ARR Wo. LliK£2a NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS WARTIME REPORT ORIGINALLY ISSUED December 19kh as Advance Restricted Report lAK22a THE COHFORMAL TRAHSFORMATION OF M AIRFOIL HWO A STRAIGHT LIKE AHD ITS APPLICATIOH TO THE mVERSE PROBLEM OF AIRFOIL THEORY By William Mutterperl Langley Memorial Aeronautical Laboratory Langley Field, Ya. NACA WASHINGTON NACA WARTIME REPORTS are reprints of papers originally issued to provide rapid distribution of advance research results to an authorized group requiring them for the war effort. They were pre- viously held under a security status but are now unclassified. Some of these reports were not tech- nically edited. All have been reproduced without change in order to expedite general distribution. L - 113 DOCUMENTS DEFARI'^€t^J Digitized by tlie Internet Arcliive in 2011 witli funding from University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundation http://www.archive.org/details/conformaltransfoOOIang I I ^ 5 Jl ^{o NAG A ARR No. Ll|K22a RSSTKICTIilD NATIONAL ADVISORY COIvITTITTSE FOR AERONAUTICS ADVANCE RESTRICTED REPORT THE CONFORMAL TRANSFORMATION OP AN AIRFOIL INTO A STRAIGHT LINE AND ITS At=PLIC?vTICN TO THE INVERSE PROBLEM OF AIRFOIL TKr^ORY By William Mutterperl 3UI.^I,IARY A nethod ox" conformal transformation is developed that j.iaps an airfoil into a straight line, the line being chosen es the extended chord Una of the airfoil. The mapping is accomplished by operating directly with the airfoii ordinates. The absence of any preliminary transformation is found to shorten the v/ork substantially over that of previous methods. Use is made of the superposition of solutions to obtain a rigorous counter- part of the approximate methods of thln-airfoll theory. The method is applied to the solution of the direct and inverse problems for arbitrary airfoils and pressure distributions. Numerical examples are given. Appli- cations to more general types of regions, in particular to biplanes and to cascades of airfoils, are indicated. INTRODUCTION In an attempt to set up an efficient numerical method for finding the potential flow through an arbitrary cas- cade of airfoils (reference 1) a method of conformal transformation was developed that v/as found to apply to advantage in the case of isolated airfoils. The method consists in transforming the isolated airfoil directly to a straight line, namely, the extended chord line of the airfoil. The absence of the hitherto usual preliminary transformation of the airfoil into a near circle m.akes for a decided sim.plif ication of concept and procedure. RESTRICTED 2 NACA ARR No. Ll;K22a The exposition of the methocl, 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 IT 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 vi^ith previous inverse methods is made. Addi- tional material that vv-ill 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 rjiethod 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 dlscixssed in appendix C. Acknowledgment is made to ?Trs. T,ois Evans Doran of the com_puting staff of the Langley full-scale tunnel for her assistance in miaking the calculations. S^ffiEOLS z = X + iy plane of airfoil t = ^ + li'i plane of straight lines p plane of unit circle cp central angle of circle Ax component of Cartesian mapping function (CMF) parallel to chord Ay component of Cartesian mapping function perpen- dicular to chord AXq, Ay^ particular CI/lF's, tables I and II T displacement constant for locating airfoil r = 2r diameter of circle, semllength of straight line ^n ~ ^n + i'bn coefficients of series for C!!?' Sjq- negative of central angle of circle, corresponding to leading edge of airfoil NAG A ARR IIo. ikliZZa. 5 prp central angle of circle minus luO , corresponding to trailing edge of airfoil c airfoil chord C7 section lift coefficient Vg velocity at surface of airfoil, fraction of free- streaia velocity v>-, velocity at surface of circle, fraction of free- ■P V J' C^ U O LA -L J. O'. v^ t; V^ X ^^ J- J. W A. ^ ^ strearr! velocity V free -stream velocity ds element of length on airfoil r circulation u-t thickness factor Uq camber factor T thickness ratio X ^ normalising constant Ic denom.inator of equation (I7) C camber, percent 6x, 5y incremental C^JP's U positive area under approximate Vq(cp) curve L negative area under approxim.ate v-q(cp) curve a angle of attack aj ideal angle of attack •Y = ex + p^^ ^, true potential ^Q_ approximate potential 9 central angle of near circle c = CD - e i|. MCA ARR No. IJ4.K22a Subscripts: N leading edge (nose) T trailing edge c camber t thickness o, 1, 2 saccessive approxlraation in direct or inverse CIIF nethods I - THE DIRECT POTEITTIAL PRGBL-"2M OF AIRFOIL THEORY THE CARTESIAN MAPPING PUNCTION The Derivation of the Cartesian Mapping Function Consider the transformation of an airfoil, z-plane, into a straight line, t-plane (fig. 1). The vector distance between conf orijially corresponding points such as Pg and Pf on the tv/o contours is composed of a horizontal displacenent Ax and a vertical displace- ment Ay. The quantity Ax + i Lj is only another way of writing the analytic function z - t; that is, z - t = {x + ij) ~ {I + It,) = (x - J) + l(y - n) = Ax + i Ay (1) By Rlemann ' s basic existence theorem on conformal mapping, the function z - t connecting conformallv corresponding points in the z- and k-planes is a regular function of either z or t everjrvirhere outside the airfoil or straight line. This f miction will be referred to as a Cartesian mapping function, or CMF. In order to map an airfoil onto a straight line, the airfoil ordi- nate s Ay are regarded as the imaginary part of an analytic function on the straigh^t line and the problem^ reduces to the calculation of the real part Ax. MCA ARR No. li+E22a The calcultition of the real part of an analytic function on a closed contour from the known values of the imaginary part is well kno'wn. It is convenient for this calculation to consider the straight line as con- forraally related to a circle, p-plane, by the ffuniliar transformation I , r = V + — (2a) P v/here the constant displacement t has been inserted for future convenience in locating the airfoil. ?or corre- sponding points on the straight line and the circle,, equation (2a) reduces to £=:T + r>COSO ;2b) r, = : Considered as a function of p, therefore, the CMF z - t is regular everywhere outside the circle and is therefore expressible by the inverse pov/er series: z - £ = ZI — (5) p"^ The analogy of equation (J) with the Theodorsen-Garrick transformation (reference 2) log El :. >~ rii Pip- v;hich relates confornally a near circle, p' -plane, to a circle, p-plane, may be noted. On the circle proper, icp where p = Re , and defining Cj^^^ e a^^ + ibj^, equa- tion (5) reduces to two conjugate Fourier series for the ClilF; namely, ^^ = a + yZ -^ COS nc? + Y^ — sin nP (i;) 1 R-""' 1 R MCA ARR No. LLi:22a = 1^0 ^^ Ih. R n cos no - / 1 R- 13: n sm ncp (5) These series evidently determine Ax fron Lj or vice versa . An alternative method of performing this calculation is possible. It is knovvn that if the real and imaginary parts of a function are given by conjugate Fourier series, as in equations (Ii) 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 Ax(cp) _1_ 2ti 2rr Ay(cr' ) cot - do' (6) Ay(cp) _ _1_ 2v t Jo 2tt 'S> ' cot: ^ .