARTS OF STATIC AND ROTARY STABILITY DERIVATIVES FOR CROPPED DOUBLE-DELTA WINGS IN SUBSONIC COMP
3、RESSIBLE FLOW 7. Author(s) John E. Lamar . 9. Performing Organization Name and Address NASA Langley Research Center Hampton, Va. 23365 2. Sponsoring Agency Name and Address National Aeronautics and Space Administration Washington, D.C. 20546 5. Supplementary Notes 6. Abstract 3. Recipients Catalog N
4、o. 5. Report Date February 1970 6. Performing Organization Cot -8. Performing Organirotion Ret L-6716 0. Work Unit No. 126- 13- 10-01-23 1. Controct or Gront No. -3. Type of Report ond Period C Technical Note 4. Sponsoring Agency Code An evaluation of a modified version of the Multhopp subsonic lift
5、ing-surface theory was made by comparing the theoretical values with experimental data. Near zero lift, the theory was found to predict reasonably adequately the lift-curve slope, aerodynamic center, damping in roll, damping in pitch, and lift coefficient due to pitch rate for delta and cropped delt
6、a planforms and also the lift-curve slope for double-delta planforms. Based on this theory, a series of design charts has been prepared for cropped double-delta planforms in subsonic compressible flow. 17. Key Words Suggested by Aulhor(s) 18. Distribution Statement Design charts Unclassified - Unlim
7、ited Cropped double-delta wings Static and rotary derivatives I 9. Security Classif. (of this report) M. Security Classif. (of this page) 22. Pric Unclassified Unclassified $: For sale by the Clearinghouse for Federal Scientific and Technical Information Springfield, Virginia 22 151 Provided by IHSN
8、ot for ResaleNo reproduction or networking permitted without license from IHS-,-,-DESIGN CHARTS OF STATIC AND ROTARY STABILITY DERIVATIVES FOR CROPPED DOUBLE-DELTA WINGS IN SUBSONIC COMPRESSIBLE FLOW By John E. Lamar Langley Research Center SUMMARY An evaluation of a modified version of the Multhopp
9、 subsonic lifting-surface theory was made by comparing the theoretical values with experimental data. Near zero lift, the theory was found to predict reasonably adequately the lift-curve slope, aerodynamic ten ter, damping in roll, damping in pitch, and lift coefficient due to pitch rate for delta a
10、nd cropped delta planforms and also the lift-curve slope for double-delta planforms. Based on this theory, a series of design charts has been prepared for cropped double-delta plan-forms in subsonic compressible flow. INTRODUCTION The National Aeronautics and Space Administration has programs underw
11、ay to pro vide aerodynamic design information on aircraft configurations and components for speeds ranging from low subsonic to hypersonic. Information has already been published that indicates the consideration which has been given to both fixed and variable geometry wings for use in the design of
12、supersonic transport and military aircraft. (See, for example, refs. 1 and 2.) A class of fixed wings, the cropped double-delta wings, has found recent application in the design of these aircraft. Examples of proposed and actual aircraft using the con cept of cropped double-delta wings are the lates
13、t Boeing supersonic transport (ref. 3), the Swedish SAAB 35 Draken and SAAB 37 Viggen (ref. 4), and the Lockheed A-11, also designated YF-12A and SR-71 (ref. 4). A search of the literature has indicated the exis tence of some design data, but only a few systematic investigations have been performed
14、on cropped double-delta planforms in the subsonic and supersonic speed regimes. (See, for example, refs. 5 and 6.) In order that a part of this void might be filled, a systematic investigation using the modified Multhopp subsonic compressible lifting-surface approach of reference 7 (after it was sho
15、wn to be applicable) with the appropriate boundary conditions was undertaken Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-to determine the lift-curve slope, aerodynamic center, damping in roll, damping in pitch, and lift coefficient due to pitch r
16、ate for nine families of cropped double-delta planforms. The purpose of the present paper is to present the results of this investigation in design-chart form. The design charts presented are for the attached-flow condition only and do not include the effects of leading-edge separation, which are di
17、scussed in reference 8. SYMBOLS b2aspect ratio, -S wing span, feet (meters) Liftlift coefficient, qoos lift-curve slope, -per degreea lift coefficient due to pitch rate, -per radian rolling-moment coefficient, Rolling moment goosb damping-in-roll parameter, -per radian Pba-2v pitching-moment coeffic
18、ient about F/4 point, Pitching moment q.p damping-in-pitch parameter, -per radian - CYF root chord -c= Mean geometric chord of the total wing b/2 subsonic free-stream Mach number 2 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-m number of span stat
