A+M+O+Smith+HIGH+LIFT+AERODYNAMICS.Pdf

6 . VOLNO , 12 . J. AIRCRAFT JUNE 1975

37th Wright Brothers Lecture* High-Lift Aerodynamics

A. M. O. Smith McDonnell Douglas Corporation, Long Beach, Calif.

Nomenclature f = chord fraction, see Eq. (5.1) H = shape factor of the boundary layer, d*/0 chord = plate£ lengt h section profile drag coefficient lif= t L m local skin-friction coefficient T w/(l/2) puj m = exponent in Cp=x flows, also lift magnification section lift coefficient factor (5.1) CL lift coefficient Mac= h numbeM r Cn conventional pressure coefficient (p— p « )/ (1/2) p = pressure dynami= q c pressure pressure coefficient when local flow is sonic Q = flow rate

canonical pressure coefficient (p —p 0 )/(\/2)pu$ R = Reynolds number (= u Ox/v in Stratford flows) c. suction quantity coefficient Q/u Ox R6 = Reynolds number based on momentum thickness blowin= CM g momentum coefficient (2uft/u^ c),in- uee/v compressible flow Stratford'= S s separation constant (4.10); - alspe o C ^ = blowing momentum coefficient referred to momen- ripheral distance around a body or wing area tum thickness of the boundary layer at blowing / = blowing slot gap, also thickness ratiboda f oo y location, uft/ujB, incompressible flow u = velocity in x-direction initia= u0 l velocit t starya f deceleratioo t canonican ni l Presente Papes da r 74-93 AIAe th Aircraf h t A9a 6t t Design, Flight Stratford an d flows Test and Operations Meeting, Los Angeles, Calif., August 12-14, v = velocity normal to the wall 1974; submitted Augus , 197414 t ; revision received Decembe, 27 r 1974. V = a general velocity Index categories: Aircraft Aerodynamics (including Component x = length in flow direction, or around surface of a body Aerodynamics); Boundary Layers and Convective Heat Trans- measured from stagnation poin f i uset n coni d - fer—Turbulent; Subsoni Transonid can c Flow. nection with boundary-layer flow

A. M. O. Smith, Chief Aerodynamics Engineer for Research at Douglas Aircraft Company, Long Beach, Calif. s bor wa Columbian ,ni Juln o , 1911 2 y. e Mo ,H . majored in Mechanical and Aeronautical Engineering at the California Institute of Technology, receivin e M.Sgth . degre botn ei h fields. After graduatio 1938n i e ,h joined Douglas as Assistant Chief Aerodynamicist. During this period, he worked on aerodynami preliminard can y design problem DC-5e divD th f esSB ;o bombere th d ;an A-20, DB-7, and B-26 attack bombers. He had prime responsibility for detailed aerodynamic design of the B-26. Becaus f earlieeo r work with rocket Caltecht saskea s . Arnol wa Geny H de b . h , H . d o organizt head e Engineerinan edth g Departmen f Aerojeo t s theia t r first Chief Engineer leavn o , f absenceo e from Douglas 1942-1944n i , . After expandin departe gth - ment from 6 to more than 400 and seeing the Company into production on JATO units, he returned to Douglas and Aerodynamics. There he handled aerodynamics of the D- 558-1 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 he has developed powerful methods of calculating potential and boundary-layer flows, culminating in a book co-authored with T. Cebeci entitled, Analysis of Turbulent Boundary Layers.publishes ha e H d ove papers0 r5 . wors hi Aerojett kr a Fo receivee h , Robere dth t H. Goddard Awar Americae th f do n Rocket Society. For his early rocket work at Caltech, he is commemorated in bronze at the NASA Jet Propulsion Laboratory. In 1970, he received the F. W. (Casey) Baldwin AwarCASIe th f do , whic commemoration i s hi firse th t f nCanadiao airn a -y fl o nt plane. He is a Fellow of the AIAA.

Presente Papes da r 74-93AIAe th Aircrafh t A96t a t Design, Flight TesOperationd an t s Meeting Angeless Lo , , Calif., August 12-14, 1974; submitted Augus , 197414 t ; revision received Decembe , 1974r27 . Index categories: Aircraft Aerodynamics (including Component Aerodynamics) Boundary Layer Convectivd san e Heat Transfer—Turbulent; Subsoni Transonid can c Flow. *Note 36te hTh : Wright Brothers Lectur Hermay eb n Schlichtin publishes gwa Aprie th n dli 1974 AIAA incorrectls Journalwa t bu y identifies da 37te th h Lecture. 501 502 A. M. O. SMITH J. AIRCRAFT

Greek a = angle of attack 6 = flap deflection 6* = displacement thickness of the boundary layer momentu= 8 m thicknes boundare th f so y layer v = kinematic viscosity mas= s pdensit y MODEL CHARACTERISTICS T = shear stress <: = M.A.C.= II.02 IN. \l/ = stream function SPAN=8.00 FT. ASPECT RATIO = 8.69 Subscripts 5=7.37 FT2 edg= e condition e s Je= t J £ = lower referenc= o e conditions Stratforn i s a , d flows u - upper w = at the wall = referenc°° e conditio t infinitna y

1, Introduction and Purpose V^i/HE NI firs t bega e nDougla th wor r fo ks Aircraft T T Company in 1938, Donald W. Douglas, Jr., was in school. Several times, in connection with his studies, he came helr fo pe witm o ht aerodynamic homework problems thao s , t knoo t t wwego each other. Years late 25-yeaa gave h re em r service pin and asked me what I was doing. For once I was quick-witte d answerean d d "still tryin o t understang d aerodynamics." That is the primary purpose of my lecture—the understanding of one aspect of aerodynamics. It is not a "how ito "d lectureo t rathet concernebu e , on r d with "whysd "an Fig 1 . GALCIT boundary-layer-control wind-tunnel model. principles researcht n directl.i Sincno m a m ea I y I ,involve d wit down-to-earte hth h design problem aerodynamif so c hard- example, over an airfoil surface. The only convenient ware, but instead am in a position of giving help to others. method r analyzinfo s g inviscid flow abou n airfoia t l were That reminds me of the definition of a mathematician I once those of thin-airfoil theory and the flow about certain simple heard l aboual w mathematiciau teln .A ho t lyo ca o wh e on s ni conformally mappable sections. Although Theodor sen's to solve a problem but cannot actually do it himself. metho beed dha n developedtediouo to s r practicasfo wa t i , l It was my privilege as a graduate student at Caltech to listen use. To calculate the pressure distribution about an arbitrary brilliane th o t t beginning f thiso s series. MelvillB y b , e Jones airfoil amounted almost to a "stunt." Furthermore, all these in 1937. It is a great honor, a privilege, and a difficult challenge to follow him and his many illustrious successors. examples were for single airfoils. Essentially nothing could be don r slotteefo d airfoils. wondeHenco n s i t ei r that effectn i , , introductios Inhi saie nh d "... instructem Ia d tha Wrighe tth t people justested an t . Performanc ddree it shapea ey p wu y b e Brothers' Lecture should deal with subjects upon which the of airfoils was correlated in terms of certain obvious lecturer is engaged at the time, rather than with a general sur- geometric parameters: camber, position of maximum camber, vey of some wide branch of aeronautical knowledge." To a great extent plan,y thacentrae m s th ;i t l interest wil sube b l - thickness, etc. Transition as an explicit phenomenon in the development of jects with which I have been closely associated. a boundary layer was only vaguely recognized. For instance, Becaus ehavI eaeronauticae beeth n i l fiel r somdfo e time at the GALCIT tunnel, surely as progressive as any, tests were now I fin, t i interestind o loot g ke stat th bacf o et a k aeronautical knowlege when I was in school in the 1930's. I often mad modeln eo commerciaf so militard lan y airplaneo st thin wilu kyo l I proceedsee s a , , that whil probleme eth e ar s lear e effec th nf Reynoldo t s numbe n drago r . Drag coefficients were measure seriea t da f tunne o s l speeds. Results far from being completely solved r capabilitieou , s hav- ead were plotted on a log scale and then generally extrapolated by vanced tremendously firsy M .t proble f msubstanco y m s ewa straight-lina e extensio fulo nt l scale curve Th . e usualla d yha M.S. thesi sperforo t projects wa m t I powerea .test n o s d downward slope with Reynolds number, but not always. Even Boundary-Layer-Control Model in the GALCIT Ten-Foot when the slope was positive, the line would be extended the Wind Tunnel. Figure 1 is a drawing of the model. same way; the effect was dismissed as a "poor Reynolds num- The project was inspired by German work of the time. In r extrapolation.be knoe w ww thamodee "No th t l must have desige winge th th philosophy—se f nth ,o knew—waI s a r ofa s been in the rapidly varying transition region, which must as simple as this: 1) select conventional airfoils of the time for surel e lefyb t long before full-scale Reynolds numbers are the wing, 2) put a slot (square edged) somewhere in the upper reached. Fixing transition was essentially unheard of. B. surface of the wing (70% chord seemed as good as anything), Melville Jones f courseo , , greatly increase r awarenesdou f so and 3) suck air and see what happened. There was insufficient the transition phenomenon. knowledg boundary-layef eo ranythin o flod wo t g much more sophisticated. At the end of the tests, the writer had a vague 2. Some History uneasiness that there should be some more scientific approach leadina n problei e s th go wa t f designt mi o y wa . e Thath s i t Because, unlik birte e th f Venus ht o burs ideano w o tne ,sd school s undetha e wa leadershit th r e eminenth f n o p vo t forth fully matured or fully recognized, or even in one place, Karma Clard n an . Millikan kB . authoritative establishment of history is difficult. We know t thaA t time, Pohlhausen's approximate methof o d better tha attempo nt t herei t . Nevertheless, some revief wo calculating general laminar greaa developflow w s ne twa s - highlights seems desirable. ment. Only the most rudimentary method was available for In a search of the older literature, one of the strongest im- estimating turbulent flows in general pressure gradients, for pressions gaine thasenss a di n i t e ther "nothins ei undew gne r JUNE 1975 HIGH-LIFT AERODYNAMICS 503

the sun." Nearly all of the basic principles for influencing a Figure 5 shows lift coefficient vs angle of attack for the ex- flow to develop high lift have been known from the very early perimenta s modifielwa airfoit i ds o eight a l froe t on m days of the airplane. The defects in knowledge were two: elements t generallI . y shows thagreatee th t numbee th r f o r first, althoug havy ma eh beeo whad no t known reasone th , s element e greatet seemi th s d e lifto confirt th san r ; e mth for doin t wergi e only dimly understood; second, quantitative author's deduction, which was made three years ago, quite in analysis of a flow could rarely be accomplished. ignorance of these tests. The seven-element airfoil reached a Prandtll had already conceived and demonstrated the prin- lift coefficien f 3.92o t . Tests were mad chora t ea d Reynolds ciples of suction boundary-layer control in 1904, and by 1913, numbe f abouo r t 250,00 wina n f 6~in0o go . chor 36-ind dan . accordin o Weyl,t g e notioth 2 f blowino n o controt g e th l span. boundary laye beed ha rn advanced concepe On . authoe th t r Handley Page appear havo t s e followe empirican a d - ap l believes to be new is that of the jet flap, which seems to have proach in his efforts. Concurrently Lachmann7 at Gottingen been conceived and developed in the 1950's. Hagerdorn and was studyin problee gth m theoretically. Lachmann used con- Ruden3 tested forms of the jet flap in 1938, but they did not formal-transformation method represented san dsla a vor y tb - understand what they were finding. The ancestry of flaps can be traced back to the early days of flying. British R&M No. 110, dated 19144 contains one section entitled, "Experiment Aerofoin a n so l Havin gHingea d S§Xs> ~~j+ Angle Rear Portion." Figur showe2 shape sth e tested largA . e num- f flao pr be settings were examined. Accordin o Weyl,t g 2 variable camber had been used even earlier. The LeBlon Fig. 2 RAF 9 airfoil with a 0.385c plain flap tested in 1912-1913. monoplan a variable-cambe d ha e r win- gad formen a y b d justabl e trailine th par f o tg s exhibite edgee wa th t I .t a d London Olympia Show in March, 1910. knowledg e idee th f slatTh a o d effectivenes e san th f eo f o s slots are nearly as old. In an important lecture given before the British Royal Aeronautical Society on February 17, 1921, Handley Page5 describe yearn developde te f worth o s n ko - men airfoilf o t s thaslotsd wors ha t Hi . k deal t onlno t y witha single slot or slat but also with multiple slots. Figure 3,6 shows one of his models, which by current standards would be considere a drelativel y modern configurationf . o Pag 5 e42 Ref. 7 shows a good photograph of a 1920 airplane fitted out with slats. Later in this paper, the author attempts to prove that an airfoil havin +n g1 element n develoca s p more lift thae on n having n elements. Handley Page investigated this problem, 5 up through 8 elements. Figure 4 shows one of his extreme Fig. 3 A wing tested by Handley Page as part of his effort at airfoils vera , y highly modifiesection9 1 F ,dRA positione t da developing slots and slats. angle th maximur efo m lift.

Fig. 4 Handley Page's eight-element airfoil modifie 9 section1 F d .RA fro n ma The model is at 42° angle of attack, the angle for maximum lift. Pressure distribution e theoreticalar s . They were o correspont ° 36 = mada o locadt t a el win6 = dangl R tunneA f attace eo th lf ko model. Theoretical c( of ensemble is 4.33. 504 A. M. O. SMITH J. AIRCRAFT

4.0 Speaking of biplanes, history seems to be repeating itself. In the early days, because they were biplanes, they had thin wings/These suffered from leading-edge stall, for which nose droo r slatpo s wer cureea . The advancee nw monoplaneo dt s 3.2 d thickean re leading-edg th wings d an , e problem almost disappeared. Later, when jets and higher speed entered the picture, wings again becam e leading-edgeth thind an , e problem returned. 2.4 Original flap development was motivated by three desired 3 benefits: 1) slower flying speeds, hence shorter takeoff and landing runs ) reductio2 ; f anglno f attaceo k near minimum 2 SLOTS flying speed; 3) increase of drag, or control of drag, in order steepeo t n glide angl approacn ei reducd han e floating ten- dencies. Currently, because of large aircraft noise problems, NO SLOT the emphasis under the third item has changed. We are trying

0.8 to reduce flap drag in order to reduce thrust requirements and hence the noise. The split flap was conceived and partly developed in the period 1915-1920 as a means of satisfying these desires. Klemin, Schrenk, and Etienne Royer2 are important con- 0 10 20 30 40 50 ANGLE OF ATTACK (DEG) tributors to its development. Orville Wright and J. M. H. Jacobs obtaine dbasia c subjece patenth n 1924n o i tt , after Fig. 5 Ci vs ct data for the RAF 19 broken up into different numbers of elements, as indicated by number of slots. they filed it in 1921. Their discussion in the patent gives evidence that they had a good qualitative understanding of the basic flow processes drawinge reproductioA . th f o e s on f no from the patent is shown in Fig. 7. We end this historical survey of mechanical high-lift devices by mentioning the Fowler Flap invented by Harlan D. Fowle r 1927n 1i 0 . Becaus extensiod ha n slota t ei ca d t i , nan be considered to be the first modern high-lift mechanical flap. In early NACA wind-tunnel tests, H it developed a maximum lift coefficient of 3.17.

Aug. 12, 1924. 1,504,663 O. WRIGHT ET AL AIRPLANE Filed May 31, 1921 3 Sheets-Sheet 1

Fig. 6 Variable camber Albatross Biplane9 model, showing approximate range of flap angles tested.

tices in front of a circular cylinder. His work established the gross feature interactiod an s maie nth effect nslad a r an t fo s airfoil. Later, Lachman d Handlenan y Page joined forces. Further basic understanding and appreciation of the beneficial effects e gainea Page,slaL f wa o to y sb dwh 8 systematically investigated the forces on two airfoils placed in tandem, t Another work that merits mentio entitles ni d " Mode- Ex l periments with Variable Camber Wings, y Harrib " d an s Bradfield,9 which was published in 1920. The report includes both test dat studie d significance aan th f so result e th f n eo so airplane performance. Figur showe6 biplane sth e cellule that was studied. tLouis Stivers and R.T. Jones of NASA Ames inform me that Chaplygin had done work similar to Lachmann's in 1911 and 1921. e SelectedTh e Se Works n Wingo Theory f Sergeo . ChaplyginA i . Fig. 7 Sheet 1 of the U.S. Patent 1,504,663, by Orville Wright and English translation available through Garbell Research Foundation. J. M. H. Jacobs, illustrating their concept of a split flap. JUNE 1975 HIGH-LIFT AERODYNAMICS 505

