1、I A CONCEPT OF THE VORTEX LIFT OF SHARP-EDGE DELTA WINGS BASED ON A LEADING-EDGE-SUCTION ANALOGY Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-TECH LIBRARY KAFB,NM OL3042b NASA TN D-3767 A CONCEPT OF THE VORTEX LIFT OF SHARP-EDGE DELTA WINGS BASED
2、ON A LEADING-EDGE-SUCTION ANALOGY By Edward C. Polhamus Langley Research Center Langley Station, Hampton, Va. NATIONAL AERONAUTICS AND SPACE ADMINISTRATION For sale by the Clearinghouse for Federal Scientific and Technical Information Springfield, Virginia 22151 - Price $1.00 Provided by IHSNot for
3、ResaleNo reproduction or networking permitted without license from IHS-,-,-A CONCEPT OF THE VORTEX LIFT OF SHARP-EDGE DELTA WINGS BASED ON A LEADING-EDGE-SUCTION ANALOGY By Edward C. Polhamus Langley Research Center SUMMARY A concept for the calculation of the vortex lift of sharp-edge delta wings i
4、s pre sented and compared with experimental data. The concept is based on an analogy between the vortex lift and the leading-edge suction associated with the potential flow about the leading edge. This concept, when combined with potential-flow theory modified to include the nonlinearities associate
5、d with the exact boundary condition and the loss of the lift component of the leading-edge suction, provides excellent prediction of the total lift for a wide range of delta wings up to angles of attack of 20 or greater. INTRODUCTION The aerodynamic characteristics of thin sharp-edge delta wings are
6、 of interest for supersonic aircraft and have been the subject of theoretical and experimental studies for many years in both the subsonic and supersonic speed ranges. Of particular interest at subsonic speeds has been the formation and influence of the leading-edge separation vor tex that occurs on
7、 wings having sharp, highly swept leading edges. In general, this vor tex flow results in an increase in lift associated with the upper-surface pressures induced by the vortex and an increase in drag resulting from the loss of leading-edge suction. Although, in general, it is desirable to avoid the
8、formation of the separation vortex because of the high drag, it is sometimes considered as a means of counteracting, to some extent, the adverse effect of the low lift-curve slope of delta wings with regard to the landing attitude. In recent years, the interest in the vortex flows associated with th
9、in delta and delta-related wings has increased considerably as a result of the supersonic commercial air transport programs that are underway both in this country and abroad. Even though sev eral theoretical methods of predicting the effects of separation vortex flows on the lift of delta wings have
10、 been developed, there appears to be no completely satisfactory method -especially when angles of attack and aspect ratios of practical interest are considered. The purpose of the present paper, therefore, is to present a concept with regard to vortex flow which appears to circumvent the problems en
11、countered in the previous methods, In i Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-this concept, the pressures required to maintain the flow about the leading-edge vortex are related to those required to maintain potential flow about the leading
12、 edge. SYMBOLS A wing aspect ratio, b2/S b wing span cDi theoretical induced-drag coefficient CL total lift coefficient, CL,p + CL,v cL,P lift coefficient determined by linearized potential-flow theory (present application does not include leading-edge- suction component) CL,V lift coefficient assoc
13、iated with leading- edge separation vortex cN,P normal-force coefficient determined by linearized potential-flow theory cN, v normal-force coefficient associated with leading-edge separation vortex cP upper -surf ace pres sure coefficient CS leading-edge suction coefficient (in plane of wing and per
14、pendicular to leading edge) CT leading-edge thrust coefficient (in plane of wing and parallel to flight direction) Ki induced-drag parameter, 8 CDi/k 2 KP constant of proportionality in potential-flow lift equation KV constant of proportionality in vortex lift equation L lift N normal force S wing a
15、rea T thrust force (in plane of wing and parallel to flight direction) V velocity in flight direction wi average downwash velocity induced by trailing vortex sheet (perpendicular to wing chord) 2 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-(Y ang
16、le of attack r total effective circulation A leading-edge sweep angle P mass density of air DISCUSSION AND RESULTS 1 Nonlinear Lift Characteristics Wind-tunnel studies of sharp-leading-edge delta wings have shown that even at rela tively low angles of attack the flow separates from the leading edges
17、 and rolls up into two vortex sheets or cone-shaped cores of rotating fluid, as illustrated in figure 1. Flow attachment lines have been observed inboard of the vortex sheets and indicate that air is drawn over the vortex sheets and accelerated downward. An however, this method departs radi cally fr
