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?JAcR L ARR No. L6A05b NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS WARTIME REPORT ORIGINALLY ISSUED February 1946 as Advance Restricted Report L6AO5b DETERMINATION OF INDUCED VELOCITY IN FRONT OF AN INCLINED PROPELLER BY A MAGNETICANALOGY METHOD By Clifford S. Gardner and James A. LaHatte, Jr. Langley Memorial Aeronautical Laboratory Langley Field, Va. 'N A CA .. ''. ,.  NACA... WASHINGTON NACA WARTIME REPORTS are reprints of papers originally Issued tu 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 154 DOCUMENTS DEPARI M Nf ..' hi... Digitized by the Internet Archive in 2011 with funding from University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundation http://www.archive.org/details/determinationofi001ang ()' NACA ARFR ?'o. L6AO5b ".'jTTIO 7.AL ADVTSOY CO( TI'Tl FO AE, O ',,,T S A:' ACE PESTRaT. .EPOGT E'TI T 0' UT F r1DUCED :)CITY 11T rT AT rCUED PFOPLLER BY A AG7"CTCA. A7 v `1 D By Clifford S. Gardner and James A. LaHatte, Jr. Th: horizontal and vertical cor onents of the 'iniced velocity in front of an inclined propeller in a ori zontal stream were obtained by a magnnticanaloy method. The problem was formulated in terrs of tie linear theory of the acceleration potential of an incrc ressible ronvisc.as flui d. The propeller wa3 assu'od to be an actuator dis h. The horizoi.tal cororent of the induced velocity was found by a numerical calculation. unerical calculation of the vertical comoonent, however, was not praticable; th:ere fore the vertical c rroonent was ootained from electrical measurements by' use of the analogy between the accelera tion potential of an incompressible nonviscous fluid and the potential of a magnetic field. An alternative formulation of the protlen in terms of the trailingvortex sheet is sho,?. to be equivalent to the accelerationcotential formiulation if the thrust coef ficient is assumed so small that the sliostrea: is not deflected and undergoes no contrsctior. 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 vwash  I1 .,"'DUCTIOT The recent dI.velc: rent of airplane designs with pusherpropeller installations has occasioned several inquiries regarding the nature of the flow in front of an inclined propeller and the correspor.i., aerody7naic effects on the ''.*!... cause of the difficulty of the calculations, little effort has heretofore been made to 2 :AC: ARR bh. L6A05b compute the flow. Some exper imental work, however, has been cone in connection with the pr olem of the lift increment on the wing (fir exa:rle, references 1 and 2). Further development of the theory is considered desirable to serve as a 'asis for correlation of these and similar da b. The purpose of the present paper is to give detailed theoretical data on the induced velocities in front of an inclined oro eller. Only the components important to the problem have been obtained; namely,the component parallel to the freestream direction and the component nonal to the free stream and in a vertical plane, which will be designated horizontal and vertical components, respec tivelv. T.: determination of these components is based on the linear theory of the acceleration potential of an incomoressbole nonviscous fluid, and the propeller is assumed to be an actuator disk. Because the theory is valid only for small perturbations, the results, which are presented in dimensionless form independent of the thrust coefficient, are valid only if used for propellers operating at low thrust coefficients. The horizontal component of the induced velocity was determined by numerical computation. The computation of this comioonent was practicable because. of certain simpli fications due to symmetry. erical calculation of the vertical com:.ponent, Io :ever, is excessively laborious and tirmeconsru'riina: consequently, the vertical component was obtained from electrical measuree. ents b use of the analo) bet',een the acceleration potential of an incom pressible nonviscous fluid and the potential of a magnetic field. An alternative formulation of the oroolem in terms of the trallingvortex sheet is shown to be equivalent to the accelerationpotential formulation if the thrust coef ficient is assume so small that t e slipstream is not deflected anc undergoes no contraction. er. 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 wash angle of the slipstream. lNACA ARR o. LoAO5b 13 laT TE D17j p 4 U, V, VZ u, v, vi V u vT', w local static pressure static pressure at ov:::strea. face of pr:Opeller disk air density tie ,e rectangular coordinates (fig. 1) freestre.n.T velocity cornMonents of ~ erucurbation velaoity in ,Y, ,:,' ZdirectCins respectively electric current, c;,s electr ... ;nstic units radius of proplcler diameter of proneller thrust of propeller thrust coefficient 2 \ / u cimensionlcss veocities  respectively ) V N Vi ,. y, y z' x, Ey, HZ dimensionless pressure 1  dimensionless coordinates (x/R, y/R, respectively) :nitic scalar potential, cgs elect notic units components ,of n 'neticf).elc strength, electroi:agnetic units :.eri:..entally measured voltage, volts z/r, cgs L 'A ARR io. LA05b ratio of w' to na1o oo3 seasurec. volt. e E a a:ile of inclinat r n )f pr oeller disk to Zaxis, degrees as ;eued constc.nt dov:nwiash an I e of slipstreaT, ra l ans : mrass rate of ('clw ucross propeller disk Subscripts: 2 in ultiLmate. wal:e 1 at downstream face of Droneller dis) T H E,'1 F. I .L Linear t@ ter of the acceleration potent a 1. The E ler ,1. 