jOftcfV L^l^'l ARE No. L6A05t NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS WARTIME REPORT f ORIQNALLY ISSUED February I9U6 as Advance Eestrlcted Eeport L6A05b DETERMINATION OF DTOQCED WLOCITY IN FRONT OF AN INCLINED TROPELIER BT A MAGNETIC -ANALOCT METHOD By Clifford S. Gerdner and James A. LaHatte, Jr. Langley Memorial Aeronautical Laboratory Langley Field, Va. WASHINGTON NACA WARTIME REPORTS are reprints of papers originaUy 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. DOCUMENTS DEPAHlMtN I Digitized by the 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/determinationofiOOIang MCA ARR No. L6A05b MATIOITAL ADVISORY CO'I'MTIEE FOR AERONAUTICS ADVANCE PESTRTCTID REPORT DET5R:/:INATI0rJ OF I INDUCED VELOCITY IN FRONT OF A^' If>'CLI.-[SD PFOPZLLER BY A ^^AGMETIG-A: ALCGY METHOD By Clifford S. Gardner and Jarnes A. LaHatte, Jr. SUf^MARY The horisontal and vertical cor.ponents of the induced velocity ?n front of an inclined propeller in a horizontal streaiT' were obtained by a rragzietic-analogy method. The problem was formulated in terms of the linear theory of the acceleration potential of an inccr.-pressible nonviscous fluid. The propeller vas assumed to be an actuator disk. The horizontal coiroorent of the induced velocity waa found by a num.erical calculation. I'uir^erical calculation of the vertical comoonent, however, i?v'as not practicable; there- fore the vertical component was ootaiijed from electrical measurements by use of the analo&y between the accelera- tion potential of an incom.pressible nonviscous fluid and the potential of a m.agnetic field. An alternative form.ulation of the problem in terms of the traili ng-vortex sheet is shown to be equivalent to the acceleration-potential formulation if the thrust coef- ficient is assum.ed so small that the sliostream is not deflected and undergoes no contraction. From the results presented, induced velocities of greater accuracy are shown to be obtainable from a modification of the vortex theory based on the assumption of a constant finite down- v-ash angle of the slipstream.. INTRODUCTION The recent developm.ent of airplane designs with pusher-propeller installations has occasioned several Inquiries regarding the nature of the flow in front of an inclined prooeller and the corresponding aerodynai'^ic effects on the wing. Because of the difficulty of the calculations, little effort has heretofore been m.ade to l6a05^ conipiite the flov/. Some experi:uental work, hov/ever, has been done in connection with the problem of the lift Increment on the Vv'ing (for exair.ple, references 1 and 2). Further development of the theory is considered desirable to serve as a basis for correlation of these and similar cata • The purpose of the present paper is to give detailed theoretical data on the induced velocities in front of an inclined propeller. Cnl^r the components important to the problem have been obtained; nar;iely, the component parallel to the free-stream direction and the component noriTial to the free streaun and in a vertical plane, which will be designated horizontal and vertical coiuponents , respec- tively. The determination of these components is based on the linear theory of the acceleration potential of an incompressible nonviscous fluid, and the propeller is assurr.ed to be an actuator disk. Because the theory is valid only for small perturbations, the results, which are presented in dlmensionless form independent of the thrust coefficient, are valid only if used for propellers operating at low thj^ust coefficients. The horizontal component of the induced velocity v;as determined by numerical computation. The computation of this component v/as practicable because of certain simpli- fications due to syiTimetry. Niomerical calculation of the vertical coii.ponent, nosvever, is excessively laborious and time-consur:iing ; consequently, the vertical component v;as obtained from electrical measurements by use ox" the analogy betvi-een the acceleration potential of an incom- pressible nonviscous fluid and the potential of a m.agnetic field . of the trailing- vortex sheet is shown to be equivalent to the acceleration-potential formulation if the thrust coef- ficient is assiimec so small that the slipstream is not deflected and undergoes no contraction. From the results presented, induced velocities of greater accuracy are shown to be obtainable from a modification of the vortex theory based on the assumption of a constant finite down- wa^h angle of the slipstream. NACA ARR No. LoA05b ' J p local static pressure Pi static T)re8sure at :".ov;nstrear.- face o.f proDellt disk p air density t ti.iie X, J, z rectangular coordinates (fig, 1) V free-strean. velocity \x, V, v/ co.