1、.a71-= REPORT No. 116APPLICATIONS OF MODERN HYDRODYNAMICSTO AERONAUTICS TWO PABTSBy L. PRANDTLGWiugen UniversityTrandatedby theStaffof theNaffonaIAdvisoryComndtteefor Aeronautics 167Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-.REPORT No. 116.APPL
2、ICATIONS OF MODERN HYDRODYNAMICS TOBy L. PEANDTL.AERONAUTICSPREFACE.I have been requested by the-United States National Advisory Committee for Aeronauticto prepare for the reports of the committee a detailed treatise on the present condition of thosa-.-applications of hydrodynamic which lead to the
3、cakulation of the forces acting on airphmewings and airship bodies. I have acceded to the request of the Nat.ionrdAdvisory Committeeall the more wiIlingly because the theories in question have at this time reached a certain con-.clusion where it is worth while to show in a comprehensive manner the l
4、eading ideas and theresults of these theories and to indicate what contlrmation the theoretical remdtahave receivedby tests.The report WN give in a rather brief Part I an introduction to hydrodynamics which iswhat happens is, thi finrd fluid layer immediately in cent act with the body isattached to
5、it (ii at rest relative to it), and all the friction of fluids with solid bodies is therefore”an internal friction of the fluid. Theory and experiment agree in indicat there-fore vortices are formed there very soon after the flow begins, and consequently the resultswhich are ftihed by the theory of
6、nonviscous flow can not be applied. On the other hand,in the rear of very tapering bodies the retardations are often so small that thereis no noticeableformation of vortices. The principal successful results of hydrodynamics apply to this case.Site it is these tapering bock -which offer specially am
7、alI resistance and which, therefore,have found special consideration in aeronautics under simik applications, the theory can bemade useful exactly for those bodies which are of most t the difference therefore isvery great. Wth water it is nearly always allowable to neglect the changes in volume aris
8、ingfrom the pressure differences due to the motions, and therefore to treat it as absolutely incom-pressible. J3ut also in the case of motions in air we can ignore the corppressibility so long asthe pressure dHerences caused by the motion are suiliciently small Consideration of compressi-bility in t
9、he mathematical treatment of flow phenomena introduces such great difficulties thatwe will quietly neglect volume changes of seved per cent, and in the calcrdations air will belooked upon as incompressible. A compression of 3 per cent, for instance, occurs in front of abody which is being moved with
10、 a velocity of about 80 m./sec, It is seen, then, that it appearsallowable to neglect the compressibility in the ordinary applications to technical aeronautic.Only with the bladw ofo or dthe coordinates Z-red y, a velociy perpendicular to the plane AllF, of-the amount,-*l)l=rz; The image of the airp
11、lane furnishes an equsJ amount perpendicular to the plane ABF.If P k the angle betwikn the plane ABF %=+I+aw=2 2 d.J( ) 2+8in which a= .9(30)If the flight velocity is V, this gives for the inclination of the downward sloping air-currenttan P= $“ We proved this relation in the year 1911 and found an
12、approximate agreementwith observation.The principle made use of above his been applied with profit tti the calculation of theinfluence of one wing of a biplane upon the other wing and has given a method for the calcula-tion of the properties of a biplane from the properties of a single wing as found
13、 by experiments,The fundamental idea, which is always applied in such calculations, is that, owing to tho vortexsystem of one wing, the velocity field near the wing is disturbed, and it is assumed that ti wing “experiences the same lift as in an undisturbed air stream if it cuts the stretimlincs of.
14、 the flowdisturbed by the other wing in the same manner as a monoplane wing cuts the straight stream-lines of the undisturbed flow. As is easily seen, the wing profile must in general bo slightlyturned and its curvature slightly altered, as is shown in figures 38 and 39, By the rotation ofthe wing t
15、he direcion of the redtant air force acting on it is turned through an equal angle.If the magnitude of the velocity as welI as its direction is dso changed, this must be expressedby a corresponding change in the resulttmt air foce.Provided by IHSNot for ResaleNo reproduction or networking permitted
16、without license from IHS-,-,-APPLICATION% OF MODERN HYDRQDYXAMICS TO AERONAUTICS 185 . . .-.As an illustration we will treat”briefly the case of a biplane without stagger. The mostimportant component of the disturbance velocity w is again the vertical one; in the plane ofthe mean lift lines of the b
17、iplane it is affected only by the pair of -rortices running off the wings,since the transverse vortex of one wing causes ordy an increase (or decrease) of the elocity offlow at the other TV. We are concerned here only with the calculation of that downwarddisturbance veIocity clue to the vortices frg
18、m the wing not under investigation, since the othervortex system is present with the monoplane and its uence has already been taken intoaccount in the experiments on a monoplane.The total velocity due to a portion of a vox proceeding to infinity b. one direction, inthe plane perpendicular to the vor
19、tex at its end, is, qs maybe deduced easiIy from the formulain section S, exactly half of the corresponding velocity in the neighborhood of a rectilinearvortex fiIament exten to infid.y in both directions. This can ah be easily seen from thefact that two vortex flaments, each extend- ,ing to infinit