7. ... . _. le nin visc;us fluid ::a e ,v;riltten in .he following, form: pD(V + u)  DC x Dv o t ~ p These eq.t io:.s are in g. ner.l nonlinear in the velo cities. TiLe equations r. ay be 1do linear if the o: .v.nents of the perturbation veloci by are assu 'o to be s:ail :. Sn. ed. i th bhe frpestreami v(locity. (See reference 3.) If t. S assur'otio)n is valid and if terrs of the second order are neglected, r(V + u) Du .c : v V Pv CV v Dt 7 x NACA ARR No. L6A05b 5 By virtue of equations (2), equations (1) become 6u p x (3a) 6x by pv =  (3o) ox 6z If equations (3) are differentiated successively with respect to x, y, and z and are added, the result 2 66 u IV ow v p = pv + + o= (4) 6x Tx 6y 6z/ M "netlc an 1 vy. Since by equation (4) 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 analo.:, of the present oaoer. For low thrust coefficients the three boundary condi tions for the pressure p in the problem of the actuator disk are: (1) i:.e pressure has some constant value pl uni formly over the upstream face of the disk and a value pl uniformly over the downstream face. (2) ihe pressure has no singularities other than the jump discontinuity at the disk. (5) At great distances from the disk, the pressure is uniform and without 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 6 'NACA ARR :To. LbAC, currentcarryin, loon p actually ensure. the appropriate behavior o' the magneticc field at the disk since the .magnetic .'. entfa2l 1 such a 1)op has the value 2rTI uniforr.ly over olie face of the loc nnc the value 2IT over ,e othr face if cas electron ., tic units are used (referer.ce ). L hbe second condition is satisfied if D obher mina;rtic fielc's and no ma.anetic materials are in the nc.. crhod of che loop. .e third condition is satisfied .c.ot atically, since tle maonetic potential of the loo approaches z ro at great distances from the loop. Basis of keter.nination of horizontal velocity. If equa_ ~. ' '~ "".".. g t :c ' : ,.d t 0 x, The result is pvu = p (5) Tnasmiuc as the iensionles s velocity u' and the dimen sonless pressure o' are defined by VTc ana ) equation (5) beco::es u' = p' (6)  value f the dimr n onless n pressure T' at the :;n st.r e i faoe Gf )he I1 1 L 1 C S  = (7) NAA . 7o. L6A05b 7 Since this boundary value is a universal numerical con stnt, the values of the :. ensionless velocities th.r ht space, which are determined oy the oounfdry values )_ p', are also universal numerical constants. y means of the dimensionless coefficients u', v w', and pr, therefore, the robler:. is stated in a nn dimensional forn: that is indepenient of all relevant variables such as the dis: radius, density, thrust, and freestroami velocity. In acccrd.nce ':.ith tIle 2 ... tic enalogy, the ; .tic potential is directly ancl~gous to the opr sure p' that is, = i (d) where r is the ".. ttic potential measure at a ooi.nt cf ';i ch the dirienslonless coordinates x', y', nd z' are the same as the dimensionless c or. xnates of the o at heree p' is :.easuec., and. _ure w: is "o, con stant of roortonaity hat Ieenis on th!e dinerni oins of the elect: ..etic sste:n.  alue of a a hence, u' :as thus obtained r. c alating u a nd u ltinly the constant k .. ay be determined by ". ....':.g. ".r; the value of p' "at cae & isk, which h b equation (7) is 2/7r, with the value of f at the disk, which is 2TI S. calculation of > :as effected b nu.cerical inteGration of a form' la o v " Slycte (reference 5) for the _.. tic field f a circular loco. potential j and, consequently, the i1orizontal velocity ul at ar. noint cepend on.l en the position of this point relative to the disk; thus, he results f)r u at positions given in terms of coordinates fixed in the disk are the sane for all & : lea of inclination. C cause of rotational sn entry, moreover, the values of u' at corresponding points in any two planes throi.;. the a.:is rf the disk are the sa me. It was thus necessary to calculate u' over only one axial plane. The linear theo of the .'esent oar gives values of u' that are valid for low thrust coefficients; VTc that is, the results for u' are essentially the deriva tives of u/V with respect to Tc at Tc Th momentura theory of the propeller (reference 6) gives the inflow velocity at the disk as ACO ARR No. LCAO5b u  ,1 +  i a  ), a 'r d4 T 2rt / ee ..... e ..... .. u at t / .U \' ' C T=0 c S : l ..... tin'. c.:.,._ ,,_., l .i! ... value iven ty the linp the..r; of the resent anr eu atcl'ns (6) and (7)). tO sis o te r.i i O 0 erticzal velocity. If to x, te result is X piw = I  5 Z If eqs:tin (j) is divided br p' an if dimensionless Sor:; not  are int4o ced, tre r..suit is (10) <..2\ In acc^ri nce :.vith th "'netic ...l", if euation ( ) iS used, eq.rat n (10) ':e cor.c C ; '~' t I ) = k f  d i =!