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ARR No. L5D23 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS WARTIME REPORT ORIGINALLY ISSUED May 1945 as Advance Restricted Report L5D23 AN E1LECTROMAGFTICANALOGY METHOD OF SOLV3IG LIrESISURFACETHEM RY PRCBiE By Robert S. Swanson and Stewart M. Crandall Langley Memorial Aeronautical Laboratory Langley Field, Va. MACA U)l *.A "'.:'.". .'', k .. . ." . ~' '*' ~I ~ ; 3. .*,. WASHINGTON NACA WARTIME REPORTS are reprints of papers originally issued to provide rapid distribution of advance research results to an authorized group requiring them for the war effort. They were pre viously held under a security status but are now unclassified. Some of these reports were not tech nically edited. All have been reproduced without change in order to expedite general distribution. L 120 DOCUMENTS DEPARTMENT 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/electromagnetica001ang 71 L, I 1 315' TTACA ARR No. L5D25 IIATIOllAL AD'JSORY COi','ITTEE FOR AE.RO.AUTICS ADVAIiCE RESTRICT REPORT All ETEC'T'ROM AI' 1 RETICAINALOY '.'{HI.,' OF SOLV\1riG3 LIFTIjiGSUPRFACETHEORY PFCELTEYFS By Robert S. Swanson and Stewart i.I. Crandall SUL!"'iARY A method is suggested for maikine liftin;jsurface calculations by merns of magnetic rieesurements of an electrompgneticanrlogy model. The method is based on the pelrf,'ct ?nalocy t3ec':een the strl"_nt.h of the rmannetic field around a conductor and the strenratl, of the induced velocity field around 9 vortex. L:lectlic conductors are arranged to represent Uche vortex sheet. The magnetic field stren.tth is determined by mieecsiurn, with an tlec tronic voltmeter, the voltage induced in a si:call search coil by the alternrtin. current in the wires reorecenting the vortex sheet. Solutions of nonlinear liftinisurfPce ;problems mayT be obtained by ,lacing the conductors repres ntin; the trailing vortices along the fluid lines (Helmhoitz con dition). A potentialflov. solution for the distortion and rolling up of the trailingvort.e: sheet r,,ay be obtained. By use of the Prandtl,Glsuert rule, the liftingsurfaca theory may be adapted to include firstorder ccmpresisioility effects. A comparison wTs made of the dovnmiesh determined by means of e preliminary electrcrnisneticanalocy model with the dowrnwash obtained by calculation for an .lliptic wing having an aspect rtio of 7. The sccurecy of the magnetic measurements compared satisfactorily with the accuracy of the downwash calculations. IIITRODUCTI'Oil There are many important aerod:neamic problems for which solutions by liftingline theory aire inadequate. These problems can be solved much more satisfactorily by 2 NACA ATR No. L5D23 a liftingsurface theory; that is, a theory in which the lift is assumed to be distributed over a surface instead of along a line. The calculations necessary to determine solutions by liftingsurface theory, however, are rEther laborious even for the simplified case in which the vari ation in incremental pressures with the effective camber or the angle of attack of the surface is linear. A more exact nonlinear solution is very nearly impossible to calculate except for a few special cases. A few of the aerodynamic problems for which solutions by liftingsurface theory are desired are: the planform corrections necessary for the prediction of finitespan hingemoment characteristics from section data; the determination of spanwise and chord wise load distributions of wings with low aspect ratio, wings with sweep, wings in sideslip, wings in roll, and wings in turning flight; more exact solutions for the unsteady lift of finite wings; and an improved theory of the field of flow near propellers. In reference 1 it was shown that the planform correc tions determined from liftingline theory are inadequate for hingemoment predictions. The planform corrections determined by a linear liftingsurface theory (reference 2), however, were shown to be quite satisfactory for the pre diction of hinge moments at small angles of attack. For wings at larger angles of attack, especially for wings with square tips, a nonlinear liftingsurface theory is required. The electromagneticanalogy method was developed in an attempt to make calculations by both linear and non linear liftingsurface theories practical. The time and expense required to build and test an electromagnetic analogy model of a wing and wake were expected to be small compared with the cost of applying other methods available at present, even for the linear case. The electromagnetic analogy method is based on the fact that the magnetic field around a wire carrying electric current is perfectly analogous to the velocity field around a vortex. It has also been shown (reference 5) that the lifting surface and wake may be represented by a vortex sheet and may therefore be replaced by conductors arranged in the configuration of the equivalent vortex sheet. Simple measurements of the magneticfield strength then replace the difficult inducedvelocity calculations. For nonlinear solutions of liftingsurface problems, the trailingvortex sheet represented by the wires is rolled up and distorted instead of lying in a plane as it EACA ARR lio. L5D25 ' is usually assumed to do. In figure 1 is shown a sir lified picture of a rectcni glar '; in of low asp..ect ratio at a l1rge ancle of attack with a rolleduo ?nd distorted trailin&. vortex sieet. Of tie various features of the distorted vcrtex sheet that contribute to the