1、REPORT No. 651DOWNWASH AND WAKE BEHIND PLAIN AND FLAPPED AIRFOILSIly Am SILVERSTEH,S. KATZOFF and W. KEWVIITE BULLIVANTSUMMARYExtenske experimentalmeasurementsham been made ofthe downwash angles and the wake characterietioebehindairfoils with and un%houiflaps and the. data hare beenanalyzed and corr
2、elatedm“ththe theory. A detailedstudyUXMde of the errors inooked in applying lifting-linetheory, such as the effects of a .j%de wing chord, theroflingp of the trailing vortex sheet, and the wake.The downwaehanghw,as computedjrom the theoreticalspan load didribution by means of the Biot4ammt egua-iio
3、n, werefound to be in saitifactory agreementwith heexperimental results. The rolling-up of the trailing vor-tex sheet muy be neglected, but the nn-ticaldisplacement ofthe oartexslwetrequirtwoonsiderabn.By the uae of a theoretical treatment indicated byPrandtl, it haa been poseible to generalize the
4、arailableexperimentalresults so thutprediciione can be made of theimportant wake parameters in term8 of the di8tance be-hind the airfoil trailing edge and the projlledrq coefi-oient.The method of applicaiwn of tlu theory to design andthesatisfactoryagreementbetweenpredictedand experimen-tal results
5、when appltid to an airplane are demonstrated.INTRODUCTIONRational tail-plane design depends on a knowledge ofthe direction and the velocity of the air flow in theregion behind the wing. Numerous investigations,both oxperirnental and theoreticrd, have been devotedto the determination of the downwash
6、for wings with-out flaps. The agreement between theory and experi-ment has, as a rule, been only partly satiefaotory, andthe oompariaons have been inadequate as bases forgeneraJizationa. The existing empirical equatione fordownwash angles make allowuce neither for variationsin plan form nor for the
7、use of flaps. Only scant atten-tion has been given to the important problem of thewake behind flapped wings.As the first part of a comprehensive study of tail-plane design, the air flow in the region behind the winghas been studied for the purpose of developing generalnethods for predicting the dowm
8、vash and the wake.kluch of the -workon downwmh was concerned with the:elation of the induced fieId in the region behind the.irfoilto the theoretical span load distribution and tothe corresponding vortex system. The bafi for tietheoretical calculations is the Biot-fbart equation forLheinduced mlooiti
9、es in a vortex field. Some of theiata were particularly useful in investigating the rate ofrolJing-upof the trading vortex sheet.The wake constitutes a not altogether separateproblem. Its position and the velocity distributionBcrossit must be known in order to predict the tailefficiency for casesin
10、which the taiIis within it. Do-ivn-wash and wake generally require simultaneous treat-ment because the downwash determines the position ofthe wake and the wake has, in turn, an effect on thedownwash.The data used in this analysis were obtained mainlyin the N. A. C. A. full-rode wind tunnel with airf
11、oilsand airplanes that -wereusually so small that the jet-boundary corrections either were negligible or could beaccurately _ (1+JS2+X2+2 )due to the two trailing vortices.+.=.=:*,(F+2+F-+2J+2+7mdue to the U-vortex.By means of equation (1) a computation of the down-wash angle behind a monoplane airf
12、oil can be made ifthe load distribution, or circulation distribution, alongthe airfoil is known. The wing is replaced by its liftline, where the bound -rortex is considered localized,and the vortex sheet that is shed from its trailing edgeis considered to originate at the lifting line and extend,unc
13、hanged, to infinity. The strength I of the boundvortex at any section is related to the section lift co-.eficient cl by the equationin which T is the free-stream velocity and c is the chordlength. The intensity of vorticity in the trding vortexsheet is dl/dy.In a separate paper (reference 5) are pre
14、sented, foruse in tail-plane design and stability studies, the re-sults of extensive downwaah computations based on theforegoing idealized picture. These computations beingfor wings of various aspect ratios, taper ratios, and flapspans, it is essential to investigate the generality of themethod and
15、to justify its application.The validity of the foregoing concept as a foundationfor the computation of downwash angles is, indeed,subject to objection in a number of particular, whichhave been separately studied and are discussed in thefolIowing sections. The cases of wings without andwith flaps are
16、 separately treated.WINGS WITHOUT FLAPSFlow about an airfoil section,-h obvious objection-to the proposed method of calculation is that a vortex.at the lifting line is an inexact substitution for them142-i+l3actual airfoil. In order to investigate the order ofmagnitude of the discrepuq, the theoreti
