1、NASA TECHNICAL NOTEZk,-ZNASA TN D-6243CHARTS FOR PREDICTING THE SUBSONICVORTEX-LIFT CHARACTERISTICSOF ARROW, DELTA, AND DIAMOND WINGSby Edward C. PolhamusLangley Research CenterHampton, Va. 23365NATIONAL AERONAUTICSAND SPACEADMINISTRATION WASHINGTON,D. C. APRIL 1971Provided by IHSNot for ResaleNo re
2、production or networking permitted without license from IHS-,-,-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-1. Report No. 2. Government Accession No.NASA TN D-62434. Title and SubtitleCHARTS FOR PREDICTING THE SUBSONIC VORTEX-LIFTCHARACTERISTICS
3、OF ARROW, DELTA, AND DIAMONDWINGS7. Author(s)Edward C. Polhamus9. Performing Organization Name and AddressNASA Langley Research CenterHampton, Va. 2336512. Sponsoring Agency Name and AddressNational Aeronautics and Space AdministrationWashington, D.C. 205463. Recipients Catalog No.5. Report DateApri
4、l 19716. Performing Organization Code8. Performing Organization Report No.L-755810. Work Unit No.126-13-10-0111. Contract or Grant No.13. Type of Report and Period CoveredTechnical Note14. Sponsoring Agency Code15. Supldementary Notes16. AbstractThe leading-edge-suction analogy method of predicting
5、the aerodynamic characteris-tics of slender delta wings has been extended to cover arrow- and diamond-wing planforms.Charts for use in calculating the potential- and vortex-flow terms for the lift and drag arepresented, and a subsonic compressibility correction procedure based on the Prandtl-Glauert
6、 transformation is outlined.17. Key Words (Suggested by Author(s)Slender wingsVortex liftSubsonic compressible flow18. Distribution StatementUnclassified - Unlimited19. Security Classif. (of this report)Unclassified20. Security Classif. (of this page) 21. No. of PagesUnclassified 10oFor sale by the
7、National Technical Information Service, Springfield, Virginia 2215122. Price*$3.00Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-CHARTS FOR PREDICTING
8、THE SUBSONICVORTEX-LIFT CHARACTERISTICS OF ARROW, DELTA,AND DIAMOND WINGSBy Edward C. PolhamusLangley Research CenterSUMMARYThe leading-edge-suction analogy method of predicting the aerodynamic character-istics of slender delta wings has been extended to cover arrow- and diamond-wing plan-forms. Cha
9、rts for use in calculating the potential- and vortex-flow terms for the liftand drag are presented, and a subsonic compressibility correction procedure based onthe Prandtl-Glauert transformation is outlined.INTRODUCTIONThe leading-edge vortex lift associated with the leading-edge-separation vortexwh
10、ich occurs on slender sharp-edge wings has, during the past decade, become morethan an aerodynamic curiosity with airplanes such as the Concorde supersonic transportand the Viggen fighter utilizing this flow phenomenon as a means of eliminating the needfor flow control devices and high-lift flaps. (
11、See refs. 1 to 3.) Although many analyticalmethods of predicting the aerodynamic characteristics associated with leading-edge vor-tex flow have been developed (some of which are reported in refs. 4 to 8), they have beenlimited primarily to delta planform wings or wings with unswept trailing edges. B
12、ecauseof the increased use of slender wings exhibiting leading-edge vortex flow, at least in themany off-design conditions if not at the design condition, analytical methods applicable toarbitrary planforms are needed. The leading-edge-suction analogy, described in refer-ences 8 and 9 appears to pro
13、vide an accurate method of predicting the vortex-lift charac-teristics which, at least in concept, is not limited to delta planforms and has been shownin reference 10 to provide accurate estimates for a fairly wide range of fully taperedwings. Although the subsonic analysis was limited to incompress
14、ible flow, an appropriateapplication of the Prandtl-Glauert transformation should provide a subsonic compressi-bility correction. The purpose of this paper is to present, in chart form, the potential-flow and vortex-flow constants, including subsonic compressibility effects, for a wideseries of arro
15、w-, delta-, and diamond-wing planforms.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-SYMBOLSA wing aspect ratio, b2/Slongitudinal distancefrom root trailing edgeto wing tip station, positiverearward (seefig. 1)a/z wing notch ratio, positive for arr
16、ow wings and negative for diamond wingsb wing spanCDCD,oAC DCLCpedrag coefficientdrag coefficient at zero liftdrag-due-to-lift coefficient,lift coefficientCD - CD, opressure coefficientleading-edge length of wing (see fig. 1)e YfMKpKvleading-edge length of transformed wingcompressibility factor (see
