NATIONAL ADVISORY gblVIMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM No. 1185 SYSTEMATIC INVESTIGATIONS OF THE INFLUENCE OF THE SHAPE OF THE PROFILE UPON IHE POSITION OF THE TRANSITION POINT By K. Bussmann and A. Ulrich TRANSLATION "Systematische Untersuchungen ilber den Einfluss der Profilform auf die Lage des Umschlagspiinktes" Technische Berichte Band 10, Heft 9, und Vorabdrucke aus Jahrbuch 1943 der deutschen Luftfahrtforschung, lA 010, pp. 1-19 Washington October 1947 UNIVERSITY OF aORIDA GAINESVILLE. FL326t1-70tl USA NATIOML ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM NO. II85 SYSTEMATIC INVESTIGATIONS OP THE. INFLUENCE OP THl^ SHAPE OP THE PROFILE UPON THE POSITION OP THE TPJiNSITION POINT*' By K. Bussmann and A. Ulrich The position of the beginning of transition lamJnap/ turbulent as a function of the thiclcness and the camber of the profile at varlovxs Reynolds numbers and ' lift coefficients was investigated for s> series of Joukovslq/- profiles. The calculation of the boi-Uidary layer vas carried out according to the Pohlhausen method vhich m.ay be continued by a simplified stability calculation according to H. Schlichting (^), A list of tables is given which perm.its the reading off of the position 'of the transition point on suction and pressure side for each Joukdwsky profile. • OUTLIIME I. Statement of the problem. II. Extent of- the investigation III. The calculation of the potential velocity and the practical application of the boundary la2rer and stability calculations: \t\ a) Potential flow b) Boundary layer and stability calculation *"Systematlsche -Untersuchungen uber den Elnfluss der Profilform auf die Lage des Umschlagspunktes ." ' Zentrale; far wissenschaf tliches Berichtswesen der Lilf tfahrtforschimg des Generalluf tzeugmelsters (ZV/B) Berlin-Adiershof , Technische Berichte und Vorabdrucke aus Jahrbuch 1q43 der deutschen Luf tfahrtforschung^, Band 10(1943), Heft 9, Sept. 15, 19^3, lA 010, pp. 1-19 . NACA TM Wo. 1185 IV. Results: (a) Influence of the Cg^-value and of the ReynoD.ds number b| Influence of the camber of the profile f/t c) Influence of the thicloiess of the profile d/t (d) List of tables for the separation and lnstabilit2/' points for all Joukowsky profiles (e) Mean value of the laminar-^ low- distance of suction and pressure side for all Joukowsky profiles V . Summary VI. References SYMBOLS x,j rectangular coordinates in the plane s profile contour length starting from the nose of the profile t ving chord t' length of the profile contour from • nose to trailing edge (different for press\ire and suction side) Uq velocity of incoming flow Uj^(s) potential velocity at the profile 5pi| boundary layer thickness according to Pohlhausen P4 5* displacement thickn.ess of the boundary layer . . , . nondimensional boundary layer thickness form parameter of the boundary— layer profiles according to Pohlhausen P4 W Uo Zl| = ■0 t hk t dUni = % Uo ds MCA TM No. 1135 Tv^/r ■■ form parameter according to ^° Pohrnausen.P6 ^V^vh.)^ ^V^vh.) universal fimctions of the boundary- \Fy \ F V layer calculation Sqp;!^^ position of the instability point, measured along the contour of the nose of the profile s_^pg position of the separation point according to ?6 method I, S'TATEICFT OP THE PROBLEM^ The position of the transition point laminar/ turbulent in the frictional boundary layer is of decisive importance for the problem of the theoretical calculation of the profile drag of an airfoil since the friction drag depends on it to a high degree. The position of the transition point on the airfoil is largely dependent on the pressure distribution along the contoiir of the profile and J therefore, on the shape of the airfoil section and on the lift coefficient. A way of theoretical calculation of the start of transition (instability point) J that is, the point downstream from which the boundary layer is unsta,ble, wa.s recently indicated by H. Schlichtlng U.3,^) and J. Pretsch (2). According to present conceptions the turbulence observed in tests develops from an unstable condition by a mechanism of excitation as yet little known; therefore, the experimental transition point is always to be expected a little further back than the theoretical instability point. Knowledge of the theoretical instability point is, nevertlieless, important for the research on profiles, in particular for the drag w'ohlora. 