nonlineari ty, the moist inm'ortnrt s the vertical sncifnj of the trailin:. vortices. The increase in vertical spacin; as the anTle of' ttck'.. is increased results in a decrease in uh: vertical component of induced velocity at the surface, especiillj near the wing tips; thu3 the sloce of the lift curve is increased as the an7le of attacil: incre,:.ses (sr rsfernc.c 4) arnd theU slones of hinurnortent curvs 're imoi; necative Treffr enc: 1). The present re:,ort describes thie basic theory of tbe electroma.neticsnaloy method ind the general procedure by which various aerod/ynamic problem;. mna be solved by this Penlogy. A few preliminlsrr results for the linear case are presented for en ellintic rin. hav:in: n as. ct ratio of 3, as well as a coIImpIPiscn of th; results obtained by the present method and the results calculated by the method of refsrence 2. S'I FO L r vortex strenth 'maX< mri;ax;imjm vortex strength \p pressure difference across lifting sirf'c.2e V freestream velocity V. mach number, ratio of freestre.a velocity to sonic velocity p fluid. dens ity x distance along freestream direction from leading ede of winz y spanwise distance z vertical distance above pine of vortex sheet H msagneticfield strength NACA ARR No. L5D23 i current in conductor e induced voltage 1 length of conductor or vortex r distance from element of conductor or vortex to point in question v induced velocity w vertical component of induced velocity (downwash) u horizontal component of induced velocity (freestream direction) K constant t time b span c chord Bar above symbol indicates a vector, as h, T, r, and v. BASIC THEORY Solution of Aerodynamic Problems by Available LiftingSurface Theories The distribution of lift over a lifting surface cannot, in general, be expressed in any simple mathematical form such as can be obtained by liftingline theory. This state ment is especially true for nonlinear liftingsurface problems. Except for a few special Dlan forms (references 5 and 6), the method of determining the induced downwash for a given lift distribution also is too complex for expression in mathematical form. In order to obtain an exact, complete analytical solution, however, such expressions must be known. The determination of the surface to sustain an arbi trary lift distribution may be accomplished by means of the electromagneticanalogy method described herein or, for the linear case, by the semigraphical method of refer ence 2. The inverse problem, determining the lift distri INACA ARR No. LSD25 5 button over an arbitrary surface, msy then be solved by a process of success: ive 3 n rpro:imtti'r.s. A reason eole distri butzon of vortici O t is 0is0newa or' calculated from the simr:le liftin:line and thin:sirfo*il theories, nd the induce. vel.:cities corrscondlin, to that vortex distribution aere sIeccrniin.d by i,,ikin an electLromiagnetican1 lo' y mod..l of the vortex sheet and measurin, the Imrneticfiid strength. If the induced velocities do not satisfy the .cundiry7 cor. ditions that is, the slp=s of the lifting urf,.ce the vortex sheet is sutLbl, elt.i'red lan the process Irepest.Ed until the boundasry conditions are s&tified. For the non linear problem, not only nmu i: tha induce velocities satisfy the boundary conditions of the wink shane but Piso the trailing vortices must satisfy the Helrrdmhiltz condition, namely, that the vertices must trail long Cllid lines, (See reference 7.) In nrfrctice, satisfying these simple conditions ;nqy, require a considerable a ounit of work unless the first sop:ro:imttion is fairly accuraPe. In order to obtain a somewhat Irore general solution, for the lin5?r case qt lesst, the surfaces required to su::,oort s3'veral different li't distributions aiy be determined so tiht the shepe for s'. .Erticualar lift distribution may be estimated by a process of interpolapti n or superposition. There are, however, several nrobleris for which a com plete ootentialflow solution of the inverse problem: is not necessary. For exeimple, in order c. include the i. iin effects of viscosity, the estimation of the hin,erioment parameter for fini esoan wins should be made by applying theoretical aspectratio correction to experimental section hingemoment parameters. For such problems the additional aspectrgtio corrections may be de Lc rmineid simply and accurately from the surface required to support a .iven lift distribution (reference 1) as found by liftingsurface theory. The results of the Qlectror.saneticnallory solution of the liftin.surf ce tlieorv may be corected for first order compressiti litty, effects by a sir:le ap::licstion of the PrandtlGlauert rule (reference 8). Th m.nethod consists of determznin t the incomjnressibleflvo characteristics of an equivalent wins the chords of which are increased by the 1 factor It is therefore necessary only to build '41 ,2 an electromagnzticanalogy model of a Wfinl: of this slightly 1. o t1 \ lower asoect ratio (lower by thi factor ' or to ~\test moel s of sverl i2t test models of several aspect ratios and inter'olato. The NACA ARR No. L5D23 pressures (or vorticity) acting upon this incompressible equivalent of lower aspect ratio, however, must be increased 1 by the factor In order to find the lift, these increased pressures are referred to the original wing and integrated. Vortex Sheet Inasmuch as the equivalence of a lifting wing and wake to a vortex sheet may be considered to be well established (reference 5), only the important