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'PccA 1 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS WARTIME ORIGINALLY December 1 i:,: Advance Restricted THE CONFORMAL TRANSFORMAT SA STRAIGHT LIKE AND ITE REPORT ISSUED 944 as i Report L4K22a 'ION OF AN AIRFOIL INTO APPLICATION TO THE INVERSE PROBLEM OF AIRFOIL THEORY By William Mutterperl Langley Memorial Aeronautical Laboratory Langley Field, Va. WASHINGTON L. WARTIME REPORTS are reprints of papers originally issued to provide rapid distribution of ie research results to an authorized group requiring them for the war effort. They were pre ,. heldi under a security status but are now unclassified. Some of these reports were not tech yfifed. :Al hav eep. produced without change in order, to expedite general distribution. ili iff:ll fe" all . R.:2 : i;L; B6.'..h: '17"" _"""L 1. 1.. 1..... ::'.. .':,'.." =.i=, .. : r$, = . '0 ARE No. L4K'C2a Digitized by the Internet Archive in 2011 with funding from University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundation Illip: www.archlive.org details conlormallranslo00ilang I I ) 1Z I d ' ITACA rL.FR lie. LL4:.2 a F:R.7 !, T 'iL . IUiTIODIIAL .ADVISOr(Y COif'TlITTEE FOR MERO'iu".'TICS ,A.D.ATC E RESTRICTED :E?D T THE CO'IFCRMAL TR.iiF'ORi'.IT :, ,F ._.II AIRFOIL IlITCO A. STRAIGHT LIIJE AIL ITS A ."LIC .'I .Ii T,, THE IIIVES3E PRDBLEZI OF AIRFO'"IL TH.FOT.. By W'illiam liutterperl S Ji' 1? 1. ,y e. t!.nhd c coni om:.l"m trans'orm, tion is developed that maps an riixf'oil i.nto a strSitt line, the line being chrse:i cs the e.:tenrdeCd chore' ii. r.'o the airfoil. The flapping ic acconls?..*. by opcratini d'irec ti ul thi the airfr:.il ord'inates. The .Abs.:nce of an' prell.inary transformation is found to slh: irt.ii1 the .c.r:: subs taint l ally over th' t :'f pfrev c.Ius :;.ethods. Use is imaie c'f tile superpos tion o of obtain I rti~.:irous counter part of the appro:xirmate rmeth!os of thin:!irfoil theory. The method is applied to the solution nf the direct and inverse problems for arbitrary airfroils and pressure distributions. Ilumeiricsl ex.,mples are ti' n. appli cations to mrcre genereal types f regions, in i.articular to biplanes ,rid to cascades of airfoils, are indicated. Il iTRODUCTIOc: In an attempt to set up an efficient nurierical method for finding the potential flow, through an arbitrary cas cade of airfoils (reference 1.) a I;et.jod of conriformal transf.oriimation "was developed that w:.s found t. apply to advantage in the cass ,.f isni,t:~e.d airfoils. The method 7.onsists in t ans.rlrmin the isolated airfoil directly to a straight line, nmrely.. the ec:tended chord line of' the airfoil. The absence of the hitherto usual preliminary transfoi.j,ation of the airfoil into a near circle rakes for a decided simplification of concept and procedure. RESTRICT PD NACA ARR No. L4K22a The exposition of the method, followed by its appli cation to the direct problem of the conformal mapping of given airfoils, is given in part I of this paper. In part II the method is applied to the inverse problem of airfoil theory; namely, the derivation of an airfoil sec tion to satisfy a prescribed velocity distribution. A comparison with previous inverse methods is made. Addi tional material that will be of use in the application of the method is given in the appendixes. In appendix A cer tain numerical details of the calculations are discussed. In appendix B extensions of the method to the conformal mapping of other types of regions are indicated. The relation of the methods used for the mapping of airfoils to the Cauchy integral formula is discussed in appendix C. Acknowledgment is made to "T rs. Iois Evans Doran of the computing staff of the Langley fullscale tunnel for her assistance in making the calculations. S', ; OLS z = x + iy plane of airfoil S= ( + ir plane of straight lines p plane of unit circle P central angle of circle Ax component of Cartesian mapping function (CMF) parallel to chord Ay component of Cartesian mapping function perpen dicular to chord Ax0, Ayo particular CiT's, tables I and II T displacement constant for locating airfoil r = 2R diameter of circle, semilength of straight line c = an + ibn coefficients of series for CPT' PN negative of central angle of circle, corresponding to leading edge of airfoil ,TA, A ARR Io. LLI22a 5 p. central an:le :,f c'rci? niinu.s t.0D corresponding to trailing ed;e cf c Lirfo'i c aircil chord c z section lift coeffrciernt vz vslocity at surface cf air.cil, fra.tion of 'ree streaf!,, velocity vp velrocity =t surface of circle, frazt3on of fr'ee strear, velocity V fre.estr':am vlccit7 ds element of leni.th on airf: jl F c irculstion ut thiA.kness fact. tor U1 cal' t.'; r' .1':a i;ctr Stici:.Lnress rati, X nror'n lizling c~nr tarnt Sdenomin;: tor :f equation (17) C ccarib r, rentcet x, 6y incler.ental C:i?'s U positive areas. under a;pro::imate vp(C) curve L negative arei unler s;prroxImat a p( ) curJve a angle of attack aI ideal angle of attack Y = + Ft true potential 0a approximate potential 8 central anole of near circle c = e 4 NACA ARR No. Li4K22a Subscripts: N leading edge (nose) T trailing edge c camber t thickness o, 1, 2 successive approximation in direct or inverse CIIF methods I THE DIRECT P, T.ITIAL PRQBTLT OF AIRFOIL THEORY THE CARTEST/. iI. PPING FUNCTION The Derivation of the Cartesian Tapping Function Consider the transformation of an airfoil, zplane, into a straight line, tplane (fig. 1). The vector distance between conformally corresponding points such as Pz and Pt on the two contours is composed of a horizontal displacement Ax and a vertical displace ment Ay. The quantity Ax + i Ay is only another way of writing the analytic function z t; that is, z = (x + iy) (S + irn) = (x ) + i(y r) SAx + i Ay (1) By Riemann's basic existence theorem on conformal mapping, the function z t connecting conformally corresponding points in the z and cplanes is a regular function of either z or t everywhere outside the airfoil or straight line. This function will be referred to as a Cartesian mapping function, or CHF. In order to map an airfoil onto a straight line, the airfoil ordi nates Ay are regarded as the inam in'iry part of an analytic function on the straight line and the problem reduces to the calculation of the real part Ax. IACA ARR Ho. L4?K22a 5 The calcul:t.ion of the reil part of an analyrtic function on a closed contour fror the kno.jrn x\"lues cf the imaginary: part is well kI1,own. It is convenient for ti.is calculation to consider thc straight line as con forimall. iy reited to i circle, ppl ne, by: tihe f'aiiili'ar trains ormrat ion p2 S = + 2a) P where the constant disp:.laceraent T has been inserted for future conver.ience in locating the ai:fcil. 7or corr.e sponding points on tnh str:irht line ant the circle, equation (2a) reduces to S= TT + r cos 0 : (2b) Considered as a function of p, therifor: the C.i  is re ular e E',.'r.'her.o ,uats Li t.fi .m i;'le ?,nd is therefore expressible b:: thle inverse iov' sEriEcs z : " T he analogy, of equation (3) with the T'h:eo:idr.senGlar'ic1' trarnsfornation (referencer. 2) n log =  P 1i pa which relates *conforr.mall, a near circle, ptplane, to a circle, pplane, ay be noted. Oini thie circle proper, where p = Re' and deF'ning c = + ibn, equa tion (5) reduces to two conjugate Fourier series for the CF; namely, Ax = a + a cos nc sin nQ (I) a n n 1 0 1 R NACA ARR No. LLK22a CO bn ao y = bo + cos nO n sin nc (5) 1 Rn 1 R These series evidently determine Ax from Ay or vice versa. An alternative method of performing this calculation is possible. It is known that if the real and imaginary parts of a function are given conjugate Fourier series, as in equations (4) and (5), with the constant terms zero, two integral relations are satisfied. (See, for example, references 2 and 5; also, appendix C.) These relations are 1 2Tr ( Ax(cp) = Ay(c) cot  de' (6) 27r 2 Ay(p) = L f Ax(p') cot ~' dp' (7) S2n0 2 Before the detailed application of the CT' z  to the solution of the direct and inverse problems of airfoil theory is made, some necessary basic properties of this function will be discussed. Airfoil Position for Given ClIF It is noted first that the regions at infinity in the three planes are the same except for a trivial and arbitrary translation; that is, by equations (1), (2a), and (5), lim z = AxM + i Ay = co o ao + ib, Z,  co (8) lima = p + T Secondly, if an airfoil is to be mapped into a straight line, it becomes necessary to know the point on the straight line corresponding to the trailing edge of the airfoil. For a given CMF, ax(C), Ay(P), and straight line of length 2r located as in figure 1, ITACA ARPR To. 4 tV22a the sa loil coorinates ::, y eir obtained frcrn e':Iuia tion, 1) and (2L) as x = T + r c0 c,' + ~ ::L(L ) i" ) ((1_7) The leadin' and trallinr ed.lfes of' the airfoil will be taken as the points '...rles.:ontin. to tl.e e:treriities of the airfoil abscissas. Thie ccrrs[oonr'irng locations on the circle ::.e therefore deter.miirned by tma'riniziiig x with respect to C in equav Li (). Thius dx ,. d :x = j r sin + d 01, d', d c" sin = (11) 2 d,:" The condition ( I'1) yelds, usuallyy, by gr hical deter rin action; tne an:.le.1 corrOs. po;nL,...ng to the leading and trailing edp:es (iig. 1 S} (1 ) q T _ n + TS It will be found, convenient to so alter the position and scle of a derived airfoil that, for ex:aple, its chordwise extremities ars .ocgt.cd. at : 1 and tle trailing edge has the ocrdiinate : :r 0 L to be peflrred to as the normal for:ni). The cinrd c of a dei'ived air foil is by definite on the difference in airfoil abszissa extrer:ities, r b: equ.tionrs (12) an:d 1'), c = r ,os ,. ?) + x L(w) (;1) The increase in scale fr'om a t': sone des red co is obtained simply by ,milt iplyin r, x, d i by the facto c..,/c. The translation neicessar to bring t the trailing edge of the airfot'c, to its desired location is then accornrplished by adjusting the trinslation constants T and ho. NACA ARR No. L)K22a Velocity Distribution on Airfoil Once the C.77 Ax(p), Ay(c) and the diameter of' circle r of an airfoil have been determined, the velocity at a point on its surface is obtained in a well known manner as the product of the known velocity at the corresponding point of the circle and the stretching factor from the circle to the airfoil that is, vz() = r p() (14) where vp(P) is half the velocity on the circle (since r = 2R) and ds is the element of length on the airfoil. The velocity on the circle v,(C), which makes the point c = n + PT corresponding to the trailing edge of the airfoil a stagnation point (Eutta condition), is vp(c) = sin ( + a) + sin a + T) (15) where a is the angle of attack. TIe velocities vp and vz are expressed nondimeAsionally as fractions of freestream velocity. The stretching factor ds/dC is obtained from equations (9) and (10) as d2 d_ r sin 2 (16) ds (dx) + Ody=) The velocity v,(cp), equation (14), therefore becomes z sin (cp + a) + sin a + ) (17) Vz (() (17) dL sin c + d  Sd (r dcp/ This equation is the general expression, in terms of the CF?, for the velocity at the surface, equations (9) and (10), of an arbitrary airroil. The denominator depends only on the airfoil r sometry, while the numerator depends also on the angle of attack. Equation (17) is similar to the corresponding expression in the TheodorsenGarrick method except for the absence of the factor representing a preliminary transformation from the airfoil to a near circle. FTACA ARR I!o. LiK22a 9 The exrressions for the lift coefficient and ideal angle ci attack : nay b. not. ci. The circul;tion P around the airfoil is (V is freestream velocity) r = ir RV rin (a + 3,) (1) The lift coefficient c, is defined by 1 Scc V = Hence = !,nr sin r (a + '3 (19) where the airLfoil cird .c is gven