aiaa 59830 918 amosmith high lift aerodynamics

8/8/2019 AIAA 59830 918 AMOSmith High Lift Aerodynamics

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VOL. 12, NO. 637th Wri g ht Brothers Lecture*

J . AIRCRAFT JUNE

High-Lift AerodynamicsA. M. O. Smith

McDonnell Douglas Corporation, Long Beach, Calif.Nomenclature

chordsection profile drag coefficientlocal skin-frictioncoefficient T w/(l/2) pujsection lift coefficientl i f t coefficientconventional pressure coefficient (p p )/ (1/2)pressure coefficient when local flow is soniccanonical pressure coefficient (p p0)/(\/2)pu$suction quantity coefficient Q/u OxC M = blowing momentum coefficient (2uft/u^ c),in-compressible flowC^ = blowing momentum coefficient referred to momen-tum thickness of the boundary layer at blowinglocation, uft/ujB, incompressible f low

Presented as Paper 74-939 at the AIAA 6th Aircraft D esign, FlightTest an d Operations Meeting, Los Angeles, Calif., August 12-14,1974; submitted August 14 , 1974; revision received December 27 ,1974.Index categories: Aircraft Aerodynamics ( including ComponentAerodynamics); Boundary Layers an d Convective Heat Trans-ferTurbulent ; Subsonic an d Transonic Flow.

CLCn

c.

f = chord fraction, see Eq. (5.1)H = shape factor of the boundary layer, d*/0 = plate lengthL = l i f tm = exponent in Cp=xm flows, also lift magnificfactor (5.1)M = Mach numberp = pressureq = dynamic pressureQ = flow rateR = Reynolds number (= uOx/v in Stratford flows)R6 = Reynolds number based on momentum thicku e e / vS = Stratford's separation constant (4.10); also

ripheral distance around a body or wing area/ = blowing slot gap, also thickness ratio of a bodyu = velocity in x-directionu0 = initial velocity at start of deceleration in canoand Stratford flowsv = velocity normal to the wallV = a general velocityx = length in flow direction, or around surface of a

measured from stagnation point if used innection with boundary-layer f low

A. M. O. Smith, Chief Aerodynamics Engineer for Research at Douglas AirCompany, Long Beach, Calif., was born in Columbia, Mo. on July 2, 1911majored in Mechanical and Aeronautical Engineering at the California InstituTechnology, receiving the M.S. degree in both fields. After graduation in 193joined Douglas as Assistant Chief Aerodynam icist. During this period, he workeaerodynamic and preliminary design problems of the DC-5; SBD dive bomber; anA-20, DB-7, an d B-26 attack bombers. He had prime responsibility for detaerodynamic design of the B-26.Because of earlier work with rockets at Caltech, he was asked by Gen. H. H. Ato organize an d head the Engineering Department of Aerojet as their firstEngineer, on leave of absence from Douglas, in 1942-1944. After expanding the dement from 6 to more than 400 and seeing the Company in to production on JATO he returned to Douglas an d Aerodynamics. There he handled aerodynamics of th558-1 Skystreak and the F4D-1 Sky ray, both of which held world speed records.

In 1948 he moved into the research aspect of aerodynamics. Since then hedeveloped powerful methods of calculating potential and boun dary-layer fculminating in a book co-authored with T. Cebeci entitled, Analysis of TurbBoundary Layers. He has published over 50 papers.For his work at Aerojet, he received the Robert H. Goddard Award of the AmeRocket Society. For his early rocket work at Caltech, he is commemorated in bronthe NASA Jet Propulsion Laboratory. In 1970,he received the F. W . (Casey) BaAward of the CASI, which is in commemoration of the first Canadian to fly aplane. He is a Fellow of the AIAA.


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502 A. M. O. SMITH J. AIRCRGreeka = angle of attack6 = flap deflection6* = displacement thickness of the bou nda ry layer8 = momentum thickness of the boundary layerv = kinem atic viscosityp = mass densityT = shear stress\l / = stream functionSubscriptse = edge conditionsJ = Jet = lowero = reference conditions, as in Stratford flowsu - upperw = at the wall = reference condition at infinity

1, Introduction and PurposeV^i/HEN I first began work for the Douglas AircraftT T Company in 1938, Donald W. Douglas, Jr., was inschool. Several times, in connection with hi s studies, he cameto m e for help with aerodynamic homework problems, so thatwe got to know each other. Years later he gave m e a 25-yearservice pin and asked me what I was doing. Fo r once I wasquick-witted and answered "still trying to understandaerodynamics ."That is the primary purpose of my lecturethe un-derstanding of one aspect of aerodynamics. It is not a "howto do it" lecture, but rather one concerned with "whys" andprinciples. Since I am in research, I am not directly involvedwith the down -to-earth design problems of aerodynamic hard-ware, but instead am in a position of giving help to others.That reminds me of the definition of a mathematician I onceheard. A mathematician is one who can tell you all about ho wto solve a problem but cannot actually do it himself .It was my privilege as a graduate student at Caltech to listento the brilliant beginnings of this series, by B. Melville Jonesin 1937. It is a great honor, a privilege, and a difficultchallenge to follow him and his many illustrious successors.In his in troduction he said "...I am instructed that theWrigh tBrothers ' Lecture should deal with subjects upon which th electurer is engaged at the t ime, rather than with a general sur-vey of some wide branch of aeronautical knowledge." To agreat extent, that is my plan; the central interest will be sub-jects with which I have been closely associated.Because I have been in the aeronautical field for some timen o w, I find it interesting to look back at the state ofaeronautical knowlege w hen I was in school in the 1930's. Ithink you will see, as I proceed, that while th e problems ar efa r from being completely solved, our capabilities have ad -vanced tremendously . M y first problem of substance was myM.S. thesis project. It was to perform tests on a poweredBoundary-Layer-Control Model in the GALCIT Ten-FootWind Tunnel. Figure 1 is a drawing of the model .The project w as inspired by German w or k of the time. Inth e design of the wing, th e philosophyso far as I knewwasas simple as this: 1) select conventional airfoils of the t ime fo rth e wing, 2) put a slot (squ are edged) somewhere in the upp ersurface of the wing (70% chord seemed as good as anything),and 3) suck air and see what happened. There was insufficientknowledge of boundary-layer flow to do anything much moresophisticated. At the end of the tests, the writer had a vagueuneasiness that there should be some more scientific approachto the problem of design. That is the way it was in a leading

MODEL CHARACTERISTICS


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JUNE 1975 HIGH-LIFT AERODYN AM I CSth e sun." Nearly all of the basic principles fo r inf luencing aflow to develop high lift have been known from th e very earlydays of the airplane. The defects in knowledge were two:first, although what to do m ay have been k n o wn , the reasonsfo r doing it were only dimly unders tood; second, quanti tativeanalysis of a flow could rarely be accomplished.P r a n d t l l had already conceived and dem onstrated the prin-ciples of suction boun dary- layer control in 1904, and by 1913,according to W e y l ,2 th e notion of blowing to control th eboundary layer had been advanced. One concept the authorbelieves to be new is that of the jet f lap, which seems to havebeen conceived an d developed in the 1950's. Hagerdorn an dR u d e n 3 tested fo rms of the jet flap in 1938, but they did notunderstand what they were f inding.The ancestry of flaps can be traced back to the early days off lying. British R&M No. 110, dated 19144 contains one sec-tion entitled, "Experiments on an Aerofoil Having a HingedRear Portion." Figure 2 shows the shape tested. A large num -ber of flap settings were examined. According to Weyl ,2variable camber had been used even earlier. The LeBlonmonoplane had a var iable-camber wing formed by an ad-justable part of the trailing edge. I t was exhibited at theLondon Olympia Show in March , 1910.The idea of slats and the knowledge of the effectiveness ofslots are nearly as old. In an important lecture given beforeth e British Royal Aeron autical Society on February 17 , 1921,Handley Page 5 described ten years of work on the develop-ment of airfoils that had slots. H is wo rk dealt not only with asingle slot or slat but also with multiple slots. Figure 3,6shows one of his models, which by current standards would beconsidered a relatively modern configurat ion. Page 425 ofRef . 7 shows a good photograph of a 1920 airplane fitted outwith slats.Later in this paper, th e author attempts to prove that an air-foil having n +1 elements can develop more lift than onehaving n elements. Handley Page investigated this problem,up th rough 8 e lements .5 Figure 4 shows one of his extremeairfoils, a very highly modified RAF 19 section, positioned atth e angle fo r maximum l ift .

Figure 5 shows lift coefficient vs angle of attack for thperimental airfoil as it was modified from one to elements. It generally shows that th e greater the numbelements the greater the lift; and it seems to confirmauthor's deduct ion, which w as made three years ago, quignorance of these tests. The seven-element airfoil reaclift coefficient of 3.92. Tests were made at a chord Reynumber of about 250,000 on a wing of 6~in. chord and 3span.Handley Page appears to have followed an empiricaproach in his efforts . Concur ren t ly Lachm ann 7 at Gottwas studying the problem theoretically. Lachmann usedformal- transformation methods and represented a slat by

SXs> ~ ~ j + AFig. 2 RAF 9 airfoil with a 0.385c plain flap tested in 1912-19

Fig. 3 A wing tested by Handley Page as part of his effdeveloping slots and slats.

Fig. 4 Handley Page's eight-element air-foil modified from an RAF 19 section.Th e model is at 42 angle of attack, theangle for m a x i m u m l i f t . Pressu redistributions are theoretical. They weremade at a = 36 to correspond to localangle of attack of the AR = 6 wind tunnelmodel. Theoretical c( of ensemble is 4.33.


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504 A. M. O. SMITH J. AIRCR4.0

3.2

2.4

0.8NO SLOT

32 SLOTS

40 501 0 20 30ANGLE OF ATTACK (DEG)

Fig . 5 Ci vs c t data for the RAF 19 broken u p into different num-bers of elements, as indicated by number of slots.

