REPORT No. 563 CALCULATED AND MEASURED PRESSURE DISTRIBUTIONS OVER THE MIDSPAN SECTION OF THE N. A. C. A, 4412 AIRFOIL By Roser M. Poceatox SUMMARY Pressures were simultaneously measured in the cariable- density tunnel at bf orifices distributed over the midspan section of a 6- by 80-inch rectangular model of the N..A. ©. A. 4412 airfoil at 17 angles of attack ranging from| 20° to 90° at a Reynolds Number of approximately $3,000,000. Accurate data were thus obiained for stuily- ‘ng the deviations of the results of potential-flow theory from measured results. The resulta of the analysis and} discussion of the experimental technique are presented. It is shown that theoretical calculations made either at the effective angle of attack or at a given. actual lift do na accurately describe the observed pressure distribution over anairfoil section. There is therefore developed a modified theoretical caleulation that agrees reasonably well with, the measured results of the tests of the N. A. O. A. 4412) section and that consists of making the calculations and evaluating the circulation by means of the experimentally] obtained lift atthe effective angle of attack; i. ., the angle| that the chord of the model makes with the direction of the flow in the region of the section under consideration. In the course of the computations the shape parameter « ia modified, thus leading to a modified or an effective profile shape that differs slightly from the apecified shape. INTRODUCTION Pressure-distribution measurements over an airfoil section provide, directly, the knowledge of the air-forea| distribution along the chord that is required for some purposes, In addition, such data, when compared with the results of potential-low (nonviscous fluid) theory, provide a means of studying the offects of viscous forees| on the flow about the airfoil section. ‘Tho results of experimental pressure meesurements for o few miscellaneous airfoils may be found in various publications. ‘The general application of this method of obtaining design data, however, is limited because of tho expenso of making such measurements, A method of calculating the pressure distribution is developed in references 1 and 2. ‘This method, based ‘on the “ideal fluid” or potential-flow theory, gives the local velocities over the surface; the pressures aro cal- jeulated by means of Bernoulli’s equation. Although this method provides an inexpensive means of obtain- ing the distribution of pressure, the results may not be in satisfactory agreement with measured results. Such disagreomont, however, is not surprising since the theory does not account for the effects of the viscous, boundary layer. ‘A reasonably accurate method of calculating the pressure distribution over an-nirfoil section is desirable and might be obtained by two procedures. First, such ‘a mothod might be found by the development of 2 com- pleto theory. Such a theory, however, must take into account all the factors or phenomena, involved and must give satisfactory agreement with actual measure- ment. A second procedure, tho most feasible one at present, is the development of a rational method of correcting the application of the potential-low theory to minimize the discrepancies between the theoretical and measured results. ‘Tt was realized, however, that unusually reliable ex- ‘perimental pressure-distribution data for comparison with calculations were not available. ‘The experi- ments to obtain such data consisted of pressure measurements at 2 large number of points around one section of an airfoil. Because the investigation was primarily intended to study devintions of the actual rom tho ideal, or potential, flow, the tests were made in ‘tho variable-density tunnel over a range of values of the Reynolds Number, representing varying effects of viscosity. In addition, tests wore made in the 24-inch high-speed tunnel at certain corresponding values of ‘the Reynolds Number obtained by means of high speeds, thereby bringing out the effects of compressibility. Parts of this experimental investigation outside the scope of this report are still incomplote. ‘Tho present report, which presents the most impor- tant of the experimental results (those corresponding to the highest value of the Reynolds Number), is divided into two parts. ‘The first part comprises the descrip- tion and discussion of the experimental technique: Materials that are essential to establish the fact that the ‘measured results are sufficiently accurate and reliable to meet the demands of the subsequent analysis, The 305366 second part presents e comparison of theoretically cal- culated results with mensured results and an analysis of tho differences and probable causes. A mothod is developed to modify the application of potential-flow theory in order to minimize discrepancies from the measured