1、NASA Contractor Report 3449 Calculation of Vortex Lift Effect Cambered Wings by the Suction Analogy C. Edward Lan and Jen-Fu Chang . GRANT NSG-1629 JULY 1981 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-TECH LIBRARY KAFB, NM NASA Contractor Report
2、 3449 Calculation of Vortex Lift Effect for Cambered Wings by the Suction Analogy C. Edward Lan and Jen-Fu Chang The Utriversity of Kumas Center for Research, Itzc. Lawrence, Katlsas Prepared for Langley Research Center under Grant NSG-1629 National Aeronautics and Space Administration Scientific an
3、d Technical Information Branch 1981 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Summary An improved version of Woodwards chord plane aerodynamic panel method for subsonic and supersonic flow has been developed for cambered wings exhibiting edge-s
4、eparated vortex flow, including those with leading- edge vortex flaps. The exact relation between leading-edge thrust and suction force in potential flow is derived. Instead of assuming the rotated suction force to be normal to wing surface at the leading edge, new orientation for the rotated suctio
5、n force is determined through con- sideration of the momentum principle. The supersonic suction analogy method is improved by using an effective angle of attack defined through a semi-empirical method. Comparisons of predicted results with available data in subsonic and supersonic flow are presented
6、. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-1. INTRODUCTION In references 1 and 2, an improved panel method was shown to be capable of predicting accurately the leading-edge and side-edge suction forces. In the method, a specific set of control
7、 point locations is obtained, based on a two-dimensional theory. All three-dimensional results presented for this method have been for non-cambered wings exhibiting edge-separated vortex flow. For highly swept cambered wings in subsonic compressible flow, a simplified method (as compared with that t
8、o be developed in this report) has been developed based on reference 3. That method uses the vortex-lattice method and suction analogy (VLPI-SA) and is applicable only to subsonic flow. Its application to analysis and design of slender cambered wings has been reported in references 4 and 5. It shoul
9、d be noted that in the existing suction analogy method, as exemplified by reference 3, the edge suction forces predicted for attached flow are rotated so that they are normal to the cambered wing surface along the leading and side edges to produce the vortex lift effect. This is a direct extension o
10、f Polhamus suction analogy originally developed for a flat wing (reference 6). However. experimental evidence (references 7 and 8) indicates that the leading-edge vortex on a slender wing tends to migrate inboard as the angle of attack is increased. This implies that its suction force orientation de
11、pends on the local camber and the angle of attack. and is not always normal to the camber surface at the leading edge, as it is assumed in the existing method of suction analogy. Therefore, the migrating behavior of leading-edge vortex can not he predicted without modification of the current concept
12、 of suction analogy. In addition, the exact relation between the predicted thrust forces and edge suction forces has not been derived for a cambered wing. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-The main purpose of this report is to present a
13、n improved method of suction analogy for slender cambered wings in subsonic and supersonic flow. The aforementioned deficiencies of the current method will be resolved, and comparison of experimental results with various existing methods for a variety of configurations will be given. Provided by IHS
14、Not for ResaleNo reproduction or networking permitted without license from IHS-,-,-2. LIST OF SYMBOLS A b C C cA C AV d AcD cD i cD ii =R cL cL a C m C m C P P(PM) AC P C S Ct C tip t f 1, II, i: -c i S K P K v,lle K v,se aspect ratio span chord reference chord total axial force coefficient axial fo
15、rce coefficient due to leading-edge vortex sectional induced drag coefficient = D -(Dlc =. of non-cambered wings) L total near-field induced drag coefficient total far-field induced drag coefficient in attached flow sectional lift coefficient total lift coefficient lift-curve slope at small c1 secti
16、onal pitching moment coefficient about Y-axis total pitching moment coefficient about Y-axis based on s pressure coefficient pressure coefficient calculated by Prandtl-Meyer theory lifting pressure coefficient sectional leading-edge suction coefficient sectional leading-edge thrust coefficient tip c
