AEE No. L'*G15 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS WARTIMK REPORT ORIGINALLY ISSUED October 19^4-^ as Advance Restricted Report IAGI5 THE FLOW OF A COMPRESSIBI^ FUJID PAST A CIRCULAR ARC PROFILE By Carl Kaplan Langley Memorial Aeronautical Laboratory Langley Field, Va. -^ WASHINGTON NACA WARTIME REPORTS are reprints of papers originally issued to provide rapid distribution of advance research results to an authorized group requiring them for the war effort. They were pre- viously held under a security status but are now unclassified. Some of these reports were not tech- nically edited. All have been reproduced without change in order to expedite generail distribution. ^^6 DOCUMENTS DtHAKlML»f Digitized by tine Internet Arcliive in 2011 witli funding from University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundation http://www.archive.org/details/flowofcompressibOOIang -3 f 17^7Y NACA ARR No. l1|G15 NATIONAL liDVISORY COMMITTEE FOR AERONAUTICS ADVANCE RESTRICTED REPORT THE PLOV'V OF A COMPRESSIBLE FLUID PAST A CIRCULAR ARC PROFILE By Carl Kaplan SUMMARY The Ackeret iteration process is utilized to obtain higher approximations thsn that of Prandtl and Glauert for the flow of a compressible fluid past a circular arc profile. The procedui*e is to expand the velocity poten- tial in a power series of the camber coefficient. The first tv^o terms of the development correspond to the Prandtl-Glauert approximation and yield the well-known correction to the circulation about the profile. The second approximation, involving the square of the camber coefficient, improves the velocity and pressure fields but yields no new results with regard to the circulation, since the circulation about the profile is an odd func- tion of the cainber coefficient. The third approximation, involving the cube of the camber coefficient, permits the use of higher values of the camber coefficient and furthermore yields an improvement to the Prandtl-Glauert rule with regard to the effect of compressibility on the circulation of the circular arc profile. Numerical examples with tables and graphs illustrate the results of the analysis. INTRODUCTION The calculation of the two-dimensional steady flow of a compressible fluid past a prescribed body can be pei-'forraed by a method independently discovered by Janzen (reference 1) and by Raylelgh (reference 2), which con- sists in developing the velocity potential or the stream function according to powers of the stream Mach num.ber. The first approximation is the incompressible case and the succeeding approximations represent the effect of compressibility. The method has in recent years been 2 NACA ARR No. Li|G15 successively improved by Poggi (reference 5)> by Imai and Aihsra (reference i|), £.nd by the present author (refer- ence 5)« Although the method can be applied to an arbi- trary profile, it suffers from the practical restriction to small stream Mach numbers, because approximations beyond the second or third entail a prohibitive amount of labor. For the flov; past a profile of smell thickness, camber, and angle of attack, Prandtl (reference 6), Glauert (reference 7)> ^^^ Ackeret (reference 8) obtained by various means an approximation that applies to the entire subsonic range of velocity. The present author (reference 9) extended the method of Ackeret by an itera- tion process that takes into account the effect of thick- ness and applied the method to a particular family of symmetrical profiles. In the present paper, the effect of camber is investigated oy a similar application of the method of reference 9 to a circular arc profile. In the application of the method, it is desirable to avoid stagnation points so that the variation of the local velocity from thst of the undisturbed stream can be made small. For this reason the direction of the undisturbed stream is chosen parallel to the chord of the circular arc (ideal angle of attack) and the circulation about the profile is determined in accordance with the Kutta condi- tion; nfar.ely, that the flow past the profile leave the trailing edge tangentially . The flow is symmetrical fore and aft and the velocity remains finite at all points. The circulation in a com.pressible flow will be seen to be an odd function of the camber coefficient. In order, then, to obtain an improvement of the Prandtl-Glauert rule, it is necessary to carry the Iteration process through three approximftlons . T?IE ITERATION PROCBSS The velocity potential 0{X, Y) of the two- dimensional, steady, irrotetional flow of a com.pressible fluid satisfies the following differential equation of the second order; e-"^)§--.^;^(=^-^^)§=° MAC A AER No. L4GI5 3 whe;re X, Y i-ectangular Cartesian coordin&tes in pl£^ne of riow u = ^— , V = 77- fiuid ve.;.oclty components elong X and Y ^■^ O- fezes, respectively c local velocity of sound The local velocity of sonnd c is expressed in terms of the flaid velocity q by means of Bernoulli's equation d 3 X p ■J-^1 the equation deflnir.g the velocity of sound up and the adiabstic relation betv.een the press;u"-e anl the density In equations (2), (5), and (ii), p static pressure in fluid p-[_ static pressure in undisturbed stream at infinity p density of fluid p-i density of undisturbed stresni at infinity q magnitude of velocity of fluid Y adlabatic index (apprcx. l.li. for- air) h IT AC A fCU No. L.LOip Per the adiabatic case, equatioxi (3) yields ^ P C) By means of equa-ujons (ii) t^nd (3) Bernoulli's equstioii, equation (2), yields the following relations: P 2 - = C-]_ . - ^ .^ / ? _ ^ Pi 1 - y - 1 p/j2 o Pi i 1 . ■L.::^ u^ ( '-t . 1 1 V. (6) /i / where U velocity of lindlsturbed streexii at Infinity ^1 Ht velocity of sound in undisturbed strs;ai.-i at infinity Mach nui:iber of undisturbed strea^n at infinity Now, if the pre .e is held fixed in the uniform stream of velocity U end if a characteristic length s is assumed to be the unit of length and the stream velocity* Tj is assumed to be the unit of velocity, the f undainental differential equation (1) and the first of equations (6) become 2, (7) and 1 - ^-^ K /(.= - l) (8) NACA ARR No. Ll^ai5 where X, Y, u, v, q, £nd J^ novi denote, respectively', the nondiirisnsional quantities x/s, Y/s , u/U, v/u, q/lf, ar.d ^/Us . The itereticn process consists in developing the velocity potential in pov^ers of a parameter h, the carnber of the circular arc profile. Tl.us = . X - y4\ - h-/^2 - '-^Vz - £-.nd . > / u = - V = " 1 - h -^^ - h^ ---^ - ly — .'- - dX 6X o:'. h £iL - h^ i!^. - h5 lii. 6Y dY 6Y V ^ (10) together VVlien these express ion-? for 'p , '."/ith the expression for c'-/c-| given by equation (8), are introduced into the fundamerxtal differential equa- tion (7) snd when the coefficients of tiie various po'cver of h are equated to zero, the following dlff erei-itial equations for '^^ , ^o, 'Y'v , ... result: (^ T. 2\ 6 (l ■■ r.V? + .2 oY^ (11) Ml (y- 1) cx ax^ >2; \'^'i ( Y - 1) 14. ^4-2 ^ ^ cX 6y2 cY dXdX ( 12 ) EACA ARR No. L'jJI^ ^^ (1 - .,2) -^ . 1^ = dX^ cY' 2 \ 6Y/ 2 L Y - i.) V?:9f ^X"^ + r N +1) — ^ dy^ I ,/, 6X ^2 - 1) i^i cY cX 6-^ (y + D— ^-+ (y-D— ^ gX'^ cY- ^3 VoX 6Y oXdY 6Y cX6y dY 6X6Y/ ( (1?) These differential equations may be put into niore f&ir:iliar forms b^- the ir.troductior of a new set of independent variable? x r.nd y. where X : ^ > (l.!-!.) and f 1 - K,-) (1 1/2 For \vr^ < 1, equation (11) tl'^en becomes a Laplace equa- tion and equations (12) and (IJ ) beccir.e Pcisjon equations. Equation (11) replaces the fundamental differential equation (7) for flous that differ only slightly froirt the NACA ARR No. l]|u15 7 undisturbed stream, s.nd Its solution yields the well- known Prandtl-Glauert result. The solutionc of equa- tions (12) and (15) provide successive improvements in the approximation to the solution of a compresoible-f ].oy/ problem. For the present problem, the procedure to be fol- lowed in solving equations (11) to (13) is first to obtain the velocity potential for the incompressible case in the form of a power series in the camber coeffi- cient h of the circular arc profile. The solution for the first approxim.etion 0'-, of the compressible flow is then obtained by analogy from the form of th-r:- coefficient of h of tlie Incom'precsible velocity potential. Th.-j solutions of equation.