1、iTdTIOTAL ADVISOZT 2OKI:iITTEE ?OR AZBONaUTICS NftC A-Th- No. 856 -_- THT LIFT DISTRIBUTIOB OF VIHGS BITH END PLATES - Luf t f ahrtf or schmg Verlag yon R. Oldenbourg, Pfunchen und Berlin VoL. 14, KO. 11, XovemGer 20, 1937 “_ - REPRODUCED BY NATIONAL TECHNICAL INFORMATION SERVICE U S DEPARTMENT OF C
2、OMMERCE SPRINGFIELD, VA. 22161 -_ I 1vc s h ing t on April 1938 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NOTICE THIS DOCUM.ENT HAS BEEN REPRODUCED ._ FROM THE BEST COPY FURNISHED us BY THE SPONSORING AGENCY. ALTHOUGH.IT IS RECOGNIZED THAT CERT
3、AIN PORTIONS ARE ILLEGIBLE, IT IS BEING RELEASED IN THE INTEREST OF MAKING AVAILABLE AS MUCR INFORMATION AS POSSIBLE. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-HATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM NO. 85 6 THE LIFT D
4、ISTBIBUTION OF WINGS VITH END PLATES* By If. Mangler s u ME11519 Y This report describes the lift distribution on wings with end plates for the case of minimum induced drag (in- duced donnivash constant over the span). The moments on the end plates are also determined. It is found that moving an end
5、 plate of certain length uy from the symmetrical position, is followed by a slight increase of the total lift. As a marked increase in the moments of the end plate about its attachment also results, the symmetrical end plates are most advantageous if a higher circulation on the wing only is contempl
6、ated. But, if the end plates.are to serve for producing hori- zontal control forces, a pronounced unsymmetrical arrange- ment nil1 be preferred. The magnitude of the ensuing side force for end plates of any length and position is illustrated in figure 13. The ratio of coefficient-s of side force and
7、 of lift can be interpolated from figure 14. For the application of the side force, an empirical formula (22) is given. The mriter wishes to express his appreciation to Miss 1. Lotz, for the many suggestions in the preparation of the study. NO TAT I ON b, wing span. h, height of end plate. k, ratio
8、of height of the upper part of the end plate to s e mi sp an. -_-_c_I_- *“Die Auftriebsverteilung am Tragf1;gcl mit Endscheiben. “ Luftfahrtforschung, vole 14, no. 11, November 20, 1937, A* DD. 564-569, Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,
9、-2 B.A.C.A. Technical Memorandum No. 856 k, ratio of heiKht of the lower part of the end plate to b/2. k = -, k0 ratio of height of unper to height of lomcr part plates). k of the end plate (k = 1 fur symmetrical end h ko f IC, - - - 3 , ratio of height of end ?late to span. b L. b , coordinate (ref
10、erred to = 1) along the span. V, coordinate (referred to - = 1) along the end plate. b 2 - = + iq, coordina.tes in a plane through the vortex sur- face shed by the wing (referred to b/2.= 1). r , circulation. A, lift. c, lift coefficient. Tiy induced drag. cwi, drag coefficient. mi, rrte of damnwash
11、 induced by the vortices leavins the “lr$y. 2TJi = !I rate of clownwash inciuced by the vortices at great c:.ist,ziice pehind the wing. V, air sTeed. p, 3iT density. Pa 2 q = - V , dparnic pressure. F = b t, ning area. So (or SUI, side force on upper (or loner) part of the end plate. Provided by IHS
12、Not for ResaleNo reproduction or networking permitted without license from IHS-,-,-N,A.C .A. Technical Memorandum No, 856 3 . EOTATIOIJ (Con t , Fo (or Pu), c (or c ), side force coefficient referred to Fo (or SO %l area of upper (or lover) part of end plate. FU) Mo (or I the section nhich is at rcs
13、t is :.nFroached fron bclov at thc constant speed m = 219. If the complex otential I = 0 +- i$ of this flow - is knonil, the ckrculation strength 2,t a point of the 1Vilie or !?oint Tl of the end plate (fig. l), is equal to -the ju:i:3 suffered by the yotcntial - nhen changing at poilit E, correspon
14、ding to point or point 1 + iq, corre-. npondilig to v, respectively, from one side of the vortex layer to the other (figs. 1 and 2). T.he section being a streanline (reference 4), $ is constant. - t:.lcrc Q1 and $2 are the potential ve.lues for point or 1 + iq). Then the lift becomes: -1 Vith this +
15、1 -1 or, mctking I? dimensionless with w - b .: 2 +1 Tkus, me havc: The in4uced angle of attack ai becomes: Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-1J.A:C.A. Technical Memorandum No. 856 which nith -1 gives t:ie drag coefficient 5 In this for
16、m the factor K = - gives at once the departure of the drag coefficient from that of the ellip- tical wing. For equal lift and dynamic pressure a wing of span b nith end plates has the same induced drag as a wing of - span without end plates. 25 fi For the determination of the side forces and moments
