1、NASA TECHNICAL 1 NASA TR R-428 REPORT 00 N EXTENSION OF LEADING-EDGE-SUCTION ANALOGY TO VVINGS WITH SEPAKATED FLOW AROUND THE SIDE EDGES AT SUBSONIC SPEEDS NATIONAL AERONAUTICS AND SPACE ADMINISTRATION WASHINGTON, D. C. OCTOBER 1974 Provided by IHSNot for ResaleNo reproduction or networking permitte
2、d without license from IHS-,-,-1. Report No. NASA TR R-428 2. Government Accession No. 3. Recipients Catalog No. 15. Supplementary Notes 4. Title and Subtitle EXTENSION OF LEADING-EDGE -SUCTION ANALOGY TO WINGS WITH SEPARATED FLOW AROUND THE SIDE EDGES AT SUBSONIC SPEEDS 16. Abstract This paper pres
3、ents a method for determining the lift, drag, and pitching moment for wings which have separated flow at the leading and side edges with subsequently reattached flow downstream and inboard. Limiting values of the contribution to lift of the side-edge reattached flow are determined for rectangular wi
4、ngs, The general behavior of this contri- bution is computed for rectangular, cropped-delta, cropped-diamond, and cropped-arrow wings. Comparisons of the results of the method and experiment indicate reasonably good correlation of the lift, drag, and pitching moment for a wide planform range. The ag
5、ree- ment of the method with experiment was as good as, or better than, that obtained by other methods. The procedure is computerized and is available from COSMIC as NASA Langley computer program A03 13. 5. Report Date October 1974 6. Performing Organization Code 17. Key Words (Suggested by Author(s
6、) Suction analogy Reattached flow Vortex flow Cropped wings 7. Author(s) John E. Lamar 9. Performing Organization Name and Address NASA Langley Research Center Eampton, Va. 23665 12. Sponsoring Agency Name and Address National Aeronautics and Space Administration Washington. D.C. 20546 18. Distribut
7、ion Statement 8. Performing Organization Report No. L -9460 10. Work Unit No. 501 -06 -04 -01 11. Contract or Grant No. - 13. Type of Report and Period Covered Technical Report 14. Sponsoring Agency Code Unclassified - Unlimited 19. Security Classif. (of this report) 20. Security Classif. (of this p
8、age) Unclassified Unclassified STAR Category 01 21. NO. of Pages . 22. Price 71 $3.75 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-CONTENTS Page SUMMARY . 1 INTRODUCTION 1 SYMBOLS . 2 THEORETICAL DEVELOPMENT . 5 BEHAVIOROF Kk se 11 Unswept Wings .
9、 11 Straight trailing edge . 11 Notched trailing edge . 13 Swept Wings . 14 Cropped deltas 14 Cropped diamonds and arrows 15 MODELS AND TEST CONDITIONS 16 EFFECTOFEDGESHAPINGONRECTANGULAR-WINGRESULTS 16 COMPARISONS WITH EXPERIMENTS 17 Unswept Wings . 17 Straight trailing edge . 17 Notched trailing e
10、dge . 18 Swept Cropped Wings 18 Taper -ratio variation (deltas) 18 19 Arrowwing 19 COMPARISONS WITH OTHER THEORIES 19 OtherTheories . 19 General . 19 Bollay . 20 Flax 20 Gersten 20 Belotserkovskii 21 Bradley 21 Comparisons 21 Zero Aspect -Ratio Limit 22 CONCLUSIONS . 22 Trailing-edge sweep variation
11、 (diamond and arrow) . iii Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Page REFERENCES . 24 TABLE 26 FIGURES 27 iv Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-EXTENSION OF LEADING-EDGE-SUCTION
12、 ANALOGY TO WINGS WITH SEPARATED FLOW AROUND THE SIDE EDGES AT SUBSONIC SPEEDS By John E. Lamar Langley Research Center SUMMARY This paper presents a method of determining the lift, drag, and pitching moment for wings which have reattached flow around the leading and side edges. The method is an ext
13、ension of the leading-edge-suction analogy of Polhamus applied to the side edges. The value of the term associated with the lift contribution of the reattached flow from the side edges Kv,se has been found to exceed that of the leading edge for rectangular wings of aspect ratio less than 2. Limiting
14、 values of Kv,se have been determined for rectangular wings. Comparisons of the results of this method with experiment indicate reasonably good correlation of the lift, drag, and pitching moment for a wide planform range. The agreement of the method with experiment was as good as, or better than, th
15、at obtained by other methods. INTRODUCTION Many current and proposed aircraft and missiles designed for high-speed flight employ highly sweptback and tapered low-aspect-ratio wings with sharp or thin edges. These planforms exhibit flow separation along the leading and side edges followed by subseque
16、nt reattachment downstream or inboard, respectively, over a large angle -of - attack and Mach number range. However, the effect of this separated-flow (commonly termed vortex flow) phenomenon is more important at subsonic speeds because of its larger contribution to the total aerodynamic characteris