-^ /l--j and (15) for the second and third approximations ^2 ^'"^^ i-^^ follow by a straight- forward, procedure. The boundary conditions - that the flow be tangential to the profile and that the disturb- ance to the miSin streajii vanish at infinity - are satisfied to the same power of the camber coefficient b that is involved In the approximation for the velocity poten- tial ^ ■, The calculations are laborious when m.ore than two stejjs in the iteration pi'ooess are involved bvit the third step is necessary to obtain results that extend present-day knowledge. Most of the details of calculation are given in appendixes in order not to obscure the presentation of the main results. RESULTS OF TK3 ANALYSIS Expr ession for the velocity potential.- The choice of the circular arc as the solid boundary was made for two reasons: (1) the solution of the incompressible flow can be easil;/ expressed in a closed form, and (2) when the circular arc is fixed in a uniform strevm floving In a direction parallel to the chord and when the Kutta condition - that the flow leave the trailing edge tangen- tially - is applied, the velocities at the nose and the tail are finite and different from zero. No stagnation points occur, therefore, on the boundary or in tlie field of flov; and a greater degree of accuracy in the iteration process is a.-'sured. Appendix A contains the calculation of the incomxpressible flow part tiic circular arc profile and appendixes B, C, and D contain thi detailed calcula- tions for ^-) , 02> ^^'^ 5^7' respectively. The final 8 FACA ARH "Mo. lLG15 expression for the velocity potential takes the foil ovv ing f o rm : ^ - co£h S cos r, - h(^^ - h^J^p " ^^^^ ~ ••• ^^5^ where, from equation (D9), )^1 "^ p \^ ^^^ ^^J " -^v frcni equation (CIJ), 0'2= f2D|_(Y+l) D+OSe"^+ 2De"'^ + 2 -p - D+ D [( y + l) I>+ 'i] ? e'^y^os t^ -f~(Y+l) D^e"-+|<12+12D-D[(Y+l)lJ+^j}' e~— J cos 5^1 and from equation (Di8), 0^ = a^d) sin 2r, + G2(|,) sin l+r] + G(£)(J- -— — -e -^ sm ^To-e -= sm [i.Ti) \^ cosh 2e, - cos 27^ / - r2G-i(0) + liC-2,{d)\v, In these equations Gn (£,), Gpls), E'-nd G(|) functions of S given by equa- tions (D12), (DIJ), and (Dl9)j respectively §,, T] elliptic coordinates i-elated to rectangular Gf-rtesian coordinates X, Y by equations of transformation: KA^A ARR No. L]lu15 9 X = X = cosh 'g CCS T] y = py = ainh q sin r) Tii e_c li' cul a t icn _g oi r e c t io n T c rniulj2_= - 3qu f 1 1 c n ( 1 5 ) I-" e pre Toll t^inie sol.utioh 'o'7~tTie fiindaiiisutf-l differe-rtiai equation {!) tiiat sctist'iss the bo^'ndai-y condit.lona at the surfsce of the circi3.1ar arc pixfile end .■•t infinity Insofar es tlie t'err.s inclusl^'-e of the third yower of ti'ie camber coefficient h £.re concerned, i'.t-.ch of the expressions ^-]_j ^p, and ^- are obtained in closed form and are fini^-e for all ■'.-alu-js of the stresin Mech niunber ¥i from ze.ro up to t'Ut not includinp- unit: v. rhe Kutta condition, .vihicxi deter:ulnes the circulation uniQuelv by stioulttin;?; a finite velocity at the sharo trail ine: edce of the circular vields thu f ollowln.a'; circulation correct icn f crrauj a (s-'-e equf.-.tion (LJb) + 10 1 £5 + 1) ^ -:.:.:.. s - ) -- (J -f o2.\ Jv 1 ' . ^^2 ii..::ii)! ^^, , .z. 2l! (y + 1) a( r) (16) where P^. and rj_ are, respectively, the circulations in the compressible ^nd incompressible flows. The incom- pressible circulation T^ is proportional to the first power of h sc th o.t the compressible circulation V Is an odd functioii of h. The second approKimation cf P Is therefore Identical with the first a\)proximatlon and ho departure fi-om the Prandti-Glauei't rule is obtained until the third povi^er of h is incJuded. rhie result explains Vvhy th.e simple Prandtl-G-la\\ert rule for the effect of compressibility on the cii'culatlcn ov lift of aji ajrfoil has been very satisfactory. For comparison, a foi'muln analogous to equation (l6) has been obtained by &ppl7'inf,^ the von Ka'nvian-Tsien velocity corroction formula to the circular arc pr':)file. 10 ^ACA ARR HO. l;!.015 From reference 10 1 -p. 1 - fiq.^ where q velocity of compressible fluid q^ velocity of incompressible fluid 1^ = 1 . (l - .,2) .1/2 2 By s.n elementary integration ai'ound the circle, corre- sponding conformally to the circular arc, the following reletlon is chen obtained: 1 - f 1/2 ti^ ± - 'i ' cos 5 Ti ,1/2 , 2= 'i i+p. ' 3in^£ h /P !l - 2tLV2 ^n + sin^s) + p. cos^^sl-/' 1/2 + a" ' cos OS -5 Ft ^ o 1/2 ^n ^ . 2^^ [1 + 2[j, ' \^1 + 3:n-5yi + u CO :^5JV2[ (17) v.'here the angle 5 (see. fig. 1) is related to the camber coefficient h by me ens of the equation tan 5 = 2h Table I gives values of bl"e ratio V.^^V. for various values or the stream Kach nuiROtr an„, the cambo]-' coeffi- cient h, calculated by moans of equations ( l6 ) and (17). Figure 2 shows the graphs of T /T. as functions of T, •1 'or various values of h. The curves based on the von Karman-Tsisn velocity correction formula lie between the Pr"-andtl-Glauert cui=ve and th.e curves based NAG A ARR Ko . Li^-G15 11 CO P CO a o •r-i -P O Kl o o w d -p o CI 'J ; •H ro CD •H Hi &1 rH K^ CO H > CC o CQ > f^ M CL O CD -p ^ O O O hOi p 43 •H G O G O -rH r-\ -P tH u -p a', A_. 00 CO .-a CO l--^ I OJ i .—1 + 1/3 O. + II + ^* a •H o P o V'l vii I 03 ^ -P ^H O 03 •rH CO fl, ^x 1-- CU T^ fi, -H. ~r» -J 1 ^- '0 + -1 r-j :0 -p O.i. + u c^ o 3-^ ,-~, a Ti p "H G + c— CO i^ CO __zt' , VII •V ' L + ^-^ o o o ^ M r 1 '-■Q v_-' C',3 — ^'j' OvJ Vi! + -_- Cs O c\J o T-J K^!-J- r-.l + 'Ti C J P H 3 r\ b _+Ik> + ^ ' ^0 r-\ + >- a rri — • X •H i-i. K\ • + ; J C '.J P (\, 1 CtX II Sh rH ■~X ■ J ^7^ •i> + rH :- CO o ---MCL c> _j o (~i 1 + 'v •.) N■^ ■13 II ^ .i:: + H-) o u • — ' A o H '^ l-H o + r-C ,d 4J rH ■p ■ ■> •\ -P 0-} OJ c-J rH CO !>^ cs + -P o •H i- f3 O rH K> rH JD P O .H Cju > CO CO hI i^ O ..C; ?-! 1 -P P. f-1 ^ s. <;h o n U C) ai. q NN O 'I"! ''C! •r ■Zi •P ^ Cj. •H 43 CNJ r--l bb 'd 4- cJ c 2 c3 H K'.:- (1? , ^ .-1 H r-\ -p ,n .rH + •> CO t>i C) ^■>- rH O ^ — fn c\i ■ P> Oh 1-5 •H J:- 43 O r-i!OJ O f3 I ri. ro ,ci ^ — \ 4^ Ci. f-< f! 4- •V •^^ rHl<^J- 43 r3 •v^ • O rH P CI "-i C'J (.-A ■xi O 1 Sh a 1. •r-i r.A O' 0.3 •■^i 1 43 'O rHiC'i. C fM 1 Cj O ' 1 03 I i C3 O ,-^ D' ca o <■-! -— ' ^■^ Ol C-i 3 C3 Hi CO 12 NACA ARR No. L1;(:t15 _ct- vTs o r-t r\J r- H r 1 OJ OJ a,^ 'O (1) V. --• C M -P d {-J O pL, f\l • r.O 05 O O :0 r i ~~~i ^ 'H U 1 , — , I +3 A^ -IJ o o c- 3$ <*r fs .— 1 rH &.S c O ':':• O a. 'A CO l-iC + + o '^ i 1 -- s -'"*• o a, ro, -H CJ^ LTn UTn fi, CO t t3 r-l 1 — i 1 4- + T,-! CO X> "H <^f a) ^ <^ (' , ca rj -p r-, .^ J 1 : OJ 1 1 \ 4J w >- ( J._. .1 ' ' J O* Tl O 4J K^ •\ hA \ c r_1 <; _i- rvi X? pIaj S^ O ■.'3 cr -$. '-'"-I .^J" K~\ . J' >r\ W OJ r>J + x: 1 A^ •> r-l •-•1 • -"77^^ 11 ■.:0 II CO TJ I j 1 - • 1 o j; CO '--'. fvJ I Of ■f\J + '-H O 1 'H + .H ^ -^ >. 4J O • H r.j .-1 C\' tj (D '^ c 1 ■^ I —J f^ c: •H c :. -< C!> + 51 « /i -P CQ -p CM o o 4J C-. K o :;- ^ o <"^ .q ?u w 3 _J K^ r-{ !<\ ---J ^--1 --1 •H 3 cr i a- + + + 1 O IV. cti + ;H OJ ^^ Cv] Q 0) rH e t\ .H &• ^ "s r-J 'D A1 U 1—! o 1 1 r rH «-y rH :J +i ^ u H 4- Fl 4. i-i U 1 1 s-l O •rH .^: ■M UJ ^ o o* 0-' K rHi'^ C ^ v,^ > O (^ o •H t3 CO ^1 OJ •r-l <;-* ^ rH a; H F! V. ^ o IM C N-\ K\ > •> o •rH Q-! + "^-H + ♦H f>, •H Ci •> O o tj .H -P -p i^ C. ; (X- O O ^ n 1 1 . c 1 1 > > O 11 Q L. 1 1— i 'H •H ru _ OJ 4^ f^ K <^ -P J.J ,"1 C O O 1-1 •H , "H -L +i u> O C'J 6) in. >> D^ O C ft i- o H ^"^ M to JJ + 1 ^ 'H o (D 1 03 O 0) O ^: rH X^ r-l 0) .» O ?^ -p II -t-^ 11 rH -^ r-! CJ II 11 ..O CO (D XI -p O 1 T-" 4' O 1 .H a OJ f> t. ^ < o- Ic? «i O' lo^ H --^ NACA ARR ITo. l1j.G15 IJ shows the graplis of — - - 1 , a.? functions cf };h. V-i y,;ax for* the three approximations Tor various vrlues of the camber coefficient h. The critical velocity ^r-r* defined as the value for which the velocity of the fluid equals the local velocity of sound, is obcrined fro/ti the first of equr- tions (6) by puttinf; q == c = 'j^^. Thus %.•. = ( — +— — r- ) (25) The values of c^^^ are given in table V In thc> coluirn for which the local lilach number is unity. The ratio ^cr/^1 -^^ easily calculated :^or the various approxinjo- tions. The graphs of only the third rpproxixnaticn of Q "^^ - 1 are includ-d in figure li. Table VI li3ts the ^i first, second, and third approxiniste values of the critical stream Mach niunber M-i , and figure 5 shows cr the correspcndixig graphs ac functions of the CPi:iber coefficient h. The graphs of the third approxli;aation of the ma/.imun and minimum values of q,, obtained frcia tables III and IV, are shov/n in flfure 6 as fanctionb cf the stream Mach nviniber Mi. The constant local Mech number lines shown in f igiire 6 are obtained from equation (3) by introducing the local F.ach number ",.1 in place of the local velocity of sound c. Thus + —- ;-!y.