characteristics of the equivalent vortex sheet and the relations between the lifting wing and the vortex sheet will be given. The part of the vortex sheet representing the lifting wing consists of a sheet of bound vortices. The strength of the vortices is directly associated with the lift distri bution of the wing. The product of the air density, the freestream velocity, the vortex length perpendicular to the freestream velocity, and the vortex strength of each elementary vortex equals the lift contribution by that elementary vortex (KuttaJoukowski law). If the lift distribution of the wing is known or assumed, therefore, the equivalent vortex distribution may be easily obtained. A continuous lift distribution (as measured by pressure distribution Ap) may be integrated to give a continuous vortex distribution. The integration formula (reference 2) for obtaining the vortex distribution is r = p dx (1) 0J P where pV is the product of the density and the free stream velocity and the integration is made in the free stream direction. Equation (1) sives the chordwise rfunction at each section. The values of F at the trailing edge of the wing at each section also give the spanwise vortex distribution of the wake. The bound vortices may be assumed to lie along a mean surface, half way between the upper and lower surfaces of the wing. The part of the vortex sheet representing the wake consists of the socalled trailing vortices. As the name implies, these vortices originate at the trailing edge of the wiing and merely trail behind the wing. These vortices are free to move and thus lie along the local stream lines, or fluid lines. This simple kinematic condition, the Helmholtz condition (reference 7), determines the configu ration of the trailingvortex sheet. NACA ARR 1To. L5D25 The treiline ve:rtices for lirhtli leaded wings usually lie very near a plane; that is, these vertices ter.vel almost straight back: fror their origin at the trailin. edge of the wvinj. For highly loaded wings, however, the trsilingvortex iheet is known to be considerably distorted, rolled up, and inclined with respect to the freestream direction. (See fig. 1.) The charscteristics of the air flow behind wings are described in mior' detail in refer ences Q and 10. ElIc tromagneti c An lo"y The perfect nnaloy that exists bztw:en the strsinth of the magnetic fieli round conductors and the strenalth of the inducedvelocity field around columnrer (finite diameter) vortices is Lxell ';nowrn. In fact, the rnhenomena of the induced ve locities Laroiund vortiCeS 9re usually explained in aerodynamiic textbookls by the anr:lojy with electromagnetic phenomena. Both hain ormens are potential flowss. The vector form of the diiferentitl equation for the ma.gneticfield strength dH at any point caused by the, current i flowing in an infinitesimal len.th dT of wire is (from p. 2)2 of reference 11) .jl x r dH = I (2) where r is the vector from the current element to the ooint in question. This equation is u ually called the BiotSivsrt law in aerodynamic te::tboo!s. The sarme forr. of e.uation (2) but '.iitii different constants arplies to the induced velocity dv at any point caused by an infinite i!nal length dT, of a vortex of strength re (reference 9); that i, P ,td x r V= = () .Tr ir13 The units in which the various quantities in equations (2) and (5.) are usuiily meia.ured are widely different. In equation. (2), for exasr.ple, H is usually given in gauss, i in aba:srperes, and I and r in centimeters. In equation (?), v is usually in feet per second, d in feet squared per second, and 1 and r in feet. o NACA ARR No. L5D23 Small search coils are used to measure the strength of the magnetic field. These search coils must be cali brated in magnetic fields of known strength for example, in a Helmholtz coil (fig. 2). If the vortex equation (3) and the usual vortex units are used to commute the induced velocity in the Helmholtz coil (p. 269, reference 11)  that is, are used as the calibrating unit and if the ammeter measuring the current in the Helmholtz coil is considered to read vortex strength, conversions of electro magnetic to aerodynamic units will not be necessary. All conversions and constants become merely a part of the over all calibration constant of the search coils. CONSTRUCTION OF ELECTROMAGNiTICANALOGY MODELS Approximate Representation of Continuous Vortex Sheet The replacement of a continuous vortex distribution by a finite number of conductors must, of course, involve some approximation. Two procedures for constructing models have been tried. For the preliminary electromagnetic analogy model, a set of 50 circular electric wires carrying the same current (connected in series) representing 50 columnar vortices of equal strength were distributed over the wing and wake, CSee fig. 5.) The arrangement of the wires was determined as follows: Fifty contour lines of P were calculated from equation (1). (See reference 2.) Each wire was placed halfway between two adjacent Pcontour lines and thus represented APF/max = 0.02. In order to illustrate the degree of the approximation involved, the continuous chordwise rfunction at the plane ofsymmetry and the stepwise distribution of wires used to represent the continuous distribution are given in figure 4.. The possibility of errors resulting from the use of an incremental distribution is, however, more serious than simply the possibility of not.obtaining a good represen tation of the distribution of