pressure distributions. EXPERIMENTAL PRESSURE DISTRIBUTION ‘The experimental investigation described herein was made in the variable-density wind tunnel (reference 3). ‘The model used was astandard duralumin airfoil having prea cress the NAO, A. aft. . C. A, 4412 section and a rectangular plan form with a span of 30 inches and a chord of 5 inches. It was modified by replacing a midspan section 1 inch in length with a brass soction in which the pressure orifices were located. ‘The 54 orifices, each 0.008 inch in diameter, wore drilled perpendicularly into the air- foil surface and placed in 2 rows about the airfoil. The method and accuracy of construction of the model are described in reference 3. In order to evaluate the pressure force parallel to the chord, a relatively large number of orifices were located at the nose of the airfoil (Gg. 1); well-defined distributions of pressure long normsl to the chord were thus assured. The locations of the pressure orifices aro included in table I. Brass tubes were connected to the orifices and carried in ‘grooves in the lower surface of the airfoil to the planes of the supporting struts where they were brought out of the model, After the model was assembled, the grooves were covered with a plate carofully faired into the surface, ‘The tubing extended through the tunnel wall into the dead-air space and the part exposed to the air stream together with tho support struts was faired into a single unit (fig.2). ‘The tubes were connected by rubber tubing to a photorecording multipletubemanomt- eter mounted in the dead-air space. Figure 3 shows the 60-tube manometer, composed of 30-inch glass tubes arranged in a semicircle and con- nected at the lower ends to a common reservoir. ‘The total-head pressure of the air stream was chosen as the reference pressure and was measured by e pitot head, mounted as shown in figure 2, to which four equally spaced manometer tubes were connected. The dynamic pressure of tho air stream was determined by two tubes connected to the calibrated static-pressure orifices used in the normal operation of tho tunnel. One tube was comnected to # set of four orifices spaced around the inner wall of tho return passage and the other tube to a set of four orifices spaced around the entrance cone REPORT NO. 563 NATIONAL ADVISORY COMMITTEE FOR ABRONAUTICS near the test section. ‘The remaining 54 tubes, used to measure the pressuro at the orifices on the airfoil, ‘wore connected to the tubes leading to the airfoil model. ‘A lighttight box mounted on the flat side of the semicircle contained drums for holding photostat paper and the necessary operating mechanism. ‘The ma- nometer was arranged so that it could be operated from outside the tank that houses the tunnel. ‘The manometer characteristics determined by trial included the time required for the meniscuses to bo- come steady and the proper exposure of the photostat paper. ‘A record of the heights of the manometer fluid in the glass tubes was taken at cach of 17 angles of attack ‘roons 2-—Prewmredebaton modal manta nth tunes from —20° to 30° at » Reynolds Number of approxi- mately 3,000,000. In order to Koop tho results as accurate as possible, it was necessary to obtain large deflections of the ma~ nomoter liquids, which was accomplished by using two liquids of widely different specific gravities. Liquid: Spetegrarty Meroury.. 136 ‘Tetrabromosthane.. 30 ‘The proper choice of the anglo-of-attack groups and of tho liquid enabled the uso of largo end comparablePRESSURE DISTRIBUTIONS OVER THE MIDSPAN SECTION OF THE N. A. ¢. A. 4412 AIRFOIL 367 deflections throughout the angle-of-ettack range. Re- peat tests, using the samo and different manometer liquids, provided data on the procision of the tests. ResuLTs A copy of a sample photostat record is shown in figure 4. The pressures in inches of manometer fluid were measured to 0.01 inch. All measurements were made from a reference line obtained by drawing a line connecting the meniscuses of the four reference tubes. ‘The quantities thus obtained from the photostat records were: Ap=H—p where His the total-head pressure of the stream and P, the pressure at the airfoil orifice; and g=fectorX Ap, where gis the dynamic pressure and Ap, is the difference in pressure between the static-pressure orifices in the entrance cone and those in the return passage, ‘The factor was previously determined by comparing values, of Ap, with simultaneous values of tho dynamic pres- sure obtained with a calibrated pitot-statio tube ‘mounted in the air stream in the absence of a model. Finally, the pressures on tho airfoil were computed as ratios to the dynamic pressure, thereby making the results independent of manometer liquid. ‘Bernoulli's equation for the undisturbed stream becomes Pat KeV?