- \ -/ Note that equation (2];.) becomes equation (23) when = ( _L.-__id j V ll;- FACA ARR Mo. LhG.15 ?'I = 1. Table V contains values of q for various values cf M and M-,. A comparison of the results of reference 9 on the compressibility effect cf thickness and the results of the present paper en the compressibility effect of camber Is ■-if interest. For this purpose, a sy;ruaetrical shape of reference 9 wc.s compared with a circular arc profile with the same incompressible maximum speed at the surf&ce. Results of this comparison for several corresponding thickness and camber coefficients are ^iven in table VII. The dashed curves in figure 6 are asso- cisted with the various symmetrical shapes. For moderate values of CM'.ber and fnicknoss the difference may be seen to be negligible ever the entire subsonic range. This observation indicates thet, at least to a very good approximation, the effect of compressibility in the subsonic range co^n be considered to depend explicitly only on the incompresaible fluid velocity and the stream Mach m;ur;ber and to be independent of the shape of the profile. This result therefore substantiates the use of velocity correction f o^'mulas such as the prandtl-G-lauert, the von Karman-Tsicn, uhe Temple-Yarwood, and the Garrick- Kaplan (reference 11) formulas, which depend only on the incompressible fluid velocity and on the streair. Mach number. In general, the velocity q at the surface of the circular arc profile may be written as follows: P/ q = 1 + a-^h sin ^ + h-( a^,-, + a^ cos Z^) + hy^a,, sin ^ + a^ sin ^^') + ... (.^S) where, from eqiiation (l3). Cl-| — — [^ 1 - P' NAG A ARR No. LkG15 15 ^'^ .- 2(,- - 1){'^-V^) ai^ = 4 -- + u. (0) + 2G, (0 n Sr ::: c i ~ r -^ 2p(2D + 3) + Cb(0; ^ L I- :i Values of ai J a^.. Bz; ai , and a,.-: £o^ various values of the stream Maoli nujibor M- are giv^n in teble VIII. As an example of the b:!hfvior of ths velocity dlati'lbu- ticTx over a circular a.::'c oroflls fis the stream Mach nuTuber is vai-'ied, the cfi^ae of r.i - 0.0^ wi iVo 0.5, 0.5* ^''^'^^' '^'7 is calcalatcd and oorr'pared with the incom- pressible case. The calculated values of tno velocity at the upper and lov*er surf ..ices jf the circulai- arc profile, h = O-O^j, for the various values of V; are given in table IX and the corresponding velocity- distribution curves are sho.vn in fi^iiure 7. The pressure coefficient.- Jr. the crse -of a uniform, flow past a fixed bound;: ■"3" , the -ni^essure coefficiert is defined as C„ 0,1,:. •' o U^' From the third of eqvietions (6) it follov.s easily that G n Vi ~-^ %) - < - 1 + cr "I'l'-i + (y - 1)M- Y ]7=T For the limiting case or absolute v&cuur,, - 1 , 1 \l/2 ( 2 and q = \ \ ■'--2 Y 2 V 'J S oo ^. r.,^ Table X gives assooiatad valuts of the velocity and the pressure coefficient C., rr i'^'^ various values of the stream Mach number M-i , fcnd figure 8 shows the corre- sponding graphs. By rrioajis of table X and fi^^are 8, the volccity readings fi-cm figures 6 and. 7 "^^^'^ ^- replaced by the corresponding proasuro coeff icleiits . Langley Memorial Aeronautical Laboratory National Advisory Comrr.lttee for Aeroi-.aubics Langley Field, Va., NAG A /J?K No. L-'i-G15 1 APPSNLIK A DETERMINATION OF TIIE COMPLEX POTENTIAL FUNCTION w TIio Inconprss.". j.cle Flow past a CIi'ouIst Arc profile Consider the luapping of a circle C' ir: the Z' -plane Into a circular arc C in the Z-plano. (See fig. 1.) If the cental- is at (0,m) on tlie Y'-cxls end uhe circle passes through the poixjts (a,0) and (-s,0) on the X'-axis, then the Joukows'-cl transformation a2 S = Z' + --- (Al) maps the circle C in the Z' -plane Into a circular arc C in the Z-plsne. The equation of the clrciuar arc is / 2 -^V / P a - m 1 / •n' ... The parts of the circle C' lying aoove and belov/ the X'-axis correspond, respectively, to the upper and lower surfaces of the circular arc C. The end points / and B of the circular arc are the points X -■ ±3 a and the iTiaximum ordinate is Y ■- 2a tan 5 - 2m The camber coefficient h is defined as the ratio of the ma.ximum ordinate to the chord, or •u _ 2m = I tan 5 (..5) 18 II AC A ARR No. Li+ril5 The complex potential of the flovv psst a ci-^cular radius R i'Jxed in a uniform flow of velocity U at zero angle of attack and with a circu- lation r is given by w / ^2\ -;-p 7 It where Z" = Z' - l£ tan 6 Fcr the purpose of the present paper the circulation V must be so choc en that the stagnation points en the circle C' lie at the points X' ~ ±a corresponding- to the leading ana trailin.j; edges of the circular arc C; tii^t is » r = 8r,Uah = IrrUR G5n 6 (A5) With this value of the circulation inserted in equa- tion {jr.!\.) axid with Z" replaced hj Ke , the ccinplex velocity at the surface cf the circular arc C becomes dw „.,^ -i9 sin 8 + vx-ci 5 ( l^ ^ • • ^\- dZ -, .. i9 . _ 219 ■ ' 1 - 2_e •^xr o - e The raagnitude of the velocity is g-.v dv. dZ dZ tTA.l/^ '='-V = l"/l + 2 sin a sin 5 + sln'^o) (a6) It is recalled thab the upper surface of the circular arc is traversea in a clockwise sense as 6 goes from -6 to rr + 5 and the lower surface, as 9 goes fr^om -(tt - 5) to -5. The velocity at the nose or tail is then given by Nr.CA iiP.R No. Li,G3 5 nose '^t ai. U C O 3 - The mar.ixri.iin. sand nlniin'-u:; ve^or^itlGS o^oii.r -Trt 6 = at Q = - —, respectlvo] y , &rd ai-e given by 2 i •- ana %iax ^;nij.n = U(l + f-in s: = U(l - sin o)^ (A?) Equation of CiT'oi-la-:- Arc a^' rojv-r Sei'-ics in h The equation of the cii'culax' z.v'^ , obtained froTi equation ( a2 ) for the entir-e cir-cle, ir; y = 2i;i - V + \r- 1/2 '- y (a8) v/here r = -'■ '^— 1.: the radlu.? of the clrci2. Expan- sion of t'A°- radicrl in equation (he) acco;cdirg to powers of x/r yields Y = 2111 - — 1 ::' 1 Xjl; ^ _1_ J^ 8 r5 " l6 p^ (A9) Ey use of n m Or- + I+h'^ 31^ = 2h - 3h^ + 52h^ rhen equation (AQ) b:.:;cci:''e ( AlO) 20 RACA Af:R Ko. L[^015 . Kow, put -— = cos ^ and replace — and — by X 2a 2a 2a and Y, respectiv3ly. Equation (AlO) then becomes y=:2h sln^^ +2h5 sin^2^ + Sh^ 3ln22^ cos 2-^ + ... (/-li) and — =-Uh cos •^-l6h^ cos -^ COG 2^-l6h^ cos -"^ (l+J cos h^ - ... (Ai2) dX Equation of w as a Pcv. er Series in h Consider equatior. (aIl) vvitl". r ~ Girljaii and R^ = a^(i -t- [ih'^-j. Then = -Ijfz" + w = -UIZ" + a^ -^1 - UilJah lo^ V^ Z" r! (Al$) Now Z" = 2' - ia tan 5 = Z' - 2i-ih Then by expanding the right-hand side of equation ( AIJ ) according tc powers of n and replacing Z' hy ^2 — ^— — obtained fro:n the J"oukcwcki transfor- mation (Ai), it follows that /o1 2 1/8^ w = -7Z + 2ial"h La^ J If w/2aU and z/2a are v/ritten, respectively, w and Z, then w = -z + ih Jl-|z-(z2.1 N' r - 2 log ! Z + (z^-lj j> + ... (AlU) i NAG A AKR !To . L'!.G]5 21 Proin equ^itlcn (/>ll-! ) for w aad s corrsspondlrig equation fo^"" the con.plex conjugate w, the nono'liaenslont 1 volocit; potential becones "--0 , r- , . + ! Z - Z I ^l/2"^ , - z .fz2 - i) '-'■'- J.) 1-2 iog ^ ^ — .— > (AI5) Z + V- -0 22 NACA ARR No. L;|ai5 APPENDIX E DETERMINATION OF TliE FIRST APPRCXIIJATICN p^3_ By means of transf orm£>tion (Ilj.), equation (11) for ^-j becomes ^2/, 5 2^, dx' (Bl) oy A comparison of the expressions for ^ given by equa- tions (9) and (AI5) suggests the assumption r _ _ 1 - 1 z - v^ - V i^i = Mh - G^ - ! ■ i (^- - 1) > + 2 log ^ ~— -\ z + i^z - - Ij ! (B2) ^ where z = x + ly, z = x - iy, and k is an arbitrary constant. Since this expression for 0-, is the sura of a furi'^tion of z only and a function of z only, it satisfies Laplace's equation (51). The arbitrary con- stant k is to be determined from the boundary condition t^ G_7 _ 6/ cX dX ~ 6Y or 60' dy ox dx (EJO The expression for ^, insofar as the first power in h is concerned, is = -X - h0^ NAG A ARxR No. l4^'^15 and, to the flr^t povi^er in h, fron equ:.\t:'on (A12), ^- = -tph cos -'^ ax The boundary condition, equation (BJ), then becor.es i+ph cos ■% ■ en 6y = Sikhp'^ < /^IP / p ■\l/2!'- I _ /-o M/? r \ / .J J >= ii ,/_ -J ^ f'.2 - 1 t' '- 72 — r (EiO By definition z = cos ^, and fr-ora equation (All), to the first power in h, y = 2j:h sin -V. Hence, to the first pov/er in h, z " c 3 ■^ + 2 1 jHh 3 in ■^ z = cos ^^ - 2i;ih sin"^-^ "> z- = CC3'"'?- + Ij.iph sin^^ cos ^ cs^'i - 1^.1 ph sin-^^ COS -^ —? z- - c chsn (z2 - ij-/"^ = 1 sin .^ (1 - 2if,h cos ^) z- - l)^/^ = -1 sin -^ (1 + 21pli cos ^^ O r2l-> _ c--^!--' i|.ph cos ^ = -2ikhf/ ~ i sin ■^ sm -> ■BlkhB'^ cos * ^h NACA ARil No. Li|G15 or 2p Ths exfression for the first apprcxiraation cf / is then :>i Gf = -X - — < 2p i - (- - ^r' (^ - !)