Powered lift augmentation y suctiob s d blowinga ,an n , showe ar FigC n i d B. an ,10 fourtA . limitinr ho g cas juss ei t received considerable attention in the 1920's but never quite straigha t line. Accordin Joukowsko gt i airfoil theoryy an r fo , prove efficiene b o dt t actuaenougn a n i ljustife o ht air us -s yit of those circular-arc mean lines, regardles f degreso f cameo - plane. Betz Ackeretd an 2 Germann i 7 y were leader thin i s s ber, line of development. For those interested in further research into the early his- tor f high-lifyo t work mentioe w , n several references. Firss i t tha Alston,y b t 12 generaa whic s hwa l lecturRoyae th o et l In that equation, c is the length between the ends of the arcs, Aeronautical Society in 1934 entitled, "Wing Flaps and Other angle th f a.attacks ei o measur a s /d 3cambei e an th , f eo s a r Device s Aida s n Landing.i s t I survey" e statth f so e indees i c chor e , dth B s d a show r arcan Fo sA Fign . ni 10 . knowledge in 1934, but does not particularly deal with earlier conventionally defined. But for arc C, the chord c is not now history especialle . Thosar o ewh y intereste histore th n df i y o the longest dimension; the diameter is. In fact, it is evident high-lift devices and the origin of concepts and applications fro figure mth e tha 0^90°s a t , c-+0. Hence define w e f i ,eth papee shoul th Weyl,y e b r dse 2 whic vera s hi y broad sur- lift coefficient in terms of the longest dimension, we have vey—it has 116 references—of the entire subject, with in- [from Eq. (3.2)], teresting sidelight e developmentth n o s n exampla s A .f o e (3.3) such a sidelight, Weyl notes that in connection with the developmen f splio t t flap Frenca s h scientist, Lafay 191n i , 2 observed "tha unsymmetrican a t l roughenin a cylinde f go r 1 whic exposes hairflowa n a o dt w resulte aerodynamin a n di c - AWRIGH T FLYER H - 749 LO. - BSPIRILST F TO JIS 1 - 1049 force whic directes hwa d toward smoote sth h side acrose sth C- C-4 7 J - C-130 flow (lift force)." D - 23012 AIRFOIL 4 MA - K E - B-32 L - L-19 F - C-54 M- 727 mose Th t comprehensive reference examine authoe th y db r THEORE1 ICAL 13 G - C-124 N - C-5A is a report by A. D. Young, "The Aerodynamic Charac- " LIMIT- 47T teristic f Flaps.o s t dealI " s wit e aerodynamith h c charac- 12.0 I.E. DEVICES & teristics—lift, drag, moment, etc.—o l typeal f f flapso s s It . TRIPLC ESL DTTER DO > < FOWLER FLAPS extensive bibliography cover l aspectsal s , including general (SWEPT W NGS) 4.0 reviews, history, theory d investigationan , e variouth f o s s —i SPLIT FLA PS V type f devicesso alln I . , 138 reference givene sar papee Th . s ri K certainly not obsolete, although it was published in 1953. 3.0 A paper that amounts to an updating of Alston's 12 is one J^.-- ^ s CL/ \RK Y AIRFOIL 14 J by Duddy, which was also presented to the Royal C fH^ Aeronautical Society. It compares and evaluates various types 2.0 r^ \-& -t of flaps analyzed an , s their benefit termn i s f landinso d gan A D o—— _—— ^ - -^ FOWLER OR takeoff performance. Effects of sweep are included. 1.0 -SLOTTED F A useful historical summary of the gradual improvement of MAC A AlRFOILS gross lift coefficient s showi s o t n Fig i n , e whic8 .du s i h 15 0 Cleveland. From about 1935 to 1965—a period of 30 1930 1940 1960 years—we have advanced from coefficients of roughly 2 to YEAR roughly 3 on important service-type airplanes. By 1995 will we have advanced to 4? Fig. 8 Growth of maximum lift coefficients for mechanical lift system functioa s sa timef no , accordin Refo g. t 15 . 3. Some Lift Limits 1 Limit3. Potentian si l Flow Jus ideal-cycls a t e limit usefue sar thermodynamicsn i l o s , are the theoretical limits of lift useful in aerodynamics. Knowledge of those limits helps give us a perspective as to attainable willinb e wha d y ar an e tma gw w f ei no where ar e ew to seek without compromise the maximum possible lift. First conside e limitth rf n inviscici o s( d flow, where separation wilt occurno l . Conside classicae th r l circulatory flow abou circlta e show differeno Fign Tw i . .9 t level cirf so - culation are shown; that in Fig. 9b is of the greatest interest, for ther e circulatioth e o strons s i ng that frond reaan rt stagnation points coincide. Greater circulation moves the stagnation point off the body. Although that is entirely Fig 9 . Flow abou circlta e witdiffereno htw t degree f circulationso , possible—a Flettner Rotor generates such flows—it is not a a) moderate circulation, b) circulation so strong that the two realistic analog for an airfoil, which in general has two stagnation points coincide. stagnation points ,surfaces botit referenc e n hth o f I . e chors di taken as the diameter of the circle, it is easily shown that flow Figf o representb .9 lifsa t coefficient,

C( = 47T (3.1) where the reference length is the diameter. Now let us consider mean lines. We shall consider only circular arcs, because they are easily obtained from Joukowski's transformatio r flofo nw abou circlea t e eve vien th I .r f wo present specte separationf ro undun a s i t ei , refinemen cono t - sider either the effects of thickness or the effects of other Fig 0 Thre1 . e circular-arc mea , stemminnC , linesB , A ,g froe mth kind meaf so n lines. Three possible circular-arc mean lines, A , Joukowski transformation. 506 A, M. O. SMITH AIRCRAF. J T

and Observe tha. (3.8 tEq nearl s )i . (3.5) sam e cire Eq yth -s Th .ea cular-cylinder case is represented by f=l, and again we (45°

brings the two stagnation points together again, and we ef- High values of CL cannot be maintained indefinitely as fectively recover the flow about a circular cylinder, as in Fig. speed is increased, for soon surface pressures less than ab- 9b. solute zero woul indicatede db loos u t k int Le .proble e oth m extrem n lase a s ti Th e mean- linehalf-circlex e o Th .s t no s ei briefly, and search especially for the limits of lift rather than treme factn I . , modern multielement flap system t fulsa l flap f lifo t coefficient usuae Th . l equatio lifr s ni fo t deflection begi o approximatt n e half-circlth e e mean line (e.g., Fig. quantit29)e Th . y c = 47r/(2) a . (3.6 s Eq ha )1/n i 2 2 f L = V2PooV (x>CLS (3.10) valu f nearlf ecourseo O . yvalue 9 th , f ceo ( depende th n o s length used as a reference. If arc length were used instead of An alternate form, one that uses a different expression for conventional chord, as above, the values would not appear to dynamic pressure, is higho straighe s th facte n I r b . fo ,t line, half circle fuld an l, circle, the theoretical values would be 6.28, 5.65, and 4.0, (3.11) respectively. The mean lines that have been considered approximate With 7 = 1 .4, we can rewrite it as flows about airfoils having sharp trailing edges r themFo .e th ,

Kutta condition setcirculatioe sth suco nt h strength thae th t Lip 0.7(MiC= oo L)S (3.12) rear stagnation poin s alwayi trailine t th t a s g edge. Such a flow might be called a natural flow. To round out our Sinc a functios know i e e produc« C b ,eM th , f o t no n t discussion shale w , l mentio ncasa whicn ei h thersharo n s ei p L quantite th s Li C y f reai thao M s li t significancee w o s d an , trailing edge ellipse th : e family. Assum circulatioe eth e b o nt seek to make statements about its value. Observe that for a controlled in such a way that the rear stagnation point always give proportionaw n- no i lifvalue at Cs i M tth e f eth o o t l jc-axise th same f remainth o ,reat e d thaa pointh , r en t tis s a t L 16 mospheric pressure. as in symmetric nonlifting flow (see Fig. 11). Thwaites s cannon tensioni ga e A b t . Henc e limitinth e g suction proposed airfoils that followed that principle. He used area pressure is a perfect vacuum over the entire upper surface. suctio reae eliminatth o t r t na e separation. Whe r connou - The limiting pressure on the lower surface is stagnation dition for the circulation is met, the lift coefficient based on pressure. the length of the x-axis becomes exactly By definition, with 7=1.4,

c(=2ir (1 + t) sin a (3.8) ______f-^ f f w (3.13) If ce were lengt e baseth the^-axisn f dho o woult i , e db p~ l/2p „ VI 0.7p , Mi

(3.9) If the flow is assumed to be isentropic, Cp can be written in the form

REAR STAGNATION 1 + 0.2MI POINT (3.14) 1 + 0.2M2

Before proceedin determinatioe th o gt liff no t - limitsin s i t ,i terestin pauso gt consided ean limite th rf Co sp valuesn I . high-lift testing, maximum lift is often found to occur when Fig. 11 Lifting flow about an ellipse with circulation adjusted to highese th t local velocities reac hMaca h numbe f 1.0e o r Th . maintain the rear stagnation point at the end of the Jt-axis. Cp correspondin thao gt t condition wil callee lb d C*pera f .I - JUNE 1975 HIGH-LIFT AERODYNAMICS 507

If the entire flow on the lower surface is a stagnation flow, M?=0, and hence with a vacuum on the upper surface we have (3.17)

Equation (3.17) represents the absolute lifting limit. The quantity increases with Mach number because lower-surface pressure increasn sca e with Mach number. Equation (3.17s )i optimistic, because isentropic compression is assumed, whereas in reality there surely will be a shock when M> 1. Table 2 shows values of Mi CL for several flow conditions, including Mayer's MiC l limitLe maximu= Th . m value possibl subsonin ei c fligh contributione th 2.71s i t d an , s from each surfac aboueare t equal Mayer'If . s value Mis i C-1 p= accepted limie th , t decrease «- 2.28o M t s=0.5 ab t A e . th ,

Fig. 12 A/i C values as a function of Mach number for four con- solute maximum value is only 1.70 and Mayer's value is 1.27. p ditions. 0.7 vacuum corresponds to M\, Cp = -1. Those are the limiting values, regardless of the kind of high- lift devices that are used. At M «, =1.0 and below, the assumption of isentropic recompression on the lower surface is

Table 1 Limiting Mac valuew lo t h a f C numbers o s p very good, as is well known. As Mach numbers become large,

Moo 0.10 0.15 0.20 0.30 0.40 0.50 pressure lowee th n rso side become great largd an , e valuef so C*(M = 1) -67 -29.3 -16.3 -7 -3.7 -2.1 i CM L develop—values tha t^ M excee =2.0t a 1 d1 . Mayer (^(perfect considers the case where under-surface pressures in supersonic vacuum) -143 -63.6 -35.8 -15.9 -8.9 -5.7 flight correspon normaa o dt l shock givee H . s considerabl- ein formation on maximum-lift values in supersonic flight, in- Cp (0.7 vacuum) -100 -44.5 -25 -11.1 -6.3 -4.0 cluding correlation f theorso experimentd yan t sincr Bu . eou interest is principally in subsonic flight, we shall not discuss wors hi k further.

feet vacuum were reached, we would obtain, according to Eq. 17 3 Demonstrate3. d Lifting Limits (3.13) i Cvalua ,M p f equaeo --1.43o t l . Mayer looked highese foth r t possible negative Cp value experimenn si y b t examining hundreds of NACA test data points from all sorts s interestini t I g thae questioth t f aerodynamino c lifting of tests. He found the remarkable empirical result that the limits in subsonic flight was posed to the author in 1946 by E. highest experimentally measured values correspond with . HeinemannH problee Th . fightes mwa r maneuvering—what lift coul dwina g really develop available th l ?Al e flighd an t reasonable accuracy to Mi Cp=—l. That value corresponds to 0.7 of a vacuum [see Eq. (3.13)]. Figure 12 shows the test wind-tunnel data were examined examinatioe Th . n included dat usee a h supporo dt findinge tth . the large supply of German World War II data, which covered Since r interesmuc ou high-lifn i f ho s i t t testing, whics hi testgreaa f so t variet airplanf yo missile-typd ean e wingse .Th done at low Mach numbers, we convert some of the results results were reported in Ref. 18. The study, which was strictly from Fig. 12 into C form and present them in Table 1. one of observation, concluded that the maximum possible lift p was about 1/2 atm. That number corresponds to a value for At low Mach numbers, quite high values of Cp are ob- tainable before compressibility becomes very importantt bu , Mi CL of about 0.7. Since that time e desigth , f swepno t wing s advanceha s d t highea r Mach number limitine sth g Cp t vervalueno ye sar high. Note, however, that Mayer's data in Fig. 12 do not considerably. Now highese th , t i valuCM Le f eseeth o y nb cover the lower Mach numbers; hence their validity there is autho swep1.2s a ri n 0o t wing usin aft-loaden ga d airfoil section with no auxiliary lift devices. The value corresponds to t trulno y established. t reall *C/no M 1.2 =0.975,= s a maximumyt a 6 wa t I . - e authoTh r o theoreticaknown f o s l explanatior fo n lift value but, rather stoppina , g poin wind-tunnea n i t l test Mayer' vacuu7 s0. m correlation t doe I .t correspon sno a o dt determinethas wa t strengty db stine th gf ho support . Hence constant local Mach number. By the use of Eq. (3.14), it is the value can surely be exceeded. It should be pointed out that easily deduced that Mi Cp = -\ corresponds to Mlocal = 1.43 this is not a buffet limit, which is usually considerably lower. atM,, =Oandl.55atM00 =0.5. If one went all out to maximize M£ CL, without regard to Now conside airfoie rth l problem hole w constan dM f I . t a t low-lift performance, much better could surely be done. uppe e value th on n reo surfac constand ean t anothea t r value Leading-edge and trailing-edge devices undoubtedly would on the lower surface, Cp represents the CL for each surface. raise the limit. Variable-camber wings are currently receiving The total CL is the difference. Hence from Eq. (3.14), attentio anothes na r possible mean raisinf so limite gth . Some further data related to limiting lift conditions are ,.,~ 1 r , 1 + 0.2MI , ,, 0.7 1+0.2M/ Table 2 Maximum-lift limits assuming isentropic compression, and uniform chordwise loading -( 55 (3.15) 1 + 0.2M* ) ] M^ M lvlM, Mic MiC upper J 00 Lupper lower Llower Ltotal wher d e/JL an £denot e lowe d uppean r r surfaces perfecA . t 0.5 00 1.43 0 0.27 1.70 0.5 1.55 1.00 0 0.27 1.27 vacuuuppee th n rmo surface correspond r Mo ^0 o p t s/x = 0.5 1.5 0.97 0 0.27 1.24 oo= than I . t case secone th , d. (3.15 terEq mzeron s i )i . With 1.0 00 1.43 0 1.28 2.71 respect to atmospheric pressure, the load carried by the upper 1.0 1.86 1.00 0 1.28 2.28 surface when there is a perfect vacuum is [see Eq. (3.13)] 1.0 1.5 0.69 0 1.28 1.97 2.0 oo 1.43 0 9.75 11.18 2.0 4.97 1.00 0 9.75 10.75 1/0.7=L= 1.43, constana t (3.16) 508 A. M. O. SMITH AIRCRAF. J T

-17 for any trailing edge, even the cusped type, the flow decelerate beloo st w freestream values. Hence t morge eo t , lifte w , -15 need higher velocities over the airfoil, but that in turn means greater deceleration towards the rear. The process of deceleratio o severeto s i ,t i s separatiocriticali nf i r fo , n develops e sciencTh . f developineo g high lift, therefores ha , componentso tw analysi) 1 :boundar e th f so y layer, prediction of separation determinatiod an , kinde th f flowf o sno s that are most favorable with respect to separation; and 2) analysis inviscie oth f d flow abou givea t n shape wit purpose hth f eo finding shapes that put the least stress on the boundary layer. The two parts amount to a kind of applied-load and Fig 3 Experimentall1 . y measured leading-edge minimum pressure allowable-load problem analysin .A s canno valide tb unless sit

coefficient at cimax for a variety of two-dimensional high-lift con- elements are sound, so let us first look at the problem of figurations. Test Mach numbe 0.20r= . predicting the onset of separation.

shown in Fig. 13, which is taken from Ref. 19. In tests of a 2 Accurac4. Predictinf yo g Separation Points considerable variety of two-dimensional models, it was ob- Because this question is so vital to many aerodynamic

served tha lowesnearle s modele th t wa th ya tn t co Ca , (max p design problems, the accuracy of four leading methods was constant equa C*o t l p. Ther considerabls ei e scatte date th an ri studied recentle cas th f turbulene o r Refn yfo i , .23 t flows, of Fig. 13 becaus teste eth s wer t planneeno investigato dt e Cp f interesto e e flowon Th whic.e sth consideres hi d involved t a C(max t therbu , considerabls ei e evidenc criticaa f eo l con- simple rear separation. Laminar bubble or laminar leading- dition compressiblf I . e Cp calculatevaluee b n airn sa ca r - dfo edge separations wer t consideredeno maie Th .n reasone ar s f estimatino foil n y easa , wa y g c(max s suggestei e th y b d that proper design can eliminate them and that at high figure—just find the c( at which Cp becomes sonic. All the Reynolds numbers transition has occurred by the time adverse tests wer airfoin eo l sections where rear separatiot no s nwa gradients appear. Fortunately, the problem we consider here critical. is technicall e moryth e important, becaus e probleth e f mo separation after a laminar bubble and reattachment cannot be Load-Carryine Th . 4 g Capacit Naturaf yo l analyzed nearly as well. That is a problem for the future. Boundary Layers There is no difficulty with simple laminar separation. It can be calculated to three- or four-place accuracy if the pressure 4.1 Introduction input data have sufficient accuracy. nexe th Thit d sectiosan n constitut paper e bode th e th f yo . Four leading methods were examined: those of 1) In combination, they expound the basic aerodynamics of Goldschmied, 2) Stratford, 3) Head, and 4) Cebeci-Smith. mechanical high-lift systems. Until the 1960's, our analytical One case representative of the fairly large number examined capabilit insufficiens ywa mako t t e quantitative calculations. in Refs show i e separatio 3 .Th 2 n Figi n . 14 . n poins wa t Now, since the tools are here, it is time to put the whole story carefully measured. The predicted separation points are together bes s knoe a , thay w t wsa t nearlo itt .t Thal no y al s i t marked on the pressure-distribution curves, which were sup- problems have been solved, but to a certain extent the plied by the experiments. Goldschmied's method was erratic. remaining problems amoun juso t t irritating details. The other three were in reasonable agreement, both in what is We go into considerable detail, and often rather elementary show complete th n i herd e ean study, 23 Stratford's method exposition intereste th n i , f makino s g thi papee s th par f ro t tended to predict separation slightly early. The Cebeci-Smith complete. The approach is motivated by the many miscon- method appeared best, with the Head method a strong ception e author-hath s s encountered r instanceFo . , with second. respect to slotted flaps, it has been suggested that if one just Hence e Cebeci-Smitth , h metho s beedha n chosee th s na shapo t kne airfoile w eth w ho , higher lift coul obtainee db d basic method, although s madi muc e f Stratford'o eus h s with a one-piece airfoil, because the leakage from bottom to allowep slote to th sy db amount kina f o shorst do t circuit.

