1、NATIONAL TECHNICALINFORMATION SERVICEU. S. DEPARTMENT OF COMMERCESPRINGFIELD. VA. 2216!Provided 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-,-,-REPORT No. 121THE
2、 MINIMUM INDUCED DRAG OF AEROFOILS. + sy M_X M. M-_NKNatiomd Advisory Committee for Aeronautics_/72-_1- lProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-.;“ - . _.:-:_; : . ,. rt “ _ *. : . t r ._. _ -,: , ,_o ,r. , . , ,. ,., ; * ,Provided by IHSNot
3、 for ResaleNo reproduction or networking permitted without license from IHS-,-,-i_i/LNo. 121.THE MINIMUM INDUCED DRAG OF AEROFOILS.By MAx M. MuNx., , ._i! ,.: ._: _“ ., : ,_ ,-. ! ,., ,_, . . INTRODUCTION. The following paper is a dissertation originally presented by the author to the Universityof G
4、oettingen, It was intended principally for the use of mathematicians and physicists. Theauthor is pleased to note that the paper has aroused interest in other circles, to the end thatthe National Advisory Committee for Aeronautics will make it available to a larger circle inAmerica. The following in
5、troduction has been added in order to first acquaint the readerwith the essence of the paper.In the following development all results are obtained by integrating some simple expressionsor relations. For our purposes it is sufficient, indeed, to prove the resultS for a pair of smallelements. Tlie qua
6、lities dealt with are integrable, since, under the assumptio_q we are allowedto make, they can not be affected by integrating. We have to consider only the relationsbetween any two lifting elements and to add the effects. That is to say, in the process of inte-giat_g each element occurs twice-first,
7、 as an element producing an effect, and, second, as anelement experiencing an effect. In consequence of this the symbols expressing the integrationlook somewhat confusing, and they require so much space in the mathematical expression thatthey are apt to divert the readers attention from their real m
8、eaning. We have to proceed upto three dimensional problems. Each element has to be denoted twice Coy a Latin letter andby a Greek letter), occurring twice in a different connection. The integral, therefore, is sixfold,six symbols of integration standing together and, accordingly, six differentials (
9、always the same)standing at the end of the expression, requiring almost the fourth part of the line. The meaningof this voluminous group of symbols, however, is not more complicated and not less elementarythan a single integral or even than a simple addition.In section 1 we consider one aerofoil sha
10、ped like a straight line and ask how all liftingelements, which we assume to be of equal intensity, must be arranged on this line in order tooffer the least drag. If the distribution _s the best One, the drag can not be decreased or increased by transferringone lifting element from its old position
11、(_) to some new position (b). For then either theresulting distribution would be improved by this transfer, and therefore was not best“ before, orthe transfer of an element from (5) to (a) would have this effect. Now, the share of one elementin the drag is composed of two parts. It takes share in pr
12、oducing a downwash in the neighbor-hood of the other lifting elements and, in consequence, a change in their drag. It has itself a drag, be ix_._tuated in the downwash pr_tu_ by the other elements.!_Considering only two elements,Fig, I _ that inthe case ofthe liftingstraightlinethetwo-downwashes, ea
13、ch produced by one element in the neighborhood of the other,are equal.For.thisreason the two drags of the two elements each produced by the other are equal,too,and. hence the two parts of the entire dz_:o_ the wings due to one element. The entire dragProvided by IHSNot for ResaleNo reproduction or n
14、etworking permitted without license from IHS-,-,-_ _ REPORT NATXOlCAL ADVISORY CO_TAJ_ FOR A_NAU_OS. :_: produced by one element has twice the value as the drag of that element resulting from the - downwash in its environs. Hence, the entire drag due to one element is unchanged when theelement is tr
15、ansferred from one situation to a new one of the same downwash, and the distribu-: tion is the best only if the downwash is constant over the whole wing.In sections 2 to 6 it is shown that the two parts of the drag change by the same value inall other cases, too. If the element8 are aituated intern
16、transverse plane, the two parts areequal. A glance at Fig. 2 shows that the d?wnwash produced by (1) at (2), (3), (4), and (5)t. r,_. _ J-is equal But then it also equa_ the d0wnwash dueto (4), say, produced at (1). This holdstrue evea_for the oomponeatof the downwash in the direotion of the lift if
17、 the elements ave nor-.realtoeach other (Fig. 3.); for,t_ componan_ is proporfioaal _Ly/r s, _ to the _y_nbols _ _._.of the figure. Hence, it is p_ved for lift of any inclination,horn, ntal and verfic_i,e_mtsable,by combination, to.produce liftinany direction “ _ . ;. Th. _.,_.onl_the_uestio_wheat t
18、he _o p_ of th_ -_,o_if _e“ :elem_m_ are situated one behind the other-that is to _ay, in different longitudinal position_._ii_. iTh_y!ar .not;but their sum. is independent of the longitudinal distance apart. T o. prove?- linearlongitudinalvorticesin the inversedirection. The reader observes that i_
19、hetra_.versevortices (2) and (3) neutralize each other; the lon_tudinal linear vortices, however, have the_ “ “ _me aign I and all four vortices form a pair of vortices .running from infinity to _ty. The.drag,.pr_ducect_by the Combinetion of. (1). and this pair, is obviously independent of the iongi