■ - I z - U^ - 1 u ^ + 2 IC:? z + id^^l (b5) This expression for ^ can te simplified considerably by introducxrs elliptic coordinates ^ and n- Thus, let z = cosh (b6) wher^ Then t = S + in X + iy = cosh (^ + Ir, ) = cosh £, cos "^j + i s:*nh £, sin r^ so that I X - cosh ^ cos Tj y = sinh f sin rj 1 Equation (.35) can then be vvritten $2f=-|(cosh t +co3h I) -^-^(r^Le"^^ +2 log e^"^) 2p or = - cosh £ CCS r, - — i o - sxn dr. - -) (37) (33) (B9) NAG A i\R.R Ko. l1uG15 25 Fro'n a coi.ip&riscn oT equations (/-I5) fnd (B5) note that, if rj_ and p denots the cii-culat :' on in fie incoKiprs^p Ible C8 38 and the coiiiprassrhl'S cas?, then ^_ 1 Equation (PIO) is tlie v';eli -known Pi'Stidtl-Glauert rule connecting the circulations (or lijrts) in the incom- pressible and compressible cases. In order to ubilize equation (39) for the celcula- tions, the equation^ of transi or;-nation (BY) ifi'-ist be invf^rted. Thus, 1 1 i x2 cosn"£ -1- sinh c x2 - p 1 J cos'~rj 3 in r; y (BID Fron equations (Bll). ,.^.,2,^ _ _- , (-2. 1, 2 r 2 sinh c, = -0 + id"" + I4 2 sin r, = b + (b- + Jiy^ 2. _. K 4- y?- -. ),.r^') ] — u ^ \u ^ ay I \V2 f (E12) where b - 1 - (^- ^ r-) By means of transf o..'mation (iLj, 2^ _ ,. ^ ^.2 ^ i,^2^2y/^ 2x1/2 2 slnh^^l := -b + [b'^ + k^ - J 2 sin'^'n = b -;- (b- + i+p"^ '/"_)' 26 '^lACA Aim No. LL[.C1.3 v/here b = 1 - (X + p ,^ ;V^ In terrr.s or the corr.plex variables t, and i, the velocity components in the direction of the coordinate axes are u = 62f 60 d0 "N 6X sinli i ^-i siritj. t, ct, 6/ 1 60 \ 6Y \sinh i 6t sinh i 6i J Let 0^ be given by equation ( BS ) ; then. (B15) u = - e -> sm T] V = i4l"e - cos T] y (BlM IJow, to the first power in h, at the boundary, S = T^ = ^ Hence, if q and q.; denote the maj^nitudes of the velocity at the surface of the circular arc profile for the compressible and the incomprsssible case::., respec- tively, then •> hh q = 1 + ±L. sin ^ ^ P [ q= = 1 + '|li sin ^ or, when h sin ^ is eliminated, 1 (315) ^^ p \) qi ( Bl6 ) NAG A ARR No. Li|G15 27 where P = (. - :..^)^/^ Equation (Bl6) refj-resents the velocity-correction formula for the Pri-ndtl-O-lauert approximution. Equations (BIS) can also be written as foilo\vs: qi - 1 ^ Since the Prandtl-Glsuert approximation is strictly true for infinitesimal disturbances to the uniform stream, equation (Bl6) may be replaced by tne differential coef- ficient (from reference li) ^ ^'^q.=l (Bl8) 2^ KACA /J^R No. Ll+G-15 APPENDIX C DETERMINATIOK OP THE SECOND APPROXIMATION 0^ By means of transf or-.nation (11+), the symbolic relations , 6x 6z 6; v2 + 2 ^00 Q Ox dz^ bzcz cz"" 6^^ ^:V 2 6 ^ oz oz^ + 2 i^ .2 V 2 . X - s -2 oz ozoz oz N d x6 y A-^ v-2 > (CD ana the equation of transformation (b6) a = cosh i or z = (;osh I differential equation (12) for 02 '^^^■^ '^-- expressed in terms of the complex variables i and i as follows: NA"A ;J^R ^:o, L.'|G]5 2-. CM O I ■H • T-t o I J»_ri o liv^ ■-_/i| ._0 00. I z' CD til 1 ^_ 1 ro ^ — - f'i— o C 1 O .-1 i-O o ^^ P ^.^ •iH cd rvj -M ^' X d "J '■^i, „ cr + ;:5 0) o^ -u i 1 »-fl o c? r-.j 1 -i-O ' 1 o_0 C 1 Ch Ik. ^ (D o —1 . -H r , " ^ 1 ^ C3 C -1 ! Q ol M ■ci rj ■-> i M ^. •>^, cr\ «e C f'-J n n + ..pJ ] d^) Ti <1> . •^.i r->. ■^■'s-L. -P Q 1 J f; W "^^' --0 O •H 1 '=^ .-! G> 'O ^ la) 1 M -P 1 1'-^ r-t II .-M Ka' -'Sk f-'.0 (M - -3 r-H •H^iS. >S;. M •H Ph ! W o O Jj 1 ^M ^ o a o O J5 -H -P o CO a -' in •H -1-5 CJ a Cj T3 n; b- 0) O'' >< O CLi c O «i S': !^ -r! -)-• -P JJ C s:; r: •rH C; (D f: ^ m (u ■^""^ •H iVh 50 FACA ARR I'lo. L4GI5 CCCt bU '- - ~ Hoi pi 8p^ t~^ -• e~')\o'^ sinh t + e"^ sinh t) > Finally, by putting I = E^ + ir] and ^ = J, - Ir], ^2 2 + (y + 1) n - G^^ e~^ cos 31] ! / (C3) The right-hand side of equation (CJ) suggests a solution of the form 02 - ?l(?) cc'S ri + F^(|) C03 3rj (Ck) By substituting this expression for ^n into equa- tion (C5) and by equating the coefficients of cos t] £ nd cos 3*^ to zero, the following differential equations for F^d,) and Fz(£,) are obtained: a F d§ T - Fi = 4 r o I r 1 - P"^ I .1 ( r+i) - ( Y - 5 ) PI e"- + !+f2e-5S> (C5 ) J d'^F, d; = U(y + i) (c6) The solutions of these equations are Pl = 2 L, IJACA ARR No. LV'il5 51 where A2_ and A^ ar'e arbitrary constants to be deter- irjlned by the boundery condltlou at the surface of the profile. The other tvo arbitrary constants are taken equal to zero since F-j_ and F„ must vanish at infinity. In terms of the v&riables £ and r], the boundary condition (Bj) takes the form /. , » b'f , ^ . ^'^\ dy „?/ , V . 6/ isinl'i t cos n ~ — cosn c, sm T, -—- I —^ — 8 ( cosn s sm -n r-^ + sinh £ cos rj v^ ) (C9) where th3 velocity potential 0* has the f orm yf = - cosh £, cos r\ - — (e~'~- sin 2r, - 2t] ) - h^^F^ cos r; + F:. cos ^ri + r2rj) (CIO) and where Fp is en arbitrary circulation to be deter- mined by the Kutta condition at the ti-ailing edge of the circular arc profile. In Oi'der to m.ake use of the. boundary equation (C9)» the various functions of c, and r^ appearing in equa- tion (CIO) must be expressed as functions of •^ evalu- ated at the boundary. From equations (All) and (A12 ) , the boundary and its r.lope aro now given by y = 2ph sin^^ + 8ph-^ sin'^-'^ cos^^ + ... dv , ^, > — !^ = -iiSh cos ^ - , » , dx 52 NACA ARR No. L[|.G15 At the boundary then, with x = cos ^, when powers of h above the second f-x'e neglected. b = 1 - (^ * /) Then, fror. equations (5l<'?) sin'^'Tj = sin^'^ \\ + kp h cos ^) cos~Ti = cos'"'* f 1 - iip h sin ^j sin T' - sin ^ \\ + Sp^h"^ cos^^) f 2 2 2 ^ cos Tj = cos ■* \^1 - 2p h sin ^J slnh i, = iip h^ sln"'^ cosh"^^ = 1 + l^p'^h^ sln"^^ s inh I, = 2 ph s in ^ 2 2 -^ cosh £ = 1 + 2p h sin ^ e~^ = 1 - 2ph sin ^ + 2p^h'^ sin^.^ £ = 2ph sin ■^ Vifhen these expressions, with equations (G7), (C8), and (CIO), are utilized in the boundary equation (C9), the follov/ina results are obtained: NACA ARR No. L]4G15 55 2 ^-^ H = - 6 A - p2\ + 2(y + i)(i !- ) .2 P'^ ^ -^(^^5-|^4^--)?7l^) V P /' V p^ ^^ (cii) y The value of the arbitrary constart T-) i^ deter- Kiined in the following w&y. The magnitude of the velocity, when terms containing powers of h higher then the second are neglected, is given by = 1 + h _a + h-|- P^'-^l + -^ dx or, in the variables I, and r), 2h /. , ^ 6;?^^ q = 1 + .. Sinn c, cos ri — —- cosh 9 sm r) — — ] cosh 2£,-G0£ 2r: V ^^ f^' 2h'^ cosh 2c, - cos 2t] V (smh 9 cos T, -x~-~ cosh c, sm ri —t. \ 6£ 2e'-h'^ icosn i-, sm t' -— J^+ sirih s (cosh 2£,- cos 2v) ' \ •^cosh i^ s • 6-^ 6r] / S cos r; :!LL] + ... (C12) From equeticns ( B9 ) and (CIO), /i = - (e"'-- sin 2v, - Zrj p ■ ' and (C15) ':h NAG A ARii L'o. LhCl^ < ^ LTN M3 .p l>. JO •r-l «^ rH ._ A ^ Ch rH CO o C\J ^ ^ q a) • CD 'J VII 'PT""^ (1^ c p (0 p ,•' ^ t: tS •H r 3 1-1 t=^ "^ <"' a -— . fl X3 !^ P- l>» cJ .♦ 00 CQ -H CD •H rH -— « o CD W 5 .:) > rH "^"^ r" 1 !h in 'H a) rH i-H -H 1 C3 •v OJ s =S Ph CD P i- ''"^ Ci, 0' p, 1 > *— ^ f-i 1 ■ 1 1>- OJ 0) 55 —1 11 H- Cj (\J o 1 CD. f-i +J 0) ft r-l '*^ ■~' ^_ _^ P Q} , ^K, OJ d II rH Oi ^^ J= (D Cj ^ 1 1 Jh ^ 1 a (~H ^ ?-i "•h .-:d- CO •H o 1 II -P Ph - + y^ ' VJ •H >- 0} •\ CU. ■p « — •=*■ --cr. CO r\l ^ S "3 rH c =H t) ■H + H a— ^ 'M x: uc' CO 1 •H CO i^ CO r! CO 1 rH M (D Q) CJ ca iH cf VJ 0) H P ^ :•■-< ^ r P cO. C S ^c ^ :ci c? s:^ -P Oh — - rCj u ^^ •H o o r-i P « CO Q) 1 -TT OifH Eh l^^__ __ 1 a OJ -p q) 1 u P ^Nx + ■:i d ^ x> ^ >- t) a* q bO o ,-^ 1 x: OJ pr-H 0) C .H 0) 1 r-l CD •H 1 (D OJ <^ -=k »r-' ^ T-* OJ P 'O r-\ U CO p ri CO V. ' A ?■; Lj II '' ^ CO V CD — 1 e P •H •H ^q rH CM G ^ ^ O' f^l OJ Cm p Ol 'H K> C * ^ ' > ■H r-l B x: a xy lo^ &H o •H <;_, rH r^J rH + 'Vh ^ •M P C! T3 'H ■;i t^ P o CJ •H s:-! fn q CO 3 ^c T! iti x; 'd vi. C '.0 t; t, VII ai t^ C :3 + c r ■ — fi •-a CO cr -* P> Sh 'Vh •^^ Ci- •H •h 0) e; dU. en CO c t ^ Q d* 'H C vii •v rH •\ -H c« -^'• •P 5^ u c: ^ '>i <0 -p P P -H 9 Sh 5K P t! CO "H P + ^ . tH (D 'd + i^ 'ri 3 .C D r-l rH S-i c t-1 C.V P r-l t ?-l a) .— ■ rH (D ff u (D CD ^ o 11 rC <1) 'r-l a 0! •H ^\ ^ U X) c r iH ;a 'H O^l O' t» P NACA ARR No. L^ai5 55 rr For the position of rnaxLmm velocity, ^ = p-. Qi 2 \.^^ / (1 + zhY (ci?) TT For the position of minimiim velocity, •^ = - p-> 1 - __v_ (1 - 2h) 2 \ ^ liii,i,sLg,y4.iT-.. 5/^-1) VP 2 (CI8) 56 NACA ARR No. l1|.G15 CO .-H o M Q a p-i Si o M EH < m M x: o Ph -n 9 M Eh O n S5 ID ro CQ Q) !h a x; ® d o r5 C o • H 4J CO rH CD o rH o r/3 a; r H 4J o -d- ■•,H . -M —I O O — I -P 0) ■p d o <^ -4 O O ro d 1.0 (D S E-H Its) X3 ^1 + + •^s C\J !c\J CO.; CM In ^ o d CO •H *. ^ — . Lr\ QQ — ' d N~ IKI o ^Sl. • O •H 04 ts) +J ^ 'O III O" (P In M S rO\ o "tSL s:-i Cm TJ © d • r-t OS +J ^ o CD, .