-nponents of perturbation velocity in X-,Y-, and Z-directicns J respectively I electric ci-rrent;, c^.s electroriiagnetic units R radius of propeller T> dia-neter of pro;neller T thrust of propeller T^ thrust coefficient f — \ '.V ! /' u V u', v'-, w' c.iTr.ensionless velocities ' , , respectively ] / ■''^ \ p' dimensionles s pressure ^ — ■■ \ :i ' , y', z' ciT.ensionless coordinates (x/R, y/R, z/R, respectively) ^ inarrnetic scalar p'Otontial, cgs electronag- netic units Ex, Hy, H„ ccniponents of jntisnetic-f :'.eld strength^ cgs electrojiiagnetic units E experi'.^entally ineasured voltage, volts KACA aRR ':lo . L6A05b a ratio of w' to an£.lo:;ou3 r.ieasured voltage E an:i,le of inclination of prooeller d.i sk to Z-ci-xls, ' as^iUired c^^nstant downwash an^^le of slipstrearri, radians V: mass rate of flew across propeller disk Su"bscript3 ; 2 In i;ltii:iate '.vaiie 1 at downstream; face of propeller disk TH30RY Oy THE METHOD Linear theory of the acceleration potential . - The 2uler equations for che flovy of an incon.presslble non- vise dus fluid -ay be written in the followir-e; form: D(V + DC u) _ : ±_ GX P Dv _ Dt . 6p P Dw Dt . 6p bz (1) These equatri.ons are in general nonlinear in the velo- cities. The equations r;:ay be !naae linear if the coniponents of the perturbation velocity are ass-oined to be snail coiT.- pared with the free-streaTi velocity. (See reference J.) If this assu!;-:ption is valid and if terras of the second order are neglected. Dt Dt ^x Dv Dt v|2 OX Dw _ dj^ (^ (2) NACA ARR No. L6A05b By virtue of equations (2), equations (1) become PV — = ~ (pa) ox ox PV — = --^ (5o) bx oj PV^.:-^-^ (3C) OX bz If equations (J) are differentiated successively "-vitii respect to x, y, and z and are added, the result ^ u 2 ,, b fbn bv 6w\ ^ ,, , -V p = pV — — + — + — 1=0 ik) bx \bx dy cz/ Magnetic analogy .- Since by equation (i-j.) p satisfies Laplace's equation, and since the scalar potential of a magnetic field also satisfies Laplace's equation, it follows that for similar boundary conditions p is directly analogous to the scalar potential of a magnetic field. This fact is the theoretical basis of the magnetic analogy of the present paper. For low tlirust coefficients the three boundary condi- tions for the pressure p in the problem of the actuator disk are: (1) The pressure has some constant value -pi uni- formly over the upstream face of the disk and a value p]_ uniformly over the downstream face. (2) The pressure has no singularities other than the jump discontinuity at the disk. (5) At great distances from the disk, the pressure is uniform and v/ithout loss of generality may be assumed to be zero. For the magnetic potential the first condition is sat- isfied by using as the source of the magnetic field a circular wire loop carrying a current. The use of this NkCA ARR llo . L6A05b current- oarryinr^ loop actually ensures the appropriate behavior or the ijiagnatic Tield at the disk since the magnetic potential o± such a loop has the value -2rri uniforrr;ly over one face of the loop unc the value Zrrl over the other face - if cgs electromagnetic units are used (reference I^) . The second condition is satisfied if no other magnetic fields and no r.agnetic materials are in the neishhorhood of bhe loop. The third condition is satisfied automatically, since the raagnetic potential of the loop approaches zero at great distances from the loop. Basis of deter.-aination o: equation fTs-T is integrated ;/ith respect horizontal velocit y . - If to X, the result pVu -o Inasmi.ich as the dimensionless velocit;^ sionless 'oressure d' are defined by (5) u • and the dimen- u' = anc p' = ■D PV^T, equation ( -S ) beco;;ies u -P' (6) The value of the dimensionless pressure p' at the streai-a face of the disk is ■ ■vvn- ^i' P- pv=-^ -c PV 2 TT r2. ^(^■y^o.\-\ 'I ; PV^ J (7) NAG A ARR I'o . l6A051d Since this tsouncs^Vf value is a universal nuTxerical con- stant, tlie values of the Girr.ensionless velocities throughout space, which are deter.-ninec oj the boundary values or p', are also universal numerical constants. By means of the dirnensionless coefficients u', v', W, and p', therefore, the Droblem is stated in a non- dimensional form that is independent of all relevant variables such as the disk radius, densitT, thrust, ana f ree-streaj!. velocity. In accordance with the magnetic analogy, the magnetic potential is directly analogous to the pressure p'; that is, O- = k;/ (8) v;here is the magnetic potential rieasured at a point of v/hich the dinensionless coordinates x', y', and z' are the same as the diinensionless coordinates of the point v/here p' is iieasured, and where k is sonie con- stant of proportionality that depends on the dimensions of the electromagnetic system. The value of p- and, hence, u' vvas thus obtained oy calculating and ir.ultipljT-iiig by the constant k- v;hi;',h n.ay be detenr^ined by comparing the value of p' at the disk, kvhich by equation (7) is 2/tt, with the value of at the disk, which is 2Tri . The calculation of was effected by nu.T:erical integration of a formr.la jjiven by Smythe (reference 5) -^^ the xr.agnetic field of a circular loop. The ootential and, consequently, the horizontal velocity u' at any noint depend only en the position of this point relative to the disk; thus, the results for u' at positions given in terms of coordinates fixed in the disk are the same for all angles of inclination. 