1、NACA-TR-151REPORT No. 151GENERAL BIPLANE THEORYIN FOUR PARTSBYMAX M. MUNKNational Advisory Committee forAeronautics99:76 - 22 -1 eRpR01)i.tC D BYNATIONAL TECHNICALINFORMATION SERVICEU.$. OEPARIMNI Of COMMRCSPRINGFILD, YA. Z216iProvided by IHSNot for ResaleNo reproduction or networking permitted with
2、out license from IHS-,-,-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-INDEX.Page.1. Introduction 5THE Two-DIMENSIONAL FLOW NEGLECTING _ISCOSITY.2. General method 73. The biplane without st_tgger and with equal and parallel wings . 94. The biplane
3、with different sections, chords, and with decalage 155. The staggered biplane 17THE INFLUENCE OF THE LATERAL DIMENSIONS.6. The aerodynamieal induction 20THE DETERMINATION OF THE WING FORCEfl BY THE DESIG.XER.257. The absolute coefficients .8. Determination of the drag . 26279. Determination of the a
4、ngle of attack I0. Determination of the moment 281I. Conclusion . 29TABLES AND DIAGRAMS.1. Two-dimensional flow without stagger . 312. Two-dimensional flow with stagger . 323. Aerodynamical induction . 324. Table for the calculation of horsepower 335. Table of the induced drag coefficients 346. Tabl
5、e of the induced angle of attack 387. Table of dynamic pressure . 423PrecedingpageblankProvided 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. 151.GE
6、NERAL BIPLANE THEORY.By M,_x M. MU_KSUMMARY.The following report deals with the air forces on a biplane cellule.The first part deals with the two-dimensional problem neglecting viscosity. For the fLrst timea method is employed which takes the properties of the wing section into consideration. Thevar
7、iation of the section, chord, gap, stagger, and decalage are investigated, a great number ofexamples are calculated and all numerical results are given in tables. For the biplane withoutstagger it is found that the loss of lift in consequence of the mutual influence of the two wingsections is only h
8、alf as much if the lift is produced by the curvature of the section, as it is whenthe lift is produced by the inclination of the chord to the direction of motion.The second part deals with the influence of the lateral dimensions. This has been treatedin former papers of the author, but the investiga
9、tion of the staggered biplane is new. It isfound that the h)ss of lift due to induction is almost unchanged whether the biplane is staggeredor not.The third part is intended for practical use and can be read without knowledge of the firstand second parts. The conclusions from the previous investigat
10、ions are drawn, viscosityand experimental experience are brought in, and the method is simplified for practical applica-tion. Simple formulas give the drag, lift, and moment. In order to make the use of the simpleformulas still more convenient, tables for the dynamical pressure, induced drag, and an
11、gle arcadded, so that practically no computation is needed for the application of the results.1. INTRODUCTION,The appearance of a treatise on the aerodynamics of the biplane cellule, including themonoplane as a particular case, needs hardly any apology at the present time. For the wings,which primar
12、ily enable the heavier-than-air craft to fly, are its most important part and deter-mine the dimensions of all the other parts. The knowledge of the air forces produced by thewings is of great practical use for the designer, and the understanding of the phenomenon isthe main theme of the aerodynamic
13、al physicist. In spite of this the present knowledge on thesubject is still very limited. The numerous empirical results are not systematically inter-preted. The only general theory dealing with the subject, that is, the vortex theory of Dr. L.Prandtl and Dr. A. Betz, gives no information concerning
14、 the influence of different sections.nor on the position of the center of pressure. This theory is indeed very useful, by giving aphysical explanation of the phenomena. But the procedure is not quite adequate for obtain-ing exact numerical information nor is it simple enough. The theory of the aerod
15、ynamicalinduction of biplanes, on the other hand, is developed only so fax“ as to We the induced drag,but not the individual lift of each wing.I hope. therefore, that the following iv- ;stigation will be favorably received. I try in itto explain the phenomena, to calculate the numerical values, and
16、to lay down the results insuch a form as to enable the reader to derive practical profit from the use of the given formulas,tables, and diagrams without much effort.The problem of the motion of the fluid produced by a pair of tmrofoils moving in it is a three-dimensional problem and a very complicat
17、ed one. The physical laws governing it are simple,indeed, in detail, as long as only very small parts of the space are concerned. But the effect on6PrecedingpageblankProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-REPORT NATIONAL ADVISORY COMMITTEE F