'Recently^ a report ■"■An extract of this report was given in a lecture of the first-named author at the Lilienthal meeting for the discussion of boundary-layer problems in G?3ttingen on October 28 and 29, 19^1. it NAG A TM No. II83 was made about airfoil sections which, due to a position very fai? back of the irstabjlity and transition i)oint, have surprisingly small drag coefficients (laminar profi.les). Thus far no systematical investigations of the i.nfluence of the shape of the profile upon the position of the transition point have been made cither experimentally or theoretically. The following calcu- lation of the theoretical instabjlity point is, therefore, given for the first time in a sufficiently large range of c^^— values and Reynolds numbers to achieve a greater s^rstematJ zation of airfoil sections. In order to keep the extent of calculations within tolerable limits only the two most important profile parameters,, thickness and camber were varied. A rather convenient and accurate mode of calculation of the potential flow for the profjleo is important for these investigations and the selection of a series of Joukowsky profiles was, therefore, natural. It was not advisable to take for instance the MCA series as a basjs; the calculation nf the potential flow for such profiles according to the methods at present available does pxOt achieve the accuracy which is required here , II. EXTT]NT OF THE. im'ESTIGATION A series of ordinary Joukowsky yjrofiles of the relative thiclmesses d/t = 0, 0. 05/ 0.10, O.I5, 0.'?0, 0.'?5 and the relative cambers f/t = 0, 0.02, O.C^, 0.08 V7ere taken as a basis, (See fig. 1.) For instance, J 415 stands for the Joukowsky profile of camber f/t = C.04 and the thickness d/t --^ 0,15. The c -region which was examined is c„ = to 1 and the Re-number range Re - _-i_ = 10"^ to 10 . The complete calculations •D were carried out only for the following 'orof iler-'- : 000, 003, 015, 025, 215, ^'^0, hl3, ^'<5, -300, 3I5, and 825. The. results for the remaining proiiles could be obtained by incerpolation„ Thus it was possible to obtain a result with- tolerable loss of time in spite of the very extensive program (four parameters )j a cei'tain amount of accuracy had to be neglected since the Jnterpolation sometim-es was carried out .over three points. NACA T^M No. II85 5 ..■ Ill .'THE CALCULATION OF THE POTENTIAL VELOCITY AND THE PRACTICAL APPIJCATION OP TIIE BOUNDARY LAYER AND STABILITY CAI.CULATIONS ■(.a) The Potential Plow The calculation of the potential velocity with its first and second derivatives along the profile contour forms the basis for a boundary layer and stability calculation. The potential flow about a Joukowsky profile is obtained by conforrnal mapping of the flow about a circular cylinder. (See fig, 2.) A short list of the most important symbols and for'inulas for the profile contour and for the velocity distribution follows: z = X + iy j >= Coordinates in the comnlex plane I = ^ + ill I " . mapping function: ■ „2 ? = 2 + ^ 2 circle K^^ — ^ mean camber line of the profile circle K — ) cambered profile a radius of the imit circle in the z~plane R ■■ radius of the circle to be mapped in the z-plane t wing chordv t* ■ length of the profile contour from nose to trailing edge (different for suction and pressure side) ^n* Yr, coordinates: of the center of the circle to be mapped in the z-plane (circle K) o MCA TM No, 1185 o, 71 a a g center coordinates of the mapping circle of the mean camber line of the profile (circle Kii.) varying angular coordinate of the conformal transformation zero lift direction (See fig. 2.) angle of attack of the airfoil referred to the theoretical chord geometrical an^le of attack referred to the hi tangent (See fig. 2.) Profile nose: cp = Tr + 6 Trailing edge: 9 = -^ ^o ■ - — = e^ = thickness parameter j k = 1 + e, ^1 ■gf = 62^ = camoer parameter 1 ' See table I. P = arc cos f = arc s.m + e 1 + e The profile parameters ^^, e and 3 can be found in table 1. Profile contour: X _ a t ~ t -A > ^ 1 + e 1,^ cos 9 ~ e Z = a t t kfe^ + ^ll + e^2 g..^ ^A ^^ 1 '; N y (1) MCA TM No. 1185 7 W = k^/l + 2ej^^j+ e^^ + 2k>jl + S^ ( " ^ t °°s ^ ^ ^li^ '"^-^ ^ / ^. 1 +lk + €. ) a - ^ + k + €^ (1(a)) Fose radius P/t; Pop symmetrical profileG the equation ■2 P t 2e. -*- . 4- (2) Is valid exactly. This formula may vith a good approxi- mation also he applied to cambered profiles. The numerical values in tahle 1 show that the nose radius of the Joukowsk^'' profiles is only little larger than for the NACA profile family according to NACA report 460 for which P/t - l.l(d/t)^. Velocity distrlhution: - m r ~] ~ = 2 sin (9 — a) + sin (a + 3) P-, (9) (3) Stagnation points: Back 9 =■— 3 i'l(cp) = Front cp = TT + 3 + 2a N (^) pi - if + 4k2 A^ 4-^ o \o 1- + ^ u'- sin cp '"^ ,y NACA TM No. II85 Velocity at the trailing edge: lira 'U. cp — ^ --p \ U, m\ co s (g + p) (5) Arc leiipjth: ds dcp W <4" (6) s as a function of cp is to be ascertained from (6) b^ graphical Integration or can he seen directly in an enlarged presentation of the profile contour (t ^ Im) , Velocity gradient: IT" -— - = ^ < — TT^ CO- (cp -. a) o dcp 1 B + " ■ 0/2 js in (9 — a ) + s in (a — 3 ) N A = 2k (n., + v + ^ si^ 9) N^ = (N - 1)^ + A^ B N^ N' - N Rn - 1) N' + 2k A 1 + c - A cos 9] N' = ?k Jl + €.2 /e sin cp + e k cos q)J (7) MCA TM No'. 