17、cal two-dimensional flow about a Clark Y airfoiI at CL= 1.22was obtained by a confornd transformation of the. .-FIGURE3.Theoretlml downwash-angle mntonrs for tmdfmenskmal flow about aClark 1“alrfofl wctfon. q 6.43; CL, 1.23.flow about a circIe. The transformation was effectedby the method of Theodor
18、sen (reference 6); the ClarkY airfoil was chosen because much of the experimental .work w-asdone with Clark Y airfoils and aIso becausethe transformation had already been partly performed=-. .-.-. .FIGURE4.Theoretical downwash-angle mntonrs for a mrtex In a uniform stream(two-dimensional flow). Vort
19、ex strength corresponds ta CL-1.22.in reference 6 for this section. Four complex Fouriercoefficients were used, -which,inasmuch as they suflicedto transform the circle with good accuracy into theClark Y section, necessarily sticed to transform theflow at distances from it.The results are plotted as
20、downwash+mgle contoumin Qure 3. Comparison with the corresponding mapfor an equivalent vortex placed at the quarter+hordpoint (g. 4) shows that, at a distance of about one _chord length behind the trding edge, the difbrence isless than 0.3. It appears reasonable to assume thatthe difference wouId be
21、 of this order for other airfoils.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-182 REPORT NO, 651NATIONALADVISORY COhIifITTEE FOR AERONAUTICSThe conclusiveness of this result may be open toquestion inasmuch as the actual flow about an airfoilsecti
22、on ordy approximates the potential flow, owing tothe finite viscosity of air; the difference is probablyFIOIXII 5.-Theoretieal downwash-m.ule wntmzrs for twodbnemtoti flow about aClark Y alrfoll Wton. a, -6.6fi CL,O.slight, however, exceptin the vicinity of the wakeitself.Figure 5 shows the theoreti
23、cal stream a.ngleaCCU-lated for the Clark Y airfoil at zero lift. The simplifiedtheory for this case predicts zero downwash at everypoint in the field; and it can be seen that, at one chordlength behind the trailing edge, the differmce fromzero is small.Distortion of the trailing vortex sheet.The sh
24、edvortex sheet does not extend unaltered indetitelydownstream but, as a result of the tiirmotions that thevork sYstem itaeIf creates, is rapidly displaced do-Qw-fw-chwo IhePositions ofVortex shedFmurrE 6.Isometrio drawing showtng the U. S. A. 45 afrfotl and the dfstortsdtmiling vortex sheet. C (b) t
25、he lift coefficient is quite.high, in fact, nearly the maximum for this airfoil; and(c) the 26-foot survey pkme is considerably farther.3it is the line across which there is an abruptchange in the lateral component of the velocity. It isSISOthe line where the wake intersects the surveyphme, as was v
26、erified by the dynamic-pressure surveys.The circles indicating the positions of the tip-vortexcores are somewhat more arbitrarily located; they ampoints that appear to be the approximate centers ofrotation of the air flow near them.Provided by IHSNot for ResaleNo reproduction or networking permitted
27、 without license from IHS-,-,-DOWNWASHAND WAKEBEHIND PLAIN AND FLAPPED AIRFOILS2. %th of integrat,%n.,/ -+:$?r1!j: .1”111I t I/QSco/e of vecfors2 /Vo;tex1“o“ 10”20”30”(core #+(-_ - _ - - 1!-4. . -j+:ww: of vorfex shed NALQ+0 , B ,cwhere dA- and d; are, respectively, vector elements ofarea and of bou
28、ndary, and is the velocity -ector.IQTOW,the circulation around the airfoil. at, say, 18feet from the center is known from the experimentedspan load distribution and the air speed. This circu-lation constitutes the amount of orticity thtit mustbe shed from the trailing edge. between this point andthe
29、 airfoil tip. If the trailing vortex sheet extendedunchanged indefinitely downstream, the value of$ .d along any path that enclosed the edge of thesheet and cut the sheet 18.feet from the cer as, forexample, the path SWQR in figure 9; should equal thiscirculation. Owing to the rolling-up,. owelrer,
30、theamount of vorticity within such a path exceeds thisamount, especially when the point at which the pathcuts the sheet is well back of. the trailing edge. Theexcess indicates the extent of the rolling-up.Such integrations were performed along tile rec-tangular paths shown in figures 8 and 9. The in
31、tegra-tion along these paths is particularly simple, for, nlougthe vertical sides, .d= IT sin 8 dr and, along tilehorizontal sides, .d;= 17sin # ok, where 8 nnd 4 methe experimentally determined pitch and yaw mgles,V is the local air speed, and dr is the length of the pathelement.From these integrat
32、ions it was found that, at 10fee.behind the quarter-chord line, the total vorticityin the area of integration was 1.024 as much m thecirculation around the airfoil at the 18-foot station;whereas, at 26 feet behind the quarter-chord line, itwas 1.13. as much. These values correspond, respec-tively, t
33、o the circulations tht existed on the airfoilat 17.2 feet and 14.3 feet from the center. It followsthat the vortex sheet leaving the trailing edge rolls upat such a rate that, in the first survey plane, the vor-ticity originally between 17.2 feet and the edge has mlbeen concentrated between 18 feet