<0z (11) But since / is the n: ten lai o:' a , r..tc field, .:.A ;.. No. LbA05b 9 i 'z 6z where Hz is the zcoimoonent of the .,.:n ticfield 3tre'n h. '. .tion (11) now becomes w, = k Hz dx (12) 1:i order to find the value of w' at a point (xI, T', z') t is therefore sufficient to measure the integral f H. aloi,' a path parallel to the XE.xLs, extending from minus infinity to the point (x, y, z). An alternating 6. netic fiell in air _"t.ces in a coil of wire a volta:e proportional to the total flux linking the coil, whichh in turn is proportional to the inte.al over the face of the coil of the normal cor no nent of the ri.. .ticfield stren th. 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 narrow and, consequently, the field tre:.gth is almost constant across the width of the coil, this surface integral is prooortional to the line integral along the length of the coil. ie 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 Saxis, which extends parallel to the Xaxis from the point (m, y, z) the noint (x, v, z), as shown in figure 2. Since the mr.gnetic field dies out rapidly with distance, a search coil of practical length actually suffices to obtain accurately enough the infinite integral. Thus the follc.ving equation holds: w = KE (15) 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. NACA ARR No. L6AO5b APPARATUS A.T. ':?TTDS Field coil. A field coil of 61p turns of Brown and Sharpe o,. 18 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.&7) inch by 0.575 inch. The coil ;ias supported by pivots about its horizontal diameter in such a way that its angle of inclination to the Zaxis could be varied and the support could be moved up and down. The arrangement is shown in figure 2. Search coil. As previously explained, a long narrow search coil was used to perform the integration of the magneticfield strength indicated in equation (12). The search coil was made up of 110 turns of cr:wn and Sharpe ,. O copper wire wound lengthwise on a glass rod 72 inches by 0.225 inch by 1.2 inches. The coil rested on Lucite supports at the two ends. The supports were scribed with crosshair linc,, to aid in setting the posi tion of the coil and were sunoolied with leveling screws so that the face of the coil coul be turned exactly 900 to the flux being measured. voltage developed in the search coil was fed through a filter eliminating O0cycle pickup to an electronic voltmeter by which the voltage was measured. PoTwer suoply. Current was supplied to the field coil from :a motorgenerator set delivering 5.0 amperes at 590 cycles per second. The WardLeonard speed control system was used ss that the frequeue and output voltage could be adjusted by rheostats. The output volta ~e was continuously adjusted to maintain through the field coil a constant current of 5.0 amoeres, as measured on a standard highfrequency ammeter. The output of the generator was connected in parallel with the input of a cathoderay oscilloscope, and a 60cycle line volt., .was connected across the sweep circuit. The resulting Lissajous pattern was held stationary by continuous adjustment of the frequency control rheostat; thus the frequency of the current was maintained constant at 390 cycles per second. The arruin ' Test procedure. In order to measure the int?;.r l in equation (12) at the point (x, r, a) when the disk NACA ARR "!. L6A05b was inclined to the Zaxis by an angle a, the field coil was set so that its center was a distance z below a horizontal tale; then it was set at the angle a Cit) a protractor and the search coil was placed on the table parallel to the Xaxis with one end at the noint (x, y, z) and the other end away from the field coil. Tne arrange ment is shown in figure 2. For each setting of a and z, readings of the voltage were made at the 170 vertices of a rectangular grid 6. inches by )0 inches that was made up .f lines parallel to the Xaxis and the Yaxis soaced at intervals of )L inches. The arrangement is shown in figure 1. Zero hei ht adiustment. In order to locate the height _..... eac oll corresponding to a value of z = 0, a was set at 00 and the height of te field coil was then adjusted for zero voltage in the search coil. The voltage in the search coil is zero when a and z are zero, so that the search coil is on a plane through the axis of tihe field coil, since the component of the magnetic field normal to such a plane is zero. Leveling: adjustment for the search coil. The com nonen ,.et,.: 1 .L  i:. .:::ic about the AZolane for all values of a; consequently if bhe search coil actually measures the component z of the r,:.:.etic 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 ofi y have equal magnitudes but opposite si i'. If, however, the search coil is not level so that the component Hy also contributes to the induced voltage, the voltage readings will not be the same at symmetric points since H has opposite signs at symmetric points. When the com ponent Hy is strong, the error in the voltage reading may be lar if the search coil is not level; hence at each setting of a and z the coil was leveled by adjusting the leveling screws until the readings were the sam:e for a oair 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 .... tric positions; hence average values were used for the