1185 ■ 9 The first derivative of the potential velocity vith respect to the arc length -^f- was calculated numerically from equations (6) and (?), and from tha d2Un, graT)hically the second derivative — — . ds2 Relation between c^^ and a: R c == 8'/r '^-sin (a. + P) R k {1+ e|^2 ■^ - . , ^ . compare tables 1 and 2, 3 ^ 2e^_ ^ _^^„ (b) Boundary-Layer and Stability Calculation After calculation of the potential velocity with its first and second derivatives along the profile contour there is a boundary— layer and stability calcu- lation to be made for each profile. The boundary—layer calculation according to Pohlharv.!:'eter accopd.inr: to Poiilhausen Pi| [! f (kpj and s(?.p, J =: . .li 1 i -re r 3 a 1 f one t i o n s f (\) = — ^ -^. :-- ^— ' 630 OTO It^ip.;. 3. , 2\, lr£2 ::2I2 Initial condltioncs Jit the &tagnati->n point that is.. X =: 7,0S2 o - '12. ~ 7»052 - . ).1.0 " U' "^ u' 19^ MCA TM No. 1185 11 Besides, ^l[0' = > U" - 5.391 ~72 (10) The isocline method vas selected for the solution of the differential equation. The particular advantage of this method Is that not only the initial value zuq is knoTm, but that the initial inclination at the stag- nation point ZhQ*. also can be determined. The latter • value is obtained by exact performance of the limiting dz process lira g;^ in (8) (Kovarth(6) ) . With z^-q' known u-^o the integral curve passing through the initial value z^q is easily found which otherwise is not Immediately possible because of the singularity of the Pohlhausen equation at the stagnation point. For the profiles of the thickness d/t = 0, that is, for the flat plate and the circular-arc profiles, the case vheve the flow does not enter abruptly (a = O) is exceptiginal :s5.nce there exists no true stagnation point: . the velocity at the leading edge has a finite value'.different from 0, The initial value of the thickness of the boundary?' layer is here zero, that is, at. the- leading edge there is: ■ . ^^.0 ^ ° (11) .. Profile c not abrupt flow entrance ■ 200 •-.25 - - 4oo , '■ -5 800 1 -12 Ni.CA TM No. 1185 Tho velocity near the leading edge of circular- arc profiles takes the same course as v's: that is, U* "becomes vith s— ^ infinite like 1/ 'fW^ the velocity has a perpendicular . tangent vhich always occurs when the contoiir of the profile shows a sudden change in cin'^vature as it does here (v, Koppenfels (8)), zi^ near the lead.ing edge for a circular-arc profile behaves like zj, for the flat plate, that is, zii goes in a 'linear relation to s toward 0. Taking these facts into consideration there results at the leading edge: It has proved advantageous to calculate the line elements zj,' directly from the equation (8) "by means of a plotting of the curves f(^piij ail's- sf'^-pi J ^ (See ^IS. 3.) This method is superior to the calculation of the line elements by means, of the often used nomograms of Mangier (7) with respect to accxiracy and its equal with respect to loss of time. Generally it will be sufficient to determine the line elements for each value of the abscissa s/t at two ordinate values only. The bound.ar3''-layer. calculation yields for each profile for a given c^ value the nondimensional boundary.' layer thickness Zj, and the form parameter Api^ as a function of the length of the arc s along the contour. The distribution of velocity u(y) in the laminar boundary layer is then obtained from: <5 For theflat plate z^ = 34.03 s/t (according to Pohlhausen (5) ) . "^ lUCA TM No, 1185 ^~ = ^"(£) + >^^i G- ^'^O 15 (13) with ^k y ) h %~ 6 5 7_ _ lA? > J (ll^) The results of the boundary-layer calculation for the profiles J OOO and . J 025 hsve been plotted as examples in figures it, and 5* the form parameter A, and the nondimensional displacament thick- Pk ness Uot with 5-''- standing for the displace- ment thickness* The following relation exists between the displace- ment thickness and • the boimdary layer thickness according to Pohlhausen: 6'"" = ow Jrli 0.3 12 cy (15) The displacement thickness of tiie flat plate in longl^ tudlnal flov; ('/Vpii = 0") is represented ■gra;Dhicall7/ in figures h. and 5 -fo^ com'oarison. The following equations are valid: _ _6S. ^Pk ^Pk ,/ \l^ - 0.3 f[ = 1.7^ n (16) ih li^^c:. Ti: No. 11O5 The profile J 8OO (fig,. It.) shows clearly that the displacement thickness" for accelerated flov/ (suction side) is sinaller "than- the disp la cement thickness of the flat plate whereas it is larger for retarded flow (pressure side). (Compare also fig, 16.) Prom the boundary-layer calculation there result also the laminar separation points. According to the foup-terti method of Fohlhauson separation occurs at 7\pjj_ = -12, according to the six-term method (see below) at "x / = -10 corresponding to T^pi = S*^'^*- Flow photographs have been talcen in a Lippisch smoke tunnel for a part of the calculated profiles of models of 50-centiraeter wing chord and at Re-numhers — jj- of about 2 X 10^, The points of separation have been ascertained from the flov; graphs (figs. 6 to 11, appendix). Figure 12 shows the experimental and theoretical separation points for various profiles for ooraparison. Compare . also table y. The agreement is rather good. After 7\pi has been ascertained a.