18、OR AERONAUTICS.the fluid at large can not be predicted with safety _ithout reference to experience. Tlle vis-cosity of the fluid plays a strange part, though not quite without analogy with the frictionbetween solid bodies gliding along each other or with the behavior of structural members. Forunder
19、certain conditions the forces produced by a mechanical gear can be calculated withoutpaying much attention to the friction. But often this can not be done. as in the case of a screwwith narrow thread which does not turn its nut if a force in the direction of its axis is applied.as it would do withou
20、t friction. The deformation of structural members follo_ a certain lawonly up to a certain limit; then another law suddenly replaces the first one. The behavior ofthe air around a biplane also can be investigated independently of the viscosity under certainconditions only, and it is not yet possible
21、 to express these conditions. If the viscosity can beneglected at first, its small influence can be taken into account afterwards by making use ofempirical results. This case alone is the subject of the following report. It is the most impor-tant one. But this paper also refers to the more difficult
22、 part of the problem. This can notbe solved without systematic series of tests, but for the interpretation of these tests, to be madein the future, the following results are hoped to be useful. For the influence of friction isalways associated with the influence of other variables, and it can not be
23、 separated from themunless the original and ideal phenomenon without friction is known.The phenomenon in a nonviscous fluid is still three dimensional and complicated enough,and we are far from being able to describe even this completely. Consider a single aerofoil.In the middle section the directio
24、n of the air indeed is parallel to the plane of symmetry. Atsome distance from it it is no longer so, and so far as it can be described approximately by atwo-dimensional flow, this flow is different at different sections. Near the ends the flow isdistinctly three dimensional. On the upper side the d
25、irection of the air flow near the surfaceis inclined toward the center, on the lower side it is inclined toward the ends and finally flowsaround the ends. It is a fortunate circumstance however that along the greatest part of thespan the flow is almost two dimensional. Moreover, most of the variable
26、s are linearly connectedwith each other, and hence the effect can easily be summed up to an average. Hence, the con-sideration of the two-dimensional problem is a very useful method to clear up all questionswhich refer to the variables given in the two-dimensional section; these are not only the dim
27、en-sions of the wing section but also chord, gap, stagger, and decalage. The truth of this pro-cedure is felt. intuitively by everybody who considers the wing section separately. This prob-lem will be discussed in the first part of this paper. The results are useful however only bycombining them wit
28、h the effect of the dimensions in the direction of the span. This effect isdiscussed in the second part. The third part will contain the consideration of the viscosityand the final results for the use of the designer, developed not only from the preceding theorybut also by taking into consideration
29、the results of experience. The fourth and last part con-tams a list of the important formulas and the necessary tables.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-TWO-DIMENSIONAL FLOW NEGLECTING VISCOSITY.2. GENERAL METHOD.In order toinvestigate
30、the influence of two aerofoils on each other, I take into accountthe fact that the dimensions of the wings at right angles to the chord are generally small whencompared with either the chord or the gap. It can not be assumed, however, that the chordis small when compared with the gap. On the contrar
31、y, it is often greater than the gap. Thefirst assumption reduces the problem to the consideration of the influence of two flat plateson each other, or, as I will generally express myself throughout this part, the mutual influenc(,of two limited straight lines. This does not mean, however, that I hlt
32、end to confine mys,lfto considering the effect of this particular section only, as for one particular case has beendone by Dr. W. M. Kutta (ref. 5). The flow around a straight line is by no means deter-mined by the genera conditions governing potential flows, but in addition to these th(, charactero
33、f the flow near the rear edge is to be taken into account. I do not intend to choose this lastadditional condition indiscriminately, and the same for any wing section; besides, th, (lecisi,*nas to the direction of the straight line to be substituted for the wing section must be made.The effect of th