data at other positions. (1iibrt1 ' of 'ten n, manti h nl'_Y anartus v In ,order "ci.1 re ,' i L, t,. nit 1 Ter by use NACA ARR No. L6AO5b of equation (13) determination of the value of the con stant K was necessary. This value was obtained by calibrating the aooaratus; that is, by comparing values of E Ieasured at a series of calibration points with values of w' calculated for those points. The values of w' were calculated from equation (12) by use of an electromagnetic formula (reference 5). The method is similar to the method previously discussed by which the horizontal velocity u' was calculated. The values of w' wert calculated for values of a =90 and z = 0 at a series of points along the Xaxis. A comparison of calculated values of w' and measured values of E is given in table I to show how the cali bration constant K was obtained. Accuracy. In order to estimate the accuracy of the experiment, values of w' were computed at several points for a value of a = 00 and were compared with the corresponding experimental values. The experimental values were found to be low, some by as much 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 vas of finite length. The error due to the finite length of the search coil could be calculated by means of the ass'i .ition 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, aoout half of the error may be attributed to the finite length of the search coil and the other half, to inaccuracies in measuring distances and alles. Te error of the calibration reading ,s (table I) may be seen to be less than the 8 percent error mentioned previously. This greater precision is probably due to the fact that for values of a = 900 and z = 0 the voltage reading is insensitive to s:All errors in the alinement of the field and search coils because the m u.!tic field is symmetric about the origin and is a maximum relative to both a and z. NACA ARR c. L6A05b ' mLTS The horizontal veloc ty field of the actuator disk is presented in figure L as a ap of contours of constant di:ensionless horizontal velocit. u, in a lane ;thrIo. the axis of ... etry of the disk. The horizontal velocity at any point for any value of a is then the sar.e as the horizontal velocity at the corresponding point that is in this plane of ..t ,try and has the same position in te:'.3 of coordinates fixed in the actuator disk. The vertical velocity field of the actuator isk is presented in figures 5 to 9 as a series of r:.sa of contours of constant dirmensionless vertical velocity w' for five different values of a at nine verti cal sections that are parallel to the free stream and spaced ot intervals of 1/3 radius from values of = 0 to y = 2.. radii. In order to plot each contour nap, aer...e. of the volteD readi. s on the to sides of he. ....olane were used. )n each section the contours of constant velocity are drawn throui. ut a rectan ular area extening horizontally 5 radii unstream from the actuator disk and vertically 1 radius above and below the center of she disk. The nine sections on which contr)r raors are drawn are labeled a to i fr.m the *1.. of syr3met out:ards, as sho' in figure 1. VORTEX TREk .i"I OF "T ,7CTA D1T P70. L Equivalence at low thrust coefficients of vorte: anI a cce l( 7' :n ' : .t .:n ,. ': "~l" i  ..:. . ": with an actuator disk at low. thrust :)fficieiLt a cyli n r c a 1 sheet of circular vortex ri ns hi h 1 ave the disk an travel c ownstream in the reestr3* direction. T.:s vortex pattern is approximated ';a propeller operating at low thrust coefficient and rotational speeds high in cornarison with the freestrean velocit and ha .:.z blades alon: which the ":;und vortex strength is uniform. Because the bound vortex strength is uniform, tra. .n vortices leave the blades only at the tip and at the center of the propeller. The tip vortices travel downstream in helical paths and the vortices from the pr'.ller center travel downstream in a straight line. PACA ARR 'o. L6AO5b At any instant the density o the bound vortices and of the trailinC vortices fr.. the pro Iller center is necr ligible compared with the density of the helical vortices, since the density of these tip vortices is pro .rtional to the high speed of the blade tios. The velocity field of the propeller is therefore the induced velocity field of the infinite cylindrical vortex sheet shed from 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 inuus r.: 'f circular vortex lines. It may also be shown that the induced velocity field of a propeller operating at low rotational speed and low thrust coefficient and having an infinite number of blades along which the bounv: vortex strength is uniform is the same outside the slipstream 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 rl. s to vhich must be added another vortex system composed of the radial bound vortices together with straight trailing vortices from the nrooeller center and from the blade tips. The Induced velocity field of the system of rings is the sane as that of an actuator disk. . induced velocity field of the remaining vortex system.;, however, may be sholn to be zero outside the slipstream. rotational induced velocity is zero because of