20、y in only one direction-but II ,oppositely in the two casesform, if com.- ? bined, a single filament extending to irity .4Iin both directions. The total velocity ,caused at the point P by the -rortx A, -1see figure 40, is where r= -/ its kvertical component isr, x . .-.=kr T F1.w . .The vertical com
21、ponent due to thevortex B isr, 1,x11=G7 .-7-where r= /( Z)S +h:. -Therefore the vertical component dueto both ortices is“w+=) 3”If we assume that the lift is uniformlydistributed over the effective span l, whichagain we shall take as somewhat less thanthe actual span, then, since every element ofthe
22、 wing must be turned through me anglep according to the formula tan p=% theI /direction of the air force must be turiudalso! which means a negligible change in - FIG.mthe lift, but an increase in the drag of this Intiuenca of one wLngof t bIplrme upon the other; rotation of wLngprofile,wing which mu
23、st be taken into account.alteration of M curvature.“It is essential then in this calculation that we pass from a condition for a monoplane toone in which the wing when part of a biplane has the same lift as when considered as a mono-plane.- The angle of attack for whhfh this condition will arise can
24、 be es.lirnated afterwardsfrom the average of the angles w19. The contribution of vortex A to the increase of the drag of the upper wing in re 0is evidently.=,-. .,.-.-. . - -.-.,.-.-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-,186 REPORT NATIONA
25、L ADVISORY COMMITTEE FOR A.ERONAUTIOS.The contribution of vortex B is, by symmetry, the same. In accordance with equation (20),we can put r- remembering that tan w= ; -(35)outside of the vertical plane, owing to the disturbing wing, three vortices contribute tothe magnitude of w. A side vortex contr
26、ibute, at a point at the height h and the distance y( 4)in front of the transveme pltine, a velocity v perpendicular to r“ of the amount 1-r ?and therefore its share of w is“,”W-$)=W-+OThe transverse vortex contributes -in which the meaning of r“ and r” may be seen from figure 41. The total w is, ac
27、cordingly,S5(U-,-+J , . . us=2usj+loz 4* f prf= . . . .1 ,=. . . . . -. .1The mutnid action of two Mngs pfnesdetdeby eIdo can el.w be mlsuhted from the conskferations stated above, and rcstdta lma decreaee ofthe drag. TMS dwaese is of a ebntler nd to that which arks tn the theory of a mOMOneby an in
28、cR3Se hr the SSJW2tratio.By cbenga in corveture of the wing according to equation (35), .(36)CaIcuIations of the preceding nature were made in 1912 by my assistant, A. Betz, so asto compare experiments with monoplanes and biplanes and to study the influence of difTerentangles of ahtack and different
29、 degrees of stagger of the two wings of a biplane upon each other.The influence upon the drag was not lnmwn to us at that time, and the calculation was carriedout so as to obtain the changw the Iift due to w, to v and to the curvature of the streamlines.In this connection the change of the lift of a
30、 monopkme when flying near the earths surfacewas aIso deduced, by ctdculatm the influence of the “mirrored. wing” exactly as was that ofthe other wing of a biplane. AII that wu necessary was to change some algebraic signs,because the mirrored vchig had negative lift. The theory of thesecalculations
31、was given by Betz in the Z. 1?.M., 1914, page 253.The redts of the the the litter is in an causedb trimmerse rmd Up rortkee.intensikl descending current due to the forward wing at the same time the ratio t experiences adeorease for the forward wing and a marked increase for the rear one.For a wing i
32、n the n&uhborhood of the grom”d, owing to the influence of v there is a decreaseof lift, and conversely there is ati increase of lift due to the influence of w, provided the angleof attack is kept cunstant, but as the result an evident decrease in the ratio =*. owing tothis last it is seen -why in t
33、he early days of aeronautics many machines could fly only near theground and could not rise far from it. Their lo-iv-powered engines were strong enough to omr-come the diminished drag near the ground but not that in free air. D. THEORY OF THE MONOPLANE.20. If -we xtend the principles, which up to th
34、is point have been applied to the influencof one wing upon another, to the effect upon a single wing of its own vortices, it can be said inadvance hat one would expect to find in that case effects similar to those shown in the influenceof one wing of a biplane upon the other, i. e., the existence of
35、 lift presupposes a “d=cendingflow in the neighborhood of the wing, owing JO which the angle of attack is made greater andthe drag is increased, both the more so the C1OSWto the middle &e vorticies flowbg off atthe epds are, i. e.,”the smaller the aspect ratio is. One might propose to apply the theo
36、ry pre-viously given for biplanes by making in the formulas of this theory the gap ecpd to zero. Apartfrom the fact that the formulas developed do not hold for the immediate neighborhood of the.vortex-producing wing, but must be replaced by more accurate ones, this certainly is not theproper path to
37、 follow, for, in the earIier treatment, we have taken the msturbed monoplaneas the object with which other CSSEZSare to be compared and have asked what drag, what changein angle of attack, etc., are caused by adding a second wing to this monoplane. To proceed- i :-Q-. . -.-. -. .:.-.-.-.-._. - - -Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