3ecause of rotational s^Tnrr.etry, moreover, the values of u' at corresponding points in any two planes through the aj-;is of the disk are the sarxe. It v/as thu.s necessary to calculate u' over only one axial plane. The linear theory of the present paper gives values of u' = that are valid for lov/ thrust coefficients; that is, t:ie results for u' are essentially the deriva- tives of u/V with respect to T^ at 1^=0. The mcmenturn theory of the propeller (reference 6) gives the iriflov/ velocity at the disk as c Iv'AGa ARR Nc. LIrAOyo The valxie of \ it the d.?lsk should therei'cre be u' - /•' -U \ TT =0 a value that checks exactly with the value given by the linear theory of the -oresent paper { equations (6) and (7))' Basis of deteririination cf vertical velocity, - equati en ( j: c ) is inte^^rat; to Xj the result is ;a "/.j-t^i resoecL co x ■'o;.-! _co pVw = ~ / '-^— - GJC (9) oz If equation (9) is divided 'oy PV'^T^ and if diinensionless coordinates are introduced, the result is .x' (10) '.■■ -c CS In accordance with the r^agnetic analogy, if equation (C) is used, equation (10) becoraes n X -/ •dx' iix , , ^ k / — ^ tix (11) But since is the potential of a magnetic field, NACA ARE No. l6a051d 60 _ ^ bz where H^ is the z-cora-oonent of the magnetic -fie Id ;th. Equation (11) now beco:]ies = k I H^ dx (12) I U-co In order to find the value of v/' at a point (x', y', z' it is therefore sufficient to measure the integral of H^ along a path parallel to the X-axis, extending from minus infinity- to the point {x, y, z). An alternating Triagnetlc field in air induces in a coil of wire a voltage proportional to the total flux linking the coil, '.vhich in turn is proportional to the integral over the face of the coil of the norr:al compo- nent of the magnetic-field strength. The voltage induced in a long narrow search coil is proportional to the sur- face Integral of the normal field over the area of this coil; since the coil is nar^row and, consequently, the field strength is aLnost constant across the width of the coil, thiis surface integral is prooortional to the line integral along the length of the coil. The line integral in equation (12), therefore, is proportional to the voltage induced in a long narrow search coil, the plane of which is perpendicular to the S-ajcls, which extends parallel to the X-a:ci3 from the point (-ix>, y, z) C-: the t->olnt (x, y, z) , as shown in figure 2. Since the magnetic field dies out rapidly with distance, a search coil of practical length actually suffices to obtain accurately enough the infinite Integral. Thus the following equation holds: w' = I^E (13) where E is a measured voltage proportional to the voltage induced in the search coil and K is a constant of proportionality to be determined by calibration. 10 NACA ARR No . L6A05b APPARATUS A:TD METHODS Field coil .- A field coil of 61+ turns of Brown and Sharpe IJo . l8 copper wire wound on a circular wooden form was used to simulate the actuator disk. The mean radius of the coil was 12 inches and the cross section was a square 0.873 inch by 0.875 inch. The coil «vas supported by xjivots about its horizontal diameter in such a way that its angle of inclination to the Z-axis could be varied and the support could be moved up and down. The arrangement is shown in i'igure 2. Search coil .- As oreviously explained, a long narrow search coil was used to perform the integration of the magnetic-field strength Indicated in equation (12). The search coil was made u-o of 110 turns of Bro'wn and Sharpe No. 1|.0 copper wire •.voui:id lengthwise on a glass rod 72 inches by 0.225 inch by 1.2 inches. The coil rested on Luclte supports at the two ends. The supports were scribed with cross-hair line 3 to aid in setting the posi- tion of the coil and were suDolied with leveling screws so that the face of the coil coulc be turned exactly 90° to the flux being measured. The voltage developed in the search coil was fed through a filter eliminating 60-cycle pickup to an electronic voltmeter by which the voltage was measured. Pov-.'er supply .