-I a o (D r-H ~^ o <;h c! o "H W CG (D fU ft EH NAG A ARH No. l'+G15 57 ^1=2T^ ':-(/- Oi Z - I z •n - ^r r 2 + 2 lot if..) Introduce nevv complex variables ^- aiid A., where (D2) X = z + (z^ - l) ,1/2 X = z - 2-.)- /, = 3 + \z''- - 1 (^ - ^) 1/2 (D5) The relations between the complex variables X and X and the complex variables ^ and ^, respectively, are >. = e W = e i 2g \ (dU) ^1 ^i-VX \ d - 2 log-^ X/ (D5) Similarly, the expression lor /^ , obtained from equa- tions {Ck) , (C7), (CS), and (Cll'T, is X + \ i^^-. = 2 (D - E) ^— ^ lo/: XX + D V% + ("^C + D - 53) XX "■ X'X^ XX X2x2 X5x5 (d6) 58 KACA ARR No. U\Xrl^ where -1 - 2L^£.i,,. i;A-^^> p2 12 ~ • 2 D = 1 - (b^ ■nd ■2 E C--D-7-S=l 5 ^ From eauations (D5) and (Do) with the use of equa- tion (D5), the following relations are obtained: 0. - = _ -'4- -ZZ iP \" - i 5^: v^ - 1 log A.A.)- 2D \ + ?X 1 ^ >^^ - ^\^ J: — + 2S : 6C ?.^X2(^\'i - l) 1 ^ ^ "1 x2k2(^2_,>) - X^(>2.i) KACA A:'J-{ l^J'o . LkG^Sj 35 ^2zz=8(D-E) -J^!_ log Xl - g_\!(^^'M, UX-~X) x' ^'■^' -^)- x(x^ - ly x(x^ - if I + 8d -— ^^"^ ".- - 8(30 - D + 3E) ^ X^X^ - 1^^ (.^ - 1/ 8e -^- 3^i!^ijL^l + 2),C i^O^^ - 1/ 2\2 _ 1 x(x^ - l)' ^:>„= = 8Df-4--i ; - A, + X, ;zz 8S (X+X)(>.-\) A"^ "'^ (X^ - l)(?.2 _ i) XX(X2 - l)(x2 - l) and expressions for the corJ^espondlng conjugate complex qu&ntities . When the foregoing expressions are Introduced Into equation (Dl), and when equations (dIl) are used to express the various quantities in terns of the variables £, and T;, the follov.lng differential equation for ^-^ Is obtained: 5 i+0 N/.CA A^:^ N<^' lJfG15 <- -^ P pi r ^' CJ CO + + CO. CM I ON + o ITN + ax. 00 ON I CM CM u NACA AP.R No. l!lG15 kl k^ = 80pD(D + .1) + 8iiD^ (y + 1)(6d + 1) + pD^ (y + 1)^ A^2 = -192pD^ - 52pD^ (y + 1) + 12[;D^(y + 1)^ ^■lo' = 96 PD^ + 1+8 pd5 (y + 1) + 6pD^ (y + 1)^ .p^ = Pd5 (y + 1)^ (15D + S) + I|.pD^ (y + i)(7D + 9 ^u^^ . it "6 Aell- 'Ho'' A ,J4- 12 R.2 B2 f = = 96 PD - 16Pd5 (y + 1) - 10pD^(Y +1)^ L8pd5 (y + 1) - epD^ (y +1)^ -22)+Pd2 - 16 pD^ (y + 1) + 22pD^(Y + 1)^ : -BI^PD^ (y + 1) - 2ipD^-^(Y +1)^ -- i2SpD^ + 6^pd5 (y + 1) + 8pD^' (y + if -epD^fly + 1) D + 14]^ -12 pD^ (y + 1) ijY + 1) D + UJ i6pr/ Ry +i) d + lij^ i2pd5 (y + 1) |1y + 1) d + U] -i6pd2[(y + 1) d + [^ 'Q li -20pd5 (y + 1) JJY + i)b + li] 21iPd2[(y + 1) c + I'f '42 KACA ARR lie. L[lG13 C-, = C, = ^2 Di D2 D;^ = b(:D lai^r^ [( Y + 1) D + 2 } I] Y + 1) D + U] SpD^jjr + 1)D + [J I^PD2[(y + 1) D + 2j PD' (y + 1 ) Ij Y + 1) D + 2 .c.^2 Di = 2Pd5 (y + 1) Note tlir. t ,^ 2 i| 2 „ ^ — w 6^ = -232^ = -B^-^ = fB^ = 2A U _ 1 „ 2 ^].2 " 6 1'^ and "4 7 -'-^ 5 B,^2 ^ _B^4- = 5 .^^^ ^^ ^^^^, ^ ^ ^ The right-hand side of equabion (D7) suggests a solution of the forir 0^ =G]_(£) sin Zri+Cr^C^,) sin Kr, + ^ ^'^(S) sin 2nr; (d8) n=5 When tills expression for 0-^ is inserted in the left- hand side of equation (D7) and the coefficients of sin 2ri, sin Ur,, and sin 2nrj are equated to zero, the following differential equations for G-l(£,), G^Cc;), and Gj^(£,) result: d^Cr df + Ap e - + A-]_Q'^e (D9) NAG A AF:;R No. L.'iG15 15 H/) ^-» O vO 1—1 1-1 i-H 1 Q P ,5^ "1- + OJ HJ) 1 m OJ + 1 .i"^ + Q ON CM 0) OJ co Hi! H/) CM 1 Q \0 f ^n' + + o + 1 r-H a + 1 1 — ' o ^, 1 '"'CM^ o + u I OJ % 1 1— 1 1 + 1-1 flj pq + 1-H o I + -J + r-H + 1 0) C\J •, o 1 CvJ V ' OJ ■'"Tj^ Oj + G + CM + -t- 0) CM o CO aj !^ ai vO + + i-H o + cu o OJ ^ 1 G -^ N + 1—1 .—1 -<- + CM 1—1 1 ai 1 1 (r- + CVI 1 fl5 + r-H}r-N + CM 1 CO + CM CO 1 1— 1 1 -* + OJ + 1 1 1 CM + CM + O 1-1 -> + «« I :hJ) K/) CM CO ViJ 1 to CM , CO no '-' r^ >U1 + HiCO I Hil O rH I CD J. O 1-1 IHil I CD C3 I rH CT) r-J CO SH.O GJ CD ' C: i CO --d- I OJ SHi) 1 rH I CD _d- c\J H CM 1 H + KJ) o sn CS MAC A ARR No. L^G15 ii-5 id) CM 1 -d- •xS) sn_n rH ^ fi P. I fM "-^ K/) a CM 1 1 frvj 1 1 ^^ ! ca (0 t , 1 H 'cm Q >- 1 ca ' ^ CM KA CM rv) + Q CO. CM a CM 1 H + 2; o ca -d- + CM_^ H - CM P ca _d- + smJI CO CM + (M H H + rH CO. ^ 1 + + 1 ^t + >- + -d- P 1 ca_i a X >- >- + + r^ + #• o ca o + rH|_d- iK/l HJ"! CO 1 ■d •H p ca V3 p ca i KA , s-J) r; o ca 1 iKjl (M *,H 4^ + + + rvi rH|f\J + ■Kf) CM CM 1 (D l^.Nco, + CM 1 © pi d a) + C\J + ' rM ' P ca + 2: + H/l a CM V.J ' Q CO. 'oj 1 P (ja VO r-H + > cci >- >- P 1 ^i 1 VO ai 1 rH iH ca ~P ca { j + CD t/] 1 + , — , + d -J- ■ -d- H rH >- o •H II ^S) f" _ 1 II + >- Q ca + >- rH + + P ca -P H CM U 1 CM o + (D rvj + >- i + CM + >- O Gi o CO. (X) + CM + >- P + CM + ca CO + CM + -d- + CM H + >- -d- P M •H O -H rH O •h © iH O r-l o r-Hl_:t kaI -t -J- -J: P i "^ 1 1 ^J J3 1. '■^ II Q p rH M ^-v 1 1 i ! '^J 1 '^i rHJCM l^-^ + + ' + + ¥ K/:CA AHR Ho. I,aG15 Di - Dj = Ii-pD^ (y + 1 ) D2 -D|^ = PD^(y + 1)^ D;l + ^2+ Dj +D|^= PD^(y+ 1)^ + 8PD^ (y+ 1) + l6pD^ = ^(c-^ + C2) ^'1 + D2 - D5 - Dl| = PD^ (y + 1 )^ + l+PD^ (y + 1 ) = ^(Ci - C2) The arbitrary constants k-^, k2, and k^(n ^ 5) are to be deter;nined by the boundary condition at the sur- face (the boundary condition at infinity is taken care of by putting equsl to zero the other set of arbitary constants that normally appear in the solutions of linear second-order differential equations with constant coef- ficients). It Is now anticipated that the arbitrary constants kj^ are independent of n say. Then and equal to k. sin 2r, ^ cosh Zg - CCS 2r] S-2S • o -Us - - e sm 2r] - e ^ si n I4.T] (DI5) where NACA APR TTOo l!lG15 ^'^7 CJ H;l „„,..., '— • H/) H/) « ^ ^ O OJ c .'„ r-i HJT 1 -^O S4j) ft OJ I (D H — t- r ^"^ a) r J + _ >- -i- CO. >- K> K\ >- r-i n Q "-^ -4- CCL JO. K, + Q -t --^1 ft ■XL + + Go. CO i-i 1 OJ OJ + + • • SI 1) >- • r\l H -J CO " — " -* "^ H'\ ^ 1 + + rH ft ca O hf> 1 V. 1 r-> >- < — + CO CQ >3C + Pi -I- + •H ^ >- , t:a. , , ^- 1 J" OJ 1 1 1 1 1 ft j^-~^ >SI. Q C-L .-4 l-rr 1 1 + ^J OJ CO >- 'cj ^:i + Hj) 1 H/1 ~ — •H 1 OJ iD G5 '"n, 1 'o' "i ft ■u r-J CM M h oX JJ >=a. a i OJ 1 Cli VJ rHlOJ o ^ + p vij H ^i. - •rH ^1J- + I CO 'i~ c\l + 1 ixj) $^ K!) J- 1 .— ^ C'J •■H C\! tt ^ — . rH 1 O m 1 ca ,— . rH O O o v^y f-NUt r-H + + + >- K/) r-H > Hl'-a- 1 ^-- 1 1 ' 1 ■ - ~—^ r-J 11 rH >— • ^f^ + >- ■'.'> r-H ■f N'A ft -P m o •\ K/) a CJ. CO ^^ o cr\ -^ ca C') K> (^ cc. CD -1- ft o 1 ' — CM___ + + C_L Ch o rH OJ OJ OJ — J C II -J + o ^•■s. •rH + .-H r i + OJ •H Vy. + + >^ r-\ ^ o" -^1 >^ ~ — + ;h (L) -:. Q ~'Q >- x' S Q . ^'-I rft e - I cos T) - ( 1 f ^- + 1) D^e"^ + is]2+ 12D-Dr(Y+ 1)D +Ii'l >e"^^ ) cos Jil (Dl?) where r.) = _ 1 - ^" sm ^^T] and^ from equations (d8), -^ - '^ \'^ cosh 2d, -cos 2t] - e"'^-^ sin 2r, - e"^^^ sin kr^) + TyVi (DI8) / 5 where Gt(£,) and G^i^) ere t"iven by equations (D12) and (D13)» respectively, and G(|,) can oe v/ritten C-( I) = j- jM --,/jK 5e2£ + (- J + 1 v'?i^-") e"^' - Kk - 1; v'Jk)^ + K^2 - Je"2^ - VJK £e"^^ - I ^ - 2 J - i^. V^ ) e~^^^ ■i "^ ij lo (DI9) with 2 J = pD^i (y + 1) K = Pd2 [(y + 1)D + iij VJK = pD? (y + 1)[(y + DD + 111 NACA ARR No, l);G15 The arbitrary conr-tants k, , kg, and k appearing in tha expressions for Cn , '^2> ^^^^^ '^* respectively, are deteriTiinsd by the boundary condition at the surface of the circul'ir arc profile. The yplue of tne erbitrary circulation Fz is determined by the Kutta condition at the trailing edge - that the velocity there be finite. In order to evaluate the various terjns appearing in the boundary condition, equation (C9)> the f'ollovi/ing relations are necessary: FroKi equations (All) and (A12) y = pfzh sin^^ + Pjb} sln~^ cos'^^j + . . — ^ = -ii-ph cos ^ - 16 ph-^ cos ^^ cos 2^ - ... d:i From equation (312) b = sin^^ - p^(l+h" sin'^.^ + 52h^ sln"^-^ cos--^) sinh £. = 2ph sin ^ 1 + ]spf' cos"^^^ '■"Y. (2 cos'^A + sln^^V, + (2 7 7' ^ cosh ?, = 1 + 2p~h~ sln"-'> + ... e- = 1 + 2ph sin -^ + 2p'^h^ sin -^ r 2 fihy 3 in o o Z' o 1 \\ cos~-^ - p'-i2 003"^^ +sinU 1 - 2ph sin ^ + 2p^h^ sln^^ - 2eh-^ sin ^ .'4. cos-^^ - ^(?L co3-^ + 3irA^)| + . \\ zoz h 2ph sin ^ -I- 2ph'^ sin -^ - p"^ {j- sin"^-^ + 2 cos'^'^ + sin'-'^) + ... sin T', - sin ■^ cos r, (1 + 2p'^h'^ Gos'-^^ + . . . CC3 h (^1 - 2p'^h- sin^--^) -:- ... 50 NACA /3:R No. lI|.C-L5 - Y'^ien the e::pre3??i'^n for y^ given by -equation (Dl6) is substituted into the bounaarv condition, equation {C9)> the coeffloienb of h- on the left-hand side is cos 3^ r ,- -, ^ + Al.y'.'lng (5ca'.a;;\cr,3 .for the fc-rbxtrary constants k-^, k2, tnd k: are obtained; Ki.CA ARR IIo. TJ+GI5 -7 p -- ?0 -s'J T iP (y + 1 ) + 2D-^ (y + 1 )'' + fi^ D^^ (y + 1 )^ (D25 ) 4d i-f-k. +2k2 -k)=- ^^r(Y+l)D+l|l + 4--lC -^ D + f d2 P p^ ^ 3 C P ?^ D" (y + 1 ) - ^ iJ^ (y + 1 ) - ^ I^^ (y + 1 )'" 9 nr 16 d^(y-^i)'- (D2i;) -r--2k2 + ^k) = -?