Fig . 6 Variable camber Albatross Biplane9 m odel , showing ap-proximate range of flap angles tested.tices in f ront of a circular cylinder. His w o r k established thegross features and interaction effects for a slat and the mainairfoi l . Later, Lachmann and Hand ley Page jo ined forces .Further basic unders tanding an d appreciation of thebeneficial effects of a slat was gained by Le Page,8 w hosystematically investigated the forces on two airfoils placed intandem, tAnother work that merits mention is entitled "Model Ex-periments with Variable Camber Wings," by Har r is andBradf ie ld ,9 which was published in 1920. The report includes

Speaking of biplanes, history seems to be repeating iIn the early days, because they were biplanes, they hadwings /These suffered from leading-edge stall, fo r whichdroop or slats were a cure. Then w e advanced to monopand th icker wings, and the leading-edge problem aldisappeared. Later, when jets and higher speed enteredpicture , wings again became thin, and the leading-problem returned.Original flap development w as motivated by three debenefits: 1) slower flying speeds, hence shorter takeofflanding runs; 2) reduction of angle of attack near min iflying speed; 3) increase of drag, or control of drag , in oto steepen glide angle in approach and reduce floatingdencies . Currently , because of large aircraft noise probthe emphasis under the third i tem has changed. W e are tto reduce flap drag in order to reduce th rus t requirementhence the noise.The split flap was conceived an d partly developed inperiod 1915-1920 as a means of satisfying these deKlemin , Schrenk , and Etienne Royer2 are impor tan tt r ibu to rs to its development. Orvi l le Wright and J. MJacobs obtained a basic patent on the subject in 1924,

they filed it in 1921. Their discussion in the patentevidence that they had a good quali tative understanding obasic f low processes. A rep roduct ion of one of the drawf rom th e patent i s shown in Fig. 7.We end this historical survey of mechanical high-lif t deby mentioning the Fowler Flap invented by HarlanFowler 10 in 1927. Because it had extension and a slot, itbe considered to be the f irst modern high-lif t mechanica lIn early N ACA wind-tunnel tests, H it developed a maxilift coefficient of 3.17.A u g . 12, 1924. 1,504,663O. WRIGHT ET AL

AIRPLANEFiled May 31, 1921 3 Sheets-Sheet 1


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JUNE 1975 HIGH-LIFT AERODYNAMICS

Powered lift augmentation, as by suction and blowing,received considerable attention in the 1920's but never quiteproved to be efficient enough to jus t i fy its use in an actual air-plane. Betz 2 and Ackere t7 in Germany were leaders in thisline of development.For those interested in further research into the early his-tory of high-lift work, we mention several references. First isthat by Als ton ,1 2 which was a general lecture to the RoyalAeronautical Society in 1934 entitled, "Wing Flaps and OtherDevices as Aids in Landing." It surveys th e state ofknowledge in 1934, but does not particularly deal with earlierhistory. Those who are especially interested in the history ofhigh-lift devices and the origin of concepts an d applicationsshould see the paper by W e y l ,2 which is a very broad sur-veyit has 116 referencesof th e entire subject, with in-teresting sidelights on the development. As an example ofsuch a sidelight, Weyl notes that in connection with th edevelopment of split flaps a French scientist, Lafay, in 1912observed "that an unsymmetr ical roughening of a cylinderwhich was exposed to an airflow resulted in an aerodynam icforce which was directed towards th e smooth side across th eflow (lift force)."

The most comprehensive reference examined by the authoris a report by A. D. Y o u n g ,1 3 "The Aerodynamic Charac-teristics of Flaps." It deals with the aerodynamic charac-teristicslift, drag, moment, e tc .of all types of flaps. It sextensive bibliography covers all aspects, including generalreviews, history, theory, and investigations of the varioustypes of devices. In all, 138 references are given. The paper iscertainly not obsolete, although it was published in 1953.A paper that amounts to an updating of Alston's 12 is oneby D u d d y ,1 4 which w as also presented to the RoyalAeronautical Society. It compares and evaluates various typesof flaps, and analyzes their benefits in terms of landing an dtakeoff performance. Effects of sweep ar e included.A useful histor ical summ ary of the gradual improvement ofgross lift coefficients is shown in Fig. 8, which is due toCleveland. 15 From about 1935 to 1965a period of 30yea rswe have advanced from coefficients of roughly 2 toroughly 3 o n impor tan t service-type airplanes. By 1995 will wehave advanced to 4?

3. Some Lift Limits3.1 Limits in Potential Flow

Just as ideal-cycle limits are useful in thermodynamics, soare the theoretical limits of lift useful in aerodynamics .Knowledge of those limits helps give us a perspective as towhere we are now and wha t may be attainable if we are willingto seek without comprom ise the maximum possible l ift .First consider th e limits of c ( in inviscid f low, whereseparation will not occur. Consider the classical circulatoryflow about a circle show n in Fig. 9. Two different levels of cir-culation are shown; that in Fig. 9b is of the greatest interest,fo r there th e circulation is so strong that front an d rearstagnation points coincide. Greater circulation moves th estagnation point off the body. Although that is entirelypossiblea Flettner Rotor generates such f lowsit is not arealistic analog for an airfoil, which in general has twostagnation points, both on its surface. If the reference chord istaken as the diameter of the circle, it is easily shown that flowb of Fig. 9 represents a lift coefficient,

C( = 47 T (3.1)

B, and C are shown in Fig. 10. A fourth or limiting case isa straight line. According to Joukowski airfoil theory, foof those circular-arc mean lines, regardless of degree of ber,

In that equation, c is the length between the ends of thea. is the angle of attack, an d /3 is a measure of the cambshown in Fig. 10. For arcs A and B, c is indeed the choconventionally defined. But for arc C, the chord c is notthe longest dimension; th e diameter is. In fact, it is evfrom th e figure that as 0^90, c-+0 . Hence, if we definlift coefficient in terms of the longest dimension, we[from Eq. (3.2)] ,

1 2 . 0

\ side0.0463 'IR"5

After substitution from (4.20) into (4.19), we obtain0.075R 1/5

1-Cn

(4.20)

(4.21)where is the length of the flat plate an d R(=u0t!/v. Figure31 shows the results. The two sides of Eq. (4.21) are plottedseparately. When the left-hand side exceeds the right-handside, instability develops. Them =1curve is a special dividingcase. It begins with the value of 1.0. Any higher value of mbegins with the value 0; any lower value of m, the value .Hence, according to Gartshore's criterion, th e wake is alwaysinitially unstable fo r m < 1.0. The figure shows a strong effectof wake thickness upon the stability. For our problem ahigher Reynolds number is favorable because it reduces < 5 *an d hence increases th e value of the right-hand side of Eqs.(4.19) or (4.21), but the effect is weak. According to Eq.(4.21), for the Cp=xm flows th e left-hand side ultimately ap -proches infinity, an d therefore any of the wake flows shouldfinally go unstable . Hence, with m < 1, theoretically w e have asituation where the wake flow is unstable to begin with, butquickly becomes stable an d finally becomes unstable again ifcarried far enoug h. Whether that indeed is true is not k now n.However, according to Gartshore's analysis, a boundarylayer is more prone to separation than th e wake is to in-stability. Consider the f=0 .25- f t plate in Fig.31. For any ofth e ra-values, th e final instability is at x^O.8 ft . However ac -cording to Fig. 20, for most values of m, th e boundary-layerflow has separated earlier.

20r

Gartshore cites experimental evidence that confi rmsdeductions. It was obtained by examining flow over defleplain flaps. Without doubt , th e effect can and hascurredseparation off the surface. But since there has bvery little study of that kind of f low, th e validity of rela(4.21) is in question. Especially in question are some ofderived consequences, such as the initial instability for sfractional values ofm. For example, when m - 1/3 and 1boundary- layer flow would not separate, but the wakewould be unstable. The principal purpose in presenting expl e applications is to g et the concept into the open, and at lexhibit in a qualitative way the effect and the interacbetween w ake thickness and pressure gradient.In practical applications, with their lack of infinite advgradients through which a wake must flow, the winstability problem should rarely be critical, which meanswakes can end ure pressure rises that boundary layers caendure. Wha t a multielement airfoil does then, in effect,use two m ethods of pressure recovery: the conventional,is, on-the-surface pressure recovery, an d off-the-surpressure recovery. On a slatted airfoil, for instance,history of the flow is this: Air flows over the slat, reachpeak velocity, then decelerates in contact with its surfleaves th e surface and continues to decelerate until trailedge pressures ar e reached, after which it graduallycelerates back to freestream conditions. By the off-the-surdeceleration, recovery from very high negative Cp valuesbe made in much shorter distance than can be made whethe deceleration is in contact with a surface.On the right-hand side of Fig. 17 is a canonical presscale based on the maximum velocity on the slat. Accordinthat scale, the final velocity-squared ratio at the trailing is only 0.025. If all the deceleration had been in contactth e surface, th e flow would surely have separated, accorto Figs. 19 an d 20.Nothing so far has been said about the case when bounlayers merge. Lockheed 30 '31 has studied the problem anfact, has developed a general method fo r analyzing conflboundary layers. But because of the complicated nature oflow, bold simplifications had to be made. Like Gartshanalysis, the me thods need development and further checkdetailed experimental work . Improved methods fo r bothmerging and the nonmerging wake flows are problemsfu ture wo rk .It is our observation, as well as that of Foster et al.32,gaps between airfoil elements should be so large that wand boundary layers do not merge for, if they do, separation will set in. In fact, Foster et al. in discussingsubject makes th e following statement:4 ' . . .when th e flap is in the optimum position fromviewpoint of obtaining the highest maximum lif t, theteraction between the wing wake and the flap boundaryis comparatively mild, with the two layers retaining separate identity almost to the flap trailing edge."Hence, we are somewhat for tunate . Optimum designoutside th e region of merging boundary layers, andthermore, flow re tardation rates are rarely so great thawake becomes unstable. The boundary layer on an elebeneath the wake is nearly always in worse trouble . Howth e merging of boundary layers helps establish th e opt imIf th e boundary layer were much thinner an optimum spamight be less. Hence, because of scale effects on the bounlayer, th e optimum spacing for an aircraft may be appreciless than indicated by a small scale wind-tunnel model.problem is worst for the slat because it s wake has the lonrun .