vorticity in the continuous wake. The induced velocity resulting froman incremental vortex pattern may vary greatly about the mean value that would be obtained. by thecontinuous sheet, because the ma gnitude of the induced velocity varies inversely with distance..from. the voctex core and becomes higher than the mean value on one side of each steopwise vortex increment and lower than the mean value on the other side. The I]ACA A7R lFo. L5D23 effect is illustrated in figure 5. It is thus necessry: to us3e as :rsrny vi .es ,.s prac tic 9ale to rp}:ro ach as nfe a1l as pc'sitle a c.on inuo:us sheet and to measure the iniuced velocities at a. .reac number of' ,points so that the m,::an !.vlue of the induced velocicy: may be obtained nim:r 1 easily and :nore accurate ly, from f aired curves. The pri 1 iminriary mLodel, co.ns trictcd of ,, wiree, w'ive satisfactory results ird it is believed trit ',0 is slboi.t the ootiTmuji niWnber of wires. Cojs truth ion difficulties are too great if mnoie wires are used,, &rd the accuracy is not great eini.ouh if fe.wer wires s:e used. Another, minor complex method of construction was adopted for a few models out the results obtained were not much mnor scc.urate than those co'tsined with the siimoi.r ir cors truction met.iod. The other cons tru.ccion met.1ho was to use t ii alLuminum sr:trips resultingg in "flat 'ipr'.es") rather than circulsa copoer wires (_io. 6), the recs:n for th,, typ of construction bein thT t a m:ore n': rl continuous distri bution of vorticit:' Fnd f c:,res':or:in mnoothr induced velocity:; field could be obtained. The" ffec t cf oddy currents in the a.uninum s tri.s proved to be more im.ort nt thln w:as originally expected, hol.e'e and the induc,:d :narrietic field was only sli,t.:tly simoother th n with circul ar wires and wvs nert so revular" thus difficulties in fairing the measured induced .na nli.Lic fieli proved to be about the sarre for both t:,es of model. Correction for Finite Thic,'ness of ires calculations by the liftingsurface theory are usually made for points in the plne of the ';rtex sheet. Eec :use the wires raeoresentin the vortices nre of finite thicl:ness, the mnagneticfield strein.th :..ust tb me asre al at severals i vertical heights and extra~,olated to cr., that is, to the center of the wires. Z:cept ne:r t[li.: a ina, tips rnd the leading z edge of the wini, tnis et:.r;.;:olation is u'ually linetr and can thus be rrmde quite 9ccur:ely. The f'ct that ,,'eesurements rce mauc above thi wires sir'.:lifies the fasiinn problem, because th: vari: ion, in tne meanitude of the vertical co:niponent of induced :elocicy about the m sin vsluu (fig. 5) aecre iaes with .nccrease in vertical height. NACA ARR No. L5D25 Correction for Finite Length of TrailingVortex Sheet For the steadystate condition of a finitespan lifting surface, the trailingvortex sheet extends from the trailing edge of the wing infinitely far downstream. It is necessary to determine corrections for the finite length of trailing vortex sheet of the electromagneticanalogy models. The wires representing the incremental vortices were connected in series; the closing loops were about 0.7 span behind the wing for one model and about 2 spans behind the wing for another model. (See fig. 3.) Because the correction for the approximation of the infinite length of the trailing vortex sheet is fairly small about a 5percent correction for a 0.7span wake and less than 1percent correction for a 1.5 to 2.0span wake it appears that wakes need not be longer than 1.5 spans. More dovwnwash is contributed by the closing loops than by the missing trailing vortices. The correction is simply the difference between the down wash contribution of the closing loops and the contribution of the missing part of the trailingvortex sheet. The corrections are small and can usually be estimated by assuming that the span loading is a simple rectangular loading or the sum of two rectangular loadings. The corrections may also be determined from measure ments of the induced field ct a distance behind the closing wires equal to the length of wake represented. That is, if a measurement is made at the corresponding spanwise point I wake length behind the model, the effect will be the same as if a measurement were made on the wing of the downwash due to a mirror image of the model reflected from the closing loops. Such an image would cancel the effect of the closing loops and would double the length of the wake. The remaining error is relatively small. SUGGESTED METHOD OF MEASURING MACNTICFIELD STRENGTH Several methods of measuring the magneticfield strength were investigated. The fundamental principle of the method selected as being the simplest and as requiring the least special equipment and the least development work is given herein. This method consists in passing alternating currents through the wires representing the vortex sheet and measuring the magneticfield strength by means of the IHAC1. ARR lo. L5D23 I voltage induced in a small search coil. A discussion or other possible methods of rCeasuringf the rrnanet.cfield stren..Lh is piven in the sppen i.::. B3sic Principles According toi the nrincoln:s of _electrxomPqnetLic induction (refe.cnce 11), the electrcriotijve force inluced in a fixed circui t (the search c.il, in tlhis case) by chaning. the irqenetic flu.