by eqi ation (1). The ideAl angle j att.) (,'e;er.:nce is defined as that i.n tle of aftut ':k fnpr '!:..a; a sa r'.r t onr poin.:t exists at te leaeiifn edge; that :., vz = 0 fnr =  in equation (i1 ). ence, CT = (2,) 2 Superp.oition c' Solutions The sun of tn;, analytic functions is an analytic function; therefore, for a given f.plane ci:cle, the sun of two Ci's is itself a CI S' a ,. also evident froc equations (4) to (). Thus, s i'x + i A and Lx2 + i Ay2 of tw" crompionent airfoils rC, f'or the sae r, be added tog thr: to .'ive a CiT (xjl + A:' + i (y + Y and thence, by equation (17), an exact velocity distribu tion for a resultant airfoil. The resultant profile and its velocity distribution is a superposition in this sense of the component profiles and velocity distributions. Thus, without sacrifice of exactness anC: w.itl no grerat increase of libor, airfoils nma oe analyzed and synthe sized in termr? of cnorionent syrn_:etrical thicl;rkess distri butions and mean camber lines. This result provides a rigorous counterpart of the v,ellknoin approxiinate super position methods cf thinairfoil vrrtex and sources inI potential theory. NACA ARR No. L4K22a As a particular case of superposition, a known CM? Ax + i Ay may be multiplied by a constant S and the resulting CMF S Ax + iS Ay determines a new profile by the new displacements S Ax, S Ay from points on the original straight line. It is evident that, except for the corrections (S 1) Ax to the airfoil abscissas, this new profile is increased in thickness and camber over the original profile by the factor S. The effect on the velocity distribution is that of multiplying the derivatives in equation (17) by S. By virtue of a reduc tion in scale by the factor 1/S this profile may also be regarded as obtained from the original one by using the same Ax, Ay but a length of line 1/S times the length of the original one. The use of superposition as well as the application of the Ci':' to some particular airfoils will be illustrated next. Application of the C:.F to Some Particular Airfoils Symmetrical thickness distributions. The Cartesian mapping function was calculated for a symmetrical 30 percent thickness ratio Joukowski profile from the known conformal correspondence between a Joukowski profile and a straight line. I1; CMF is given in normal form in table I. The associated constants To and ro are given in table II and the profile itself, as determined either from the standard formulas or from equations (9) and (10), is shown in figure 2(a). The symmetry of mie profile required only the calculation of Ax('), Ay(P) for 0 cp 1800. i"ih corresponding velocity distri bution (fig. 2(b)) was obtained from equation (17) by use of the computed values of the derivatives. At the cusped trailing edge the velocity as given by equation (17) is indeterminate; however, the limiting form of equa tion (17), determined by differentiation of numerator and denominator, is limr v = Icos + a) (21) cpT 2 )2 / \2 P` )T d' x d\ c 0 r dp2/ \r dcP2 FTACA APR Po. LT.r22.a i It is sen f'rcomi this cx::res.: ) tih"t tle v'locit ,t c.i.S id e..d depends on the co0 ,.1' der' .:.ti' .af tn. napping function, that is, On tii' cii.r'aturl e t tlie i .:: . The ,c:,:1.uT ed second d, riv'.tives : :,:.. 'i d' L,,t/',IY of the CiF : .of table I are plo tt.d in i 'u. f'r '. ' of valuL:3 of. ear 1.' f "l..' . T.=. ".s '.oi' .Tru L" etr2 ,;O i ",r 11il, :' do:if'r: nt JTu.. D Ii .,,f'.i1. s. in;i tzJ cr. : i. tIe j tin us uperI siti..n i Soltion::" The L_ f ih L) p,  '7 L ,' 1 I .to sultil to obtain .: r 1i .: tf in rotio _. ,'btrin,: fLon Ut Ayo T it .. 1)  whe re s, i t r.,a.. i:jl i: rlinat t e :an CIlF (t:. 1 an.. 1 the c.l n eintd o tl. r. s Lt the seil'icnrd o; the der.tve'l ,., i... T"i t2. ri ltt. ic.:n s ,w o r o 1 n iL .1 sn tde t A D + T  .,alues ,.._ uat ','e e calcul..ti e,t r'"oo;., th,_,z f' rru.,i _a ;or th ic :ne i ,.ic f' ,if pe ;. e ,_t ,n,ni 12 p. re ^ 1 .t /,'l E.,rIe riven in t hble II. 1i'9 re ,Itin.7 C.'. '_ .ere. i:!.,,rn n,:.ir mali zd 's indici t, d i. i l: e s e t, .i.n ".kr.c' i P,:."ti,.,n for Gi'n C ' So that th actual f;tfrs :." i.t :h t1o ..iultipi: the 'o n ia n:l L. .,,, w ' ThE: "aSes are 1'i ' , t be II, t. eth: ith ith.:. : s oci9tcd. cons tants T sail r T!h r profi i S thus l d. et ,'r Li'd 1 i.e slhown in i,.:.e ( .) a nd tin e c,'l,y.Spri.InL' t V eloc'.it' dIS tribut LonWrs in Ii 're 2() . Th. ds .'ved : rci '.: re n t TJ.u';:,. s;'i pi, ..fi1e . Th pr'oint o x" M x.v':,, thI' :nl'ss is s i ftt..d ",il.: 'long the chord so r'. e' t as the Lhi.C':1.EZ rv tiao .i.ctaes. CZo v'erseli,,, the point of irxi.u. thi l i,ess v.OLid C e shifted for ar ,d by !oin.i fr'omi a thin Jout:. vsW i pr.fr l' to a J:CA ARR No. L4K22a thicker one. (This result was the reason for starting from a thick section.) The C;',F for the 12percent thick derived profile is illustrated in figure 4. It is to be noted that the horizontal displacement function AXot(CD) is symmetrical about CP = whereas the vertical dis placement function AYot(p) is antisynretrical about CD = iT. Mean camber lines. The CIMF was next calculated for a circZuTarc profile of 6percent camber from the known conformal correspondence between a circular arc and a straight line. The normalized C7 and its derivatives are given in table III. The C:7T is illustrated in fig ure The symmetry in this case is with respect to CD = 900 and C" = 2700, the Ax c () being antisymmetrical and Ayoc(c~) symmetrical. The circulararc mean camber line is shown in figure 5(a) and the corresponding velocity distribution in figure 5(b)' Derived mean camber lines were obtained front the CI'F for the circular arc in a manner similar to that for the symmetrical profiles. The expression determining the factor uc for a desired percent camber C is Uc 7r uc Omax 2jro cos CPN + uc Ax9) with the solution for uc 2Cr. COs CN Uc = cs (25) Lgy 2C Ax( c) max I The angle cP in equation (25) (as in equation (22)) corresponds to the extremity of the derived mean line. Because the factor uc is to multiply the derivative dAxo(c )/dc, the angle CN as determined by the maxi mum condition (11) depends on u,. One or two trials are sufficient to determine uc simultaneously with C from equations (23) and (11) for a given desired camber C. Values of uc and T.r (also CT by symmetry) are given in table IV for derived cambers of 5 and 9 percent. The actual multiplying factor to obtain the derived CIiF's in normal form is given in table IV as Xuc. NWACA ALFRR T. TL.K22a The derived camber lines. are shown in figure 5(s) It is seen that the derived canmb'er lines have been separated into distinct upper and loiee surfaces. Fur thermo.re, for the qoercent camber line the "lower" surface, that is, the surface correspondin3 to the lower part of the strai:iht line or circle, lies above the "'upper" surface. Lltnough such a scanber line is physi cally meaningless b: itself, nevertheless its Ci.T can be compounded with that for a thickness distribution t. giEve a physically real result (if the resultant profile is a real one). The velocity distribution of the ,percent camber line is given in figure 5(b). The velocity : dis tribution" of the 9percent camber ilne is included in figure 5(b) for arithlm!etical com;arison although it is physically meaningless for the reason just mentioned. The velocities at the cusped rnxtremities of the camber lines are given by equation (21). The second derivatives of the CiF of table III were computed. They are plotted in figure 5 as d2Lx ./dc'r2, d2A' /d2 for a range of c near 1300'. These second derivatives, in combination w ith those for the s LnMe trial profile, can be used to !.ive a more accurate determination of the velocity at and near a cusped trailing edge than is obtained by using equation (17) near the trailing ed.e. Combination of s3y.L.i trical nrcfile and. mean .camber line. The CliT's derved for the symcLnetrical .pro'files and for the mean camber lines can nowv be combined in ,'ryving proportions to produce airfoils having both thickness and camber. These airfoils may be useful in themselves or, as in the folloviing sections, ray be used as initial approximations in toth the crect and inverse processes. As an illustration of such combinations, the CDiF of the 12percent thic!: s:y.inetrical profilee ,;f fi.giure 2(a) and the CMF of the 6percent a iber circular arc of figure 5(a) were added together. The airfoil profile thus determined is shn.;n in figure 6 (a). For comparison, the airfoil obtained in the mannerr of thinairfoil theory (see,for example. reference 4) by superpDsition of the same symmetrical profile and a 6.5percent camber cir cular arc (in order to duplicate the cg,,mber of the exact airfoil more closely) is indica ted in the figure. The velocity distribution of the dotted airfoil should, according to thinairfoil theory, be. the sum of the syrrmetricalprofile velocity and the increment above the NACA ARR No. L4K22a freestream value of the camberline velocity. This velocity distribution, determined from the two component exact distributions at zero angle of attack, is shown dotted in figure 6(b). The exact velocity distribution of the "exact" airfoil of figure 6(a) was determined for the same lift coefficient (cL = 0.88, a = 10131) from the known CIJ. This distribution is shown in figure 6(b). The two velocity distributions differ ap preciably, although in the directions to be expected from the differences in shape of the corresponding air foils. It appears that the CMF's of a relatively small number of useful thickness distributions and camber lines would suffice to yield a large number of useful combi nations of which the (perfect fluid) characteristics could be determined exactly and easily in the manner indicated. The superposition of solutions can also be used with the airfoil mapping methods based on the conformal trans formation of a near circle to a circle. There is a decided advantage, however, in working with the airfoil ordinates directly, both in the facility of the calcula tions and in the insight that is maintained of the rela tionship between an airfoil and its velocity distribution. THE DIRECT POTENTIAL PROBLEM FOR AIRFOILS The direct problem for airfoils is that of finding the potential flow past a given arbitrary airfoil section situated in a uniform free stream. This problem can be solved by a CMF method of successive approximation some what similar to that in reference 2. Method of Solution Suppose an airfoil to be given as in figure 6(a). The chord is taken as any straight line such that perpen diculars drawn from its extremities are tangent to the airfoil. For example, the "'longestline" chord, that is, the longest line that can be drawn within the airfoil, satisfies this definition. The xaxis is taken along this chord and the origin is taken at its midpoint. Suppose, in addition, an initial CMF Axo and Ayo, ITACA ARR ITo. LTJY22a straight line ro, and choi.ovwise translation constant To to be oven such that the corresponding airfoil has the sa5e chord and is similar in shape to the given airfoil. (At the worst the initial airfoil could be the given chord line itself.) At the chordwise locations ;:r(cfO) of the initial airfoil, corresponding to an evenly spaced set of  values by equationl (9), the differences 6yl(i() between the ordinates Aytl) of the given airfoil and Lyo ( '') of the initial airfoil are measured. The ordinate dif ferences 6y1 (Z,) determine a conjugate set of abscissa corrections uzx1 ( r1 in accordalii c either withl equa tions (4) and (5) or equation (6). The details of this calculation are given in appendix A. The initial sernilength of straight line r corre sponding to the initial airfoil is then corrected to r and the translation constant To adjusted to T SO that