17、 at the end ?late, the sense of rotation of the bound. and free vortices is given in figure 3. Seen from above, the bound vortices of the right-hand end. glate turn in the ug- per art in yositive direction; in the lover part, in neg- ative direction; so that the air speed V causes a force perpendicu
18、lar to V and to the elid plate. In the upper part this force is inwardly, in the loner part outwardly, directed. The forces created by the right and left end plates cancel each other, but exert a moment about the point of attachment of the end platcc. For the side force So, in the upper part of the
19、end plate, ne have : I) kO 0 rlvhere ?7e write: Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-6 N.A.C.A. Technical Menornndum No. 856 TLGl IT) (7) Jo -, b - 0 To w2 and. find from equ,tions (2) and (6) in conjunction cith equction (4), the ratio of
20、 side force to lift at: nnd correspondingly for ,the lower part: -u S - - JU A 25 (9) nhere and 0 The side force coefficient c is introduced through the r e 1 at i on SO P2 so = cso - 2 V Fo nhere Fo clecotes the area of the upper part of the end plat e. Assuming the wing to be rectangular (F = b t
21、tvith elligtical end plates we find: 71 TT (F0 = 8 ko b t; I?, = - k, b t) 8 and Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-J NJ,A,C.A. Technical Memorandum No. 856 7 The moment Mo of the side force So, is given by 0 Vith equations (3) and (5) t
22、he moment about the upner part of the end nlate becomes: M4 J* ,L = ,D b 2J A- 2 and a3out the lower part: if 1778 nut: * JO and A h 25 2 0 0 b 2 m- Finally, equations (lo), (8), and (11) yield for %he lever arm at the upper nart of the plate, the formu12,: and at the lower ?art: * J U Tu = - JU Pro
23、vided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-8 I I I . TRANSFOSllATION The flom potential 67 (5) in the (e, T) plane is determined by so transforning the (e, T) plane through an analytical function f: = f(z) into the z plane (z = x f iy) that the rig
24、ht half plane (e 0) is trans- formed into the upper z-half lane (y 0) c?,t the same tinc RS the right-hand part of the section a.nd the ina.gin- ary c-axis changes into the real z-axis (fig. 4). Then the C2.o: Potential C1J (z) an6 consequcntly, also, Ci (5) can be readily established. The desired t
25、ransformation could be easily expressed m i t 11 the S ch172r z- Chr i s to f f e 1 t ran s f o rma t i on f o rmul a, tv h i ch would qivc 2.n elliptical integral nhose parameters - the noints of the z-pla1:e for thc! corners of the l-plane, could h?.ve to be dctermincd through the Geometrical quan
26、- tities of the C-Flanc. But as this nould entcil the so- lution 02 a system of five transcendent equations nith five unlcnonn quantities, a, difyerent nay, an approximcte method, is preferable. The desired transformation is bnilt up in three stages: T:?c half nlme 0, to which we can confine our- se
27、lves for rczsons of synmetry, goes into the zt-plane (fig:. 5 and 6). I The straight pieces BC and ED become the pnrcbol- ic crcn atCr and EtDt. It is permissible to assume th.r,t thc rz,tio k of the distances BC and ED is 2 1:k= 50 ? - 1. In gddition, since me 17e chose the most convenient one.) Th
28、is menns thst the investigation tcnds toward end plates with sm,o.ll outnsrd curvature rather than flat ones. This curva- ture, nhich depends on h/b, is negligibly small in our case Po ill t Dr = 41 - ku) - 21 ku A linear transformation trp,i.sforns the point Lt into point 03; L1 clle circle thcrcfo
29、rc bccomcs a straight line. Toint Ht = OD bcconcs Faint HI1 = 4 + ko2, and the end points C* nnd Dt bccornc the points 2 ,/ 1 + ;? 1 (ko2 - G2) The treatment thus reduces to the flow of a source and sink at point HI1 ;Lbout the surfaces, illustrated in figure 7. Up to tcrms of higher order and a rea
30、l factor, the gotentin1 in the vicinity of point 3“ = ;lett has the f 0 rm - 1 . This is casilg treated with the Schnarz- Christoffel formula. J z“ a It 3d stage: Tho outside of polygon A“ B“ C“ D“ E“ F“ G II H It L It is traniforned into thc upper z-half ?lane (y 0) (fig. 8). Thc effcct of the appr
31、oximate method is that the polygon nof; contains only two angles of mngni- Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-10 N.A.C.A. Technical 1,lenorandurn KO. 856 tudc /2, and three angles of magnitude 2n. The inte- gral follorrigg the apglicatio
32、n of thc Schnarz-Christoffel fornula thcrcforc lends itself to evaluation nithout el- liptical integrals : w1 deilotcs CL scale factor so chosen that point z“ = 1 corresponds to point a,. Oaing to the condition that the integral should be real for real z a5 = 0 (fig. 81, and the further condition th