rotational *..etr and the fact that the total circulation around a closed path exterior to the slipstream is zero (since the total included vorticity is zer ). The radial and axial components are zero, since only the radial bound vortex elements in the plane of the disk could contribute to such components and the contribution of these elements vanishes because of synmetry. The velocity field of an actuator disk may be cal culated by interatin the effect of the infinite row of circular vortex lines. Since the velocity induced by a vortex line is directly analo_as to the ri n'tic field of a current filament, the induced velocity of the infinite vortex sheet is analo.us to the integral of the n1tic field of an infinite row of circular current filam::ents. This inte :r_, in :Iwhich the point at which the field is evaluated is fixed and the position of the source is variable, evidently has the sa e value as a related integral of the ma ..tic field at a variable roint due to a fixed source. T related inte r!, however, as .::.AA nRR 10o. LbA05b 15 has been shown oreviov.sly, gives the induced velocity according to the accelerationootential formulation of the prUolem. Tncer the assumption of 1_ perturbations, the altenative treatment of the orobem of the act ator disk us" thIe trai ling vortex sheet, Whic hay obe con sidered to be th treatment in terms of vhe v' locit potential, 'elds the sa_e results as the treat&.ent in terms of t!e acc aeration ot znia! 4 >i : t rf< t f 1 t t_ i "" 1i . i JL J e 1 )s^ Us'tut o results are valid cilq at 1,;: tl.2ust coefficienc:. In the cGcelerati n o:e liol a... I n s urc of tii s liO it atl o ccc b_1 0.in c in t` J L; 1 "/  turbation velocities are s"ull c.r ... .ith free streaD.x veloc ty; in the tr lin vorte:. for.ulatin, Le limitation occurs in the assur1'or..ons thoat ;he slipsttieam under es no contraction and that it travels dI:nstrea in the freestrean direction. Ile t ery .' the vertex .frmulation may ee T:o3 ified to _ive greater acc racy at .'... ^ cU1ust coefficienos :' as: : tLt ce sli str is oelc.ted o.n;.;ard J o. u free scorer. : a constant finite angle E. T1e Cn znwash an!e i. te l .lti:tLte ...... c si pl4 c vcu aati cn 'd ay taken as the valuevilo of n. ince, however, ohe 'axi . influence is exerted b the traili". vertices just behind the propeller, it bi.i.t be more acerate ) ise as tie value of F the dKc:n:/ash an le L ...i. tely .ehind th .Doeller, which is aoout onehalf of e I. ".e integration S .1en :erformed in the section n of the trai v! vorte t rather than in tn. free stro d..irecton, e orc S the ans ('7. 1). _, the 'i xis and the ..s are rotated through an ar!le about th axi S a new' set of c rcinte &>s ,), Y, a), Tn i .:ooration in the "Irection of t o slipstreamr. is i.te ra tl alonen te axis of the field of a coil .in a .: Ie a 57.xE in I" )v:ith the a:i s. (?ee fi,. .,10 ... orionents of the o~,rturbatiorn velocity orall1e to t.: .a:is and the :axis cuan be obtained fito. the results .r :,nteda in _.,'es ; to 9 .r t eI hori ontal and vrt: cal perturbation velocities or th an le a 7 .3s F'r the x and 5corponcnts of t e perturbation velocity, . x and zcoroonents :lay then be Iound by a si ole 2 !hulation. 16 NACA ARR :10. L6AO5b Calculation of slipstream. down',aash an le. The do;:n wash ... 1. ,i . ;, "..: l, r :.. ,. the co".,po nents of the porpeller thrust parallel and normal to the free stream to the corresponding components of the rate of change of ::o;:entur of the air flow. When the normal thrustm:oentum equation is set up, care must oe exercised to inclun0 the iomentum of the flow about the slipstrean; Lhat is, the morentuu of its virtual mass (references 8 and 9). :I normal thrustsmomentum equation therefore is T ;(V + u2) + V 2 ( .) 57.3 2 here is the mass rate of flow across the ropDeller disk, and u2 is the velocity increment in the slipstream in the ultimats ;:ake. ; term X(V + u2)E of equa tion (1) is the vertical r;rentuem in the slipstreanm; the term I'VE D is the vertical r:o:ientui of the virtual mass of the slimstream. :..e thr'i.' is .iven by the equation S = .7i (15) ',ubstit ut n of equation (15) in equation (1L.) gives u2a1 57.3" 2 = (10) u2 + 2V In order to aply equation (16) the value U2 = V + 1 1 (17) derived from the :omentum theory of the pr'p'eller with no inclination (reference 6) ';ay be used. In practice, the correction angle e is very small. For exar9ple, if a = 10' and T. = 0.2, equation (17) becomes r x 0.2 u2 V 1 + T 1! iT/ u2 = 0.22,V yACA ARR "3. LIA05b and, consequently, equation (16) becomes 0.228V x 10 57.3C2 0.228V + 2V ich ives for e a value of 1 o0.5 e = 5 E = 57 radian 0.50 2 57.5 Iv ODF PUSHER PROPLL.J, 0 LIFT AYD PITC .': hOV"::_ OF :'. Tha incremental horizontal and vertical velocities induced b a usher Droeller ayv be expected to cause an increase in the lift of the wr. (reference 1) and a decrease in the pitching mor.ent, inasmuch as tnese incre mental velocities increase toward the trailing e(f c The results of the present n:xuer indicate that the induced vertical velocities (figs. 5 to 9) in the region directly ahea: of the oropeller are siall in comparison with the induced horizontal