- Current was supplied zo the field coil from a motor-generator set delivering 5*^ a.mperes at 390 cycles per second. The Ward-Leonard speed control system was used sd that the frequency and output voltage could be adjusted by rheostats. The output voltage was continuously adjusted to m^aintain through the field coil a constant current of S.O am.peres, as measured on a standard high-frequency ainmeter. The output of the generator was connected in parallel with the input of a cathode-ray oscilloscope, and a 60-cycle line voltage was connected across the sweep circuit. The resulting Lissajous pattern v/as held stationary by continuous adjustm^ent of the frequency control rheostat; thus the frequency of the current was maintained constant at 390 cycles per second. The arrangem.ent is shown in figure 5. Test procedure .- In order to measure the integral in equation (12) at" the point (x, y, z) when the disk IIACA ARR No . L6A05b 11 was inclined to the Z-axis "by an angle a, the field coil was set so that its center was a distance z below a horizontal table; then it vv'as set at the angle a v.ith a protractor and the search coil was placed on the table •oarallel to the X-axis with one end at the point (x, j, z) and the other end av/ay from the field coil. The arrant;,e- ment is shown in figure 2. For each setting of a and z, readings of the voltage were made at the I70 vertices of a rectangular grid 6Ii. inches by J+O inches that v\fas made up :f lines parallel to the X-axis and the Y-axis spaced at intervals of [;. inches. The arrangement is shown in f i gur e 1 . Zero height adjustrrxent . - In order to locate the height of the field coil corresponding to a value of z = 0, (1 v/as set at 0° and the height of tlie field coil was then adjusted for zero voltage in the search coil. The voltage in the search coll is zero when a and z are. zero, so that the search coil is on a plane through the axis of the field coil, since the component of the magnetic field nomal to siich a plane is zero. Leveling adjustment for the search coil .- The com- ponent E2 of the magnetic field is s^T:riv.a"Gric about zlie XZ-plane for all values of a; consequently if the search coil actually measures the component Hg of the magnetic field, the voltage readings should be the same for two positions of the search coil in which the values of X and z are the same and in which the values of y have equal magnitudes but opposite signs. If, however, the search coil is not level so that the ccm.ponent Hy also contributes to the induced voltage, the voltage readings ¥ifill not be the same at sym'ietric points since Hy has oionosite signs at sjnrunetric points. When the com- ponent Hy is strong, the error in the voltage reading may be large if the search coil is not level; hence at each setting of a and z the coil v/as leveled by adjusting the leveling screws until the readings were the san.e for a pair of symiietric positions. Because of some uneveness of the table top on which the search coil rested, the readings were not exactly the same for other pairs of syiiir.ietric positions; hence average values were used for the data at other positions. Calibration of the magnetic analogy apparatus . - In order to calculate w' from the voltage readings by use 12 NACA ARR No. LbAO^b of equation (15) determination of the value of the con- stant K was necessary. This value was obtained by calibrating the apparatus : that is, by comparing values of E iieasured at a series of calibration points '.vith values of w' calculated for those points. The values of w' were calculated fron equation (12) by use of an electromagnetic forraula (reference 5)» '^^^^ inethod is similar to the method previously dise..ussed by which the horizontal velocity u' was calculated. The values of w' v/ert calculated for values of a =90'^ and z = at a series of points along the X-axis. A coiiiparison of calculated values oT V;? ' and measured values of E is given in table I to show how the cali- bration constant X was obtained. Accuracy .- In order to estimate the accuracj'' of the experiment, valxies of v; ' were computed at several points for a value of a = 0^ and were compared with the corresponding experimental values. The experimiental values were found to be lovir, som.G by as iriuch as 8 percent. This inaccuracy in the data can be attributed to errors in the measurement of distances and angles and to the fact that the search coil used was of finite length. The error due to the finite length of the search coll could be calculated by means of the assumption that at great distances from the field coil the magnetic field could be approximated by the field of a magnetic dipole. This error was found to amount to less than 5 percent at great distances from the field coil, where the magnetic field falls off slowly with distance. In general, about lialf of the error may be attributed