18、om experiment at low aspect ratios despite the fact that slender-body techniques are 1.0 Brown and / /Mangler and Michael -v , Smith .8 .6 CL 0 .5 1.0 1.5 2.0 A Figure 5.- Comparison of experiment with results determined by previous theories. a = 15. 5 Provided by IHSNot for ResaleNo reproduction or
19、 networking permitted without license from IHS-,-,-. used. Reference 16 describes some preliminary studies of a theoretical approach, requiring a high-speed computer, in which vortices shed from the leading edge are allowed to interact and roll up. This general approach might ultimately provide a me
20、thod of predicting details of the flow; however, in the initial application described in refer ence 16, this theory appears to depart from experiment to a degree similar to that of reference 10. Present Method The present approach assumes that if flow reattachment occurs on the upper sur-B face the
21、total lift can be calculated as the sum of a potential-flow lift and a lift associated with the existence of the separated leading-edge spiral vortices. First, the potential-flow lift will be examined with regard to the effect of high angles of attack and modified leading-edge conditions. Then, the
22、vortex lift will be determined by a method in which the vortex flow is assumed to be related to the potential flow about the leading edge. Potential lift.- Inasmuch as potential-flow theory is usually presented in a form applicable only for wings at low angles of attack, the development of the theor
23、y in a form more applicable for the high angles of attack of interest in the present study will be used. In addition, the potential-flow theory must be modified for application to the leading-edge separation condition for the sharp-edge delta wings considered in this paper, In order to account for t
24、he leading-edge separation, a Kutta type flow condition is assumed to exist at the sharp leading edge and, therefore, no leading-edge suction can be developed. It is further assumed, since the flow reattaches downstream of the separation vortex, that the potential-flow lift is diminished only by the
25、 loss of the lift component of the leading-edge suction. Although, logically, the loss of the lift component of the leading-edge suction should be charged to the vortex-lift term, it is more convenient in the present concept to consider this loss as a modification to the potential-lift term. For the
26、 condition of zero leading-edge suction, the resultant force (neglecting friction drag) for the planar wings is perpendicular to the wing chord and is equal to the potential-flow normal force. The potential-flow lift coefficient for the zero-leading-edge- suction condition is given by The normal for
27、ce can be determined by applying the Kutta-Joukowski theorem using the velocity component parallel to the wing chord; thus, N = prb(V COS 0;) where is a.total effective circulation. The distribution of circulation in the lifting system is determined by satisfying the boundary condition which require
28、s that the veloc ity normal to the chord plane induced by the total vortex system be equal to V sin a! at points on the wing planform. The total effective circulation can then be written as 6 i Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-r=Kpzsv
29、sin a! (3) Since, for the wings of interest herein, departures of the potential-flow vortex system from the wing plane and its extension would be expected to have negligible effect on r, it is assumed that Kp depends only on planform. Substituting equation (3) into equa tion (2) and reducing the nor
30、mal force to coefficient form results in CN,p = Kp sin a! cos a! (4) f The potential-flow lift coefficient for the condition of zero leading-edge suction1 can now be determined by substituting the expression for CN,(eq. (4) into equation (1);thus, . cL,= K sin a! COS! (5) Since Kp depends only on pl
31、anform and equation (5) reduces to CL,= Kpa! for small angles of attack, it is apparent that Kp is equal to the lift-curve slope given by small-angle theory and can therefore be determined from any suitable lifting-surface theory. In this paper, Kp is determined by a modification of the Multhopp lif
32、ting-surface theory of reference 17. The variation of Kp with aspect ratio is presented in figure 6. Vortex lift.- The major problem associated with the prediction of the lift of sharp-edge delta wings is, of course, the calculation of the so-called vortex lift associated with the leading-edge separ
33、ation spiral vortex. This problem arises pri- Kp marily from the difficulty in determining with suffi- 2 cient accuracy the strength, shape, and position of the spiral vortex sheet and it is in this area that the 1present method differs from previous approaches. In the present method an attempt to a