-r(Y+l)I^ + U~i + ^+o + l|D- uD^ ( y + 1 ) 1 , _2 . . -rP lY + - ) - ^ r^^ ( Y + 1 ) " - ^ i^"^ (y + 1 )'^ (1^25 ) Note that the si-iti of equations (D2li.) and (D25) ^J-ields equation iL2$;, so that these equations for k-j , kp, and k ai'e not indsnendont . rfence, one oi' the constants, say k, ^^. eni:,.U^ely arb.lt-:-ary. It \;ill be seen in the follov.ins discussion ohat the arbitrary disposal of k is n-'cessai^^r In cr-dei- that no ini-'inite velocities occur on che cl'T'cula-" src profile. The velocity components along the X and Y axes are given by .¥_ 1, = -' - 2 / • u r ^9^ X y • ^^ u =- — = STuh £, COS T] — — • cosh 9 sm r\ —- OX coah c£^l-cos 2ri \^ 6c, 6 ^) (D26) .6} d 2P ■^'•^ cosh 2c,- cos 2ti \ cosh c sm -n r^-+ cmh c cos n r — j ^5^: ^) J NAG A ARR No. L:VG15 55 A.long the chord of the circular arc profile, 5, = 0; equations (D26 ) therefor'^ bscoine u = V = bC^ •\ o sin ri tt] i3 S^ sin r 62, ^ (r27) J By means cf equation (Dlo) for ^ and the expressions for 0^, ^2> ^-^^^ 5^7, J ^t follov'/t; esslly froiii equa- tions (D27) thot u = - 1- %- sin T' + ^'' < 12 COS 2t] + D \^i + i)'D+ k] (2 P -- - - [_ i2G]_(G) cos 2n + LG2(0) cos Ur+T^ COi -1)^ rP m r L_ b3 r !in ri G(0) -2 cos 2rj - li cris .^rj - cos rj c:OB 2n ! , + ^ + . . . 2 sin'^'T, I (D26) 1 1 2 'S V = L|h cos rj+i4.;'h (2d + 3 ) sin 2ri-2|3h''^ cos Tj \a:-/Q \>'i:7jVj+ sir J^. sxn^ri - 1 - 2 cos 2r) + 2G(0)(1 + !i cos 2r,: + . . , (D29) |,t the trailing edge, r, =^Tr or sin tj = C). ilence, according to equations (d2G) ana (C^;-;), an infinite velocity seems to occur thore. The Kutta condition at 51+ KACA AHR lio. i,L;:;i5 the trailing ed^e, hcvvev^r, denands that the vel.oclty be rinite. ?rom equal lous (D£2) 11 is seen that (^] = so that the velocity component v is finite on the boundary. The velocity oomponent u c itn be rendered finite by showing th^it the coefficients of — ~ — in 3 in r, equation (D28) nan be made to equal zero when r; = or TT. Thus, since the constant k occurring in equa- tion (D13j is a^'bitrary. It can be chosen so that G(0) - 0. Again, if the fii'st coefficient of . 5 in equation (D2G) vanishes for r. = t^, then the sin T] circulation consuant r^ = -20,J0) - kG.^{0) (D50) wher:; G;l(0) and Go ( ) ai-e given by equations (d22). The arbitrai-v constant k lias b'\'.cn determined by the condition G(0) = 0. ji^'rom equations iD'^2) , therefore. p - — r D [^ + 1) - ~ O (y + 1; + D 1- Ic ^ (Dipl) and from equations (D25) and (D-~'5)> respectively. 20 .. L 2 k-] I". - -. I + ^ d2(y+1) + i^ D^ (y + 1 ) + 2D? (y + 1 f' + ^ r'"' (y + 1 (D52) k. T=--iL<->-" D + 3 .2 , -iT,2 2D + D"^ + 2D^ (y + 1 ) - D^ (y + 1 ) + 7 D^ (y + 1)' +^ d'-Cy + i) IB (D53) KACA APR No. L'i-Gl^ y-j Note that, Lf d tLe Inccmprossnble flow past a cir-oular arc pj:=ofll'.i been det3I>l.^in3d according tc the methods of the present paper, a discasaion oiriiilar to the foi*egoing v/culd have been necessary, w:'th che re.^ult that k;j_ = 8, kg - ~k, k = 0, and Tz, = 0. Substituting from equaticns (D21) for C-i(O) and Gp(0) :nto equation (DJO) gives r - - f • pD - ^ PD^ - ^- pd" (y+ 1) - — pd' (y + 1) ■^ y t 'J 2,1 I. ? 8 ~ (^54) The circulation T- in the incorapre;ss ible case, obtained from equation (>.5)> -^ l4.TTTJa = c:n The circulation To in the ccinpresslblo case, inclusive of terras containing tiie third povrer of h, is obtained by adding the circulation term froii: equation ( E9 ) to the vaxue 01 given by equation (D^li) and iTiuitiply/ing the result oy iiTrUa. Tnus, If D is replaced by liTrUa P h + + -^ (y + 1) 12 I 1 _ •^^- 1 .-^-- A^ ^ P-) I h^ V _i The clrculetion correction forpula then becomes ^ 3 1 — r_ -f. _ ( Y + j_ ; -^^ — ~-~i~- ( o + :; p ) P^' ^4 p ^ '' n r2 ' (r55) (D56) 56 NACA mR No. Lii.Gl The first term on th*:? right-hand side is the familiar ?r&ndtl-Glauert terra sc that the second term represents the first dep&rbviie from tht. Prandtl-Glauert rule. The magnitude of the velocity at the surface cf the circular arc profile is calculated by the use of equa- tions (D26). T]iUG ^■^1 2 q = 1 + h --^ + h ex :^ /a ^ \2 "2 V^y / ^ ^x + h^ p2 '^)2^i /c^^; 2 dx 2 ^i 5i^2 + ^^'v ^>/ cy c; oy (1:^57) v/here, symbolically. d_^ 2 dx cosh 2cl - cos .ir. 3 Inh S cos T] T o^: - cosh b sin r\ ^ — ( cosl"; c sin Tj — + sirih c cos rj t — oy cosh 2£, - CCS 2t] V "S C^r, and the expressions for 0,, 0^, and 0^ are given by equftions (By), (Dl?), and (Dl8). \Vhen all the functions of t and T] are expressed as functions of ■^ at tlie sa^fece of tht; profile and terrus In-volving pov.'ers of h higlier thf'n the thxrd are reglocted, the expression for q becomes NAG A ARR IJo . l1i-U-15 57 a? •r-l .^^ o f^i o !m ^-^ OS P •H CO OJ I + ruWii rj 0) OJ + + + I i _^_, + m o J ! + 0-] H + ail ^ ,_.i_.J cts aj V:, c CD i£L \_rr + II + CD O (1^ C\l O ro> o C5 n + f. ) + \1 + OJ n I -|J o u p + H CM + CAJ P lOvJ c xl .a £1 ■P .f:l -p ;>-. H P 0) ..H > O -IJ H O > ft (D P O Od . m (D Cd ■T3 O -P a .o 0) © (H ^1 ft 4-' Si O CI) o 4J C 0) OJ 'tJ .q OJ CO CO H ■^J CD 11^ Cd .-O ?-; •'-t O O 0) a o^ CD C! ?.-, o fiH & 4J MOO O CD i^ Q) Ph P O Cf to CO I O O CO I rj W 58 NACA ARR No. lLG15 u -. X Q) f-i -P aj .a -H -H s 4J CJ O o ■"-» ^1 !=I O 1 ^_^ 1 •Vh (U*H fr^ o .a -P -P -H • — VII ty w r-: "J o 'J3 -^ .^! •> p. . n t: VII 0) ' — • fn II 4:; IJ^ 0' -P ■ \. s ■=* fH P © iH -d- ?H M fc • — ' •« CO f^ n VII® 'H \ ■JU — - tiO-P 1 — ! «♦ 13 cd 1 1 fj '-Dp VII _ 0^ r-J ^4. ^'^ .■H hO 0) ^ -.-d^ r-l p j= a X -^3" as 1 -H (>» >- ^ -H ^/ 10. + a* '^ H- 1 Ti 1.0 C rx G !h 0) rj^ £^ Ci -P > 'x "-^ ■ — i + J^ J-i bO + VH ?H I— 1 CO G 1— ' en. 0) fn rt Cd "•-^ ^ E! (.d -H oO .-TN •H rH TD CS 4- i3 ■:$ C3 •» -P 0) OT t^lro UJ ,0 ;< rH -H II t ' -.1 C ^5J <^_ on ..;:3 "H Of ^ •H 1) -p cr" -P in 0* O* 4i^ + 5 ii. 0^ 1 CT* o I 11 4- O o C! o •H -P t\ m o P, O (X, ro» N. CO KACA ARR No. L;lG15 59 REFERICNCES 1. Janzen, 0.: Boitrag zii eiiier Tlieorie der stationaren Stro-fnung komnressibler Fliissigkeiten. Phys . Zeltschr., ll; Jahrp;., IJr. ll^, 15..July I913, pp. 639-6^3. 2. Rayleigh, Lord: On the Flow of Goxnpressible Fluid past an Gttstacle. Phil. Mag., ser. 6, vol. 32, no. 187, J'aly 191''^> P^« 1-6 • 3. Fcggl, Lorenzo: Campo di velocitk in una corrsnte plana dl fluido conpresslbxle. Parte II.- Gaso doi proflli ottanuti con rapprssentazione conforne dal cerchlo ed in partlcolare dei profili Joukowski . L'Aerotecnica, vol. XIV, fasc, 5» I'ay 193^, pp. 532-5I19. '(.. Imai, Isao, and Aihara, Takasi : Cn the Subsonic Flow of a Compre3sibl3 Fluid past an Elliptic Cylinder. Rep, No. 194 (vol, XV, 3),\A3ro. Res. Inst., Tokyo Imperial Univ., Aug. I9I1O. 5. Kaplan, Carl; On the Use of Residue Theory for Treating the Subsonic Flow of a Gorapressible FlLild. NAG A Rep. lie. 726, 19k2. 6. Frandtl, L. : Remarks cn paper by A. Busemann entitled "Profilriessungen bei Geschwindigkeiten nahe der Schallgeschv/indlgkeit (im Hinblick auf Luf tschrauberO," Jalirb. Vv.G.L., I928, pp. 95-99. 7. Glanert, H.: The Effect of Compressibility cn the Lift of an Aerofoil. R. & M. No. 1135, British A.R.G., 1927. (Also, Froc. Roy. Soc, (London), Per. A, vol. ll£, no. 779, March 1, 1923, pp. 113-119). 3. Ackeret, J.: ITticr Luftkrafte bei sehr grossen Gesch'.virdigkeiten insbescndere bei ebenen Stroin^ungen. Helvetica Physica Acta, vol, 1, fasc. 5» 1923, pp. 301-322. 9. Kaplan, Carl: The Flow of a Gor.pressible Fluid past a Curved Surface. NAG A ARR No. 5KC2, 1943. 60 l^ACA ARR No. Li;G15 10. vcn l^armdn. Th.; Conpressibilltv Effects in Aero- dyr.airics. Joui'. Aoro. Sc;i,, vol. 8, no. 9» July IQlfl, pp. 3,^7-556. 