to the finite length of the search coil and the other half, to inaccuracies in measuring distances and angles. The error of the calibration readings (table I) may be seen to be less than the S percent error mentioned previousljr. This greater precision is probably due to the fact that for values of a = 90° and z = G the voltage reading is insensitl ve to s:iall errors in the allnem.ent of the field and search coils because the magnetic field is symmietric about the origin and is a maximum relative to both a and z. NACA ARR ITo . L6A05b 13 RESULTS The hcrisontal velocity field of the actuator disk is presented in figure k as a raap of contours of constant diraensionless horizontal velocity u' in a plane txxrough the axis of syzuhietvj of the disk. The horizontal velocity at any point for any value of a is then the sair.e as the horizontal velocity at the corresponding point that is in this plane of syimnetry and has the sane position in terras of coordinates fixed in the actuator disk. The vertical velocity field of the actuator disk is presented in figures 5 to 9 ^-s a series of naps of contours of constant diraensionless vertical velocity v\f ' for five different valiaes of a at nine vertical sections that are parallel to the free stream and spaced at intervals of l/$ radius froin values of y = d to y = 2-fr. radii. In order to plot each contour map, averages of the voltage readings on the t.vo sides of the XZ-plane v/ere used. On each section the contours of constant velocit;/ are dravm throughout a rectangular area extending horizontally 5 radii upstream froin the actuator disk and vertically 1 radius above and oelovi uhe center of the disk. The nine sections on which contour maps are drawn are labeled a to 1 fron the plane of s^^iynetry outwards, as shovvn in fi:::ure 1. VORTEX IRZkl^iE'ST 0.F ACTUATOR-DISK .PR03LE/I Equ ivalence at low thxr u s t coefficients of vortex and acc eleration- oten;:laT formulations of actuator-disk prooleiti.- It i s shovvn in reference 7 tha elated with an actuator disk at low thrust coefficient a cylinorical sheet of circular vortex rings, which leave the disk an:l travel downstream in the free-streani direction, This vortex pattern is approxinated by a propeller operating at low thi'-ust coefficient and rotational speeds high in comparison vjith the free-stream velocity and having blades along viihlch the bound "/ortex strength is uniform. Because the bound vortex strength is uniform, trailing vortices leave the blades only at the tips and at the center of the propeller. The tip vortices travel downstream in helical paths and the vortices from the propeller center travel dov.'nstreani in a straight line. lir KACA ARR I'Jo. L6A05b At an7 instant the density o"' the "bound vortices and of the trailing vortices from the proraller center is neg- ligible compared with the density ox" the helical vortices, since the density of these tip vortices is proportional to the hJ.gh speed of the blade tips. The velocity field of the propeller is therefore the induced velocity field of the infinite cylindrical vortex sheet shed fron the blade tips. Since the rotational velocity is high, the pitch of a helical vortex is small and the sheet'~can therefore be considered to consist of an infinite con- tinuous row of circular vortex lines. It may also be shown that the induced velocity field of a propeller operating at low rotational speed and lov/ thj-ust coefficient and having an infinite number of blades along vvhxich the bound vortex strength is uniform is the same outside the slipstreaiii as the velocity field of an actuator disk. The vortex pattern of such' a propeller may be considered to consist of a system of vortex rings to which must be added another vortex system composed of the radial bound vortices together with straight trailing vortices from the propeller center and from the blade tips. The induced velocity field of the system of rings is the sa-ne as that of an actuator disk. The induced velocity field of the remaining vortex system, however, may be shown to be zero outside the slipstrearri. The rotational induced velocity is zero because of rotational syKimetry and the fact that the total circulation around a closed path exterior to the slipstreaia is zero (since the total included vorticity is. zero). The radial and axial components are zero, since or:ly the radial bound vortex elements in the plane of the disk could contribute to such com.ponents and the contribution of these eleraents vanishes because of svmmetrv. The velocity field of an actuator disk may be cal- culated by integrating the effect of the infinite row of circular vortex lines. Sixice the velocity induced by a vortex line is directly analogous to the riagnetic field of a current filament, the