34、void the various problems associated with the calculation of 0 1 2 I3 4 Athe strength, shape, and position of the spiral vor-Fiaure 6.-. Variation of $ with A for delta wings. tex sheet is made by relating the force required to maintain the equilibrium of the flow over the separated spiral vortex (p
35、rovided that the flow reattaches on the upper surface) with the force associated with the theoretical leading-edge singularity for thin wings in potential flow. Figure 7 shows pictorial 1 sketches, in a plane normal to the leading edge, of the potential flow about a sharp leading edge and a round le
36、ading edge and the separated vortex flow about a sharp leading edge. 1For full leading-edge suction, the nonlinear effects in potential-flow theory can be determined from L = pn(V - Wi sin o$, where r is obtained from equation (3) and wi sin a accounts for the reduction of the relative velocity in t
37、he flight direction due to the inclined vortex system. 2The modifications to the Multhopp theory include the extension to higher order chordwise loading terms as developed by Van Spiegel and Wouters (ref. 18) and refinements in the numerical integration procedure developed by John E. Lamar of the La
38、ngley Research Center. 7 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-For the attached flow condition, the flow ahead of the lower-surface stagnation point flows forward and is accelerated around the leading edge to the top surface. The pres sure
39、required to balance the centrifugal force created by the flow about the leading edge results, of course, in the leading-edge suction force. As the leading-edge radius is increased, the potential-flow suction force remains essentially constant since the suction pressures vary inversely as the leading
40、-edge radius. (See ref. 2.) As shown in the sketch representing the separated flow about the sharp leading edge (fig. 7(c), the flow ahead of the stagnation point flows forward but separates from the wing as it leaves the leading edge tangentially and rolls up into the previously discussed spiral vo
41、rtex sheet. (See fig. 1.) Air is drawn over this vortex sheet and accelerated downward to an upper-surface flow attachment line. Since the flow over the vortex sheet reattaches, the basic assumption of the present method is that the total force on the wing associated with the pressures required to m
42、aintain the equilibrium of the flow over the separated spiral vor tex sheet is essentially the same as the leading-edge suction force associated with the leading-edge pressures required to maintain attached flow around a large leading-edge radius. The flow pattern in both cases would be somewhat sim
43、ilar, as indicated by com paring figures 7(b) and 7(c); however, for the sharp-leading-edge condition, the force acting on the wing will act primarily over the upper surface rather than on the leading edge. Therefore, a normal force occurs which is equal to the theoretical leading-edge suction force
44、. (See fig. 7(c).) Since the theoretical leading-edge suction force is essen tially independent of leading-edge radius, the normal force associated with the separated spiral vortex sheet should be equivalent to the leading-edge suction force as predicted by an appropriate thin-wing lifting-surface t
45、heory. (It must be kept in mind that this force is the resultant perpendicular to the leading edge and not that component parallel to the flight direction.) Both suction forces are in the wing chord plane, and the component in the flight direction will be referred to as the leading-edge thrust coeff
46、icient CT and the resultant suction force perpendicular to the leading edge will be referred to as the leading-edge suction coefficient Cs. (See fig. 8.) (a) Potential flow 1(sharp edge). (b) Potential flow (round edge). I Attachment Spiral (c) Separated flow (sharp edge). Figure 8.- Relationship be
47、tween Figure 7.- Leading-edge flow conditions. CT and Cs. 8 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-r 1 .I t Based on the concept of this paper, the resultant suction force is rotated into a direction normal to the wing chord plane and theref
48、ore the normal-force coefficient associated with the leading-edge vortex is given by The lift coefficient can then be given by cos CY cL,v = cN,v cos CY = cT The problem now reduces to the determination of the thrust coefficient associated with potential flow around the leading edge. By applying the
49、 Kutta-Joukowski theorem using velocity components normal to the wing chord plane, the leading-edge thrust is given by T = pr“b(V sin CY - Wi) (7) where wi is an effective downwash velocity induced normal to the chord by the trailing vortex system. The magnitude of wi is that which will act on the total effective circula tion r to produce the same