< throu h cho ccil is equel to the time oate o1 chan;,e of ux linkaces ,'.ich the c:,il. If ai altern rtinc c.irrent i2 crassed Lhrouh the wij.res represen in 1 the I iftin p w nr cid t.h '. !k.e, the _ne:iic field strengtth will 1iso :b alte'nristinc sat thz car.e frequency and with the saie wave Sr;ac Thiis nct may be seen from equti.n (2), bercuse tkL j ]stantFneous value of d7 is ':roor orti.:n to th3 itnstsnt. ne.ous' v lue o i. A sinimle method cf der"mnirC. c.he :niticfi li strength there 'ore, is to ireasure i'. ith en el3ctrcric voltmeter the, v;lte, : indrc.d in a smll su,rch coil (fig. 7) when altern.tino cuirent is ras.sd through the wires representing the vorticl c (1Ji 5). ''Thk sEerch .:cil is direction il; thlai is, thi coil measures only the componrrnt of maneriticfiela intensity alonr thl Sxis of the ccil. Practical Problems Upp*rfr sque.nc limit. E5ca.use the volts a: i.duced in the sc.rch coil is proportions] to the :t. ,.f char..e of flux, it would sa'=: desir.bla to uise a vry ~: high frequency curr"nt so th.t the volts ,: inrd'cd in th.e search coil would be very large 5nd thus sromrewhiL lesser to measure e. The rn exinimu frequency tnst :ir,': te uszd, however, is cocut 50j cycles, because the cC';acitance between the closely s.sced wires in the rmouil re3:'r'sent ng the vortex s:.ae3 becoI.es ;i'crtt 'ove this frequency. If greater frequencies re ued, csnacitstive rectence becomes sufficiently 1i 0 to have ,rcc, t ibli eff'oct. Vhen the copecitative rea ctnce between r't res becom::es smali, the currnt is not thle s5e ,i in all wvir.es even tl.ou.h they/ are connected in series. The :ne.:imun allo'vble frequency wIE dt,,rmined from mee aurrimnts of tlhe cs:pecitntive reactance of the el?ctrrmnane ticenaloey model Lsed for the pr:liminsry tests, to b.: described in the section "Prl ri:in.sry. Tests w ,th Electrama. neticAn12 1lcy olod3l." The cDoscitative, rectnce, end thus the iea ,ags' currPnt of this model, became reasu'aible abovZe bout 5300 cycles. 12 NACA ARR No. L5D23 Several other models tested have had about the same or a higher limiting frequency; therefore, a 300cycle limit is believed to be conservative. Wave shape. The fact that the frequency is limited to ab at , cycles also requires that the wave shape of the current in the vortex sheet be very nearly sinusoidal; that is, any departure from a simple sine wave means that higher harmonic frequencies are also present. Usually the most nearly sinusoidal wave shp.e is obtained by putting enough condensers in series with the model to balance the inductance of the model; in this way a pure resistance load is put on the generator. The easiest way to determine the wave shape is to look at the voltage output of the search coil by means of an oscilloscope. This wave shape is not that of the current in the wires but depends on the rate of change of current i with time t that is, the induced voltage e is equal to di K . The amolitude of the third harmonic as seen on the dt oscilloscope is, then, three times the amplitude of the actual third harmonic of the current. In other words, the wave shape of the voltage output of the search coil looks much more irregular than the wave shape of the current in the wires actually is. If the filters that are added to the circuit make the voltage output of the search coil appear satisfactory, no further test is necessary. Practically all alternators have wave sh'.pes that are not sinusoidal and that change with load. The wave form of the current in the Helmholtz coil, used to calibrate the search coil, must be identical with that of the current in the electromagneticanalogy model. Probably the easiest way to make these wave forms identical is to connect the Helmholtz coil and the electromagneticanalogy model in series. The Helmholtz coil and the electrormnetic analogy model should be placed as far apart as possible and should be oriented in such a way that downwash measure ments on the analogy model are unaffected by the field of the Helmholtz coil and that the Helmholtz field is unaltered by the analogymodel field while the search coil is being calibrated. It would also be advantageous to effect an arrangement such that the searchcoil calibration could be checked frequently. A fairly satisfactory arrangr.iernt was employed for the preliminary tests. (See fig. 8.) The leads from the search coil were long enough to go to either the Helmholtz coil or the analogy model. 11ACA ARR No. L5D25 Search coils. The optimum size of the search coil depends uon two factors. First, the search coil must be small enough to make "point" measurements possible. If the coil is too ler,' and the inmnetticfield strength varies nor.1 nearly v'ith ;osj tion, the effective center of the coil muy be too far from the gometric center. If the coil dimensions :re kept small relative to the win, little error due to this cause will result The size cf the search coil terefmrc devondis upcn the size of the electromae neticanalc.ry model. The e'conid cotnsider'StionL is that, for accurte v:ltvCe menasurmenit, crn voltPs outout of the ser.rch ccil should be at least 0.0001 volt and a minimuLTr value of 0.0O1. volt 3i crreftrsble. Several sizes of coil w.re tri:i with the prelimiinary modil, and it is felt tht the iaximur.size sarch coil that gives measurements fairly close to point rmeasurer..nts is one with a Jiemater about 1 percent of' the Viin ei:.i span and a height about J.5 or.rcent of th wini. semispen. In order to get the desired searchcail voltage output, the model semispan must be at leasL 5 to f..t and the search coil