11. GarricV:, I. E., and ifeplin, Carl: On the Plov; of a Co"mpre3siblo Fluid by the Hodograph Method. I - Unification and Extension of ?resent-Day Results. NaCA ACR I'o, rJ+C^li, 19i|lj.. NACA ARR No. L4G15 61 TABLE I RATIO OP OIRCnLATIONS FOR COHPRESSIBLB IMB MCOMPRBSSIBIB PLOWS Q/R \v. Ml Approxi- mation^-^ 0.10 0.20 0.50 0.40 0.48 0.80 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 Prandtl- Qlailert 1.0050 1.0206 1.0483 1.0911 1.1198 1.1547 1.1S74 1.2500 1.3189 1.4003 1.5119 1.6667 1.8983 2.2942 b = 0.010 Third von Karman 1.0050 1.0050 1.0206 1.0206 1.0483 1.0483 1.0912 1.0911 1.1200 1.1199 1.1550 1.1548 1.1978 1.1975 1.2508 1.2503 1.3172 1.3165 1.4029 1.4013 1.5170 1.5135 1.6788 1.6692 1.9348 1.9025 2.4603 2.3030 h = 0.015 Third Ton Karman 1.0061 1.0060 1.0207 1.0207 1.0484 1.0484 1.0913 1.0913 1.1202 1.1202 1.1553 1.1553 1.1984 1.1983 1.2517 1.2511 1.3189 1.3175 1.4059 1.4026 1.5234 1.5152 1.6940 1.6720 1.9804 1.9078 2.6680 2.3142 h = 0.020 Third Ton Karman 1.0061 1.0051 1.0207 1.0207 1.0485 1.0485 1.0915 1.0915 1.1S06 1.120S 1.1658 1.1557 1.1992 1.1987 1.25SO 1.2520 1.3212 1.3187 1.4102 1.4043 1.5323 1.6179 1.7153 1.6763 2.0441 1.9153 2.9588 2.3301 h = 0.025 Third Ton Karman 1.0061 1.0061 1.0207 1.0207 1.0489 1.0485 1.0918 1.0916 1.1309 i.uoa 1.1565 1.1560 1.2002 1.1996 1.2547 1.2531 1.3242 1.S20S 1.4158 1.4066 1.5438 1.6213 1.7427 1.6818 2.1262 1.92S0 h = 0.050 Third von Karman 1.0081 1.0061 1.020e 1.0208 1.0487 1.0486 1.0921 1.0920 1.1214 1.1812 1.1572 1.1570 1.2015 1.2006 1.2568 1.2544 1.3278 1.3222 1.4226 1.4094 1.5578 1.5256 1.7762 1.6885 2.2264 1.9370 h = 0.055 Thli;d ^ Ton Karman 1.0061 1.0061 1.0208 1.0208 1.0488 1.0486 1.0926 1.0922 1.1230 1.1316 1.1581 1.1575 1.2030 1.2017 1.2595 1.2560 1.5321 1.3245 1.4307 1.4127 1.5744 1.5306 1.6187 1.6966 2.3449 1.9514 h = 0.040 Third von Ki^rman 1.0051 1.0061 1.0209 1.0209 1.0490 1.0488 1.0929 1.0926 1.1336 1.1230 1,1592 1.1688 1.2047 1.2031 1.2621 1.2579 1.3371 1.3271 1.4400 1.4166 1.5936 1.5364 1.8613 1.7060 h = 0.045 Third , von Karman 1.0061 1.0061 1.0210 1.0210 1.0492 1.0490 1.0934 1.0930 1.1334 1.133a 1.1604 1.1699 1.2066 1.2046 1.2653 1.2600 1.3487 1.3301 1.4806 1.4209 1.6153 1.5430 1.9130 1.7168 h = 0.050 Third von Karman 1.0081 1.0061 1.0210 1.0210 1.0494 1.0492 1.0939 1.0936 1.1343 1.13S8 1.1617 1.1611 1.2087 1.2063 1.2689 1.2624 1.3490 1.3336 1.4623 1.4288 1.6396 1.6505 1.9708 1.7290 h = 0.060 Third , von Karman 1.00S2 1.0052 1.0212 1.0210 1.0499 1.0496 1.0952 1.0950 1.1363 1.1260 1.1648 1.1640 1.2137 1.2102 1.2775 1.2679 1.3636 1.3413 1.4698 1.4373 1.6958 1.5681 h = 0.070 Thljd , von Karman 1.0052 1.0062 1.0214 1.0212 1.0505 1.0500 1.0967 1.0960 1.1285 1.1281 1.1685 1.1673 1.2196 1.2148 1.2871 1.2744 1.3808 1.3607 1.5218 1.4511 1.7622 1.5895 h = 0.080 Third _ von Karman 1.006S 1.0052 1.0217 1.0215 1.0512 1.0510 1.0984 1.0976 1.1312 1.1289 1.1727 1.1711 1.2265 1.2202 1.2988 1.2820 1.4007 1,3616 1.5589 1.4673 .... 1 h = 0.090 Thljd ^ von Karman 1.008S 1.0082 1.0219 1.0217 1.0520 1.1003 1.081S 1.1001 1.1342 1.1S40 1.1775 1.1756 1.2342 1.2263 1.3144 1.2907 1.4232 1.3741 1.6011 1.4861 h = 0.100 Third von Karman 1.0064 1.0065 1.0222 1.0220 1.0528 1.0S26 1.1025 1.1020 1.1S76 1.1S70 1.1828 1.1804 1.2428 1.2332 1.3268 1.3004 1^.4484 1.3883 1.6482 1.5076 NATIONAL ADVISORY OOMMITTEB FOR AKRONAUTICS NACA ARR No. L4G15 62 TABLE II RATIO OP VELOCITIES 4T LBADIHO OR TRAILING EDCE FOR COMPBBSSIBLB AND INCOMPRESSIBLE FLOIIS 'e/'l Approxi- mation \ 0.10 0.20 0.30 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.96 b ' O.OlOj (ll',j;,5t ' 1/1.0004 First Third 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 .9999 0.9999 .9999 0.9999 .9998 0.9999 .9998 0.9999 .9997 0.9999 .9996 0.9998 .9994 0.9998 .9991 0.9S97 .9985 0.9996 .9973 0.9996 .9939 0.9991 .9757 h « 0.015; ( m a o a o > o 9 fi ca < ■< ^ ti > !a H r: CO o B > O > d tvj OJ t- o 5 ^ OO CO CO o 4 00 (M (M Q Q t- ir. 5j lf^ KN CO O ^e r-t o ITS CO 00 u~. t- t\l ir» K\ "f ir» -=f -3- i\j KN r- o i3 CO K\ u> iH O <\j O t~- (H CO CO K> H\ "^ ON N> UN o f- -d- K\ o o *^ ON cr- iH ^- Ov ON o tvi IfN -d- si> if\ a o t- C7N ir\ iH :3 • ON ir\ CM o CJN CO C^ • J ° t- • rr» tTv K\ K\ K\ CM CO CJN o o o -d- CJN ON NO NO ^ -=1 rO c~ 00 K\ o -3- ^ KN a CO CO t- •H J- o cr oj w o ocotH_d-a-rr\K\c^mt\j crc^o OO ir\ir*o ^--4N^-HOO OC (T* OlX^K^OJC\J»-^»-^r^rH^Hr^r^OO iH OOOOOQOQOOi'^OU-sO r-trvifrN-d^irN^^r-mo-OOr-ti-iry O .H -I rt .-H rH NACA ARR No. L4G15 66 TABLE VI VALTJBS OF CRITICAL STREAM MACH NUMBER FOR VARIOUS VALUES OF CAMBER COEFFICIENT h Ml Approximation First Second Third 0.02 .04 .06 .08 .10 0.848 .770 .716 .670 .625 0.832 .746 .682 .628 .585 0.825 .758 .672 .620 574 TABLE VII VALUES OF MAXIMUM VELOCITY FOR CORRESPONDING BUMP AND CIRCULAR ARC PROFILE «lmax M Camber coefficient h Itilckness coefficient t 0.02 0.04 0.06 0.08 0.10 0.052 0.100 o.iij5 0.186 0.226 1.0615 1.1659 1.2527 I.5IJI5 1.4520 I.C816 1.1660 1.2527 1.5414 1.4520 .2 1.0834 1.1701 1.2597 1.5520 i.[lIj.66 I.C854 1.1701 1.2595 1.5515 1.4454 5 I.C859 1.1759 1.2695 1.5668 1.4675 1.0859 1.1757 1.2689 1.5651 1.4641 4 1.0899 1.1851 1.2855 1.3915 1.5024 1.0900 1.1647 1.2840 1.5876 1.4950 5 1.0960 1.1997 1.3116 1.4324 1.5627 1.0959 1.1988 1 . 5084 1.4245 1.5467 6 1.1056 1.2259 1.5572 1.5078 1.6780 1.1052 1.2217 1.5492 1.4879 1.6-575 7 8 1.1225 1.1594 1.2705 1.5='79 1.4550 1.6780 1.1215 1.1557 1.2640 1.5701 1.4298 1.6197 9 1.2055 1.1960 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS NACA ARR No. L4G15 67 § I M a o tn fa CQ O s »9 Eh « 01 CO M o M O o fa o M CO * CM •* CM 00 o 00 00 CM tO 00 ^ CM 00 r-i CM oo m 0> i» 00 c^ in 0> KJ 00 * c- > ^ t- e- o to rH 0) * UJ W lO «o CM •* Ol o C3) lO •'f CM 0> Oi w o t- o 00 t-t 00 o CJ> to t- CO m to rH o 1 iH CM to 00 w o> rH o 01 to * to rH 1 t 1 1 rH rH fi w 00 c^ 00 to CM at 1 t 1 1 rH 1 to o rH 1 1 1 o * 00 . o CM (0 00 w CM a* o to 10 c~ 00 00 00 lO o 0> lO w rH CO o M to ^ a* 01 CM 01 01 o CD t- C- (O •* rH CM CO o CM t~ CM lO to 00 o> • 1 1 1 • CM ^ CO rH rH CM to to to to 00 O at o * O lO 8 ^ lO ^ rH 01 o CM CM > O CM r- r— » o 00 fi t- a> rH to to C- O to 8 1-1 iH 00 00 o o CM 00 rH t- o 01 lO (O o CO ^ lO 0> •* o C- r-i OI rH CM o» to ■<<' CM 5^ o o (0 CO to KJ CM lO O rt rH CM to to to to ■««< ^ ■* ■* Mi «0 t- CO O CM to 01 lO t- lO o> 1 1 1 1 1 1 1 ' rH 1 rH * rH 1 01 1 CJ in 1 CM rH CD 1 ^ to c>- o CM o e- 00 C- (0 to o to rH ^ «o 00 o s ^ o rH to Ot lO rH to to lO to o o o> o o to o> o 00 O to to rH o CM •* t- rH ^ c~ o oo o rH to fH CJ> 00 > to 4 o iH "^ O) rH rH to 01 O to 01 CM o «o 00 CM CM CM 00 CJl o to rH t- to to to O <-l o CM CO (O t- t-i CO o to O <* to o> f rH « o o CJ iH to ■* «o W «o in o> o rH to CM ^ 0> CM CM 8 c- w rH o t- lO o> rH O lO 0> rH to to CM o (Q «o o lO CM to CD lO to t~ ^ lO C~ 00 ^-^ o Ki K) ■«t* c- (0 00 ■* c- rH o to rH rH a> CM O o o H « > rH «o tfj o> Ui >♦ CJ> in o» t- o> N ■>r * ■* ^ V lO lO to t- o J5 lO CM f- 00 O O 1 1 1 • 1 1 1 rH 1 rH 1 01 1 1 10 rH 1 lO « •0 CM o 00 o (0 00 CM rH lO lO ^ o> 00 CM •* CM Ok o •* ■* w c- OJ o CJ> flO to 0» lO O to o (O c- CO (O t- a> to to CO H CO 01 o> lO •♦ ^■^ o 0» CM ^ c- ^ t- w 10 t- <» to on ■4< lO 01 O o o •>«< t-{ to ■♦ CM lO .-1 (0 CO 00 o» o CM ^ c- to •* «o CM to o 0> o cs rH rH rH rH CM to lO o rH CM 01 to CM CD to to to o C- o 00 CM to o> o o 00 rH 00 o to >-i l-i -^ C» to «o IC to c- C~ c- to rH <*• o rH CO o lO •O to CM rH o tf) c- to to to p rH •* a> o> in lO lO o (0 lO o O o rH CM to •««• m t- 0> CM c- to CM CM o rH l-t CM ^ 0> o o> O ■* CM 10 to to o to •* ^ o CO at lO o CJ> CD CJ> lO o O iH 8 o rH * 8 r- CO CM o •* OJ lO If rH «) to CM ca o o> c- lO rH at (£> 10 o lO rH to o CM to rH o • 0» • • • o> • 00 • 00 • 00 • CO • • • to • to • • • n • lO 2 lO o lO o to o to o to H CM w ^ * lO «> to e- £- 00 00 o i> o CO o Ola CO o §■ I-IO <;fa Is O O NACA ARR No. L4G15 68 •0 o @ • o g II M a (4 § « X M g M Cl, b s & o s H 0< e-« •4 p M 53 o M p^ e 3 PQ ^ H C") « « f- H V} O M a o ^ M 8 0> (O CVJ l> CO in CM l-i t- c;> ■<1< CM t- •* rH ^ o» IO c- IO CM (0 c- •^ 8 • c~ o t- IO <-\ OJ 00 E- to it> IO o • o 0> • CD • CO • CO • • • « • • • • o t- t~ 00 ^ c- 00 00 CM r- o CM lO (O cu Ki r-* c> •* CM fi t- CM s o> o • CO 1-t Oi «o •* Oi r-\ o o> c» 00 u o o> c» CO 00 CO CO CD CO t- c- t> > d • • • • • • • • • • • ■ • V4 o (< 9 n o o c» t^ ^ IO CM CO IO »o o> o ^ 10 0> to 0) M CM t^ to c- rH c- * •* 3 • 00 CM g t- lO 8 CM t-i r-i o o o o Ok Ok (0 CO CO CO CO 00 00 00 o o CO rH 00 t-\ «0 e- ■* ^ CM CO o o o t- t- 00 IO CM * CO •^ r-i rH 0» 01 o t- ■o •^ IO CM r^ l-i r^ r-i = o 0> • o • • co • CO • 00 • CO • 00 • 00 • CO • 00 • CO • o> 00 rH o f-K IO ■«1< o c- to Ot ^ c- ■* w «D 00 to lO c- fi CM to «# o* • c- (O o to rH IO 00 r-i f> "* IO «o o • o o • r^ • rH • H CM • CM • CM • V) • H • rH • rH IO • H »0 • rH o * o r\ fO ^ CM t- •0 ■* CM 00 • M) w o C4 «o CO •0 •0 O) s 0) 10 s o • s « » •o r^ CM * Hi m o o o rH f* l-i M CM CM CM CM CM Y o iH rH r-\ f-t r-{ rH rH r-i rH rH r^ 3 (X o U) CM (O CM ^ O CO ^ r-i O lO P< •o 0> • CO K) CO CM to t> 00 o r-i rH CM CM o w o o rH 1-1 r-i r-t CM CM CM CM CM o f-\ rH rA r-t rH r-i rH rH r-i r^ rH o iH iO to IO O IO CM O CM CO O o •* o «o CM CM c» OS OU ■<«< t- 0> o 0» lO CO rH ^ to t- 00 c» o o O o» o o rH <-* i-t t-t r-i r-i CM CM CM o l-l rH r^ t-i r-i t-t r-i r-i H i-i r^ >-*/ / lO O s o O o O o O o 0> o> c- lO CJ lO lO « OJ <0 iH M to iH to r-l lO M rH O O 0> to O) to OJ OJ U3 W CO > 'J' lO ^j- I^ to r-H C0i-l»O lO > CDCO t^ (D lO rH O rH OJ C- 'J' OJ to 01 to OJ O lO CO Oi CD 0> 0> Ol (7> ^ to CJ rH O m ^ iH ^ f-H OJ > O O O Ol CD o o en o> rH CM CM (0 (J> to CO 00 lO to r-t (7> lO CO ■^ o» o o> ^ a> CO (D to lO ^ lO to C*- {D I