=H where Pa is the pressure and V the velocity. ‘The pressure of the fluid at the wing orifice is given by p=H—Ap Substitute for H from the previous equation and remember that %pV?=g, the dynamic pressure, then P=Patg-Ap Consider p., 08 the datum pressure. ‘The pressure coefficient then becomes P. where Ap and g aro quantities obtained from the Photostat records as previously described. Values of P at each orifice on the airfoil and for all angles of attack are tabulated in table T. Figure 5 (a, b, 6) presents plots of P against orifice position along the chord and against position perpendic- ular to the chord for each anglo of attack. Large-oale plots similar to those presented here wore mechanically integrated to obtain tho normal-foree, the chord-force, and the pitching-moment coafficionts, which are defined by the following expressions: 5 rT ROR a ‘iowas 3~Photrourding mallet manomee. whore ¢ is the chord, z is the orifice station along the chord, and y is the orifice ordinate measured from the chord. ‘The lower-case symbols ¢y C Gay designate368 section, characteristics and refer respectively to the normal-force, chord-foree, and pitching-moment, efficients for tho midspan section of tho airfoi Plots of these coefficients (see table II) against geo- meiric angle of attack are given in figure 6. ‘The geo- metric angle of attack a is measured from the mean direotion of the flow in the tunnel. This direction is dofined as the zero-lift direction of a symmetrical airfoil in the tunnel and was found to be equivalent to 20’ of upflow. In order to have true section characteristics @-dimensionsl) for comparison with theoretical cal- culations, a determination must be made of the effec tive angle of attack, i. e., the angle that the chord of REPORT NO. 563 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS where wis the induced normal velocity produced by the vortex system of the airfoil, including tho tunncl-wall interference, and V is the velocity of tho undisturbed flow. In order to calculate the induced velocity w, the distribution of the lift (or circulation) along the span of the airfoil must be determined. A theoretical method of obtaining this distribution is given in refer- enco 4 and, when applied to this problem, gives for the induced angle of attack of the midspan section 584 where ¢; is the lift coefficient for the midspan section. ‘This lift coofficiont is obtained from the prossure ee Arr oe iin ey ‘ovwe &—Copy of sample red. N, ann odg xo ta; 8, alls prarae ter, lg ge ein tube and Z, rence eer tubes tho model makes with the direction of flow in the region | measurements by means of the equation of the midspan section of the model. ‘The effective angle of attack, corresponding to the angle for 2-dimensional flow, is given by 2 where ar is the angle that the flow in the region of the Airfoil section makes with the direction of the undis- turbed flow. ‘The amount of this deviation is small and can be calculated from e164 008 a6, sin a ‘Values of cy, ay, and ay are given in table TI. -PRRCISION ‘The reliability of the results of the pressure mensure- ‘ments reported herein may be determined by considera- tion of the technique of obtaining and measuring tho pressure records, of the deviations of the pressure diagrams obtained from several tests at the same angle of attack, and of the method of calculating the effective angle of attack,PRESSURE DISTRIBUTIONS OVER THH AUDSPAN SECTION OF THE N.A.C.4. 4412 amor, 369 ‘Phe method of obtaining the pressure records is a | to becomo steady and by delaying the taking of the direct, simultaneous, photographic recording of the |record at each angle of attack until sufficient time had height of the liquid in the manometer tubes. Since | clapsed. As a further check, a zero record was taken the pressure coefficients used in the analysis are ratios | at the end of each test run under the samo conditions. ah aera! force +L crore force ermal force Qnord force al : oe | N, ; Peters te bee “A = : a, ase rhe a a 9 a - 4 ene . _ r D a a i P , / 4 | : - 7 2 F a / a ane a 'o 50 wo 8 wo 0 50 Wo 0 0 Percent chord oy ‘loonr &—Espemontal and theraal prsnredietstbatin dagen fr the NA. A? el a sea ange of tack, of quantities taken from the same record, the primary | In addition, the tubes were checked for leaks before soureo of error therefore lies in the unequal damping in | and after each run. In order to minimize any possible the tubes connecting the airfoil orifices to the manom- | error in reading the photostatic records (ig. 4) measure- eter. This source of error was minimized by deter-| ments of the recorded pressures were made independ- ‘mining the time required for tho liquid in all the tubes | ently by two persons. ‘The readings were then com-‘puvont NO. 609 NATIONAL ADVISORY COMMOTTER FOR AERONAUTICS 370 ~4| ‘Nermo! force -3t \ p 20 9 0 0 6 Percent chord remeron daa tx a NA. OA nate sv