17、hord length unit vectors along X-, Y-, and Z-axes, respectively a unit vector normal to the wing leading edge (fig. 3) planform lift curve slope per radian at CI = 0“ leading-edge suction coefficient at one radian angle of attack side-edge suction coefficient at one radian angle of attack Provided b
18、y IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Nc -+ nQ NS -f n m r + r u, v, w V vz % x7 Y, = X2 Y2 22 an a-correction factor for the supersonic flow (eq. 54) Mach number, or number of integration points a unit vector normal to the wing surface a normal vect
19、or number of chordwise panels a unit normal vector to the wing surface at the leading edge number of spanwise strips on right wing a unit vector normal to the freestream velocity vector streamwise distance of suction force center from the leading edge as defined in figure 7. Radial distance in figur
20、e 6. position vector local semi-span leading-edge suction force vector leading-edge thrust force vector a unit vector along the leading edge a unit vector along the freestream velocity vector induced velocity components along X-, Y-, and Z-axes, respectively velocity magnitude z- component of veloci
21、ty in the vortex flow field (figure 7) circumferential velocity component in the vortex flow field (figure 7) rectangular coordinates with positive X-axis along axis of symmetry pointing downstream, positive Y-axis pointing to right, and positive Z-axis pointing upward a rectangular coordinate syste
22、m obtained by rotating the XYZ system through an angle $I about X-axis a rectangular coordinate system obtained by translating the xlylzl system along Yl-axis a rectangular coordinate system obtained by rotating theXIYIZl system through an angle CI tw about Y -axis 1 Provided by IHSNot for ResaleNo
23、reproduction or networking permitted without license from IHS-,-,-XR (Y) Zc(X,Y $X,Y) Z,(Y) a - a x-coordinate of the leading edge camber surface ordinate measured from X-Y plane camber surface ordinate measured from the mean chord plane z-coordinate of the leading edge measured from X-Y plane angle
24、 of attack average local angle of attack including twist and camber Au angle of attack correction in supersonic flow (eq. 53) atw b) 6 wing twist angle at y = sin-l(+) A, Al, A2, A3 percent of elemental panel chord by which a control point on a leading-edge panel is moved downstream. See equations (
25、47) and (48) 6 6 C Y deflection angle = tan -1 d; (2) 2 ratio of specific heats (=1.4 for air) yX x streamwise vortex density taper ratio /I leading-edge sweep angle dihedral angle = tan - (aZ,/ay2) control surface d;f 9 elemental area vector = tan- (2) ax Subscripts f flap te, R leading edge P pote
26、ntial flow r root se side edge or tip chord 6 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-te vRe vse m trailing edge leading-edge vortex flow side-edge vortex flow freestream Provided by IHSNot for ResaleNo reproduction or networking permitted wi
27、thout license from IHS-,-,-3. THEORETICAL DEVELOPMENT For the present purpose, the wing is assumed to be thin, and is cambered and twisted. The flow field satisfies the linear compressible governing equation which is solved through the use of pressure doublets (references 1 and 2). To calculate the
28、pressure distribution and other aerodynamic char- acteristics, the boundary condition of flow tangency must be satisfied. This condition and others will be developed in the following. 3.1 Boundary Condition Assume that the wing surface (fig. 1) can be described by z = Z,(X,Y (1) where z (x,y) is the
29、 ordinate of camber surface measured from the X-Y plane. C Introduce a function f such that f = z - zp,y (2) Therefore, a unit normal vector on the wing surface can be defined as a “c * a “c + + ?f -ax=-j+i: n=-= ay l$fI aZc 2 a l+(c) +2 The boundary condition on the wing surface requires that the t
30、otal velocity component normal to the surface should vanish. Hence, (Vmcosa + u)I + vf + (Vmsina + w)Z*Z = 0 (3) (4) Using equation (3, equation (4) can be expanded to give az 2Z -Vmcosa $ - v c + Vmsina + w I/ az ay 1+($)2+($)2zo (5) where u is assumed small in comparison with V cos a. co To simpli
31、fy equation (5), it is assumed that the pressure doublets are distributed on a mean chord plane which is defined to be a non-twisted plane inclined with a dihedral angle (9) to the X-Y plane. Since conventionally 8 Provided by IHSNot for ResaleNo reproduction or networking permitted without license
32、from IHS-,-,-the airfoil camber is defined with respect to its chord line (denoted here as Zc(x2, y2) and shown on fig. 2) and the twist angle (otw) is measured relative to the mean chord plane, it is convenient to express the camber slopes, azc and az, dz,dz, ax ay 3 in terms of dx 2 dy2 otw and 4