should have sbout 1000 Liurns nrid be wound with about No. 13 wire (0.002 in. diam.). If a largeri model is used, larger wire may be ued to wind the c..ji, although wire as small as 0.001 inch was wound satisfactorily for the coil used for the p.elimrinr.ry tests. It is probably desirable to have s number of search coils (fig. 7) of various sizes, however, to meet saiy s.eciFl conditions . Smaller coils are desirable near the leaiing ed'e rnd the tios of the wing and in other places v'hore the flux field varies raidly. The search coil must be wound rather carefully so that all looos are Ps nearly ,erotndiculsr e possible to the axis of the coil in order to m,7.c, tain the directional properties of the coil. The coil must then be carefully mounted on the survey apparatus. A test fc the correct mounting (or alinement) cf the coil is to read the voltrge output of the sn:mll coil when rrmountd with its axis vertical at a position on the electroiriencneTSicn i, ogy model where the horizontal component of the magnetic field is strcnger than rth vertical component. aegzadles. of how the search coil is turned about its vertical axis, the voltage out put should be the same if the search coil is kept &.t the same lace on the model. The leads from th3 search coil to the lectronic voltmetr must be twisted so that no 1'.g:e loops sre present. i..l'A ARR No. L5D23 Otherwise, the voltage measured by the electronic volt meter will not only be the voltage induced in the search coil but will also consist of the voltage induced in these leads. The errors resulting from pickup in the leads may be made negligible by tightly twisting the leads, by making the number of turns in the search coil large so that the lead pickup gives a small percentage error, and by bringing the leads in perpendicular to the vortex sheet so that the leads will be in the region of low magnetic field strength. Extraneous fields. One of the most important problems in measuring the magneticfield strength is to filter out all extraneous fields. It is desirable to make the tests in a wooden building fairly far from electrical distur bance.s such as electric motors and computing machines. Reasonably satisfactory results may be obtained in spite of these disturbances,if necessary, by the use of an electrical filter in the circuit of the search coil and electronic voltmeter. Suc. a filter suppresses any voltage of frequencies other than the one passing through 'the vortex sheet. Because the filter may act as a search coil itself, it must be located some distance from the model. Teasuring equioment. No current should be drawn. in measuring searchcoil voltage, which is of the order of only 1 millivolt. Commercial electronic voltmeters combine the high sensitivity and high resistance demanded. Small amounts of powersupply ripple usually exist in these voltmeters. This powersupply ripple interferes with measurements made at integral multiples of the line fre quency, and careshould be taken to avoid use of these frequencies in testing. PRET'7ITUARY TESTS WITH ELECTROMAGNETICANALOGY MODEL In order to check the accuracy of the electromagnetic analogy method of solving liftingsurface problems, a vortex pattern for which the induced velocities had already been calculated by the method of reference 2 was inves tigated. Calculations were made of the vertical component of the velocities induced by a plane vortex sheet repre senting the lift distribution estimated from the two dimensional theories for an elliptic wing having an aspect ratio of 5. NACA ARR No. L5D23 Equi rentt A small model of th plane vortex sheet was constructed of 50 wires rooresenting 50 incremental vortices. This model (fic. 5) reTresents about the srt.lles t model (winT scan of 5.5 ft) that will yield satisfactory results. A srr.gll search coil (fig. 7(7)) of 1j30 turns was used to .measure the mifgneticfield strength. A lemlnholtz coil (fi;. 2) ws used to celebrate the search ceil. Thi ;.ower supply was a 500Ccycle alternator dJiven by a l re t current cnotor, the s,.ed of ~iaich 'was controlledc so lhat the frequency out'ut vi.s iheld nat 270 cycles. The itrect current field of tn alterntor u\s r .ud L to give a current o it':ut of ea:. 'r.re*. The .va,' shcipe wcs ver. nearly sinuscidal (third harmoni c, cles then 4 percent of the first harmoniic). Some trial tests v.nee n:de in the w'or :sho': of the Lsnley Atmncao'so:ric W'ind T'unnel SecLion. The results proved uns9tisfactory however, bee 9e or excessive elec tric and m rnetic interference. The acunertus .a.s then moved to a lre vcenDier. .i.din', in which there was very little electric i a .iacnetic interference. The setup is shown in figure The smnll snmouiint of oCcycle electric interference that was still present was elinineaed by using a 130cycle hi1rhp5ss filter in the electrornic voltmeter circuit. Rssults A comn lets survey was made of the inducedvelocity field (both the horizontal and vertical comr:ronents) st from 50 to 100 chcrd,;ise points at each of 1; spcanjis locations on the rodal and at several vertical heights. Near the leading 6edg and the tips of the .:o'el, the effect of v4rticai height was vy l=r a en, at all locations, the effect &was lrige cnoizh to r.quire surveys at several heights in order that t,he r::sults could be extrapolated to zerc vertical hs~lnt. The most imnortanc co..7onrent of induced velocity computed by liftingrsurfce theory is the vertical con:.onent. The only reason for r.eqsuring the