-j a function of the length of the arc 3 there results the instability point (s/t) .^ from a stabil.ity calculation ' crito '' (H, Schlichting (k) ) base '^ on the six-term, method of Pohlhausen, The P6-metho'-.l is based on a one-parameter group (parameter Ap^) of bounda.ry-layer profiles Vi/hich can be i^epresented by polynomials of the sixth degree, An investigation of stability v/as carried out for a number of these bbmidary- layer profiles in (.'.l); first, the critical Re-nui:iber of the boundary layer 1 "":" '' crit, as a function of 7^ /- was obtained* The critical Re-number of the laminar layer \ — ^3 — j as a function ■'crit, of T^pji (fig. 15) is then immediately kv:iov/n also because of a vmiversal relation between ^^p/ and j^-pL indicated in (I;.), NAG A TM ¥.0. II85 15 Once Xpi (s/t) has been ascertainod from the bouTidary-layer calculation accorlihg to Pohlhausen's /U Q*\ me tho d a 'critical He - nurab c r \ 1 may be \u irit. coordinated to each point of the profile by means of fi^urs 13, '.loreover the Re-mmiber of che boundary layer — — can be calculated for each point of the ■^ TJ^t nrofile at a ceytsln — — 1 y • ni -'i , ; ^ V *-' / T ■•■7 ^ The location of the instability point is then given by / crlt. . IV. r^ESIJLTS (a) Influence of the Cj^-Value and the Re-number The results of the stability calciilatlon , that is, the position of the theoretical instability point I—) ^'■fcrlt. foi- the sample profiles J 80O and J 025 are plotted in figures ik and I5 against c v/ith the Re-nuinber ^ o t as Daraineter and furtheririore against — : — v/ith tho v c^-.value as par.ametez''. The characteristic course of the curves is the sane for all profiles | the following statements are v;--lid: the Instability point travels, v/ith increasing c at a constant Re -number, forward-. ^^ N'ACA TI,I No. 1185 on the suction side, bac^r/zr-rd on the pressure side^ tho Instability point travels forward on both suction and pressure side with increasing Re-number at a fixed c^-value, TIiIe behavior is demonstrated very clearly m figiires 16 and I7 which represent the velocity distributions .for the two profiles J SOO afid J 025 for the various Cg_- values with instability and separation points. One can see in particular that the instability points of the suction side for Re-nuiT:bers r, n - 10-^ to 10! lie near the vblocity maximumi position of tho instability point for Re = 10 agrees well with the j.ocation of the velocity maxiiiiujii. The pressure side of J 80C in the case where the flow does not enter abruptly (c£;_ = 1) is an exception amonfi' the cbovo :nLention3d examples, s 5 ne'e the flow from the leadini^ edge to the center of the profile is con- siderably increased so that no relative velocity maxlr.iujia exists. Measurements concerning the dependency of the transition point on the c^- value were taken by A, Silvorster'n and J. V. Becker (9)« Ihese tests showed (as a result) the sarr.e dependency of the transition point upon the lift coefficient as the present theoretical investigations, (b) Influence of the Camber of the Profile Th-e influence of the camber upon the position of the instability point can be described as follov/s; the instability y;oint travels with increasing Camber, at const.ant thickness, for all c -values and Re-numbers bac];;:i'/Grd on the sixction side, forvv'ard on the pressure aide, Tiiis influence of the camber can bo vuiderfetood from the fuct that the stagnation point and therefore the region of the accelerated stabilising flow travels, with Increasing camber, backward on the suction side whereas b-^causu of the flow aroiuid the nose of the profile a region of considerabl3'- retai"ded destabilizing flov/ originates immediately behind the nosi;: on the pressujr'o side. Figure 18 represents as an example the results for profiles of the thickness d/t = 0,15, with variable cariber f/t i'or c^ = 0.25 ^^'^ again the Re-number as parameter. The curves for all thick- nesses and all c„-values have the same characteristics, a NACA TM No. II85 1? (c) Influence of the Profile Thickness The dependency of the instability point on the thickness cannot be described in such general terms as the influence of the caraber since this influence depends in the following way on the Cg_ value ; A .certain "c^ not abrupt flow entrance"* ^^^^ Is, the c^ value that corresponds to the not abrupt entering of the flow (a = 0) for the circular arc profile with the given camber, is coordinated to each value of the camber f/t. The curves (s/'t)crit. versus d/t at a constant f/t shov/ on principle two different types (fig. 19}: • I, With increasing tliickness, the curves {s/t) ^^^.^ versus d/t start from a finite value and have a flat minimum: ^ • On the suction side for Cg^ - Cg_ for not abrupt flow changes. . On the pressure side for c^ = Cg. for not abrupt flow changes, II, The curves (s/t) .