20、-.tudinal positions of (1) and (2). But the added element (3) has no_ changed the drag, for (1.)and ,(3) are _itua_d symmetrically and produce the same mutual dow_wash. The directiono_,_t,h_wever,:ismve_,and_.f- _etwo_,_.have_e_-_,_.d_th_,-iszero.Ifthe two liftingelements are perpendicularto eac_ Ot
21、her (chapter_),a Similarproof canbe given.Sections 6 and 7 contain the conclusions. The condition for a minimum drag does notdepend upon the lo.ngimdinal coordinates, and in order to obtain it the downwash must beassumed to be constant-at all points in a transverse plane of .a corresponding system o
22、f aero-foils. Thin is not surprising; the wings _ct like two dimensional objects _accelerat_ug the airpassing _n an _n_nlte transveme plane at a particular moment. Therefore the calculationleads _o _he consideration of the two dimensional flow _bout -theprojeotion of the wings on .a transverse plane
23、.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-_U_ INDUCED DRAG OF AEROFOILS. 5Section 8 gives the connection between the theory in perfect fluids and the phenomenonin true air. It is this connection that allows the application of the results to pr
24、actical questions.I. THE LIFTING STRAIGHT LINE.A system of aerofoils moving in an incompressible and frictionless fluid has a drag (in thedirection of its motion) if there is any lift. (perpendicular to the direction of its motion). Themagnitude of this drag depends upon the distribution of the lift
25、 over the surface of the aerofoils.Although the dimensions of the given system of aerofoils may remain unchanged, the distribu-tion of the lift can be radically altered by changes in details, such as the aerofoil section or theangle of attack. The purpose of the investigation which is given in the f
26、ollowing pages is todetermine (a) the distribution of lift which produces the least drag, and (b) the magnitude of thisminimum drag.Let us first consider a single aerofoil of such dimensions that it may be referred to withsufficient exactness as a lifting straight line, which is at right angles to t
27、he direction of its flight.The length or span of this line may be denoted by Z. Let the line coincide with the horizontal,“orz axis of a rectangular system of coordinates having its origin at the center of the aerofoil.The density of the liftA-dAwhere -4, the entire lift from the left end of the win
28、g up to the point z, is generally a function ofz and may be denoted byf (z). Let tim velocity of flight be re.The modern theory of flight I allows the entire drag to be expressed as a definite doubleintegral, if certain simplifying assumptions are made. In order to find this integral, it is, neces-s
29、ary to determine the intensity of the longitudinal vortices width run_from any lifting dementto infinity in a direction opposite to the direction of flight. These vortices are generallydistributed continuously along the whole aerofoil, and their intensity per unit length of ther,L-_.p . dA-_ (2)Wher
30、e p is the density of the fluid. Now, for each lifting element d,_, we shall calcUiate the down-wash _v, which, in accordance with the Iaw of Biot-Savart, is produced at it Dy all the longi-tudinal vortices. A single vortex, beginning at the point z, produces at the point z-_ thedownwa_h_ , “clw- dA
31、 _-z (3)Therefore the entire downwash at the point _ is “: “ “ “ - +I , i: . ., ,w-_- dz , (4)JThe integration is to be performed along the aerofoil; and the principal value_ _ _-_be takenatthepointz-_. This_e _I_oappliestoallofthefolo_ in_ _follows that tha drag according to the equation : 1“ : _ ,
32、 _ “ :“ “ “ r “ “ _ “is+Z t +l Ad_ . _ . t-s _ L. Prsndtl, Tragflllg_lth_ie, I. Mittellun|. Nachrichten d_ G_. _L WL_. zu G6t_, 1918.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-;._.“7_:_2,_,_z _-_._-_-_._ - L_“_ _j_ -_;6- :_ _ _. _: -_:,_. = L_“_
33、 _ _.-_. ? = ._ _ =;fi_ m-.J “ _L :r; , . 3_POBT NATIONAL _DVISOBY COMM1TTY_ FOlt _gRONAUTI_.- ? 4 , (e)f here _zxi_es the ,derivative of f with respect to z or .“The entire lift is represented by ;:Hence the solution of the problem to determine the best distribution of lift depends upon “the det_fi
34、on of the f_ctionfso fl_t the double integral ;, shall have a value as smaU as possible; while at the same time the value of the simple integralJ,- f/ (=)_-eomt, _, _ ._ _ . :, (9) The _n_t step towards the solution of this problem k to_cnm the .first var_tion of-Jr _,-:, , The second integral on th
35、e right side of (10) can be reduced to the fu_ By exche_ng the :symbo_ z and _ and by partial integration with respect to z, consideringf(_)as the integrabletsctor, there is obtained +,1 _g “ ,i,1 ! +It _ :_ ,:.,/The second member disappears since f-O at the limits of integration.“ Further, the righ
36、thand part o! (11) “ _,1 _ upon substitution of the new varisbles z and t-z-_ for z and ;is transformed intO _i;_,.,. . - “. :. “ : - L I 1 I _ . - I i If thk ws_ aet trus, them weuld be laaaltt v_e/tlss st them palnla. _ .Provided by IHSNot for ResaleNo reproduction or networking permitted without
37、license from IHS-,-,-J“ _ MINIMUM II_DUC_D DRAG OF AEROFOILS.NOW ._“ .so that, finally, , /Z f/ Z ,_I.+. i z _ .z*_ ,+or,sincef disappearsat the limitsofintegration,. . , , “:z_/ s+y ,which, upon the replacement of the originalvaxiables,becomes+_;f(0d_J+; +_ “ .+_ “ +t “ .- _ “_ _ -_S_tituting thin.