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JUNE 1975 HIGH-LIFT AERODYNAMICSis possible without causing separation. Provided that bound-ary-layer control is not used, our only means of obtaininghigher lift is to modify th e geometry of the airfoil. Con-siderations that guide th e modif ication, then, are the subjectof this section. It is helpful to th ink in terms analogous tothose of structural designapplied and allowable loads. Theairfoil applies the loads, and the boundary layer determinesthe allowable. Separation of a boundary layer may be likenedto reaching th e yield point in materials testing. Initialseparation rarely coincides with the maximum lift that an air-foil develops, for the lift usually continues to increase.Likewise, the yield point in a m etal is not the point of ultimateload. That usually is the rupture point . Hence, the point ofmaximum lift coefficient may be likened crudely to the rup-ture point of a material. The analogy is quite rough, ofcourse, but it is mentioned because the interaction betweenboundary layer and shaping of an airfoil is not very widely ap -preciated. We close these introductory remarks by observingthat aerodynamic science has advanced to the point where weca n satisfactorily predict the point of initial separation (theyield point) but not the condition of maximum lift (the rup-ture point). The second pr oblem is still beyond the s tate of theart , but it is assuming high priority because of successfulsolution of the simpler problems.5. 2 Single-Element Air foils

The subject of single-element pressure distribution typesan d consequent airfoil performance has been indirectlycovered by the material in Sec. 4. The canonical pressuredistributions show general limits to the pressure rises, and thepair of charts can be used for visual checks prior to carefulexamination of more nearly final designs by detailed bound-ary-layer calculations. If separation is indicated for theproblem at hand, some change must be made in the shaping inorder to remedy the defect. In m aking the change, it must beborne in mind that definite bounds on what can be ac-complished have already been established. For a one-piece air-foil, there are several possible means fo r improve-mentchanged leading-edge radius, a flap, changed camber,a nose flap, a variable-camber leading edge, and changes indetail shape of a pressure distribution. A pressure rise may beimproved by changing from convex to concave in the direc-tion of Stratfor d's typ e. Even changing the trailing-edge anglefrom finite (e.g., 10) o cusp may be useful because a cuspedtrailing edge imposes less pressure rise at the trailing edge.Furthermore, a cusp increases th e slope of the lift curve, sothat a given CL is reached at a lower angle ofattack, which lowers nose pressure peaks.Simple hinge systems as on a plain flap, even though sealed,can have a significant adverse effect on the separation pointan d hence on lift. Figure 32 shows the adverse effects of the

- DEFLECTED 25*-VARIABLE CAMBER

-2.0

-1.6

-1.2

-0.8

-0.4

0.4

0.8

o T E S T D A T A- T H E O R Y

MACH NUMBER = 0.26A N G L E O F A T T A C K =1.76NOMINAL R E Y N O L D S N U M B E R = 2.0 x I06

20 40 60PE RCENT CHORD

80 1

Fig. 33 Pressure distribution for a typical aft-loaded aPressure distribution is corrected for boundary-layer effectsshaded area on the airfoil represents the displacement thickness.

L O W E R0.2 -

20 80 10 60P E R C E N T C H O R D

Fig. 34 Pressure distribution of Fig.33 in canonical form.break at a hinge line. The airfoil is an NACA 63A010 ahinged on the lower surface at the 75%-chord point. shown for comparison is a variable-camber flap whoseterline is a circular arc. Its final slope is 25, so thattrailing edges have th e same final angle. Both flapsseparation, but the variable-camber shape ha s attachedover 85% of the flap, while the simple flap has attachedover only 71 %. It is interesting that th e simple hinged flap


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518 A. M. O. SMITH J. AI RCRdistribution. Where is separation most imminent? Near th efront on top? Near the rear on top? Or on the bottom? Figure34 is the canonical form of Fig. 33. Comparison with th ecurves of Figs. 19 and 20 indicates that the nose is far fromseparation. The rear upper surface is marginal, but onlybehind about 97 % chord. T he lower surface retards to aboutCp = 0.5 in a rather short distance, and the retardation beginslate. Figures 19 and 20 indicate that a C^-value of about 0.6could be reached in this distance. Hence there seems to besome margin , but not much. Provided that other constaintswould allow it, the faintly S-shaped pressure distribution af tof 50% chord on the bottom could have been improved byreplacing it with one that is concave all the way. In problemsan d approaches like this, the need for inverse methods becomeevident. First, we would like to have an inverse boundary-layer method that for arbitrary initial conditions could tell uswhat pressure distribution w e could have that contains a cer-tain margin against separat ion. Then, given th e specifiedpressure distribution, the method should find th e airfoil shapethat produces it. Definite progress is being made along theselines.5. 3 Multi-element Airfoils General

There seems to be a great deal of ignorance an d confusionabout the effect of gaps in properly designed multielement air-foil systems. One misconception was mentioned in the in-troduction to Sec. 4. The most common misconception is thata slot supplies a blowing type of boundary- layer control. Theidea can be traced at least as far back as Prandtl,33 w ho said,1 The air coming out of a slot blows into the boundary layeron the top of the wing an d imparts fresh momentum to theparticles in it, which have been slowed down by the action ofviscosity. Owing to this help the particles are able to reach th esharp rear edge without breaking away." Abbot t and vonDoenhof f 34 merely make the safe comment , *'Slots to permitth e passage of high energy air from the lower surface to con-trol the boundary layer on the upper surface ar e commonfeatures of many high-lift devices." Here again, boundary-layer control is implied. Perkins an d Hage 3 5 make th eharmless and noninformative statement: "The ai r flowingthrough the slot in Fig. 2-49 [sic] is accelerated and movestoward th e rear of the airfoil section before slowing down andseparating from th e surface." Lindfie ld, i n Lac hm an n ,7 has abrief paper on the slot effect. Part of the article describesstudies by Lachmann, in 1923, who represented a lifting slatby several vortices located near a circle. The circle was thentransformed into a Jouk owsk i airfoil . Lachmann ' s theoreticalstudies seem to have been largely forgotten and not really ap -preciated. However , he considered only half th e problemtheeffect of a slat on an airfoil. H e did not consider the effect ofthe airfoil on the slat. Lindfield's article is correct as far as itgoes, but it is not very factual. He points out the need for bet-ter analytical methods before the slot effect can be wellanalyzed. (Now 14 years later , most of the methods areavailable.)A remark about th e action of slots comes from a recentNASA report30: "It is well recognized that the usual functionof the slot is that of a boundary-layer control device per-mitting highly adverse upper surface pressure gradients to besustained without incurring severe separation. This stabilizinginfluence results from th e injection of the high energy slotflow into the upper surface boundary layer." A still morerecent NASA repor t 36 states, "This leading-edge slat gives th efluid which passes through the gap between the slat and themain airfoil a high velocity. Consequently, a boundary layerwhich grows on the upper surface of the main airfoil has moremom entum than i t would have in the absence of the slat."

that each component develops its own boundary layer uth e influence of the main stream, and there is no merwithin th e slot. Topologically, th e process of boundary-ldevelopment is no different from that on a biplane. Subjetheir particular pressure distributions, the two bounlayers on a biplane grow, trail of f downst ream, diffuse,finally merge. That is jus t th e process for an airfoil systetwo or more elements so long as merging does not owithin th e slot.The next paragraphs will elaborate on and confirm whahave just said. There appear to be five primary effectgaps, and here we speak of properly designed aerodynaslots.1) Slat effectin the vicinity of the leading edge of a dostream element, th e velocities due to circulation on a forwelement, fo r example, a slat, ru n counter to the velocitiethe downstream element and so reduce pressure peaks ondownstream element.2) Circulation effectin turn, the downstream elecauses the trailing edge of the adjacent upstream element tin a region of high velocity that is inclined to the mean linthe rear of the fo rward element. Such flow inclination indconsiderably greater circulation on the forward element.3) Dumping effectbecause th e trailing edge of a forwelement is in a region of velocity appreciably higherfreestream, th e boundary layer /'dumps" at a high veloThe higher discharge velocity relieves the pressure risepressed on the boundary layer, thus alleviating separaproblems or permitting increased lift.4) Off-the-surface pressure recoverythe boundary lfrom forward elements is dumped at velocities apprecihigher than freestream. The final deceleration to freestrvelocity is done in an efficient manner. The deceleratioth e wake occurs out of contact with a wall. Such a methomore effective than the best possible deceleration in conwith a wall.5) Fresh-boundary-layer effecteach new element sout with a fresh boundary layer at its leading edge. Tboundary layers can withstand stronger adverse gradithan thick ones.Those effects will now be explained an d discussed in tu rsome length. Laminar bubbles, merging boundary layers,th e like m ay complicate th e problem; but when Reynnumbers are high and at design conditions, such side effshould not be important . Therefore, only conventiboundary- layer effects ar e considered.5.4 Slat Effect