^^^ versus d/t rise starting from with increasing thickness j hence, the transition point moves backward as follows; On the suction side for Ca > Ca for not abrupt flow changes. On the pressure side for Ca < Ca for not abrupt flow changes. The results for the symmetriqal profiles at c^ = 0,25 are represented as an example in figure 20. For- the symmetrical profiles Ca for not abrupt flow changes = 0* that is, the dependency of the instability point on the thickness d/t for all Ca > is of type II on the suction side, of type I on the pressure side. The flat miniraum^ in curves of type I does, in some p' cases, not exist at high Re -numbers (Re = 107 to 10'°)," and Is/t) Qy^^^ versus d/t rises from the finite value d/t = 0, ■ 18 NACA TH No, II85 (d) List of Tables for the Separation and Instability Point in all Joukowsky Profiles The total result of the boundary-layer and stability calculations is represented by a graph of the curves (s/t)^p / = const, and (s/t) ,. = const,, respec- tively! in a system of axes thickness d/t - camber f/t, (See figs, 21 to ^0. ) A profile corresponds to each point of the plane. In t)articuiar, the symmetrical profiles are coordinated to the noints of the d/t-axis, the circular arc profiles to the points of the f/t-axis, and the flat plate corresponds to the zero point. Lift coefficient and Re -number are considered as parameters. One has therewith a catalogue of Joukowsk;/ profiles that make it possible read off, for every profile in the region = d/t = 0,25; 0%/t = O/OG, the position of the separation points for = c = 1 (figs,' 21 and 22) and the position of the ins ta Dili ty point for = ca = 1 and lo5 = Re "^ 10^. Figures 25 to JO represent' the curves (s/t. .. j, = const, for the crx 1-. , U„t c, Q Reynolds numbers from Re =■ -~^— = 10^ to 10 at the Cj3^-values , c^^ = 0, 0,25, 0«5* -^^d. 1 for suction and pressure side, For instance the values indicated in the following table for profiles of the camber f/t = 0.02 and the thickness d/t = 0.10 to O.I5 at Re = 10*^ and lo''' are taken from these rerDresentations, (See page 19. ) The most remarkable matter in this graphical repre- sentation is the location of the curve (s/t)^p / = 0, and (s/t) ^^ = Of respectively, at the various c^-values. The poslti'^n of this zero curve in the catalogue for the instability points will be discussed; the same is valid for the separation points, ^^A^crit. ~ '^ ^^^ only appear for the flat plate and the circular-arc profiles on the suction side for Cg^ > c^^ for not abrupt flow changes, on the pressure side for ^a "^ °a ^°^ ^°^ NACA Til No. 1185 19 ! * H II w m a) • -p --- \.C\\0..r-A ,-< • . • 000 c -p -p JJ rH • • • ' tion side •:— f ICO rH rH • « • U^ ■ H t-i • Jo 000 -p* 4^ +:< .p *,H ■i-A -p H H V— X • • • r-l < rr.! CD ,ii It v-^ xi C— H VO •*i m -r-l r-i r-i CO to DJ . • . • f^ '^ . 000 ■P -P +3 !l -p' 'LA -P U \A CO CO "'-^ _^o f'i ^^ «H ^0 -p • • • 000 11 a) « fU •H -H 4-^ m P' CO ■ -p ■ir\uA CM CM r-H • . • ■.■.; 000 +; -p -p LTN ^- 1/ s cvJ CNi H • • a 000. I •LTN rf CJ I, A 1 • • 1 ^0 MAC A Td No, llo5- . The curv0 a a for not abrupt ilov/ changes for al], circular-arc profiles. For the pressiu^e side, on the other hand, (s/t) . - on the whole f/t-axis, crit. Tliero follows in the saine way for c^ = 0.25 that (s/t) .^ =0 for = f/t < 0.02 on the suction ■ cr3.t. side and for f/t > 0.02 on the pressure side. Pressur'e and suction side, therefore,. alv/'".ys ccnplorient each other. 'Jho point vhlch corresponds to the circular-arc profile v;ith c=c^ ,, j.-.-. -u ^ a a for not abrupt ixovr nhangcs (for instance J JjOQ at c„ - 0.5, conparo figs. 2^ to JO), that is, the end ooint of the distance (v/t) - crit. is a singular point in the follov/lnfr sonS'.; '.uie point itself assiufies a certain vaj.ue (s/b) (different crit. for pressure and suction side), but an infinite nu:nber of curves (s/t) = const, v/bich are crov;dlng crit. together asyrriptotically tcv/ard (s/t) ,, = run C X^ X V « , into it. It is true, these relations for the ver^'- thin profiles give only qualitative results from the present investigations. An additional series of thin profiles would have to be investigated in order to ruaice piore accurate statements possible. However,, only profiles with thiol-cnesses d/t > 0,05 v/hich can^ be analyzed quantitatively, are of practical interest. For c =0, (s/t) ' is the sa:ne on suction and a crit. pressure side for the symmetrical profiles. Therefore the curves (s/t) = const, for suction and n'.-»es3ure crit. side would adjoin at Cg^ -=-0 in a joint representation of the suction and pressure side where for the pressure side the neasm^e of the csmber is directed down'-vard. For values Co r also the curves (s/t) = const, crit. NACA TM No. II85 21 have contini;!.at?