Properly designed multielement airfoil bettere ar s wils a ,e b l 6 shown. Another common statement is that a slot provides a 2.0 /?^=3.3xl0 a =10.5° kind of blowing boundary-layer control by the jet of *'fresh" moro n r tha s thateo ai d ha t t t flowi ?I n sca througw Ho . hit total head tha ambiene nth t stream. Admittedlyf , o tha t je t 1.6 "fresh" air has more energy than the boundary layer, but it only has freestream total head. At best, it may be a question of semantics. Later we shall see that the principal effect of a slot or slat is to reduce negative pressures, rather than to 1.2 blow. Sections 4 and 5 are a revision and, we hope, an im- provemen pape a f Ref , o t 20 .r giveAGARn a t na D meeting. There are many papers and reports on the subject of high-lift 0.8 PREDICTION OF SEPARATION BY devices t theibu , r emphasis usually e "hotendb o wt s to." EXPERIMENT Ours is "why." One useful document is Ref. 21, which con- CS METHOD HEAD tains a large collection of lectures on the subject of high-lift 0.4 devices. Another is Thain's,22 which is an extensive review of GOLDSCHMIED experimentalle th y observed effect f Reynoldo s s numben o r STRATFORD higs mori o he to lift concernet I . d with overall results than with explanatio fundamentae th f no l flow processese b t i t Le . noted thadiscussionr ou t thesn i e section confines si twoo dt - dimensional flow. vien I Bernoulli'f wo s law surfaca f i , lift velocito e t s th ,ei y Fig. 14 Comparison with experiment of predicted separation points over the upper surface must be speeded up. But we know that for Newman's airfoil. JUNE 1975 HIGH-LIFT AERODYNAMICS 509

method and ideas in this paper, mainly because of their great 100.0 RMS ERROR FOR convenience. Figure 15 summarize predictioe sth n accuracf yo 7 POINTS*7.186 % the Cebeci-Smith method. The data consist of the results from Ref togethe3 2 . r with other, unpublished studies shoule On . d not conclude from Fig. 15 that the general accuracy of turbulent-boundary-layer calculatio goos t appeari a s s ni da o st 10.0 . Neabe r separation accurace th , boundarye f manyo th f yo - layer quantities, such as the momentum thickness 6 and the velocity profile, become poor. Nevertheless, accuracy of predictin e locatioth g f separatioo n n remains goode Th . (EXPERIMENTAL) Cebeci-Smith method predict spoine separatioth t ta e b o nt 1.0 where cf, the local skin-friction coefficient equals zero. In view of those results, the method of predicting separation appear accurate b o st e enough engineerinn a n i , g sense, to be used in determining relations, limits, and the like. It should be noted that the Cebeci-Smith method can handle O.I with great accuracy the effects of Reynolds number or Mach O.I 1.0 10.0 100.0 foue r moreth o r f numbe5 o onle .o e t th Also yon p s i u r t i , methods that can analyze the case of axisymmetric boundary- —— X I06 (CALCULATED) layer flow. Fig 5 1 Accurac. f predictinyo g turbulent separation e pointth y b s Cebeci-Smith method. 4.3 Canonical Pressure Distributions In almost all design work, pressure distributions are presen- tetermn i d f Co sp = (p-p » )/(l/2p«i) than I . t kinf o d presentation, high negative values of Cp invariably look bad, and one is unable to tell by inspection much about the margin of safety of the boundary layer against separation. Yet we know from basic scaling considerations that if two pressure distributions can be made congruent by proper scaling in the x- and Cp-directions, then the two flows are identical except for the Reynolds number effect, which is weak. Then, if separation occurssame th t e a t wil scalei , e b l d poin botr fo t h flows. A particular 2-in. airfoil model at 100 mph will have very high values of velocity gradients, but a similar 200-in. o x ———

model at 1 mph will have extremely low values. Yet the flows Fig. 16 A canonical pressure distribution, Cp (x). e exactlar y similar because their Reynolds numbere th e ar s same. It is the dimensionless shape that counts. Hence it is particularly useful to scale out the magnitude of the velocity e chordd als th o scalan t ot .ou e Where separatio- im s i n portant e besth , t scaling e velocitfactoth s i r y just before deceleration begins. Because all pressure distributions are put in a standard form, it is natural to call the_m canonical pressur variablew distributionne a ee Cus typicapA .d an s l one is illustrated in Fig. 16, which shows the idea and basic relations. The exact details of the normalization may well e e e pressurproblenaturdepenth th th f n d o eo dm an e distribution. The canonical pressure distribution, together with a Reynolds number, completely describes the flow. A meaningful Reynolds number is Re at the beginning of pressure rise. The left-hand part in Fig. 16 might represent the airfoiln nosa f eo . Distanc measures i ex d alon surfacee gth , but the origin of x is a matter of convenience. Often it is convenien beginnine locato th t t a t e i f pressurgo e th e n risei s a , 0 6 ,jrO 0 5 0 9 8Qj 10/ 0 04 JiO__- 0 3 ^ 0 2 10 ______figure. Separatio occuy nma t som a r e poin noteds a t . Because PERCENT TlNBROKEN CHORD " of the very simple relation between pressure coefficient and velocity ratio e terth ,m pressure distributio s usei n- in d Fig. 17 A conventional theoretical Cp plot of a four-element airfoil. discriminately for plots of either pressure coefficient or right-hane Th d canonicascale th s ei l form, referre peao dt k velocitt ya velocity ratio, provided that the flow is at low speed. the nose of the flap. a= 15°, 6/ = 25% ds = 25°, CF = 4.08. In the canonical system C^^O represents the start of pressure Cd ris e maximu+p= an e 1th m possible,_tha, is t u Q.e= Normall ynegativo n ther e ar e e value f Co sp . Fur- thermore, if two pressure distributions can be made congruent 2 by proper scaling floa , w havin deceleratioga f (uno e/u x ) from 20 to 10 is no more and no less likely to eparate than deceleratine on g0.7o t fro 5 r eve5o m 1. n from 0.1 0.050o t . The canonical plot is the one that is meaningful in separation analysis. With magnitude effects scaled out, much more can be told by a simple inspection than by a conventional plot. We canonicae dwelth n o l l pressure distributio t somna e length,

because most working aerodynamicist t realizno s o it ed s °0 5 0. 5 0 1.5 00. 1.0 0 1.0 0 1.0 0 value. Figured 18an sho7 1 s w conventiona d canonicaan l l pressure typicaplota r sfo l high-lift airfoil basie Th . c charac- Fig. 18 Airfoil Cp of Fig. 17 plotted in canonical form. 510 . SMITO . M H . A J. AIRCRAFT

ter of the four pressure distributions is much more apparent in 4.5 Location of Separation in Two Families of Canonical Fig. 18. Figure 17 includes a canonical scale at the right, for Distributions the complete ensemble. To exhibit the limits on the ability of a boundary layer to flow into region f higheo s r pressure presene w , plotso tw t , 4 Relatio4. n betwee Canonicae nth l Pressure Distributiod nan the Conventional covering families of flows at two different Reynolds numbers, e flowTh s. consis20 lengta d f Figsan o tf constant 9 ho 1 . - velocity flow followe a pressur y b d e rise describee th y b d The relations between canonical and conventional pressure equation distributions are very simple, but because there has been con- subjecte fusioth n o n , they wil discussee b l t somda e length here canonicae Th . l pressure distributio easils ni y relateo dt (4.7) e conventionath l form, wher e referencth e e velocitd an y 6 7 pressure are u «, and/? „ . Because the square of the velocity is The unit Reynolds number is 10 per ft in Fig. 19 and 10 per never negative, it is simpler to deal with velocity than with ft in Fig. 20. Pressure rise is set to start at ;c = 0, but forward pressure coefficient fundamentae Th . l relatio quits ni e simple. of that point are various lengths of flat-plate flow. It seemed It is more convenien construco t t plote th tterm n i s f d feeo s an t «u Iv rather tha termn ni f Reynoldo s s number, although (u /u )2(x)=(u /u )2(x)-(u /u )2 conversion into Reynolds numbe easys i rtotae Th . l regiof no e (K e 0 0 OD (4.1) pressure rise is seen to be 1 ft. Four lengths of flat-plate run The last factor is just a constant. In many design problems, it initiae werTh . el ft studied1 d an :, ft 1/6 4 , 1/14ft 1/ , 6ft is desirable to start with a canonical pressure distribution, flows then developed boundary layer f variouso s thicknest sa beginnine th f pressurgo e rise indicates a , valuey n di eb R f so t necessarilthe 0no /u u ( ns i „) y know musd nfoundan e b t . the figures. The flat-plate flow is assumed to be entirely tur- Construction of a canonical pressure distribution from a con- bulent. If it were mixed laminar and turbulent, values of R 9 at ventiona mean. (4y b . Eq trivial1e s )i f so on l , becaus 0e/u_u x x = 0 would be less and different. is known. The method of converting velocities into Cp or Cp m form is obvious from the given relations. Calculation flat-plate th f Cso d p ean -x flo e partwth f so were then Cebeci-Smite madth y eb h method until separation probleA f sommo e importanc s thai ef applyino t a g m was reached. Lines cutting acros Ce pth s=x curves mark canonical pressure distributio desige airfoin th a f o ny t no b l separation points for the four lengths of flat-plate runs. The inverse methods. Somewhere alon pressure gth e distribution straight-line or concave pressure rises permit the greatest separatefloy e th wma . adaptatioe Theth n nairfoili e th o nt , recovery before separation occurs. Also curvethe , s show that e separatioth n point e forwarshoul b e trailin t th f dno o d g edge. From Eq. (4.1) we can get the following equation: amoune th f recovero t sensitivs yi lengte th f flat-plato eht o e run before the beginning of recovery. The separation loci, while indeed functions of the length of flat-plate run, are 2 J p I se J = I\ —— } \ —— V

-0.0625 -0.015625 (276)^ (108) 2 Solvin (ur gfo 0 /u^) substituting into Eq.(4.1)t ge e w ,

r ]2

2 From general airfoil theory, we know that (u e/u „ ) at the trailing edg abous ei t 0.8, correspondin Co gt p = 0.2. Then,

for instance, if the canonical pressure distribution separates at

(u/u) 2 = QA, the factor is 0.8-^0.4 = 2.0. Of course Eq. 0 e POINT SPACING FOR FLAT PLATE RUN => I FT (4.3) is valid, no matter where separation occurs, but an air- I I i I I I I I IIIIMIIIII Illlllllll foil is not likely to be deliberately designed to have separation. I I I I I I I assume Ifw e separatio trailine th t always na i ge b edg r o st efo airfoiy an l designe r maximudfo specializn mca lifte w . , eEq Fig. 19 Separation loci for a family of canonical pressure fore th m (4.3o t ) distributions. Point spacin gboundary-layee useth n di r calculations e 1-ffoth rt rooftos i noted n ru p. Value n parenthesei s s under origin. 0 = f flox svalueo t e a ew ar R f so ["-*- (4.4) un

Observe that if by some artifice we could greatly increase (u e 2 lu oo )-f E., we could greatly increase (ue/u „ ) (x) and hence vere th yr samfo lifte l eth al , canonical pressure distribution. More will be said about that later. Equation (4.4) is easily converted to Cp form. Writing

Cp = l-(ue/u o. )*, C^E =1- (ue/u .

2 and Cp = l-(ue/u0) (4.5)

we have

l-Cr (4.6) Cp(x) = l- 0 2 SeparatioFig. n loci for_a famil f o canonicay l pressure

l-Cr distributions. Loftin's criterion noted0.8 = Cis 8 . p JUNE 1975 HIGH-LIFT AERODYNAMICS 511

more fundamentally function e boundary-layeth f o s r he can get, the pressure-rise curve for ra = 1/4 or 1/3 is the thicknes e beginninth t a s f pressuro g e rise. Hencee th , best, because that give greatese sth highese t th rati d otan mean separation loci could jus s wel a tidentifiee b l notee th y ddb valu Cf eo p alon uppee gth r surface. valuebeginnine th Rf t so a d f pressurgo e rise. Then Figs9 1 . Reference 20 gives two charts for a different family of 0 becom2 d an e applicabl y combinatioan r fo e f risino n g pressure distributions; they are arcs of circles. Results are not m pressure and transition. For any arbitrary forward flow, one significantly different from those of the Cp^x family. Fur- would then interpolatcalculatd an 0 t x= a d eR e betweee nth theremore, Ref contain0 2 . additionan sa l pai f charto r r sfo m functioloca sayins e briefa ein .ar R I e f ngo w , thafloy wan t the case where Cp -x at M0 = 1. The separation loci are con- e th thad s f value o an eha t R t thaf se so te developth f o e son siderably different difference t morth bu ,f eo e appeare b o st sam t floeaf w will hav same eth e separation poin thas a a tf o t eliminated if we consider the flow forward flow developed by an equivalent run of flat plate. 2 The equivalence is not rigorous, because a separation point is l-(ue/u0) =x™ (4.8) functioa t onl nshape f no initiae y o t th als th f f bu eo o lR profil dominane describes eth a s i e R t H.y td b Bu parameter , The expression for Cp when the flow is compressible is and, considering the accuracy of turbulent boundary-layer calculations onle thae th yon s t i neet i , consideree db d here. With u0/v specified 6 converts to Re as the interpolation yMj w+ parameter. (4.9)

These comment face s pointh t t thaou t t extensive laminar If ue/u0 = 0, C^ = 1.28 at M0 = l; and if Cp = 1.0, 2 flow in the forward part helps to delay separation con- (we/w0) = 0.18. Hence t C 1.0a , pflo= e ,stilth ws fror i fa l m siderably, becaus t reducei e s Re. Furthermore, boundary- stagnation. Therefore, the better canonical compressible flow 2 layer suction forward of the pressure rise would delay consideo t e flo th termn w i s i rf (uo s e/u0) . Some points separation. Another fact that is surprising is that if the calculated thu r floshowe t fo sMar wa Fig0 f n 0ni o -1. 8 . pressure rise is not too great, it may be extremely steep; in e e answequestioth Th Ref. o t f rwhethe 20 o n. r com- fact, nearle dCb y pldxy ma infinite f o e . us (Owin e th o gt pressibility aggravates separation depend referencee th n o s . finite steps e slopth ,f Co s enevei p r quite infinite th n i e Certainly, compressibility aggravates pressure gradients on a Cebeci-Smith method t werei f I . constana , partiae th n i t l dif- given bodyL which implies earlier separation. When con- ferential equation woul methoe e infinitth db d dan e would sidered p C form n i , compressibilit s favorableyi . Hence eth e nexth fail. t n I section )e tha e infinitse th t e w ,e ratf o e most fundamental approach consideo t appear e b e o th t rs 2 pressure ris confirmes ei Stratford'y db s equations, whice hh problem in terms of ue/u , which is a measure of the kinetic derive analytin a y db c approach. energy of the flow that remains. In that form, at least to M0 Loftin and von Doenhoff24 studied a large number of thin = 1.0, the effects of compressibility are minor when con- airfoils and arrived at a separation criterion that in our terms sidered from the basic canonical standpoint. is Cp = 0.88. It is plotted in Fig. 20. A thin airfoil at angle of In closing this section e remarw , k that because of_the attack has an effective forward flow about the same as that of analytic naturr canonicaou f eo l pressure distributions Cp = the x=l/64-ft distance, and the pressure-rise region is ap- x m, it is very easy to apply Stratford's criterion. That has been r m-curveo 4 proximate1/ 3 =m 1/ . e Hencee th se y e db w , done Stratford'f I . s prediction beed sha n addeplote th f so o d t that Loftin's criterion is rather closely predicted by the charts. Figs. 19 and 20, the differences in separation loci would ap- r fro fa als e thame s i w o se t universalt t i Bu . pear considerable. Stil smalchartlo e s the th e f lyar so thae us t f charto t se s e provideTh n "eyeballa s " methor fo d e finding t th negated no d s an i s . Additio f Stratford-typno e estimating separation points e shalW . l illustrat y conb e - loci would only cause confusion. Therefore shoe w , w onle yon siderin relato t canonicae y ge th FigTr thaf . o .14 e t flolon wo t typ f calculationo e e Cebeci-Smithth , , whic n generai h s i l plotspressure th f I . e distributio replottes f Figi n o 4 1 . un di 2 mose believeth e t b accurate o dt . Furthermore t shouldi , e b /«r canonicao i, l form reae th ,r flow looks approximately noted that Figs. 19 and 20 differ from their earlier forms in like the x1/3 flow and the front flow is effectively quite short. Ref. 20. The main reason is that here we used the analytic m Transitio s measurenwa t *=1.16da . Henc9ft e logicaeth l nature of the Cp=x flows to compute certain necessary equivalent flat-plate length is 0.015625 ft. Which chart should derivatives.. In the earlier work, finite-difference formulas be used? figurese Probablth e us , yo conver T Fig . e 20 . th t were used. Disagreements suc thoss ha e just indicat state eth e chord of the airfoil to 1 ft. Because it is so small, there is no of the art of turbulent-flow calculation. concernee neeb o dt d abou 0.01562e th t 5 forward - partor n I . der to hold Reynolds number at 3.3 x 106 with the reduced 4.6 Limiting Canonical Distributions 6 chord neee w ,increaso dt e x 10 th 3 t /ft3. ^eu Bu .Ivo t m charts are in terms of ujv. Now from Fig. 14, The Cp =x families are very useful for almost all practical 6 flow problems f courso t shapee Ce bu e,th th (x) f so curves u0/u(x>=2.l. Hence, u00 3./vx = 310 correspondo t s p 6 are arbitrary; the shapes are selected as a matter of analytical u0/v = l x 10 . Then Fig. 20 seems to be the better of the two to use. Of course, interpolation between the two is possible. convenience . mann i Jus s a yt other problems,e theron s i e From r m1/3 fin= e Fig fo w ,d0 .2 that separation occurt sa shape that is "best," Stratford's solution. We do not mean to (u /u )2 = Q.\l or u /u = OAl. In Fig. 14 imply that the solution is exact, but as is indicated by Fig. 14, e 0 e 0 acceptabls iha t e accuracy. Stratfor s derivedha dformulaa , (ue/u mao) o2.1 x= . Hence separation should occu t aboua r t 0.41 x 2.1=0.87, a ratio that is satisfactorily close to the Ref. 25, for predicting the point of separation in an arbitrary measured separation point. Mor e eanalysi th car n i e s decelerating flow: probably would increase the accuracy. For instance, a careful C [x(dCp/dx)] 1/2 calculation of R e at x = 0 by Truckenbrodt's method would be p ___ o (4.10) better tha nguessa . Also exponene th ,pressure-rise th n i t e (10 -6 region coul determinee db d g morlo f eo accuratele us e th y yb graph paper. wher f dei 2 p/dx2>0, then S0.39f d= i 2 p/dxr o ; 2l/3. The left-hand 512 SMIT. O . M H. A J. AIRCRAFT