ace tas © ‘rovux so —Espurtmeatl and thetaPRESSURE DISTRIBUTIONS OVER THE MIDSPAN SECTION OF THE N.A.0.A. 4412 atrron, 371 pared and a compromise was made where differonces | from soveral tests at the same angle of attack. Figure 7 occurred, The differences between any two such| presents such diagrams at two angles of attack, —4° independent readings rarely exceeded 0.01 inch except | and 8°, Totrabromoethane, because of tho larger in the case of obvious errors. Possible errors duo to | deflections, gave more accurate results, which agreed = : [sms ee so —— a a a wh Ey | -1 el fb . « wor « ¥loune 62—Brprimenat end hurt! premre-deeaton Alarms for thy N, AO, A 42a a over al fata. shrinkage of the records were avoided by the uso of the | very closely with the mean values obtained from ratio of two pressures obtained from the same record; | repeated mercury tests, of which the greatest devie- namely, the ratio of the pressure at s wing orifice to | tion from the mean values was approximately <3 per- the dynamic pressure, cont of the dynamic pressure. ‘This deviation is not a ‘The precision of the measured results is indicated | random scattoring of points from say given test but is by the variations of the pressure diagrams obtained | e consistent difference between repeat tests and may372 be partly accounted for by @ possible small difference inangleof atteck. Figure 7(b) elso includes the results of tests made before end after carefully polishing the midspan section of the model. The change in surface smoothness and « slight change in fairness had no dis- cemnible effect on the distribution; the differences were i " ; 7 ‘ tH oe i gu i = ‘« 8 Lo oN i °. oan apd KH oe a 4 z 24 Bibie of ottocks a, deortos ‘locas &—Nerm nd chord oe eles nt ptchlag mementos ‘stg ie Toso manot fr s less than those obtained by repent tests of the same surface. ‘The determination of the effective angle of attack of the midspan section entails certain assumptions that are subject to considerable uncertainty. First, the angle of attack of this section may be in error because of the assumption that the deviation of the air-stream axis from the tunnel axis is uniform along the span of the model; i. e., that the geometric angle of attack a is the same for all sections along the span. Actually there is some variation of the airstream direction across the tunnel. Because of the interference of the support strats, the deflection of the stream in this region might reasonably be expected to exceed the deflection at the midspan seetion; hence, the deflection at the midspan section is probably less than the effective mean value. Furthermore, a zero deflection of the stream at the midspan section would bring the angle of zero lift obtained from the pressure tests into agree- ment with foree-test results, ‘A second and rather large source of error lies in the determination of the induced angle of attack. The method used probably produces erroneous results REPORT NO. 568 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS because of the fact that the tips of a rectangular wing carry a. larger proportion of the lond then is indicated by thé theoretical calculations on which the mothod is based. ‘To make an accurate experimental determina- tion of the lift distribution on which to base the induced angle calculations would require pressure measure- ments at several sections along the span, especially neat the tips. An estimate can be made, however, of the possible error in the induced angles of attack givon herein by comparison of the deduced slopes of the lift curve for infinite aspect ratio obtained from these tests and from the best force-test data available, Such a comparison indicates that the induced angle of attack may be approximately two-thirds of tho calculated values given herein, which would moan possible error of approximately i° for # lift coofficiont of 1. It is evident, therefore, that the effective angles of attack aro subject to a considerable error of uncertain magnitude. Approximate possible errors have been (al + Mercury [before polishing) on” otter ey Steer SR, Tefrobremoathane a t 00 30 Percent chord ‘rooms 7—Precuredteaton arate rom evurl stat two ants oft: estimated and summarized as follows: Tho values of the angles ns given may be too largo by a constant error of approximately ¥° because of a possible error in tho assumed direction of the stream. On the other hand, the angles may be too small by approximately ei/2°, owing to the error in tho indueed-angle calculations,FRESSURE DISTRIBUTIONS OVER THE MIDSPAN SECTION OF THE N.A.C.A.4412 amrom, 373. THEORETICAL PRESSURE DISTRIBUTION mined by means of the seme transformations. Refer POTENTIAL-FLOW THEORY ences 1 and 2 present detailed discussions of the under- A theoretical determination of the distribution of | lying theory and the