34、and the tip; and, the rear survey plnne, the part origindy outboard of14.3 feet has been concentrated between 18 feet andthe tip. Integrations about the inner parts of thevortex sheet showed, as expected, that the inner part,lost as much vorticity as tho tip gained.The rest of the sheet must distend
35、 corrpondingly.Thus, the portion extending origimdly between the cen-ter and 14.3.feet from the center has, at the 26-footsurvey plane, become so distended. that it reaches tu18 feet. Further evidence on this distention is foundin the surveys made in the vertical line 8 feet from thecenter (figs. 8
36、and 9). The rate of outward displnce-rnent of the vortex filaments 8 feet from tlio center is”roughly given by the average of the yaw angles justabove and just below-the sheet. M the 10-foot and tho26-fooL survey pkmes, these average nngks are 2.5 nnil3.5, respectively. The mean along the path being
37、 thusnbout ion, because tlm pressure-orifice measure-ments showed anomalous Iift distribution near tl.ocenter of the airfoil.It “riiiij be remarliccl that, although the distortionand distention found are not inconsiderable, the rolling-up process, by which the vorticity is evontunlly concen-trated i
38、nto a pair of tip vortices, appears to be still farfrom complete. This result is evident from the positionof the tip-vortex core, located at the approximate centerof rotation of the air flow near the tip. It has moved inordy to .96 percent of the sernispan whereas, for com-plete rolling-up, it would
39、 be at 7.8percent of the scmi-span. “Figure 10 shows similar survoya behind the tipof the 8- by 48-foot Clark Y airfoil. For complete roll-ing-up, the tip vortex would be tit about 87 percent. ofthe semispan, whereas the survey shows it ulmost titthe tip.Further evidence of the displacement of the t
40、ip-vortes core is found in the photographs obtuincd insmoke-flow studies and reproduced in figure 11. Thevisible flows were obtained with kerosene vapor flowingpast the tip of the tapered half-wing mounted on a re-flecting board. The core of the tip vortex is easily dis-cerned as far back in the mou
41、th of the exit bell ns canbe seen, n total distance .of about 50 feet. _This vimethod was not capable of yielding quantitative re-sults although, qunlitat.ively, it was clear thnt tho in-ward displacement of the tip-vort ox core vms small.Kadens theoretical calculations (reference 7) indi-cate tl th
42、e rolling-up process is not complete until adistance of 0.56/CL semispans behind the trnihgedge of the airfoil. For a lift coefficient CLof 1.35 andan aspect ratio A of 6, this value is 2.5 se.mispnns,orabout four times as far back as is the usual locationof tk tail. It-rnay be remarked thut, in fif
43、uttrayswork (reference .8), the rolling-up, as determined bythe inward displacement of the tip-vortex core, wasfound to be slower than is given by the foregoingexpression,Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-.DOWNWASHAND TAKE BEH NDThe eff
44、ect of rolling-upon the downwash.-It will berecalled that, in the tentative scheme for calcuhitingclownwash angles, the trailing vortex sheet is assumedto originate at the Liftingline and to extend unchangedto infinity. The question now aria as to bow seriouslysuch rolhg-up, distortion, and distenti
45、on of the trailingvortex sheet as were found in the preceding studyaffect the accuracy of the results computed on this. IPLAIN .4NDFLAPPED AIRFOILSbasis. For purposes of comparison,185calculations weremade of the-do-rewash angl c, mme as b, but d!splamd verUeSUYby SIIamount equal to the dlspkmment a
46、t the middle of tbe dlstortad sheet.and the error involved in assuming the distortion to beuniform appears to be negligible.(b) The vortex sheet assumed to be neither dis-torted nor rolled up but to extend straight behind thequarter-chord line to intlnity.The curves in figure 12 gi,o.foot U. S. .A.
47、46 tmsred tifoL I% 1.175.10% I IIJ6 . 2-bJ 12-fk oirfoi-F . - -5.64 A - _ _ _ -_$b= ow calculated- Expsrimentilfl” m . 5-by 30-in. airfoilI ! 1$ .74. , -. - - _ . - -5w -, _ _+ .25 . _ _ .o10-by 30-irt. airfoilj JO.91 “= - .- -. - -.- _ _5.21 %. _I t0 .5 Lo f5 2.0Longiiuditwl dis%nce fiwn l/4-chwd p
48、o% swnispcmsbnh”nd M nsmlnr. IrfnP-FIGCBE17.Theoretieal displsmd downwash-arrgle rantoure in the #yrnrnctry piano.+: .“ of the Iztit U. S. A. 45 tapered airfoil. CL,1.1?S.ii#y, , ,1-.6H b):8j .4 T:J :yLongibdinol disfunce from -chord ,Dwh semis,nons, .?.f- 1 I I t I I.8 20 I!2 i4. -Lsh) cL= CL-O.74.
49、(0) U= Q.i?;cL-1.iYllFmuru !uL-Downwseh-sngle snd dynemfc-premre mntoors fD smmekyph Oftbe S-by Wlhcb CIerk Y SkfO Dynemk Pressures SWSIIGWDss flWtiOl19of kes-stresro dynerdo wessrue.It is assumed here that the trailing vortices extendunchanged inddnitely downstream. The resultingdown-wash-angle contour map
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