ARE No. L5D23 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS WARTIME REPORT ORIGINALLY ISSUED May 1914.5 as Adrance Restricted Eeport L5D23 M EnEXm?(»!AG]!IETIC-AMLOGfr METHOD OF SOLVTHG LrFTURJ-SIIRFACE-THEOHI PROBLEMS By Roteii; S. Svanson axA Stewart M. Crandall Langley Memorial Aeronautical Laboratory Langley Field, Ta. NACA WASHINGTON NACA WARTIME REPORTS are reprints of papers originally issued to provide rapid distribution of ■advance research results to an authorized group requiring them for the war effort. They were pre- viously held under a security status but are now unclassified. Some of these reports were not tech- nically edited. All have been reproduced without change in order to expedite general distribution. 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/electromagneticaOOIang NACA ARR No. L5D25 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS ADVANCE RESTRICTdlD REPORT M ELECTROMAGNETIC -ANALOGY METHOD OF SOLVniG LIFTING-SURE ACE-THEORY PROBLEMS By Robert S. Swansoji and Stewart M. Cr-andall SUMMARY A method is suggested for making lifting-surface calculations by means of magnetic measurements of an electromagnetic-analogy model. 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 stieet. The magnetic- field strength is determined by m.easuring, with an elec- tronic voltmeter, the voltage induced in a small search coil by the alternating current in the wires representing the vortex sheet. Solutions of nonlinear lifting-surface problems may be obtained by placing the conductors representing the trailing vortices along the fluid lines (Helraholtz con- dition). A potenti al-f lovv solution for the distortion and rolling up of the trailing-voi'tex sheet may be obtained. By use of the Prandtl-Glauert rule, the lifting-surface theory may be adapted to include first-order com.pressiblllty effects . A comparison was miade of the downwash determined by means of a preliminary electromagnetic-analogy mxodel with the downwash obtained by calculation for an elliptic wing having an aspect ratio of J. The accuracy of the magnetic measurements compared satisfactorily with the accuracy of the doT/vnwash calculations. INTRODUCTION There are many im.portant aerodynamic problems for which solutions by lifting-line theory are inadequate. These problems can be solved m.uch more satisfactorily by NAG A ARR No. L5D25 a lifting-surface 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 lifting-surface theory, however, are rather laborious even for the simplified case in which the vari- ation In incremental pressures v^rith the effective camber or the angle of attack of the surface is linesr. A more exact nonlinear solution is very nearly impossible to calculate except for a few special cases. A few of the aerod^rnaraic problems for which solutions by lifting-surface theory are desired are: the plan-foi-'m corrections necessary for the prediction of finite-span hinge-moment characteristics from section data; the determination of spanwise and chord- wise load distributions of v/ings 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 shoi/vn that the plan-form correc- tions determined from lifting-line theory are Inadequate for hinge-moment predictions. The plan-form corrections determined by a linear lif ting-sui'^f ace theory (reference 2), however, were shov/n 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 lifting-surface theory Is required. The electromagnetic- analogy method was developed in an attempt to make calculations by both linear and non- linear lifting-surface 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 shov;n (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 magnetic-field strength then replace the difficult induced-velocity calculations. For nonlinear solutions of lifting-surface problems, the trailing-vortex sheet represented by the wires is rolled up and distorted instead of lying in a plane as it MAC A ARR No. L5D25 5 is usually assumed to do. In figure 1 is sho^^m a simplified picture of a rect&ngular wing of low aspect ratio at a large angle of attack with a rolled-up and distorted trailing- vortex sheet. Of the various features of the distorted vortex sheet that contribute to the nonlinearity , the most iniportsxit is the vertical spacing of the trailing vortices. The increase in verticel spacing as the angle of attack is increased results in a decrease in the vertical component of induced velocity at the surface, especially near the wing tips; thus the slope of the lift curve is increased as the angle of attack Increases (see reference Lj.) and the slopes of hinge -moment curves are more negative (refer- ence 1). The present report describes the basic theory of the electromagnetic-analogy method and the general proced\are by which various aerodynamic problems may be solved by this analogy. A few preliminary results for the linear case are presented for an elliptic wing having an aspect ratio of 3, as well as a comparison of the results obtained by the present method and the results calculated by the method of reference 2. SYMBOLS vortex strength