hcrizontal comionent is to check the arrancemren of the wires reoresentin1: the vortices. According to the assunmpticns of thinairfoil theory, the horizontal component of induced velocity is proportional to the pressure disLribution. ,lues of the NACA ARR No. L5D23 horizontal component u of the induced velocity measured at a relative vertical height of 0.008 b/2= are shown in figure 9 along with the theoretical velocity distribution that the wing was built to represent. The agreement is satisfactory and indicates that the model was constructed with sufficient accuracy. Chordwise surveys of the measured value of the ver tical component w of the induced velocity are presented in figure 10 for several spanwise locations and vertical heights. (Not all of the data are presented.) These and similar data were extrapolated to zero vertical height (fig. 11) and corrected for the finite length of the trailingvortex sheet (correction, about onehalf of 1 per cent) and are summarized in figure 12. Included in fig ure 12 are the calculated values. The agreement between the two sets of results may be seen to be satisfactory. The time and labor involved in obtaining the solution were considerably less (approximately onethird the manhours) by the analogy method than by calculation, after the proper experimental technique had been determined. CONCLUDITDIG E R1 iRKS A method for making liftingsurface calculations by means of magnetic measurements of an electromagnetic analogy model has been developed. The method is based on the perfect analogy between the strength of the magnetic field around a conductor and the strength of the induced velocity field around a vortex. Electric conductors are arranged to represent the vortex sheet. The magnetic field strength is determined by measuring, with an elec tronic voltmeter, the voltage induced in a small search coil by the alternating current in the wires representing the vortex sheet. A comparison was made of the downwash determined by means of a preliminary electromagneticanalogy model with the downwash obtained by calculation for an elliptic wing having an aspect ratio of 5. The accuracy of the magnetic measurements compared satisfactorily with the accuracy of the downwash calculations. Other applications of the method include solutions of nonlinear liftingsurface problems obtained by placing NACA ARR No. L5D25 17 the conductors representing the trailin, vertices alon; thi fluid lines (hnelrmholtz condiiticnr). A potent l9flow solution for the diEtortion and rolling up of the trailing vortex sheet nmy be obtained. By use of the Prendtl GlIuert rules the liftit,'.sur' fce theIry Q aT y be a.: 'pted to include fi rstor:der coipressib, lity effects. Langley ,emori l Aer'on sliical Labor~tory Ni&tiorin l Advisor' Comnr'ittee fo,r Acr.,r,'autics Ln;ley Field, Ve. NACA ARR No. L5D23 APPENDIX ALTE'RATE EZMTHODS OF i:'A3URI \G MAGNETICFIELD STRENGTH By R. A. Gardiner Several methods could be used to make the measurements described in this report. The methods that were considered are: Pickup Device 1. With dc. field on wing a. Search coil (flip coil or collapse of field) b. Rotating search coil With slip rings With commentator c. Saturatedcore magnetometer d. Torsiontype magnetometer 2. With ac. field on wing a. Saturatedcore magnetometer b. Bismuth bridge c. Search coil Measuring Instrument Ballistic galvanometer Ac. electronic voltmeter Dc. amplifier and voltmeter Suitable electronic e equipment Suitable optical equipment Suitable electronic equipment vheatstone bridge Ac. electronic voltmeter The considerations that led to the choice of the method used (method 2c in the foregoing list) were the simplicity, the sturdiness and availability of the equipment, the development work required, the probable success and accuracy, the magnitude of field strength required, the minimum possible size of the pickup device, and the free dom from interference of stray fields. The outstanding IACA ARR Ho. L5D23 advaiite es or dissiavantages of the various methods mayo be surmmrarized as follow;: Dtet:od l. The use of a search coil and ballistic galvanonmeter is the established method of maaneticfield measuren.ent. In ord.r to secure the necessary sensitivity/ , however, a rthner delicate galvsnorieter would hlve to be used 2nd would probably require r se,cial vibrationfree support. The flip coil would require ter readinfgs of the earth's m!a.retic field. ith a stationa.rv coil end collapsing field, the inuictance of the hingi would prevent the desired instantanreous colls cse, the galvPnometer would not be used in a true ballistic manner, ndri errors would result. Method lb. A rotstini search coil and, associated equipment have been used to ma.:a marietic messuren.ents; however, the induced voltpees to ba ;.eesured are lower than those usually measured by this Irh: t"d. The necessary sliding contacts woulG probbl' introduce th;rmal electromotive forces ar v:ariabl r.s isitan.ce and would be subject to corrosicn. P'cision rr hire worl would be necessary to minimize difficulties from the r.otor and besarins. Metihos Ic end 2a. It is ka.o'wn that a coil corntlSninig a highasrmeability .:netal (Per::alloy or M lu.etl) will have a larsce induce volt.oae across its rermirinais as the core becomes mnsanetic l lly saturate. This induced voltage is due to the great change in induct .nce that occurs at the saturation point. This principle has been used in the measurement of magnetic fields; the field to be measured is superimposed upon s. fisld set up in the ccre and the chasne in voltage due to saturaticn is measured. Although this method can be made very sensi ti' e, a large amount of electronic equipment is necessary and the nresssne of a ferromarnnetic substance might cause distortion of the field. ivMethod 13. The use .of torsion mri:.netometer is n accurate method of mneasurin2 the eearh's fiel'.. A sall manet suspended by a suitable fiber is deflected by the earth's field. This deflection is measured and, from th; known magnetic moment of this ma r et, the earth's field ray be determined. The mreasuring eleiient is very sensitive and the time necessary to take one reading is rather long. In addition, such an instrument would probably be of delicate construction. 