VALUE F ffnO S recoverie vere b yy rapidsma . Whe initialone s i d n th n gan ru l boundare th y laye s thicki r allowable th , e average pressure U 'V 0 gradient is much less. Or conversely, thick boundary layers A- - FT. I06 7 0 I0 muce ar h more likel separato yt e than thin. 1/64 108 517 e uniTh t ) Reynold4 s number effec s rathei t r small [see 1/16 276 1511 Fig. 21 and Eq. (4.1 la)]. 1/4 759 4490 5) Theoretically, 100% of the dynamic pressure can be 1 2196 13930 recovered distance th t bu , e require infinites di . 6) Aside from theorye erroth n curvee ri th ,e Figf ar so 1 2 . shortese th t possible pressure recoveries "en—e th the f de o yar the line." Nothing better can be done except by boundary- layer control. 7) The Stratford pressure distribution is the path of least resistance connecting two pressure points A and B, as will no showne wb . If we accept quadrature formulas such as Spence's or Fig. 21 Stratford limiting flows at two values of unit Reynolds number. Truckenbrodt's, then Stratford's flow is found to be the minimum-drag flow. Conside floe th rw situation sketchen di Fig. 22. We wish to go from some point A to another point B. side then grows. Whe t reachei n e limitinth s g valu f So e, Variou possiblee s b path y s sketchedma sa , Stratfore Th . d separation is said to occur. The study of Ref. 23 showed the path is the path of lowest drag. We shall show this by means constant slightle b o st y different r purposes t hereou r bu , fo , , of Truckenbrodt's formula.26 For purposes of demon- we accept Stratford's values further Refe fo 5 Se .2 . r details. stration, we can write it as follows: If S is held at its limiting value of 0.39 for d2p/dx2>0, Eq. (4.10) amount n ordinara o t s y differential equatior fo n B 333 C(x). It is evident from Eq. (4.10) that the equation de- = [eA+K\* \ dx] (4.12)

scribep s a flow that is ready everywhere to separate. Stratford J*4 presents the following solutions: The path that developes the minimum value of the momentum 1/5 1/5 defec pate minimuth wiltf B ho 6e t b Bla m drag obvious i t I . s Cp -0.645(0. 435 R 0 [(x/x 0) - by inspection that the Stratford path is the answer. For it, the forC ,<(/i-2)/(n + l) Truckenbrodt-type integral is minimum. As has already been / shown, any curve below Stratford's would suffer from and separation. Accordin thio gt s metho f analysisdo e besth , t pat Stratford'ss hi , eve t meani f ni suddesa n jump upwarn di n-2 velocity as sketched at B. Exactly how this flow can be applied C =1- (4. lib) to airfoil design is uncertain, in general, because of all the in- n + 1 teracting t effectsleas a shoule t w thay t Bu able sa .t d b o et whe nStratfora d flo appliews i souna n di d fashion draw lo g, In that two-part solution, x 0 is the start of pressure rise, R 0 can be expected. distanc«e = 0jtth 0s /j>i x e, measured fro vere mth y starf o t The first part of Eq. (4. 11 a) of the pressure rise is quite the flow, which begins as flat-plate, turbulent flow. The num- rapid, but the final part, Eq. (4. lib), is quite slow. It is constana s i be n r t that Stratforde aboue findb Th . o t 6 st enlightening to compare with a lossless two-dimensional dif- quantities a and b are arbitrary constants used in matching fusor as sketched in Fig. 23, which provides a simple physical values and slopes in the two equations at the joining point, Cp picture canonican I . l form, (with velocitys a V , pressure )th e = (n — 2)/ (n + 1). Of course, Eq. (4.11 a) describes the begin- coefficient referre conditiono dt s i 0 = x t sa . (4. Eq flow1e ninIbfinad e th )f th an ,g o l part flon e a ws .Th i equilibrium flow that always has the same margin, if any, C=1-(V2/V2) (4.13) against separation. familieo Tw f suco s h flows have been computed; thee yar show Fign ni . The 21 . y correspon f cone samo th t -ese o dt m ditions that wer eCe useth p =xn di alreads flowsha s A y. been mentioned, they represent true limiting flows—the slightest increas adversn ei e gradient anywhere should cause separation curvee .Th s assum lengte eth f flat-platho e run- sin

Ce =xdicatedth r families jusfo t s bu ,a t , whamorf o s i te

p fundamentam l significance is the boundary-layer thickness at the beginning of pressure rise. Hence, a table of initial values of R 9 is included. With the aid of those R e values, the curves Fig 2 2 Possibl. e pressure distributions connectin. gB pointd an sA coul appliede db arbitraro t y forward pressure distributions having mixed laminar/turbulent flow. Withi accurace n th theory e th f yo , which tests have proved correce b somewhad o t an t t conservative, those curves show fastese th t possible pressure ris obtainee eb than ca t d froma natural boundary layer. Also, when compared with Cebeci- Smith predictions theore th , y appears conservative. Figure 21, together with Eq. (4.11), exhibits the following features: e initia1Th ) l slope dC /dxs infinitei o thas , t small

pressure rises can p be made in distances from very short to zero. STRATFORD 1/3 2) It is easy to show that Cp ~ x in the early stages. dominane 3Th ) t variabl x/xs ei 0 [se. (4.11)]eEq . Hence, when xsmals i 0 l (i.e. boundare th , y laye thin)s i r , pressure Fig. 23 Two-dimensional diffuser. JUNE 1975 HIGH-LIFT AERODYNAMICS 513

The width of the diffuser is w = b + 2x tan a. When that ex- idealized airfoil-upper-surface pressure distributionsy ke A . pressio combines ni d wit equatioe hth continuitf no subd yan - parameter in the conversion is the effective velocity at the stituted into Eq. (4.13), we have trailing edge effective , th tha , is t e edge velocit whico yt e hth forward flow must decelerate when a boundary layer exists. (4.14) Withou boundare th t y layer, airfoils that ended wit hfinita e = /~" [l+(2x/b)tana]2 trailing-edge angle would always have zero final velocity. The a formula whose structure is generally like that of Eq. (4.1 Ib) effective velocity we are talking about is really a correlating sense inth e tha infinitn a t e distanc r fulefo l recover- in s yi velocity. Edg r 98-%o e - velocitie97 chore th t da s poine ar t dicatedexponene th denominatoe t difth e Bu .f th - o tr fo 2 s ri good values. Becaus e floth ew ac ovee b n airfoi a ry ma l fuser, as compared to 1/2 for Stratford's flow. celerated, retarded, heated, sucked, blown, reenergizedd an , A Stratford flow has a continuous margin of safety over its finally discharged "overboard," the author tends to think of entire length with respect to separation, whether at the begin- the boundary-layer air as going through a sort of physical nin f pressurgo e ris r fareo r downstream,fa margie Th . f no proces thed san n being "dumped" overboard. Hence likee h , s safety can be adjusted by changing the constant S in Eq. to refer to the effective leaving velocity by a homely (4.10). name—dumping velocity. Both Eqs. (4.lib (4.14d )an ) show tha infinitn a t e distance Observation of various pressure distributions shows that a

is required to bring a flow in the boundary layer to complete typical effective trailing-edge velocit r "dumpinyo g velocity"

e thius s n forie svalueo ca w O.S.m (utw = e f /uI ) w 2 , 00 rest. Now if a round-nosed object is placed in a stream, it e generate stagnatioa s n point. Then round-nosea f i , d object main families of limiting upper-surface pressure distributions, lik flooe wina ecylindea th f o rn d o tunnel placed s ri en n d,o r fulfo l e laminaon r flo wr ful fo alonl e rooftoe gon th d pan turbulent flow. Some result presentee sar 5 2 d Figsn dan i 4 2 . situatioe th sucs ni h tha boundara t y laye broughs ri reso t n ti 6 a finite distance, with separation sure to occur. (It is believed for a chord Reynolds number of 5 x 10 . Figure 25 amounts that the three-dimensionality of the flow does not negate that to a direct scaling of the standard Stratford flow, because the conclusion.) Airplanes often have similar intersec- rooftop is assumed to be entirely turbulent. In Fig. 24, the tions—wing-fuselage r examplefo , e inevitabilitTh . f o y rooftops are considerably longer, because with laminar flow a separation in such cases does not seem to be fully appreciated. longer distanc requires ei develoo dt same pth e values oa e fR The inevitable separation is one of the justifications for that fullf developeo n y turbulenru a y db t flow. filleting. hav5 2 e propert e flowth ed th f an Figsl o sf 4 o Al y2 . producing the same final velocity without any separation but 7 Limitin4. g Suction Lift wit mose hth t rapid possible pressure rise from various roof-

Cp -levels. to rooftoe th f I p ver s Ci y high pressure th , e rise p By means of the relations in Sec. 4.4, the Stratford p canonical distributions of Fig. 21 can be transformed into must start early f lowstarn I . ca tt , i late f areaI . s withie nth curves are calculated, it is found that a maximum exists and that the initial negative Cp for this condition is not very high. Valueupper-surface th f so e lift coefficient e notee th sar n do

Fig 4 2 Suctio. n side pressure distributions using Stratford pressure recovery to €^=0.20 at trailing-edge. Laminar rooftop, R=5 x

10 . Values beside each curve indicate the lift that is developedc . In Fig. 25 Suction side pressure distributions using Stratford pressure the6 M oo c form M^ is that value which just makes the peak velocity recover o Ct y =0.2 t trailin0a g edge. Turbulent rooftop. R =5 2 ? p c sonic accordinKarman-Tsien vo e th o gt n formula. xlO6. 514 . SMITO . M H . A J. AIRCRAFT

LAMINAR ROOFTOP 6 Rc = 5xl0 =2.31 =0.0055 cdu =0.0053 cdt =0.0002

--I.OH 0.2 0.4 0.6 0.8 1.0 5 optimizen A Fig 6 2 . d high-lift Liebeck airfoil that use Stratforsa d Liebece Th Fig k7 2 .airfoi l that produces maximum lift with full- yat

pressure recovery. Values noted are theoretical. S is peripheral distan- tached flow, design R 10 x =.5 Pat. Pend.

c ce measured from the trailing edge. Pat. Pend. 6

figures r r turbulenlaminafo Fo . d an r t approach flowe th , maximum suction-surface lift coefficients are 2.03 and 1.00, -3.0 respectively. Higher Reynolds numbers would increase these TRANSITION values. It is interesting to note the very strong theoretical effect of laminar flow on lift. Usually, we think that the main benefit of laminar flow is reduction of drag. We have been discussing lift coefficient analysis. A second -2.0- STRUT and perhaps more important subject is the maximum force or M^Cf thatalkee w t d aboue proble Th Secn i t. 3 .m cannot no e solvewb s accuratelda r incompressiblfo s ya e floy wb means of Stratford's analysis. A compressible Stratford-type -i.o solution is necessary as a minimum. Nevertheless, nothing prevent fros su m determining critical Mach number eacr sfo h pressure oth f e distribution Karmay sb Tsien'd nan s ruld ean

assuming that the pressure distributions are still valid. If so, valuee th t wf Mparenthesen so ^ci ege 2 25d Figsn .an si 4 2 .

Agai( n there is a maximum value. For laminar flow it is 0.33. The maximum occurs at much lower lift coefficients. In the two figures, thejy-ordinate was scaled to produce (ue lu « )2=0.8 at the trailing edge. Suppose that we could by some means rais numbee eth 1.2o t r . (4.4. Thel Eq al )y nb values would be increased 50% and a 50% gain in lift would result. Therefore, a high dumping velocity is very important.

8 Example4. s Within the accuracy of the theory, we have just shown in an idealized form the maximum suction lift that can be developed with fully attached flow when the flow is that in a thickese Th Fig 8 t 2 .strut s having fully attached flow. natural boundary . layerLiebeckH . R .greatl s 2wa 7 y respon- sible for the basic studies. He has recently applied them to ac- tual airfoil design, where lower-surface pressure, upper- surface pressure d shapan , e considered b mus l al t . the design point. One of the most remarkable features of the Furthermore, he formulated the problem exactly, working airfoil is the extremely low theoretical drag of the lower sur- in term f surfaco s e distanc circulatiod ean n instea f presdo - e seeb nn froca facee pressurs mth A . e distributione th , sure formue sTh projecte. -25 planed a Figsn an n i do s 4 a 2 ,. velocit lowee th acceleratin s i n ry o t surfac bu t onl w no ys lo e i g lation leads to an inverse airfoil design problem that can be continuously from nose to trailing edge. Hence, it is likely to solved with great accurac Jamesy yb ' method other 28o r good be laminar all the way. The upper-surface drag is relatively inverse methods. low too, partly becaus e minimum-drath f o e g propertf o y One such solution is given in Fig. 26. To be sure of Stratford flows (Sec. 4.6, property 7). developing lamina e upper th flo n rwo surface e forwarth , d Liebeck's work was begun in an attempt to answer the part was given a favorable gradient instead of just a constant question, "What is the maximum lift that can be developed by velocity flow, as in Figs. 24 and 25. Theoretical values of c( single-elemena t airfoil with fully attached flow whad an ,s i t 6 and cd are shown on the figure. The section LID is 420. Tests, the shape required to develop it?". At Rr = 5 x 10 , for all airfoilo type th tw e f f o o showsRef , 27 Fign .n i sho6 .2 w practical purposes, Fig. 27 gives the answer. The airfoil has almost perfect agreement between theor experimend yan t a t substantially zero thickness, because at design conditions JUNE 1975 HIGH-LIFT AERODYNAMICS 515

thicknes s harmfuli s a littl s a e, thought will indicate. This limiting airfoil develops a section LID of 600! Conventional airfoils rarely exceed 180. Althoug t relate hhigh-life no the th e o dyt ar t probleme w , shall close this section with some" interesting examples of what e accomplisheb n ca y varioub d s kind f boundary-layeo s r flows showiny B . g some limiting shapes, Fig demonstrate8 2 . s e abilitth f variouo y s type f boundary-layeo s r flowo t s withstand adverse pressure gradients. All three shapes are cuspe orden di eliminato t r reae eth r stagnation point that theoretically accompanies a finite-angle trailing edge. All boundary-layer calculations were made for the naked shapes. A slight relief would be provided if the effects of boundary layer were taken into account. Hence bodiee th , s shown here have a slight degree of conservatism. The first cases represent INCREASING PRESSURE - the thickest symmetrical Joukowski strut that can have com- STATIO N0 X —- STATION1 plete laminar flow without any laminar separation. It is only Fig 0 3 Flo. waka f wo e int pressuroa e rise. 4.6% thick Reynolde Th . s numbers mus lowe b t ; otherwise the flow would become turbulent. The intermediate case is a Joukowski strut having mostly turbulent flow with transition as shown in the figure. The chord Reynolds number is 107. that leave pressure slate th sth f I . e ris greas ei t enoughn ca e w , Finally, if we pull out all stops and so shape the forward have flow reversa streame th n i l , entirel surfacee th f e yof Th . gradients as to be favorable to laminar flow and then use the phenomeno s easili n y demonstrate y resortinb d g to Ber- Stratford pressure recovery in the rear, we obtain the thickest noulli's equation. Conside flora illustrates wa t A Fig. n di 30 . shape, which is more than 53% thick. The theoretical Michel station 0, there is a flow in which static pressure is constant transition poin s markei e figuret th n o d. Wit e stronth h g acros streame sth t ther wake-lika bu , s ei e portio whicn ni h favorable pressure gradients, there should be no trouble in ob- velocitie deficiente sar wake Th . e flows int oregioa highef no r taining the required laminar flow, even at the rather high pressur t statioa e . Wha1 n t happen e wakth eo t s region? Reynolds numbe f 10o r 7. Lik high-life eth s t shapeswa o to t i , Becaus considerine ar e ew g regions very thin with respeco t t designed by the James method. Because of the extensive any overall curvatur flow e reasonabls i th t f i ,e o assumo et e laminar flow, the theoretical value of cd is only 0.0077, in that the static pressure p l at station 1 is also constant across spite of the great thickness. the boundary layer. Because gradients du/dy will be small, the shear stresses will be low and then it is a good approximation 4.9 Off-the-Surface Pressure Recovery assumo t e that each streamline maintain s totait s l hea- dbe Until now, our concern has been with fluid flowing into tween station 0 and station 1. Such an assumption has been regions of higher pressure while it is in contact with a surface, confirmed in numerous analyses, including Stratford's. that is, a decelerating boundary-layer flow. But there can be Hence, using U for potential regions and u for energy another kin f flowf d o o e flo t th f wake,w o ou e b s thay ma t deficient regions, we can write, for incompressible flow, contact with any wall, into regions of higher pressure. Such a flow occurs degreemultielemena y o an t , n o , t airfoil slaA . t (4.15) develop boundarn ow s sit y layer, which flows oftrailins it f g edge, formin a gwak f low-energo e r floww thaai yno t s and alongsid downstream n airfoie o th resd e f eth an o tl . Consider geometre th r instance fo streamline f Fig, yo th 17 r . o , e pat- (4.16) tern of Fig. 29. Each forward element produces wake components over its downstream partners. Solv obtai d U?V\r d an e fo an n their ratio, thus: The theory of that kind of wake flow is not nearly so well developed as the theory of boundary-layer flow. Therefore, we shall be conten o givet t n onl a ybrie f discussios it f no (4.17) features, purpose chieflth r yfo f calline o g attentio themo nt .

kindo tw f wake o se Therb n s eca flowin g int oa regio f no Now introduce our canonical pressure coefficient Cp—(p — higher pressure. One is separated from the adjacent boundary p0)/ (1/2)pL^into (4.17). We obtain layer by a region of potential flow. That kind occurs when gaps are large. The other kind is so close to the adjacent boundary laye flowo rtw thase finallth t y merg becomd ean e eon (4.18) thicker boundary layer. That kind occurs when gaps are l-Cn small somy calles .B i t ei d confluent boundary-layer flow. The equation shows that as C increases, u?/U? can reach Because a wake is usually near the main airfoil surface, the p zero well before CD reaches +1, provided that u%/U%<\. pressures impresse little ar et i differen n do t from e thosth n eo 2 2 r exampleFo f i u, 0/U 0=l/2, u]IU} = § when C l/2.p= airfoil surface; for example, consider the streamline in Fig. 29 velocity-defece Thath , is t t rati magnifieds oi flod an w, rever- sal can occur in the main stream. Viscosity, of course, helps to smooth out the wake velocity defect, which means that a Ber- noulli approac unduls hi y conservative. According to Eq. (4.18), if flow is into a region of higher pressure, the velocity defect ratio always worsens. Opposing it is the effect of viscosity, which tends to smooth out the defect. Gartshor derives e 29ha approximatn da ewhetheo test s a t e rth wake grow decaysr so s i t I .