derivation of the necessary equa- pressure about an oirfoil section has beon developed | tions for the calculation of the characteristics of the for potential flow and assumes an ideal fluid that is| potential field about the airfoil. ental How, some. nate oleate fom, Sane Drover §—Presur-vecor dgramsfor tN. AO. A AUtabfoatavvenangls attach. nonviseous and incompressible. Briefly, the method] ‘The general equation for the local velocity about an consists of the conformal transformation of the aixfoil| airfoil section in a potential low as given in raference 1 section into a circle. ‘Then, inasmuch as the flow |is about the circle can readily’ be calculated, the flow vile r characteristics about the airfoil section ean be deter- oovi{ sno tete)+ze7] ®374. 7 (+8)ou Vis the velocity of the undisturbed stream. a, the angle of attack (2-dimensional). T, the circulation. 4, ¥,6 Parameters that are functions of the airfoil coordinates, Yo, the mean value of y. R=aets, tho radius of the conformal circle about which the flow is calculable. In order to calculate the velocity field from equation (2) the circulation must be evaluated. ‘This evalus- 2 ae 2) p o” 1 2 ff s 4 Re Lieb es : SSE ECE “? Etrectwe onde of offack, a segre icone O-Lit and peblagmeest wecon characte fo the N. AC. As “eid aid tion is done by the use of the Kutta condition, which requires that the velocity at the trailing edge (9) be zero so that equation (1) becomes = Vkfsin(@-+e+a) +sin(a-ter)) where er is the value of « ut 0x (trailing edge). ‘Th angle of zero lift is equal to —er. ‘The necessary equations and a step-by-step description of the calculation of the velocity field aro given later. ‘Tho pressure coefficients are computed by means of Bemoulli’s equation, ot —h) (a) P. @ REPORT NO. 563 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS where p is the pressure at the airfoil surface and p= the pressure of the free stream. ‘COMPARISON OP THRORY AND BXPRRIMENT ‘The theoretical distributions of pressure have been calculated for the 2-dimensional angles of attack corre- sponding to the measured distributions on tho N. A. C. A. 4412 airfoil. Comparisons of the calculated and measured distributions aro presented in figure 5 (ox- cluding the diagrams after the airfoil has stalled) and in figure 8, Figure 5 presents the usual normal- and chord-component pressure diagrams and provides a means for a goneral study of tho differences bobwe tho theory and experiment as a function of anglo of attack. Figure 8 provides a more detailed study at a fow angles of attack and presents vector dingrams for tho angles of —8°, —4°, 2°, 8°, and 16°, ‘Those dia- ‘grams were obtained by plotting the pressure coofficionts normal to the airfoil profilo; the perpendicular distance from the profile line represents th magnitude of tho coefficient. The experimental pressures are represented by the drawn veetors and the theoretical pressures by ‘the solid contour line. ‘The other contour lines repre- sent certain modified calculations to be discussed Inter. It is immediately evident that the theoretical results do not satisfactorily agree with the actual measure- ments except for angles of attack nest ~8°, correspond ing approximately to tho anglo at which the expoi ‘mental and theoretical lifts aro the samno (fg. 9). ‘The comparisons in figure show, moreover, that with in- creasing anglo of attack tho differencas between theory and experiment become larger as predicted by the higher slope of the theorotical lift curve. A detailed study of the vector diagrams (fg. 8) shows how theso differences vary around the profile of the sirfoil. ‘Tho largest differences occur in the regions of low pressures, or the high-velocity areas, and as previously stated they increase with incrossing angle of attack. Furthermore, the percentage difference in pressure is largor near the ‘railing edge than in the region of tho nose, indicating a progressive influence on the flow as it moves over the sirfoil surface. ‘The effect of theée differences in the pressure distri- bution on the pitching-moment characteristics is shown in figure 9. ‘Tho theoretical pitching moment about the quarter-chord point was obtained by intograt theoretical pressure diagrams, ‘The results show an in- creasing diving moment with increesing angle of attack, wheroas the diving moment actually decreases. ‘The comparisons have thus far been made at the samo relative angle of attack, that is, for the angle of attack in 2-dimensional flow. Another condition of comparison that has been used more or less regularly in previous studies is suggested; it allows » comparison at the samo lift and consists in comparing the theo- retical distribution calculated at an angle of attack that gives a theoretical lift equal to the experimentalPRESSURE DISTRIBUTIONS OVER THE MIDSPAN SECTION OF THE