-^ dcp' (7) Before the detailed application of the CT-LP z - [> to the solution of the direct and inverse problems of airfoil theory is made, some necessary basic properties of this function v/ill be discussed. Airfoil Position for G-iven CLIP It is noted first that the regions at infinity in the three planes are the same except for a trivial and arbitrarv translation; that is, by equations (1), (2a), and (5),^ - ^ E Ax^ + i Ay, 1 im z z, ^-^ ^ lim t = p + T i, p->-w L + ih ■■0 -^^o > (8) Secondly, if an airfoil is to be mapped into a straight line, it becomes necessary to knov/ the point on the straight line corresponding to the trailing edge of the airfoil. For a given CI/IF, £ix(c), Ay(<^), and straight line of length 2r located as in figure 1, FACA ARR No. lk.l\Z2Si the airfoil coordinates tions (1) and (2b) as are obtained from ecma- T + r cos c,i + Ax(cr) y = Ay(¥) (9) (10) The leading and trailing edges of the airfoil will be taken as the points corresponding to the extreraities oi the airfoil abscissas. The corresponding locations on the circle are therefore determined by maxinizing x with respect to C" in equation (9)' Thus or dx dcf sm + sin Q^ = dAx r dq^ d&x dcp 11) ?he condition (11) yields (usually by graphical deter- corrssoondmg to the leading and 1) mination) the angles edges (fig". trailing = -8 - TT N J (12) It will be found convenient to so alter the position and scale of a derived airfoil tiiat, for examiole, its chordwise extremities are located at x = il and the trailing edge has the ordinate y = (to be referred to as the normal f orn) . The chord c of a derived air- foil is by definition the difference in airfoil abscissa and (9), extremities, c = r ^cos 3- N or by equations (12) cos Th£ increase in scale ii^ori to obtained simply by m.ultipl^ring Ax(^cnrj) some desired c i\x, and Ay (15) o ,-1,. ii^- , v an exact velocity distribu- tion for a resultant airfoil. The resultant profile and its velocity distribution is a superposition in this sense of the conp'^nent profiles and velocity distributions. Thus, v/ithout sacrifice of exactness and with no great increase of labor, airfoils nay be analyzed and synthe- sized in terms of component symrrietrlcal thickness distri- butions and nes.n canber lines. This result provides a rigorous counterpart of the well-known approximate super- position m.ethods of thin-airfoil v^'rtex and source-sink potential theorv. 10 NAG A ARR No. Li|K22a As a particular cass of superposition, a known Clip Ax + 1 Ay may be multiplied by a constant S and the resulting ClvJP 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 nev\r 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 (1?) by S. By virtue of a reduc- tion in scale by the factor l/S this profile ma7,' also be regarded as obtained from the original one by using the same Ax, Ay but a length of line l/S tines the length of the original one . The use of superposition as v^fell as the application of the CMF to some particular airfoils will be illustrated next . Application oi the CI;1P to Some Particular Airfoils Symmetrical thickness distributions.- The Cartesian mapping function was calculated for a sjTnmetrical JO- percent thickness ratio Joakowskl profile from the known conformal correspondence between a Joukowski profile and a straight line. The CMF is given in normal form in table I. The associated constants t^ and r^ are given in table II and the profile itself, as determined either from the standard formulas or from equations (9) and (10), is shov/n in figure 2(a). The syrametrj'^ of the profile required only the calculation of Ax('P), Ay((?) for '^ r. The correction for r is necessary because if the chordwise locations of the first approximate airfoil were ■ computed by equation (9) v/ith tiie original values of r and r, Ax-j (cp) being used instead of ZiX^(cp), the re- sulting chordv/ise extremities would in general not be at X = ±1. It is therefore necessary to adjust r such that v;ith the derived L\x^, Ay]_, and X- X- -r%) = ' = -1 (21^) where <^j and c,o„ are the axigles 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 . (.2ii.) applied to equation (9) yields 1 = T^ + r3_ cos (Pj. + Ax]_^cp Ni) > -1 T3_ + ri cos together with those corresponding to one approximation by the Thecdorsen-Garrick method (refer- ence 5)« J^hs second approximation velocity distribution differs appreciably from that of the Theodorsen-G-arrick method on the UDper surface but agrees faiz-^lv well on the lov/er surface. The discrepancy for the rearmost S percent of chord on the lower surface appears to be due to lack of detail in this region in the Theodorsen-Garrick cal- culation. The convergence of the CMP method is seen to be rapid, considering the approximate nature of the initial airfoil, although two approxim.ations are required for a satisfactory result. The second approximation could probably have been made unnecessary by suitably adjusting the first increment 5y-| (cp) near the leading and trailing edges on the upper surface before calculating 5xt_(CD). The direction in v/hich to adjust the increm.ent is obtained by comparing the thicimess of the initial airfoil v:ith that of the given airfoil in these regions. Because a thicker section has a greater concentration of ohordv/ise locations tcv/ard the extremities, for a given set of cp points, than does a thinner section, the chordwise stations woiald be expected to be shifted outward as the thickness of the section is increased. The ordinates 'i^yi(^-P) should therefore have been chosen at chordwise stations slightly more toward the extreir^ities than those given hj equation (9)- The accuracy of the velocities is estimiated to be within 1 percent. It was expected, and verified by pre- liminary calculations, that the results v/ould tend to be more Inaccurate tov/ard the extremities of the airfoil than near the center. This result is evident fromi equa- tion (17). A given inaccuracy in the slopes dax/d^c and dAy/dcp can produce a large error in the velocity near the extremities, v/here sin Cp approaches zero. This disadvantage does not appear in the Theodorsen-Garrick