the use of ri with the first appro':imate CorF Ax1 = Axo + 6x1, L:.1 = Lo + by1 yields a first approxi mate airfoil of v'w.ic.h tlhe chordwise e.:tremnities coincie with those of the given airfoil. This correction is described in detail presently. If the first approximate airfoil is not satisfactorily close to the give r airfoil, the procedure is repeated for a second appro.r;:inmate air foil, and so on. The successive :irfoils thus deter mined provide a very useful criterion of con.vergence to the final solution; nai.iely, the given airfoil. Evidently, the fundamental relation between an airfoil nd its mapping circle c1 c2 Z p = + + *+ can be used in the manner indicated to effect directly the transformation of an airfoil into a circle. It appears preferable, however, to subtract RB/p from the second term on the right and thence to introduce the 2 straightline variable = p + F. Ths exact velocity distribution of any of the "approximate" airfoils (hence the approximate velocity 16 NACA ARR No. L41K22a distribution of the given airfoil) may be obtained from equation (17) using the derivatives of the corresponding CMF. The zerolift angle PT to be used in equation (17) is determined for each approximate airfoil along with the corresponding correction for r. The correction for r is necessary because if the chordwise locations of the first approximate airfoil were computed by equation (9) with the original values of r and T, Axl () being used instead of Axo(c), the re sulting chordwise extremities would in general not be at x = 1. It is therefore necessary to adjust ro and To such that with the derived Axl, Ayl, xl(i N = 1 (24) Xl( T1 = 1 where a' 1 and CPT1 are the angles on the circle corre sponding to the extremities of the desired airfoil. This operation was mentioned in the section "Superposition of Solutions." It may be termed a horizontal stretching of the given airfoil. The condition given by equations. (2;,.) applied to equation (9) yields 1 = Ti + r1 cos N1 + AXl(o1) (25) 1 = T1 + r1 cos PT1 + Ax(QcT) J Subtraction of the second of these equations from the first gives for r! i + 2 rl = (26) cos PN1 cos T1 11 1l !TACA ARR ITo. L4!22a Addition of equations (25) gives for T cos Fp,1 + cos0 A1xQ 1) + L.: i TT) T = r1  + (7) 12 2 The angles Ce and mTD in eq.uations (24) and (.27) correspond to the extremities of the desired airfoil. They are civen b. G.rphic'4l solution .,f equation (11) sin =  ( ) r1 d * Equation (11) mn'st be ol'e,1 sirlultaneoinl ,;ith equn t on (2) fonr r~ 1 and In prr.'tice only a few siccessive trials re ijecess"r. Thence T is obtained by equation (27). The argle ':T determined in this process 12 equivalent to the zerolift anwic of the airf:,il, equa.tion (12). Illustrative Example of Direct Meti:od AS a numerical illustration cf tihe direct method the velocity distribution of the NA"i. 6'512 airfoil was cal culatea. In order to obtain an initial airioil, the C1!F of the 6percent camber circular arc (tabls III and IV) was added to the CI.F of the 12ocrcent thii: sri.Ltrical profile, derived from that of table I as inrdicalted ii a previous section. Before this addition was r;wade, the C0.hF for the circi.lar arc was increased in scale (multi plied) by 1.092S/1.0072 to correspond to the same length of straight line r as the sy'.m nrtrical pr.file GrFT'. The normalized result:nt CLIF andl the associated constants are given in tables V(a) and VI, respectively. The initial airfoil is shown in figure 7(a). The given airfoil, il'CA 6512, was so rotated through an angle of 0.88'" (nose .do n) as to be ta:;1ent to the initial airfoil at the leading edge. The cornvergence near the leading edge was thereby accelerated. The given airfoil is shown in this position in figure 7(a). Two approximations wore then carried cut in accordance with ITACA ARR No. LKK22a the procedure given in the preceding section. The numeri cal results are given in tables V and VI. The first approximate airfoil is indicated by the circles in fig ure 7(a); the second approximate airfoil was indistin guishable to the scale used (chord = 20 in.) from the given airfoil. The velocity distributions of the initial, first, and second approximate airfoils are given in fig ure 7(b), together with those corresponding to one approximation by the TheodorsenGarrick method (refer ence 5). The second approximation velocity distribution differs appreciably from that of the TheodorsenGarrick method on the upper surface but agrees fairly well on the lower surface. The discrepancy for the rearmost 5 percent of chord on the lower surface appears to be due to lack of detail in this region in the TliodorsenGarrick cal culation. The convergence of the CTF method is seen to be rapid, considering the approximate nature of the initial airfoil, although two approximations are required for a satisfactory result. The second approximation could probably have been made unnecessary by suitably adjusting the first increment 6yi(p) near the leading and trailing edges on the upper surface before calculating 5x!(9). The direction in which to adjust the increment is obtained by comparing the thickness of the initial airfoil with that of the given airfoil in these regions. Because a thicker section has a greater concentration of chordwise locations toward the extremities, for a given set of T points, than does a thinner section, the chordwise stations would be expected to be shifted outward as the thickness of the section is increased. The ordinates Ayl(P) should therefore have been chosen at chordwise stations slightly more toward the extremities than those given by equation (9). The accuracy of the velocities is estimated to be within 1 percent. It was expected, and verified by pre liminary calculations, that the results would tend to be more inaccurate toward the extremities of the airfoil than near the center. This result is evident from equa tion (17). A given inaccuracy in the slopes dAx/dr and dAy/d can produce a i&lee error in the velocity near the extremities, where sin p approaches zero. This disadvantage does not appear in the TheodorsenGarrick method, in which sin C is replaced by one. Exces3lve error in these regions can be avoided in various V;ys. N.ACA ARR ,To. LiLT22a If the initial airfoil, for which the slopes dxo,/d?' and dAyo/dc have presumably been computed accurately, is a Zood approxiriatioln in these regions, as evidenced by the smallness of o5l, 6y1 compared to Ax0, 7y, the effect of inaccuracy of the slopes d5xl/dC", dbyi/dc, will be reduced, since the:T are added to. the initial slopes dAxo/dr', dAyo/dC. It was to reduce the magnitude of the incremental CIF near the leading edge that the TACA 6512 airfoil was drawn tanoent to the initial air foil in this region. The error in the derivatives can also be avoided by computing tlher, front the differentiated Fourier series for 6x1, 6:.i. (See appendix A.) This calculation was made in the illustrative example, after it was found that an error of about 5 percent in the velocity on the upper surface leading edge could be caused by unavoidable inaccuracy in measuring the incremental slopes. The fact that the computed derivatives do not repre sent the derivatives of the CIRF but rather the deriva tive of its Fourier expansion to a finite number of terms may introduce inaccuracy. (The derivative Fourier series converges more slowly than the original series.) A comparison of the computed derivatives which the measured slopes will indicate the limits of error, however, as well as the true derivative curve. The importance of knowing the CMF derivatives ac curately may make it desirable to solve the direct problem from the airfoil slopes, rather than from the airfoil itself, as given data. This variation of technique enables the CMT' derivatives rather than the CLIP itself to be approximated initially. Further details are given in reference 1. II THE INVERSE POTEITTIAL PROBLEM OF AIRFOIL THEORY The inverse potential problem of airfoil theory may be stated as follows: Given the velocity distribution as a function of percent chord or surface arc of an unknown airfoil to derive the airfoil. Before the questions of existence and uniqueness of a solution to the problem as thus stated are discussed, several CP'T methods of solu tion will be outlined and illustrated by numerical NACA ARR No. LLK22a examples. Various previous methods of solution will then be described briefly and their inherent limitations and restrictions on the prescribed velocity distribution will be compared with those of the CMF methods. The prescribed velocity distribution is assumed to be either a doublevalued continuous function of the percent chord or a singlevalued continuous function of percent arc. (Isolated discontinuities in velocity are, however, at least in the percentchord case, admissible.) CMF Method of Potentials This inverse method is based on the fact that, if the airfoil and its corresponding flat plate and circle are immersed in the same freestream flows and have the same circulation, conformally corresponding points in the three planes have the same potential. Consider first the case where a velocity distribu tion corresponding to a symmetrical airfoil at zero lift is specified as a function of percent chord. If an initial airfoil is assumed, the prescribed velocity can be integrated along its surface to yield an approximate potential distribution as a function of percent chord. This potential increases from zero at the leading edge to a maximum value at the trailing edge. Of fundamental importance to the success of the method is the fact that this potential curve depends mainly on the prescribed velocity distribution and only to a much lesser extent on the form of the initially assumed airfoil. The chord line of the initial airfoil taken as the xaxis is next sufficiently extended that, in the same freestream flow as for the airfoil, the potential, which in this case is simply V1, increases linearly from zero at its leading edge to the same maximum value at the trailing edge as exists for the approximate potential curve derived initially. Horizontal displacements Lx between these curves are then measured as a function of the straight line abscissas and, hence, as a function of the central angle c ofthe circle corresponding to the straight line. These horizontal displacements Ax(f), together with the conjugate function Ay(p) computed therefrom and the length of straight line previously determined, constitute a CMF fnr an airfoil that is a first approxi mation to the unknown airfoil. The approximation is based on the use of a more or less arbitrary initial NACA ARR No. L4K22a airfoil to set up the first approximate potential. The exact veloc.it: distribution of tie derived first approxi nate airfoil can now be computed and compared with th,e prescribed velocity. If the agreement is not satisfac torily close, the procedure is repeated, with the airfoil just derived taking the place of thie one initially assumed. The complication introduced in the general case in which the prescribed velocity distribution corresponds to an ur.synmentrical airfoil w\lth circulation can be resolved as follows: It is convenient in this case to discuss the potentials in the circle plane. The pre scribed velocity distribution is trEansferred to thi, circle plane by means of the stretchingr facctor, pre.Eured l:nowrn, of the initially aszuLmead airfoil: that is, equ.tion (lI) is solved for v,.) (. The first appr?.:irtate potential distribution as a function of the cn.trAl 9.n, lo is obtained b integrating Vo(c) throut1h a (rr..n ne of 2n radians (around the ni.rfoil), starting from. tile v'.ilue of C) near zero for which vp(CD) is zero I the front stagnation point). This apnroxirmate potential curve has a r.inim.un ialue of