N. A.C. A. 4412 AIRFOIL value. ‘This mothod has been used for the diagrams in figure 8 and the distributions thus caloulated are represented thereon by the long-and-short-dash con- tour lines. Again the differences are too large to be neglected, especially at angles of atteck where a large lift is obtained. At —8° the curve coincides with the previously described contour, since the engle and the lift are the same, while at —4° the distribution cal- culated on the basis of tho same lift is approximately the same as the dashed contour representing o third calculation presented herein. At the higher angles of attack the calculated distributions depart progres- sively in shape from the measured distributions. It may therefore be concluded that, on the basis of these comparisons, the usual celculations from the potential theory do not give an accurate determination of the distribution of pressure about an airfoil ‘The inaccurate prediction of the forces on an airfoil by the usual potential-flow theory is not surprising since the theory neglects the frictional force of the viseuous fluid acting on the sirfoil. The direct effect, of this foree, which acts tangential to the direction of the local flow, is important only on the drag and contributes what is known as the “skin‘friction”” drag. Because of the small magnitude and the direction of this foreo, the component in the direction of the lift is probably negligible, the lift being determined en- tirely by the pressure forces. ‘The indirect effect, how- -ever, of this friction foree is the deceleration of the air in a thin layer near the surface of the airfoil and the production of the so-called “boundary-leyer” phe- homens, which are important in the development of Jift by an airfoil. In tho boundary layer tha velocity changes rapidly from zero at the surface of the airfoil to the value of the local stroam velocity at the outer Jimit of the layer. ‘The loss of energy involved in over- coming the friction forees results ins cumulation of slowly moving air as the flow moves back along the sirfoil; hence the boundary-layer thickness increases toward the trailing edge. This cumulative effect is indicated by the progressive incroase in the differences between tho theoretical and measured pressures. ‘From this discussion it is not to be presumed that -agreement betwoon the measured and caloulated results should oceur at zero lift, except approximately for symmetrical airfoil section. ‘Tho velocity distributions over the upper and lower surfaces of an asymmetrical section are not the same, even at zero lift. ‘The viscous effects on the flow over the two surfaces at the caleu- lated angle of zero lift are therefore different and a lift is measured, which is negative for most sections. Actually, then, the experimental and theorotical angles of zero lift are not the same and for normal sections the two lift eurves intersect at a negative value of the Lift cooffcient, Outside the boundary layer the viseous forces can probably be considered negligible and the flow a 190002—27—25, 375 potential one; probably the pressures may also be considered os being transmitted undiminished through the thin boundary layer. ‘Tho actual flow might there- fore be replaced by a potential flow about » shape slightly different from that defined by the airfoil coordinates, which would require the determination of ‘the boundary-layer thickness to define tho effective profile shape. ‘The pressure about the new shape could then be computed by the potential theory. Boundary- layer calculations, however, are at presont subject to uncertainties that would cast doubt on the validity of the results and, in addition, the computations are difficult and tedious. [MODIFIED THEORETICAL, CALCULATIONS A simpler and more practicel method of calculating the pressure over an airfoil section has been developed aA * —a 5 ey a aed eee 25 a result of the foregoing analysis, Tho analysis shows that theoretical distributions calculated at the ‘true angle of attack are similar in shape to the true distributions but give too high lift. Conversely, when the theoretical distributions aro calculated at an angle of attack thet gives tho same lift as the experi- mental distribution, the two distributions aro dissimilar in shape. ‘The modified calcwation is mado at the offective angle of attack but the circulation is determined from the experimentally measured lift instead of by tho Kutta-Joukowsky method. The preliminary calcula- tions made on this basis resulted in an excessive velocity and © consequent high suction pressure at the trailing edge, as shown in figure 10. This unsutisfactory result Ghowa by the dot-dash lino in fig. 10) was finally avoided bymeans of a further modificationsubsequently described.376 Sinco a change in the effective profile shape has been predicted by boundary-layer