method, in v.fhich sin cp- is replaced by one. Excessive error in these regions can be avoided in various ways. MCA ARR No. LliK22a I9 If the initial airfoil, for which the slopes dAx^/dCp and dAyo/di? have presumably been computed accurately, Is a good approximation in these regions, as evidenced by the smallness of 6X]_, 5y]_ compared to Ax^, Ayo> the effect of Inaccuracy of the slopes dox^L/'^^^j d5y-j_/dcp will be reduced, since they are added to the initial slopes dAXp/dT', dAy^/dcp. It was to reduce the magnitude of the incremental CI'.IP near the leading edge that the IIACA 6512 airfoil was drawn tangent to the initial air- foil in this region. The error in the derivatives can also be avoided by computing them from the differentiated Fourier series for 5x-]^, Gy] . (See appendix A.) This calculation was made in the illustrative example, after it was found that an error of about 5 psrcent in tlie velocity on the upper surface leading edge could be caused by unavoidable inacciiracy in measuring the Incremental slopes. The fact that the comptited derivatives do not repre- sent the derivatives of the CMF but rather the deriva- tive of its Fourier expansion to a finite niurriber of terms may introduce inaccuracy. (The derivative Fourier series converges more slovvly than the original series.) A com.parison of the computed derivatives vn.th the measured slopes will Indicate the limits of error, however, as v.'ell as the true derivative curve. The importance of knov/lng the CMP 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 CMF derivatives rather than the ClIF itself to be approximated initially. Further details are given in reference 1. II - THE INVERSE POTENTIAL 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 CMP methods of solu- tion will be outlined and illustrated by numerical 20 MCA ARR No. l4K22a 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 m.ethods. The prescribed velocity distribution is assiomed to be either a double-valued continuous function of the percent chord or a single -valued continuou.s function of percent arc. (Isolated discontinuities in velocity are, hov:/ever, at least in the percent-chord 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 sane free -stream flows and have the same circulation, conformally corresponding points in the three planes have the saine potential. Consider first the case where a velocity distribu- tion corresponding to a s7,Tm-fietrloal 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 3^ield an approximate potential distribution as a function of percent chord. This potential increases from, zero at the leading edge to a maxira,um value at the trailing edge. Of fundamental importance to the success of the method is the fact that this potential cm'^ve 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 sirfoil taken as the x-axis is next sufficiently extended that, in the same free-stream flow as for the airfoil, the potential, which in this case is sim.ply V£, increases linearly from zero at its leading edge to the same miaximtim value at the trailing edge as exists for the approximate potential curve derived initially. Horizontal displacements ax betv-/een these curves are then measured as a function of the straight- line abscissas and, hence, as a function of the central angle cp of ■ the circle corresponding to the straight line. These horizontal displacements /ix(cp), together with the conjugate function Ay(cp) computed therefrom and the length of straight line previously determined, constitute a CMP for an airfoil that is a' first approxi- m.ation to the unknown airfoil. The approximation is based on the use of a more or less arbitrarv initial NACA ARR No. li+K22a 21 airfoil to set up the first approximate potential. The exact velocity distribution of the derived fii'-st approxi- mate airfoil can now be coraputed and compared vn*.th the prescribed velocity. If the agreement is not satisfac- torily close, the procedure is repeated, with the airfoil just derived taking the place of the one initially assumed. The complication introduced in the general case in which the prescribed velocity distribution corresponds to an ursyimrietrlcal airfoil with 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 transferred to the circle plane by means of the stretching factor, presiomed known, of the initially assumed airfoil; that is, equation (li;) is solved for v,-^((P). The first approximate potential distribution as a function of the central angle cp is obtained by integrating Vq(o) through £i <^P-range of 2.rr radians (around the airfoil), starting from the value of s (:(p), and solving for Ay(cp), by wMch the contour coordinates are obtained as single-valued func- tions of <9. The necessity for closed contours does not, however, exclude the possibility of deriving physically unreal shapes; namely, contours of figure-eight 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 figure-eight type contours that adinits 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 l\j is determined from the given data - that is, the airfoil - and a Ax is computed therefroFi, 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. Slm.ilarly, just as the direct problem can also be solved by deriving dAy/dCp from the given airfoil slopes and thence computing dAx/dcp, so, conA-ersely, can the Inverse problem be solved by deriving dAx/dcp from the prescribed velocity dis- tribution and thence computing dAy/dcp. This inverse method of derivatives will be discussed after som.e numerical examples are presented. Illustrating the m.ethod of potentials. Exam.ples of GI.TP Method of Potentials Symmetrical s ection . - The m.ethod of potentials was applied first to~tKe derivation of the s^/mmetrical profile corresponding to the prescribed velocity distribution shown in figure 9(a). As an initial airfoil the 12- percent thick profile derived from the 5'3-percent thick Joukowskl profile in part I was used. The initial CLIF and associated constants are given in table VII. The initial airfoil and its velocity distribution are shov/n in figure 9. The first increment CMP and the resultant first approximate airfoil and its exact velocity distri- bution v^rere calculated by the procedure of the preceding section. The increm.ental slopes d5x]_/d(p, d5y]_/dcp were computed and found to approximate the measured slopes I'TAGA ARR No. Ll|K22a 25 very closely. The results are presented in table VII and figure 9. It is seen that the change in velocity and profile accomplished by one step of the Inverse process is large; that is, the convergence is rapid. The high velocity of the first point on the upper surface (cp = 15°) is due to lack of detail in the calculation. (Twelve points on the