Figure 35 displays the slat effect. A slat that is liftingcirculation in the direction sketched in the upper part o

o .oa -15 eg TOTAL = 2.47


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JUNE 1975 HIGH-LIFT AERODYNAMICSfigure. To a first approximation, a slat may be simulated by apoint vortex. A more accurate approximation would accountfo r thickness by means of several sources an d sinks, sources infront an d sinks behind, th e total yield being zero, of course.For our purposes, it is sufficient to consider only the vortex.From th e sketch it is evident that the velocities induced on theairfoil by the vortex ru n counter to those that would developon the nose of the isolated airfoil, especially at high angle ofattack. Pressure peaks would therefore be reduced. Liebeckand Smyth 37 studied the effect by means of a specialgeneralized Joukowski airfoil program combined with com-puter graphics. Figure 35 is an example of the work. In themethod, the vortex could be moved around at will. The bestlocation found for the vortex was typical of the location of aslat. The "best" location is not a precise statement. It is thelocation where th e nose suction peak was largely eliminatedand resulting pressure distribution w as smooth. A t a = 15,th e plain airfoil developed a very strong suction peak. As isnoted at the bottom of the f igure, c( for the airfoil alone is2.37. With a vortex located as shown, the pressure peak wascompletely eliminated, the airfoil c( fell to 1.88, but the totallift, including the vortex, increased to 2.47. In Fig. 35, thecurve called "vortex" shows the velocity induced on the air-foil by the vortex in the absence of the main stream. For bestcancellation of the isolated airfoil suction peak, vortexposition and strength were so adjusted as to generate anegative velocity peak that was roughly the mirror image ofthe isolated-airfoil suction peak. Flow near the rear of the air-foil was almost unaffected, as can be seen by examination ofeither the airfoil pressure distribution or of velocities inducedby thevortex.That then is the slat effect . Contrary to the implication inth e various quotations cited in Sec. 5.3, th e velocity on thenose of the airfoil is reduced. Peak nose velocity ratio was re-duced f rom about 4.4 to under 2.0. Obviously, th e bound-ary layer is much better able to negotiate th e modifieddistr ibution. With a vortex operating, there is only a verysmall increase in total lift, a fact that is consistent with wind-tunnel observations. With slat extended, th e main effect is todelay the angle of stall, not particulary to shift the angle ofzero lift. In the beginning of this discussion, it was mentionedthat thickness effects could be accounted for approximatelyby sources in front an d sinks behind, to approximate th eshape of the f lap. Since total sink strength must equal totalsource strength and since the sinks are closest to the airfoil,the net effect of the combination is the induce a small forwardvelocity component on the airfoil. Hence thickness effectsshould generally reinforce th e circulation effects.

30 40 50 60PERCENT CHO RD

There is an interesting and more physical way of ex plaithe slat effect. Again consider Fig. 35 . When the airfoilan angle of attack, flow whips rapidly around its nose, whas a small radius. High centrifugal forces are develoWithou t outside help, high negative pressures around the ar e needed to balance out the centr i fugal force. But the vois a turning aid, and a t rue slat also would be one. Tshould be a close correlation between the amount ofcarried by the slat and the average Cp-reduction overnose. The author looked at Fig. 35 and made th e followcrude estimates: Area modulated= 10% chord, mean Cth e airfoil alone for this region =-8. Mean Cp for sregion with vortex present =-2. Net exchange = 6. Tapproximate lift force c ( that must be supplied by the vois 6x0.6. The actual lift on the vortex is the differbetween c ftotal and c fa i rfo i l . According to the numbers attom of Fig. 35, c F = 0.6 is indeed the load carried by thetex. Probably, the analysis cannot be made quantitative, bdoes add to the understanding of the flow process.problem would be a good one for further study. The fexerted by the vortex is not jus t vertical; it has a drag cponent too. According to the f igure, because net drag muzero, it s c^-value is 0.3680, a negative force that causesto extend automatically. A more careful analysis wouldsider this component .The slat effect on a real airfoil is illustrated well by Figwhich represents a well-developed, practical design. pressure distribution for the airfoil withou t slat is also shThe quantity (ue/u00)2 reached a maximum value of 1(C^-9.04). With the slat in place, the value was redremarkably, to just over 3. The effect is strikingly similthat of the point vortex in Fig. 35. The figure also showupper-surface pressure distribution in canonical fo rm. Agsigns of a jet blow ing effect ar e conspicuous by their abseThis and most of the other pressure distributionscalculated by one version or another of the Douglas Neumprogram38'39 that has been tested by so many hundredcomparisons with wind-tunnel and other data that its prtions can be accepted as being essentially exact, at leasour current purposes.Figure 37 was specially synthesized for purposes ofpaper. It consists of three identical airfoils, all at the angle of attack. For comparison and indication of the dof interference, the pressure distribution for the isolatedfoil is included. Observe the extreme round ing of the peathe two rear airfoils, which again is the slat effect . A samount of the rounding is undoubtedly due to the decrevelocities that accompany a converging flow, as at a traedge. As in the point-vortex example, th e rear of each aiis covered with a surface sink distribution, according toNeumann program. The surface sinks induce a counter-

10 A 18.00

jj / ON SIMPLE AIRFOILON I s AIRFOILQON 2" AIRFOIL\C ON 3 AIRFOIL

TOTAL SYSTEM


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520 A. M. O. SMITH J. AIRCR

a = ( 5 cj OF I S O L A T E DA I R F O I L = 2.5937

c/ T O T A L =3.4937Cj AIRFOIL = 3.3631C. A I R F O I L = -0.0763

Fig. 38 Point vortex used to sim ulate a slotted f lap. Vortex increasesC g of airfoil at a = 1 5 from 2.59 to 3.36. Leading-edge pressures ar egreatly increased b y action of the vortex.

Fig. 39 Airfoil-circular-cylinder combinations studied to learn effectof an obstruction on circulation.over the following airfoil. Although it was not done, it wouldbe easy to learn the thickness effect by replacing a fo rwardelement by a much thinner section while keeping the samemean line.5. 5 Circulation Effect

The slat effect ha s been recognized before; in fact, it wasclearly recognized as far back as 1923 by Lachmann.7 But thecirculation effect does not seem to be explicitly recognized.Figure 38, which shows the effect in its simplest form, wasproduced by the computer graphic setup used in Fig. 35. Sincethe vortex could be placed anywhere in the field, it was nowplaced near the trailing edge, as indicated in the figure. If itwere to represent the lift of a slotted flap, the vortex cir-culation would be as indicated. The flow effectively places thetrailing edge at a high angle of attack, and if the Kutta con-dition is to be met, th e circulation must increase. The effect islittle different from that of deflecting a small plain flap on theisolated airfoil. In that case, the onset flow would be ap-proaching the rear of the airfoil at a considerable angle. B utfo r Fig. 38 we did not tu rn the trailing edge; instead, w e used adevice to turn th e f low. Figure 38 shows a slightly differentairfoil from that in Fig. 35 , but again the angle of attack is thesame. The vortex has a drastic effect on circulation, in-creasing Cg f rom 2.59 to 3.49. In Fig. 38, the vortex-alonecurve shows th e velocity distr ibution around th e airfoil that isinduced by the vortex. That distr ibution does not meet th eKut ta condition, but the airfoil vortex combination does, ofcourse . It is obvious from th e figure tha t an y means ofchanging the flowfield near th e trailing edge would changethe lift. Hence, a source, properly positioned, is apt to be aseffective as a vortex. The effects on circulation that are due to

discharges boundary layer at the trailing edge into a strthat is locally of h igher velocity. That is the "dumping effthat has already been mentioned an d that will be discufurther in Sec. 5.6.We have ju s t indicated that an y method capable otroducing cross flow at the trailing edge will influence th eculation. A n obstruction, properly placed, can be a powfactor for controlling the circulation. To illustrate suchtrol, we used a circular cylinder, which is about as neutbody as can be selected. Figure 39 shows the system stuTw o circular cylinders of different diameters are centeredray from the trailing edge. The gap is constant at 0.1 chThe angle of the ray, 6, was varied from 0 to 90. The lith e airfoil and on the airfoil-circle combinationcalculated. Figure 40 shows some results for the case wth e cylinders were directly to the rear of the chord(6 = 0). Large increases in lift ar e indicated. The cylindself carries a large amount of lift, even though i t hatrailing edge. Figure 41 shows the effects in more detafunct ions of d. As might be expected, an optimum defleangle is found . The most effective angle is about 60 orAt 15 angle of attack, th e lift coefficient of the isolatedfoil is ce= 1.75. With th e 0.50c circular cylinder set at 6 =ct = 3.35 fo r ju s t th e airfoil, nearly double the value fo r aalone. Hence th e effect is very great. Figure 39 is drawn5 = 60. It appears that this most effective position is sito the position found most effective fo r slotted flaps an din combination with airfoils . Considered as a control surth e control effectiveness dc f/dd is not much less than thatplain f lap .Figure 42 shows calculated pressure dis t r ibut ions focases with 6 = 60 and with ct held constant at 1.5corresponding values of a are noted in the figure.velocities at the nose are considerably reduced by the cylinIf th e pressures were plotted in canonical form, the onesth e cylinder would appear considerably more favorVelocity ratios at 0.975c, which amount to the dumvelocity, are tabulated. The cylinders double the dumvelocity ratio. However , according to Figs. 19 andseparation should still occur in all cases. That is notpris ing, in view of the specified c f an d shape of the airfoil

2.0 -

,0=

Q'/D-0

=0


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JUNE 1975 HI G H-LI FT A E R O D Y N A M I C S

a= 5 I S O L A T E D A I R F O I LD-Q.25C

= 10 ISOLATED AIRFOIL0.25


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522 A. M. O. SMITH J. AI RCRsails on the same boat. A useful an d very interesting study ofthe problem has been made by A. E. Gentry. ^ By using theelectrical-analogy technique and conducting paper, he ex-plained correctly for the first t ime th e j ib-mainsail interactioneffect.5. 6 Dumping Effect