considerations, an arbi- trary modification of the shape parameter c is made so that the velocity becomes zero at @=. (See equation ().) The shape is thus altered to satisfy again the Kutte-Joukowsky condition. In order to maintain the continuity of the curvo, a study has been made of the manner in which ¢ should be modified. ‘The indicated cumulative effects of the viscous forces toward the trailing edge show that most of the change in « should ore S45 6 REPORT NO. 563 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS tions obtained by means of the modified calculations ‘are given by the dashed lines, Tho relative merit of the unaltered potential theory and the modifiod motibod for the calculation of the pressure distribution about an airfoil section is shown in figures 5, 8, and 9. ‘The following step-by-step description of the compu- tations required to obtain the calculated pressure dis- tribution is given in sufficient detail to enable the calou- ations for any airfoil to be made. ‘The local velocity about the airfoil is computed by means of equation (1) 7 8 9 Wail le 1s 14 15 16 17 18 1S 20 Flovur 11—Thcratealprsmaters requ compute the thea peur oa the. A. 0. A. ial probably be made in that region. Inesmuch as the effect of changing ¢ is not critical for different. dis- tributions of the change, provided that most of the change is made near the trailing edgo of the airfoil, a purely arbitrary distribution is chosen that pormits ready application, namely, a sinusoidal variation with 0. ‘Tho ¢ curve and subsequently the other parameters must be modified for each angle of attack. This modi- fication has been made and the corresponding pressure distributions determined for soveral angles of attack. (Geo figs. 5 and 8.) At —8° the distribution is the same as that shown by the solid line representing the unaltered theory. In the other diagrams the distribu- modified as indicated by the preceding discussion. ‘The detailed forms of the modifications are introduced as they appear in the course of routine computations, In order that the transformation from the airfoil to its conformal circle may be of # convenient form, the coordinate axes are selected so that the profile is as nearly as possible symmetxical about them. (Seo refer~ ence 1.) The x axis is chosen as tho line joining the centers of the leading- and trailing-edge radii. ‘The origin is located midway between a point bisocting the distance from the leading edgo to the center for ‘the leading-edge radius and the corresponding point at the trailing edge; the coordinates of theso points arePRESSURE DISTRIBUTIONS OVER THE MIDSPAN SECTION OF THE N. A.C. 4. 4412 AIRFOTL, respectively (2a, 0) and (—2a, 0). In the following discussion the coordinate scale has been chosen so that ais unity. (For practical purposes it is probably suffi- cient to choose the chord joining the extremities of tho mean line as the z axis.) ‘The following equations express the relationship between the airfoil coordinates previously described and the parameters and y. 2 cosh ¥ cos @ 2 sinh sin @ @ In order to compute values of 0 corresponding to any given point on the airfoil profile, equations (4) are solved for sin’. Sint 3 (4 FP) © (77-47 @-@) A similar solution for sinh*y can be obtained but experience has shown that s more ustble solution is given by the equation below where hem ri sinh vas 6) A plot of ¥ as a function of @ for the N. A. C. A. 4412 airfoil is given in figure 11. ‘The function yo is given by BJ and can be determined graphioally from the y curve or by a numerical evaluation. ‘The value of Ye for the N.A.C. A. 4412 airfoil is ve Y= 0.1044 ‘The parameter ¢ as a function of @ is given by the definite integral, aH "4 cot Ghia ® where the subscript n refers to the particular value of 6 for which the corresponding value of ¢ is to be deter- mined. A 20-point numerical evaluation of this inte- gral is derived in reference 1 and is included here for convenience. The integral is evaluated at 20 equal intorval values of @, namely, %=0=0_m 377 The value of « at 6, equation. 7 is given by the following — 2[ 5G) 4.001 Gated) +0496 (Yuta—Yors) 0218 $0217 Fo.58 to.15 0.0804 0.0511 $+0.0251 (Yass Yord | ® where the subscripts designate the particular @ at which the named quantity is taken. A plot of eas a function of @ for the N. A. C. A. 4412 airfoil is given in figure 1, ‘Thus far the calculations aro identical with those made for the potential theory. As stated in the discussion of tho modified theoretical calculations, the circulation is evaluated by the experi- mentally known lift of the airfoil section. ‘The well- known equation relating the lift and the circulation is L=evr Also by definition L=}pVee: “Expressing the circulation in terms of the lift coeflicient, roo, and finally roe eer eee © Substituting the numerical values