38、in (10).therefinallyresults,: _,- s/(x)#a a_- -.(12). . , , _.(is)From which ,_ e_mditio_ forthe minimum amount of drag,takingintoconsiderationthesecond oondition (9),is,_(0d_+_= 0 04)or,when equation (4)istal_enintoconsideration, “- _“ “_eon_t:-_ : - - (15)“ _ _sa_ o_u_ for the m_nim_m of _ for a l
39、i/t_n_ stra_J_ Z_ is _ th_ doum-u_sh ?ro_ce_ by thelon_zl _,_s be constant alo_ O_ _re line.That this necessary consideration is also sufficient results from the obvious meaning ofthe second variation, which represents the infinitesimal drag produced by the variation of thelift if it alone is acting
40、, and _herefore it is always greater than zero., .,- : ;. 9_p_s._u_ U_L_G _ LrmO m A sv_ss_:_ “ The method just developed may be applied at once to problems of a more general nature.If, instead of a single aerofoil, there are several aerofoils in the same straight line perpen-dicular to the directio
41、n of flight, only the limits of integration are changed in the development.The integration in such cases is to be performed along all of the aerofoils. However, this isnone_mtiql for all of the equations and therefore the condition for the minimum drag (equa-tion 15) applies to this entire system of
42、 aerofoils. . - ._. _ _ .1._._.,_ _, -_: ,+,_-_ “_:. “,_:_!_I “ _:_ :,._- .:_ :,;-. ;.,_. J_, ., ,_ _,. _- “ -.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-REPORT NATIONAL ADYIS0_Y CO_CM3_TEE FOR AEIONAUTIC_Let us now discard the condition that al
43、l of the lifting lines ai-e lying in the same straightline, but retain, however, the condition that they are parallel to each other, perpendicular tothe line of flight as before, and that they are all lying in a plane perpendicular to the line offlight. Let the height of any lifting: line be designa
44、ted by z or l. Equation (3) transforms intoa similar one which gives the downwash ,.produced at the point x, z by the longitudinal vortexbeginning on the lifting element at the point _:JA b _ - (E-_P +-(.I-z)_: (Sa)JThe expression, which must n_wbea minimum,-Is “-i._J,- . r) z (swith the unchanged s
45、econdary conditionThese integrals are to be taken over all of the aerofoils.This new problem may be treated in the same manm_ as the first. _,.-:L._,;i ,:. ,_(.I;. !-,.: , , . * m, _1is always to be substituted for _-_- It may l_, _hbwn .thkt,this sulmtitutkm :dom_mot-: affect the correctness of equ
46、ations (10) to (15). Therefore:“-. _, _I) ,.- !._, _:,.,- _ :S iS again obtained as the necessary condition for the minimum of the entire drag. “ “., , -.- “_ “ Finally,this_L_o halds true for the limiting ease in .which, over a limited.portion of,_transverse plane, the individual aerofoils, like ve
47、netian, blinds, lie so ,closely togethel_._thatthey may be considered as a continuous lifting part of a plane. Including all cases whichhave. been considered so fax, the condition for a minimum of drag can be stated:Let the dimenM,on_ of a sya.em of aerofat_s be #t_m, _tXo_ in the dlreaion of flight
48、 bdn 9 smallin _ml_r_on with those in o_er direet_on_. Let t_ lift be everywhere directed ve_ca_y. Underthese_ndi_iom, the dawnwa_h praduc_ by the longitudinal vort_ m_t be uniform at al_ pair_on th_ amofage in order that there may be a minimumof.drag for a ._ total:lifl_ ;:. , , _,_“ (!_ swhere “ “ “andThe entire drag is“ _4rJ ewds; e - (_-zP + (,_-s)+ _- z. (21)Now, in the double integral (22) the variables z, _, z may
