NASA-TR-D-121-1923 The minimum induced drag of aerofoils《翼面的最小诱导阻力》.pdf
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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
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