zero at the front stagnation point, rises to a rmaximr,aui for the v;ilue o'f 0 near T: corre sponding to the reer staRn tion point, then f:lls to a minimum for the final vrlue of 0 (the front stn 'net ion point), which is ar. anrle 2n radians from the starting cpoint. The difference between the final and the initial potential minirurms is a firrt approximation to tle circu lation r. A circle of such diameter is now derived which, with this circulation and the same freestrearl flow as for the airfoil, yields a potential distribution (henceforth called true potential distribution) th1,t hAs the sane mrnaimlul and minimuiri values as the approxiimats potential curve just derived. If the maximur approximate potential is denoted by roU and the decrease of pote ntial (considered positive) from the maximum to the final value by r,L, where ro is the diameter of tne circle corre spending to the initial airfoil, the para.:eter y is first com puted from T U L = J (28) 2(Y + cot y) U + L NACA ARR No. LK22a by means of figure 8. The desired diameter r is then given by ro(U + L) r = (29) 4(cos y + y sin y) The parameter y is actually the sum of the angle of attack and zerolift angle of the unknown airfoil, to a first approximation; that is, y = a + PT (50) It is related to the circulation P by equation (18). This procedure for the calculation of the diameter (see, for example, reference 6) follows easily from the expression for the potential distribution on a circle, obtained by integration of equation (15) as t(r) = rof v(CP)dCP = r [cos y+ y sin y cos ~+1 a) + (C+ a)sin y (51) If the diameter r of the derived circle is much greater than the diameter ro of the circle corresponding to the initial airfoil, it is desirable to increase the CMF Axo, Ayo of the initial airfoil by a factor suffi cient to modify the initial airfoil such that it corre sponds to a circle of diameter r. A new approximate arid true potential distribution is then obtained as described but by using the modified initial airfoil. The first approximate horizontal displacement func tion is now determined as the sum of the horizontal displacement Axo(cP) corresponding to the (modified) initial airfoil and an increment 6x1(0) produced by the noncoincidence of the approximate potential distri bution 1a and the true potential distribution Dt. This horizontal increment may be measured between the two potential curves, both considered plotted against chordwise position in the physical plane. With sufficient accuracy this incre~,n nt may be computed as the vertical distance between the potential curves divided by the !..C. JalR Ho. iL4:22a ,TC'2 slope of the g ppro.ximnt potential cu?.ve nmely,, the Drescribe.d velocity '. If, therefore, all quantities are considered as functions of ' Ax1 = Lxo + 6x1 = Ax, + (32) The ordinste function on yl(C') conjugite to LAx1 ) can nc'., be computed and, together with Axp ) a nd the diameter r obtained previously, determines the first approxir'mae airtoil by: ecuations (Q) and (10). CaiLu lation or measurenent of the CfIF derivatives dx1//dcQ, dAyl/drm and the use oi equations i11) and (17) tL.en determine the zerc lift unjlecle 3 arnd the ex:ict velocity distribution of the first approxim'te airfoil. The angle of attack, to a first &approximation, is given by equa tion (50), tihe value of y derived fror., equ.,tion (23) being used. This exact velocity distribution is compared with that prescribed and, if the agreement is not close enough, the procedure can be repeated with the first approximate airfoil as the initial 9aircil. In the case where the prescribed .velocit:y is speci fied as a function of percent t;c, then b:" line integra tion of the prescribed velocity along the percentt are, the true potential distribution of tio: urInknnon airfoil is known as a function of qrc (e;:ceet for a trivial scale factor). The maximum and rniniir'auja values of this potential distribution then pern.it the .uniqc..e determination, by the calcui.tion previously described, of the circle corre sponding confornr.All: to the un.:nc..n airfoil. Correlation of the 'potential distribution of t:is circle with the potential distribution as a function of arc initially calculated therefore ields exactly ti, potential distri bution of the un:n'own airfoil a. a furn tion of Lhe central angle 0 of the circle. This fsct has been noted by Gebelein (reference 6). The calculation of the diameter r as outlined above for the percentchord case is thus unnecessary. The remainder of the proced'lre is the sar.ie, the successive approximate airfoils no." being adjusted to correspond conformally to tnis circle before corre lating their percentarc lengths vitn the prescribed velocity distribution in preparation for the next approri nation. :T.AC ARR No. LlK22a The successive contours determined by the method of potentials are, of necessity, closed contours, whether or not the sequence of contours converges to a solution satisfying (mathematically) the prescribed velocity dis tribution. The closure of the contours is a consequence of the method of setting up the horizontal displace ments, Ax(C), and solving for Ay(cp), by which the contour coordinates are obtained as singlevalued func tions of 9. The necessity for closed contours does not, however, exclude the possibility of deriving physically unreal shapes; namely, contours of figureeight type. This point will be discussed at greater length later but it may be remarked here that it is the extra degree of freedom introduced by the class of figureeight type contours that admits the possibility of a unique solu tion to the inverse problem treated here. It will have been noticed that, whereas in the direct method a Ay is determined from the given data that is, the airfoil and a Ax is computed therefrom, conversely, in the inverse method of potentials a Ax is determined from the given data that is, the velocity distribution  and a Ay is computed therefrom. Similarly, just as the direct problem can also be solved by deriving dAy/dp from the given airfoil slopes and thence computing dAx/dc, so, conversely, can the inverse problem be solved by deriving dAx/dc from the prescribed velocity dis tribution and thence computing dAy/dc. This inverse method of derivatives will be discussed after some numerical examples are presented, illustrating the nethod of potentials. Examples of CMF Method of Potentials Symmetrical section. The method of potentials was applied first to the derivation of the c.,r.etrical profile corresponding to the prescribed velocity distribution shown in figure 9(a). As an initial airfoil the 12 percent thick profile derived from the 30percent thick Joukowski profile in part I was used. The initial OMF and associated constants are given in table VII. The initial airfoil and its velocity distribution are siLown in figure 9. The first increment CMF and the resultant first approximate airfoil and its exact velocity distri bution were calculFted by the procedure of the preceding section. The incremental slopes d6x1/dp, d6yl/dp were computed and found to approximate the measured slopes ;HACA ARR Io. L4K22a very closely. The results ;re presented in table VII and figure q. It is seen that the change in velocity and profile accomplished by, one step of the inverse process is large; that Is, the coe.vErgnce is rapid. The high velocity cf the first point on the upper sLrlface (c = 153') is due to lackl of, detail in the calculation. (Tvw.elve points on the upper surface wEre calculated.) 'or niprcticnl puposes the nose could be easil modified to reduce this velocity if desired without rcing through a conmlete second a.ppro::imati .on. ean car.ber line for ,aniforri ';elocitnr increment. As a second examule of tihe inverse CM F rethod, the profile producing uniform equal and opposite v:elocit: increments on upper and lower surfaces .was derived. 3By the methods of thinairfoil theory this elocity distribution yields the socalled logeriitihnic carlberi line. The prescribed velocity distribution is indicated in figure 10(a). The velocity peaks at the extre:iities of the F:rescribed velocity curve were assumed in o.d'er to coLm:ensate for an expected rocuding .f. of the velocity, in this region in worl:ing up front the initial velocity' distribution. The convergence to the prescribed i.'nifcrrm v.,locity dis tribution would there" be accelerated. The initial airfoil was taken as tle Spercent camber circular arc, discussed in part I. The initial C;.TF and its associated constants are given in tables III and IV. The circular arc and its velocity distribution are shown in figure 10. A first approximation was calculated d as outlined in the previous section. A n.u.ierical difficulty appeared in the process of solving equation (11) for the zero lift angle of the first al:.rcrxinmate airfoil. It appeared that a 2Lpoint calculation (12 points by symnetry) did not give sufficient detail in the range TT < C < 1 T to yield a reliable solution of equation (11) for the zerolift angle. This res.ilt was a consequence of the prescribed velocity discontinuit Pt the extremities with the consequent lrge but local changes in CIF and profile shape required in these regions. The solution obtained for the zerolift angle wan ;T = 6.1', which by equa tion (19) with r = 1.3045 and al = 0 yielded cz = 0.67. The desired cl, however, is .80,' which would correspond to = 7.27. It w/s considered that a relatively minute change in the shape of the extremities of the derived caliber line v.ould alter the ,TACA ARR No. L4K22a slope dAxl/dcp in the desired range sufficiently to yield a zerolift angle of T = 7.270. On the other hand the effect of such a local change on the CMF as a whole would be small. The velocity distributions of the derived profile were therefore computed for both zero lift angles quoted previously. The results are given in table VIII and in figure 10. Included for comparison in figure 10(b) (vertical scale magnified) is the logarithmic mean line of thinairfoil theory, computed for ct = 0.80. The velocity distri bution of the derived shape as calculated for the desired lift coefficient of c = 0.80 is seen to be a satis factory approximation o the desired rectangular velocity distribution. The profile itself is seen to be one of finite thickness:as.compared with the single line of thinairfoil theory. Airfoils obtained by superposition of this type of camber line with thickness profiles would therefore be increased in thickness over that of the basic thickness form. The changes in velocity distribution and in shape of profile are again seen to be large; that is, the con vergence was rapid. As is to be expected, the rapidity of convergence of both the direct and inverse methods in comparable cases is about the same. CMF iethod of Derivatives Instead of approximating by the method of potentials to a CiiF that, when differentiated, yields the prescribed velocity, the CMF derivatives may be obtained directly. The controlling equations are equations (17), (9), and a modification of equation (7). Vsin (co + a) + sin (a + T) (17) jz(C ) = (17) dVy 1 dAx ,p' d A sin (P ) A (  = cot dp' (7a) dcp 27T dcp' 2 Jo ITACA ATr" TTo. ILLY?2a x Q + Ax' Ao) (9) = CS CO + A ) (9) r r These equations, together with the au:iliary equations (11) and ('1), constitute a set cf si:rultaneoius equati.ons from which, the CIT derivative d.x/d'3 may be determined from a prescribed volcity distribution v. The corresponding airfoil is determined by integration cf dLAx'/dr and its conjugate dy//de. Consider first tie case wnre the velocity is speci fied as a function of percent arc. As explained in the previous section, the constants r and y of the final circle corresponding to the Lunl.aiowrn airfoil can in this case be deter.nined initially.. Points