17、tics. 2, and 3 has shown that, for a variety of pointed-tips planforms and Mach numbers, the contribution of the leading-edge vortex to the lift and drag can be accounted for by what is termed the leading-edge -suction analogy. only that the attached-flow leading-edge suction, available from invisci
18、d theory, be known accurately. Polhamus in references 1, The application of this analogy requires It is evident that the separated flow around the tips or side edges of swept and unswept lifting surfaces has essentially the same behavior as that around the highly swept leading edge. Consequently, it
19、 should be possible to predict the effect of side-edge Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-vortices on the aerodynamic characteristics of low-aspect-ratio planforms by an analy- sis similar to those used to treat the leading edge. The pur
20、pose of the present paper will be to detail how one such extension to the leading-edge-suction analogy can be effected. The one new element required in this extension is the attached-flow side-edge suction force and its derivation is presented. Total-force and total-moment predictions, including bot
21、h the leading- and side -edge suction-force contributions, have been made for a variety of planform shapes and are com- pared with previously published data and new subsonic wind-tunnel data. sion of some other methods which attempt such predictions (refs. 4 to 11) will be given and comparisons with
22、 the present technique are made where possible. Also, a discus- I SYMBOLS Values are given in both SI Units and U.S. Customary Units. The measurements and calculations were made in U.S. Customary Units. PR aspect ratio Bj (xi) b span, cm (in.) coefficient of q2(j-l) at Xi of the spanwise curve fit o
23、f equation (12) CD Drag drag coefficient, - qdref CD, 0 experimental value of drag coefficient at CL = 0 ACD drag coefficient due only to lift, CD - CD,O lift coefficient, - CL q-Sref - aCL a aa, Lift - Cm pitching-moment coefficient about the reference point, unless otherwise 4 q,Srefcref Cref Pitc
24、hing moment stated it is located at - normal-force coefficient, Normal force q,Sref CN 2 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-C streamwise chord, cm (in.) 417) streamwise half chord at 17, cm (in.) 417) x-location of local midchord with re
25、spect to half root chord, cm (in.) F tip suction force from one side edge, N (lbf) G(x) defined by equation (2), m3/2/,ec (ft3/2/sec) j index ranging from 1 to p Leading-edge suction force from one qdref a sin2 a! %,le = (2 Tip suction force from one side edge) a - q-sref J a sin2 a! Kv,se = Kv,tot
26、= BKv,le + Kv,se M Mach number of free stream N maximum number of chord loadings in modified Multhopp solution n index ranging from 0 to N - 1 coefficient of chordal loading function in modified Multhopp solution, m free -stream dynamic pressure, N/m2 (lbf /ft2) (ft) qn(V) q, q, R Reynolds number Sr
27、ef reference area, m2 (ft2) 3 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-s (4 U free-stream velocity, m/sec (ft/sec) distributed edge suction force (see eq. (l), N/m (lbf/ft) U induced velocity in the X-direction at a point (x,y), m/sec (ft/sec)
28、 induced velocity in the Y-direction of a point (x,y), m/sec (ft/sec) distances from a coordinate origin located at the leading-edge apex; x positive downstream and y positive toward right wing tip V X, Y Ax distance along tip chord, a angle of attack, degrees cm (in.) r Y 6 accumulated circulation
29、at a point (x,y) (see eq. (5), m2/sec (ft2/sec) distributed bound vorticity at a point (x,y) (see eq. (6), m/sec (ft/sec) distributed trailing vorticity at a point (x,y) (see eq. (4), m/sec (ft/sec) 9xi A x nondimensional spanwise variable, 2y/b chordwise angular variable (see eq. (9), degrees 8 val
30、ue which yields Xi in equation (9) leading-edge sweep angle, positive for sweepback, degrees taper ratio, Ct/Cr fraction of local chord density, kg/m3 (slugs et3) trailing -edge sweep angle, positive for sweepback, degrees 4 Provided by IHSNot for ResaleNo reproduction or networking permitted withou
31、t license from IHS-,-,-Subscripts: C centroid i particular item of location le leading edge n notch P potential or attached flow r root ref reference se side edge t tip te trailing edge tot total vle vortex effect at the leading edge vse vortex effect at the side edge THEORETICAL DEVELOPMENT The con
32、cept embodied in the leading-edge -suction analogy of Polhamus is developed in reference 1. mary ideas are briefly reviewed. However, to aid in illustrating the application to other edges, the pri- Wings which have attached flows develop suction forces along their leading edges if the stagnation sur
33、face does not lie along that edge. This suction force can be envi- sioned as arising by either of two processes: (1) the pressure near the leading edge acting over the edge thickness or (2) the product of the square of the induced tangential velocity and the distance to the edge, For a wing of infin