34、e direction of this wing section-that is, of the angle of attack-is ,xpressedby the moment of the air force produced about the center of the whag. If the angl, of attackof a section shaped like a straight line is zero. this moment of course i_ zero. The most suc-_- - ;_FIG. I,-_ecLion flo_ _ ithouI
35、cilcul:t: ion._ Anq/e zFI,. 2.c_,ssful proceeding is th_,reforc to ,hoes(, th, dir,tion ,)fthe substitut,d straight line s() as togive always the saint, moment around the center as th, repla(:ed section does. An easy methodfor the calculatiou of this momcut is discussed by me in a former paper. (Ref
36、. 3.) For thepresent discussion it is not essential whether the moment is determined in the way describedthere or by any other thdoretical or empirical method. The direction of the straight linedetermined according to this precept always becomes nearly parallel to the chord of the section.This is pa
37、rticularly true if the section is not S-shaped; but even then the angle between thechord of the section and the substituted straight line will seldom exceed 2 . This angle is2/rC,_ where (_o denotes the coefficient of the moment about the center of the section at zeroangle of attack. It is always sm
38、all. The assumption of a straight line not exactly parallelto the chord is thus justified, as it will always run near the points of the chord. (Fig. 2.) Onesuch isolated substituted straight line at the angle of attack, zero, thus experiences no moment,but the air force due to the physical straight
39、line in that position wouhl still be different fromthat of the replaced wing section, for the lift of the straight line is zero, too, but this is notso in general for the actual wing section, in consequence of its curvature.Consider the theoretical flow of smallest kinetic energy around the wing sec
40、tion insteadof the flow actually occurring. (Fig. l.) The former flow has no circulation around the wing;7Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-REPOIIT.X-ATIO_AI, ADVISORY (O_I3ITTEE FOR AER()._:AITI(S.that is to say, the velocity integral
41、is not increased if a closed path around the section is taken.Hence the lift is zero and a straight line at the angle of attack, zero, can be taken as the mostperfect substitution among all straight lines, for the air produces neither lift nor momentin either case. The effect of the wing section on
42、the flow at some distance is very small in thecase of this flow without circulation. It can be assumed, therefore, that two such wings, pro-ducing individually neither moment nor lift, have the smallest influence possible on each otherat the usual distance and continue to experience no air forces wh
43、en arranged ill pairs. Theinfluence, indeed, can be entirely described by sources and sinks, and I have shown in a formerpaper (ref. 4) that such influence is always exceedingly small. I have thus arrived at twostraight lines replacing two sections in the particular case that the moment is zero in c
44、onse-quence of the particular angle of attack, and the lift is zero in consequence, of the flow arti-ficially chosen without circulation. Now it is easier to fix the thoughts if the different thingsoccurring are designated by particular names. I will call this particular flow around the sectionwitho
45、ut lift and moment the “section flow.“ (Fig. 1.) It differs from the flow around thetwo straight lines only in the neighborhood of the section, but there it differs very much, forat the rear edge the velocity of the section flow (which we remember is only imaginary) isinfinite. This infinite velocit
46、y near the rear edge, which I will call “edge velocity“ for sakeof brevity, is the reason why the pure section flow generally does not really occur but hassuperposed on it a second type of flow with circulation (Fig. 3) in such a way that the edgevelocity becomes finite. The “circulation flow,“ as I
47、 will call the second type, possesses anFIG. 3.-Longitud|nal flow. Fxc,. 4.-Vertical flow. FxG. 5.-Circulation flow. Flo. 6.-Countcr-circutaton flow.infinite velocity also at the rear edge, but opposite to the previous one, and the superpositionof section flow and circulation flow makes the infinite
48、 velocity vanish.The magnitude of the infinity of the edge velocity can still be different in different cases.for it is infinite only directly at the edge. Near the edge, in this assumed case of an angle of 360 of the edge, it is proportional to 1/_/_-, where _denotes a small distance from the edge.
49、 The mag-nitude of the edge velocity at each point is given by an expression rn/v/-_ where m is a constantnear the edge; and for two different conditions the edge velocities, though infinite both times, candiffer from each other by different value of the factor m. The superposed circulation flow isdetermined by the condition that its edge velocity i