velocities (fig. '). The effect of the induced vertical flow on the lift and pitching moment of the wing may consequently be expected to be small in co.narison with the effect of the induced horizontal flow. .lculation of the magnitudes of the increments of lift .0i pitching moment due to the presence of the pusher rpeller is however considered impracticable, inasmuch S(1) the available liftiy surface theories require a .hibitive amount of labor, especially for a flow field 7 nonuniform as that in front of a propeller, and i' the c .. .s wrought by the pressure field and velocity "iZld of the propeller in the boundary layer, vmhich cannot : taken into account in the liftingsurface theory, are '..'ected to cause increments in lift and moment comparable ih the total increments due to the presence of the r...:her propeller (reference 1). It nay be concluded, n, that further work both to clarify the physical ..nor'.ena and to improve the computational methods will If8 ,'..CA ARR No. L6A05b be required before the effect of the propeller on the winrg can be accurately predicted. Lang . Memorial Aeronautical Laboratory N.ltional Aivisory Com2ittee for Aeronautics Langley Field, Va. .. AR . LcAC`b 19 1. ~.~lt, R., and 3mith, P.: "ote on Lift Cl 2nge Due to an Airscrew mountedd behind a '.:... Rep. .:'. B. l1 .ritish R.A.E., Dec. 1953, and _cdencan, hep. Do. 3.A. 151a, April 1939. 2. .. n, J. S., Selt, R., ;vison, B., and Smit 7. Co,'.arison of '..sher a:. Tractor Airscrevws oun te on a ;>l 3. 1R V ",. B. o. 1610 ., 'Jitish R.A.E., JTne 1949. . rndtl, L.: ccnt .,ork on Airfoil . ... No. 962, ,Lo. .F.e, LeT ., and odam, !oitan lisley, Jr. Principles of Ilectrieity. i an ostr: .. Inc., 1 $ 1, *', 7, 21 5 27 257 5. 3 'he, 1illioa R.: Static and Dnaic Electricity. I;cr..:ill ok Co., Inc., pp. 26271. 6. Gl uert, H.: The Ele sents of Ae1ofoil and airscrew Thor". Cajbriage Univ. Press, 19)7, P. . 7. von an, .., anrd aE ".ers, J. I.: neral Aerac,' nar.ic ..ry Perfect ?luids. T.thematical Far. .:.tin o2f the T , ry of .. .with Finite : ,.. Vol. i! of Aerod'n. c Theory, riv. E, ch. ITI, sec. 3, F.. F. Durand, ed., Julis . ':. er (.rlin), 1355, 1 1051' brer, Herbert S.: Pro ellers in Yaw. NACA .RR ..e., ': ". .:ntals of Fluid Dynamaics for Aircraft Designers. Te Ronald Press Ca., 1929. p.. 15. NACA ARiR Yo. L6AO5b TABLE I CO PRISON OF COPUTED VALUES OF w ~ N ME.:"D VOLTAGE E a = 90; y = 0; z = 0 x wi v E = w'/E (radii) 2.0 0.0: ll 0.04225 1.367 2.5 .55 .01891, .65 5.0 .01690 .01230 1.571 5.5 .0 q .00855 1.58  .0 .00872 .o06l6o 1.565 4.5 .oo66 .oo90 1.5 35L Average value f I = 1.366 :'.ATIO 0:AL ADVISORY COD 'TEE FOR AERONAUTICS NACA ARR No. L6AO5b c^H C) s I l0 C.)" Fig. 1 0 r. 0 .4 * 0 D0 ,4 '9 0 Fig. 2 I '~1 4', NACA ARR No. L6A05b S 4 0, >m 4z 0 *3 u' 0 1 to I A: NACA ARR No. L6A05b Fig. 3 S 4a 4 0 oo 0 40 A 0 q4 0* 00 rd 0 21o I .5.0 I NACA ARR No. L6A05b 3.25  3.00 ,__ __ 2.75 , 2.50 O / ________6 2.25 /_ _ S/75 //1 2 0 450 __ 125 _' _1_ .50 S60 .. 06 \0\ 0 2 .5 70 .75 /00 125 /50 175 2.00 2.25 2,0 2.75 300 3.2S Ax1 c/ /is n7ce from center of prolpe//er, rad/l NATIONAL ADVISORY CONNITTEE FOI AEDMNAUTICS Figure 4.~ Conifours ,' ?orzorfo/ ve/ofIy. Fig. 4 NACA ARR No. L6AO5b Fig. 5a 0 0 N1_______ ______. NI ~ + 4 ~ 4 4 4 4 + ~ S I I /1 I/ + 0, T1 4 Z  I. Z 8 cJ C 0. I CJ w a 0 r 0 LL. 100 n  w 0 zi + r)qr C+ C CDYto ^sx~zz0 I 0 0 0: z I w U, 0 2 F u1 a 1 0 _ SOZ 0 N 0 to 1 N01103 O d 31713dOSd ____________ ___________ ___in 0 0O 0 U) 0(' C\ I 0 IIGV 'a3713dOEd 0O 831N30 W08I 30ONV1SIO 1VOIIa3A *1 \I 0\ II t 0 t f/ l: z/ j NACA ARR No. L6A05b q a i/// PROPELLER PROJECT ION 0n o i 0\O o0 0 Ln 0 1 INL4 aIIOV '377113dO8d dO 831N30 i8OJd 30ONVISIG IVOiId3A Fig. 5b In 0 r'. cJ cin oJ C'4 5 0c 0 (m cr w J J CL 0. CL 100 LL 0 wO LC)I w Jz I.Lj LJ 0 LL Z z _J 0i o Ir 00 0 (T 0 in OJ 0 in NACA ARR No. L6AO5b ro1 + iO 0 PROPELLER PROJ SECTION o\ \ / X r ">. \ ()DOD i  ^ ^   ^   _ ______________ \ ______ \ ______o 4___ __ ________ ^"SlZ^ T^ \ / / ^Sto q s.  v t   f r  ^ o CM O O N30 3 0SI OI3A IIGlV 8I'83773dOE~d JO 831N30 V108 4 3ONViSI(] 7VOIIi?3! Fig. 5c V) 4 4g. 2  8 < Z Q I' or NACA ARR No. L6AO5b 0 0 O ci +0 P 0 PROPELLER PROJEClION O 'I 0 wt WM 0,' IIOVy '83113dO8d o0 I31N30 INO84 "4'WISIO iVOIII3A Fig. 5d 0 in LP 04 NACA ARR No. L6A05b r 0 / 0 0 ?. 9 ____ _______ / 0 N. 0 Lc) 0 C4 0 _________ If)/ __ __ +, Ln 0)C 0 0 rv__\___ \__ /_ 0o P0 + 0 r 00 PROPELLER PROJECTION _. ___ /)~ 0 LOn 0 0 0 p: L 0N 0 .0 I" IIGVa '837713dOad jO I31N30 IAOId BO3NVISIG 1VOIW3A Fig. 5e 0= 0 8 a .J Li 8 o Qr LL (oK 0 01i NACA ARR No. L6AO5b \ /+ 0i 0 c\J 0 U)0 ui 0 0+ ) to \ L+'0 q 0 Lo \   /7 0* 0 _\__ _z __U o o. r, \ \I I PROPE 0 LLER PROJECTION 0 U) '1* 0 .4. II 0 z 81 zI li a aJ UJ Ll Q. 0 L 0 U oU) 0I T IIOV'd3113dO8d io0 3iN30 A0O8A 3ONVIJSI1 71VoI3A U) 0. I. Fig. 5f NACA ARR No. L6AO5b o OO 0 \ J_ 0 LO N +0 o o q' __ s^ \ ___ / / ^o ^^ ^^^72     1NL ^ NO19POI \ l 1 / /dO /d 0 \ \0 \ 'C 0 10 0 \ \o \ N / / >1 2 tI S 0 of o4 0: w Ld. 0J J 0. w 0~ 0 0a i U 0 a: w 0cr LiL 0 z 4 I (I j 4 z 0 0 m Fig. 5g NACA ARR No. L6AO5b o o Um o to o0 Q i.. 0 cq ,n rI. IIGV8 '3113dO8d dO 831N3O INOd8 3ONVISIOG VOiOU3A Fig. 5h NACA ARR No. L6AO5b 0 0 I_ \ 0 Si 0 (D ' 0 0 I00 00 __ PROPELLER PROJECTION 0 I _ I _ U) 1,) 0* 1c~ (M 03n 9' O I" _.. IIOVi'83113dO8d O 831N30 VY08d 30NVISIO 1VOI1I3A a L. 0 S ZR a: J LJ 0 a w " Z c. cUJ o z I 0 Fn z 0 N x: 0 T Fig. 5i Fig. 6a NACA ARR No. L6A05b 0 +/ .  