upper surface were calculated.) Por practical purposes the nose could be easily modified to reduce this velocity if desired without going through a complete second appro::imatlon. Mean camber line for uniform velocity increiaent . - As a second example of the inverse CMF method, the profile producing uniform equal and opposite velocity increments on upper and lov;er surfaces was derived. By the methods of thin-airfoil theory this velocity distribution yields the so-called logarithmic camber line. The prescribed velocity distribution is indicated in figure 10(a). The velocity peaks at the extremities of the prescribed velocity curve ivere assumed in order to compensate for an expected rounding off of the velocity in this region in working up from the initial velocity distribution. The convergence to the prescribed uniform velocity dis- tribution would thereby be accelerated. The initial airfoil was taken as the o-percent camber circular arc, discussed in part I. The initial CIvIP and its associated constants are given in tables III and IV. The circular arc and its velocity distribution are shov/n in figure 10. A first approximation v/as calculated as outlined in the previous section. A niAi.ierical difficulty appeared in the process of solving equation (11) for the zero- lift angle of the first approximate airfoil. It appeared that a 2k-polnt calculation (12 points by sjTmetry) did not give sufficient detail in the range tt < cp < 13. tr to yield a reliable solution of equation (11) for the zero-lift angle. This result was a consequence of the prescribed velocity discontinuity et the extremities with the consequent large but local changes in CliP and profile shape required in these regions. The solution obtained for the zero-lift angle was Pm = 6.1°, v/hich by equa- tion (19) with r = I.OOJ4.5 and a3_ = yielded cj = 0.67. The desired c,, hov.'ever, is O.8O, which would correspond to ^^ = 7.27°. It was considered that a relatively minute change in the shape of the extremities of the derived camber line vvould alter the 26 MCA ARR No. l4K22a slope dAx-[_/dcp In the desired range sufficiently to yield a zero-lift angle of Pm = 7.27°. On the other hand the effect of such a local change on the CMF as a whole would be snail. The velocity distributions of the derived profile v/ere 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 thin-airfoll theory, computed for c-, - 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 to the desired rectangular velocity distribution. The profile itself is seen to be one of finite thickness as compared with the single line of ■ thin-airfoil theory. ■ Airfoils obtained by superposition of this type of camber line with thickness profiles would therefore he 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 j 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 Method of Derivatives Instead of approximating by the method of potentials to a CMF that, when differentiated, yields the prescribed velocity, the CMP derivatives may be obtained directly. The controlling equations are equations (17)j (9)j s^^d a modification of equation (7)« I s In ( CO + a ) + s in (a + Pm) i vz(cp) = ' _ ^ n^ (17) \ri^ . 3ln cpf . (-^f V\r dcp / Vr dcp/ nSrr dAy _ 1 I dAx , 0, and in particular at the nose of the airfoil v/here dAy/dcp is comparable to dAx/dcp in m^agnitude, 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 Theodorsen-Garrick transf ormiation. An example of the use of the ClIF 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) slu air- foil and its velocity distribution are assumed known (fig. 11) and a desired- velocity is specified as a func- tion of percent arc. The nev/ velocity and length of arc are specified in such a way that the extremities of potential are the sam.e as on the known airfoil. . Both known and unknown airfoils then transform into the same MCA ARR No. lJiK22a 29 circle and, in particular, the velocities at points of equal potential on the two profiles can be found. In order to deterruine the profile corresponding to the new VGlccity, the complex displacement Zo - z-, between points of equal potential on the two profiles is expressed as a function of the CGrr3sponding complex velocities (denoted 07 Vg ) thus. dz- I J dZj_ d^.'/dz^ ^Zj_ aw '7dz 2 Vr. Hence Zo - z .33) where the integration is cerrled out elong the known pro- file from the trailing edge, which is taken es coincident for the two airfoils, to the point z-i. The complex I'unction V, 1/^-2 known rstio V. is determined approximately from the corresponding to the points of equal potential by the argument that, inasmuch as the two pro- files hsve nearly the s sme slope at corresponding points. the real iDart of V. V, n 1 is given by ■1 v. - 1. (This assumption, like the approximations in the CMP methods, is least valid at the nose of the airfoil. The function Zp - z-| is in fact a Cartesian mapping f sanction. ) The imaginary part is then computed as the conjugate function, equation (7)» In addition to the restrictions on the velocity dis- tribution mentioned initially, further conditions must be met in this method, if closed contours are to be ob- tained. Thus, the condition for closure of contour. n 1/ '^ d^Z2 - z^ - 1 (M 50 IIACA ARF. No. I,i|I-r22a and the recuired coincidence of v„^ and v^, at infinity, lead to the follovi.'ing three restrictions on the real part R(cp) of the integrand In equation (3)+) considered as a fionction oi' cp in the circle plane, JT2rr ' /02tt p,2v R(c)dcp= ( R(cp) COS CD dcp = R(C|)) sin cp dc,o = (55) JO Jo Method of Wei nig and 3ebeleln The method of Weinig and Gebeleln (reference 6) nay- be described essentially as follovv's: The given data are the same as in the 3etz method. Consider the function V, log V ^ = log ^1 V. Vr -K^- ^^1) {56) where P„ and P„ are the slopes at corresponding points of the tv;o airl'oils (fig. 11). Since jvg^j and Iv^ij are knov/n functions of cp with the data as given, and since log — — is regular outside the circle. V z- S^ _ - f5^^ can be calculated as the function conjugate to V, Z2 ^zi log V. a The ane-le P^ being known, S^_ is thereby ^ Z-i o J ^2 "^ determined and hence, by simple Integration, the unknov/n airfoil coordinates are obtained. As in the Betz method, the condition for closure of the desired contour 'dz = ''' ^^'''^'^ ap Vr C ^z dp = (57) leads to the additional restrictions on the prescribed velocitv distribution. FAOA AF:R No. IiiK22a 51 _1_ 2tt A 2Tr log Vg ( c)| dcp = u ri2Tr lOK V (cp)| sin cp dcp = -sin 2y > 1 P"^^ - I log V2(cp)| COS c? dq^ = -Tr(l - cos 2y) vJO v/here y is given by equation (5'3)« (38) Discussion of the Various Inverse I'lethods The methods of 3etz and of Weinig-Gebelein may be somewhat narrower in scope