0.007 Fig 9 2 .Calculate d streamline flow fiel airfoir dfo l with leading edge (4.19)

doublslad tan e slotted flap, a = 0°, 3.70 c= . l-Cn l-Cn t J. AIRCRAFT 516 A. M. O. SMITH

displacemene wherth s i * e6 t thicknes poinwake e th th t f eta so Gartshore cites experimental evidence that confirms hi s being considered left-hane th f I . . d(4.19 Eq sid f s lesi )eo s deductions. It was obtained by examining flow over deflected than 0.007/6*, the wake decays; if greater, it grows. plain flaps. - Withouoc s ha t d doubt an e effec n th , ca t To study the structure of Eq. (4.19)_, we shall apply the test curred—separation off the surface. But since there has been m floa o wt thavariatioa s i Ce t pth =x f no flows shown ni very little study of that kind of flow, the validity of relation Fig. 20. Referring to that figure, we shall assume that we have (4.21) is in question. Especially in question are some of the flat-plate flow f variouo s s lenths, from , 0.01562ft 0 1. o 5t derived consequences ,initiae sucth s ha l instabilit r somyfo e that develop boundary laye x t botn . =o a_ r0 hd sidesen l Al , fractional values of m. For example, when m - 1/3 and 1/2, a m They then dump their boundary layers inte Cth op=x boundary-layer flow would not separate, but the wake flow pressure region. Will the wake defect amplify or decay? First would be unstable. The principal purpose in presenting exam- we must know the initial displacement thickness of the wake. ple applicationconcepe th t ge t o intt t leas opene a s si o th d t an , finaAssume equae b th lo o t t ldisplacemen t ei t thicknesf so exhibit in a qualitative way the effect and the interaction boundare th y layer from both plate e sideth convenienf s.A o t between wake thicknes pressurd san e gradient. formul Blasius ai s 1/7-power formula In practical applications, with their lack of infinite adverse gradients through whic a hwak e must flow e waketh , - 0.0463'I instability problem should rarely be critical, which means that (4.20) > \ side R"5 wakes can endure pressure rises that boundary layers cannot endure. What a multielement airfoil does then, in effect, is to After substitution from (4.20) into (4.19), we obtain use two methods of pressure recovery: the conventional, that is, on-the-surface pressure recovery, and off-the-surface 0.075R1/5 pressure recovery a slatte n O .d airfoil r instancefo , e th , (4.21) historfloe th thiss w i r flowf yo Ai : s ove slate th r , reachesa 1-Cn peak velocity, then decelerates in contact with its surface, leave surface th s continued an e decelerato st e until trailing- where £ is the length of the flat plate and R(=u0t!/v. Figure edge pressure e reachedar s , after whic t i graduallh - ac y 31 . show(4.21 sideo Eq resultsplottee e f tw sth )ar so e dTh . celerates bac freestreao kt m conditions off-the-surface th y B . e separately. When the left-hand side exceeds the right-hand deceleration, recovery from very high negative Cp values can side, instability develops. The m = 1 curve is a special dividing be made in much shorter distance than can be made when all case. It begins with the value of 1.0. Any higher value of m the deceleration is in contact with a surface. begins with the value 0; any lower value of m, the value « . right-hane th n O d sidcanonicaa f Figs eo i 7 .1 l pressure Hence, accordin Gartshore'o gt s criterion wake th , always ei s scale based on the maximum velocity on the slat. According to initially unstable for m < 1.0. The figure shows a strong effect that scale, the final velocity-squared ratio at the trailing edge of wake thickness upon the stability. For our problem a is only 0.025. If all the deceleration had been in contact with higher Reynolds numbe s favorabli r e becaus t ereducei s <5* surfacee th floe th ,w would surely have separated, according and hence increases the value of the right-hand side of Eqs. Figso t . 19 an. d20 (4.19) or (4.21), but the effect is weak. According to Eq. m Nothing so far has been said about the case when boundary (4.21), for the Cp=x flows the left-hand side ultimately ap- layers merge. Lockheed30'31 has studied the problem and, in proches infinity, and therefore any of the wake flows should fact, has developed a general method for analyzing confluent finally go unstable. Hence, with m < 1, theoretically we have a boundary layers. But because of the complicated nature of the situation where the wake flow is unstable to begin with, but flow, bold simplifications had to be made. Like Gartshore's quickly becomes stable and finally becomes unstable again if analysis methode ,th s need developmen furthed tan r checky sb carrie enoughr dfa . Whether that indee t known trus dno i s ei . detailed experimental work. Improved method r bote fo s hth However, according to Gartshore's analysis, a boundary merging and the nonmerging wake flows are problems for laye s mori r e pron o separatiot e - n in tha e wako t th n s i e future work. stability. Conside f=0.25-fe th r t f plato Figy n ei an .r 31Fo . r observationou s Ii t wels a ,thas a lf Foste o t t al.e r 32, that the ra-values, the final instability is at x^O.8 ft. However ac- gaps between airfoil elements shoul largo s e edb that wakes cordin mosr fo Figo gt , t value20 . boundary-layef m,e so th r and boundary layers do not merge for, if they do, early flo separates wha d earlier. separation will set in. In fact, Foster et al. in discussing the subject make followine sth g statement: 4 20 r '...when the flap is in the optimum position from the viewpoint of obtaining the highest maximum lift, the interaction betwee wine nth flae g th wakp d boundarean y layer is comparatively mild o layer, tw wite s th hretainin g their separate identity almost to the flap trailing edge." Hence, we are somewhat fortunate. Optimum designs are outsid e regioth e f mergino n g boundary layers, and, furthermore, flow retardation rates are rarely so great that the wake becomes unstable boundare Th . y elemenn layea n o r t beneat wake hth nearls ei y alway worsn si e trouble. However, the merging of boundary layers helps establish the optimum. If the boundary layer were much thinner an optimum spacing migh lesse b t . Hence, becaus scalf eo e boundareffecte th n o s y layer optimue th , m spacin appreciablaircrafe n b a r y gfo ma t y less than indicated by a small scale wind-tunnel model. The problem is worst for the slat because its wake has the longest run. 5. Loads Applie Boundaro dt y Layer Airfoily sb s 1 Introductio5. n Fig. 31 Evaluation of Eq. (4.21) for the decelerating flows of Fig. problee Th f obtaininmo g high lif thas i t f developino t e gth lift in the presence of boundary layers—getting all the lift that JUNE 1975 HIGH-LIFT AERODYNAMICS 517

is possible without causing separation. Provided that bound- -2.0 ary-layer control is not used, our only means of obtaining higher lift is to modify the geometry of the airfoil. Con- siderations that guid modificatione eth subjece , thenth e ar t, of this section s helpfui t I . thino t l termn ki s analogouo t s -1.6 those of structural design—applied and allowable loads. The airfoil applies the loads, and the boundary layer determines o TEST DATA - THEORY the allowable. Separation of a boundary layer may be likened -1.2 to reaching the yield point in materials testing. Initial separation rarely coincides wit maximue hth m lift tha airn a t - foil develops, for the lift usually continues to increase. Likewise yiele th , poinde poin th ultimatf metat a to n no ti s li e -0.8 load. That usually is the rupture point. Hence, the point of maximum lift coefficien likenee b y dma t crudel rupe th -o yt ture poina material f o t e analogTh . s quiti y e roughf o , -0.4 s mentionei course t i t bu , d becaus interactioe eth n between boundary laye shapind rairfoi an n t vera no f gyo s i lwidel - yap preciated. We close these introductory remarks by observing that aerodynamic science has advanced to the point where we satisfactoriln ca y predic poine th tf initiao t l separation (the yield point) but not the condition of maximum lift (the rupture point). The second problem is still beyond the state of the 0.4 s assumini t i t artbu g, high priority becaus f successfuo e l MACH NUMBER = 0.26 ANGL ATTACF EO K =1.76° solution of the simpler problems. NOMINAL REYNOLDS NUMBE I0x 60 2. R=

5.2 Single-Element Airfoils 0.8 e subjecTh f single-elemeno t t pressure distribution types and consequent airfoil performance has been indirectly covered by the material in Sec. 4. The canonical pressure distributions show general limits to the pressure rises, and the 20 40 60 80 100 pair of charts can be used for visual checks prior to careful PERCENT CHORD examinatio f morno e nearly final design detailey sb d bound- Fig3 Pressur3 . e distributio a typica r fo nl aft-loaded airfoil. ary-layer calculations f I separatio. s indicatei ne th r fo d Pressure distributio s correctei n r boundary-layefo d r effectse Th . problem at hand, some change must be made in the shaping in shaded area on the airfoil represents the displacement thickness. orde remedo rt defecte yth makinn I . changee gth t musi , e b t born n mini e d that definit- e ac bound e b n whao n s ca t complished have already been established. For a one-piece airfoil, there severaar e l possible mean r improvefo s - LOWER ment—changed leading-edge radius, a flap, changed camber, a nose flap, a variable-camber leading edge, and changes in 0.2 - detail shape of a pressure distribution. A pressure rise may be improve changiny db g from conve concavo xt direce th n ei - tio Stratford'f no s type. Even changin trailing-edge gth e angle from finite (e.g., 10° usefue b cus)o y t l pma becaus ecuspea d trailing edge imposes less pressur e trailine th ris t ea g edge. Furthermore cusa , p increase life slope th t sth f curveeo o s , tha a givet n s Ci reacheLa lowe t a dr anglf o e attack, which lowers nose pressure peaks. Simple hinge system plaia n o n s flapsa , even though sealed, can have a significant adverse effect on the separation point hencd an liftn eo . Figur show2 e3 adverse sth e effecte th f o s

- DEFLECTED 25* 20 40 60 80 100 -VARIABLE CAMBER PERCENT CHORD Fig 4 Pressur.3 e distributio canonicaFigf n i o 3 .3 l form.

break at a hinge line. The airfoil is an NACA 63A010 airfoil hinged on the lower surface at the 75%-chord point. Also shown for comparison is a variable-camber flap whose cen- terline is a circular arc. Its final slope is 25°, so that both trailing edges have the same final angle. Both flaps have separation variable-cambee th t bu , r shape s attacheha d flow over 85% of the flap, while the simple flap has attached flow over only 71 %. It is interesting that the simple hinged flap has concava e pressure distributio camberee th d nan dconvexa n I . Fig. 32 Comparison of two kinds of flaps on a NACA 63A010 air- canonicae th l sens plaie eth n flap pressure recover mors yi - eef ficient variable Th . e camber flabettee b p o turnt r t onlsou y plaie th inviscinfoild r flaan Fo . p° d0 a = c 1.78 = tairfoi e .Th l with

variable cambe t a1.06a = t orden ri °se flas obtaio rt pwa sam e nth e because of its drastic reduction in the suction peak. Cg. Separation point markee sar arrowsy db . 10R= . 7 Transitiot a ns i Canonical pressure distributions are useful for preliminary

forward suction peak. c scanning. Figur 3 show3 e a conventionas l pressure 518 A. M. O. SMITH J. AIRCRAFT

distribution. Wher s separatioei n most imminent? Neae th r that each component develop boundarn ow s sit y layer under fron topn o t bottom e ? Neath rean e topo n rth ro r ? ?O Figur e the influence of the main stream, and there is no merging 34 is the canonical form of Fig. 33. Comparison with the within the slot. Topologically, the process of boundary-layer indicate0 curve2 d f Figsan o s s9 1 tha. r fro nose fa th tms ei developmen differeno n s i t t from biplanethaa n to . Subjeco t separation. The rear upper surface is marginal, but only their particular pressure distributions, the two boundary behind about 97% chord. The lower surface retards to about layer biplana n so e grow, trai f downstreamof l , diffused an , Cp = 0.5 in a rather short distance, and the retardation begins finally merge. That is just the process for an airfoil system of late. Figures 19 and 20 indicate that a C^-value of about 0.6 tw r moro e element o lons ss mergin a g g doet occuno s r coul reachee db thin di s distance. Hence there seeme b o t s withi slote nth . some margin, but not much. Provided that other constaints nexe Th t paragraphs will elaborat confird an n emo whae w t would allow it, the faintly S-shaped pressure distribution aft have just said. There appear to be five primary effects of bottoe chor% th o50 n fmd o could have been improvey b d gaps, and here we speak of properly designed aerodynamic replacing it with one that is concave all the way. In problems slots. approached an s like this neee inversr th , dfo e methods become 1) Slat effect—in the vicinity of the leading edge of a down- evident. First, we would like to have an inverse boundary- stream element, the velocities due to circulation on a forward layer method that for arbitrary initial conditions could tell us element, for example, a slat, run counter to the velocities on what pressure distributio coule nw d have that contain cera s - the downstream element and so reduce pressure peaks on the tain margin against separation. Then, given the specified downstream element. pressure distribution methoe th , d should fin airfoie dth l shape 2) Circulation effect—in turn, the downstream element that produce . Definitsit e progres beins i s g made along these cause trailine sth g adjacene edgth f eo t upstream elemene b o t lines. in a region of high velocity that is inclined to the mean line at 3 Multi-elemen5. t Airfoils—General the rear of the forward element. Such flow inclination induces considerably greater circulatio forware th n no d element. Ther egreaa seeme b t o deast f ignoranco l confusiod ean n 3) Dumping effect—because the trailing edge of a forward about the effect of gaps in properly designed multielement air- element is in a region of velocity appreciably higher than foil systems e misconceptioOn . s - mentionein wa n e th n i d freestream boundare th , y layer /'dumps higa t "a h velocity. troduction to Sec. 4. The most common misconception is that The higher discharge velocity relieves the pressure rise im- a slot supplies a blowing type of boundary-layer control. The pressed on the boundary layer, thus alleviating separation idea can be traced at least as far back as Prandtl,33 who said, 1 problem permittinr so g increased lift. The air coming out of a slot blows into the boundary layer 4) Off-the-surface pressure recovery—the boundary layer on the top of the wing and imparts fresh momentum to the from forward element s dumpei s t velocitiea d s appreciably particles in it, which have been slowed down by the action of higher than freestream finae Th .l deceleratio freestreao t n m viscosity. Owing to this help the particles are able to reach the velocit efficiens n donyi a n ei t manner deceleratioe Th . f no sharp rear edge without breaking away."n Abbotvo d an t 34 the wake occurs out of contact with a wall. Such a method is Doenhoff merely make the safe comment, *'Slots to permit more effective tha bese nth t possible deceleratio contacn ni t passage th f higeo h energ r fro yloweai e mth r surfac cono et - wit wallha . troboundare th l ye uppe th laye n r o rsurfac e commoar e n 5) Fresh-boundary-layer effect—each new element starts features of many high-lift devices." Here again, boundary- 35 t witou a hfres h boundary s leadinlayeit t a r g edge. Thin layer control is implied. Perkins and Hage make the boundary layers can withstand stronger adverse gradients harmless and noninformative statement: "The air flowing than thick ones. through the slot in Fig. 2-49 [sic] is accelerated and moves Those effect explainee b s wilw discusselno d dan turn di t na toward the rear of the airfoil section before slowing down and some length. Laminar bubbles, merging boundary layersd an , 7 separating from the surface." Lindfield, in Lachmann, has a the like may complicate the problem; but when Reynolds brie fe slo papeth t n effecto r e articl. th Par f eo t describes numbers are high and at design conditions, such side effects studies by Lachmann, in 1923, who represented a lifting slat e importantb shoul t no d . Therefore, only conventional by several vortices located near a circle. The circle was then boundary-layer effects are considered. transformed into a Joukowski airfoil. Lachmann's theoretical studies see mhavo t e been largely forgotte t reallno - d yap nan 4 Sla5. t Effect preciated. However consideree ,h d only hal problem—the fth e Figure 35 displays the slat effect. A slat that is lifting has t consideno airfoild n effec a effece sla di a n rth f e f o to o tH t. circulation in the direction sketched in the upper part of the the airfoil on the slat. Lindfield's article is correct as far as it goes, but it is not very factual. He points out the need for better analytical methods before the slot effect can be well analyzed. (Now 14 years later, most of the methods are available.) A remark about the action of slots comes from a recent NASA report30: "It is well recognized that the usual function of the slot is that of a boundary-layer control device per- mitting highly adverse upper surface pressure gradients to be sustained without incurring severe separation. This stabilizing influence results from the injection of the high energy slot flow into the upper surface boundary layer." A still more recent NASA report36 states, "This leading-edge slat givee sth fluid which passes througbetweep e ga th e slad e hth nth an t main airfoi higa l h velocity. Consequently boundara , y layer o.o which grows on the upper surface of the main airfoil has more

momentum than it would have in the absence of the slat." a-15° eg TOTAL = 2.4741 thingo Thertw e s ear wron g with these statements. Firsf o t ISOLATEF O cj D c^ AIRFOI = 1.878L 9 all, the slat does not give the air in the slot high velocity. If AIRFOIL = 2.374I c, AIRFOIL = 0.3680 anything, it gives the air low velocity. Secondly, the air Fig 5 3 Velocit. y distribution airfoin a n so l witwithoud han vorteta x throug sloe hth t cannot reall callee yb d high-energy airl Al . locatelengtc ar e showns hda th aroun s i airfoi5 .e dth l surface begin- thr outsideai actuae eth l boundar ysame layerth es totasha l e trailinninth t ga g edge, measure clockwisa n di e direction. Total head. Properly designe spaced dan enougr d fa slat e sar h apart perimeter is unity. JUNE 1975 HIGH-LIFT AERODYNAMICS 519