34、itesimal thickness the induced 5 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-tangential velocity approaches an infinite value of u(x,y) as shown in sketch (a); however, its product (described above) is still finite. LEADING- EDGE NORMAL FORCE I I
35、 THRUST rb LL NORMAL FORCE SIDE- EDGE SUCTION FORCE SIDE- EDGE SUCTION FORCE Sketch (a) If the flow separates from the wing in going around the leading edge due to its sharpness or thinness, or due to a combination of thickness and angle of attack, the suc- tion force in the chord plane is lost. vor
36、tex which causes the flow to reattach to the leeward surface of the wing, then the energy redistributes on the upper surface near the leading edge and consequently the force acts in the normal-force direction. mal force can be generated at almost all angles of attack. However, if this separated flow
37、 forms into a shed By making the edge sharp this additional nor- According to the analogy, the reattached line or details of the pressure field need not be known in advance in order to determine the reattached-flow force. However, if pitching-moment estimates are needed the distribution of the reatt
38、ached force must be known. The centroid of the leading-edge suction has been used as the longitudinal loca- tion of this force. This assumption does not have provision for angle-of-attack effects on the location of the reattachment line or vortex core, hence the core is assumed to remain stationary
39、near the wing leading edge. From the above outlined ideas it can be seen that the conditions which lead to this additional normal force would not necessarily be limited to wings with separated flows around the leading edge but could be applied to any similar situation where, in potential flow, an ed
40、ge suction force would be produced. Sketch (a) also shows, for example, that along the side edge of a finite streamwise tip chord large values of v(x,y) are produced 6 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-due to flow around the side edges.
41、 These in combination with the infinitesimal thickness that lead to them would produce a finite suction force in the Y-direction (side-edge suc- tion force, plus or minus depending on the edge). Hence, all that is required to employ a generalized suction analogy to the side edges is to determine the
42、 attached-flow side- edge suction force. A mathematical procedure for computing this side force is developed in the follow- ing steps. Figure 1 illustrates graphically selected steps for a typical wing at a partic- ular chordwise and spanwise location. (1) The suction distribution along ar. edge per
43、 mit length is obtained from refer - ence 12 to be of the form s(x) = pG(x) (1) where for side-edge or tip suction the term G(x) is interpreted as where v(x,y) is the perturbation velocity in the Y-direction. (2) The velocity v(x,y) is related to the trailing vorticity by 1 V(X,Y) = 5 %Y) where (3)
44、The trailing vorticity of a particular x-position xi is determined by See figure 1. (4) The subsonic bound-vorticity distribution is represented herein by (3) which is employed in a new version of the modified Multhopp lifting surface solution of reference 13. 7 Provided by IHSNot for ResaleNo repro
45、duction or networking permitted without license from IHS-,-,-(5) Upon obtaining solutions for the - qn(rl) terms from the method of reference 13, qca equation (5) be comes or where x=- c(v) cos 8 + d(q) and dx = c(q) sin 0 d9 Upon integrating, the result of equation (8) is (6) Knowledge of the I(xi,
46、Y) distribution at discrete points is not sufficient for the present analysis since it is the a77 q (77) knowledge of the continuous variation of the spanwise distributions of these terms are assumed to be composed of a sine series, which is expressible as an even power series in 77. Hence, the - 17
47、) values corn - puted by equation (11) will be curve fitted at each variation which is needed. terms. In the Multhopp solution the This requires a q, U xi location with where the Bj(xi) values are determined in the fitting process. Four terms in the series were determined to represent adequately the
48、 sense. r(xi,v) distribution in a least-squared U 8 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-(7) Hence, the differentiation of equation (12) with respect to 17 leads to As 17 - 1 the first group of terms - 0 and the second group - -m. Therefore, it is only these last terms in combination with their multiplier which can contribute to the suction. Setting = 1 in all the terms except those multiplied by 1 results in (8) Substituting equation (14) into equations (4) and