Closely related to the circulation effect is the dumping ef -fect. The favorable interference of a downstream element in-duces cross flow at the trailing edge that enhances circulationof the upstream element. But the interference m ay also in -crease velocities in a tangential direction so that th e flow froma forward element is discharged into a higher velocity region ,thus reducing pressure-recovery demands. The effect is quitefavorable to the boundary layer . According to Eq. (4.4), thesuction lift can be increased in proportion to (u T E )2 for thesame margins against separation.Does th e effect really exist? Yes, indeed! It can be seen inan y properly designed multie lement airfoil. It is clearly showntheoretically in Fig. 42; values of the dumping-veloci ty ratioar e tabulated in the r ight-hand columns of the table in thefigure. For the three casesplain airfoil, airfoil plus 0.25c cir-cle, an d airfoil plus 0.50c circlethe velocity ratios ar e 0.88,1.147, and 1.186, respectively. The canonical ratio nearlydoubles; it increases from 0.043 to 0.085.Figure 36 is a theoretical case for an airfoil with slat. Forthe main airfoil, th e dumping-veloci ty ratio squared is 0.85.On the slat, it is much higher: 2.35.The pressure distr ibutionsare also shown in canonical fo rm , and according to them theslat is less severely loaded than th e main airfoi lin the senseof margin against boundary-layer separation.Figure 44 shows experimental data for a three-element air-foil. Again the dumping-velocity effect is clearly displayed,this time confirmed by experiment. For the three surfaces,starting with th e flap, the dumping-velocity-squared ratios(ue/u o o )2 ar e 0.67, 2.0, an d 2.28. The canonical plots in -dicate that, for the conditions shown, the main airfoil is lessseverely loaded than th e slat and the flap. The enhanceddumping velocity shown is typical . The pressure dis tr ibut ionof any properly designed mult ielement airfoil shows it; oneonly need look for it .It is interesting to conjecture w hat might be done with th eeffect if an inverse design method fo r multielement airfoilsbecame available, assuming that the complete airfoilrequirements permitted. Start with some desired pressuredis tribut ion for a hypothetical three-element airfoil, for

example, that of Fig. 45. The basic pressure distributionwe shall use is shown as the dashed line. Then suppose wea three-element airfoil that has that canonical predistribution for all three elements. Fur the rmore , assumethe rear element induces a dumping-velocity-squared( H e / t i n ) 2 of 1.5 on the middle element. Becauseelement now has higher circulation, it can induce even greffects on the front element. Assume the induced velratio is the same for each element with respect to its velocities, which is a bold assumption. With that assumpth e pressure distr ibution for the three elements in Fig. 45sketched. The load carried is far greater than that oequivalent single-element airfoil, yet the several canopressure distributions are all the same .The effect, as jus t described, is readily quantified. It wdone only for a three-element airfoil, but the analysis is egeneralized. Let c =chord of the ensemble; /]c =chord ofairfoil (element 1) ; f2c = chord of center airfoil (elemenand/3c = chord of front airfoil (element 3). Then for the elements /!+/2+/3 = l. Also let m2 = magnificationfo r velocity at trailing edge of element 2 due to being inhigh-velocity field of the nose of element 1; m 3 = m agnification ratio for velocity at trailing edgelement 3 due to being in the high-velocity field of the noelement 2.Let the upper-surface lift coefficient be ce for the basic pressure distr ibution. Then for the rear element thetion-side lift is/7cfo. On element 2, the wind speed ovewhole element is greater by a factor m2. Hence fo r elemeth e lift isf2c(om2. For the front element, the apparentspeed is increased by a factor m3. But the speed on the melement has already been increased by the factor m2. Hon element 3 the lift isf3(m2m3)2c(o. Then th e lift of thsemble is

and the ratio is(Ct/c{o)=f!+f2m2+f3(m2m3)

For the example of Fig. 45,fl =0.30, f2 = 0.55, andf3 =Also, m2 =m3 =(1 .5) 1/2 . Then, according to (5.2),(c,/c,o) =0.30+ 7.5(0.55) +2.25(0. 15 ) =1.463

Fig. 44 Typical three-element ashowing dumping velocity effect. Adenote dumping velocity. NACA 2301foi l , a = 8. Line marked trailing edge ltrailing edge dumping velocity dividemaximum velocity on m ain element.


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JUNE 1975 HIGH-LIFT AERODYNAMICSthat is, a 46% gain, with no more tendency for separationthan on the base airfoil. The effect definitely exists, but (5.1)is as muc h hypothesis as it is statement of fact. In any real air-foil design, so many more factors enter into th e problem thatis is not nearly that simple. Compressibility problems wouldultimately set in, for instance.5.7 Off-the-Surface Pressure Recovery

Off-the-surface pressure recovery has already beendiscussed in Sec. 4.9. Without its assistance, boundary layerswould often be unable to meet all pressure-rise demands .Consider Fig. 44 , which represents theory roughly confirmedby test data. The scale on the right-hand side shows th e totaldeceleration ratio from the peak velocity. At the trailing edgeof th e flap, th e velocity-squared ratio has decreased to 0.075.Figure 17 shows a theoretical pressure distribution that can berealized experimentally. Its final velocity-squared ratio is0.025. According to Figs. 19 and 20, the flow would neverhave reached such very low velocity ratios if it had bee n a sim-ple boundary-layer flow tha t was always in contact with awall. But in mu ltielement flows, th e front element's boundarylayer, which undergoes the greatest deceleration ratio, islargely decelerating off the surface. Figure 4 illustrates an air-foil that shows a theoretical total deceleration ratio ( U T E /u m a x ) 2 equal to 0.0045.It is opportune to mention something that has been almosttaken for grantedthat any accurate calculation of pressuredistributions must take into account the displacementthickness of the boundary layer. That assumption is wellcovered in Ref. 39 . In this paper, we have not been consistent,because consistency is not necessary to illustrate our point . Ingeneral, our simple theoretical calculations do not considerboundary layers. In other cases, we note in the captionwhether or not boundary-layer corrections were included.A wake alongside a body has only a slight effect on the in-viscid theoretical lift. If a wake has a square profile, it can beapproxim ated by sheets of vorticity, positive vorticity on oneside and negative on the other. Obviously, the wake has someeffect on circulation, although the two sheets almost canceleach other. Furthermore, if the wake flow is curved, there is acentrifugal force of the same sort that is developed by the cur-ve d sheet of a jet f lap, except that here the momentumamounts to a defect instead of an excess. Foster, Ashill, andWil l iams,32 an d Ashil l41 have carefully examined those ef -fects in both theoretical and experimental studies. Theydevelop formulas for the various effects. In general, theyfound that, for properly designed multielement airfoils, th eonly effect of importance is that of the displacementthickness. Moser and Shol lenberger42 also have studied the

problem theoretically and experimentally. They foundth e theoretical inviscid effect of the w ake is to reduce liftsmall amoun t , typically a few hundredths in c f. Their this in satisfactory agreement with experiment. In one partistudy tha t used an NACA 4415 airfoil, they found that creduced by an amount 2.3 c ds, where c ds is the drag cficient of the slat. All the findings have, in general, beenfirmed by practical experience.To summar izewithout th e displacement-thickness cotion the errors are greater than desirable. Withdisplacement thickness considered, the errors are so smala design engineer would hardly know what to do withaccuracyin the case of fully attached flow, of course.

5. 8 Fresh-Boundary-Layer EffectOn each element of a properly designed multielemenfoil, th e boundary layer starts out afresh. It is well knowcourse, that th inner boundary layers ca n sustain grpressure gradients than thicker ones. That fact is destrated well by Figs. 19 and 20. It can be displayed in a analytic form by Stratford 's formula, Eq. (4.10), whichere repeat.

] V2=(10-6R)1/WS (where 5 is a constant. The term (10-6R)1/10 varies so sthat it is almost constant. But x, a measure of distancethe origin of the flow, enters to the 1/2 power , andviously, longer boundary layers admit less adv erse_gradViewed in another way, we see that if x is halved, dCp/dxbe doubled with the same safety against separaTherefore, breaking up a flow into several short bounlayer runs is favorable with respect to the delay of separand hence to the increase of lift. It might be arguedbreaking up the surface into a number of elements steegradients just as much as length is shortened, leaving x/dx) unchanged. In view of the favorable interferenceshape of the total pressure distributions the quantity doedeed reduce.5.9 Are Two Elements Better Than One?

Our previous discussion indicates that several elemenbetter than one eleme nt. But there has been no proof . Heshall attempt a proof. The proof will be for a two-elecase, but by the nature of the development it will be seenthe proof could also show that three elements are bettertwo , four better than three, etc. We make the follo

Fig. 45 Illustration of compounding of liftby using multielement airfoils. For this / ^6 \illustration the factor m is (1.5)1/ 2 . \ U < )


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524 A. M. O. SMITH J. AIRCR

DESIRED PR E SSUR E DISTR IBUTION- C A L C U L A T E D PRESSURE DISTR IBUTION

5 I TE R A TIONS

Cp

Fig. 46 Load diagram for proof that a two-element airfoil ca ndevelop more lift than a single-element air foil.assumptions: 1) the lower surfac e plays only a weak role , an dhence only the upper surface needs to be considered; 2)Reynolds number effects ar e weakwhich they are at largescale. Hence th e lift on an airfoil whose chord is unity is jus tth e same as the lift on two of the same airfoils whose chordsare each 1/2;3) by some means , perhaps an inverse m e thod ,th e elements can be so shaped that both develop the samecanonical upper-surface pressure distr ibution. The shape ofthe pressure distr ibution is not restricted; and 4) bounda r ylayers and wakes of forw ard elements do not degrade th e lift-in g capacity of following elements.Consider Fig. 46. Let ABCDE be pressure dis t r ibut ion onthe upper surface of an airfoi l of chord A E. That same loaddistr ibution could be applied twice, to two 1/2-chord elementsas loads ABC'D 'E" an d E"B"C"DE. The sum of those tw oareas equals area A BCD E . B y proper inverse methods , th eshapes of two elements that could generate th e same pressuredistr ibution can probably be found , although that is a weakpoint of our proof . For the two half-size pressuredistr ibutions, the basic dumping velocities squared are E"D'and ED, which ar e equal. Now on any properly designedmult ielement airfoil, th e flow off the trailing edge of the for-ward element can be made to discharge into th e high-velocityregion of the rear element. Hence for the front element thedump ing veloci ty squared is increased from E"D' to E"F. Theincrease is the key to ou r p r oo f . Tw o cases now arise:

a) the maxim um veloci ty has some kind of limit; an db) the max imum velocity has no limit.Consider case (a ) first . Perhaps the velocity limit comes fromMach number cons idera t ions . Because E/ /F