of equal potential along the arc and circle are then found, which yield vz as a function of :. The angle of attack a in equa tion (17) is taken is some re'ascnable value and dA:'/r d, determined by successive ayproxlmration. In the first approximation dLy/ir dg may, for exrample, correspond to some lnowLn Cio ,L some known CiO?. Equation ( 17) is then solved for dAx/r dr, for which the conjiuate dAy/ji dC is calcu lated next and used as a basis for a better determination of dAx/r d(. The airfoil coare6spondin; to any approxi mation is obtained by intePraLion of d'x/d.d and its conjugate day/, 'd. ('The nethod: of derivatives may be regarded as based on the use of the function p z This function is regular ev.'erywhere outside do the circle p = Re approaches zero at infinity, and reduces tc + i on r;te circle itself. ) QQ dfo In general the d':/d' as det.ermincd in any approxi mation will huve an aver geuevalue other than zero. The Ax(c) obtained, sa, by integration of .ts Fourier series .;ould therefore contain a tern pr.oportiornal to cO in addition to a Fourier series. Thus, iQ() would not be a singlevalued function of rc anr. the resulting contour w;iould not clcse. Si.iply subtracting the average value of dAx/dcr (the constant term in its Fourier series), however, vill close the derived contour. If the method converges, this average value approaches zero in the suc cessive approxisrat ons. A preliminary r overall adjustment of an initially chosen CIiF ma; be desirable. Thus, if dx1./aQ is NACA ARR No. L4K22a calculated in terrs of the dAyo/dcP of a previous approxi mation and is found to be larger than dAxo/dp by some factor, dAyo/dc can be multiplied by this factor and the calculation of dAxl/dc repeated. Although the angle of attack may be arbitrarily set initially in this calculation it should be so chosen that the final airfoil will coincide approximately in position with the initial airfoil. After each calculation of dx/dcp, the zerolift angle PT can be calculated, equation (11), which thereupon fixes a, since y= a+ T is known. If the prescribed velocity distribution is speci fied as a function of percent chord, vz(c) must be determined in the successive approximations by use of equation (9). The quantity y = a + PT may be deter mined in each approximation as in the method of potentials or, in physically real cases, by equation (19). The diameter r is so determined that the successive airfoils are of a standard chord length. It is evident from the structure of equation (17) that near the airfoil extremities where sin T)0, and in particular at the nose of the airfoil where dAy/dp is comparable to dAx/dcp in magnitude, the convergence by this method (and by the method of potentials) will be comparatively slow. If modifications to the airfoil only in the immediate neighborhood of the nose are required, it may be more expedient to apply a preliminary Joukowski transformation, that is, to use these methods with the TheodorsenGarrick transformation. An example of the use of the CMF method of deriva tives to solve an inverse problem is given in reference 1 for the case of a cascade of airfoils. Method of Betz In the inverse method of Betz .(reference 7) an air foil and its velocity distribution are assumed known (fig. 11) and a desiredvelocity is specified as a func tion of percent arc. The new velocity and length of arc are specified in such a way that the extremities of potential are the same as on the known airfoil. Both known and iLujknown airfoils then transform into the sare IJACA ARR No. I4K22a circle end, in particular, the velocities 't points of equal potential oi. the two rrof?les can be found. In orDier to determine the prcfile co:responrdirn to the new v: icity, the corjplex dris;lIcerien t z2 z between points of equ.i'l potent al oi the tw'o p00i1' les is expressed d !s a fur.t on o:" th c :rr.sp:cni : l omp;;lex velocities (denoted b:' vz) thus, d z2 d/dz V71 ._2 z rI = dzI ') z7 _c/ v Hence z V_ z z = ( 1 dz1 (55) T v where the ".ntegr tion is cerrie,; ,jout 10o.: tn? knl.wn pro file fro.n the tr'silin. ?dge, w.ich ia t::>ein as coincident for the t'c Sirfoilq to the point The complex function v, / is Cdet r.rinecd pi:.rjo: ximr.tel' from the 1 , known rctio corresrond.In to the :points of equal v,_ potential by the er.ig',ent tbt, inssi..uch as the t,.o pro files have nearly tie s,,n :ilo6e. s corr E:n ondirng points, z t i S the real Dret of 1 is iven by 1. (This Vz2 v2 assumption, like the rppr'oxiinations ins tip Ci'.iF methods, is least valid at the nose of the airf'oil. The function z2 z1 is in fact a Cartesisn i:npping f nction.) The imaginpsr part is then computed as uhe c:nnjug~te function, equation (7). In addition to the resztrict ions on the velocity dis tribution mentioned initially, futrtier con.'iit ions must be nmet in this ;::ethod, if clossI co:iturs are to b? ob teined. Thus, the condition for closure nf contour, d(z z1 = f( 1) dzl 0 (V, 1(.5 NACA AE No. I41<22a and the required coincidence of v 2 and vz1 at infinity, lead to the following three restrictions on the real part R(p) of the integrand in equation (54) considered as a function of p in the circle plane, p2Tr =21T p21n R(C")dp= R(c) cos C dcp = R(cp) sin cp d( =O (55) Method of '.loinig and Gebelein The method of Weinig and Gebelein (reference 6) may be described essentially as follows: The given data are the same as in the Betz method. Consider the function log log i 6) log v = zl (z2 )(6) where Pz2 and "Z are the slopes at corresponding points of the two airfoils (fig. 11). Since !vz21 and I vzl are known functions of p with the data as given, and since log 2 is regular outside the circle, vz1 2 P1 can be calculated as the function conjugate to log V2 The angle Z1 being known, z2 is thereby determined and hence, by simple integration, the unknown airfoil coordinates are obtained. As in the Betz method, the condition for closure of the desired contour dz w/ dp = dp = 0 (7) JC Cj dw/dz V JC Vc /c z leads to the additional restrictions on the prescribed velocity distribution, ITA'A A'"7 ?To. LLTj22a 1 log vz(o) dcp = 0 J2 log vz( )I sin ? dp = sin 2y (58) o2 n log IVz()1 cos Q dr;: = '(1 cos 2') where y is given by equation (3B). Discussion of the Various Inverse Hethods The methods of B3etz and aof V'einigGcbelein may be somewhat narrower in scooe than the CLF :ncthods. The use of mapping functions such as in equations (.55) and (56) is based on the ability to srecify dz2/dz1 unambiguously in the corresponding regions. This requirement appears to restrict the contours obtainable by these methods to those bounding simply connected regions, l'urther investi gation of this point is necessary, however. By the CIIF methods, figureeight contours Lave arisen in the course of solution of both the direct and the inverse problems. (See the 9percent chamber derived mean line (fig. 5(a)) and the illustrative examples in reference 1.) Such con tours were first encountered as preliriinary results (unpublished) in using the method of potentials with the TheodorsenGarrick transformation. The CI.F apparently makes no fundamental mathematical distinction between simply connected and figureeight contours, for although z  must be a singlevalued function of z, t, or p, the coordinate z itself is of the same character as t and the latter has two Riemann sheets at its.disposal in consequence of the Joukowski transformation front the I to the pplane. The methods of Betz and of VcinigGebelein require the numerically difficult closure conditions equations (55) and (38)) to be satisfied in advance. If the methods are worked through for prescribed velocity distributions which do not satisfy these conditions, it appears that 32 TACA ARR No. LT4K22a open contours result. In the CNF methods, however, there is either no closure condition (method of potentials) or a numerically simple one (method of derivatives): )2T, dAx d 2m = 2T dAy J0 d d 0 d c [This simple closure condition in the method of deriva tives is fundamentally a consequence of the fact that the required absence of the constant term in the inverse power series for the C1F derivative mapping function ipd(z t), mentioned previously) automatically ex dp cludes the inverse first power (the residue term) from the power series for d(z t)/dp. Thus, physically impossible velocity distributions lead to open contours in the BetzWeinigGebeiein methods and to figureeight contours in the CMF methods (if the latter converge). From the practical point of view in these cases, it may be easier to obtain the airfoil corresponding to the "best possible" physically attainable velocity distri bution by the CI. methods than by the others. If the succession of airfoils determined by an inverse CTU method is seen to tend toward the development of a figureeight, the successive approximations can be stopped at the "best possible" physically real airfoil. As to the existence and uniqueness of a solution to the inverse problem as stated, a rigorous discussion of the solutions, for a prescribed velocity distribution, of the controlling equations (17), (7a), and (9) is lacking. For physically possible velocity distributions, however, specified as a function of percent are, the ' !inigGebelein method shows that there is one and only one airfoil as a solution. If, however, the velocity is specified as a function of percent chord, some further condition is necessary. This requirement is evident from the fact that one velocity distribution for an airfoil can, for differently chosen chords, be expressed as a different function of percent chord in each case. One chord with a given velocity as a function of percent chord can therefore have more than one corresponding airfoil. There is reason to suppose that the further condition for uniqueness of solution in this case is, the chord being defined as in the section "The Direct Potential Problem for Airfoils," that the ordinates to the airfoil at the chordwise extremities be specified. "TACA ARR o. L4122a From the experience '.vith tle C".TF nmethds g hiredd to date, it i tblleved thst tc a velocity Jictrio.ution specified as at the buFginnrirg j part II, ard w'.th thie further condition nentione in tLe opecentchordl case, there corresponnds onie end jnly ,one clsed ccnt'iur satis fying the rlF s:ste::a of elqu,tioni s. t is "urth .r ,o e believed .hat ihe Ci' r nethcr. s are fle::ble enough to converge t.: thi.s ~C: it on ir. at lea:t tntsc cas.s of aerodyn n.1ic intrest. CO'ICILUSIC S 1. Ti: onfiorrmal trarsformtion of in airfoil to a strain t lin7 bTr the Carteslia;, m i: .' ir. lfunc t io (CIlT) method results ir. single.' nu."rical olitions of tnr direct and invrse potential problems for .irfriWls than have been hitherto. availa'.le. 2. Tlhe us of super.osition w tb th, CrIF t.iethod for airfoils vrovi].ds a ri orous coLunter;'t of the approx.mae micl cds o' thinairfoil th. or7. Langley :vleriorial Aerorautical Laboratory national A.dvisory, Coru.:ittl.:. for Aeronautics Lariley Field, IF. INACA ARR No. L14K22a APPENDIX A THE CALCULATION OF CONJUGATE FUNCTIONS BY TE RTUNGE SCHEDULE The basic calculation for the type of mapping func tion treated in this paper and in reference 2 consists of the computation of the real part of an analytic func tion on a circle, given the imaginary part, or vice versa. To this end the conjugate Fourier series, equa tions (4) and (5), or the conjugate integral relations, equations (6) and (7), are available. This type of cal culation appears to be fundamental in many kinds of twodimensional potential problems. For example, the solution of the integral equation relating normal induced velocity to circulation in liftingline theory can be solved easily by a method of successive approximation if the transformation from the