*.ons wuich correspond to the cxiT'ves (s/t) ., = const, for the orescure and crit, suction side, respective.ly, at the appertaining Gg_-value with inverted sign, (e) Mean Value of the Laninar-Flow Distance on Suction and Pressure Side for all Jcuiiowsky Profiles In view of the development of larainar profiles the me ail value of the lamlnar-f lev/ distance on suction and pressure side is interest_in,f. Figures 51 and 32 shov; the curves mean value (s/t) = const, in crit. the d/t~, f/t-plane for various lift coefficients and IJ, " the Re-nui-nhers -- — = 10^"' and 10' s = 0.5 (s ., ^, + s ,, \|.. Generally the V orit. suction: crit, oress-ar'e Ji , followia^^ concluslons_are valid: The profiles vith the smallest mean valu-j (s/t) „. for a certain -va3.ue ' crii;. a lie near the circular- arc profile to Vvliich this value is coordinated as c , a lor not aorupt ilov/ cnanges Ti'iis -orofile v;rill be for c =0 the flat plate, a for c„ " 0,2"-) the Drofile J 200, for c =0,3 the a ' '- ' a profile J [{.00 and finally for c^ = 1 the profile J 8OC. There seems to be an excentional case at he • — — ■- 10'-' u and Cg,^ = 0,5 (fig« 31) which can he explained as follows"! T/ae circular- ai'-c profile for w^hich at the co3isidered c^ -value the flo^"; enters "not ahructly" (for instance J lj.00 at c^ = O.5) is a singular point in the f/t-, d/t -diagram? Approaching this profile on the f/t- axis from, two different sid'.-s one obtains tv:o different limJ.t values (s/t) ,, , since once only c r XT' • the siiction side' and once only the pressure side contributes to the m.ean value, Onlj; for the singular -point itself suction and pressure side coiitribute so that this profile has a higher (s/t) than the crit. 22. MAC A TM Fo. II85 profiles on the f/t-axls_ near it. If one now considers the curves mean value (s/t) , - const, for values ' crit, higher than the two li'-nit values (ran^e I), These curves enclose the singular- point and end at two points on the f/t-axis. The remaining smaller mean values (s"/t) generally cover only a small region crit • near the singular point (ran^e II) where, with the present investigations as a basis, raore accurate statements are not possible. Only for the TJq^ ( ^ case - 10-^ and c - 0,5 the ran.'-e II com"orises all profiles of the series considered here since on the pressure side the -orofile J JlOO at c^ = Q,5 an^-i He = 10 has no transition ^oint (s/t) ., = l| and j crj/G, _| therefore^ the point f/t = O.Ok obtains a high mean value (s/t) >'0.5. For this case there are closed _ , crit , curves (s/t) =; const, and there exists a profile (J II5) with thu smallest mean value ("/t ) .^ = O.ISS at / -; I ' crit. -"^ Re = 10°. ■ ■ ■ Moreover, the following results are obtained from figures 3I s-^^. ^2; All Joukowsky profiles have small mean values (s/t) ., ; for instance, the mean values' ' crit. ' for practically . Important lorofiles vi^ith the carr.ber f/t - 0.02 and t^ie thickness d/t = 0,10 to 0.20 at Re-numbers of IC^ to 10' are between 0,08 and 0,2, These mean values are only to a sm.all degree dependent, on the lift coef fie lent | for instance, the mean values for the profile J 215 at Re = 10^ and at .lift coef- ficients c = to 1 are between 0,1 6 and 0.175 r v.- SUKMARY A series of Joiokowsky profiles with thick- nesses d/t = to' 0.25 and caiiibors f/t = to O.O8 was investigated with respect to the position of the instability point foi' various lift coefficients and Re-nvanbers, llie following result was obtained 1 "vVith NAG A TI.I No. 1185 25 increasinr* Re-nxAmber, the instability point moves forward on suction and pressure side; with increasing Cg^-value it moves forv-rard on the suction side, bacliward on the pressure side. The position of the instability point as a function of thickness and cainber of the profile is represented. in the shape of a graphical list of tables v/hich permits the readin.-^ off of the position of the instability point on suction end pressuj^e side as well as of the mean vaJ-ue of the laminar-flow distance on suction mid pressui'e side for each profile of the series. Translated by Mary L, Mahler National Advisory Committee for Aeronautics 24 ' HACi'. TH ITo. IIC5 VI. REFERENCES 1. Schllchtlng, H.: Tiber die Berechnung der kritlschen Reynoldsschen Zahl einer Heibungsschicht in "beschleunigter imd verzogerter Stromung, Jahrbuch 19^0 der deutschen Luf tfahrtforschung, p. I 97. 2. Pretsch, J.: Die Stabllitat der Laminar stromung bei Druckgefalle vmd DruckanGtieg. Jahrbuch 19^1 der deutschen Luf tfahrtforscliung, p. I 58. 3. Schllchting, H.: Berecimung des Urns chlags punk tes laminar/turbulent fur eine ebene Platte bei kleinen Anstellwinkeln. Nicht veroff entlichter Bericht, 4. Schlichtingj H., and Ulrich, A.: Zur Berechnung des Umsohlagspunktes laminar/turbulent . Preisausschreiben 194o der Lilienthal- Gesellschaft flii' Luf tfahrtforschiong. Jahrbuoh 1942 der deutschen Luf tfahrtforschung, p. I 3. 5. Pohlhausen, K.: Zur naherungsweisen integration der Differentialgleichung der laminaren Reibujigs- schicht.Z.angev. Math. u. I!ech. Bd. 1, p. 252, 1921. 