figure. To a first approximation, a slat may be simulated by a There is an interesting and more physical way of explaining point vortex morA . e accurate approximation would account the slat effect. Again consider Fig. 35. When the airfoil is at for thicknes meany sb severaf so l source sinksd san , sourcen si an angle of attack, flow whips rapidly around its nose, which front and sinks behind, the total yield being zero, of course. a smal s ha l radius. High centrifugal force e developedar s . r purposesou r sufficiens i Fo t i , consideo t t r onl vortexe yth . Without outside help, high negative pressures around the nose From the sketch it is evident that the velocities induced on the are needed to balance out the centrifugal force. But the vortex airfoil by the vortex run counter to those that would develop iturnina s gtrua aidd e an sla, t also woul onee db . There on the nose of the isolated airfoil, especially at high angle of should be a close correlation between the amount of load attack. Pressure peaks would therefor reducede b e . Liebeck e averagth carried e slath ean t y Cb dp -reduction ovee th r and Smyth37 studied the effect by means of a special nose. The author looked at Fig. 35 and made the following generalized Joukowski airfoil program combined with com- crude estimates: Area modulated = 10% chord, mean Cp of puter graphics. Figure 35 is an example of the work. In the the airfoil alone for this region =-8. Mean Cp for same method, the vortex could be moved around at will. The best region with vortex present =-2. Net exchange = 6. Then location vortee foun th typica s locatioe r xth wa dfo f a o l f no approximate lift force c( that must be supplied by the vortex slat. The "best" location is not a precise statement. It is the is 6x0.6 e actua Th e .vortee differenc th th l n s lifi xo t e location where the nose suction peak was largely eliminated between ccd ftotafairfoilan l . Accordin numbere th o gt t bota s - and resulting pressure distribution was smooth. At a = 15°, tom of Fig. 35, cF = 0.6 is indeed the load carried by the vor- the plain airfoil develope dvera y strong suction peaks i s A . tex. Probably analysie ,th s canno made tb e quantitativet i t ,bu noted at the bottom of the figure, c( for the airfoil alone is does add to the understanding of the flow process. The 2.37. Wit hvortea x locate showns da pressure th , e peas kwa problem r furthewoul fo gooa e e db rdon study e forcTh . e completely eliminated, the airfoil c( fell to 1.88, but the total exerte vortee t justh no ty s xd b i verticaldraa s gha comt i ; - lift, including the vortex, increased to 2.47. In Fig. 35, the ponent too. According to the figure, because net drag must be curve called "vortex" show velocite sth y induce aire th - n do zero, its c^-value is —0.3680, a negative force that causes slats vorte e absence th mai th foie y n th xb i l nf e o stream r besFo . t to extend automatically. A more careful analysis would con- cancellatio e isolateth f o nd airfoil suction peak, vortex sider this component. position and strength were so adjusted as to generate a slae tTh effec reaa n lo tairfoi illustrates i l d wel Figy , b l .36 negative velocity peak that was roughly the mirror image of which represents a well-developed, practical design. The isolated-airfoie th l suction peak. Flow air e nea reath e - f rth ro pressure distributio airfoie th r nlfo without sla alss ti o shown. 2 almoss foiwa l t unaffected seee b examinatioy nb n ca s a , f no quantite Th y (ue/u00) reache dmaximua m valu f 10.0eo 4 either the airfoil pressure distribution or of velocities induced (C^-9.04). With the slat in place, the value was reduced by the vortex. remarkably, to just over 3. The effect is strikingly similar to That then is the slat effect. Contrary to the implication in poine thath f to t vortefigure Th Fig. n xei als35 . o showe sth the various quotations cited in Sec. 5.3, the velocity on the upper-surface pressure distributio canonican i l form. Again, nose of the airfoil is reduced. Peak nose velocity ratio was re- signs of a jet blowing effect are conspicuous by their absence. duced from undeo about 4 r4. t 2.0. Obviously bounde th , - This and most of the other pressure distributions were ary layer is much better able to negotiate the modified calculated by one version or another of the Douglas Neumann distribution. Wit ha vorte x operating, ther s onli e ya ver y program38'39 thas beeha t n mantesteo s y ydb hundredf o s small increase in total lift, a fact that is consistent with wind- comparisons with wind-tunne othed an l r data tha predics it t - tunnel observations. With slat extended, the main effect is to tions can be accepted as being essentially exact, at least for delay the angle of stall, not particulary to shift the angle of our current purposes. zero lift. In the beginning of this discussion, it was mentioned s speciallFigurwa 7 3 e y synthesize r purposefo d f thio s s that thickness effects could be accounted for approximately paper t consistI . f threo s e identica same l airfoilsth t ea l al , by source n froni sd sink an t s behind o approximatt , e th e angl f attackeo r comparisoFo . indicatiod nan degree th f no e shape of the flap. Since total sink strength must equal total of interference pressure th , e distributio isolatee th r nfo d air- source strengt sincd sinke han eclosesth e airfoile ar sth o t t , foil is included. Observe the extreme rounding of the peaks of combinatio e t effecth ne f e o t th induce th s ni esmala l forward reao tw r e airfoilsth , which agaislae th t s neffecti smalA . l velocity component on the airfoil. Hence thickness effects amount of the rounding is undoubtedly due to the decreased should generally reinforc circulatioe eth n effects. velocities that accompany a converging flow, as at a trailing point-vortee th edgen i s A . x example reae f eacth , o r h airfoil is covered with a surface sink distribution, according to the Neumann program. The surface sinks induce a counter-flow

10 A 18.00

jj £/ ON SIMPLE AIRFOIL = 1.69 ON I si AIRFOIL = 3.1 4 Q ON 2"° AIRFOIL =0.98 \ C£ ON 3«° AIRFOIL =0.49 TOTAL SYSTE M1.5= 4

0 6 0 5 0 4 0 3 PERCENT CHORD

Fig 6 Typica3 . l theoretical pressure distribution two-elemena r sfo t 2.0 2.2 2.4 2.6 2.8 3.0 airfoil, correcte e boundarth r dfo y layer e dumpinTh . g velocits yi denoted by the arrows. Also included is the canonical pressure Fig. 37 Pressure distribution on a three-element airfoil formed from distributio uppee th r rn fo surface wels a ,pressurs a l e distributior nfo three NACA 632-615 sections, arranged as shown. All are at the same the main airfoil alone lattee Th . r exhibits clearl favorable yth e effect angl f attackeo , 10°. Show npressure alsth s oi e distributioe th n no oslat a f13.15° = a , two-elemene ,th 1.4 cr = t 5fo t airfoil. basic simple airfoil at 10° of attack. Slot gaps are 1% of each chord. 520 . SMITO . M H . A J. AIRCRAFT

discharges boundary trailine layeth t a rg edge int ostreaa m that is locally of higher velocity. That is the "dumping effect" that has already been mentioned and that will be discussed further in Sec. 5.6. We have just indicated thay methoan t d capabl- in f o e troducing cros strailine floth t wa g edge will influenc cire eth - culation. An obstruction, properly placed, can be a powerful facto r controllinfo r circulatione gth illustrato T . e such control, we used a circular cylinder, which is about as neutral a body as can be selected. Figure 39 shows the system studied. circulao Tw r cylinder differenf so t diameter centeree sar a n do ray from the trailing edge. The gap is constant at 0.1 chord. The angle of the ray, 6, was varied from 0° to 90°. The lift on the airfoil and on the airfoil-circle combination was a=(5° c/ TOTAL = 3.4937 calculated. Figur show0 e4 s some case resultth e r wherfo s e cj OF ISOLATED Cj AIRFOIL = 3.3631 e cylinderth s wer ee chor th directle rea f th do r o lint y e AIRFOIL = 2.5937 C. AIRFOIL = -0.0763 (6 = 0°). Large increases in lift are indicated. The cylinder it- Fig. 38 Point vortex used to simulate a slotted flap. Vortex increases self carrie a largs e amoun f lifto t , eveo nn thougs ha t hi f airfoiCo g=15 a t a l° from 2.5 3.36o 9t . Leading-edge pressuree sar trailing edge. Figure 41 shows the effects in more detail, as greatly increase actiovortexy e db th f no . functionmighs A . expectede d b t f o s optimun a , m deflection angl founds ei mose Th . t effective angl r 70°abous o ei ° . 60 t angl° 15 f attack et o A life th t, coefficien isolatee th f o t d airfoil is ce= 1.75. With the 0.50c circular cylinder set at 6 = 60°, ct = 3.35 for just the airfoil, nearly double the value for airfoil alone. Henc effece eth vers i t y great. Figur draws i 9 e3 n with 5 = 60° t appearI . s that this most effective positio similas ni r to the position found most effective for slotted flaps and slats in combination with airfoils. Considered as a control surface, controe th l effectivenes st muc dcno f/dds hi les s than thaa f o t plain flap. Figure 42 shows calculated pressure distributions for the Fig 9 3 Airfoil-circular-cylinde. r combinations studie learo dt n effect cases with 6 = 60° and with ct held constant at 1.5. The obstruction a f o circulationn no . correspondin e figuree notegth ar valuen i da . f Peao s k velocities at the nose are considerably reduced by the cylinder. over the following airfoil. Although it was not done, it would If the pressures were plotted in canonical form, the ones with be easy to learn the thickness effect by replacing a forward e cylindeth r would appear considerably more favorable. elemen muca y b th thinner section while keepin e samgth e Velocity ratios at 0.975c, which amount to the dumping mean line. velocity e tabulatedar , e cylinderTh . s doubl e dumpinth e g velocity ratio. However, according to Figs. 19 and 20, 5.5 Circulation Effect separation should still occur in all cases. That is not surprising, in view of the specified c and shape of the airfoil. The slat effect has been recognized before; in fact, it was f clearly recognize bacr fa 192s ks a da Lachmann.y 3b e th t Bu 7 circulation effect does not seem to be explicitly recognized. Figure 38, which shows the effect in its simplest form, was produced by the computer graphic setup used in Fig. 35. Since the vortex could be placed anywhere in the field, it was now placed nea trailine th r g edge indicates a , figuree th t i n df i I . wer o represent e a slotte e liff o th tt d flap e vorteth , x circulation would be as indicated. The flow effectively places the ,0=0.50 trailing edghiga t ea h anglKutte th f attackf eo i a cond an ,- mete circulatioe ditiob th ,o t s ni n must increase effece Th .s i t little different from tha f deflectinto smalga l plaie n th fla n po isolated airfoil than I . t case e onseth , t flow - woulap e b d •Q.25C proaching the rear of the airfoil at a considerable angle. But '/D-0.5C t tur trailine no n th d di fo ge r edgew Fig8 3 . ; instead usee w , da =0.25C devic turo et flowe nth . Figur show8 e3 slightlsa y different 2.0 ———————- airfoil from tha Fign ti . t agai35 bu angl,e nth attacf eo e th s ki same e vortea drasti Th s . ha xc effec n circulationo t - in , creasing Cg from 2.5 9o 3.49t e vortex-alon th n Fig.I , .38 e curve shows the velocity distribution around the airfoil that is induce vortexe th y db . That distribution doe t meee no s th t Kutta condition, but the airfoil vortex combination does, of course. It is obvious from the figure that any means of changin e flowfielgth d nea trailine th r g edge would change the lift. Hence, a source, properly positioned, is apt to be as effectiv vortexa s effectea e Th .circulatio n o s o t e n du tha e ar t aft distortion floe thee th w ar f n so wha calcirculatioe e tw lth n effect. According to Fig. 38, the final upper-surface velocity is increased considerably over the upper-surface velocity for the

isolated airfoil. Because of the additional velocity caused by Fig. 40 Cg vs a curves for airfoil-cylinder combinations, showing adding the vortex to the general translational flow, the airfoil strong effect circulationn so . Deflectio cylindef o n5 0°r= . JUNE 1975 HIGH-LIFT AERODYNAMICS 521

0° SOLUTION

CIRCULATION SOLUTION 90° SOLUTION

Fig. 43 Three fundamental solutions used for calculating flow about a lifting airfoil at angle of attack.

° ISOLATE^a\5 = D AIRFOIL culation factore Th . s canno isolated—there b t mano to e year D-Q.25C interactions. Fortunately, accurate multi-element-airfoil analysis methods are available, together with much experience. But perhaps factors such as those discussed here will be of value in endeavors to improve design. Rarely do people = 10° ISOLATED AIRFOIL involve design di slottea f no d flap understan bese th ty dwh 0.25

30 60 Becaus smale th f leo radiuleadine th f so g edge, local velocity S —DEGREES ratio than si t regio vere nar y higd h an indee ° botr 90 de fo hth Fig1 4 Airfoil-circular-cylinde. r combinations. Effecf o t 6, and e circulatioth n solutions positivA . e lifting circulatios i n diameter on lift coefficient. shown in Fig. 43c. It is clear that its disturbance velocities add solutio° 90 e thoso positivr t th n fo f eo e angle f attackso r Fo . ordinarn a y airfoil, increasin life gtth requires increasine gth procese anglth f attackn eo i s t undesirable morBu th . f eo e 90° solution enters into the summation. Now if the airfoil anglw kepe lo attackf b noset o a tn eca gaia , n wil made b l f ei circulation can be produced by other methods. A plain flap in-

CONFIGURATION creases that circulation because when it is deflected, a con- ISOLATED AIRFOIL 12.8642° 20.2548 0.8800 0.043 0.25C CIRCLE 6.5950° 13.3444 1.1472 0.086 siderable cross-flow componen onsee th f to t velocity existt sa 0.50C CIRCLE 4.O929 0 14.0289 1.1855 trailine th g edge vortee obstacln a Th . trailin e r xo th t ea g edge has the same effect. To avoid nose peaks, the nose should be at a slight negative angle in any real design, in order that the ° solutionvelocitie90 circulatioe e th th sn n i si oppos d nan e each other and cancel out, or more precisely, the linear combination of the three solutions substantially cancels out. In short, considering the separation problem, high lift is best obtained keepiny b nose gth e angl inducind f attacan eo w klo g 9 0. 1.0 8 0. 7 0. 6 0. 5 0. 4 0. 3 0. 2 0. I O. 0 circulatio meany nb s other than pitching. Figur 7 illustrate3 e e circulatioth s n effecte c(Th . reae th r n airfoilo , devico whicn s augmeno et hha s cirit t - Fig. 42 Pressure distribution at c£=1.5 for airfoil-cylinder com- culation onls i , y 0.49reae rTh . airfoil boost circulatioe sth f no binations. 6 = 60°. middle th e airfoil reao , tw r f whic 0.98ca e airo f s Th -.h ha foil t togethe frone ac s th tn o rairfoil , givin f 3.14ca o ( t gi . The velocity ue on the surface of an isolated circular cylin- There is some feedback, of course. Circulation on a forward der in crossflow is ue/u ^ = 2 sin j3, where /3 is angle from the element effectively reduce e anglth sf attaco ea rea f ro k problemr noseou r velocite Fo . th , y component tha normas i t l element, as well as the velocities on its surface, and hence to the airfoil is 2 sin 0 cos /3 = sin 2/3. The maximum value of reduces its lift. That is one reason for the very low lift of the that component, which occurs at 0 = 45°, is not much less rear element. than the 6 = 60° maximum seen in Fig. 41. The rough coin- In Sec suggestes . 4.1wa t i , d that slots amoun shorta o t t - cidence justifie assumptioe sth n tha change th t circulation ei n circuiting of the lift. That effects seems to exist if the elements ifunctiosa crose th f sno flo w induce trailine th t da g edgy eb are not properly poisitioned. Observe the drop in pressure other bodies. near the slots in Fig. 37. It is interesting to note that at 10° Anothe thinkinn i d rai g abou problee tth modifyinf mo e gth angle of attack, according to inviscid-flow calculations, the circulatio consideo t s ni reae th rr stagnation poin nonliftn i t - single-element airfoil carries slightly more load tha threee nth - ing flow. Without circulation reae th ,r stagnation poinn o s i t element system. the upper surface at some distance forward of the trailing e samTh e kin f interactioo d n occurs between a sail n o s edge farthee moveTh .s i t i r d forward whatevey b , r meanse ,th sailboa r eveo t n between sailboat closn i s e proximity- Ac . greate circulatioe rth n require movo dt trailine t bacei th o kt g cording to Fig. 37, if the airfoils correspond to sailboats, the edge. The two guidelines—cross-flow strength and nonlifting last two boats, which are drawing up on the leader, would stagnation-point location are the best that can be proposed as find it very difficult to pass him. They are augmenting his lift mean f understandino s designind gan maximizo gt cire eth - and reducing theirs. The effects are even stronger between 522 . SMITO . M H. A J. AIRCRAFT

same sailth en so boat usefuA . verd an ly interesting studf yo example basi,e tha Th f Figc o . t pressur.45 e distribution that the problem has been made by A. E. Gentry. ^ By using the we shall use is shown as the dashed line. Then suppose we seek electrical-analogy technique and conducting paper, he ex- a three-element airfoil s thathaha tt canonical pressure plained correctl firse th tr timy fo jib-mainsai e eth l interaction distributio l threal r e nfo elements . Furthermore, assume that effect. the rear element induces a dumping-velocity-squared ratio (He/tin)2 of 1.5 on the middle element. Because that 5.6 Dumping Effect element now has higher circulation, it can induce even greater Closely relatecirculatioe th o dt n dumpine effecth s i t - gef e froneffectth tn o selement . Assum e induceth e d velocity fectfavorable Th . e interferenc downstreaa f eo m elemen- in t ratio is the same for each element with respect to its own duces cros strailine floth t wa g edge that enhances circulation velocities, whic bola s hi d assumption. With that assumption, of the upstream element. But the interference may also in- pressure th e distributio three th r e nelementfo s a s Figi n si 5 4 . crease velocities in a tangential direction so that the flow from sketched. The load carried is far greater than that of the a forward element is discharged into a higher velocity region, equivalent single-element airfoil, yet the several canonical thus reducing pressure-recovery demands effece Th .quits i t e pressure distributions are all the same. favorable to the boundary layer. According to Eq. (4.4), the The effect, as just described, is readily quantified. It will be 2 done three-elemena onl r yfo t analysiairfoile th t bu easil,s s i y suction lift can be increased in proportion to (u T E ) for the same margins against separation. generalized. Let c = chord of the ensemble; /] c = chord of rear Doe effece sth t really exist? Yes seee , b indeed n ni n ca t I ! airfoil (elemen fchor; 2= c1) t f centedo r airfoil (elemen; 2) t properly an y designed multielement airfoil clearls i t I . y shown and/3c = chord of front airfoil (element 3). Then for the three theoreticall Fign y i ; valuedumping-velocite .42 th f so y ratio elements /! +/2+/3 = l. Also let m 2 = magnification ratio are tabulated in the right-hand columns of the table in the for velocit t trailinya g edg e beino f element eth o e n gi du 2 t three figureth er cases—plaiFo . n airfoil, airfoil plus 0.25c cir- high-velocit e ynos th f d o elemen fielean f o d; 1 t airfoid clean , l plus 0.50c circle—the velocity ratio 0.88e sar , magnificatio= 3 m n rati r velocitfo o t trailina y g edgf o e 1.147, and 1.186, respectively. The canonical ratio nearly element 3 due to being in the high-velocity field of the nose of doubles; it increases from 0.043 to 0.085. element 2. Figure 36 is a theoretical case for an airfoil with slat. For Let the upper-surface lift coefficient be ce for the given maie th n airfoil dumping-velocite th , y ratio square 0.85s di . basic pressure distribution. Then for the rear element the suc- slatmuce s i th t i , n h O higher: 2.35 pressure .Th e distributions tion-side lift is/7cfo. On element 2, the wind speed over the alse ar o show canonican ni l form accordind an , theo gt e mth whole elemen greates i t factoa y rb r m2. Henc elemenr efo , 2 t

the lift isfmc. For the front element, the apparent wind (o 2 slat is less severely loaded than the main airfoil—in the sense 2 of margin against boundary-layer separation. spee increases di factoa y middl de b speee rth th mn t do e3 .Bu elemen alreads ha t y been increase factoe th y rdb m2. Hence Figur show4 e4 s experimental three-elemena dat r afo t air- 2 foil. Again the dumping-velocity effect is clearly displayed, elemenn o life th t isf3 t 3(m2m3) c- (o.en e Theth life f o ntth this time confirmed by experiment. For the three surfaces, sembls ei starting with the flap, the dumping-velocity-squared ratios 2 (5.1) (ue/u oo0.67 e )ar , 2.0 d 2.28 e canonicaan , Th . l plot- in s dicat conditione thatth r fo , s shown maie th , n airfoi less i l s and the ratio is severely loaded than the slat and the flap. The enhanced dumping velocity show typicals ni pressure Th . e distribution 2 of any properly designed multielement airfoil shows it; one (Ct/c{o)=f!+f2m +f3(m2m3) (5.2) only need. looit r kfo It is interesting to conjecture what might be done with the For the example of Fig. 45, fl = 0.30, f2 = 0.55, and f3 = 0.15. effecn inversa f i t e design metho r multielemendfo t airfoils 1/2 became available, assuming that the complete airfoil Also, m m 2(= 1 3= .5) . Then, accordin (5.2)o gt , requirements permitted. Start with some desired pressure distribution for a hypothetical three-element airfoil, for (c,/c,o) =0.30 + 7.5(0.55) +2.25(0. 15) =1.463