induced velocity of the infinite vortex sheet is analogous to the integral of the magnetic field of an infinite row of circular current filaments. Tills integral, in which the point at which the field is evaluated is fixed and the position of the source is variable, evidently has the sarie value as a related integral of the magnetic field at a variable point due to a fixed source. The related integral, hov/ever, as J NAOA iJ^R No. L6A05b 15 has been shown -oreviously, gives the Induced velocity sccordln^ to the acceleration-potential formulation of the problem. Uncer the assumption of small perturbations, the alternative treatmient of the problem of the actuator disk using the trailing vortex sheet, which ma^," be con- sidered to be the treatment in terms of uhe velocity potential, yields the sa:rie results as the treatment in terms of the acceleration potential. llccif ica ti on of ve r tex sheet formulatio n for high thru s u CO ef f 1" c 1 ent s . - ia limitation in the t-heory as pre- sented thus far is^the fact previously noted that the results are valid only at lov.- thrust coefficients. In the acceleration-potential formulation, the source of this limitation occurs in the assumption that the per- turbation velocities are siiiall compared with the free- stream velocity: in the trailing vortex fonr.ulaticn, the limita.tion occurs in the assumptions that the slipstream undergoes no contraction and that it travels downstream in the free-stream dii'ection. The theory of the vortex formulation may be modified to give greater accuracy at high thrust coefficients ''oj assuiaing that the sllpstreajii is deflected cownvvard from the free stream by a constant finite angle e. The dovvnwash angle ir. the ultimate •>vake e^. '"-^y ^^ determined by a simple calculation and may be taken as the value of e. Since, hov/ever, the maximui.i influence is exerted by the trailing vortices just behind the propeller, it might be more accurate to use as the value of e the dov.'ni,vash angle ii-imediately behind the propeller, which is about one-half of co- The integration is then rerformed in the direction of the trailing vortex sheet rather than in the free- stream direction, the direc- tion of the X-axis (fig. 10). If the X-axis and the Z-a>Lis are rotated through an angle € _^about_the Y-axis to form a nev/ set of coordinate axes (X, Y, L), the integration in the direction of the slipstream is integra- tion along the X-axis of the field of a coil making an angle a - 57 -J^ (indeg)with the Z axis. ( See fig. 10.) The components of the perturbation velocity parallel to the ^-axis and the Z-axis can be obtained from the results presented in figures ii to 9 ^■^^' ^^'^-'^ horizontal and vertical perturbation velocities for the angle a - 57 O- • Prom the x- and s-coirponents of- the perturbation velocity, the X- and z- components may then be found by a si:7:ole calculation. l6 NACA ARR No. L6A05b Calculation of sllpstreaTn downv/ash an^l e . - The down- wash angle e^ rra;/ be calculated by equating the coMpo- nents of the propeller thrust parallel ana nornal to the free stream to the corresponding components of the rate of change of ::io;nentuin of the air flow. When the norrial thrust-iTior.ientum equation is set up, care must De exercised to include the moriientura of the flow about the slipstrea:n; that is, the moidentum of its virtual mass (references 8 and 9)» 'j^he normal tlxr-ust-iriomentuju equation therefore is Ta 57.5 v?here M is the n.ass rate of flo'w across the propeller dislc, and U2 is the velocity/ increment in the slipstream in the ultiir.ate wake. The term M(V + uo/^^ of equa- tion (iL.) is the vertical nomentuiii in the slipstream; the term MVe2 is the vertical momentum of the virtual mass of the sliostream. The thi'u^t is ^^;iven by the equation T = IuU2 (15) Substitution of equation (15) in equation ( ll.;- ) gives u,::a 57. 3^2 = — — ;:; (i^> \X2_ + 2.\[ In order to apply equation (I6) the value up = vlWl + — ^ - 1/ (17) \ Wl + — ^ - 1 \V TT / derived from the "lomsntum tlieory of the propeller A'ith no inclination (reference 6) may be used. In t)ractice, the correction angle € is very small, For exainple, if a = 10''-' and Tq = 0.2, equation ( I7 ) becoTnes w, = O.22OV !'ACA ARR No. LoAO^b 1? and, ccnse'-.uently, equation (l6) tecomes '^2 0.228V + 2V v;hicn gives for e a. value of e = ^ ep = radian -0.5° 2 ^ 57-3 EFFECT 3F PUSHER PHOPSLLSR ON LIFT AND PITCHING KOMENT OP WING Th3 incremental horizontal and vertical velocities inciiced "by a iDusher Di-oTDeller rnav be exaected to cause an increase in the lift of the wing (reference 1) and a decrease in the pitching norr.ent , inasmuch as these incre- T.