33、before equation (5) can be further simplified. This can be done through the following coordinate transformations. The original XYZ system is rotated about X-axis through an angle 4 (dihedral angle), system, and then is rotated through an angl the Yl axis to result in the X2Y2Z2 system resulting in t
34、he X Y Z 111 e atW (twist angle) about (fig. 1). In figure 1, the XiYiZi system is obtained from the XlYlZl system by a translation along the Yl axis, just as the X;Y;Z; system is related to the X Y Z 222 system. According to vector analysis, the results of such coordinate ro- tations can be obtaine
35、d by a series of orthogonal transformations (p. 413, ref. 9). aZ, It is shown in appendix A that ax can be written as: - Simi dz -sin atw + 2 cos a dx2 tw azc = .-_ ._- . ax dz cos $3 (cos a c . tw + dx, sm atw) d? -sin c az tw +$ cos atw C 2 cos eax= .- . - az, dTc cos atw + 5 s1n atw s given by d;
36、 C cos!$ . .larly, -? i ay azc sin $ cos atW t ay2 ay dz cos 9 cos a - c sin I$ tw dy2 (6) (7) (8) To simply equation (5), note that the perturbed sidewash (v) is of the first order in magnitude. Following the thin wing theory, only first order terms will be retained in equation (5). Therefore, only
37、 the zero 9 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-az order in 2 as given by equation (8) will be retained. w In equation (8), d; atw and c are both regarded as small and of the first order in magnitude. dy2 d; Hence, if the denominator in e
38、quation (8) is expanded for small , dy2 it can be easily shown that: azc ay 2: tan 9 Hence, equation (5) now becomes (9) W V azc - v, v, tan 9 =: ax cos a - sin a (lOa +- cos l$ - $ sin c$Z azc co co cos 4 ax cos a - co9 C$ sin a (lob) The left-hand side of equation (lob) represents the total induce
39、d velocity normal to the mean chord plane. Using this interpretation, equation (lob) is still applicable even if I$ = 90, azc where cos 4 ax- will be replaced with equation (7). 3.2 Relation between Leading-Edge Thrust and Suction After the lifting pressure distribution is calculated, the leading- e
40、dge thrust coefficient ct(in the negative x-direction) can be determined by using the pressure distribution as described in references (1) and (2). To calculate the suction coefficient (c,) from ct, the following steps are taken. Let Is be the unit vector along which the leading-edge suction force (
41、121) acts in attached potential flow,? L is a unit vector along the leading edge and g II is the unit vector normal to the wing surface at the leading edge. They are indicated in figure 3. It follows that: f 1 =;: xz S II R (11) where g R is given by equation (3) evaluated at the leading edge. To fi
42、nd -+ 5 note that along the leading edge, equation (1) can be written as 10 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Z,(Y) = Zc(X,Y , x = X.,(Y) (12) The position vector of any point along the leading edge can be written as 2Y = X,(Y)? + Y3 +
43、Zk(Y)r: (13) The tangent vector is determined from d: dxR t dz -1+ dy - dy dy where % - = tan A dy dzll az az -cc dy ax tan A + c ay It follows that the unit tangent vector ;6 is given by where I ,: ,; = S !L R (14) (15) (16) (17) (18) -azC -azC J a, ax ay 1 1+c*j2 dz tanA 1 -L d y azc dzll aZ az 11
44、 + fr - - ax dy + tanA + “t-a$ + $ tan/ (19) Since the suction force z is in the direction of Ts in attached potential flow and the thrust $ is the component of d in the X-Z plane, it follows that 11 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Th
45、us, from equation (20), the sectional (c,) can be obtained from the sectional (c,) as follows: I az -5 tan*/cz)2 + ()2 I$1 ay (20) leading-edge suction coefficient leading-edge thrust coefficient J 2Z 1+($)2+ (2)2 C =c .-._=L ._._._ S t s 2 + 112 + - 2 + 2 tan*212 (21) where all quantities are evalu
46、ated at the leading edge, azc/ax, azc/ay are given in equations (6) and (8), and dz E is given in equation (16). dy 3.3 Orientation of Rotated Suction Force In the method of suction analogy (reference 6) for a plane wing, the suction force predicted in the attached potential flow theory is rotated b
47、y 90 so that it is normal to the wing surface to simulate the vortex lift effect. If the same concept is used for a cambered wing, such as a delta wing with conically cambered leading edge (reference 7), then the rotated suction force, being normal to the wing surface at the leading edge, would prod
48、uce an increasingly large thrust component on the mean chord plane as angles of attack are increased. However, experimental data for a conically cambered wing (reference 7), reproduced in figure 4, indicate that at high angles of attack the locations of minimum pressure values will move inboard onto the planar portion of the wing. Since the experimental leading-edge vor- tex-induced suction force can be obtained by summin, ff all vortex-induced suc- tion pressure