6. Hovarth, L.: On the Calculation of Stead3- Plov in the Boundary Layer Wear the Surface of a Cylinder in a Stream. ARC Rep. 1632 (1935). 7. Ma^ngler, ¥.: Einige Noraograrnme zur Bereclmung der laminaren Reibijuigsschicht an einem Tragf liigel - profil. Jahrbuch 194o der deutschen Luftfahrt- forschung p. I lo . 8. Koppenfels^ V. v.: Two-Dimensional Potential Flov Past a Smooth "Tall vith Partly Constant Curvature. RACA TM 996, 19^1. 9. Silverstein, A., and Becker, J. V.: Determinations of Boundary-Layer Transitions on Three Symmetrical Airfoils in the NACA Pull-Scale Vind Tioniiel. NACA-Rep. No. 637 (1938). NACA TM No. II85 25 10. Holsteln, H., and Bohlen, T.: Ein verelnfachtes Verfahren zur Berechnung laralnarer Reibungs- schichten, die dera Naherungsansatz von K. Pohlhausen geniigen (noch nicht veroffentlicht) 26 NACA TM No. 1185 ■d ^ OJ-P ID ■d 'h' CO 03 r--Lr\rHCO rH iH iH rH rH O iH rH iH rH iH iH O O O O O O .-I (\J J^O o o o o o O 1-1 (\J J--0 o o o o o 49|0) ax. o J- o PL, O C\J lTNOmT^ O OO O rH iHvDCO t-J- O OO O O Mvo a\voa3 ON O O O rH C\J If^ OO oooo <^J^<^lr^l^ c^t^ O iH <\JJ-U>C~- O O O O O O O f\i U^CTNtJ"^ O O O O i-H 0\vO LTvcf rr\ o O C^^o lTv^ OOrirHH OOOiHrHrl pHVDCO t--_d- ^r^l-^ L/^_d■t^ O >-< CM J<0 O O O OO o rH rM3H C\J Ln O O OO OO I-H rH r-l H iH i-( O i-i ^r^JV)0 O O O O OO r-t r-t r-t i-{ •-< r^ o CM ir\o\i^ O O O OrH rH iH rH rH rvj J" O O O O OO rH rH rHrHrH rH t— CTNCVJCO ^-•-o tH (M_^irvt— rvj O O O O O rH voj-..j:Jcovo O C\J LfNONl-Pi OO O O rH OO OO o o OO O o o o J-cr>irij-cM r<^rvJ o^ChCO C— O-J^-t^rH LfNO> CO O O rH rH rH f-i i-i <-{ i-i r-i r-t ^O^O^^^OvD^O J-J--^- O O O O o o CO CO a:i CO CO CO o oo ooo t-i rTNrH K-xO JCOl<>CO_d- O 0>HrH CM rH Ov£) O^CO O ,^C^^C^O O J r-{ r-l r-i i-l i-l i-i CM CM CM CM CM CM CTsONCTNCTNaNaN CM CVJ C\J CM CM CM JCO K>CO J- O O rH rH CM O ITNO ITNO t/^ O O rH rH CM OJ O L/^o Lr^o L/N o o H rH rvi (\J CO CO CO CO CO cn 4> ,d « +3 r-l CJ IM +3 o •H h » ft !» IP rH x; 4^ ■p O IS IM w O IB c ID o *> •rt iH 43 3 01 t^" rH 3 . ^ a O a ■d ID o e •o CO 43 rHP u O US 3 rH d 4J • s> \ -d- ti 'd \jj « 43 •n TJ ID CO U ID m 3 rH. a, O^ xi 4h S *> O u >> d a ff « 3 ■a H »-P V «H N^|J■ •p ID II V rH-— T3 •H H •d)43 (m — n t. a •• o a o o a «H (D43 t (D a a S ■P 3 rH •• cr V) rH o >>o tH ^ O x: o •o •-I *> ■d43 •H D « iH (A ID 4^ to HI ■P <-\ •H O »H O IH •HT) n ■P O •a f< ^ u c o 0) o, •ri O O a o (. o ID in a ID •P a in 0) ti ^ a •-i C 3 CM ^ Ji! O o 043 •H C 5S NACA TM No. 1185 27 ■ TABLE 2 THEORETICAL ANGLE OF ATTACK a (UEGREE) Ca Profile 000 005 010 015 020 025 .25 .5 1 2.5 9.2 2.2 2.1 k-2 2.05 8.2 2.0 k.O 8.0 1.9 ^a 200 205 210 215 220 225 .25 .5 1 6.9 -2.5 -.1 2.1 6.5 -2.5 -.2 1.9 6.2 .2.5 1.8 5.9 -2.5 -.5 1:1 \:l Ca 1+00 ii.05 ]+10 U5 1+20 1|25 .25 .5 1 '2.5 U.6 -U.6 -2.U -.2 U.2 -U.6 5.9 -i+.6 -2.55 -.5 5.5 -U.6 -2.6 -.7 5.5 -i;.6 -2.7 5.*1 Ga 800 805 810 815 820 825 .25 .5 1 -9.2 -6.9 -U.6 -9.2 -7.0 -i;.9 -.5 -9.2 -7.1 -9.2 -7.2 -5.2 -1.1 -9.2 -7.25 n ■ -9.2 -7.3 -5.4 -1.6 28 NACA TM No. 1185 TABT.F, 5 LAMINAR SEPARATION POINTS; COMPARISON OP TEST AND CALCULATION S = Suction side, D = Pressure side Ca 0.25 0.5 0.75 1 J 1+00 The or. Exper. S D S D 0.929 .92 0.891 .88 0.835 .83 .75 J 800 The or. Exper. S D S D .8925 .88 .86 .81; .8275 .80 .76 .755 .73 J 005 The or. Exper. S D S D .997 .997 .6596 .70 .28 .10 .015 J 025 The or. Exper. S D S D .1+025 .U025 .U5 .li5 .1,? .I185 .^08 .I;9i .385 .51+ .36 .60 .252 .592 .52 JU15 The or. Exper. S D S D .686 .200 .95 .19 .650 .285 .86 .29 .570 .577 .75 .31 .65 .35 .his .1|9U .US J 815 Theor. Exper. S D S D .737 .0II27 .86 .06 .685 .0562 .78 .075 .61+8 .0951 .71 .10 .67 .12 .59U .1921+ .65 .16 NACA TM No. 1185 29 000 200 m H 1- 800 005 205 i05 805 010 ^^ 210 HO ^ 015 ^ 215 ^ H5 J IS 020 Figure 1.- Joukowsky profiles: thickness d/t = to 0.25; camber f/t = to 0.08. Profile number: for instance, J 415 stands for the Joukowsky profile with f/t = 0.04 and d/t = 0.15. g-x^/> Zero lift-axis Zero lift>a j^"^ Theoretic.l dg chord profile parameter: f/=— (thickness par. f, = ii (camber par.) Figure 2.- Explanatory sketch to the Joukowsky transformation (schemp.tic), 30 NACA TM No. 1185 tw g(xi A< 300 \ 001 250 om^ ^. \ ■am iOO s \ \ 1/ 100 SO \ A tfWi ,-' ^ ^ \ \, \ JtM flM A>0 -2 -» -e -« -10 -« stagnation point '-^T^ Figures.- Auxiliary fiinction f ( ^ p. ) and g(^p^) for the boundary-layer calculation. « 6 Pt 7052 / / / ^ ~~~- 1 1 — ■ A ■■ ■*•», y i ■2 -6 I \ / 2 J i • y "^ ^,^^j 6 S ^ \^ V N B OP N "^ y N s. \ \' \ ^^^ --_ _— - -■'' \ V \ <> V _^ C ^\, \ ps\^o 1 \ \ ^ \ ■t? ?«■ ■UPS J^U- —\ t — 1 i__: \ r — r\' v\ 1 -5? -HP*. \ W V Suction aide Preaaure aide Laminar aeparation points according to P4 Laaiinar aeparation points according to P6 "Net abrupt" entering of the flow Figure 4.