33、at the integral from a1 to a4 must be zero, it can even be expressed. by c,n algebraic fun c t ion : Here : The prefix of the square root is to be so chosen that the uu-pcr half plane (p 0) is transformed into the z“-?lane; real z and z g nl Fare to carry thc negztive prefix, z = z4 the positive sig
34、n, while for c1 r z 6 a4: Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-7.J.A.C.A. Technical I,!emorandum No. 856 11 FroE these equations a1 and a4 can be com.nuted. a3, a6 h2s the desired characteristics, .namely, that A furthnr deduction is that
35、the function (18) in the points - aztt becomes zero at these goints. For this reason, there dz is no linear tcrm at thcse uoints in the power series de- VeloFncilt for ztl; (211 - ai!) is progortional to (2- ai) . (aitt is the image point of ai; i = 2, 3, 6.) As the Fosition of points CIt, Dtt, 2nd
36、Htl is defir,ed by the position of C, D, and H Ln the -plane, thc still free parameters a2, a3, and a. nust be so chosen that point C“ corresponds to point a2, point D“ to point a3, c?nd-point HI to point a6 = 1. For a2 and a3 (which clone occur in the first tno conditions), ne can restrict ourselve
37、s to the range of The case of the ;rir,g nithout end plates (elliptical lift distribution) is obtained chen putting a, = a2 = a3 = a4 = - a6 = - 1 (ko =.ku = 0) nhile a2 a3 = a1 a4 = 2c62 = 1 gives the rrings with sym- netrical end plates (reference 3) by a different method. (ko = k,) as computed by
38、 2. Nagel IV, THE POTENTIAL Since our aim in the -plane at a great distance from the section is a flon parallel to the Q-axis cith a speed 1- (fig. 51, the development Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-12 N.A.C.A. Technical Menorandun N
39、o. 856 is viiitl for the potentin1 Cl(c) in the vicinity of = 43. thnt is, (171, and (18) in the following form: M The transforcd function in the vicinity of 5 = 43, z = a6 can be represented from equations (161, (1 -t- . )= i (1 + . D 0.) (20) -_- M ( = i _-_ z -. a6 z-1 Here Then the Fotential C17
40、() has, according to equations (19) and (20), the value: . For real z, $ = 0, i.e., a doublet with the real axis 2,s 2“ streamline. This result is anticipated (figs. 5 to 8). KO further terms occur in the development Of CIIl(z)* Since equations (16), (17), and (18) afforded no Sin- ple exprcssion fo
41、r z as function of (, several mathe- rnzticnl exzmples mere computed. First the parameters ai were d-eternined. Then the correlated image points in the c- glanc were computed for z values from the inter- val a. 5 z 2 a45 (point z,45 corresponds to point F) i Provided by IHSNot for ResaleNo reproduct
42、ion or networking permitted without license from IHS-,-,-B.A.C.A. Technical Memorandum No. 856 13 I nith the aid of equations (16), (l7), and (18). The per- tinent potentin1 C(7(z) = (2) mas obtained from cqus- tion (20), after ahich the circulation r(F) and rm), respectively, follons from equation
43、(I). iJote.- Tho circulati.on on nings nith plates (tail surfaces) of the form of figure 9 can be coaputed in ex- actly the same manner. The nuclericnl results are to be published at a future dc?te. V 0 NUIJER I CAL RE SUL T S Tipye 10 illustrates various circulation distribu- tions r() over the nin
44、g and r(q) over the end plate as obtz.ined nith equation (1). It discloses that r(T is substantizlly dependent on the height of the end plates; i.e., 011 the ratio h/b, and that shifting the point of nttcchment of the end plate has no material effect on the total circulation. But near to the end pla
45、te itself, r(f) is tependent upon the t;rpe of the plate attachment. Accordiiig to equation (l), r is the difference of tno potentials. They clnays proceed at the aing tip nith ver- tical tmgent, if the end plate does not protrude on the pcrticulcr side of the ning; otherwise, the tcngont is horizon
46、tcJ. In the first csse the flow velocity from pros- surc tonard suction side is infinite at the plate nttach- ment point; in the second casc, it is zero, Thus, for an en6 plcte at one tip k, = 01, r(E) - has at point f = 1 a vertical tangent; othcrniso, the tangent is horizon tal. (ko = 0 or - Tr (e
47、quation 5). that is, In “guro 11 the value K = - 25 CW i the ratio - is shonn agcinst k, for diffcrent vnlues of h/b, The total circulation is smallest in the symmetric21 case shifting of the point of attachment of the end platc re- sults in no appreciable change of ed from figure IO). F Ca2 ;iZ (ko
48、 = k, k = l), (equation 41, but - 2J = - 1 (as anticipct- 7T K T!,c i,oint of applicstion Qo of the side force So at thc end- plate as ne11 as 7, was dotermined from equn- Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-tions (14) an? (15). It 3s found tlizt no was apFroxi- matcly :ro:ortionnl to ko and vu to k,. (22) The va