lifting line" to the circi! is Inown. Quicker methods of calculating a func tion front its conjugate than those given in this appendix or in reference 2 would therefore be highly useful. The use of the Fourier series rather than the integral relations in the calculations of this paper was based on the following consideration. Because the func tion 1/z is regular outside the unit circle, the real and imaginary parts of 1/z on the unit circle, namely, cos Q and sin 9, satisfy the integral relations (6), (7). The substitution of sin cP for Ay in equa tion (6) and subsequent numerical evaluation by the 20 point method of reference 2 gave results that were higher than cos P by a constant error of 2.8 percent. Evalua tion by a 40point method reduced thecerror by half, or to 1.4 percent. By the Fourier series, on the other hand, the first harmonic (a onepoint method) suffices to give exact results in this case. It appears, therefore, that when the given real function is expressible in terms of a small number of harmonics, as is the case in airfoil applications, the Fourier series method is preferable to the use of the integral relations. The Runge,schedule offers a convenient means of carrying out the basic calculation of mapping functions, namely, the analysis of a periodic function into its ACA A 1R Io. L~4T22a Fourier series and the sycI.esis of a ou'rier series into a function. Ta3 theor a'.l. use of the schedule is described, for example, in reference C, herein are also given schedules for 12, 2', 56, and 72point harionic analyses. The n=cess.ry analyses and s,nthecses in the direct and inversL Ci.F methods are carr:.ed oit in accordance with equaticns (1I) and (5) and their derivatives. Table IX contains the scher!e of substitution into the Runge schedule, table X, for the various C2T^1 methods. In the dirct method, for e:xamr: e, the set of vr'lues 6y/12 corresponding to the evenly sp..ced Ovalues is substituted into the yn spaces at the beginning of the sumtable. The suns nnd differences of these qu'nitities are then obtained as directed at the left of the indi vidual tables a.nc substituted into the succeeding tables. In tnis wa tlie ntie sLuntable is filled out. Before the producttable is used, the s.ulmtah:le should ce checked. The quantities sirroun6ed b:, thie heavy lines in the suntable are ne::t rmultiplied by t!.h proi"r factors at the left of the prodJucttable al.ei, t. results entered in the apprc.pr.ite spaces as indicatdc by the letters at the left cf the individual prodLctspaces. L, heavy horizontal line at the lower le't edge of ? product space indlcictes that thle correspronding pro.luct has already been ob!': ined in a previous p.ce in the same row. A heavy vertical line Q&ong the left edge of a productspace is used to .o.iphas.ze that the negative value of the product of the sumtable quantity andr the producttable factor is to be entered. The I'ums of the producttable columns are then entered in the I, II, III, and IV spaces. ;. che.c on the v'irk of the producttable up to this point is provided by the columns at the right. The sums and differences of the I, II. III, and IV quan tities complete the producttable and give the Fourier coefficients an, bn corresponding to 'y. In order to perform a snthesis calculation front a set of Fourier coefficients an, tn to the values of the corresponding function aL the even rpoints, the coef ficients an, bn are entered in the d and D spaces, respectively, in the suntable, and the ret.ainder of the suntable and the pro .ducttable worked thro,.igh as before. The final values in the a: b spaes of the product table are then entered in thel d and L spaces at the beginning or the suntable r:nd the suns and differences 36 VACA ARR No. LQK22a obtained as indicated by the synthesis column at the left. (Note that do and d12 are to be multiplied by 2.) The resulting Yn quantities are the desired values of the function. The numerical values in tables X(a) and (b) illus trate the process of obtaining 6x1(p) from 6yl(P) in the first approximation by the direct CMF method for the NACA 6512 airfoil. RACIA ART Noo. LLi22a AP.P LDIX e, THE iiLPPIlIG OF i'ORE GEiV'AL RE"ICHIS Simply Connected Fie.Zons If the Cl,'IF nethod is app4Id to the napping of a simply coni;ected boun.dary v.ith a vertical discontinuity, such as a recta;ngtle or an infinite line with a vertical step, the nr'bigu itv "f tilh ondiJ.ite Sy at the discon tinuity will prevent an automat a nut.tc an rapid convergence of the mthi,od. ithour.h the dil'ficultv could be lessened in particular cases such as for recitn.igles. byt taking the diagonal as xaxis, thus rem'ov.ing tlie vertical discon tinuity, or by using syiu'netry,, as with squares, it is evident that in general a reference shape particulilcly suited to the contour under investigation is needed. The circle has been shown in reference 2 to be a good reference shape, for the square. It could be expected d therefore that an ellipse wruld be a ood reference shape for the re ctangile' 'urthr.mo just as the apping function based on the circle ,vs formed of an angular displacement an..d j rPdial di.slacennt, theF appin function based on the ellipse should be forried of dis placements elorn and ortho'c.ntal to the ellipse, that is, should be specified by elliptic coordinites. The speci fication of a figure byr elliptic coordinates k(P, 0) in the physical plane z is equCiralent, however, to the transformation of the figure to a t'plane by the two transformations 1 +iB z = p + where p' = e (59) t' = log p' 'where t' = W + iG and specifying the transferred figure by the Cartesian coordinates of the t'plane h(, 8). Th_ rectangle under considerat ion vrill be a near circular shape in the p' plane and a nearstrLaig t line shape in the t'plane. The mapping of the rectangle by. means of an elliptic mapping function in the physiclc:i plane 12 therefore seen to be accomplished b the Th odorseaGarck method in th nearcircle p'plane and b : the CMNF method in the NACA ARR No. L4K22a nearstraight line t'plane. Fro this point of view, therefore, the TheodorsenGarrick method consists of specifying an airfoil in the physical plane by elliptic coordinates, forming the corresponding elliptic mapping function (' 4o) it, which conformally relates the airfoil to an ellipse or Joukowski airfoil as a basic shape, and expressing' the elliptic mapping function as a regular function outside the circle. On the other hand, in the t' = log p'plane the TheodorsenGarrick method consists of the transformation of the nearstraight line *(8) to the straight line C4 = Constant by means of what is now the CMF (f xo) ic. Thus, the TheodorsenGarrick method may be regarded as a form of the CMF method, in v'hich log p' takes the place of z and log p, the place of t. The mapping of simply connected regions by dif ference mapping functions based on the curvilinear co ordinates appropriate to the particular reference hape considered is therefore equivalent to using the CTU dif ference function z t in the pl.no of the nearstraight line into which the reference shape is initially trans formed. T.,pping of the Entire Field The Fourier series representation of mapping func tions, equations (i) and (5), enables the calculation of corresponding points in the two regions to be made, once the correspondence of the boundaries has been calculated. By the latter calculation the coefficients an, bn and the radius R of the circle of correspondence have been determined. If now a larger radius R' > R be substi tuted for R in equations (4) and (5), the resulting synthesis of the Fourier series will yield the mapping function for the circle of radius R'; that is, will determine points in the given plane corresponding to the points in the circle plane at the distance R' from the origin. It is necessary, of course, to use the mapping function in conjunction with the shape in the physical plane corresponding to the larger circle. In this way the entire corresponding fields can be mapped out. It may be noted that substitution of R' < R for R in equa tions (4) and (5) enables the mapping of those corre sponding points inside the original contours for which the resulting Fouri er series converge. IPA'JA ARR To. IJ.1Ti22a 39 It appears 1: o ror ,r ':ifif'lcult to finri the point. In tlhe circle pl.an rlrre.p.nd.Ln. to. .poi.t of the L.'iv.en plane than ,ice rs:a. Ti.s Cl.clatL on ai:, hoeveCr, be acro..Tplished by r:ethod o:' uccessive apFpro':i:a' tins. For cxamrle, if the i ven pln: is that 0o' s near circle the pool r c,:.ordi. iates of tlie i.ver.; r.,int in the neal'r cirPZ; pine a~s,: azsstU,'ed to be a first approxir;ia tion to the coordir. te R' and ,' of tih '"V siired point in tihe circle rlRne. aubstitutiop f" t2 ' values into eq'.u tions (i 4 and .T I i=lds a i:" t a' prox;im j te mapping; function wh'icn can be used to *:orrl..ct the coordinates it' and rn, etc. ip lanes In the case of the biplane arrr.nge,,lsnt the CiFP may be set up directly in the physical pl._ne in the same way as for the sinw'le airfoil. In place .f the simple trans formation froi straight line to circle, !n'wever, the transforcimation f'r:o the two etend:' chord linEs )of the airfoils to twc c conentrc circles i.' cd. 'ihis trans format ion is deri'e'id in r.ef *,er,ce "'. Tie MF'' me thod for biplanes bears the se rs c relating to t.:. netho.: of ref erence 9 that the ClF rviethodl f.r onoplae ifeils bears to the ThlieodcrsenG'.rie ethod (refer r'.e 21. For biples (fi.. 12) the C.' a C, being regular in the reY4ion out i. tde the two str'ai.lht lij.es, is regular in the anrnuir region of the pplain: and consequenti" is expressible as a Laurent series in p where (Il') ,n = a, + ib, If, for the inner circcl,, the relationship; is written z = L:1 i p = Re NACA ARR No. L4K22a and for the outer circle z = Ax2 + i Ay2 p 2e R P (42) there is obtained, upon substitution into equation (40) and reduction an + an b n bn Axl() = ao + cos np + b n sin ncp (43a) R1 n RIn 1 1 Co Ax2(cp) = ao + 1 C0 an+ a_n cos nQ+ R2n 1 CO OD Sbn + bcn an an Ayl() =bo + cos n sin ncp R1n Rln 1 1 E bn + bn \ an, an Ay2(cp)=bo+ b+ n cos np \ sin nCP R2 R2 1 i 1 (43c) (43d) These equations are the generalization to the biplane of equations (4) and (5). The corresponding integral rela tions may be derived as in reference 9. The solution of equations (43) in either the direct or the inverse problem may be accomplished as before by successive approximations. For example, in the direct method the two airfoils are given. If no initial approxi mation biplane were available, the two chord lines would be taken as the initial straight lines. By the trans formation of reference 9 this fixes the chordwise loca tions on the straight lines corresponding *to a set of evenly spaced Q points on the concentric circles. The  bn sin np (43b) n2n FACA ARR io. L':22a or6din.'.tes u f ;' c ;n tnerc fore be mIasur:r hr ich deteriL:ijines Ly2() by analysis r'd yrtne.esis of equ' tion.r (5al) rand ( .d), resp ctivlyr. (The radius ratio 2/1 is fixed by the initial Ltansformation f',ron the stra.' I t lini.e tn the concentric ire'l s. ) Thes; .ly; (,') values then de teri,ine set of Lx2('?( values u'" t"he givr3n slhaprj of she second .,irfoil sand the known chlorldwise locations of its first aprt';.roximtion str.iht line. Analysis or L::( ") and su.bse.quent zynthesl s of Lx: ') by eiuationns (+4b) arid (5 ) respect el', det, 'urmine s a correction to I1 b a J. hor.izoItal s tixetch!ng proiccess (constant Lx, A;: adjustment of r ) to rtaintrin the given Esi;'f,:il i ic .rd. The r'DC. 