-.ental velocities increase toward the trailing edge. The results of the present paper indicate that the Induced vertical velocities (figs. S to 9) i^ the region directly ahead of the oropeller are s?nall in corriparison with the induced horizontal velocities (fig. L) , The effect of the induced vertical flow on the lift and pitching monent of the wing may consequently be expected to be sinall in co.n"oarison with the effect of "che induced horizontal flow. Calculation of the magnitudes of the increments of lift and pitching T.oment due to the presence of the pusher propeller is however considered impracticable, inasmuch as (1) the available lifting surface theories require a prohibitive ainount of labor, especially for a flow field as nonuniform as that in front of a propeller, and (2) the changes wrought by the pressure field and velocity field of the propeller in the boixndary layer, which cannot be taken into account in the lifting-surface theory, are expected to cause incre;Tients in lift and moment coiaoarable with the total increments due to the presence of the pusher propeller (reference 1). It nay be concluded, then, that further work both to clarify the physical ■phenomena and to imorove the coiu'outaticnal methods v.-ill l8 NACA aRR ¥.0. L6A05b be required before the effect of the propeller on the wing can be accurately predicted. Langley Memorial ii.eroxiautical Laboratory National Advisory CoTiinittee for Aeronautics Lan£ley Field, Va . KACA ARR InTo . LbA05b I9 REFERENCES Sr.elt, R., and 3:^.ith, F.: ITote on Lift Change Due to an Airscrew .'-ounted behind a Wing. Rep. No. B.A. 151-'-i-j British R.A.S., Dec. 195'^^ ^.nd iiddendiani. Rep. No. B.A. l^lLa, April 1959 . ThoTipson, J. S., Smelt, R., Davison, 3., and Smith, ?.; Coiuparison of Pusher and Tractor Airscrews Mounted on a Win^. Rep. 21o . 3. Ac l6lli., British R.A.E., June 19iiO. Fr-andtl, L. : Recent .,ork on Airfoil Theory'-. NaCa TlH No. 962, I9I0. Paji,e, Leigh, and ndams, Korvnan Ilsley, Jr.; Principles of Electricity. C, Van //'ostrand Co.. Inc., l^'-lj -D, 19, 131-155, 253-257. 5. a:iythe, Williair: R.; Static and Dyna^.ic Electricity. McGraw-Hill Book Co., Inc., 1959, P? • 266-271. 6. C-lauert, H.: The Elements of Aerofoil and Airscrevv Theory. Cambridge Univ. Press, 1937 5 P- 20i;.. 7- von Kannan, Th. , and Bursters, J. r! . : General Aerody- narn.ic Theory - Perfect ?luids. Mathematical Foundation of the Theory cf Vvings -.vith Finite Span. Vol. II of Aerodynainic Theory, civ. E, ch. Ill, sec. 5, "• ^' Durand, ed . , Julir.s Springer (3erlin), 1555. ^P. 105-lOi^. 8. Ribner, Plerbert S.: Propellers in Yaw. NACa aR?x. >. Munk, liax " . ; Funcar.entals of Fluid D;fnamics for Aircraft Designers. The Ronald Press Co., I929 . pp. 15-25. 20 NaCA ARR 7o, L6riC5b TABLE I COMPARISON OF COMPUTED VALUES 0? W AND MEASURED VOLTAOI [a =: 90-3; y = 0; 2 = ] 1 X (radii) i 1 Vv ' E K = w'/2 2.0 1 o.oUmi C.0?225 1.367 2.5 i .02503 . 01891^ 1.565 5.0 1 .01690 .01230 1.571 1 .0118I4. .00855 1.584 I;.o i .00G72 1 .006l;.0 1.563 4.5 1 .0066^ .00li90 1.351^- 1 Average va lue of X - 1.366 3'AT 1 NAL AD V I SOR Y COMMITTEE FOR AERONAUTICS NACA ARR No. L6A05b Fig. 1 —I o g& « c 60 c ■rt u 3 •-1 o 3 to Fig. 2 NACA ARR No. L6A05b 0) 0) U as a a a> si o Oi I CVi « 3 bO NACA ARR No. L6A05b Fig. 3 I 1 Electronic voltmeter o r-l 1 1 n p= O uJ 5E o o o I 3 o O O t O » H - I t X O O Til Fig. 4 NACA ARR No. L6A05b 3.2S 3jOO 2.75 5 2.50 :§ 2.25 CO ^ 2.00 - Q) /.75 to .C5 I 150 125 LOO 75 ■50 Z5 ~~~-~~ ~~~~ ii^2 X ' / // ^~~~^ -^.016 — - — —^ ^ ^ ^ ~~~^ ^\ \ ^ // / ^^ \ ll. // ^ "" \ \ \oz ■^ IIL /> ^ \ \ \ \ _,,-- - fc ^ ^ \ X V \ r "^ ^^ "^ \; \ [^ \ \ \ \ \ \ ^ \ \ \\ \ \ \ \ \ \ \03 £oy 50\4C \30 \ \2C '\ 4 \ \08 \ 06 \ Oa ? \ \ \ O .25 .50 .75 WO 1.25 1.50 1.75 2.00 2.25 250 2.75 3.00 3.25 Axfol di stance from center of propef/eTj rad/i NATIONAL ADVISOQY COHMITTCE FOI AERONAUTICS Figure 4.- Contours of hor/'z or) fo/ ve/oc/fy. NACA ARR No. L6AG5b Fig. 5a o o fO in V C 5 / q ^v \ / o (A o la in < = / cvi ig -'so in 5 " tt: tr O O UJ U_ / / o \ / + CL f- }C ^ o \ \ / "" -• _J > o _J in tr -t O ^\ r \ / . ^ o +■ -■ UJ • < H ^ O UJ Q H a"2 S q ^ \\ / / o O 5^ 1— o "- -5 < o ^ O^ \\ // /\ ^ O fQ _ 1- J 8 1' \ \\\ / // / >-• ^ a. ° o ^ to o ;5 cr II 5 \ \ \\ / // k /^ -t- II § ? ^ ? o 6 O UJ q: CD in ll \\^i 11 ' // NO llOBTOd d U3-\- i3doad •> c D u )yci do y3iN: c 3C u c iMoad 3 S ■ 30NVJ LSIQ "IV ■> o ■: C T Dlld3A CVJ Fig. 5b NACA ARR No. L6A05b C :> q\ / ' + / ^ y c>j\ Q \ 1 / / ?^ N \ j / / Q +' Q ^ 1 \ // / q Q \\ il '// / ^ O -.08/ \\i \ 1 // + -'-"■"CO ^~~~"~^ \ \\ \\\ III // / 1 * \ \\ \\ /// 7/ / +' II > \\^ all // /// > PRO PELLER PRO J EC TION o Q K^ in IS OJ u > t- o in OJ ^.; ■4 '^ Z w o- ?T in Z X 8 CVJ n < ^ cc O o - CO o: UJ _i _i UJ Q. m 9 b^ o 3 a: n O UJ N CM o a: UJ o o o z < Q in N. _l < 1- o Q O M UJ in or =) o Z X h- (M o o 1 in o UJ q: O in O M //^ y ■>r fVl ^^""-^^^^^ >,^ \ \ \ \ \ 1 / / CM 1" \ \^ . \ \ w /// // / y^ +' II ^\\ \\\\\ ///// / / II 5 \ \\\\\\ ///// / '^ $ PRO PELLER PROJEC TION o o r<1 in CM O in Cvi in _ CM — Q < in CM o o in IS o in in CM m CM I* o in in r in OCM q I I i2 o -,g CM UJ llJ Q. O ct: Q. Li. o O Q: in UJ — • I- o UJ o c/2 l.