19、ions where pressure modes are defined N number of chordal control points at each of m span stations P roll rate, radians/second q pitch rate about E/4, radians/second qcc free-stream dynamic pressure, pounds/f oot (newtons/m eter2) S total wing area, feet2 (meted) V free-stream velocity, feet/second
20、 (meters/second) X,Y rectangular Cartesian coordinates nondimensionalized with respect to b/2, where origin is in plane of symmetry at half root chord (positive x, aft; positive y, along right wing panel) Xac aerodynamic center, in fractions of F, measured from leading edge of C (positive aft), -aCm
21、 + 1. aCL 4 yb spanwise location of leading-edge break, feet (meters) a angle of attack, degrees A outboard leading- edge sweep angle, degrees A = tan-l(tan A/p), degrees x overall taper ratio, Tip chord Root chord X inboard leading-edge sweep angle, degrees x = tan-l(tan x/p), degrees 3 Provided by
22、 IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-METHOD OF ANALYSIS The method used in this paper for predicting lifting pressures, hereafter called the present method, employs the modified Multhopp approach. (See ref. 7.) Briefly, this method employs an acceler
23、ation potential (developed from a sheet of pressure doublets) in conjunction with the linearized Euler equations to relate the pressure difference across the wing to the downward velocity over the wing surface. The effects of compressibility are accounted for by using the Prandtl-Glauert rule. Formu
24、lation of the problem of determining the pressure difference across the wing leads to an integral equation to which the solution in closed form is difficult, except for certain classes of wings, because the answers sought (surface loadings) are a part of the integrand. Also adding to the difficulty
25、of solution is the presence of a second-order sin gularity which is in the integrand. For these reasons it has been necessary to resort to an approximate solution which makes use of a finite, rather than unlimited, number of boundary points over the wing at which the flow is constrained to be tangen
26、t. Coupled with this finite number of boundary points is a series representation of the lifting pres sures which have the same number of unknowns as there are boundary points and whose values are determined by a process of matrix algebra. In reference 7 the accuracy of the numerical results of the m
27、odified Multhopp method was shown to be dependent on the combination of the number of spanwise stations m where each of N chordwise control (boundary) points was located. Two methods of estimating the appropriate combinations of N and m were studied in reference 7; one was concerned with computing t
28、he correct leading-edge thrust from the chord loadings and the other with seeking the aerodynamic center for which convergence had .occurred. The converged-aerodynamic-center method was selected to determine the appropri ate pair of N and m to be used in this report because it involved less computat
29、ional labor. Upon examining the results of reference 7 and making use of the method described therein, a pattern of N = 8 and m = 23 was postulated to be sufficient for the kinds of planforms to be studied herein. In order to verify that this pattern would give results which lay in a converged-aerod
30、ynamic-center region, the aerodynamic center was calcu lated for several different sets of N and m and is plotted as a function of m in fig ure 1 for one of the families of cropped double-delta planforms for which design charts were prepared. A +2-percent E error band is plotted about the N = 8 and
31、m = 23 results, and for m 11 the aerodynamic-center values appear to lie within this band. However, the predicted results obtained with the N = 8 and m = 23 pattern are not necessarily the same as would be found for an infinite number of points, but the two answers should compare closely. (The N = 8
32、 and m = 23 control-point pair was also used in predicting the rotary aerodynamic characteristics.) 4 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-I: Although the present method was developed for steady, longitudinal, lifting prob lems, it can be
33、used to compute rotary stability derivatives because of the relationship developed in the lifting-surface theory between tangential-f low boundary conditions and the downwash produced by a sheet of pressure doublets. The manner in which computation for rotary stability derivatives is made is based o
34、n recognizing, as pointed out in refer ence 9, that the steady-state rolling and pitching maneuvers are equivalent to linear twist and parabolic camber, respectively. This equivalence can be seen by realizing that the upwash on the upgoing wing in steady, rolling flight is just py and the tangent of