EPRoPELLLER PR 0 10. SLO N0 N t)I> IIaVy '31 13dO8d dO 831N30 IYOj 30NVISIO IVOIUI3A LO 1N. u U) 0  CM 0 0~ ocr o4. i  CLM a O ai 8cr" J ULJ 00 hin u 0a:/ Z a w 0ui 00 0O.0 z (I) 0 (r o O I z Lfl _N Z zO in N ir OJ w CD U NACA ARR No. L6AO5b I.. t I I I 0 c'J 0 I. /J 0 + ^SIZIIZI j\Y \NWI pRPEOLERCPI6 O1 h )3 C'J oj :s  u = 0 __ Sj', '. . cq C < w a 0 0 0 0 1 c*'J I" 0 I" "oo I" r" 0 lq\ I _ +  0 O__,' OD ' 4: f/ / _J I z 0 4: a ('  C \\\\\ UI1/, LO N 0 I C) U) 0 N) 0 .1 0 L0 t 7j M N II AIIVU '83y1d08d 4o 83iNJ3 iV zlQid 3ONVISIO IVI1I83A '/Z Fig. 6b y / Fig. 6c NACA ARR No. L6AO5b PROPELLER PRUoJ 0 0) 0 0 0) O o ) o IICV9 '83713d08d JO 831N30 INIO 30NV1SIG 7VOIUI3A o 0i OL Q. 0 cr, O0 QUl U 0  I 0 I 0 u0 o 0 0 0 0 004 O1 Q '~ NACA ARR No. L6AO5b Fig. 6d 0 0 O ri U) c,j IN t Lo\ 2.: 0 w  O Q 4: LL ww  E RJE  TN 0 \ \ / ^ ~ I rC ^ o 0 ^\~~ L\ w ^' ^^"V~rTcc D co^~ ~. ^ S \ \ \ \ f .'r+ r> N 0 _______________________________ 4 0^ IIOV8 '83173dOad 10O 831N30 OHj1 30ONVISIO 7VOI1a3A Fig. 6e 0 0 C NACA ARR No. L6A05b 0 \ o0 \ / / 5N 4+:U . \ / / C\j 0 \0 Cc~j _______^____ U)/_____> 0 0 ___ __\__ _[ / __X__ ) (D0 0: _____ 0 __  0 L 7 + 1 p lI ,F ' oOJEcT N Nc a s .i. SSS o_ I 4z LO 0 i) 0 O O 0 IIUV8'8d3113dOd 40 831N30 IVOd 30NVISIG 1VOIU13A NACA ARR No. L6AO5b Fig. 6f 0 o 0  _ 0     O _"L C toJ 'C) \\ / + I ) ) 0 0 \ 0 i 0 /) o II ^177 dO\.  0 f jt^ NQ7 r'r) cI 0 Si I w J J w 0 0 CL a L. 0 0 l  OTc 0"L" w UJ 0 z LU w F 4 I z Z 0 N 0 " NACA ARR No. L6AO5b to a 0 0* 0 0 o zO oz N 0 LiJ J J w Z w 0 a:: a X I.L O 0 = S< ZQ 0 t! c'J LL . 0 i 0 z O 1 0 ___________ ___________ ___________ ___________ __________ I 4 \ o +  \o '"X~~t to /*' q Y  f  f  /0N \ \ I I / 7+ ______ 0^'^ VY~nLI/./f) O 1' 0Oz II PROELLER PROJECTIONS to 0 to N C un oin r^ Q (71 I/I / /_ r / Q/ 4: k IJo 0 " 11IGV 'dl1l3doyId dO I31N30 V08I 3ONVISIG 1VOIII3A 0 m u IT) Fig. 6g NACA ARR No. L6AO5b 0 0 Ko o 0 0 0 I I 4: + / LO C 00 0 o 0N0  pROPEER   U) 0\\_ + 5U*) S .0I 1 1 Ito 0 o o 0 O) 0 ) O I CQ M UO N IIaGV 'a37173dO8d O I31N30 WYO8d 30NVISIO 1VOIIa3A 0I 01" 0 wx v> z ig 2 CLL _jS LL0 m r x t  U 5 J 0 0 0 Wcr or U_ 0 z (n (E 0 I _j I z 0 0 xr Fig. 6h NACA ARR No. L6AO5b r0 0 I \ N 0 0 I a 0 I . I'I / ro / 0 _o0 + 0 0 \\ \ \ \ / / / ODQ 0, 0 + 0 /0 40 0 0 __ L _ _ ___4 4 j / 0yto. 0 n q  IlI 'III I//i 'I E ERPR 3ECTiO Si) 0 t 0 It 5' c0 LC  IIv I '3913dOad do0 31N30 IAIOMJ 30NViSIO 9VOlJI3A 0o 0 ? w 0 0 + / ut CM 0 0 LO 04 I ~ II 0c'J 0OJ 0 I" 0 0. 0 LO z 0. O F 0_ Q U 0 IF 0 b 0 [ I. [ k I  I I tl ] : I I I i p I / I A I I I f I I I F F F T J ! ' " I 1 I I i II I I I II I l Fig. 6i 4Q NACA ARR No. L6AO5b Fig. 7a 0 010 00 l' C.J.. o   pLW / +"Z 0 __ .   __ 0 > __ _i" L" I^^ I _____ I\n L _j "N, \\\\ / ^%^ ? __________J 1> i IIGVtO '*3113dO8d A0 8I31N30 IbNOI 30NVISIG 7IVOI1I3A Fig. 7b IIVa '37l713dOUd JO 831N30 VYO8d 30t.'.lSl7IG 1VDII3A NACA ARR No. L6AO5b 0 0 I 0 r d /  / 0 0 \ o _= 0 0 Lii z 0/ cr D o _L _A ^ _y ^ z O F , < \ 1 / / /+ o \ \ / ^ ^^q \ \ / ^'^wz' \ \ / / ^"^ < NACA ARR No. L6AO5b llV8 '8313dOId 30 i31N30 V408oJ 3oNViSIG 7VOI183A Fig. 7c NACA ARR No. L6AO5b II () o rd S / oC\j + o /o +N + I o t 0.V'13dOd 0 3NO 0 3NVISI VIIA 113 OdA 81 0 Y8 3N S ^ \ \\\^ /// LE" 5 p oacco oN PROPE o ____  ___________________________ T 0C I J J a 0 cr Q_ LI: a 0 1 O UJ o 0 0:: z IL _. w 0 z H  on Q _1 z 0 N o i Fig. 7d NACA ARR No. L6A05b Fig. 7e 0 0 ____ ___ ___ ___ ___ ______ 0_______ f ____S If) rd CM a 0 0 o~il /* S ZO z 0 o } / o / 2 LO J F) \ / 0= LOJ a_ \0 4w a. LiL 10 c'Jr Z 0_ 0 9+ 0 z 0.~~~~L < 'r n u c 0 < C) J I 0 PIROI X; I V4^ \ \ / / / E "'^ \\ \ ^^^ F D "s. \  1 l l   "^  0 IlOVai '37113dO8d o10 31N30 WOl_ 39NViSIO 7VOIn3A NACA ARR No. L6A05b S0O CC 0 0 0 0 II N N ci 0 N + E LOO 9") 0* +  0   N '~~~~~~    /  /  ^ ^^^T/^jp \ \ \\ //// + \ .\ \ 1I \ I i / / ^ U) CM \ 0 0 P \ N I // i/ / '4. 0 4: In $ + K__1___L_ _____I_______ _____ 0 o0 r 0i 0: 4 4 II ZR 0 j j Q r  a 0 I, Li H U, 0 U) J I 00 N CM OCJ QI' T IIGVd '31373dOd JO 831N30 VO8J 30NVISIG 7VOIII3A Fig. 7f Is NACA ARR No. L6AO5b z0 u 2 a J a IJ J J 0 0 . U_ I iavI 8 73dO8d dO 831N30 08o3 30NVISIG iVOI83A Fig. 7g NACA ARR No. L6AO5b 0 rlU CMj CM 0 0 0o 0 0 / o \____I_ ____ 0 C 0 / i I i I / I I / w/ / /  ,,l t I /1/ 1 i T p 1_E I P 0 In In N 0 ti 0 I I /0 0* +' 0 + CM 40 2!! 7  o 7Z0* CM N o ?* _6 0\ 0. 1 IIGV: '83: 13dOad dO 831N30 VYNOd 30NVISIG 1 IVOIiI3A 'A 0: 2.9 Wt o ! CME CMJ < 0 (c _J 0 LO 0 0 U, 0 0 LL 0 w 0 I z 0 0 0 IA o( 0 0 ____~~~ + 4. N 0 0 0  N\ ILLO A A Fig. 7h u 0 1 A J NACA ARR No. L6AO5b o 0 0 + .4 0 + 0+ I. 0 03 6 PeF l OJ f_ __ __ __ _ In 0 l^. In IIOV8 'W3113dOyd J0 831N30 IY081 3ONVISIO 1VOIia3A 0 0 0 oi 2 to C. .J = 0 oe0 0.0 z tu I r. J I 0 00 0 0 cr A.. Fig. 7i c ( NACA ARR No. L6AO5b o Of UJ  o 0 Q Of a Z n z OQ 0 ) IIOV8 'H3~113dO8d AiO 83IN3O IAIOi 3ONViSIO 7ivoul3A Fig. 8a NACA ARR No. L6AO5b Fig. 8b 0 ro ILn  4zr_ c'J 0 a ___ \_ _____________ 00 ___ _____ 0z If) \ \I / + C kI.y_____\ __ \__ I________________ IL o \ \\ / \ ^^ \ / /\4 ^ ^  ss. ^ ^ f  y  ^ ^ 0 1 ^\ \ \ / OD '^~ ^s" \ \ / 7 /oo0 ^\\\ 1//4: ^co===^  S. \ \ \ I/ /  *^^^0 OD ^^ \vn7 00 ^^v ffrLM4  ^si^ 7 ^\ xW^ ^   ^ ^^^ O i^  T ^ __________________ i 0 ') 0 I io ( LO'l lIGVH '837113dOld IO a31N30 VIOUj 30NV1SIG 7VOll:83A to 0 W) frk: in NACA ARR No. L6AO5b 0 c oo 1c + __ I_" 1i ^K\\ 44 ^( \X\ N ANYl N S' 3 0 N30 3NV 11(3V '8311dO~d JO 8313N30 VY083 30NV !