-> < lij a: o _ II o in o M 5 Q UJ 3 o Z X in CM o o in UJ q: o iiQva 'aaniBdoad dO a3iN30 ;/\iOd-i ^oM^ism nvoiia3A NACA ARR No. L6A05b in\ An o \ /o Q \ / o 1 \ / + o \ — I' — \- ^ \ q / ^ c\j\ ^ \- / ^ /CJ -.0 / \ \ / o 1 V \ \ \ \ 1 / / / + ro\ "^ «v \ / ' y^ o \. r \ \ w / +■ ^^^ \ \ / ^^^ o \ . \ \ / y o 1 ■ ^\ \ \ /^ + \ \ ^ 1 / / irT"---^ N. \ ^ / / J ^■"""^ O ^ r ^ ^ / o + u3^^^^ s. \ \ \ / ^--^CO o ^^ r \ \\ V y' ^ o + ^~^^^ ^v \ \ w / ^--^^'f^- o ^ ^\ \ \ \\ / y ^ o 1 X^ \ \ \\\ /// + n 5 ^ \ \ \\\ /// / ? PROP ELLER PROJECT ION Fi g. 5e o o ro u in >- K a 3 (M 02 13 -.8 < ^ Z ui o Pt in M c\i o in — , w o Z^ — CE 0) Li. ""' LiJ O o in o O N ^. a: o X in CJ Q liJ O O I iiava a3ni3doyd jo h3in30 lAioyd 30Nvisia "ivoiia3A Fig. 5f NACA ARR No. L6A05b in IT) O o O o q/ 1 +/ q\ i' \ T 1 ^ \ — / . in\ / /in r \ \ \ / / / o o \ \ \ / 1 / o \ \ 1 1 / / ^ ifK \ \ 1 / / /in CO \ \ I 1 ; / / CVJ ?• \ \ \ / 1 / O ro\ \ \ / / / /to q \ \ \ \ , / / / / +■ A\ A ' 7/ / rO / + • V \ \ \ / / / / '^ \ \ \ \ \ \ / / / /'^ O \ \ \ \ \ \ / / / / o \ \ \ \ / / / / / + u\ w W^ \\ 1 // 7/ 1 i / "d- ^ \ \ \ \ \ \ \ / / 1 1 /■^ O o !• .__ tl PROPELLER PROJECTIO ^ 5 5 o o ro in OS N CJ 22 O in Z z (M ^ in Q •M. < CM cr UJ O _l O _l CVJ UJ a. o in Q. N o cr O in UJ ■z. UJ < o n|io 2 II o tr *♦— u. UJ O o o ^ < (- en Q in N; -J z Q o in o M (T O X Z in CM o o 1 , O Ul q: 3 o o in o in in in CVJ r o tn r in r in O I lIQVa 'd31"13dOHd JO a3iN30 INOdd 30NVlSia "lV0lia3A NACA ARR No. L6A05b Fig. 5g Q LiJ O o I in UJ QC O lAiodd 30Nvisia nvoiia3A Fig. 5h NACA ARR No. L6A05b o Q \ \ o / ro >. A 1 / lo it Z ut o 1 \ A / +■ / o o +■ \ \ \ I / / q \ \ 1 / / Ad ' o ID OJ _ < \ ' \ / / +■ \ \ / / o ~ Q CC OJ UJ _J _l UJ Q. o \ 1 ' \ \ \ / / / o / o \ \ \ / / a\ \ \ \ 1 ' / / U. o o in LJ ~ 1- D z < ■" o '^ CM ^ -. -in o i. O IJ^ ^ g -c "* UJ o z < t- if) en >"• 5 o \ k\ \ ^ A 1 1 / /'J 0J\ Q \ Y \\ \\ / f ' 1 / o / + 'A O \ \\ \\ \ 1 / o \ \ \ \ \ \ \ / I 1 o\ y \ \ \ 1 '// / / /— < o t- in z ■ O N O Q III 1 ' \ \\ 1 // / -h" / / 1- 2 q\ ^ \\i // / 1 /I 5 O o 1, in UJ NOIi03rOHd y3i-i3doad in z> Li_ in IS o in m (NJ in o in in Q ■ navy 'd3i~i3dO!dd 30 a3iN33 lAioaj 30NViSia nvoiiy3A i NACA ARR No. L6A05b Fig. 5i o ^ .\ CJ / / \ o \ o / / ^ o\ o / '^ , / o\ 7- \ + / •-* / o \ -W r ^ \ 1 . / o \ i / / o \ v^ V- -j -4 ■/ 0D\ V -^ \- 4 -L CD > / o\ \ \ \ / 1 o / o \ \ \ / ^/ 1 \ \ \ ' -f / f- q\ y \1 n o / o/ / \ \ V 1 / \ \ \ t -r II CVJ / \ \ 1 1 / \ \ 1 / "O / 1 / + / cvi 1 / II 1 1 1 1 1 $ II PROPELLER PROJECTION $ 1 1 o Q in o in in in CM O in i' in O O to" CVJ g| --§ o s: in o ji CM 5S in CO ^ CVJ < cr. O O CVJ (r UJ _j _i UJ in O r^ tr a. o o m en _: LU n 1- < 2 q: UJ <■) M(K> in CVJ CVJ 5 — o >« or . — , u. 3 o q UJ o < Q in N; _J < 1- z O o ill O M n in tr Z) in CVJ -o in O CVJ O i" O O I lO UJ (r ID CD iiava 'd3"i~i3doyd do a3iN30 ^oyj 30NVisia nvoiia3A Fig, 6a NACA ARR No. L6A05b o o ro in IS w» CVJ ^ a 3 O V) S > o < < in z 8 • in c^J- ^ 1- II < z X 8 Q in q: CVJ _ o < cc >- o - (- ncr — f\ii^ o _i -J LlI o _J CL LiJ in o > — Q. u. _l o Orr < o ^. 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NACA ARR No. L6A05b in «NI — < cr o on o UJ UJ Q. o in cr o a: — n UJ Q H < Z rr LlI O CVJ in O.I S II () ^-~^ cc Ol u_ III o (■) o < H CO Q in K _l < 1- Z o o M ^. IT O X in (NJ o o u O IQVd 'y3n"13dOyd jo y31N30 IMO^d 30NViSia "IVOIidBA NACA ARR No. L6A05b Fig. 6h \ \ o / / .75 3.C BY UTICS \ A (NJ y /cu /^ cvi 21 o\ o \ / o / o > ' |S o ^ 1" \ o r \ / o / +■ f o +■ / < < '^ Z M O o tt in ij t \ \ / / o 1 I \ / / +■ in — \ \ \ / ^ S \ \ \ / / < \ \ \ / / q: o \ L \ \ / /QO o ^ o y o \ \ \ / / o 1 • \ \ \ 1 / o / + / — \ \ \ \ 1 / / / 1— ?A \ \ \ \ / / / Ad in 5 \ \ \ \ \ 11 / O J \ \ \ \ 1 / / _i < \ \ \ \ III / J\ / t- ci \ \ \ \ 1 I j / z LjJ -^V \ \ \ III / / / o o in M 3 C'\ \ \ \ 1 / / / /co z r ] \ \ \ / / 1 1 1 /o ir M \ / / / / 1 / +' o H $ 1 1 / / / / / / X in 2 O / / / / / j o/ o / / / / / / / c M 1" c u ■) r ■) o - o T CO 1" iiQva y3ii3doad jo a3iN30 Noyj jonvisiq nvoiia3A Fig. 7b NACA ARR No. L6A05b 1 1 / + o \ 1" \ 1 \ \ / /^ o +■ (-1 ^v ' \ \ / r \ \ \ \ \ 1 1 / ^ ^ o +■ o ^ 1' \ yi / ^^''Id o +■ O 1" \ \ \i W / /^ o +■ - — "o" +■ / / JO- 01- ^ ^ \\\ \ / M /^ +■ (NJ ID ~~- r ;^ \^ VA 1 1 1, ' (/ ^ CVJ •♦-" II II 5 \ \^ AW l\l Tf \6 1 1 / -^ -"^ " o o o o to' If) cvi > I- a 3 OS os; o to C\j CM < (Ni Q: or LlI UJ Q. o q: Q. 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