35、 the angle that the section makes with the free stream is just py/V. (The tangent of an angle is approximately equal to the angle in radians for small angles.) Integrating this expression along the chord gives a straight mean camber line, but one that is twisted along the span. A similar procedure c
36、an be used to obtain the mean camber line for the pitching motion about the reference point. The upwash is just qx for the upgoing part of the wing. The tangent of the angle that it makes (small angles of attack) with the free stream is just qx/V. Integrating this along the chord results in a parabo
37、lic camber. VERIFICATION In order to verify the applicability of the present method for predicting the static and rotary aerodynamic characteristics of the cropped double-delta planforms, a com parison of the results from this method with experimental data (refs. 10 to 18) and with results from othe
38、r theories (refs. 19 and 20) is presented in table I for delta, cropped delta, and double-delta wings. It should be noted that in this table no comparisons are presented for cropped double-delta wings because experimental data were lacking. How ever, comparisons of data for the other wings with resu
39、lts from the present method indi cate that this method was applicable to wings with breaks in the leading edge and cropped tips and, hence, to cropped double-delta wings. Results from the present method are also compared with experimental data in figure 2. The theoretical methods selected for com pa
40、rison are those which predict only the rotary stability derivatives since the theoretical methods which predict the static stability derivatives were compared with results from the present method in reference 7. In examining table I, it should be remembered that the theoretical method of reference 1
41、9 has an inherent restriction associated with it because its derivation was based on wings with vanishing aspect ratios. Static Stability Derivatives Generally good agreement is found between the theoretical and experimental values for CL, where the theoretical values have variations of about rtl0 p
42、ercent from the experimental values. (See fig. 2.) Also, the predicted Xac values generally agree well with the experimental values and have variations of about 40 percent from the 5 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-experimental values
43、 or *5 percent of E. No comparison between the theoretical and experimental location of Xac could be made for the double-delta wings of A = 1.60 because reference 18 reports that the pitching-moment data were questionable, and hence not presented, for small angles of attack. Rotary Stability Derivat
44、ives Comparison of results from the present method with experimental data and results from the other theories for the rotary stability derivatives is restricted to the delta and cropped delta planforms since (1)no experimental rotary derivatives were found for the double-delta wings and (2) the theo
45、retical methods of references 19 and 20 were not read ily (if at all) applicable to the double-delta wings. Damping in roll C .- The present method generally predicts values of CZP(2) about 20 percent more negative than those measured experimentally for a wing of A I 1.0. A large part of this loss i
46、n theoretical damping may be due to the wing tips of the experimental model experiencing a stall condition for which the present method does not account. The actual individual differences between the experimental data and values from the present method are smaller for all configurations than those r
47、esulting from use of the method of reference 19, which was developed for slender wings only. How ever, when experimental data and values from the present method are compared with values from the method of reference 20, it is found that the predicted values of refer ence 20 agree almost as well with
48、experimental data as values from the present method for the delta wings and agree slightly better with experimental data for the cropped delta wings. Hence, either the method of reference 20 or the present method could be applied with almost equal accuracy for delta or cropped delta wings but the pr
49、esent method is more easily applied to cropped double-delta wings. Damping in pitch (Cmq) .-The present method generally predicts values of Cm q which are about 30 percent less stable than the experimental values; however, the results obtained by this method agree better with experimental data than those obtained by the method of reference 19, which predicts results that are con
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