for the N. A. C. A. 4412, leet ieRV~oaT8" (2) ‘The prediction of unreasonable velocities around the trailing edge is avoided by altering the ¢ function so that tho velocity is zero at 0=r, ‘The altered function is designated «, and is arbitrarily assumed to be given by eat AE (1-008 0) (20) where Ae is the increment of ¢ required to give zero velocity at #7 and is a function of the angle of attack. ‘The quantity Aer is given by Neptap me where é, is determined by equating equation (1) to zero and substituting from equation (0). sin (tateen) + geper=0 Solving for tap gives,378 ‘The parameters ¢ and ¥ are conjugate functions of 8, and ¥ is given by Yemge [e cot sat ty where the definite integral can be evaluated in the same manner as equation (7). The coordinates of the profile corresponding to the modified « function ean be obtained from the new y function by equations (4). ‘Figure 12 gives the modified shape obtained by this method for the N. A.C. A. 4412 airfoil at a=8° and 16°. The profiles given in figure 12 are, of course, only effective profiles corresponding to the calculations. ‘The actual profile about which a potential flow might) bo considered as being established would be Blunt at ‘tho trailing edge and would have the thickness of the wake at that point, ‘The thickness of the boundary ayer on the upper surface, however, is greater than ‘that on the lower surface; therefore, if the trailing edge were taken as the midpoint of the wake and the after portion of the profile were faired to that point, the — a mACA 44127 ‘rooms 12—Chaoes ta profabpe anodated with th modi theoretic ele Tito ot presre resulting shape would be similar to the effective profiles in figure 12. ‘The influence of the changes in y on the value of k are found to bo negligible so that ke may be written bese ete where REPORT No. 583 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS Differentiating equation (10) Gentes Sein a Plots of 4 and kas functions of @ for the N. A. C. A. 4412 airfoil are given in figure 11, Equation (1) for the velocity at any point on the sirfoil profile is now written cave lsin O+e-tad+g ope] ab) ‘The generality of the preceding method of cal- culating the pressure distribution about an airfoil section is supported by the following evidence. First, no restricting assumptions have been made in tho development of the method. Second, the circulation is determined by a known quantity, the experimentally measured Jift, Third, the change in the effective air- foil shape is in the direction indicated by boundary- layer considerations. Finally, the computed and meas- ured pressures agree satisfactorily. ‘Lanoury Menontat AznonauricaL LABORATORY, Nastonat Apvisony Cosncrrres ror AERONAUTICS, ‘Lanouey Fiun, Va., March 26, 1986. REFERENCES 1, Thoodorsen, 7, and Garrick, I. E.: General Potential Theory of Arbitrary Wing Sections. . R. No. 452, N. A. C. A. 1088, 2. Theodorsen, Theodore: Theory of Wing Sections of Arbi- trary Shape. T. R. No. 411, N. A. C. A. 1981. 8, Jacobs, Bastman N., and Abbott, Ina H.: The N. A. C. A. Yariable-Density Wind Tunnel. ‘T. R. No. 416, N. A. Cla, 1082. 4. Millikan, Clark B.: On the Lift Distribution for a Wing of Acbitrary Plan Form in a Circular Wind Tunnel. Pub- Ieation No. 22, C. I. T., 1982.379 PRESSURE DISTRIBUTIONS OVER THE MIDSPAN SECTION OF THE N. A.C. A. 4412 AIRFOIL TABLE L—EXPERIMENTAL DATA—N. A. C. A. 4412 ATRFOUL vege prose sander tephra 1; areas Rayos Nombe: 308,08) fr leant ng of ta ‘alos pena corte, P= on we RRRRgUUYSEESE Trirere nine ir eescaauuaaeas B88 Fig 3: t TTT nee seriisuaeateateiT i a aE. 3 RRESSESE! FEET ane es q aaa: SER SS SSSLALTRLRREARSILSALEALR ISIS ESTER AARAZESES * titrineipers TCTOTT ETAT TEEPE LTE G eT SeEMUS\(eUGH: Sasa au Rana RasgnaRAZeaEANIAaAeERANANETEzeA tt t 1 ire pauseegesaae pEGSH GHG ART Ag EEIEREG=5 A HECANGEaESTBEC: oa ae Teererrertre ate SERA LESSSSESERRSSATEH ertree aso | re | xe aaaereae SIGIR are teen Saagatsedeusgazaquaegaa ~anagauegzavecasaaseceasuaases Sbaddtudevaahasier ves oo” sewnedsadannaaesesdsesased WrasaaaaTa Roan Rene RAS TeNORUTTESEER -SAgRGNSRAUAEARA "Tes, yree-doaey anna 100-4 amor gal, tateoetiane INTEGRATED AND DERIVED CHARAC- "Pe, vvibiedoty ral 08 aso, maar. TERISTICS—N. A.C. A. 4412 AIRFOIL TABLE I. Bessae: Hirt aT TTSRISeaeae jorenseeaagBteo~avancen-sessnanase GS SUT TTS se | SERERSSEESERSEESEESE " sassuegegsagas’ a2 5 SeeaeneEEE eeesee a | RERRSBE999 iBuREscaeag ulSUERGOAGGC_ gevvasavasasansazcenasae giTvreananeasesagagzenag 4412 AIRFOIL, aaaganaRnaoaaeagsg = | aeageaznaancagzeseeee ‘TABLE IV—THEORETICAL PARAMETERS—N. A. C. A. a _RRERRRRAANTEGE_ Sausangnannaasiansnegag REPORT NO. 563 NATIONAL ADVISORY COMMITTED FOR ABRONAUTICS ‘TABLE IL—THEORETICAL PARAMBTERS—N. A. C. A. 4412 AIRFOIL ote ‘ i ‘gasaneegaazeaszeeace j SSRAgEORaSSESSeeaeS i. L BeBe e. i ae Sietteanaesersaaueg 380