~1 / k!'"1 .+" cx) +4 0 4: ('J U) :  II I I 1 VI  U (A Z Z' 0! U.9 IJ5 CL cr UJ 1_1 0 ' c r0 LLI z 0 X 0. Zcn ^ I z 0 O U  Z ^ O _1 < U I. C U U C C 0 01O I" " ISIG 7I3A Fig. 8c 0  co 0 0 U, 1' ' ' I I ' i' F I / [ J I T L L 1 1 1 1 1 1 1 11 n NACA ARR No. L6AO5b Fig. 8d 0 0 00 to U)L __ __ _t _______ ___ (5 1 0 s 09 1cr oo 0 0: LL~ CM +) F C z 0 0 co 0 L 0 __________ ___ _0 01 ^ t  /  C) 0 4: j ^ e t 0~~~~ 0 \ wQ r\t I \\ 4: 0 i _ 0 ^ \ ^ 2hT^ T M 0 3: LO \ z r 0 v \\ \ \ / / ""CD 3 \ 0\ 0 ui"^<  j ^=gn  ii Pgo,^ ^\ ^ .__ to IIGVa '3113dOAd 40 83N3 IOi 33NViSIa iV3118OI3A Fig. 8e NACA ARR No. L6AO5b 0 0 \ 0 LO 0 +y o __ IC)  0^ I \ \ // Z/ 0 / / /, + 0 + U% 4.   Z 2 t 4J 0 w I_ 0 0: ILl _J 0 0 a: a U 0 I z w C" 0 a: U 0 u q.) z F 4 I (I) ' oi I z 0 0 LOl a: 0 I __________ ____ __ ____ f L 0 C 1 0 3 0 go 0' in 0 nO 0 L 0ft cO COO P. II 31N3 A 3 NVSI I A InOVa 'a311~doaid do a3iN3 l1ioad 33Nv.SIO 1LLaA CN n) tO 0 I. I~) 0 NACA ARR No. L6AO5b Fig. 8f ro O 0 / + 0 II i0 _. / . 0 I/,LO 0 "\ I / 'It 0 0 14 V +f   \ I I I I" I" O IIOV8 'a3"13dO:d JAO 831N30 WO0 30NVISIC 71V9113A (0 E tA 2 o_ 8 a Is' 0 o_ x 0 a: a: u_ w j Q w X 0 ,, 0. w z I z 0 a: Fj r 0 m 0 > z H (U) Q t H 0 a: 0 i NACA ARR No. L6AO5b 0 0 /IO 0 0 ' _' 0 0 0 q .\ __ < ^N___ (D0 65 0 4+ 0c0 0( 00 200 ,00 00 \ \ I I /// * l I ) l/ I// I /77 /77 7 *I0 V ___ / 'I o N 0 n N 0~U cl u) ,]o 0 OI 0 ,n O . ". IIOV8 '"a3113dOld JO a131N30 A1Od 30NVISIa 7VII1183A I ) E Fig. 8g tO rl C 0 0 (1^ to1! C" in U) 0 Ld 0 . t 0 a Srr w U 0 2 C wo 8 w z z 0Q I I z 0 0 0 I tn NACA ARR No. L6A05b 0 7 0) 0 00 o 0 04 + 0 00 I. \ / / co /~+J U') ^ 4 \ ^  L ^a 0 0 1. 0/ / / I. \m m^  xflS/z 1/! /7/ / +I I. + o N_ 0 4: M" 0 4: 0$ A. Lp 0~ > 2M 3 o 04 0 ow J w a. 0 to 0 0 "n a: ._.. ow u 0 z 4 N ^go z cc S U) 00 N 0 , i" ______ ._ ,______ to 0 to U 0 to O" IlOV ln CM 0 01 3 NVS 0 13A IIGV8 'a3~l13dOad 40 M31N30 NOHA 30NV1SIO lVOLia3A C c I~ I n ) Fig. 8h 0 Fig. 8i NACA ARR No. L6AO5b 0 0 0 0 0 0 0 0 0 0 z 0 00 00 749 U+ ") aO / 00 \ \ \ S S S / O 0 00/ / o 0 / \ \ \~~~ \\ / Z 0 0 *s.^ ^ /7// //// / ^LO \\\\\\ I // / 0 A \ \\ 111 III I / / / y 0 IIf i/ , /I /~ , 0 0 0n 0 0 I C N ~/X  r  At o, 40 + : __ I___ I^  ULO on iN C'  Q)L i) T iOVa '83913dOad dO 831N30 AO4d BO3NVISIG 1VOiIa3A 0 S a: C J _j _j uJ . 0 0. IL 0 w IL LzJ C, z I ctr 0 UJ w z 0 _j S 0r 0 2I NACA ARR No. L6AO5b Fig. 9a o o I in 0 o J SII \c 8 aI Q 0 9O L  ^  i ^  s t 1 S" > \ O >:  o I ^ zz o OC \ \0 L (\j. 0 C .:0 S \ \ I tj D fO0 m O LO O Nl, I IaVH '13711dOHd JO tl31N30 IYOlA 30NVISIG 7VOII13A NACA ARR No. L6AO5b S+ 0 + > \ 0_ II" 0 o C ). o 0" IIOVI 'a3713dO~d _O I0131N30 INOyI 30NV1SIO 7VOIIa3A Fig. 9b NACA ARR No. L6AO5b Fig. 9c L.) o f 0 U'). aN LO 'o 8 + S) U') 00 ___ ___ \______ _____ 4'___ 0 Xo 9  i _ _1 0 N .. .. .. .. .. . . . \ \ / / ICI 0 I0 0 0 I) iav8 'H37113dOd AO H3iN30 oaJ 3oNViSIG VOB1183A o. ' T 4 n x:: o LLJ J (r J w L 0 o " LL 0 cE ULJ w H z 0 0 m: LL z 4 Cl) 0 t z 0 N 0: I NACA ARR No. L6A05b 00 + t Cj ,oq 0 i 0 0o__ __ 0 "__________ _____ In . \D o  _0 __ __ f /X0 ^^v \ \ // ^^FO JU 0 t o 0 to Il 0 1" U I" i" I" II in C7M C0j I IIGV8 '3113dOyd dO 631N30 IoN d 30NVISIG IVOIiHJ3A 0 I I Fig. 9d NACA ARR No. L6A05b 0 9r(0 __\______+ LA c'i c'J 0 10 0 \_________________ LA IIt 0 r')) 00 00 I 0 ,0, t o ______ \ _____ ___ __ g L ^\u L' LU C.\) ILJ/^~ Fig. 9e Q a or (K _1. J o o w 0 I x a U 0 (T 0 LU UJ 0 2 cr 0U 0 z 0 I 0 m1 S0 I" I IIOV8 'a31173dOld JO 831N30 W08AI 3ONViS[O iVO9IIuA 0 N o f O.I NACA ARR No. L6AO5b 0 \o + 0 + IN ____ _c) / I< 0 0o +0 01 7 \ ___ \ ___/+ ~x N\~IIL /7/ I 7 o. 0 +   0. + C 0 Ul C4 ii 9^ 4   l ll" I f ^  0 m J I. 13. 0 a o za w Q 0. 1 z 0 0 to o t0 0 o 1IOVa 38 13dOadd do0 63iN30 1AO406 3ONVISIG I'IVOJ.83A / t\ 0 c'J 0 3~ 0 o Fig. 9f > A1 NACA ARR No. L6AO5b Fig. 9g 0 o 0 o N >a o / U) Z4 0 *~ . r 0   f1 ^ i a to S 0 o I / LO i S,"______________ i IIaV8 '13dOld .0 831N30 V08lAI 30NV1SIO VOI1i3A / +t 0 0 Fig. 9h O /o 0__ + 0. . (l Z 2 2S ) wn 0 O + 0 \ It \ // /  L    /  L  ^  7 7 7^ OD \ \ / / ' \ \ \W / g tCTUZ/SS^ \ \ f/ / / ^+ ^ \ 1 / I / / / ^ ^V^loll 0 \ \ / / / / LO I ///////'i 0 m Q:: Cr LJ J 0 Q: w Z 0 Ir U 0 z " S0: 0 1 (f) 0 z H (I) J z r 0 N 0f: 0 3z '. 0 C ,. 9 IIOV8 'a3773d0dId O 831N39 IONd 30NViSIO 7VOIJ83A NACA ARR No. L6AO5b 0  cI\ 0\ 0\ rl. LQ NACA ARR No. L6AO5b Fig. 9i OO 0 + 00 0) 0^ Q 0 40 4  C . U _0 Ul 0 0 0 04 Q UQ Cq0 Dz DS '4 a._ r)O.  0 a U_ 0 w W Zr 0 "1" Od 2; .. w 0 a: UJ 0 z Z 0 0 s o ) 0 ' r 0 in 1. IaV8 '837113dOd 10 831N30 IW08I 3ONVISIO 7V0IIy3A Fig. 10 '4, I 'U 1 / ~ N N I 6' IN, NACA ARR No. L6AO5b 0 z z1 0z UNIVERSITY OF FLORIDA 3 1262 08103 282 2 S,,T OF FLORiDA TS DEpAPTt ACT ,LL'. FL 326117011 USA i h1 .= 
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