'LL re i, n1 ow re:,Peated 'I.J th AY2(Pl aS the initial set of ietas..ur:d ordirtnate th:t. determines y S c), Lx1( ;, and A,1"`Q) as before. The radius R2 can now cb siri.ilarl corrected. Ths step completes the ~. rt appro ' ir'at ion !. Fior the second ll. d roxi mation a new corresnondence bet'erier ti e c:'rrected t'traigtht lines and the once riLi. c '.rcl is .".1cul;. ted and tlhe proce'.,ure repeated. The iP verrst problem could ls_ L.e soSied b'r methods similar. to those iv ;7n for the isolated arii oil. Sup pose, for exa'r':'le, a v;ini' section wivN e gi'"An and it were desired to der' '.e a slat of c .k 2 r'd ani giv.'n a'pro.:xi mate loctioti san,' habing a pr.scribe. velocity distri bution. Thei method of surface p>ote"t ials, for example, enailes the ca .i ilc L' _u o,'I a fij.st 'sproirte I1 i A '1 (sub'cript 1 r I ers tro sitt). The initial. correspondence of points betwreer the st: i,1't lin n:s ad concentric circles, a.nd. tli before also R2/, b inL det errined by the initially a.esi.nume straiIit i nes, thie function .. ( 4) is thlereupon o'ta inod by. analysis ajn' synthesis of e.r.a tions (i5a) and (I b), res].,e active ly. 'Te horizontal dis placemrtent L.x2 ) thence determine L :2j) by the known shape of the rmain wing section. The dieterm..ination of L y(D) br analysis anty. s ynthes of cu. tions (J4d) and (1i 3c) :.ominpites thie caicul ition of the first ap)proxi rmate slat section, for vhhLch the ex'ct vo.locit;t oistri bution can now also be calculated. If tihe main win:l section were al o unknown then the a~i n' section abo'e is regarded as an initial ap:nroxiiri.tion, the role of the two airfoils is revei:rsd, and the procIedure repea.td to com plete thn first eaproxirn nation. NACA ARR No. LkK22a The CIF method can be generalized in the same manner for multiply connected regions. The transformation from the n reference shapes (such as straight lines) to n circles being presumed known, the CMF can be set up as a series convergent in the region between the n circles, and the mapping function for each boundary explicitly expressed by allowing the coordinate vector to assume its value on each boundary in turn. A method of successive approximation for the solution of the resulting equations depending on the particular problem under consideration would then be established. Cascade of Airfoils A simplified but practically important nbody problem, namely, the cascade of airfoils, may be mentioned finally. The reference shape into which the cascade of air foils, figure 13, is to be transformed is chosen as the cascade of straight lines coinciding with the extended chord lines of the airfoils of the cascade. The trans formation from the cascade of straight lines to a single circle is wellknown, reference 10. The CMF chosen as indicated in figure 15 is therefore expressible as an inverse power series in the circle plane and the resulting procedure in either the direct or the inverse problem is seen to be essentially the same as for isolated airfoils. The detailed application of the CMF to cascades of air foils is given in reference 1. ILACA 'BR Io. L4IK22a APPENDIX C THE DETFn:t.lI'I.TIC1J OF iLAPPTIG FU OCi IOiS BY THE C4UCiY ITITTFGR\L FORITLA The foregoing nrthrods of confer, ;l t.rrsforr iation have been presented from the point of' viewl of represen tation of the various mapping functions as infinite series. In partcular, cho expression of the C'iartesian mapping function as an inverse p,;wvvr series vJi:.d every where outside aind on a circle led to the Fourier series representation for the C'?IF on the circle itsFelf. The integral formula represe.itn.taion vas ;:hin obtained from the Pourier series by the tietihod 0of refer ence 5. It is of interest to see hov, the integral rlations (6) ar.I (7) can be derived dir.ectly from the Cauchy integral formula for a function regular outside a circle. (These integral relations have also been derived 1oy 7 etZ, reference 7, by a hydie,o:Jyna!ic Al1 argiunent.) Sin:,ce the ar plicrbility of the Cauchy interpIr iforrula i.s iot rstrictertd t) circular boundaries, however, the results vill be capable of generalization, in principle at least, to arbitrary simply and multiply ccnnec;ted regions. The Caucny integral formula gives the values of an analytic function f p) within a si.t:ply connected do main D in terns of its 'ralueZ f(t) on the boundary of the domain as f(p) = rit (Ut) 2i1 t p where the path cf integration is cointerclockwise around the boundary. Consider the domain D outside the simple closed boundary C in the ppine (fig. L). This domain can be made simply connected b" an outer boundary B and the cuts betwenc thj two boundaries, as indicated by the dotted lines. The Cauchy integral formula for the func tion f(p) at an interior point n of the domain D, in terms of its values on tihe boiudary is :fp 1 = f(t) 1 + f(t)r f(p) f(t) dt +  (t dt (45) 2i i t p 2ni t p JC B ITACA ARR No. L4K22a where the equal and opposite integrals along the cuts have been omitted. The paths of integration are indicated by the arrows in figure 14. The function f(p) is as sumed to be regular everywhere outside the boundary C and in particular to approach the limiting value f, as p. If the boundary B is enlarged indefinitely, the integrand of the second integral of equation (45) approaches f,/t and thus lin f(t) dt = f (46) 02lTiB t p t BV B The variable p will now be made to approach a point t, on the boundary C, and equation (45) will consequently reduce to an integral equation for the boundary values of a function regular everywhere outside and on the boun dary. In order to evaluate properly the contribution of the remaining (first) integral of equation (45) in the neighborhood of t', the boundary C is modified as indicated in figure 14. The point p is made the center of a semicircle whose ends are faired into the original boundary. As p t', the modified boundary approaches coincidence with the original boundary. The integral over the modified boundary is now evaluated as the sum of the integral over the semicircle, which in the limit is half the residue of the integrand or f(t'), and the 2 integral over the rest of the path, which in the limit is analogous to the Cauchy principal value of a real definite integral of which the integrand becomes infinite at some point in the interval of integration. Equa tion (45) therefore becomes, in the limit, .f(t') i f(t) dt + f (47) 2 2 i C t ti In addition, there is the auxiliary condition that 1 P f(t) 2 j t dt = f (4) 2ni L1C C which follows from the fact that f(p) is regular every where outside the boundary C. Equation (47) is well known in the theory of functions of a complex variable. (See reference 11.) I:.'2 .1 ; 'o. L7K22e If, now, the function fp) is taken as the Cartesian nappin funictio.i z 0or , o: the Li., und:ary, f(t) = Lx i y, (hc9) and if, furT.Lher, thLe bounda:, C is tAe, n as a circle with origin :t the center, t = C t' = e  substitution of equations .,4) anl (') into equation 1 7 and using equation (48) (with f = ') leads to the inte gral relations (A) andr (7). if uhe poiar alL..pi: fiiunc r) tion log = i (. i i(p ) rcferencz 2)  is substtitted for ft), the Theod ) e ni:,l icl integral relations are oht.ined. The Cai.ch. ite '. 1 forrmu.la has alrsead:'r been applied (reference 12) t ro",Lcrn: .:f c :,nfornial. n ,pring ".rn the m nner just lrdi iated. E, rrnan rs included in refer ence 12 (charter XI) contributions of' two Russian authors, Gersicori and Krlo'.. In ref'e;.nce 12 the mapinnL funrc tion from a circle to s near cir.le was ta:en in the iorm log p. The resu;.itj n, nteT;.l equ'.tion does not appear to be a.s convenient a ths of te of e C'' mT.ethods. The use of forms ,ch s log  or z I are not only accurate nur:ericraliy sinr:e tr.i. e:rxpress lapnges in the coordinates of the boundaryi s, but siso th,? l.ad to pairs of integral equations wjhich contain tl: solu'..l; ions of bothl the direct and the inverss :)otential problems. Prom the anilysis given it appears possible to trans form conforrnilly i'from', one bound Jlr to another wihout requiring the. trai'nsforrl::a tion from e either bounrdary to. a circle, sine: the boundur.l C in equation (47) can be rather arbitrary and '(t) can be t:ai::en as a ma,pping function frorim this boundary. to Anoth..r arbitrary one. The resulting integral equation for the napping function is, however, not as easy, to solve numerically as v hen the bound ry C is circle. NACA ARR No. L4K22a Once the conformal correspondence between two boun daries is known, corresponding points outside the boun daries can be determined by the Cauchy integral formula (l.). It is noted that the Cauchy integral gives the correspondence of individual pairs of points rather than the correspondence of entire boundaries at once, which is given by the Fourier series representation. Further more, the possibility exists of determining pairs of corresponding points inside the given boundaries by the Cauchy integral, that is, of analytically continuing the conformal transformation beyond the original domains. For if the transformationfrom a boundary C in a pplane to a boundary C' in a p'plane were known, the outside regions corresponding, then the correspondence between a boundary A internal to C and a boundary A' internal to C', if it existed, could be determined by an application of the Cauchy integral formula to the region bounded by A and C. For example, if the boundaries A and C are taken as concentric circles and the mapping function as f(p) = log t = ie = (I X) i(( ) (51) in the notation of figure 15, the Cauchy integral formula applied to the annular region in the pplane (assumed free of singularities of the mapping function) yields, in the limit as the variable point p approaches the inner circle A, 1 2T 1 C~1 1(cl1' ) = 2 (cP) cot 2 dcl 1 ,23 co(Po) sin ( L n{21 ESCPo) sin (0 R) j(cp) sinh (XX (52a) 2JO cosh (X ) os (o0 1) 1 P2 l I ' l2 01r') ) cot C 2 dcp1 + I2n o(po) sin (c 0 ) + o(Co) sinh( X1) (52b 2J00 cosh (X X1) cos ( cq') 0 HA.CA ART 'To. L4!.22a 47 In addition, the condition of regularity of the function f(p) in the annular region yields the auxiliary condi tions 1 2TT 1 2Tr 1 1o 1 = yode 2 T 0o 0 2r (53) 1 P2* d /2, 74d P = kj eo ^ o In the problem under consideration, the mapping function P (co) io(co) for the cuter boun::aries is cnown. The radii e", e' of the concentric circles are given.. The second integral of equations (52) are th',.s k!nowvn fi.unicticns of nl'' Equa tions (52a) and (52b) thfiefore cos.titute a pair of integral equations, similar to those of TheodorsenGarrick, for the mar.ing function () i ), pertaining to the inner boundaries. It is noted that if the variable point D of the Cauchy integral forrr.ula for the annular region is made to a,,oroach the outer boundary C, then two additional integral equations similar to equations (52a) and (52b) are obtained. These equations, together with equa tions (55), are a generalization to the case of ring regions of the corresponding TheodorsenGarrick equations for simply connected regions and can be used for the conformal mapping of near circular ring regions. HACA ARR No. LK22a REFERENCES 1. Mutterperl, Ailllrmi: A Solution of the Direct and Inverse Potential Problems for Arbitrary Cascades of Airfoils. NACA ARR No. LiK 22b,, 19JL. 2. Theodorsen, T., and Garrick, I. E.: General Poten tial Theory of Arbitrary Wing Sections. NACA Rep. No. 152, 1953. 3. T1ilii.:n, Clark B.: An Extended Theory of Thin Air foils and Its Application to the Biplane Problem. NACA Rep. No. 362, 1950. Allen, H. Julian: General Theory of Airfoil Sections Having Arbitrary Shape or Pressure Distribution. IT.CA ACR Ho. 3G29, 1943. 5. Garrick, I. E.: Determination of the Theoretical Pressure Distribution for *':'nty Airfoils. NACA Rep. No. 465, 19533 6. Gebelein, H.: Theory of TwoDimensional Potential Flow about Arbitrary Wing Sections. NACA TM No. 886, 1939. 