I I I I \r> t^ IT t' 'V cy oj to c^ t^ o> O ^OJ V O) C7> ^ Cft to iH O O <7> Ol rH <-t fH to to o to e- tJ> tOtO CM CO O -H C- 03 ^ t~- O CVJ ^ to 00 CO t- to *o lO »H lo cy ^ (D O ^ ^ ■^ t- t- (O t- Ol e- 00 o> o» CJi ^ W 01 rH O Ol CM CD CO lO CM 0> r-t O rM rH CM o o o o i-l CM lO "»• > I I t I rH O CO O ^ to CJ> ^ to CD ^ CO V c- c- O 0> 0> CD t* m tn to t- 00 I I I Ol t- ^ O CJ> <-t lO lO lO o t^ C7> lO »0 01 O to 01 C- f-H •H O O 0» 0» CO ID ^ CD to rH CD Tj" 01 C- r- CT> C- O CD U3 CD*H Tji in CO > r- to u> 03 CM C7> rH to iH O to to 01 to ^ ^lOCMr-t O »H to in to tt> rH r-t r-i O CM ^ to O O Q O i-tCMrt-* O I I I I tn « to 00 CO to 01 ^ CD O r- 00 c- ^ o o o o o o lO (0 t- CD 0> I I I I I to CO 01 (D OS ■* CD CD 01 01 ■*C-0 lO lO CDt-C- to lO OlOl ^ « to OiOl O CO o OOH O lO 0» OCO OO Ok 8* O C3» OOl CD O lO « o> o oo o O I I I I 03 lO CO to O o rM CO * m lOt- 0»rHH HHH WCM I I I I I O Ol (J> c^ o C- ♦ iH C» O t- 01 «-t lO O Q »0 lO O to « -H 00 in CO CO lO CD to r- CO o» o> * to CM fH O (O c- to m O 03 <-* f i-H to CD to O O O fH »-4 01 lO ^ ) I I I I OCM"H oo * ^D^• to O c» iOr-4 e- CM t- rHOl 10 00 .H O » O 1" o to »n o CO iS >-* t- to t- o o 01 a> CO to to to CM to ^ * CO to C^ oo 0> 01 ^n OJrH o CM lO to C71 CM to Oi C- rH ^ (» to oo O >H ■-•CM lO V O I I I I o loto into CO (O 9 (O tn ■*lOOI CM^ 01lO« lOtp lO tor- 009 I I I I I in to CO o t- O CO CO CM 06 •^ O (H C- to to O to tH to o o Ol a> (D •H rH ■* O r- to ■* * to o rH o ■* n CO fH 40 CD "H 10 CD t^ to to in O Q to 'f »-t •H * c- (D m CO 10 >o to CO m t- 00 Ok 01 ^noi^o (A CD to to K) lO to 0» rH m <-( Ol O O <-( rH <- O V * Ol to en in to rH CD 'I' o in O 0> (ft 0» CO O Ol 01 * * to C- rH CD 01 CM t- 00 lO O ^ c- o to CO c- to to lO OrH Ot- to 2 CD mo) to H (O *0 CM O O rHOl •HCM (O * O I I ( I to CM « 01 <«i * OOlO Oi O »o ^o c- o» •o toe- cDoi I I I I I t- to cji 10 r- flOfOCM r- r- in * c- ^ ® O C- lO Ol * O Ol Ol 00 CD •Hcof- r-c- io o lo rH 01 10 « »H 10 Ol (3> (O C- O 01 c- r- to«)in r- rH 03 10 01 •HOllOt- CM 01 Olio Ol CO to O CO 01 Ok *(0 CMrHO 00» rH lO t* to t* in iH to « m O O rH CM rHC5 to ■* O I I I I fD<#0 rHOl Okfr-V io«o §«N Olio to t- OfcfH tDt« OOO I I I I «H 1001 oiod) a» to 01 00 10 (K O 9 00 CD 01 ^ (H «n t- S^ o to-* CD to 00 to OOOl <0 0»CM C^ t- «> *0 lO lO rH 10 Ol Q C» CO CM 01 CD '<«■«> (S C3I 01 «I0 01 rH O lO to lO 10 OOOl O Q rHC^ «>eO oo rH« rH 01 rt * O i I I i O tOOkOOM Qt-Ot'lO I I I I rH I too Ol to CM CDC- <-* to CM 01 CM r- to o coin rH c- to 0> O Ol CO O 0» * O lO (O CO CM rH V to C- CM to OlOl c^ c^ to in lO CO 0> to ^ c» 01 e^ I* CO r- rH in c7> ■*■ CM C- to oo I--IH lOOkCM r- c^ « toio rH rH O lO £0 CM CM to 'f CO lO lO O "H c^ -^tO CpAOl ^lO^i-tO CO V to o 01(0(0 01 CO CD CM OOrHiO i-tOI 10 ^ • • • • O I I I • 01 rH lO lO^ ^«*o» o o» SCO '^ tftrH 0) (» OrH 0)01 (D (X) CD 01 *0 lO c^ to lO to CM O CO to CD in r- lO ^ rH CD ^ O 9 C}» CD CO CD rH O to to V t- O * 00 rH Oi CM Ol rH Ol to rH f ^ Ol OlOk CO ooc« lOO lO o to c- o c- o r- lO rH t> ^ C3* lOrHt- or- r- tool (D 00 CM 00 Olio lO to O ■* CO rH t~ t- to lO in ^ CJ» C- to rH to to O CD to in CM ^ O CM lO O * CO rH c^ c^ to lO m lO rH O OlrH 'f C- O CM C- •H CO rH CD O m o> 'J' c- -H C^ to to lO v^ §iO lOO t- c-o » * t- rH t- tt to lOiO S« ^ rHp o» oicok rH CM OtOl ^ to r> Ok c to CJ> lO to rH 'T CM CM to to Ol rH (D O C- tO to C- Ol ot- to to t> 0» Ol ^iQOItHO to O tf^O lO e- o e-Q t- lO to C- OtOk ^lOOlrHO too « lO CM O rH to N A O «0 OOOJ to «HOI to ■* (» I" to c- (O in ^ o 01 C» rH CD O O CM to ■HOI to -<»■ (O to 01 to ■* CO 01 ^ oi o> oi o> oo 01 to rHOin « O t t t I HlOO «IQ CM imn CM Ol lO CM C- CO to o» lo in o o lO CDrH lO 01 lO tOCD Ol O I I I I rH rH rH to t* to c- c- ® in ^ rH 00 O t- 0> to 00 01 to Ol lO tDOOOt O I I I I iH I I I t rH CO ■^01 lO ^ rH O Q ^ C71 'tf t- to (O in CM lO CM to iH CD in H f- CJlOOrHCM CM I I I I I rH m o «H to CO -H ^ in o c- ^ CO C- 01 to CO to -^J" CM O C* to 01 O rH CM CM to CD o> in in o to to 01 in rH NAL AD E FOR A VISORY ■RONAUT CS /.o M, Plgxu?e 2.- Continued. NACA ARR No. L4G15 Fig. 2f 2.4- 2,2. 2.0 AS /.€ rc/n A4- A2 /,o 3 11 1 ' Prnndfl - Glouerf \ ji 2 von Kormon J Hi ISUh ^3 at pre sent po/. ■>er 1 .( f}h-- ''0.0i >- / 1 1 1 /// 1, / / V y ^ .^ NAT :oMMin ONAL Al EE FOR / IVISORY ERONAUl ICS .8 /,0 M, Figure 2.- Continued. NACA ARR No. L4G15 Fig. 2g 2.^ 2.2 2.0 /.a /.6 /^/n /.4- /.Z /.O i 11 1 Prandfl - ( Zrlouerf 1 Z von Karmon 1 3 KG. SUIL ^ or^ ores enr ^ pop er ( g) h = ox v. 1 1 /// // / / V y ^ ^ NAT COMMin ONAL A EE FOR ; )VISORY iERONAUl ICS o .8 AO M, Figure 2.- Continued. NACA ARR No. L4G15 Fig. 2h 2.^ 2.Z 2.0 /.8 /.6 ^//I- A'^ /.2 /.O 3 2 / 1 1 1 1 1 1 / Prondfi - Glouerf 2 von Karmon t 3 Results of oresenf Dooer h 1 / / / n (f i)h-- --O.Oi ^. 1 / // // / /// 7/ / / /// /^ ^ ^ f ^ p* COMMr TIONAL fTEE FOF ADVISORY AERONA JTICS .8 /.O M, Flgiore 2.- Continued. NACA ARR No. L4G15 Fig. 2i 2. i 2 / Results of pre sen f , 1 1 1 er (i ;)/.= 0.09.. 1 1 1 / / / '// / 1/ / t /// / V ^ V • ^ y^ ( NATK OMMIHE )NAL AD E FOR A i/ISORY ^RONAUT cs ^ -^ .6 ..9 /.6> Plgvire 2.- Continued. NACA ARR No. L4G15 P^ig. 2j 2.^ 2.2 - 2.0 A8 /.e /.z /.o 1 2 5 1 1 1 1 Prondfl - G/ouert von Kormon OOpi _3 I 1 o r\ Ov^ »i— » 1 1 -^ ^. f. ^§ ^'V^C ^1 II h- • j ( J)h^ ^O.IC ) , / ' » / / / / // / / h /// // ^ ^ y ^ ^ NATI :OMMITT ONAL Al :e for / IVISORY ERONAUl ICS • ^ /.^ ^/ Figure 2.- Concluded, NACA ARR No. L4G15 Fig. 3 i^lu 2.1 2.6 LS Mr ■■•%\ .6S .m / 1 / / Li 13 11 U Z.0 /.? /,8 J / / / / / / / .75- / / / / / / / / y / ^ / / 7 ^ ^ y^ .10 A 6 I.J ^ -^ .io5 14 ^_- ^___^ ■-^ .loO /•3 .55 l.l .50 .4.5 II .40 .50 ■10^. 1.0 .10 -01 .01 -01 m -OS , -Ob -07 -OS •0'? -10 h Figure 5«- Ratio of circulations for compressible and incom- pressible cases as a function of camber coefficient. NATIONAL ADVISORY COMMIHEE FOR AERONAUTICS NACA ARR No. L4(115 Fig. 4a .28 .lb •Z4 (a) h-O.OZ.. Approximation 3 Figure I4..- Ratio of velocities for compressible and incom- pressible cases as a function of stream Mach number. NACA ARR No. L4G15 Fig. 4b }}-' 'maY. .24 ■II ■10 ■18 •/^ •/4 ■II -10 •08 ■Ob '01 6 (h) 1 \ 1 Approx im of ion \^ 1 Sonic\ 1 // 2 v/ /' M / rV / X / f/ \ y y / ^ x>^ r COMW ATIONAL IHEE FO ADVISOf R AERON Y HUTICS ./ .2 J .4- .5 .b .7 .6 .9 /.O H Figure I4..- Continued. \ NACA ARR No. L4G15 Fig. 4c .52 .30 '18 'lb •Z4 ■II '10 '16 ■lb /max ■12 ■10 •06 .Ob M M App roximoti 3 on [0 h '0.0 A, , 2' Sonic // 1 1 // ^ / \ ^ /> / ^ X ^ COMM ATIONAL HEE FO ADVISOf ^ AERON Y WTICS 0.1 .13 .4 .5" .6 M, .7 .8 .? /.O Figure I4..- Continued, NACA ARR No. L4G15 Fig. 4ci 26 •24 n to 13 16 14 11 10 OS Ob 04 01 Ani^f^^ii^y^^Ti^i^ njj f-^/ \^^ 3 \ { 'd)h -Ox 18 Sor )IC j j I \ 1 1 1 A 1 y 1 ^ /\ // \ A r/ f ^ ^ ^^ COM WTIONAI fllHEE F ADVISO )R AEROI RY lAUTICS J Z .3 A .S .6 .7 .8 .9 1.0 Figure [|..- Continued. NACA ARR No. L4G15 Fig. 4e II- M .32 '30 Id lb u II 10 IS lb n II 10 08 Ob m 01 1 1 1 Approximofion J ( 'e)h ^0.1 0. j L Sonic / , / / \ h 1 f 1 // y / /, if- A ^ ^ y NA1 COMMIT lONAL / FEE FOR DVISORY AERONAL TICS .1 .5 M, ,7 .8 .9 /.O Figure k'- Concluded. \ NACA ARR No. L4G15 Fig. 5 h .11 ■10 ■09 .08 .07 .Ob ■ 05 M ■05 •01 ■ 01 \ Approx/m of J on 31 1 \ \ w \\ \ V \ \ \\ V ^ ^ \ \ k .6 .7 f.O Figure 5.- .7 .3 l^cr NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS Critical Mach number as a function of camber coefficient, \ NACA ARR No. L4G15 Fig. 6 ? 1.6 11 /.6 1.5 1.4 /J l.l /./ 1.0 .9 .8 .7 .6 .S .4 5on/c Circular arc \\ \\ i\W — Bum P , \ \\ \ A'aV r nation; COMMinEE ( L AOVIS OR AERO IRY ^WUTICS i U \ \ t=.l2^ :t / / ^■^ x% jssXL :3f Xy\ \x^ \1 ./4i-L c \ \ M -U ./oo \ ^ \3p^ s 1 \ ^5.^ \ V' ^^1- — 1 \ ^ !\\ _ \ \ :< \ ' \ \. \ V \ \l \ N ^ ^ v \' A_ -\ ■ok -V- =X 3 N ^ \ V \ \ % ^ ts^ ■^7 ^ t~^ c^v ^ X ^^6 "\ ^^^^^^ t\N N \ ^ V ^^ ■^^ \ X. \ l\"" ^^ ^^ \^ ^J -= -4. .1 .1 3 A- .5'^,. 6 7 .8 .9 l.O hi I.Z Figure 6.- Maxlmtnn and minimum velocities as functions of stream Uach number. NACA ARR No. L4G15 Fig. 7 /.S^O /.2S 9 KO ■ TS I h — j- X Figiire 7»- Velocity distribution at upper and lower surfaces of circular arc profile, h = O.O5, for various values of stream Mach number. \ NACA ARR No. L4G15 Fig. e ^ ^ _____ ^ ^ ^ ^ b ^ ■^ ^^ ^ ' s^^ Xx- ^ ^ ^ ^ ^ ■ o i Ci ^. 1 «^— ' — — . '^[^ -^^ —- " 1) I I 3 § 9 r r r O" <2> o S. ^ Q ^ QO % 9 o U > U O ai i= u A o S « u a ■p n c »-t at tio 4 1= o ■p a o <^ o o u o OS a I 00 ■p T< o o i-K O > o \ UNIVERSITY OF FLORIDA 3 1262 08 03 288 9 UNIVERSfTY OF FLORIDA 1 DOCUMENTS DEPARTMENT 120 MARSTON SCIENCE LIBRARY P.O. BOX 11 7011 ,^^.MQA GAINESVILLE, PL 32611-7011 USA