20 NACA ARR No. L5D25 6thod 2b. Resistance change due to the presence of magnetic lines takes place in bismuth. Measurement of the resistance change by use of a bridge is a possible method of measuring the magneticfield strength. At present a powerful magnetic field is necessary in order to make practical measurements. Considerations of the available power supply eliminated this method. Method 2c. The method finally selected that using an ac. voltmeter, an ac. field, and a small search coil for the pickup device appeared to be the means which would be least troublesome and which would use easily available and simple components. Details of this method are given in the report. NACA ARR No. L5D25 REFERENCES 1. S.wnsIon, Robert S., ant Gl]lir, Clarence L.: Linita sion:' of LiftingLine. Theory fo, Estimation of Aile.:'on HingeMionent Clharacterictlcs. HIAC3i CB uNo. 5L02, 15L?. 2. Cohen, Doris: A i.'othod for Determining the Camber and 'lTist of a Surface to Suvport a Given Distri bution of Lift. 1IAC. T11 lio. u55, 194. 3. von E.arian, Ph., ani Durgers, J. .i.: General Aero Jynarmp.c theory Perfect Fluids. Vol. II of Aero dynami.c Tijory, div. L, '. Durandi, b., Julius Springer ( ierlin), 1955. 4. Bolls.y, jilliam A NonLinear .in; Theo:r and Its Application to Rectan ular '.ins of Small Aspect Patio. Z.f.a...i.:., Be. 19, Ieft 1, Feb. 10.)., :p. 2175. 5. Kinner, J.: Die kreisformije Tragfl che auf potencial theoretischer Grundlage. Tnr,Archiv Bi. VIII, Heft 1, Feb. 1957, pp. .7O0. 6. Krienes, Klaus: The Elliptic Tin. Based on the Potential Theory. UL: TM Ic. 971, 131. 7. Prandtl, L.: Applications of I'odern Hydrodynamics to Aeronautics. 'NACA Rep. HIo. 116, 1921. 8. Goldstein, S., and Young, A. D.: The Linear Pertur bation Theory of Compressible Flow, with Apprlications tc ,iindTunnel Interference. Ri. & :,. ilo. 1909, British A.ri.C., 19?43. 9. Silverstein, ."ie, Katzoff, S., and Eullivant, d. Kenrneth: Downwash and vsake behind Plain and Flapped Aiirfoils. .A'CA Rep. Nio. 651, 1959. 10. Kaden, H.: Aufwicklung einer tustabilen Unstetig keitsfl.che. Ing.Archiv Ed. II, Heft 2, IIay 1951, pp. 0168. 11. Page, Leigh, and Adams, Iorman Ilsley, Jr.: Frinciples of Electricity. D. Van Nostrand Co., Inc. (New York), 1951. NACA ARR No. L5D23 Fig. 1 4Z 0) b I .0 I 1f th 6 Is O% . I I 5 s NACA ARR No. L5D23 Electrio wires wound around these hoops Di5tance between co, I equals rodius of coils NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS Fiqure Z. A Helmholt coil. The, earch coil to be calibrated is located in the geometric, center of the felm holtz, coi I Fig. 2 NACA ARR No. L5D23 Fig. 3 Co 0 : .I  oCd 0 0 r Q. o 4 cod o 4 +3 40 C t a) t Cd ...," .I , ...0  12. : ,. ,, .; ,)t!. ,ol .i : ... ... ;" '.] ;: ei ' "4 ... ...... ;: .:; ..... : , ..I ., , NACA ARR No. L5D23 %. \9 4 N 't/so.^ t/uop /Ou o /suawpo YMy 00 t W, ',3'eNvoad uqo/njjo  s o t J I I k : o I, < op 'b o s *O ^ " Figs. 4, 5 NACA ARR No. L5D23 Fig. 6 E r C 'I '..~ . o o daZ 0 . ii 0 r, ,1 ' D 0 0 Ic i .I I AI c Q .) w 0 l * 4 C70 ,i hno NACA ARR No. L5D23 Fig. 7a,b n INACA ARP No. L5f)23 Wmng 11 SE/lectron/c va vo/fmeter Ammeter He/mlho/tz co// NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS Figure General/ rra/ngemenf of e/ecfromogneo/  ona/ogy experiment . Search co// Fi g. 8 NACA ARR No. L5D23 0 . .4 .6 8 Ch/ordvwise station, x/c /'~/re 9. Chon/d'es d/s/rL /o4 of/he /7end//me,/7ona/ Yor0zot.a/ re/a/y cmpve' t 6'// 'g,,; ao/,s & o =0.008. F/otp/ate Iwoe of /oooewy. Fi g. 9 NACA ARR No. L5D23 Fig. 10a, b, c _ [ l * __ _  o II II I N O 0 >q ldlq f l) '/ a /tfgo 7o3o//ftUOp\o o 6 0. I Os C i b a 6 to.q o 'I c 0 4 b o s L4~ *d ^5 0 '0 Ii P1 C 0 k o P 9I c 0 Q t) 0x ^ ^ E b ^ '* sg 11 0 NACA ARR No. L5D23 .6k (I I I I I II (a) 0. 6h T  7 =Q7c 075 6 5 1 4 (c) =0. 4 0O hble 0 .OZ .04 Re/lafve height, (d) Y 0.60. h/z 0 .02 .04 ,Re/ofve height,  )Y = 0.95. b/2 Figure .//. Extropo/a/on to zero vertfco/ he/ghf of dofo obtained from on re/ecfromagnef/ analogy model of on e//'pL/c wing having on aspect ratio of 3 F/ofp/afe type of /oadng . XIc .n ..' *  Fig. llaf NACA ARR No. L5D23 SI X/C 0.75 C" 1 .8  ]  4 .4 x Ca/cu/ofed points NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 0 .2 .4 .. 8 /0 Span wise station, ; Figure /2 A The nonod/meens.ona downwash wb for an e/llphc wing having an aspect ratfl of 3 with f/ofp/ate type of /oad/ng. Conopar/son of results obtained by method of reference 2 with data, extrapolated to z = 0, obtained from tests of preliminary e/ecromragnetOc ano/ogy model/ Fig. 12 UNIVERSITY OF FLORIDA 3 1262 08104 968 5 UNIVERSITY OF FLORIDA DOCUMENTS DEPARTMENT 120 MARSTON SCIENCE UBRARY P.O. BOX 117011 GAINESVILLE, FL 326117011 USA 