7. Betz, A.: Modification of WingSection Shape to Assure a Predetermined Change in Pressure Dis tribution. NACA TM No. 767, 1935. 8. Hussmann, Albrecht: Rechnerische Verfahren zur harmonischen Analyse und Synthese. Julius Springer (Berlin),1958. 9. Garrick, I. E.: Potential Flow about Arbitrary Biplane Wing Sections. NACA Rep. No. 542, 1956. 10. von Karman, Th., and Burgers, J. M.: General Aero dynamic Theory Perfect Fluids. Application of the Theory of Conformal Transformation to the Investigation of the Flow around Airfoil Profiles. Vol. II of Aerodynamic Theory, div. E, ch. II, pt. B. W. F. Durand, ed., Julius Springer (Berlin), 1935, p. 91. TNAA ARR No. TLtTK22a 49 11. Hurwitz, Adol f, ard Coil.rant, ?.. AlP.l'erei ne Funl:tionenthcorie und elliptische Fur.!:tilonen, and Ceometrizche l.iirdutionerntheorie. Bd. TTi of VtchermaLischen ."!is enschaften. Jillnus ringer (Berlin), 1?29, p. 5. 12. 2erymran, Stefan: Partial Differential Equations, Advanced Tooics. Advanced Inrtructicn and Research in mechanicscs, Bron UTniv., Surmaier 1941. HrACA ARE No. L4K22a T'LTLE I C0rTESTh' r..'AP..II3 "U:DTI,,?: F?" SY'T..IETRICr,.L OPEr'F'I THI. '.iFS. JOTTj:O TI3 Pr.F'ILL TAPLTE T'SD 'cITJH COJ '] NTS II CPiF OF Profile T T u r T u _____________t (g_____) (.deg) Joukowski 0.30 1.100. 0.087 1.250 0 10G 1.000 Derived .24 .3 o .0716 1.155 0 180 .355 Derived .12 .1022 .0357 1,09231 0 1s80 .55 NATION_, ADVISORY CNITTEE FOR AE ONATI CS NATIONAL ADVISORY CO;,! ITTEE FOR AERONAUTICS TALE I IACA ARR Ho. IxK22a TABLE III CM!F FOR 6PERCENTCAMBER CIRCULARARC PROFILE (radBans) Ao dAx 0/dc dA o/dpc 6 x 0 0.120 0.108 0 12 7 .0270 .114 .0960 .04+3 8 .0482 .0955 .0658 .0371 9 .0592 .0694 .0171 .109 10 .0565 .0405 .0563 .io6 11i .oko .0160 .0o344 .0781 12 .01L2 .00169 .115 .0279 15 .0170 .0021o6 .117 .0346 14 .0o49 .019o4 .0352 .0926 15 .0587 .0490 .0239 .123 16 .0552 .0328 .0506 .125 17 .0555 .110 .113 .0756 18 o .120 .156 0 *I___ T, BLE USED ITH IV CM"l OF TABLE III NATIONAL ADVISORY COI1vITTEE FOR AERONAUTICS COISTAINTS C (P aI ] Profile (per uc Xur T r (de) (deg) (deg) c ideal cent) U ( Derived 5 0.502 0.501 0 1.0052 5.37 183.57 o 0.57 Circular arc 6 1.000 1.000 0 1.0072 6. 18 186.84 .75 Derived 9 1.502 1.199 0 1.0050 10.26 190.26  NACA ARR No. L4K22a r4 n OU0400 lOWHO 2HnV* n"mONrm'4. ' nOoooN Immop Mono mew o 0 0 ) 010i 0 00 0 I lV) OD C OLe On 0 ngOtOe OOOw OMWOO O00@ imrO' 0 C F)DI030 4'0 L 0CD I 0 0 t I m 0 S0 l 0 VOcl r) D V) CO 104 CD f? 4 Lt CI0 10 0 tn 2 o % OA OP rl CO t O So "o 0 0 1 4 i> GO 00 O00 0 ** ***** *. . CNO 0 0 H 1 u v 10 oo P ( c 8 3OO)C o O Mo C WO p 4 0 H Ix CIO O OC1 > CM C 00oooi000 00000000 1 t toCN OfrCD Q mamt tO LVO CD* 0) VO) Ol a rO 666 565O 0eOOO OOOOOOO. 14 0 1 I II 1l I I J 0 0 CO ID 40)00)6 1000(00 0 0 0 0t 00 il (000OO DC C0Ql OOO tO CO'OD CO < O3 ONr0 0e4CM OOOr 00 000 000000 5 *a  6 6 5 5*. *S* ** Ce ** 0 o l l II 111111 e 0 0)0 r4O OO ( r00 0 OOOOOC4 a 6 6 e e 5 *. .5.5* ** 9*@* * 0 11 11 1 11 1 III 0 I r a 0l 10 O0 O a 0 0 C 1OiCO 0 O .VD0IO r40 000) 0 10 iIO OlfJt' CM)OcJ 14 0 Olr40)O.2 WOI'1ri0 000000 00000000 0 11111 I I a) X in) S0 l SC In cL 0 C3 op 0 1 0 c M1 co o n N '41 oo> aoOi 0C o n O)o Ca W M 0 to to o 0 awo t io I cD a)hame o mbooo a on wno 0% 0lC C 00 C1 0 r41 0 000 C. 0 ) "I 0 M C c a 0 CM m vl v' Go Vn r ig m ca In a) cm co M rv co r4 o to o Mt>aloo Sl 0 ) OCDOIO rI 00 0 t 0 000 0 M CO C ID M CD 0 CQ) 0 0 ) v :O' O I) or = o touwool oooo oooif oo ooo coajmo CO.) 1 4 0) CC) r4 L V cc N0 0 0 S3 < ***** I* 9 1 C 0a 0 0000 00000o 00000 00 H 0 M n rI v O i)O0n N 0Il n ) N n O ri S 0 0 1 I II I 0 II I l ZC OO rI0H r0 r0C4OO0O CO00 I) CO B0)I a n C D~I 0 000 0 r0 00 O 000 000 *. .. C > C C * 0 1 III I giI S I rHOV Ht) t CD) LHOID (0WO OO OmUaV H0 C0 a oD G0)WO) to V) Ul 0Wck]O HI CO tfU io v V \L o 0 0 OmGo a t nn io O oooI 4O l 0o i Y)a)cm oo m H O m) VCJO i O3 C0000 1 HO V V)0 I* *a 9 0 C a C C G a v a Cl0a 0 1 1 1 1 0 I 1 11 1 I P 'D M CMO ) CO0w w 0m 0 CM 0 O 0O C I 01 NJ 18 Wm H 1 0 m COtotoG to 0 Lg DC 0fI 03 qI rt il to 0 le Cwrf)OVlO ti) 0 N ID Cl 0 00 4 m o lH M 000000 00000 00000 0 0000 000 0 IDOIDoV o W O o nH v) oH mW t 02I L M No 0000 O 0rr CO 00 000 00000 000 C C C* ** *C * 0 I IIII I II H HwHnv HorHEOO Hm" 01 1 14 14 NOW r O rlcynN NO rHl MOOOO Osmon rlarlr OH O O O a Oa1w b m W NACA ARR No. L4K22a 0 I 0 o C 0 .l 4. 0 0 0 (D Po 0 C) 0 4. o 0 4 C) a, u ., 0" NACA ARR No. L4K22a a o CV OD OD r 4 C O o 0 H0 LCDCD ~a Vo EM ~ 00 0 0 CM D o L7 0 E E c E L >o .* . .** . T oo0tto 11 4 wc C Oj m 0o m 10 n 1H Wo I O~)mHHC HCc J L 'U c3OlO o.c C *. . ID U) Sm %r i)0cDO vOca) CwOwma)Om 4 00oaoo o cY Mj 0D 0 Y) II *l .l o X. oooo ao)OlOCDCO 0OCrj~s..[s. cOl Ot Cr'loo r. 0 OC e00 00 00o00 V 3 ia HHHOO OOoH O0OOO OO0 OO0 ,c .0 c i Co coACio c 01 C*n o oS o t oio inoOiop r nfi CE^(O B rl Co oo o0000 o00oo 0 o L OOOO DoOO o0 0 O uO) Il D 2c M 0 O OO O4H O 0H000 O0000 0 0 0 i C ** .** ** * 4 I ogmag0 mII 1 a O I * 40 S0 0000 00000 000 0 00 00 rI v OO 0 00 000 10 O0 oo W I 0O)rtO (Y) oo O DomWo n iNoW ctO 10 C o rl rl o0 0I olr r 000 0 0000 00 o oI . I I O ; O * 6n oo 30 l0 0 D00ID K0 wrm w CO 01 w0 0HN r Or"lWl il 1OO00r 000 O Or4l iI o 0 00000 00000 000 00000 000 B 4 000000 00000 00000 000 0 000 0 Ji ll II I lO 0 00000 00 00000 0000 000 000000 0000 00000 0000 O00 S0I ii 1 110 I 1 0 S000000 OOOOO 00000 0000 00 SK 000000 00000 00000 000 o ..... ":::.. ', I C [ 00 0 X H HHHI10 HHHH 0(00 rl ilrrl~r4 irl l <1il C (MNOI NACA ARR No. L4K22a 00 om  C) 1 0 I 0 4)H to Io O U Uoo o a 04) . 00 43 Q) .4 .p4 0 O S=0 X 0) 0) *04 C r > to HDO F, d HT rO" k 0 C rl lCO CH E Co 0 0 0 0 0 0 0 0 . 0 ~ OO E. K\ 0 o o 0 0 o o o j 0 0co 0  o 0 0 I L4 3 r S000 I  0 O \ CO I h4 0 C iD4 *U )0 r .4 p ) C O 40 0I HN *P IrI SI Co ft r ul N i ft E< __ ____I rk. 0 U2 A E' 0 0 C/3 ru NACA ARR No. L4K22a o 144 4.4 c% r4lN A . I I E4 .4 i M c o 3 C. 01 0 0 D t a% 0 ,0 o % D N z c% Low 0 0 I ir o M 0 IN 1 MlI 94 O 0% "IN ma O e% On ,O O , 0 0 0 P m0u ru 4 (NJ r4 0 4 I I4 00 00 %o LINfro% iD %N _ at a N4 000NC 44to "0c K . .0040000 0 0 00 4 000 13 0 0 00 0 4.. . 0% K00% 0a0 OIN4%000'0 .4 `4 O L"tCM cM UsCOD Cu* S4 li40000000000 43 o . I #%Do 4 t>so 0 CKUr'0%.4 r4 0 r4 4^4r4 r 0 0 0l 000000000000 ..4' %O '* ,r O%.4  S0000000000 ii 0rncu, to 0 r o o N 0 44 00040000 *N 00000000000 0 O3\33 OIlOOO o .4 0 I00 0 0,'4000 0 0 H cm IV (%j 0 0 i Io 0 0 ( "0000000000 O.. ............... .. ....... 0 30 00'0,.,.4 00' 40 4,o 0 S 04 O Olh COMM OWN! 0 II I I C a o 0 1x 110 0C .O O 0 M N H q..00000 00000 ,I o r0oooo _0 330 o 0 %0 (M K OCD'0 rf0l 0 0 33 ______________ .; rcr~roro~rr LU 0% r^ N ' o cu 0 0 0 0 o .4 0 0 .4 II II II II cO  0 0 ,l 0 II II O 0 6 m 0 0 0 .4 II II 0 0 b < 0 O1 ao  V 04 00 N cM ,a (.CO <,. aI ,l ,M 4 r O COOC . . .. 4 rr O\% n\ oCO 07 0 COu pN o L coocco a c .O3t KmN'r r4 r 4 l ,4 r4 4 S'r m cO  N 0 L  0 %0 oo U 'cJo t f% .r...  , N 'oW 0 o ifr u 8 0 I I I I I 0 I uj t u'x 0 0 . W rNo oo oor % r ooo ooo olIM 0 oooo o0 ie 0 I I S I *0 N y\Cuo ,4 .rC4 o4 ______ o o 0 0000 r 00m 000O S0000 00000 OOO rO tri ~ M 14 E CO I rKff rev ooooo 0000oo oo o.4 '4Noo so M 1 0000 00000 o00 i0 14 r4 ,.M ,,D ,..d , O 0 tm S OOOO OOOOO OO O ; e ; ; o S O OOOO 0 0OO 000 oO ro aI I Cyl 0 409 r4 A: t^O r t CM N  0 rr4 11 r r4 r0 0 M N M 0 0000 00000 000 o . . 0 I r' I I I' I I0 M0',0 t r.4.r4 r4 NC& K 4 ri rI CM 00H 0000 8000 00 ~i p. 4 mN K\4 UL% 0CD S r49I r44 4 i4 'NACA ARR No. L4K22a 0 0 o Hi II '4 e a 0 II .II r  ',o rCDo'a o 141 NACA ARR No. L4K22a TABLE IX TIE USE OF THE RUNGE SCHEDULE'IN THE ANALYSIS AND SYNTHESIS OF CONJUGATE FOURIER SERIES Process Entry in schedule Result Direct method Analysis Enter a for Yn a, b 12 an' b Enter in Enter in dn spaces Dn spaces bn an 6x Synthesis nan nbn d6x/dp nbn nan d6y/dp Inverse method of potentials 5x Analysis Enter for yn an, bn 12 Enter in Enter in dn spaces Dn spaces bn an 6y Synthesis nbn nan d6x/dp nan nbn d6y/dp Inverse method of derivatives 1 dAx Analysis Enter 12 d for Yn an, bn Enter in Enter in d spaces Dn spaces bn an dAy/dP Synthesis bn/n an/n Ax an/n bn/n Ay NATIONALL ADISOr4 COMMITTEE FOD AEiRONaTCS 59 NACA ARR No. L4K22a x i x .j 1 S; .L0 i c__ T  T : 1 U CD a o ', w a J o' to 5 . S0 I _In 0u +" 1 h : 1 1 hi ? _ii , + S r j. ?? "^H  ^ NACA ARR No. L4K22a 60 :;, ~^^~^7 IS 4 4 4' . 4) !' x Px .4 4). o'~ ? ^ ^ 1 4 4 44.4 . 4 44 4 , y ) ^ .o 4 i I I I I; : **' ' '. 4. "vl 8'4i~ a ai 3 Crll *. 4.. I I I ? '*' * 4 4 >_. _ 4. C i, 4 I4 I* < I * 4) ;* Cii t i' *, a> , *, 4 4f .. 4' * *i. ~~ ~ 4, 7Q i *. ** . F 4  ; I i i ' '* "' F i tilt ,, I I) 4 ) 4 4 4. [ : '^ s < 4 ? 4 . i ' 4. a 4 *.4 4) c 4)' :  4. 4 t ? i  I I II I : 1 1' 1 1 *s ? d l  61 NAQA ARR No. L4K22a z Z q a Z o 0 W 4. S , 4 I Li fl 2! I Li LaJ ^ , _,,* i 1 m H NACA ARR No. L4K22a 62 Qla ~.I  ~ * % Io S is a I a on t . r CO C' ** 9t t s \ . I I I , I ." ^ , \ I. IP'. I 1 .: ' i < ~~YY\~~ ^^^ NACA ARR No. L4K22a 7j NATIONAL 6DvISO'l COMMITTEE FOR AE GNAUTICS Figure /  flustra/ion of the Carleskan mopping function. Fig. 1 NACA ARR No. L4K22a NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS S/ .2 .4. . .6 .7 .8 . /0 Cnord () V'ie/oct/y dsfrbuftons. f/j're 2. dbAoJi/ f/7//cAnewS forn.< derived //7/c/e75 fore'5, and ye/oe4y d/S/d/os/b/ y ettecd/ c Ctatejo'n /y/yoppinlg fs'wchor'. Fig. 2 /oj Piar!es. NACA ARR No. L4K22a /gure. Second d~ti//i^,4s of/ /~F~o. Fig. 3 NACA ARR No. L4K22a  /2percenf thickness symmetrica/ profi/e  /2prcen/t 1/tnss'6s modifiAd by inverse method  6p.erent comber circulararc orof/le  6perce'/t comrber modified by /nyerse method Figure 4. Carlesian napp/ng functions. Fig. 4 NACA ARR No. L4K22a  Upper surface Lower surface NATIONAL ADISORY (a) /'ean comber hnes. COMMiT"EE F IAMNCS 0 ./ .2 .3 .4 .5 6 .7 .8 .9 /0 Chord (b) yVeocdy d/str/ibuton5s o<0. 79ure 5 .C/rcu/ararc 7mean /ne, derived mean //'ea, and re/oc/ty d/s/'rib/to/ns by mefAod of Cot tewJ n i,,n/n4 y .f7nct/on. Fig. 5 NACA ARR No. L4K22a  Eract superpos/tion,  Approx/maoe superpos/tion CFhora' x NATIONAL ADVISORY (0o} /Pro CONNITTEE FOR AEONAUTICS I, ~"1 Upper surface . .. f Lower surface^  EXct sup erposif/o Approimrate superpos iion _= 0o88B   o J .2 .3 4 .6 .7 .8 .9 10 (b) lVe/ocity a/slr/bluton w~io/ ^/ z^y Fig. 6 NACA ARR No. L4K22a  4,ACA 65/ o/rfo// ' /n,' '/ airfoi/ ' First *approxif'i<3f/o0 If /l i l 9ii NATIONAL ADVISOQY COMMITI[ IDO AE[OMAUTICS (o) Profi/es.  Theodorsen Garr/ck retod /frst approx/latilon CMF met od, zero approximat/on o CwF method first aJproximat/on EI CMF method second approx/Imt .2 .5 4 .3J f. Chord (bj Velo fiy do'trlou t/ons. , yr. 7. aec/ C/f ,F //mweh/ or /A4C4 6S'/2 o#rAo// Rig. 7 NACA ARR No. L4K22a Figure 8. Deferm/inat/on of y = o( +6r Fig. 8 NACA ARR No. L4K22a f . ... . .. ,. Z  O             .a    Prescribed veiocity Initial ve!oc/ty __ 0 Firs t ~ADroxlfGtO//O S .3 o .6 .7 .9 Cnord (?) Ve/octgy d/ s5ir/bt/ons5. NATIONAL AOVISOVR CONMnuiT FOR IEAO auTIC  Intal orof ile Der/red profile (/iapproz) (o) Profes.. '/~y/ry 9f /r,,,.Y <'/7~ 7<7or dys/ /// 4 ess formy.  .6 ..5 d .3 .2 i _ H 2 2 4 .4 .6 .8 /J Fig. 9 NACA ARR No. L4K22a   J #"1 / C,  ! .; : _. 6 I SPrescre o c ^ _Lower surface i  Prescrioecy ve,'oc, '. '.itij=. ____ A .5 Chord /rf//la/ Vel'oc/"y (c.'cv u'r rc,c~.,6 L F/rsr approxmrr.o.f or < = .67 F/rst approXlmrnoion Imodif 'e~',.&S I I I .6 .7 .5 .9 .10 NATIONAL ADVISORY COMMITIEE FOt AIIROAUTI C  'P, .t.Z/ C'rOf//e(C,', cu/ ac~, G. 7  Fi.~ ~,c;~r,oinr;o..mal  Th..n~r~~.z.rfo..'/ t,?eory ,C7 =.8O fad ProfileJ. //iure /'. /wI .e 6r7F/77 ,7//1od f r / ?Arr ,//,, .6O. Fig. 10  '111ilr7rr 1P I~ v~/oc./5c d~c'riD~ton~. NACA ARR No. L4K22a Figs. 11,12 r, I: .. i It liz NACA ARR No. L4K22a z1o/ale NATIONAL ADVISORY COMMITTEE i0LD AEPONAuTICS Figure /13 Carlesin maop/ng funcf/on for cascades, Fig. 13 NACA ARR No. L4K22a 9q (Z '.7, to (a 0 PIt (5 1j LZ O o' Figs. 14,15   ,, Ic:' UNIVERSITY OF FLORIDA 3 1262 08104 950 3 Ij *1 S3261 USA 