than the CMP nethods. The use of mapping functions such as in equations (53) a-^d (5^) Is based on the ability to specify dZ2/dZ]_ unambiguously in the corresponding regions. This requirement appears to restrict the contours obtainable by these methods to those bounding simply connected regions. Further investi- gation of this point is necessary, hov/ever. By the CI'.IF methods, figure-eight contours have arisen in the course of solution of both the direct and the inverse problems. (See the g-per-cent camber derived mean line (fig. 5(3-) ) and the illustrative examples in reference 1.) Such con- tours were first encountered as preliminary results (unpublished) in using the method of potentials v/ith the Theodorsen-G-arrick transformation. The CMF apparently makes no fundamental mathematical distinction between simply connected and figure-eight contours, for although z - t, must be a single -valued function of z, I, or p, the coordinate z itself is of the same char'acter as t and the latter has two Riemann slieets at its disposal in consequence of the Joukowski transformation from the i- to the p-plane . The methods of 3etz and of V/einig-Gebelein require the numerically difficult closure conditions fequations (35) and (58)) to be satisfied in adx^ance . If the methods are v/orked through for prescribed velocity distributions v;hich do not satisfy these conditions, it appears that z2 -'AC A ARR ¥.0. lh,Y2Z^ open contours result. In the ClIiF methods, however, there is either no closure condition (method of potentials) or a nuriierically simple Oxae (method of derivatives): dcp J: dcp dC^ = 3 ^^' 00 [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 pov/er series for the ClIP derivative^ mapping function ( ip — — 7^ -, mentioned previously j automatically ex- cludes the Inverse first pov/er (the residue term) from the power series for d(z - ^/dpfl Thus, physically impossible velocity distributions lead to open contours in the Betz-Welnig-Gebelein methods and to figure-eight contours in the CMF methods (if the latter converge). Prom the practical point of viev; in these cases, it may be easier to obtain the airfoil corresponding to the "best possible" -oh^^sicallv attainable velocitv distri- but ion by the CI,IP methods than by the others. If the succession of airfoils deterxained by an inverse GI-IF method is seen to tend toward the development of a figure -eight, 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) ^ il'^) j snd (9) is lacking. For physically possible velocity distributions, hovifever, specified as a function of percent arc, the Vifeinlg-Gebelein method shows that there is one and only one airfoil as a solution. If, hov;^ever, 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 reas-on 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 chordv^^ise extrem^ities be specified. IIACA APlR Fo. Lifl22a 55 From the experionce with the CMP methods gained to date, it is believed that to a velocity distribution specified as at the beginning of part II, and v/lth the further condition mentioned in the percent-chord case, there corresponds one and only one closed contour satis- fying the CIvZP system of equations. It is furthermore believed that the ClfF nethods are flexible enough to converge to this solution in at least those cases of aerodynamic interest. CONCLUSIONS 1. The conformal transformation of an airfoil to a straight line by the Cartesian mapping function (CIIP) method results in simpler numerical solutions of the direct and inverse potential problems for airfoils than have been hitherto available. 2. The use of superposition with the C?ff method for airfoils provides a rigorous counterpart of the approximate methods of thin-airfoil theory. Langley Memorial Aeronautical Laboratory Rational Advisory ComLiittec for Aeronautics Langley Field, Va. J,k IIACA ARR No. iJ+KSZa APPENDIX A THE CALCULATION OP CONJUGATE FUNCTIONS EY THE RUWGE 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, eojaa- tions (i|.) and (5)j or the conjugate integral relations, equations (6) and (7), ^^Q available. This type of cal- culation appears to be fundamental in many kinds of tv;o-dimenslonal potential problems. For example, the solution of the integral equation relating normal induced velocity to circulation in lifting-line theory can be solved easily by a method of successive approximation if tho transformation from the ''lifting line" to the circ.;.i-j is ^in.O't-Jn. Quicker methods of calculating a func- tion from its conjugate than those given in this appendix or in reference 2 v/ould therefore be highly useful. The use of the Fourier series rather than the integral roj-ations in the calculations of this paper was based on the follov/ing consideration. Because the func- tion l/z is regular outside the unit circle, the real and imaginary parts of l/z on the unit circle, nam.ely, cos cp and -sin Q^, satisfy the Integral relations (6), (7). The substitution of -sin cp for A^r in equa- tion (6) and subsequent numerical evaluation by the 20- point method of reference 2 gave results that were higher than cos R be substi- tuted for R in equations ([].) and (5)^ the resulting synthesis of the Fourier series will yield the mapping function for the circle of radius R' ; that is, v/ill determine points in the given plane corresponding to the points in the circle plane at the distance R' from the origin. It is nocessary, of course, to use the mapping function in conjunction v^ith the shape in the physical plane corresponding to the larger circle. In this way the entire corresponding fields can be mapped out. It m.ay be noted that substitution of R' < R for R in equa- tions (L.) and (5) enables the mapping of those corre- sponding points inside the original contours for v/hich the resulting Fourier series converge. MCA Ar?R No. L.)iK22a 59 It appears to be r.oro C'ifficult to find the point in the circle plane corresponding to a point of the given plane than vice versa. This calculation may, hov/ever, be accomplished by a nethod of successive approximations. P'or example, if the givexi plane is that of a near circle the polar coordinates of tlie given point in the near- circle plane are assumed to be a first approximation to the coordinates R' and cp of the desired point in the circle plane. Substitution of these values into equa- tions (Ij.) and (5) yields a first approximate mapping function which can be used to correct the coordinates R' and CO, etc . Biplanes In the case of the biplane arrangement the CLIP may be set up directly in the physical plane in the same v/ay as for the single airfoil. In place of the simple trans- formation from straight line to circle, however, the transformation from the two extended chord lines of the airfoils to two concentr'lc circles Is used. This trans- formation is derived in reference 9* The CMP method for biplanes bears the same relation to the method, of ref- erence 9 that the CMP method for monoplane airfoils bears to the Theodorsen-Garrick method (reference 2). For biplanes (fig. 12) the CMF z - t, being regular in the region outside the two straight lines, is regular in the annular region of the p-plane and consequently is expressible as a Laurent series in p ^=> CO \^ pn v>;here (1^0) °n = ^n ^ i^n If, for the inner circle, the relationship is written z - ^ = Ax-[_ + i Ay-^ (liD p = R-^e ko NAG A ARR No. LL^K22a and for the outer circle z - l = Ax^ + i A p = RpO icp 72 > (142) there Is obtaD.ned, upon substitution into equation ( Lj-O ) and reduction Ax-i (q) = s.