17、 eq. (5)constant in potential-flow-lift termconstant in vortex-lift termlongitudinal distance from apex to wing tip station (see fig. 1)M Mach numberwing areaangle of attackProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-/3=_- M 2AleAleleading-edge s
18、weep of actual wing (see fig. 1)leading-edge sweep of transformed wing,All primes refer to the transformed wing.ttan Ale tan AleANALYTICAL METHODSIn references 8 and 9 it has been shown that excellent predictions of lift and dragdue to lift of sharp-edge delta wings over a wide range of angles of at
19、tack and aspectratios can be obtained by combining the potential-flow lift and the vortex lift as predictedby the leading-edge-suction analogy. The resulting equations areC L = Kp sin a cos2a + Kv sin2a cos (1)andAC D = Kp sin2a cos a + K v sin3a (2)orAC D = C L tan a (3)where, in equations (1) and
20、(2), the first term represents the potential-flow contributionand the second term represents the vortex-lift contribution.In reference 10 it was shown that equation (1) is applicable for wings of arbitraryplanform providing, of course, that the constants Kp and K v are calculated for thedesired plan
21、form. The analogy method makes it possible to use potential-flow theoryto predict both the potential-flow term and the vortex-flow term. For the arrow anddiamond planforms of interest in this paper, any accurate potential-flow lifting-surfacemethod, such as the methods of references 11 and 12, can b
22、e used. Since the method ofreference 12 appears to offer some advantages with regard to more general planformsinvolving broken leading edges, it has been programed at Langley for use in certainlifting-surface studies and was used for the present calculations of the potential- andvortex-lift constant
23、s. The constant Kp is simply the potential-flow lift-curve slopeand the constant Kv is related to the potential-flow leading-edge thrust parameter.(See eq. (3) of ref. 10.)Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-The subsonic effectsof compres
24、sibility can be accounted for by use ofthe Prandtl-Glauert transformation and the Goethert rule form (see ref.13) willbe used herein. Thisrule relates the pressure coefficientat a given nondimensionalized pointon the real wingata given Mach number to a pressure coefficientatthe same nondimensionaliz
25、ed pointona transformed wing (stretchedinlongitudinaldirectionby l/f) inincompressible flow.For a wing of zero thickness, the rule can be statedas follows:1(CP)M,a,A,Ale = _-2(Cp)M=0,af,A/_,A_eApplicationto the potential-flow-liftconstant Kp iswell known, and the effectof com-pressibilitycan be acco
26、unted for simply by determining the incompressible value for atransformed wing having a reduced aspect ratioequal to AI_ and an increased leading-edge sweep angle whose tangent is greater by 1/_, and then increasing the resultingvalue of Kp by the factor i/ft. The I/_ correction to Kp resultsfrom co
27、mbiningthe 1/_ 2 correction to the pressure and the effect of the reduced angle of attack a_.!Therefore, if Kp is the incompressible value for the transformed wing, then Kp forthe real wing at its Mach number is given byKp = _With regard to the effect of compressibility on the vortex-lift constant K
28、v, it wasassumed that the leading-edge-suction analogy can also be applied in compressible flow;therefore, the problem can be reduced to that of determining the effect of compressibilityon the leading-edge suction. Although the same transformed wing is used for the leading-edge suction and the resul
29、ting vortex-lift constant Kv, the compressibility factor thatmust be applied differs from the 1/_ that is used for Kp. This is due to two factors.First, since the leading-edge suction increases with the square of the angle of attack, theangle-of-attack reduction associated with the transformed wing
30、completely cancels the1/132 term that is applied to the pressures on the transformed wing. SeCond, since themethod used must be equivalent to applying the transformed wing pressures along thereal-wing leading edge (rather than the reference area as in the potential-flow lift case),and since the tran
31、sformed leading-edge length e does not increase as rapidly as thetransformed-wing area S, the value of Kv must be corrected to the real wing ratioof the leading-edge length to the area. In other words,SeKv = Kv S eand since S =1 andS e _1 + tan2A_-r= I_ tan2A,+ _2Provided by IHSNot for ResaleNo repr
32、oduction or networking permitted without license from IHS-,-,-K ,I-1 +-tan2AKv = vvfl2 + tan2A - KvfMPRESENTATION OF RESULTS(4)Lifting-surface solution values for the potential-flow-lift constant Kpfl as a func-tion of Aft and A_e are presented in figure 2 by the solid lines. Also presented as anaid