4 Fig4 Typica. l three-element airfoil, showing dumping velocity effect. Arrows denote dumping velocity. NACA 23012 airfoil, a = 8°. Line marked trailing edge level is trailing edge dumping velocity dividey b d maximum velocity on main element. JUNE 1975 HIGH-LIFT AERODYNAMICS 523

that is, a 46% gain, with no more tendency for separation problem theoretically and experimentally. They found that than on the base airfoil. The effect definitely exists, but (5.1) theoreticae th l inviscid wake effecreduco th t s f ei o t ea lify b t is as much hypothesis as it is statement of fact. In any real air- small amount, typically a few hundredths in cf. Their theory foil design mano s , y more factors enter int problee oth m that is in satisfactory agreement with experiment. In one particular t nearlno is si y that simple. Compressibility problems would study that used an NACA 4415 airfoil, they found that c? was ultimatel instancer fo , in t y.se reduced by an amount 2.3 c ds, where c ds is the drag coef- 7 Off-the-Surfac5. e Pressure Recovery findinge th ficienslate l th Al .f so t have generaln i , , been confirme practicay db l experience. Off-the-surface pressure recover s alreadha y y been To summarize—without the displacement-thickness correc- discussed in Sec. 4.9. Without its assistance, boundary layers e tioerror th ne greate ar s r than desirable. Wite th h would often be unable to meet all pressure-rise demands. displacement thickness considered, the errors are so small that Consider Fig. 44, which represents theory roughly confirmed desiga n engineer would hardly know wito whad ho t tmor e by test data. The scale on the right-hand side shows the total accuracy—in the case of fully attached flow, of course. deceleration ratio from the peak velocity. At the trailing edge flape ovelocity-squaree th f th , d rati decreases oha 0.075o dt . 8 Fresh-Boundary-Laye5. r Effect Figure 17 shows a theoretical pressure distribution that can be realized experimentally. Its final velocity-squared ratio is On each element of a properly designed multielement air- 0.025 .e flo Accordinth w , woul20 d Figso dan gt 9 neve1 . r foil, the boundary layer starts out afresh. It is well known, of have reached such ver velocitw ylo ybeed ratioha n sima t i f si - course, that thinner boundary layers can sustain greater ple boundary-layer flow that was always in contact with a pressure gradients than thicker ones. That fact is demon- wall. But in multielement flows, the front element's boundary displayee b n morca a strate t n I di e. d20 weld Figs.y an b l 9 1 layer, which undergoes the greatest deceleration ratio, is analytic form by Stratford's formula, Eq. (4.10), which we largely deceleratin surfacee th f gof . Figur illustratee4 airn sa - here repeat. foil that shows a theoretical total deceleration ratio (UTE/ u 2 V2 6 1/W max) equal to 0.0045. ] =(10- R) S (4.10) It is opportune to mention something that has been almost taken for granted—that any accurate calculation of pressure distributions must take into accoun e displacementh t t where 5 is a constant. The term (10-6R)1/10 varies so slowly thickness of the boundary layer. That assumption is well almosthas i t i t t constant x,t measura Bu . f distanceo e from covere Refn di . 39 thin I . s paper have t beew , eno n consistent, the origin of the flow, enters to the 1/2 power, and, ob- because consistency is not necessary to illustrate our point. In viously, longer boundary layers admit less adverse_gradient. general r simplou , e theoretical calculation t consideno o d s r Viewed in another way, we see that if x is halved, dCp/dx can boundary layers n otheI . r casese captio th e not w n ,i en be doubled with the same safety against separation. whethe boundary-layet no r ro r corrections were included. Therefore, breaking up a flow into several short boundary- wakA e alongsid eboda onls y ha ysligh a - in te effecth n o t layer run favorabls si e with respec delae th f separatioyo o t n viscid theoretical waksquarlifta a f s I . eha ee profileb n ca t i , and hence to the increase of lift. It might be argued that approximated by sheets of vorticity, positive vorticity on one breaking up the surface into a number of elements steepens side and negative on the other. Obviously, the wake has some gradients jus mucs a t lengts ha shorteneds hi , leaving x(dCp effect on circulation, although the two sheets almost cancel /dx) unchanged. In view of the favorable interference and each other. Furthermore, if the wake flow is curved, there is a totae shapth lf eo pressur e distribution quantite sth y doe- sin centrifugal force of the same sort that is developed by the cur- deed reduce. vet flapdje a shee, f excepo t t that here momentuth e m amounts to a defect instead of an excess. Foster, Ashill, and 5.9 Are Two Elements Better Than One? Williams,32 and Ashill41 have carefully examined those ef- Our previous discussion indicates that several elements are fect n boti s h theoretica d experimentaan l l studies. They better tha elemente non thert bees Bu . eproofha o nn . Here ew develop formulas for the various effects. In general, they shall attempt a proof. The proof will be for a two-element found that, for properly designed multielement airfoils, the case, but by the nature of the development it will be seen that only effec f importanco t e displacementh s i thaef o t t prooe th f could also show that three element bettee sar r than thickness. Moser and Shollenberger42 also have studied the two, four better than three, etc. We make the following

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

BASE DOMPING VELOCITY x/c 524 A. M. O. SMITH J. AIRCRAFT

DESIRED PRESSURE DISTRIBUTION - CALCULATED PRESSURE DISTRIBUTION 5 ITERATIONS

Fig 7 4 Shape. s producing Liebeck-Stratford pressure distributions similar to that of Fig. 26. Velocity-squared values on the forward elemen requiree tar magnifie e b o dt factoa y d b comf twoe ro th .f - co

Fig. 46 Load diagram for proof that a two-element airfoil can binatio 3.82ns i . t develop more lift than a single-element airfoil. assumptions: 1) the lower surface plays only a weak role, and An attempt to apply these concepts has been made using hence only the upper surface needs to be considered; 2) Wilkinson's approximate multielement inverse method.43 By Reynolds number effect e weak—whicar s ht larg a the e eyar meane methoth f searchee o se shap dw a two th f r eo - dfo scale. Hencairfoin a life n o etth l whose chor units di juss yi t element airfoil that had the same thickness distribution and same th f eo airfoil o tw life n ssame o tth whos th s ea e chords same th e canonical upper surface pressure distribution i s na are each 1/2; 3) by some means, perhaps an inverse method, distributione Th Fig . 26 . s wer applo et boto yt h elementd san the elements can be so shaped that both develop the same the dumping velocity ratio of the forward element was to be canonical upper-surface pressure distribution e shapTh .f eo (2)I/2 time rease thath r f elemeno t thao l uIs t al t valuee th n so the pressure distribution is not restricted; and 4) boundary front element woul twice db e rear e thosth .n eChordo e th f so layers and wakes of forward elements do not degrade the lift- airfoilo tw s equalwere b o et resulte . Figurth s drastii A . 7 e4 c capacitg in f followinyo g elements. change in shape is shown. We obtain a crvalue of 3.82, com- Conside t ABCDr Le Fig . pressure 46 .E b e distribution no pared wit single h th 2.3 r e1 fo elemen t airfoil. Because both the upper surface of an airfoil of chord AE. That same load pressure e identica ar se canonica th n i l l sense, there o in s distribution coul appliee db d 1/2-chortwiceo tw o t , d elements greater danger of separation on the two-element airfoil than as loads ABC'D'E" and E"B"C"DE. The sum of those two on the single-element airfoil. Only one gap size and position areas equals area ABCDE. By proper inverse methods, the was tried. Othe geometriep rga s would chang shapee eth e Th . shapeelemento tw f so s that could generat same eth e pressure method cannot be called entirely satisfactory; partly because distributio probabln nca founde yb , although thaweaa s i t k of iteration difficulties. It is noticed that the asked for o half-siztw e th r e proof ou r pressurpoin Fo f o . t e pressure distribution is measurably different from the best distributions, the basic dumping velocities squared are E"D' that coul foune db d using Wilkinson's method. Whethee th r y properlan , whicn o ED e equaly ar w hd designe an No . d asked for pressure distribution is even possible to develop of multielement airfoil trailine th floe f th ,w of gfor e edgth - f eo course is unknown. It is well known that every shape has a ward element can be made to discharge into the high-velocity corresponding pressure distribution t everno t y bu pressur, e regioreae th rf nelemento . e fronHencth r t efo elemen e th t distribution has a corresponding shape. Nevertheless, the dumping velocity square increases di d from E"D E"Fo t e ' Th . resul indicativs i t thingf dono e f whab d eo expec n o est an ca t t increase is the key to our proof. Two cases now arise: in the future. It confirms our proof that two elements are bet- a) the maximum velocity has some kind of limit; and ter than one. b) the maximum velocity has no limit. Consider case (a) first. Perhaps the velocity limit comes from Spectaculao 5.1Tw 0 r Example f High-Lifso t Airfoils Mach number considerations. Because E//FABC'D'E", which proves calculat theoreticas eit l pressure distribution t onl no ,sho o yt w r caseou . Consider case (b) f dumpinI . g velocit increases yi d powee th f modero r n computatio t alsnbu o becaus t mighei t from E"D' to E"F, the entire forward pressure distribution afford information about interactions, dumping velocity, and caratie e scaleb nth on i dE"F/E"D o obtait ' n AHIFE". other feature t clearlno s y exhibite usuae th n dli multielement Again, and even more strongly, area AHIFE" > ABC'D'E". airfoils that are so strongly hardware oriented; where they are Q.E.D. influenced by the requirements of construction and restric- We hav t assumeno e d thae basith t c canonical pressure tion. distribution be any kind of optimum-just that it be carrying Coordinates were measured from a photographic blowup its capacity load without separation. It is not necessary to the illustratioe oth f Handlen ni y Page's original paper.s i t (I 5 proof that the two elements be of equal chord. One possible hoped that his original drawing was accurate.) Figure 4 shows weakness of the proof is that for a two-element airfoil the resultse th 8-elemens Hi . t shape develope dwina g t Ca L3 —3. final velocity DE cannot be maintained at as great a value as it a = 42° (see Fig. 5). The wing was rectangular and had an can for a single-element shape, which means that all pressure aspect . ratiAccordin6 f oo wino gt g theory midspae th , n sec- levels woul depressede db . Tha problea s i t m thabesn e ca tb t tion was at an effective angle of attack of about 36°, the value answered by inverse multielement-airfoil theory, for which used in our two-dimensional calculations. No boundary-layer there is a great need and a great future. corrections were made e totaTh . l calculated sectios n wa cf JUNE 1975 HIGH-LIFT AERODYNAMICS 525

4.33. Velocity ratio instead of velocity ratio squared was used in orde avoio t r d excessively large th ef o ordinate o tw n o s 2.0 figures (Cp less than -36 was indicated). Values of c( in each velocity diagram represent the lift

carried by that element, referred to full chord. Four of the 1.0 diagrams show a secon d pressure e distributionon e th s i t I . that would develop if the airfoil was alone, at the same angle of attack. The leading element is at a considerable negative angle of attack, with a strong pressure peak on the lower side, I 2345678 which might cause separation. Nevertheless e elementh , s i t FRONT ELEMEN . NO T REAR very highly loaded increasine spitn th I . f eo g angl f attaceo k Fig8 4 Uppe. r surface dumping velocity airfoie valueth d r an l fo s

conditions of Fig. 4. The ratio (W /w) for the ensemble is E T max of each element, the lift contribution decreases continuously 2 0.0045. toward reare finae sth Th .l element carrie f onlcsa o ( y 0.08, in spit f beineo t almosga ° angl90 t f attactrailine eo th t ka g edge. If it were in isolation, it would be carrying cp of 6.80. Although not converted to canonical form, none of the 10= ' a pressure distributions appear vere b yo st clos separationo et , even that of the front element. Perhaps that is not surprising, for we are showing a condition that represents maximum lift for an airfoil whose chord Reynolds number was about 250,000, makin Reynolde gth s numbe f eaco r h element about 50,000. At such low Reynolds numbers, separation should occur quite early, which make e theoreticath s l pressure distributions look quite safe when judger currenou y db t high Reynolds number standards. Figur shoule5 reexaminede db thar calculatiow ou tno , n bees ha n made generaln I . confirmt ,i conclusior sou n that—if properly designed—the more slots, the better. But the seven- slot configuratio inferios n i six-slo e th o t r t configuratione Th . reasot e authoknownno th t s o i nt r bu ,e suspectdu s i t i s drastic modificatio e trailing-edgth f o n e contours whee nth eighth elemen formes wa t d (see Fig. Figur4) . show8 e4 e sth dumping-velocity ratio for each of the elements. There is a rather smooth and definitely continuous increase from rear to front. The rearmost element has a value of about 0.4, and the front element reaches 2.25. Figure 49 shows the extremes that can be reached in producing special-purpose high-lift airfoils. The section show designes nwa R.Hy db . Liebec Douglae th f ko s Aircraft Compan Gurney'n winDa a r s ya gfo s Indianapolis-type race carsshape resula Th . s e i f carefuo t l direct-method analysis and modification of his optimum airfoil studies. As installed racera n wine o th , rectangulas gi f aspeco d tan r ratio 2.15, with small end plates. The wing was designed theoretically, then built without benefi f wind-tunneo t l tests tested an , n do racerangle n a th t f attaceo A . f 14°ko , tufts use indicato dt e flow separation remain fully e attachemaximuth d an d m wing-section c( is about 5, according to calibrated spring- Fig. 49 Special-purpose Liebeck high-lift airfoil designed for deflection tests. Indianapolis-type racers (Pat. Pend.). yearpase Iw nth e worfe t th s d "synergistic s becom"ha e popular. Those examples, the proof of Sec. 5.9, the dumping effect discussed in Sec. 5.6, as well as the various test data, all as good accuracy as turbulent flows over impervious walls. A indicate that airfoils properly placed in tandem are fine exam- generalized eddy-viscosity formul bees aha n developed thas i t ple synergismf so . responsibl r thafo e t accuracy. Therefore - s felap wa t t i , propriate her throo et w som subjecee lighth n f delayino to t g 6. Power-Augmented Lift separation y considerinb , e theoreticath g l suction require- m 6.1 General ments. An illuminating procedure is to use the Cp=x remarkw fe A s shoul made db e abou subjece tth powerf o t - canonical plots presented earlier FigsBecausin , 20. . 19and e augmented lift. Power properly applied, as through bound- suction introduces another variable, space does not permit us ary-layer control greatln ca , y delay separatio hencd nan - ein o present t l resultfoual r fo lengths f flat-plato s e run- In . crease maximum lift. Powered lift augmentation usually stead considee w , r onl one-fooe yth t run, whice hb casy ema requires pumps and internal ducting, which amounts—at considered roughly to approximate an airfoil with a flap. least—to weight and cost complications. There is no doubt In conventional boundary-layer calculations wher transea - formed stream functio e give s usedar i ne wal e nth w l, con- that higher lifts can be obtained, but whether they make for a 7 better total airplane syste mooa ms i t question answee Th . r ditions that/ w ~ u w = 0 (no slip) and/w ~ $ w = 0 (impervious depend e efficienc th e lift-augmentatio n th o s f o y n system. wall). The problee nth find/o t ms i " calculatione w ~th cn fI . s Here we shall confine ourselves to three studies that touch on presentede b o t keep/e coursef w , o , t 0 see' w= bu ,k valuef so the theoretical performance possibilities, together with ideas fw that will maintain an arbitrarily chosen constant for improving the overall system. Cj — 0.001 e altereTh . d procedur juss ei switca t boundarn hi y conditions that is entirely acceptable according to 2 Are6. a Suction mathematical theory. Some kin f reao d l limiting suction e thorougTh h studies reporte Refn di sho4 4 . w that tur- woul e specia e founth b d r fo ld case c fStratforn i = s 0,a d bulent flows with mass transfer can be calculated with nearly flows, but because the machine program involves numerical 526 . SMITO . M H . A AIRCRAF. J T

preveno t t separatio pressura t na e jump s analysiHi . s ha s been well confirme experimeny db laminar tfo r flows, somf eo the confirmation being the author's. Let us apply the hypotheses to our problem, namely the flows illustrated in Figs. 50 and 51, except that the pressure rise region wil approximatee b l instantaneoun a y db s jump. Let ( )c designate critical conditions. Then, using our canonical flow notatio totae havth e nr lw heaefo d alony gan streamline

(6.1) l velocitIal f y hea expendes di reachinn di ghighea r pressure, Fig. 50 Effect of area suction on separation. Suction starts where Cf p c, downstream, we have has decreased to 0.001 and then vw is adjusted to maintain this value

of Cf. Initial flat plate run is 1 ft, turbulent, /vu = 10/ft. 2 0 6 Hc=pc=p0+V2Puc (6.2)

wher velocite edividine th uth s i c f yo g streamline l fluial , d 2 below being sucked off. Using Cp = (p—p0)/(l/2)pu 0 fine w d

(6.3)

This relation states that removal of fluid up to a height u c will permit a canonical pressure rise from Cp =0 to Cp . How much flow must be removed? Because thecinitial flow is flat plat conveniens i e 1/ t th 7ei e poweus o t tr formulas.