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JUNE 1975 HIGH-LIFT AERODYNAMICS4.33. Velocity ratio instead of velocity ratio squared was usedin order to avoid excessively large ordinates on two of thefigures (C p less than -36 was indicated).Values of c ( in each velocity diagram represent th e liftcarried by that element, referred to full chord. Four of thediagrams show a second pressure distribution. It is the onethat would develop if the airfoil w as alone, at the same angleof attack. The leading element is at a considerable negativeangle of attack, with a strong pressure peak on the lower side,which might cause separation. Nevertheless, the element isvery highly loaded. In spite of the increasing angle of attackof each element, the lift contribution decreases continuouslytowards the rear. The final element carries a c ( of only 0.08,in spite of being at almost 90 angle of attack at the trailingedge. If it were in isolation, it would be carrying c p of 6.80.Although not converted to canonical fo rm , none of thepressure distributions appears to be very close to separat ion,even that of the front element. Perhaps that is not surpr ising,fo r we are showing a condition that represents maximum liftfo r an airfoil whose chord Reynolds number was about250,000, making the Reynolds number of each element about50,000. At such low Reynolds numbers, separation shouldoccur quite early, which makes the theoretical pressuredistributions look quite safe when judged by our current highReynolds number s tandards .Figure 5 should be reexamined, now that our calculationhas been made. In general, it confirms our conclusion thatifproperly designedthe more slots, th e bet ter . But the seven-slot configuration is inferior to the six-slot configuration. Thereason is not known , but the author suspects it is due todrastic modification of the trailing-edge contours when th eeighth element was formed (see Fig. 4). Figure 48 shows th edumping-velocity ratio fo r each of the elements. There is arather smooth an d definitely continuous increase from rear tofront . The rearmost element has a value of about 0.4, and thefront element reaches 2.25.Figure 49 shows the extremes that can be reached inproducing special-purpose high-lift airfoils. The sectionshown was designed by R.H. Liebeck of the Douglas AircraftCompany as a wing for Dan Gurney 's Indianapolis-type racecars. The shape is a result of careful direct-method analysisand modification of his opt imum airfoil studies. As installedon a racer, th e wing is rectangular and of aspect ratio 2.15,with small end plates. The wing was designed theoretically,then buil t without benefit of wind-tunnel tests, an d tested onth e racer. At an angle of attack of 14, tufts used to indicateflow separation remain fully attached and the maximumwing-section c ( is about 5, according to calibrated spring-deflection tests.In the past few years the word "synergistic" has becomepopular . Those examples, th e proof of Sec. 5.9, the dumpingeffect discussed in Sec. 5.6, as well as the various test data, al lindicate th at airfoils properly placed in tandem ar e fine exam-ples of synergism.

6. Power-Augmented Lift6.1 General

A few remarks should be made about th e subject of power-augmented lift. Power properly applied, as through bound-ary-layer control, can greatly delay separation and hence in-crease maximum lift. Powered lift augmentation usuallyrequires pumps and internal ducting, which am oun t satleastto weight an d cost complications. There is no doub tthat higher lifts can be obtained, bu t whether they make for abetter total airplane system is a moot question. The answerdepends on the efficiency of the lift-augmentation system.

2.0

1.0

I 2 3 4 5 6 7 8FRONT ELEMENT NO. R E A RFig. 48 Upper surface dumping velocity values for the airfoilconditions of Fig. 4. The ratio ( W T E / w m a x ) 2 for the ensem0.0045.

a = 10'

Fig. 49 Special-purpose Liebeck high-lift airfoil designedIndianapolis-type racers (Pat. Pend. ) .

as good accuracy as turbulent flows over impervious walgeneralized eddy-viscosity formula has been developed tresponsible fo r that accuracy. Therefore, it was feltpropriate here to throw some light on the subject of delseparation, by considering th e theoretical suction reqments. A n i l luminating procedure is to use the Cpcanonical plots presented earlier, in Figs. 19 and 20. Besuction introduces another variable, space does not permto present results for all four lengths of flat-plate runstead, we consider only the one-foot run, which case mconsidered roughly to approximate an airfoil with a flap.In conventional boundary-layer calculations where a tformed stream funct ion is used, we are given the walldi t ions that/7 w ~ uw =0 (no slip) and/w ~ $w =0 ( impervwall). Then the problem is to find/" w ~cf. In the calculato be presented, we keep/' w = 0, of course, but seek valu


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526 A. M. O. SMITH J. AI RCR

Fig . 50 Effect of area suction on separation. Suction starts where Cfhas decreased to 0.001 an d then vw is adjusted to maintain this valueof C f . Initial flat plate run is 1 ft, turbulent, u0 /v = 106/ft.

Fig . 51 Effect of area suction on separation. Suction starts where C fhas decreased to 0.001 and then vw is adjusted to maintain this valueof c f . Initial flatplate run is 1 ft , turbulent , u0/v = 107/ft.

solution of a partial differential equation, computing an dconvergence problems arise when Cf is set equal to zero. W e,therefore , avoided those difficulties by specifying that suctionwould be sufficient to maintain cf at a value equal to 0.001.Thus the results to be shown contain a certain margin ofsafety . Previous studies have specified the suction and thenfound the resulting skin fr ic t ion. Figures 50 and 51 showresults fo r (u 0 / v ) values of 106 an d 107.The results ar e shown in te rms of values of v w/ue requiredto maintain cf at the chosen value 0.001. Suction starts whereC f ha d decreased to 0.001. At first th e suction rates are quitesmall but the rates increase rapidly as the higher pressureregion is penetrated. The final contour v w/u e = 0.05 is one forwhich a variable constant in the boun dary- layer equations isabout to exceed th e range of its validity. Furthermore, theboundary-layer equations start losing their validity when v w isnot very small compared with u e.Total suction flow rates canbe determined approximately by integrating the v w values.According to the figures th e effect of Reynolds number is notstrong.The charts show that suction is highly effective in causingsmall delays in separation. But when large delays are desired,suction loses efficiency, so that strong suction is required. Theeffect can be explained physically in a neat fashion. Assumeth e region of pressure rise to be quite short so that th e suctioncan be assumed to be concentrated at a point . Let us thenexamine the special case of a flat-plate flow entering a suddenpressure rise assisted by suction. Long ago, in connection withGriffith airfoils, G. I. Taylor considered th e same kind ofproblem. H e first assumed that the pressure rise occurred insuch a short distance that th e total head along each streamline

to prevent separation at a pressure jump. H is analysisbeen well confirmed by experiment fo r laminar flows, somth e confirmation being th e author ' s .Let us apply the hypotheses to our problem, namelyflows illustrated in Figs. 50 and 51, except that the presrise region will be approximated by an ins tantaneous juLet ( )c designate critical conditions. Then, usingcanonical flow notat ion we have for the total head alongstreamline

If al l velocity head is expended in reaching a higher presp c, down stream, we haveHc=pc=p0+V2Puc2

where uc is the velocity of the dividing streamline, allbe low be ing sucked off. U s in g Cp =(pp0)/(l/2)pwe find

This relation states that removal of fluid up to a height upermit a canonical pressure rise from Cp =0 to Cp .H ow much flow must be removed? Because thecinitialis flat plate it is convenient to use the 1/7 power formThe quantity that must be removed is Qc \ y0cudy whicoefficient fo rm iscQc- Q c i (y u- ~ = \ dyunx x J o Un

The 1 /7 power formulas ar eu/u0= ( y / b ) l l 7 an d d(x) =0.37x/(u0x/v)1/5

Upon insertion of the first, and integrat ing we haveCQc = (7/8)(d/x)(y/d)8 c"

From Eq. (6.3) and the first of Eq. (6.5) we can also write

Wh en this, together with the second of Eq. (6.5) ar estituted into (6.6) w e obtainCQc= [0.324 / (u

This formula shows tha t if the demanded pressure rise deed low the required suction C Q is very low. If u O x/v isthen CQc=0.013 C* f. If Cpc =0.c5, the required CQc is0.0008 but for reaching stagnation pressure it is 0.013. Uthe results on Figs. 50 and 51 some suction is indicateeven th e smallest pressure rise, but the four th power varishows th e demand to be exceedingly small. The mosteresting result from this analysis is the fourth power relofCQtoCp.In L a c h m a n n ,7 Wuest presents theoretical studies madPechau's approximate method. The studies were of areation applied to simple airfoils in various ways. Surequirements for avoiding separation ar e given. Geneour calculations show the same CQ= Q/uOx magnitude asthat is, of 0(10-3). We have not presented specific exambecause charts like Figs. 50 and 51 are more general. F