- Profile J 800, boundary-layer calculation: f orm parameter X p4 and displacement thickness -^ V NACA TM No. 1185 31 > —6 — ♦ — 2 \ ^i 7 stagnation point S. ^^ -^^ ^> r^- -^^ \ K^ ^ ^~- ^^ -~^^ 5 1 t 1 ''\ s^ s> ^. ^ flj OJS 0.7 — * — « —8 -to \, s^ N"", -N ^ s^ s S^ N^ s^ «* Ci -\) __^s\02)\i} t 1 ■> V RPS''-9iBi v-^ L N \ — >, -— \ 1 1 RPi,=-n t \ t C )^ \ Suction side flPi — Laninar separation points according to P4 pressure side DPf " Laminar separation points according to P6 plain plate with C, . Figure 5.- Profile J 025, boundary -layer calcualtion: form pa rameter \ p4 5 * 1/ u^t and displacement thickness — r — l^ — 5 . NACA TM No. 1185 33 ct° Cn tlipur. —4,6 —2,3 ■ 0,25 0,50') 2,3 0,75 4,6 1,00 a« Ca tlioor. —9,2 —6,9 0,25 —4,6 0,50 —2,3 0,75 1,00») Figure 6. Profile J 400, smoke tunnel photographs (Re = 2. 10^). *) not abrupt entering. Figure 7. Profile J 800, smoke tunnel photographs (Re = 2. 10^). Laminar separation points see Fig. 12. *) not abrupt entering. NACA TM No. 1185 35 a" C ^ L t ) e Figure 13. Universal relation between the critical Reynolds number ( and the form parameter X . ) crit . NACA TM No. 1185 41 li Crit 0.8 0.1 0.6 0.5 0,i 0.3 0.2 6,1 ^:^^ -^ ^^ 3 BOO ^^^ \^ ^ ^^ \^ ^ "^S^ \ ^^\ \ ^ >x \\\ \ X \ X ^v^ "Not abrupt " enter of the flow ing\ "~-- --. / — ¥X 10" ca= 0.25 0.5 10^ 10^ 10' 10^ Figure 14.- Suction side pressure side Profile J 800: Result of the stability calculation, Uot and c„. (f-)crit. versus 42 NACA TM No. 1185 0.S M(n, OA =^ 0.3 0.2 ^ 0.1 ^^--,^ •^ ^^^ — ^ ^ ■0=/ ^ ^ ^^ ^^ ~-.^ ^^ ^^ 1 ' v.t y /o* to' w^ w 10^ Suet ion side pr e s sure s ide Figure 15.- Profile J 025: Result of the stability calculation, (-r-3 ■. versus Uot and Cg^. u NACA TM No. 1185 43 Laminar separation poioti Instability points rupt ' entering he flew 'Not abrupt** entering of the flow aa 1,0 Figure 16.- Profile J 800: Velocity distribution with instability and separation points at various Re = —^ — and c^ for not abrupt flow changes; pressure side: c^ ^ c^ for not abrupt flow changes; pressure side: c^ < c^ ^^r not abrupt flow changes- ~ Suction lide -presiure lide e* 02 I Figure 20.- Influence of the thickness of the profile upon the position of the instabiUty point for symmetrical profiles with c^ = 0.25. A = laminar separation point; M = maximum velocity; S = stagnation point. 46 NACA TM No. 1185 fh Ca = Ca=0.2S UBS m t IS 5^ 19 \ \ 1 ^ / «£^ \ 1 y \ / ^ ^ Ji- i ^ \ // / y^ ^ -^ y/K \\ 1 /. / as M m // /^ ^ tU£ t SfS t ca = O.S Ca = f.O Sis t Figure 21.- Position of the laminar separation point (s/t) . ^^ ^ as a function of the thickness of the profile d/t and the camber of the profile f/t; suction side. ca=O^S (US ai ms u Figure 22.- Position of the laminar separation point (s/t)^p g as a function of the thickness of the profile d/t and the camber of the profile f/t; pressure side. NACA TM No. 1185 47 Ca=0 ca=0.25 ais oj Figure 23.- Position of the instability point (s/t)^^^^ as a function of the thickness of the profile d/t and the camber of the profile f/t; suction side; Re = 10^ . ca = Co =0.23 azs t Figure 24.- Position of the instability point (s/t)^^^^_ as a function of the thickness- of the profile d/t and the camber of the profile f/t; pressure side; Re = 10^ . 48 NACA TM No. 1185 Co =0.25 Figure 25.- Position of the instability point (s/t)^j.^|. as a function of the thickness of the profile d/t and the camber of the profile f/t; suction side; Re = 10^ o.os Jit Ca=0 Cq=0.2S J k025 ri ' / //. / / / y ^ 01^ i 00 01 o.ts ca=O.S 02S t Q2S t D.IS ti Figure 26.- Position of the instability point {s/i)Q-^ii_ as a function of the thickness of the profile d/t and the camber of the profile f/t; pressure side; Re = 10 6 NACA TM No. 1185 49 i:a=0 ca=0.2S Figure 27.- Position of the instability point ( s/t) ^^.j^^- _ as a function of the thickness of the profile d/t and the camber of the profile f/t; suction side; Re = 10 "^ . ca = Ca=025 Figure 28.- Position of the instability point (s/t) ., as a function of the thickness of the profile d/t and the camber of the profile f/t; pressure side; Re = 10 7 50 NACA TM No. 1185 cwo ca=0.25 aos 0,1 D25 t Figure 29.- Position of the instability point (s/t)(,j.j^^_as a function of the thickness of the profile d/t and the camber of the profile f/t; suction side; Hr = 10^ ca=o Ca=0.2S 025 t Figure 30.- Position of the instability point (s/t)^^^|. as a function of the thickness of the profile d/t and the camber of the profile f/t; pressure side; Re = 10 8 NACA TM No. 1185 51 Qj=0 ca=0,25 OfS t Figure 31.- Mean position of the instability point (s/t) for pressure and suction side at R$ = 10 . (s)^,^.^^^ 1/2 (s^rit. suet, side + ^crit. pressure side)' It ca-O /q.2S ^-^.225 ^^^^^ ffw ^ ^ \ff;75 s \ 0.0Z ^^ \ ootu ~ "~~-^a(;5 Ca-025 «»8 005 0/ O.IS 0,2 ms m 0,1 11,15 0.2 ozs Figure 32.- Mean position of the instability point (s/t)^j.^^_for pressure and suction side at Re = 10^ . (s)crit.- 1/2 (Scrit. suet, side + ^<,xi\.. pressure side). UNlVi^SiTY OF FLORIDA "^ DOCUMgNTS DEPARTMENT 1 20 MARSTON SCIENCE LlBRAPY RO. BOX 117011 ..^. GAINESVILLE. R. 32611-7011 USA