quantite Th y that mus removee b tudy \ whicQs di — n hi 0 c c

coefficient form isy y Fig. 51 Effect of area suction on separation. Suction starts where Cf Qc i ( ° u cQc- \-^~ — = — —y d (6.4) has decreased to 0.001 and then vw is adjusted to maintain this value unx x Jo Un

cf .o Initia , turbulentl ft flatplat 1 s i n e,ru u /v10= /ft. f 0

7 The 1 /7 power formulas are

ll7 1/5 solutio a partia f o n l differential equation, computind an g u/u0= (y/b) and d(x) = 0.37x/(u0x/v) (6.5) convergence problems arise whe t equanse Cs zeroo i ft , l We . therefore, avoided those difficultie specifyiny b s g that suction Upon insertion of the first, and integrating we have woul sufficiene db maintaio t t valua n t ca e equa 0.001o t l . f 8 Thus the results to be shown contain a certain margin of C (7/8)(d/x)(y/d)Qc= c" (6.6) safety. Previous studies have specifie e suctioth d thed nan n found the resulting skin friction. Figures 50 and 51 show alsn . (6.5oca firsFroe Eq e writ )f w . th (6.3o tm d eEq )an 6 7 results for (u 0/v) — values of 10 and 10 . resulte Th showe sar termn ni f valueso w v /u f eso required maintaio t chosee nth t ca fn value 0.001. Suction starts where Cf had decreased to 0.001. At first the suction rates are quite When this, together wit . e secon(6.5 e subhth Eq ar )f o d- e rateth smal t s increasbu l e rapidle higheth s a yr pressure stituted into (6.6 obtaie )w n region is penetrated. The final contour v w/u e = 0.05 is one for whic variablha e constan boundary-layee th n i t r equations si CQc= [0.324 (u / (6.7) abou exceeo t t s e rangvalidityit dth f eo . Furthermoree th , boundary-layer equations start losing their validity when v is This formula shows that if the demanded pressure rise is in- w 7 not very small compared wit eh.u Total suction flow raten sca deed low the required suction C Q is very low. If u Ox/v is 10 then C = 0.013 C* . If C = 0.c5, the required C is only be determined approximately by integrating the v w values. Qc f pc Qc Accordin figuree th effece o gt sth f Reynoldo t s numbet no s ri 0.0008 but for reaching stagnation pressure it is 0.013°. Unlike strong. e result th d 5 1Figs.n an o ssom 0 5 e suctio indicates ni r dfo charte Th s show that suctio highls ni y effectiv causinn i e g eve smallese nth t pressur fourte e th rise t h bu , power variation small delay separationn si t wheBu . n large delay desirede sar , shows the demand to be exceedingly small. The most in- suction loses efficiency, so that strong suction is required. The teresting result from this analysis is the fourth power relation effect can be explained physically in a neat fashion. Assume ofCQtoCp. the region of pressure rise to be quite short so that the suction In Lachmann,7 Wuest presents theoretical studies made by can be assumed to be concentrated at a point. Let us then Pechau's approximate method. The studies were of area suc- examine the special case of a flat-plate flow entering a sudden tion applied to simple airfoils in various ways. Suction pressure rise assiste suctiony db . Long ago connection ,i n with requirements for avoiding separation are given. Generally, Griffith airfoils, G. I. Taylor considered the same kind of our calculations show the same C Q = Q/u Ox magnitude as his, problem firse H . t assumed tha pressure th t e rise occurren di f 0(10-o tha , is have t 3t presente)W . eno d specific examples, suc shorha t distance tha totae tth l head along each streamline because charts like Figs. 50 and 51 are more general. For a in the boundary layer would remain constant. Then he real engineering problem, sinc boundary-layee eth r computing hypothesized tha filameno n t t whose total hea less sdwa than method is entirely general, the particular case should be and the downstream static pressure could flow into the down- e studieb n ca d carefully. Chart e thear sn jus a guidt r fo e stream regio highef no r pressure othen I . r words, unless wa t si preliminary estimation purposes. sucked off the filament would separate. These hypotheses give The results shown represent no kind of optimum. The start u procedursa calculatinr efo amoune gth f suctioo t n required of suctio arbitrarils nwa y chosenintensite th d an , y choses nwa JUNE 1975 HIGH-LIFT AERODYNAMICS 527

maintaio suct s ha valua n t ca fe 0.001 smalleA . r Cy-value would clearl beliee yth Q reducn f.C I thad wan te v earlier suction would be slightly less efficient, we started suction only when the boundary layer began needing help. Undoubtedly, m n optimuthera s i e m distributio e C r th eacp=x fo nf o h distributions, but we believe our values do not miss the op- timu vermy b y much. There surely is some best combination of suction distribution and pressure distribution for obtaining the greatest pressure rise wit minimue hth Qm.C Determining that combination amount calculus-of-variationa o st s problef mo finding the best combination of path and suction. It is interesting to note that the basic tool for the problem has re- 45 Fig. 52 Effect of tangential blowing on location of separation. cently been developed. The boundary-layer calculation Blowing slo s locatei t t poinda t where C0.001= f . Blowing velocity

method has been extended to find the pressure distribution ratio Uj/u=(W). Initial flat plate run =1 ft, turbulent; u/v = e 0

necessar o providyt a egive n c^-history, which amounto t s 107/ft, Gartshore-Newmal/2 n type of calculation. solving a nonlinear eigenvalue problem, where the eigenvalue is dp/dx. Since suction could already be analyzed by both direc inversd an t e methods havw no e e availablw , toole eth s for seekin optimue gth m combination wv (x) f o s andp(x). problee Th interestinn a ms i g mathematical challenge.

6.3 Tangential Blowing discussion e rouno th T t dou give w ,limitee a d comparison m of blowing applied to the same Cp=x flows. The wall-jet method of Gartshore and Newman46 is used. As for the suction case, blowing is assumed to start shortly before natural separation would occur, that is, at the Cy^ 0.001 locus. resultse th e ar . 3 5 Figure d an 2 s5 The conventional blowing coefficient C ^ is defined for two- Fig. 53 Effect of tangential blowing on location of separation. dimensional incompressible flos wa Blowing slo s locatei t t poinda t where Cf = 0.001. Blowing velocity

ratio Uj/u= (10). Initial flat-plate run= 1 ft, turbulent;

e 1 1/2 C M = thrust/

C=2(uj/ul)(e/c)Cit (6.10) With C^ as our blowing parameter, we obtain a direct "feel" for the strength of the blowing, for we are making up for the momentum lost in the boundary layer. Then, if the blowing is appliee b o t d over onl yshora texpecte e distanceb o t s i d t i , tharequiree th t d greatle ^valub C wilt f eo yno l different from unity. Tha indees i t case dth e . Gartshore'wit53 hd Figsan 2 5 .s program deck was used for the calculations. Several different values of Uj/ue were tried and results were found to be sen- ratioe sitivth o e.t Furthermore, sinc methoe eth onls - dywa ap Fig. 54 Illustration of the concept of lifting efficiency. Pressure proximate, hangups were encountered in some cases when the distribution airfoilo tw shown e r sar fo . 528 . SMITO . M H. A J. AIRCRAFT

to varying degrees by flow-separation requirements. If the show the specific steps that are taken in finding a shape. The figure represented were a_ design pressure distributior fo n steps, slightly paraphrased followss a e ,ar : some problem uppee th , r Cp bound migh determinee b t y db 1) Specify the desired airfoil velocity distribution and jet Mach number, for example. Two pressure distributions are momentum coefficient. shown in Fig. 54, one for a slightly cambered airfoil at angle 2) Divide the velocity distribution into symmetric and an- of attack cambereothea e r th , fo r d Joukowski airfoile Th . tisymmetric components. cambere dhighee airfoith s rha lefficiency . Obtaining lify b t 3) Determine the approximate jet-entrainment effects. angle of attack, according to the definition of efficiency, is Since the operation is noniterative, the sink distribution rep- inefficient. Because the start of pressure rise can be delayed resenting the jet is placed on an extension of the airfoil chord, mor f desigei n lifte lowar s e efficienc, th lifw tlo coeft a y - and its strength is determined by assuming that the velocity ficient oftes i s n higher than tha f shapeo t s either designer dfo jus freestreare tequas i outsidth t o je t l e eth n - valueen e Th . operatinr o higt ga h lift. trainment effects at the true airfoil surface are assumed to be Powered improvn lifca t e efficienceth y markedly. When same th thoss ea e calculate airfoie th t da l chord line. there is a curved jet, load can be carried by the jet, and it inversn a e 4Us e) metho fin o dthicknest e dth s distribution becomes possible to carry load all the way to the trailing edge. that correspond e symmetricath o t s l componene th f o t Thu e pressure-recoverth s y proble e eliminateb n mca - en d velocity specifie sten di thap2 t remains after entrainmen- ef t tirely. fects have been subtracted out. A highly accurate theory of jet-flap airfoils that makes it 5) Desig e meath n n line neede - o t producdan e th e possible to realize that benefit has been worked out in the past tisymmetric portio velocite th f no y distribution, includine gth few year co-workers Lopehis co- by s zand his of . 47>4One 8 influenc jete r thath .Fo f teo purpose linearizea , d methos di workers, Halsey,48 reviews the airfoil work in general. But used. Thi desigse parth f no t procedur onle th yes i par t thas ti here we feel we have space only to mention one aspect of it, a fundamentally different from conventional design methods. method that appears to have produced an interesting and For more details, see Ref. 48. Either: a) the vorticity at the significant advance. He has developed an inverse method for trailing edge can be specified, find the jet angle; or b) the jet t airfoilsje e .metho th Tha , is dt specifie e pressurth s e specifiede anglb n eca , fin vorticite dth trailin e th t ya g edge. distribution and the jet momentum coefficient and finds the 6) Combine cambe thicknesd ran s distributions. airfoil shape and the jet leaving angle necessary to generate 7) Usin e generath g l nonlinear direct jet-flap airfoil that shape. method, check the result. If necessary, repeat the cycle with metho e classicaa t Th no ds i l inverse method, suc James ha s pressures slightly modifie correco dt t errors revealee th y db developes ha conventionar dfo l airfoils. Instead kina s f i d o t ,i direct-method check. iterative procedure that involves finding a mean line and ob- We presen figureo tw t shoo t s w typical results. Figur5 e5 taining modified shapes until convergence is reached. Re- show e effecth s f blowingo t . High blowing rates relieve eth course can be had to the exact direct method for checking pur- camber, assuming that the velocity distribution is maintained poses t e entrainmeneffectje Th f .e o pressursth n o te constant pointe .Th check e computee sar th f so d inverse shape distribution are accounted for approximately. The procedure by meane directth f o s, nonlinear method usins inpua t gi t s beeha n foun produco dt e rapid convergenc accuratd ean e data difference Th . cn (ei thre e noteth r eo t dairfoilfo e du s si betteo n o rd than results ca quoto n t e W e. from Refo t 8 .4 e lifth t componen e jetth .f Eaco t h airfoi s showe i l th t na angle required to develop the design pressure distribution. An extreme case, one that taxes the linearized camber treat-

AIRFOIL 1, ment, is shown in Fig. 56. Here entrainment effects have been include inverse th n di e method e entrainmenTh . t effectss a , well as the entire camber-shape solution, are worked out

2.5 r

SPECIFICATION VELOCITY DISTRIBUTION O FINAL DESIGN, C0 = 5.373

SPECIFIED VELOCITY DISTRIBUTION

FINAL DESIGN, AIRFOIL I, C = 2.427 A FINAL DESIGN, AIRFOI , e2 Lg= 1.63 5 FINAL DESIGN, AIRFOIL 3, Of,= 1.32 0

ooo oooooooo

0.2 X/C Fig 5 5 Effec. f C^shapn o to f airfoileo s all designe havo dt e eth Fig. 56 A highly cambered jet-flapped airfoil designed by the inverse same upper-surface velocities. Blowing is in direction of mean line at method, including entrainment effects solie Th d.specifiee linth s ei d the trailing edge indicated an ,arrowse th y db . Each airfoi shows i l t na velocity distribution pointo e Th .s show check results fro exace mth t its design angl attackf eo . nonlinear direct method of calculation. JUNE 1975 HIGH-LIFT AERODYNAMICS 529

along the chord line. The final jet angle is at an inclination of again there is no method of prediction that can be called — chor e 74.4th o d°t airfoie planeth noss d i l an , e down 4.7° general. There is a large traffic in papers on shock-boundary- to develop the design pressure distribution. Agreement bet- laye authoe th r s interactiona r r knowsfa o probles e t th , bu , m wee exace nth t check calculatio lineae th d r ndesigan n method beenot n generasolvehas any din l sense vigorouA . s attackis is fair. Considerin extreme gth e natur f thieo s probleme th , under way that uses finite-difference methods. Possibly prac- resul satisfactorys ti . tical methods of analysis will result. One could talk indefinitely about other special problems 7. The Future related to high lift but that won't do, so I shall end by listing It was my original intention in planning this lecture to what I think are the ten most important basic theoretical devote considerable attention to the state of knowledge about problem high-liff so t aerodynamics: three-dimensional flows and high-lift flows in the transonic 1) Very general calculation of three-dimensional laminar region. However, I shall not, for two reasons. First, I am and turbulent flows. already exceeding the desired length for this paper, and 2) Calculatio f flowno s involving partial separatioe th n ni f theorsecondo y analysid yan wa , t e muctherth sno n s hei i rear. methods that can be called solid. Low-speed, two-dimensional 3) Practical calculation of flows involving forward sep- airfoil high-lift theor s becomyha efairla y "hard" science, aration bubbles. t three-dimensionabu transonid an l c theor s stil yi moro n l e 4) Practical calculation of flows involving shock- than a "soft" science, if that much. boundary-layer interaction. It is only necessary to read Callaghan's AGARD lecture 5) Calculatio viscoue th f no s flo airfoiln wo bodied san f so note srealiz o 1t 9 diversite eth f problemyo s that e arisw s ea revolutio f lengto n c hfro m about 0.95c Jo 1.05c n thaI . t come closer to real aircraft configurations and to realize the region of very rapid change and strong interaction, boundary- kind of practical empiricism that one is forced to use in ap- layer equations do not apply. With our recently acquired plied aircraft design. abilit o calculatyt e wele forwarth l d 95% r inabilitou , o yt Like the two-dimensional problem, the three-dimensional is solve the next 10% is a real irritant. Beyond 105% chord, we analysin a s involvin gcombinatioa f inviscidno boundd an - - again can usually do an acceptable job. ary-layer flow inviscie Th . d analysi nearls i s y here. Tha, is t 6) Further developmen inversf to e methods. lifting inviscid-potential-flow solutions can now be calculated 7) Drag of multielement airfoil systems. Drag predictions readily for as many as 1000 coordinate points. That number have a relative error one order of magnitude greater than that will easily solve nearly any simple wing-alone geometry and is of lift predictions. Since propulsiv d acceleratioan e - in n rather good at handling wing-fuselage combinations. 1000 formation is crucial to aircraft design, that is a very important points is marginal for a two-element wing alone, but it can problem. produce useful information handlo T . e three-element wings 8) Practical calculatio f merginno g boundary-layers, wall together with major body effects at least 2000-point solutions jetwakesd san . Although methods have been developeds i t i , are needed. In a private communication, we learn from Nor- felt that considerable further development is needed. man Donaldson that Bell Aircraft has extended the method to 9) The analysis of flows over swept wings on which a handle 3000 data points. The work is done on a modified leading-edge vortex is developed. When the vortex develops, UNIVAC 1108 computer probleA . f thimo s size requires conventional calculations are about 100% in error. more tha houro n tw machin f so e time hopes i t I . d that cosr pe t 10) Three-dimensional transonic calculations, particularly data point continue decreaseo st . for arbitrary wings and wing-body combinations. The boundary-layer solution lags behind the inviscid-flow In several of those problems, the first phase is two- solution. Although ther e formulationear methodd an s r fo s dimensional analysis, but as soon as that is accomplished the analyzing three-dimensional laminar and turbulent three-dimensiona nexe l th loom ts targeta p su . flows—and even a book49 on the subject—none is powerful As you can see from the list, aerodynamics still has a long enough to analyze the complicated problems of applied n trulo beforca g wa y e o t yw calculatee , instea f juso d t analysis that occur, for example, those of swept wings. estimate retrospectn i t tha e have Ye . se w t e e w ,progresse da Progress in fundamental work is encouraging, however. It ap- long way toward the goal in the past thirty years. Perhaps pears that extension eddy-viscosite th f so y formulations used another Wright Brothers lecture should be given on this same quite successfully for two-dimensional flows do not lose ac- subject 20 to 30 years from now. With the computer curacy when extended o three-dimensionat l flows.e th n 5I 0 revolution thahavee w te prospect th , f findino s g practical case of turbulent flow, much empiricism is involved, and the calculation methods for most of the ten problems are bright. accurac methoa e determine b f yo n dca d onl extensivy yb e We hope there will be some new "inventions" in flow and flow control. Cornish's spanwise blowing52 might be one, and comparison with experiment milestonA . developmene th n ei t 53 of the art of computing two-dimensional turbulent boundary effective application of the trapped-vortex idea might be layers was the Stanford Conference,51 in which each par- another. ticipan claimeo wh t havo dt gooea d boundary-layer method In October 1908 banquea t honos a , hi Francn n i i rt d ean was aske calculato dt elarga e variet f flowsyo l froal , m stan- after very successful flight Septembern si , Wilbur Wrighs wa t dardized input data under prescribed rules. Sometime th n ei unexpectedl wordsw fe said e a H .I knoy " ,y sa callewo t n do rather distant future, that kin conferencf do e wil needee lb o dt of onlbirde parrote yon th , , that t can'talksi very d fl t yan , establish the accuracy of methods of calculating three- high." I, too, cannot fly very high and I have talked too dimensional flow. much. Beyond that problee probleth s i mf o calculatinm g References maximum lift, as contrasted with the problem of calculating initial separation e calculationth f I . f three-dimensionao s l !Prandtl , "UbeL. , r Fltissigkeitsbewegun i sehbe g r kleiner boundary layers are forced to stop when the first separation is Reibung," III International Mathematiker-Kongress, Heidelberg, 1904. encountered, the morlikele o d yar o eyt than barely startn I . 2 predicw dimensionso no tw n t ca separatio e w , n reasonably , "High-LifR. Weyl . A , t Device d Taillesan s s Airplanes," Aircraft Engineering, Oct. 1945 . 292p , . well. With that problem out of the way, the problem of 3Hagerdorn Rudend an , . "WinH ,P. , d Tunnel Investigationa f so predicting flows with partial separation comes to the front. Wing with Junkers Slotted Flap and the Effect of Blowing Through Ideas being explored range all the way from direct numerical the Trailing Edge of the Main Surface over the Flap," RAE attempt t solvina s complete th g e Navier-Stokes equatioo nt Translation No. 442, Dec. 1953, Royal Aircraft Establishment, Farn- approximatin separatee gth d regio inviscin a y nb d flow. borough, England. transonie th n I c regime, where separation involves shocks, 4 "Experiments on Models of Aeroplane Wings at the National 530 A. M. O. SMITH J. AIRCRAFT

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