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JUNE 1975 HIGH-LIFT AERODYNAMICSsuch as to maintain cf at a value 0.001. A smaller Cy-valuewould clearly reduce v w an d C Q. In the belief that earlier suc-tion would be slightly less efficient, w e started suction onlywhen th e boundary layer began needing help. Undoubtedly ,there is an optimum distr ibution for each of the Cp=xmdistributions, but we believe our values do not miss the op-t imum by very much.There surely is some best combination of suctiondistr ibution and pressure distr ibution fo r obtaining th egreatest pressure rise with themin imum CQ. Determining thatcombination amounts to a calculus-of-variations problem offinding the best combination of path an d suction. It is in-teresting to note that the basic tool for the problem has re-cently been developed.45 The boundary- layer calculationmethod has been extended to find the pressure distributionnecessary to provide a given c^-history, which amounts tosolving a nonlinear eigenvalue problem, where th e eigenvalueis dp/dx. Since suction could already be analyzed by bothdirect and inverse methods, we now have available the toolsfo r seeking the optimum combinations of v w(x ) andp(x).The problem is an interesting mathem atical challenge.6. 3 Tangential Blowing

To round out the discussion, we give a limited comparisonof blowing applied to the same Cp=xm flows. The wall-jetmethod of Gartshore and Newman 4 6 is used. As for the suc-tion case, blowing is assumed to start shortly before naturalseparation would occur, that is, at the Cy^ 0.001 locus.Figures 52 and 53 are the results.The conventional blowing coefficient C ^ is defined fo r two-dimensional incompressible flow asC M = thrust/


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528 A. M. O. SMITH J. AIRCRto varying degrees by flow-separation requirements. If thefigure represented were a_ design pressure distribution fo rsome problem, th e upper Cp bound might be determined byMach number , fo r example. Tw o pressure distributions ar eshown in Fig. 54, one for a slightly cambered airfoil at angleof attack, the other for a cambered Joukowski airfoil. Thecambered airfoil has the higher efficiency. Obtaining lift byangle of attack, according to the definition of efficiency, isinefficient. Because the start of pressure rise can be delayedmore if design lifts are low,th e efficiency at low lift coef-ficients is often higher than that of shapes either designed fo ror operating at high lift.Powered lift can improve the efficiency markedly. Whenthere is a curved jet, load can be carried by the jet, and itbecomes possible to carry load all the way to the trailing edge.Thus the pressure-recovery problem can be eliminated en-tirely.A highly accurate theory of jet-flap airfoils that makes i tpossible to realize that benefit has been worked out in the pastfew years by Lopez and his co-workers. 4 7> 4 8 One of his co-workers, Halsey,48 reviews the airfoil work in general. Buthere we feel we have space only to m ention one aspect of it, amethod that appears to have produced an interesting andsignificant advance. He has developed an inverse method fo rje t airfoils. That is, the method specifies th e pressuredistribution and the je t momentum coefficient and finds th eairfoil shape and the jet leaving angle necessary to generatethat shape.The method is not a classical inverse method, such as Jameshas developed for conventional airfoils. Instead, it is a kind ofiterative procedure that involves finding a mean line and ob-taining modified shapes until convergence is reached. Re-course can be had to the exact direct method for checking pur-poses. The effects of jet entrainment on the pressuredistribution are accounted fo r approximately. The procedurehas been found to produce rapid convergence an d accurateresults . We can do no better than to quote from Ref.48 to

AIRFOIL 1,

SPECIFIED VELOCITY DISTRIBUTIONFINAL DESIGN, AIRFOIL I, CA = 2.427FINAL DESIGN, AIRFOIL 2, eg = 1.635FINAL DESIGN, AIRFOIL 3,Of , = 1.320

show th e specific steps that are taken in finding a shape. steps, slightly paraphrased, are as follows:1) Specify th e desired airfoil velocity distribution andmomentum coefficient.2) Divide the velocity distribution into symmetric andt isymmetric components.3) Determine th e approximate je t-entrainment effSince the operation is noniterative, the sink distributionresenting the jet is placed on an extension of the airfoil chand its strength is determined by assuming that the velojust outside the jet is equal to the freestrearn value. Thetrainment effects at the true airfoil surface ar e assumed tthe same as those calculated at the airfoil chord line.4) Use an inverse method to find th e thickness distributhat corresponds to the symmetrical component of velocity specified in step 2 that remains after entrainmenfects have been subtracted out.5) Design the mean line needed to produce the tisymmetric portion of the velocity distribution, includinginfluence of the jet.For that purpose, a linearized methoused. This part of the design procedure is the only part thfundamental ly different from conventional design methFor more details, see Ref. 48 . Either: a) the vorticity atrailing edge can be specified, find the jet angle; or b) theangle can be specified, find th e vorticity at the trailing edg6) Combine camber an d thickness distributions.7) Using the general nonlinear direct jet-flap aimethod , check th e result. If necessary, repeat th e cyclepressures slightly modified to correct errors revealed bydirect-method check.W e present two figures to show typical results. Figurshows th e effect of blowing. High blowing rates relievecamber, assuming that th e velocity distribution is maintaconstant . The points ar e checks of the computed inverse sby means of the direct, nonlinear method using it as idata. The difference in c ( noted for the three airfoils is duth e lift component of the jet. Each airfoil is shown atangle required to develop the design pressure distribution.An extreme case, one that taxes th e linearized camber trment, is shown in Fig.56. Here entrainment effects haveincluded in the inverse method. The entrainment effectswell as the entire camber-shape solution, are worked

2.5 r

SPECIF ICATION VELOCITY DISTRIBUTIOO FINAL DESIGN, C0 = 5.373

o o o o o o o o o o o


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JUNE 1975 HIGH-LIFT AERODYNAMICS

along the chord line. The final jet angle is at an inclination of 74.4 to the chord plane, and the airfoil is nose down 4.7to develop the design pressure dis tr ibut ion. Agreemen t bet-ween the exact check calculation and the linear design methodis fai r . Considering the extreme nature of this problem, th eresult is sat isfactory.

7. The FutureIt was my original intention in planning this lecture todevote considerable attention to the state of knowledge aboutthree-dimensional flows an d high-lift flows in the transonicregion. However , I shall not, for two reasons. First, I amalready exceeding th e desired length fo r this paper, andsecond, there is not much in the way of theory and analysismethods that can be called solid. Low-speed, two-dimensionalairfoil high-lift theory has become a fairly "hard" science,but three-dimensional and transonic theory is still no morethan a "soft" science, if that m uc h .It is only necessary to read Callaghan's A G A R D lecturenotes 19 to realize the diversity of problems that arise as wecome closer to real aircraft configurat ions and to realize th ekind of practical empiricism that one is forced to use in ap-

plied aircraft design.Like th e two-dimensional problem, th e three-dimensional isan analysis involving a combination of inviscid- and bound-ary-layer f low. The inviscid analysis is nearly here . That is ,lifting inviscid-potential-flow solutions can now be calculatedreadily for as many as 1000 coordinate points. That numberwill easily solve nearly any sim ple wing-alone geom etry and israther good at handling wing-fuselage combinations. 1000points is marginal for a two-element wing alone, but it canproduce useful information. To handle three-element wingstogether with major body effects at least 2000-point solutionsare needed. In a private communication, w e learn from Nor-m an Donaldson that Bell Aircraft has extended the method tohandle 3000 data points. The work is done on a modifiedU N I V A C 1108compute r . A problem of this size requiresmore than two hours of machine t ime. It is hoped that cost perdata point continues to decrease.The boundary- layer solution lags behind the inviscid-flowsolution. Although there are formulat ions and methods forana lyz ing t h r e e - d im e ns iona l l am ina r and t u r b u l e n tf l owsand even a book49 on the subjec tnone is powerfulenough to analyze the complicated problems of appliedanalysis that occur , for example, those of swept wings.Progress in fundamental w ork is encouraging, however. I t ap-pears that extensions of the eddy-viscosity formulations usedquite successfully fo r two-dimensional flows do not lose ac -curacy when extended to three-dimensional f lows .5 0 In thecase of turbulent f low, m u ch empiricism is involved, and theaccuracy of a method can be determined only by extensivecomparison with experiment. A milestone in the developmentof the art of computing two-dimensional turbulent boundarylayers was the Stanford Conference ,51 in which each par-t ic ipant who claimed to have a good boundary-layer methodwas asked to calculate a large variety of flows, al l from stan-dardized input data under prescribed rules . Sometime in therather distant fu ture, that kind of conference will be needed toestablish the accuracy of methods of calculating three-dimensional f low.Beyond that problem is the problem of calculatingmaximum lif t, as contrasted with th e problem of calculatinginitial separation. If the calculations of three-dimensionalboundary layers are forced to stop when th e first separation isencountered, they ar e likely to do more than barely start. Intwo dimensions, we can now predict separation reasonably

again there is no method of prediction that can be cgeneral. There is a large t raf f ic in papers on shock-boundlayer interaction, but so far as the author knows , th e probhas not been solved in any general sense. A vigorous attaunder way that uses finite-difference methods . Possibly ptical methods of analysis will resul t .O ne could talk indefinitely about other special probrelated to high lift but that won't do, so I shall end by liwhat I think are the ten most important basic theoreproblems of high-lift aerodynamics:1) Very general calculation of three-dimensional laman d turbulent flows.2) Calculation of flows involving partial separation inrear.3) Practical calculation of flows involving forwardaration bubbles.4) Practical calculation of flows involving shboundary- layer interaction.5) Calculation of the viscous flow on airfoils an d bodierevolution of length c from about 0.95c Jo 1.05c. Inregion of very rapid change an d strong interaction, boundlayer equations do not apply. With our recently acquability to calculate well th e fo rward 95%, our inabilitsolve th e next 10% is a real irritant. Beyond 105% chordagain can usually do an acceptable job.6) Further development of inverse methods .7) Drag of multielement airfoil systems. Drag predichave a relative error one order of magnitude greater thanof lift predictions. Since propulsive an d accelerationformation is crucial to aircraft design, that is a very impoproblem.8) Practical calculation of merging boundary-layers ,jets and wakes. Although methods have been developed,felt that considerable further development is needed.9) The analysis of flows over swept wings on whileading-edge vortex is developed. When the vortex deveconventional calculations ar e about 100% in error .