^ + ^n^^-n R- n cos ncp A ^n - ^-n -n ncp (Ij-Ja) Ax,(cp) = a +\ 5a^-^^ R, n cos np+\ — -^^ sin ncp ik-^h) 1 Ay-j_(cp) =bo -^y?. cp = bo + 1 l^n+b-n ■p. n cos ncp - \ ^n ~ ^--n R n 2 sin nCP (i^Jd) These equations are the generalization to the biplane of equations (I4) s.nd (5)' I'be corresponding integral rela- tions may be derived as In reference 9 • The solution of or the inverse prob] successive approxiria method the tvro alrfo mat ion biplane were be taken as the inlt formation of referen tions on the straigh evenly spaced Q po equations (Ij-3) Iri either the direct em. nay be accomplished as before by tions. For exaraple, in the dii'^ect lis are given. If no initial approxi' available, the tvjo chord lines would lal straight lines. By the trans- ce 9 this fixes the chordv/ise loca- t lines corresponding to a set of ints on the concentric circles. The I.IACA ARR No. L)^Y22a i^l ordlnates i\ji^{fp) can therefore be ineasured, v/hlch determines Ayp(cn) by analysis and synthesis ox"' equa- tions (i|5o) and (lijd), respectively. (The radius ratio R2/R]L is fixed by the initial transformation frori the straight lines to the concentric circles.) These Ay2((p) values then determine a set of LXpi'V) values by the given shape of the second airfoil and the known chordv/ise locations of its first appi-^oximation straight line. Analysis of Ax^ (

2TT 2tt / 1 1 2rr / ^ ^' Jo 1 ,^ 1 P2TT 2Tr ^o^^o 55) In the problem under consideration, the mapping function *o<^'o' - 1S<%' for the cuter boundaries of s lea own. -o 2^1 the concentric circles are given, of equations (52) are thus known functions of •?]_' The radii e'^^, e The second integrals jc c^xw ^...,^ ..x..^.,.. ^..ux.^^^.^.x^ ^^ . j_ . Equa- tions (52a) and (52b) therefore constitute a pair of integral equations, similar to those of Theodorsen-Garrick, for the mapping function '?^i(''?i) - ^^if'^iV pertaining to the Irxner boundaries. ^ ■ ^ •' It is noted that if the variable point p of the Cauchy integral formula for the annular region is made to a^'proach the outer boundary G, 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 of the corresponding Theodorsen-Garrick equations for slmpljr connected regions and for the conformal mapping of near circular can be used ring regions. 48 l^mCA ARR No. Ll4.K22a REFERENCES 1. Mutterperl^ VUlliRni: A Solution of the Direct and Inverse Potential Problems for Arbitrary Cascades of Airfoils. NACA aRR No. LL!-IC22b„ l^i+Jj-." 2. Theodorsen, T., and Garrlck, I. E.: General Poten- tial Theory of Arbitrary Wing Sections. IIACA Rep. No. 1^52, 1955- 3. Millikan, Clark B.: An Extended Theory of Thin Air- foils and Its Application to the Biplane Problem. NACA Rep. No. 562, 1930. Ij.. Allen;, H. Julian: General Theory of Airfoil Sections Having Arbitrary Shape or Pressure Distribution. NACA ACR No. 3G29, IQI+J . 5. Garriclc, I. E.; Determination of the Theoretical Pressure Distribution for -Twenty Airfoils. NACA Rep. No. i;65, 1953. 6. Gebelein, H. : Theory of Tv/o-Dlmenslonal Potential Plov; about Arbitrary Wing Sections. NACA TM No. 886, 1939. 7. Betz, A.s Modification of V/ing-Section Shape to Assure a Predetermined Change in Pressure Dis- tribution. NACA TM No. "Jo^( , 1935' 8. Hussmann, Albrecht: Reclinerische Verfahren zur harmonise hen Analyse und Synthese. Julius Springer ( Berlin) ,1938 • 9. Garrlck, I. E.: Potential Plow about Arbitrary Biplane V.'ing Sections. NACA Rep. No. 514.2, I936. 10. von Iva'rma'n, Th., and Burgers, J. K. : General Aero- dynamic Theory - Perfect Fluids. Application of the Theory of Conformal Tra^isf ormation to the Investigation of the Flow around Airfoil Profiles. Vol. II of Aerodynamic Theory, dlv . E, ch. II, Dt. B. vV. F. Durand, ed., Julius Springer (Berlin), 1955, p. 91. MCA ARR Fo. lJ;K22a i]-9 11. ITurwltz, Adolf, and. Coiirant, R.: All.^emelne PunktionenthGorie und slliptlsche Funktlonen, and Geoinetrische Puiilctlonentheorle . Bd . Ill of Fatheiracischen Wispenschaf ten. Julius .^.nringer (Berlin), I929, p. $35. 12. Bercrman, Stefan: Partial Differential Equations, Advanced Topics. Advanced Instruction and Research in Mechanics, Brown Univ., Summer 19^-1 • MCA ARR No. LifK22a TABLE I CARTESIAII MAPriNG rUTMCTION FOR S YI.LMETRICAL 30- PERCENT THICI CDHn*'* in o m tOH (OnOJOCMHHtO o O lO o o> in looj ■* CM o> in m in ■"I'CM iHO o>c-ioin'<* a > II «o to lO «3 «o «o » CO 00 00 CO C->t-C~C-t-lO* «-t H H r^rH p"«>«CMCMCMCMCO s o O ooH t-inc» f)r-t H-^ ID CD a> OHO oi t~ to n g M CV3 (O in to «o > ooo)Oo>c~in(OH 1a if • O • • • • • • • • • • H 6 s M § •t 5 fii Qi OO 00 o g 32 £ N iHKJOJ lOflO t-OV>0*'* HtO^'tCCO o> to t- ■ lO W t^ 0>CD to to CD 0> CO Or-llOlOt- CM OS CO 0> CM OiCMtommotocft *WO(OinoocB02 H •a; s| K o> O) CO tec 01 0> Oi Oi QD t- o fH o (M o r r r r i t t t t 1 1 rA •p in g § (0 a o •< o o o M ^ *i 00 So «p to 01 «0 m O HCM 0895 0302 0239 0410 0432 in 00 cv 8 0>«0 K5HC0 ftjioowrt CICM CV) iHO 'J'tOOt-COtOOOID CO 8- ►> fr CMHtOO»C7>inHO 00 M M < TJ H OHCM CMH OOOOOOOH • W K ■d • • • • • • • •••9««* O EH o u O 1 1 1 1 1 1 1 1 1 1 1 1 •o s ft © O Pj 4^ fc >< . •p o o H »f a (D rtoi •«J«HCO H«0 HHC- 00 00 mtoin toto ^■»i«in CM CO CM 0> CM cotoinc7>>Hior- g g «H 0 o c- CMCM 0>lOH HtOCMHa3inao> ■P < -o H O O HCV3 CM CM HH oo HHOOO OOOOOHHH •o M •H •0 • • • • • • • • • • • • ••«•«•• u s a O 1 * * 1 1 e 1 1 1 1 1 i 1 1 o H 1 a 1 o ^ o +5 EH O o M o K 00 in H CM COO'* t- Oti-iriOO C-C»0OCpWC»CO* OHHOtOe-CMtO 01WCMO(M-> o ^H to o in HO QOCM o o oo o u < o O iH H03 01 CM CV] H HO OOOOOOOO p ^ • o « • • « • • « • • « till Eh «-( in in 5| moo «D » lO 00 tOCVl (OCVl to to ■*o>«o CM 0>C» «■<*< > to «o 'if in tOtOCMtOCJiCMtOin o "* toco CM •^(O O t- '«< CM H HOJtOIOHHinO H ■p ^,_^ ♦h-O ■ o iJ c (D »=IS «5 e^ • o •d 0) X ^ 1^ o 1-103 to ^ m (o ^ 00 a> o HCM tO'i'in tDt-000»OHCM (O o H r-i r-* r-t H i-i HHHHCMCMCMCM 4 NACA ARR No. L4K22a 53 si ^ § (E ^£ ^ UJ s o •H i ■P « ID • > II o CO Hcvi c- in c- CO to (D in in moj a? to o t>cM "* in CD •* I* lOCM H CM CO ino5 ■* O CM to H O o 00 coco CO CM •*■* t0 05 00 £> C- 00 > > C^ C~ !