33、 in locating a particular wing are dashed lines which represent constant values of notchratio all. These constant notch-ratio lines are also convenient for applying the Prandtl-Glauert transformation since the notch ratio is unaffected by the transformation. Fol-lowing a constant notch-ratio line re
34、moves the need for determining the sweep angles Aleof the various transformed wings.Figure 3 presents values of the vortex-lift constant in the form Kv/fM as a func-tion of A_ and Ale. Again, lines of constant notch ratio are presented for convenience.Values of fM as determined from equation (4) are
35、 presented in figure 4 as a function ofleading-edge sweep angle and Mach number.For convenience in using the equations, table I presents values of the various com-binations of trigonometric functions needed.With regard to the expected accuracy of the method, reference 10 presents correla-tions with
36、experimental results for the incompressible case.CONCLUDING REMARKSThe leading-edge-suction analogy method of predicting the aerodynamic character-istics of slender delta wings has been extended to cover arrow- and diamond-wing plan-forms. The method of applying compressibility corrections to the le
37、ading-edge suctionhas been examined, and the resulting procedure applied to the vortex-lift constant. Chartsfor use in calculating the potential- and vortex-flow terms for the lift and drag in subsoniccompressible flow are presented for a wide range of planform parameters.Langley Research Center,Nat
38、ional Aeronautics and Space Administration,Hampton, Va., February 26, 1971.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-REFERENCES1. Maltby, R. L.: The Developmentof the SlenderDelta Concept. Aircraft Eng., vol. XL,no. 3, Mar. 1968,pp. 12-17.2. K/
39、Jchemann,D.; and Weber,J.: An Analysis of SomePerformance Aspectsof VariousTypes of Aircraft Designedto Fly Over Different Rangesat Different Speeds.Progress in Aeronautical Sciences,Vol. 9, D. K_ichemann,ed., PergamonPress,Inc., c.1968,pp. 329-456.3. Behrbohm,Hermann: Basic Low SpeedAerodynamics of
40、 the Short-CoupledCanardConfigurationof Small Aspect Ratio. SAABTN 60, SaabAircraft Co. (Link6ping,Sweden),July 1965.4. Brown, Clinton E.; and Michael, William H., Jr.: On SlenderDelta WingsWithLeading-EdgeSeparation. NACATN 3430,1955.5. Mangler, K. W.; and Smith, J. H. B.: A Theory of the Flow Past
41、 a SlenderDelta WingWith Leading EdgeSeparation. Proc. Roy. Soc. (London),ser. A., vol. 251,no. 1265,May 26, 1959,pp. 200-217.6. Sacks,Alvin H.; Lundberg, RaymondE.; andHanson,CharlesW.“ A Theoretical Inves-tigation of the Aerodynamics of SlenderWing-Body CombinationsExhibiting Leading-EdgeSeparatio
42、n. NASACR-719, 1967.7. Smith, J. H. B.: Improved Calculations of Leading-EdgeSeparationFrom SlenderDelta Wings. Tech. Rep.No. 66070,Brit. R.A.E., Mar. 1966.8. Polhamus, EdwardC.: A Conceptof the Vortex Lift of Sharp-EdgeDelta Wings Basedon a Leading-Edge-SuctionAnalogy. NASATN D-3767, 1966.9. Polham
43、us,Edward C.: Application of the Leading-Edge-SuctionAnalogyof Vortex Liftto the Drag Dueto Lift of Sharp-EdgeDelta Wings. NASATN D-4739, 1968.10. Polhamus,Edward C.: Prediction of Vortex-Lift Characteristics Basedona Leading-Edge SuctionAnalogy. AIAA Paper No. 69-1133,Oct. 1969.11. Lamar, John E.:
44、A Modified MulthoppApproachfor Predicting Lifting Pressures andCamberShapefor CompositePlanforms in SubsonicFlow. NASATN D-4427, 1968.12.Wagner, Siegfried: Onthe Singularity Methodof SubsonicLifting-Surface Theory.AIAA Paper No. 69-37, Jan. 1969.13. Shapiro,Ascher H.: The Dynamicsand Thermodynamicso
45、f Compressible Fluid Flow.Vol. I. RonaldPress Co., c.1953.6Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-TABLE I.- VALUESOF TRIGONOMETRICFUNCTIONSdeg sin c_ cos2_ sin2_ cos _ sin3(_246810121416182022242628300.0349.0694.1034.1365.16841990.2278.2547.
46、2795.3020.3220.3395.3541.3660.37500.0012.0049.0109.0192.0297.0423.0568.0730.0908.1099.1301.1511.1727.1946.21650.0000.0003.0011.0027.00520090.0142.0209.0295.0400.0256.06730842.1035.12507Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-“*- 0 -IFigure 1.
47、- Sketch defining wing geometry nomenclature.Kp,83.02.01.00i,o.J5(1:iiiiiiii:4.0Figure 2.- Variation of potential-flow lift constant with planform parameters.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-504.0!Ale=85 80“75 70“65“60“55“50“45 3.0K vf
48、M2.01.000 1.0 2.0 3.0 4.0Figure 3.- Variation of vortex-lift constant with planform parameters.9Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Figure 4. - Variation of compressibility factor with sweep and Mach number. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