10) Three-dimensional transonic calculations, part icufo r arbitrary wings and wing-body combinations.In several of those prob lems , th e first phase isdimensional analysis, but as soon as that is accomplishedthree-dimensional looms up as the next target .As you can see from the list, aerodynamics still has a w ay to go before we can truly calculate, instead ofestimate . Yet in re trospect, we see that we have progresslong way toward th e goal in the past thirty years. Peranother Wrigh t Brothers lecture should be given on this subject 20 to 30 years from now. With the comprevolut ion that we have, th e prospects of finding praccalculation methods for most of the ten problems are brW e hope there will be some new "inventions" in flowflow control. Cornish's spanwise blowing 52 might be one,effective application of the trapped-vortex idea53 mighanother.In October 1908, at a banquet in his honor in Franceafter very successful flights in September, Wilbur Wrightunexpectedly called on to say a fewwords . He said, "I kof only one bird , th e parrot, that talks, and it can't fl yhigh." I, too, cannot fly very high and I have talkedm uc h .

References!Prandtl, L., "Uber Fltissigkeitsbewegung be i sehr kReibung," III International Mathematiker-Kongress, Heide1904.2Weyl, A. R., "High-Lif t Devices an d Tailless AirplaEngineering, Oct. p.


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53 0 A. M. O. SMITH J. AIRCRAPhysical Laboratory," R&M No. 110, March 1914, AeronauticalResearch Council, London.

5 Page, F. H., "The Handley Page Wing," TheAeronautical Jour-nal, June \92l, p.263.6Glauert, H., "The Handley Page Slotted Wing," R&M No. 834,Mar. 1922, Aeronautical Research Council, London.7Lachmann, G. V. , Boundary Layer and Flow Control , Vol. 1,Pergamon Press, London, 1961.8LePage, W. L., "Further Experiments on Tandem Aerofoils,"R& M No. 886, May 1923, Aeronautical Research Council, London.9Harris, R. G. and Bradfield, F. B., "Model Experiments withVariable Camber Wings," R&M No. 677, June 1920, AeronauticalResearch Council, London.10 Fowler, H. D. , "Variable Lift," Western Flying, Nov. 1931, p.31 .11 Weick, F. E. and Platt, R. C., "Wind-Tunnel Tests of the FowlerVariable-Area Wing," Tech. Note 419, May 1932, NACA.12Alston, R. P., "Wing Flaps and Other Devices as Aids toLanding," Journal of the Royal Aeronautical Society, Vol. 39 , 1935,p. 637.13 Young, A. D., "TheAerodynamic Characteristics of Flaps,"R& M No. 2622,1953, Aeronautical Research Council, London.14Duddy, R. R., "High-Lift Devices and Their Uses," Journal ofRoyal Aeronautical Society, Vol. 53,1949, p. 859.15Cleveland, F. A., "Size Effects in Conventional AircraftDesign," Journal of Aircraft, Vol. 7, Nov.-Dec. 1970, pp . 483-51216Faulkner, S. M., Hess, J. L., Smith, A. M. O., and Liebeck, R.H., "Charts and Formulas for Estimating Velocity Fields in In-compressible Flow," Rept. LB 32707, July 1968 (AD 772 233), Mc-Donnell Douglas Corp., Long Beach, Calif.17Mayer, J. P., "A Limit Pressure Coefficient and an Estimationof Limit Forces on Airfoils at Supersonic Speeds," Research Mem o.L8F23, 1948, NACA.18 Smith, A. M. O., "Aerodynamic Lifting Potentialities of Air-craft in Subsonic Flight, Rept. ES 20930, 1947, Douglas Aircraft Co.,Long Beach, Calif.19Callaghan, J. G., "Aerodynamic Prediction Methods for Air-craft at Low Speeds with Mechanical High Lift Devices," A G A R DLecture Series No. 67, May 1974, von Karman Institute, Brussels,Belgium.20Smith, A. M. O., "Aerodynamics of High-Lift Airfoil Systems,"Fluid Dynamics of Aircraft Stalling, AGARD-CP-102, 1972.21Assessment of Lift Augmentation Devices," AGARD LectureSeries No. 43, AGARD LS-43-71 (Available as AD 720-259, 1971).22Thain, J . A., "Reynolds Number Effects at Low Speeds on theMaximum Lift of Two-Dimensional Aerofoil Sections Equipped withMechanical H igh-L i f t Devices," Quarterly Bulletin NationalAeronautical Establishment, Ottawa, Rept. DM/NAE, 1973.23 Cebeci, T., Mosinskis, G. J., and Smith A. M. O, "Calculationof Separation Points in Incompressible Turbulent Flows," Journal ofAircraft, Vol. 9., Sept. 1972, p. 618-624.24Loftin, L. K., Jr. and von Doenhoff , A. E., "Exploratory In-vestigation at High and Low Subsonic Mach Numbers of Two Ex-perimental 6-Percent Thick Airfoil Sections Designed to Have HighMaximum Lift Coefficients," RM L51F06, 1951, NACA.25 Stratford, B. S., "The Prediction of Separation of the TurbulentBoundary Layer," Journal of Fluid Mechanics, Vol. 5, 1959, pp. 1-16.

26Schlichting, H., Boundary-Layer Theory, 6th ed. McGraw-Hill,New Y o r k , 1968.27 Liebeck , R. H., "A Class of Airfoils Designed for High Lift inIncompressible Flow," Journal of Aircraft, Vol. 10, Oct. 1973, p.610-617.28James, R. M., "A New Look at Two-Dimensional In-compressible Airfoil Theory," Rept. MDC J0918/01, May 1971,Douglas Aircraf t Co., Long Beach, Calif, (restricted distribution).29Gartshore, I. S., "Predictions of the Blowing Required to Sup-

"Mathematical Model for Two-Dimensional Multi-Componentfoils in Viscous Flow," NASA Contractor Rept. CR-1843, July 1NASA.31Stevens, W. A., Goradia, S. H., and Braden, J. "Mathematical Model for Two-Dimensional Multi-Componentfoils in Viscous Flow," AIAA Paper 72-2, San Diego, Calif., 197232Foster, D. N., Ashill , P. R., and Williams, B. R ., "The NaDevelopment and Effect of the Viscous Flow Aroun d an AirfoilHigh-Lift Devices," Tech. Rept . 72227, Jan. 1973, Royal AEstablishment, Farnborough, England.33Prandtl, L. and Tietjens, O . G., Applied Hydro-Aeromechanics, Dover , New York , p . 155.34Abbott, I. H., and von Doenhoff , A . E., Theory of Wingtions, Dover, New York , 1958, p. 227.35 Perkins, C. D. and Hage, R. E., Airplane Performance, Staty , and Control, Wiley, New York , 1954, p. 78.36Chen, A. W., "The Determination of the GeometrieMultiple-Element Airfoils Optimized for Maximum Lift CoefficieTM-X-67591, 1971, NASA.37 Liebeck, R. H. and Smyth, D . N., "Study of Slat-Airfoil Cbinations Using Computer Graphics," Journal of Aircraft , Vol.April 1973, p. 254-256.38Hess, J. L. and Smith, A. M. O., "Calculation of Potential Fabout Arbi trary Bodies," Progress in Aeronautical Science, VoPergamon Press, NewYork , 1966.39Callaghan, J. G. and Beatty, T. D., "A Theoretical Methothe Analysis and Design of Multielement Airfoils," Journal o fcraft, Vol. 9, Dec. 1972, p. 844-848.^Gentry, A. E., "How Sails Work," SAIL Magazine (in seissues between April 1973 and Oct. 1973).41Ashill, P. R., "A Study of the Effect of the Wake of theAerofoil of a Fowler Flap Configuration on the Lift of the FTech. Rept. TR 72081, July 1972, Royal Aircraft EstablishmFarnborough, England.42Moser, A. and Shollenberger, C. A., "Inviscid Wake-Airfoteraction on Multielement High Lift Systems," Journal of AircVol. 10, Dec. 1972, p. 765-770.43 Wilk inson, D. H., "A Numerical Solution of the AnalysisDesign Problems for the Flow Past One or More AerofoilCascades," R&M 3545, 1967, Aeronautical Research CLondon.

^Cebeci, T. and Smith, A. M. O.,Analysis of Turbulent BounLayers, Academic Press, New York , 1974.45Cebeci, T. and Witherspoon, G. F., "Theoretical SuctionPressure Distribution Bounds for Flow Separation in RetaFlow," Journal of Aircraft, Vol. 11, Jan. 1974, pp. 61 -64.^Gartshore, I. S. and N ewma n , B. G., "The Turbulent Wallan Arbitrary Pressure Gradient," The A eronautical Q uarterly,1969. p. 25 .47Shen, C. C., Lopez, M. L., and Wasson, N. F., "A Jet-Lift ing-Surface Theory Using Elementary Vortex DistributiJournal of Aircraft, Vol. 12, May 1975, pp. 448-456.48Halsey, N. D., "Methods for the Design an d Analysis ofFlapped Airfoils," Journal of Aircraft, Sept. 1974, pp . 540-546.49Nash, J. F. and Patel, V . C., Three-Dimensional TurbBoundary Layers, S. B. C. Tech. Books, Atlanta, Ga., 1972.50Cebeci, T., "A General Method for Calculating TDimensional Incompressible Laminar and Turbulen t BounLayers; II . Three-Dimensional Flows in Cartesian CoordinaRept. MDC J6517, Mar. 1974, Douglas Aircraf t Co., Long BCalif.51 Kline , S. J., Morkovin, M . V ., Sovran, G., an d Cockrell , D"Computation of Turbulent Boundary Layers," AFOSRStanford Conference, Stanford Univ . Press, 1968.52Cornish, J. J., Ill, "High-Lift Applications of SpanBlowing," Paper No. 70-09, Seventh Congress of the ICAS, R1970.

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