> C- 00 H O in to in c- in to r-t rHiH r-IH H H H H i-l r-t H s s X cn rH to to in oj to i-t t~ t-* Oi di to* to r- t> 05(0 COCM •* HO in C- CM 00 00 C^ to lO > 05 to C7> to to CM t- in o 10 HH-<»< ^- in CO -^o 05 O HtO CO o o a>c- •* c- in •^f (0 c~ in (0 H H H 1 43 O H H n ■0 & to 05 C- tf O) CO 00 ^ CO to •* C\) to o to tor- c- '1'05 CM CM 0> OH T»« to t- 'i' to to CM CM CM 05 CO 05 CM a> (ji a> CO t~ to C- O •* HtO 00 HCO to O •* CM o to to in H 05 05 CM 00 t~ CJ> bO a a 1 o 1 1 1 1 1 1 1 1 1 1 1 o 5 g CO 0> H • o CO i-IC- ^iHCD O CO tOrHIO (MHi-tiHO • • • • • OCM CMO O ^tO H 05 in 00 to OO HHH • • • • • 1 1 • 1 1 O 005 »*< H to O 1< •* to in CM (O to HO ooo • • • • • • 1 in T*" o •* -^to ino o CO O H HO « • • o O 1 1 1 1 in CM t> 00 CO fj to to O O H « • • 1 « to H 5 M O M •0 01 H § 0)01 OHIO 05 ■* Q tool to 05 OOrlHfH 00 HCM * tOH OOO to to H CM HHO O CO ^ c- "O 00 CM lofo in 05 10 05 Tjl O O HO O O c- COCO in CM to to f J CM tO«0 O tOCM O O O OH to C- H 00 00 ID r-it-t H o o H o o 1 1 1 1 1 1 1 1 1 1 t 1 1 « ■p o o o • o 01 H lA 1^ • O HIOOHW O lO to CJ5 ■»»« iHHHO O • • • • • 1 1 1 1 1 to to c-ooot- •* o m OCM to O O HH H • • • • • *05IOCM 00 tomcoH H 00 into (O H oooo «••«■• H^ 05in to ^ to CM to in CM tO'J'O H O O O o o • • • • • 1 1 i 1 HtO CM 05 05 CM H Ti< to OOO • • • 1 ■d 4> -p •p o -d T- <0 o • o OS CM OOOCM CM 05HCM ^ ^ocM •* m oooo o • • • • • 1 1 1 00 00 coo CO 05 CO to m to to o CM in to o ooo o • • • • • i 1 O CM ^ to 05 C- tOHt- 05 CM OOOO o ooo o • • • • • lO Tt 00 ■* HO> C- to CM to CM H O o o o o o • • • • • 1 1 1 1 1 (0-<<< oc^ to H (OCM (Oin H rA t-{ t-i i-t o ooo o • • « • • 05 OOCM t- O 05 H05 to (O HCM HO O O O O O O • • • • • ^- 05 to HOO ■* t-\r-ii-t OOO • • • 1 1 1 « o a z ID ■p 2 al M P o ^D(Oloo>^- HCM •«jCM (0 to 00 CM H OH H OO OO o • • • • • 1 1 OOCM > > OHCM torn H O O O O O O OO o • • • • • 1 1 1 t- 05 00 00 t> 05 c- 00 in CM O HCM to (O O O OO o • • • • • * 1 1 1 1 HCM to o 00 to f i H O ooo • • • 1 1 1 < 00 o o •O to tOO>05 lO O lOOO o OHHHCM too om in HO £~CM C- CM CM HH O o (0 in in (OO OH oo o o o '*>in 01 CM HO to in O O o o o COHtO ooo > H o o 1 1 1 1 1 i ■p J X to C- 00 o> o H HCM (O'* in t-i t-i r-t r-t i-t to C- CO 05 o HHHH CM HCM (O CM CM CM o NACA ARR No. L4K22a 54 t3 «> o c o o H O —J o gt C O ■P as •-I K O P. a -d a o o o w in • O •* to lO'l'CM CO 'I" to HCO m c- c- ID to lo in C- H O HO CO H CM toco 'i' -^ tOCM H O 00 O C^ ■* to !> -^HO o c- 00 CO CO H'l' OOCM to CO c- to CO CO c-c- c-t- c- H O 05 in ■•^ C-- c~ in to i-l rH H t-IrS r-l r-i r-i r-t r-i i-i H to CM OD f- ■>*" in CO lO CO H H £~- H to 'f to C- > to CM to CM CM HH C- 00 CM 00 00 c- to in ■* •^f oDCM in in to c- to o to H H ■"^ C- CO 05 ■ C- lO to CM ■* ■* > in to H O H >r <0 lO <0 iHCM <0 0> Tl« to CO 05 > 0> CJ» CO to ■* CM to CM to to 05 CM •«1< CJ5 CM CM O O H •* toco tO ■* inHH CM 0) CO Oi CM O) oi 05 CO > in to C35 'J* in in CO O CO^ o •<*< CM O f J to CM to 05 05 CM CO C- 05 05 o 1 I 1 1 1 1 1 1 f 1 1 o d •d CO 1< lOiH lOiH CD 0) O O t-CM to H CM CM (HrH O O CM 05 05 •»li in to in lo c- CM o mc-H O H HHH CO to to 05-* in 05 CO ^ •<< CM in HtO to HO O O O > in 05 ■* CM o in to to o in 05 CM 00 oo OHO mm > t-to to m to O O H O 1 1 1 1 1 1 1 1 1 1 1 1 1 J? < ■d e- •d to C- > £> onto CO CM CO CM in CM C- H O O O HHCM • • • • • O 1 1 ooo CO t- 00 OVH O t- H-<* > in to o to to CO to C35 ■"J'O O HOOO • • • • • 1 1 1 1 1 CM o tn in into o to 05 to to H»0 CM oo oo H • • • • • 1 1 CM CO to CO c-m • • • 1 1 1 5 OCM to o c~ m m to ■ '1' H rH H HO O C- 00 <0 0> (0 O H OCM f- H to O CM CM O OHHH to 'i*'* OOO tO'^-i't-H Hcomtoto HO OO O OCM to CD O t- to CM in 00 lo^ in ■* H OOOOO c-o lOom CM t-H oo H O 1 1 1 ) 1 1 1 1 1 § CO COODHIO to i-ltOHO t-t O l-i r-l i-< o o ooo • • • • • o o O O C- 05 O to to in to to H H O O r1 OO O O O • • • • • 1 1 1 t^-to-* •*tOH OOO OOO • • • 1 1 OO C- tOH CM >05 O OH o oo • • • 1 O 1 o t-CM OCO HO OO • • O 1 1 CM •d ■d 'J' 00 lO CM > to 05 a> o CO OHO OH ooo o o • • • • • Olio CT>C- OH to in to ii< O O HH OOOO • • • • 1 1 1 O lOtOHHOi tOtO to tOH HO O O O O O O O O • • • • • 1 in 00 c~ to CM O to C- oo o o ooo o • • • • 1 O 1 moO'* H OOCM ooo OOO • • • 1 to j< o •* m to o o oooo o o o o oo o • • • • • • O 1 1 1 1 HCM tOC-CM to to O (0<0 O O OO o o o o o o • « • • • «30 intOCM •<1co o ^ in oo Hin H OOOOO ooooo • • • • • oomt- om-* ooo ooo • • • in C- Ooo •"J'HH O HQ O O O O O OOO o • • • • • • O 1 1 1 1 1 in in Hin 00 > •«i< o o o o o o o o o o » • • • • 1 1 1 1 1 ooooo HHH m oooo oooo • • • • O i II CMm oo oo • e O 1 1 CM < m in CO cj>co o O ■«1< 05 1< CO H O O OHHCM o CM tncM CO H H CJi to H > CM H HHO o to toin too OH oo o o • • • • o O t0 05tO >* CM H O to to OOOOO • • • • • 1 1 1 CO to 00 ooo • • • 1 1 1 uo C 0) 8- "H "d CO X O HCM «o * in to > CO 05 o H H CM to •* in i-i r-i ri r-i t-\ to C~CO 050 HHHHCM HCM to CM CM CM U O Si o I ■^ m V «0 c o H s e o h « •H u o H o u tp H at o- NACA ARR No. L4K22a 55 > < E-i EH O ra O E-i H O H CO E-< o o O ITN xi > m "(D 'H X o o o I -P CO CD , — , W^ • C to •H ex; bO O-P © = & cd • CO iH (D "H rH O tDU C ?H cd rH u^ IO> ON O t-i • iH N> rH K^ CM CJ II O o o O II LPv LfN LP. ITN r^ O "^ ir\ o -^T ON S CM 'J^ rov ca o o O o c^ J- LTN [>- * ^ QO ^ U> -=t- E-t LTN ITS ca o o o O vO c- MD l>- ^ o CTx LTn fO, rH ^ K> KN ro» O O O • • • o ^^^ OD CM fH CM O ON u O iH H H H rH r-i rH • • • • rH r-\ rH r-i C O c •H o 1 +s •H c a -P rH r-co r-i r<\vo c-fM jco rg o ON OOi-lrHr-lrHr-^O CTnONOsOO On fCv Oi NO ON OJ O O O O . • CO rH O O rH II II II 11 rH rH rH O '^ ^ ^ ^ CO f^ OJ ITN ON fCi O . CO rH O O iH II II II II COCO O rH rH rH iH i-I rH iH rH M NO rH (\J l/NrH rr\ On rH rH 00 CO NO 1 C\J ^itVO IrH K\CO t rH t\j_:tvo t--aoaDQOao t^iTNcy i 1 O 1 K O ONOJ i/NNOCONO C^ONOnOGO o O NO C— rH rH 00 J- On ffM/Nj-irNO O ONOOC^l/NOJ OrH-:lvOaOONO rH ' 1 1 1 l' l' rH • ^ ^ § ONONiTNrvi ^ hf\o o C-- ONCM-d-tvj nj Os-*ft~- -a-oVDrHjvorvU-o r-< K «3 fVJ KNKNrCirH H Or-O ON(M r-lTN ON OJ OOONO rH O ONrHNO O OnD r-* rH o CO inrg (\i uvDcoco c-NO iH rHpHO O O OO OOOO O O 1 1 1 t 1 ' 1 O 4> c • 2 o C 'H 4J D ■H fa O •a ^ J- ONiriCTNONO»>ONt~NONO rH o KN KNNO rH r-NOO r-«Mf\t^ONrH rH tt\r^ H lO ^tf\T-^ i-t i-l C3 O r-> o oooooooooooo O 1 • 1(1111 1 -^■-d'ONONfr\t^tM ONrHNO -d' UVDCO K>_d-WNNO 0Nr~O-3- O KNffstM O-^KNO O O O OOOOOOOOOOO o O 1 1 1 1 OnnO CTN^ONrH-^J-ONNQCO irMr\rr\C-fNO rH C--dT\J OrHrHOOrHrHrHOOO OOOOOOOOOOO O O ! 1 1 1 K NO ONKNONirN_d-hr>t>J ITNrH ITN O IfNNO J-e^rH 0<3-H rH O OOrHf\JC\l(\JOOOOO OOOOOOOOOOO O 1 1 1 • 1 1 1 i I 1 o o • •rt *> •H C M OONHrHr- K>P-ON rH rOiONKMrva-o mONO ONi<\ l/N -rj- rH CD lOvrHNO ONrH ONlTNrH r-f rH rHO O O OO rH O O O O • 1 1 1 1 1 1 o O < T3 t O ON JtDlTM/NrH CO O J'rH OJ OJ l/NNO OCO rfN -d-ONtM-^J-OJOO fCNrH rfNKN OOrHrHrHrHOOOOO O 1 1 1 o 41 OD-d- t^OINOC^C- 00 KNrH NO OnONC— O OJ rH rH ^r^^-o rH rH ocOno x^i-t O OOrHrHrHrHOOOOO o O ■«3 NMTNONfH f-ONrH O OnOJ ON 0VJ-:jNO O OJ CM ONi/NO\OJ rH r- oi oi o t— J-o rfMrNf-r-t-*o i/n r^ rH rHO OO OO OOOOO O 1 1 1 i 1 C s X O rH OJ KV^ITNNO t-CO OnO rH OI O M o» MO < K X ON mE O o NACA ARR No. L4K22a 57 M S ■< ^ 1 § fi o tij > m s < II « £ s s o u CO s c o CT) « 3 > «-i ^ O •H h^ II T) ■*-' o ro r- S -H O — T3 ^ < K 4 XI •o lO C e5 ON O O i-t - o oooo H ir>KN^ t^ HNnOOJiH l^O QNKNO ©■-tooo • • • • • I • I I I KVOt- OOO CU ^-Lf^K^ OOOO K>cOt~0>-:t t^H rc»r-t«y rH O O (M-5- ooooo i-»ir>o NOc-a5 OOO 00 ^-.-l^f^ .4CO00Q oooo rH t^(\jLfNON KNCVJ uSrvjNO ttSr-i rH KMCv ooooo HOO KNr-l 00 I I t I o NO r^ LTNOn 000 lOCO oooo rH NO -d-ONO (Cvir>tc\KNO 80 H rH i?\ 0000 O Oi 00 CO 00 I j_ji L-ua. t~- ITNO OnnO rH t~-C0NDCO 1^ hTNrH O <\J O OOOO OJ rcvNO OnO ptnOJ O t^lTs IC\t-i r\l~:t(M OOOOO rHt^-aO KNOD t-i rH_::tNO 000 I I I I I tr\ OnO r-f rH 04 rH rH O H O OOOO ^rH-=hOvO i?u;ti^>-OC>- rH rH rH H O OOOOO ^-0*f^ OOJ CM 000 I I I I I rg_d-OH (M ONrHNO OOOO • • • • I I I I 80 O rHCy fSJ r-l OOOO 00 t I CV NO C-COONO rHCMKV^lTN »0 t-OO NO t~ CO vO I-* • 11 II r* H re 9- o ^<^ o o O NO • I -" ° .. II II I-* f- h t- NACA ARR No. L4K22a 58 TABLE IX TIffi USE OP THE RUNOE SCHEDULE IN THE ANALYSIS AND SYNTHESIS OP CONJUGATE FOURIER SERIES Process Entry in schedule Result Direct method Analysis Enter ^ for yn ^n' ^n 9 Synthesis Enter in dn spaces Enter in Dn spaces 5x d5x/d

n nan nbn an nbn -nan Inverse method of potentials Analysis Enter || for j^ ^n' ^n Synthesis Enter in dn spaces Enter in Dn spaces 6y d6x/df d5y/d9 ^n nbn -nan -an -nan -nbn Inverse method of derivatives Analysis „ ^ 1 dAx ^ ^''^^'' 12 d^ ^°^ y^ ant ^n Synthesis Enter in dn spaces Enter in Dj- spaces dAy/dcp Ax Ay -^nA an/n -an an/n bn/n NATIONAL ADVISORY COMMITTEE FOB AERONAUTICS. 59 NACA ARR No. L4K22a z o i- < X o CL a < I- tr O DC < O < z lU T h CC o o n li. X (J M o I 1- 31 to UJ _) CD 5 4- < m 1- <) Ul n a c Q < Ul n 1 h LiJ -I r> Q UJ X o I/) Ul 2 DC Ul I I- U. o UJ Ul :3 UJ X I- * i '."■ *^ ^ « -.- ^ =? "^ T «> V >C1 O Q O -T O li ■a" .f ■n n~) ^ oo > o "■7 « «■ C? I Si <<1 s V ^ Vl u- O c a "^ Vi ^, ^ is "" 5 ■^ C' - 1 ca =5- tj •■a '^ '^ ■^ ^ ?° «J ^ ^ ^ *^ 1 ^ ^ ^ c u.. "•J- •O -' Tl ^ ■O i + * 1 tj T oa t;-* oa rfa "^ 51- ' ! ' ^ a? 1 1 Ci t" '<1 if + ■Q xs '^ i : ij S 5) i s ^ ^ ■^ J? 1 i 1 b + «j «! •< + •ft 1 rig 8 4 =0 TT ^ ^ ^ « i5 J- ^ $ ^ ^ ^ ^ • ^ "S -J + + CO ^ ^ % ■^ NACA ARR No. L4K22a 60 ; 1 ^ IN •CI 1 5s ft t? fcT ^" t^ >j e? ^ --- ■n ^ _ ^ ^ 5 JS •^ ^ 8 ^ ft ^^ 5; a Sa (o J' 1 ? N "1 «i >i ^ .* ■o ■^ *^ s s \ N « t*! > Q i, Q ^ ^ ^ ^ $ Q sj ^ >^ ■^ •-? fcj a 8' ;s N 5 1 c^ ^ .i fi S s S). 5 ? o, ^ ^ « ^ \5 VI 'J *i '1 « 6 ';^ iS 1 kj N" e «' h - -■ i - f !=(■ C\ »J 5 ij ^ h — — — — — 1 1 ■s J i-T tj- 8- a" \ „_ _ 1 ■M ■^ .-• <<> 5 \1 ^ ?! J !i Q 'S n Q 4 (■ fl -.? C ij «^ 8 a 1^ Q O O 3^ - _ 3 CVI N ftj B " "^ ^ "^^ ■N ^ = J J- ■S; ,^ N ^* ^' s* ^ «■ ^ S 'I. S 1 ^ °^ s s te s ^ ^; to ic" 1 ^ $ ^ -1 li ^ ^ .^ o !-• uj. N -^ ^ 5; vi I?! s Q S « fti V s <:? f "-~ N" ^" -S" -? h - - 1 to- =* h - — ■1 X \ to- ^ S ^ ^ ^ 7' -? \- — - -' s SI -T N^ N- -? •o- h — — — I «a *" Si 5" 1 If x Q aT ^-^ N^ N" ^ «" "1 ^ ^ i ~1 ^1 Q 11 .* ^ 5 s •6 In s ^ liT O •c. N S ■. 61 NAGA ARR No. L4K22a Ui _l CD < Q UI Q ■3 _l O z o u z o I- < i x: o cc a. < I- in d: o EC < in < < z Ld I 1- [£ O U Q O I I- LJ 5 l- u a Q u u Q liJ I O CO UJ IB Z CC LU I Ll o LJ U) 2) LlI I (- £1 I/) * "? ^i"^ I s ■5 ^ 5 SI L 1 =? i 5 % 5 <•• ^ ^ =? cj" ^ 5 c ^- «i 'S 0- ' t s «' £w t 1 1 vj c| _£ 5 1 * «a •T rO »< ^ «»J U^ "> * 'i l"? ■1 -? tti op §sj V " 5 . X s^^ ■^ ^. ;t t:^J>^^ K « N 5 5 s fl N < t- n1 >j t? = -i 5 11 ^ ^ ^ i N. 5 S + ' 't- '^ = 50 rO '^ ■Q -= ■o ^ '^ '^ 1. «3 •-0 1 1 1 i ^ ? ■* + « rie I >,|_a 1 -51 <> ? ^ ^ 1 1 1 IS ■5 ^ ^5 «a as NACA ARR No. L4K22a 62 f 5 ■a ^ ? ^ ■51 la tf «? >f >J *? ^ ^" ^ s — ^ X ^^ ^ -D 8 =1 fe !« S s a ^ 1 1 "1 ") X) •« ' ■5. <^ S' Q '^i t •? ■^ ^ »-r fci B ; h — -- - — - 1 c ci -T-. . i ^4 1 i h — — - — - -V) 1 ^ o SS. ^ 1 1-^ kj' b^° ^ 8 s <3 i: ^ ^" !S 31 :^ J o ao N ti e< i ■~ 1 ' t ts h "s S a s ^ ^S 1 1 ■fs ■s 1^ s" •I.' s~ ■o -0 ft! 2 S5 If ^ a 4 i 1 t= °^ ^ s S ". (o It 1 '^ ^ ^ o 1 1 '1 > 1 5 ■S ^ 1 s § ■^ *• 1 (5" - 1 ■y ^ \ .1 \ - - — ■ •0 'a N- t^a - - — 1 0) 1 5 ^ i- 1 - Sr Ol 1 1 <5i ki >0 i O o o o S o Ol i o o 1 0/ to % 5 8 S "s NACA ARR No. L4K22a Fig. 1 z-,K-,p-phne5 NATIONAL ADVISORY COMMITTEE FOB AERONAUTICS Figure L I/I us tr a f Ion of the Cartesian mapping function . NACA ARR No. L4K22a Fig. 2 30- percent th/cKnessJouA-ouus/c/ profi/e Z4ybercs/-y£ t/iicAcness c/er/i/eiy prof/fe- iZ-p>^/^:^nt th/c/rnesis a/er/yea py-of//e ^Oj /='/'£}/'/ /^ S . NATIONAL ADVISORY COHMITTEt Foil AERONAUTICS /wtf / __ Th/CKness 30 oercent 14 ^ ^ ~^ ^-^ /^ / — 24! perceni' . -^ ~— ^ ^ i£. yC^e-/ oc/ /t /2 r ^ N 1 ■ ^ ^ if 1 -~^ '^ §=:- \ ^ iJ^« ^ '^^ ^ ^ A » a f .1 7 ^ ? .^ 1- -w r .1 J • / r .(. i ~ i U Chord (6) ^e/oc/ty d/sfr/baf/ons. NACA ARR No. L4K22a Fig. 3 H(»i X3 .4 ^ 3 ^ ^ N, \ \. ^ d'^AXot ^ afcp z \ 1 \ ^ < ^ Qf^AXoc ^ drj y<^//'yd=is Oi^ ^A/Fi: NACA ARR No. L4K22a Fig. 4 — /2-/oerc~£A?T^ fh/ckness symmetrico/ jDrof//e — /2-/oe/-ce/7t //7/ i7 . NACA ARR No. L4K22a Fig. 6 ■ Exact 5uperposit/on ■Approximate superposition Chord /.O fo) /=^/-of// / / 26 / J6 / / i / / J^ / / / M / / M / r / ^ / COM ATIONAL ADVISOPY HITTEE FOO AERONIU' ICS o ^ 5 A t y. da •> t /( /. ? / 4 J6 Figure d. - Determ/nat/on of ]f=oc-hpr NACA ARR No. L4K22a Fig. 9 /.4 Ll / V* ""' 1-T G '—= o -- ^ If, /I -a" " -~ ^^ >i- / --^ ^ a '^' >^ Prescribed 1 yeiocitu Inttial i/e/oc/ts/ First aoDroximatlon ^, ./ w 7 ^^^ ? _i ore 5" i .< 5 ■ 7 .8 .. 9 /.O (a) Ve/ocity d/str/but'/ons. NATIONAL ADVISORY COMMITTEE FOS 4EH0NHUTICS -I/) It la I profile r^ 1 ■Derlyed profile l/^approx.) (h) Profiles. NACA ARR No. L4K22a Fig. 10 /.4- l^: /.Oh I. Upper surface Loiter surface — Prescribed velocitu —/n/ti'a/ i^efoc/ty (air cu far orc,c^^.7JJ e F/rst appro^imat/on ,c^-=.67 B First approximation modifiec/jC^SO .2. ^ A .5 .t Chord (a) Veiocify distributions. -5 iO NATIONAL ADVISORY CONNITTEE FOt AERONIUTICS Ordinates magnified—- Initiai prof lie (circuior orc,Q=.7Jj First approximation Thin-airfoi/ f/ieorc/ ,Cz=,80 rdJ Profiles. NACA ARR No. L4K22a Figs. 11,12 .o -Q NACA ARR No. L4K22a Fig. 13 -plane -plane NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS Ffgure /J!— Carfes/an mapp/ng funct/on for cascades. NACA ARR No. L4K22a Figs. 14,15 I S3 CL ^-'-- II \ ,.^^-^ \ \ \ \ \ \ I I UNIVERSITY OF FLORIDA 3 1262 08104 950 3 ,FI-