m.aximum vortex strength 4p pressure difference across lifting surface V free-stream velocity M mach number, ratio of free-stream velocity to sonic velocity n 'max p fluid density X distance along free-stream direction from leading edge of v/ing y spanwlse distance z vertical distance above plane of vortex sheet H m^agnetic-f i eld strength k NAG A ARR Ko. L5D25 i current in conductor e induced voltage I 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 (dov;nwash.) u horizontal component of induced velocity (free-stream direction ) K constant t time b span c chord Bar above sjrmbol indicates a vector, as H, T, r, and v, BASIC THEORY Solution of Aerodynamic Problems by Available Lifting-Surface Theories The distribution of lift over a lifting surface csnnot, in general, be expressed in any simple m.athematical form such 83 cen be obtained by lifting-line theory. This state- mient is especially true for nonlinear lifting-surface problem^s . Except for a few special plan forms (references 5 and 6 ) , the method of determining the induced doivnwash for a given lift distribution also is too complex for expression in mathematical form. In order to obtain an exact, complete analytical solution, hov/ever, such expressions must be knovm, The determination of the surface to sustain an arbi- trary lift distribution m.ay be accomplished by m^eans of the electromagnetic-analogy method described herein or, for the linear case, by the semd graphical method of refer- ence 2. The inverse problem, determ^ining the lift distill- KACA ARR Wo. L5D23 5 bution over an arbitrary surface, may then be solved by a process of successive aDproxlmations , A reasonable distri- bution of vorticlty is assuiaed or calculated from the simple lifting-line and thin-airfoil theories, and the induced velocities corresponding to that vortex distribution are detei'mined by making an electx'omagnetic-analogy model of the vortex sheet and m.easuring the magnetic-field strength. If the induced velocities do not satisfy the boundary con- ditions - that is, the shape of the lifting surface - the vortex sheet is suitably altered snd the process repeated until the boundary conditions are satisfied. For the non- linear problem, not only must the induced velocities satisfy the boundary conditions of the mlng shape but also the trailing vortices must satisfy the Helmholtz condition, namely, that the vortices must trail along fliiid lines, (See reference 7' ) I^ practice, satisfying these sim.ole conditions may require a considerable a:nount of work unless the first approximation is fairly accurate. In order to obtain a somewhat more general solution, for the linear case at least, the surfaces required to suprjort several different lift distributions may be determined so that the shape for a particular lift distribution may be estim.ated by a process of interpolation or superposition. There are, however, several problems for which a com- plete r3otentlal-f low solution of the inverse problem is not necessary. For example, in order to include the m.ain effects of viscosity, the estimation of the hinge-moment parameter for finite-span wings should be made by applying theoretical aspect-ratio corrections to experimental section hinge-moment parameters. For such problems the additional aspect-ratio corrections may be determined simply and accurately from the surface required to support a given lift distribution (reference 1) as found by lifting-surface theory. The results of the electromaignetic-analogy solution of the lifting-surface theory may be corrected for first- order compressibility effects by a simple application of the Prandtl-G-lausrt rule (reference 8). The method consists of determining the incompressible-flow characteristics of an equivalent wing the chords of which are increased by the factor , .. .— . It is therefore necessary only to build Vl - ?;r an. electrom.agnetic-an.alogy model of a v\fing of this slightly / ~ 1 \ lower aspect ratio / lov/sr by the factor - ] or to V ■ Vi - M2y test models of several aspect ratios and interpolate. The ° NACA ARR No. L5D23 pressures (or vorticlty) acting upon this Incompressible equivalent of lower aspect ratio, however, inust be increased by the fee tor :- . . . - . In order to find the lift, these Vl - ¥:~ increased pressures ax^e referred to the original wing and integrated. Vortex Sheet Inasmuch as the equivelence of a lifting wing and wake to a vortex sheet mey be considered to be well established (reference 3)j only the importsnt 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 s sheet of bound vortices. The strength of the vortices is directly associated with the lift distri- bution of the winf'. The product of the air density, the free-stream velocity, the vortex length perpendicular to the free-stream velocity, and the vortex strength of each elem.entary vortex equals the lift contribubion by that elementary vortex (Kutta- Joukowsicl law). If the lift distribution of the wing is tvnown or assumed, therefore, the equivalent vortex distribution ma:/ be easiljr 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 " ^'^dx (1) Vc PV where pv is the product of the density and the free- stream velocity and the Integration is m.sde in the free- stream direction. Equation (1) gives the chordwlae F-function at each section. The values of T 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 v^ing. The part of the vortex sheet representing the v/ake consists of the so-called trailing vortices. As the n.aine Implies, these vortices originate at the trailing edge of the wing and m.erely trail behind the wing. These vortices are free to miove and thus lie along the local streair. lines, or fluid lines. This simple kinematic condition, the Helmholtz condition (reference 'J), determines the configu- ration of the trail in,-?, -vortex sheet. NAG A ARR Ko. L5D25 ( The trailing vortices for lightly loaded wings usually lie very near a plane; that is, these vortices travel alraost straight back from their origin at the trailing edge of the wing. For highly loaded wings, however, the traillng-vortex sheet is known to be considerably distorted, rolled up, and inclined with respect to the free-stream direction. (See fig. 1.) The characteristics of the air flow behind wings are described in more detail in refer- ences 9 3nd. 10. Electromagnetic Analogy The perfect analogy that exists between the strength of the m.agnetic field around conductors and the strength of the induced-velocity field aroujid columiiar (finite- diameter) vortices is v/ell known. In fact, the phenomena of the induced velocities around vortices are usually explained in aerodynamic textbooks by the analogy with electrom^agnetic phenomena. Both phenomena are potential flows. The vector form of th£ dlfferentirl equation for the magnetic-field strength dH at any point caused by the current 1 flowing in an infinitesimal length dZ- of wire is (from p. Sl+S of reference 11) _ dT X r dH = 1-^^ (2) where r is the vector from, the current element to the point in question. This equation is usually called the Biot-Savart law in aerodynamic textbooks. The same form of equation (2) but v/ith different constants applies to the induced velocity _dv at any point caused by an infinitesimal length dl of a vortex of strength V (reference 9) I that is, r dl X r dv = — — iirr I r I P (3) The units in which the various quantities in equations (2) and (J) are usually m.easured are v/idely different. In equation (2), for example, H is usually given in gauss, i in abam.peres, and I and r in centimeters. In equation (5), v is usually in feet per second, V in feet squared per second, and I and r in feet. 8 NACA ARR No. L5D25 Siri'^ll search coils are used to measure the strength of the magnetic field. These search colls must be cali- brated in magnetic fields of knoivn strength - for example, in a Helmholtz coil (fig. 2). If the vortex equation (J) and the usual vortex units are used to compute tiie induced velocity in the Helmholtz coll (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 becoffi.e merely a part of the over- all calibration constant of the search coils. CONSTRUCTION OP ELECTROMAGNiTIC-ANALOGY MODELS Approximate Representation cf Continuous Vortex Sheet The replacement of a continuous vortex distribution by a finite num.ber of conductors must, of course j, involve som.e approxim.ation . Two procedures for constructing models have been tried. For the prel.imlnary electromagnetic- aiialogy model, a set of ^0 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 mrires was determined as follows: Fifty contour lines of T were calculated from equation (1). (See reference 2.) Each wire was placed halfway between tv/o adjacent P-contour lines and thus represented AF/Tj^^^ = 0.02. In order to illustrate the degree of the approximation involved, the continuous chordwise P-function at the plane of -symmetry and the stepvifise distribution of wires used to represent the continuous distribution are given in figure []., The possibility of errars resulting from the use of an incremental distribution is, however, more serious than sim.ply the possibility of no-t. obtaining a good represen- tation' of the distribution of vorticity in the continuous wake. The Induced velocity resulting from -an incremental vortex pattern mxay vary greatly about the mean value that would be obtained by the - coniilnuaus sheet, because the magnitude of the induced velocity varies inversely with -distance... from, the vortex core and becomes higher than the mean value on one side of each stepwise vortex increment and lower than the mean value on the other side. The NAG A AR^ No. L5D23 effect is Illustrated in figure 5* It is thus necessary to use as many wires as practicable to approach as nearly as possible a continuous sheet and to measure the induced velocities at a great iiumbex^ of points so that the mean value of the induced velocity may be obtained more easily and more accurately from faired curves. The preliminary model, constructed of 50 wires, gave satisfactory results and it is believed that 50 is about the optim-um number of wires. Construction difficulties are too great if more wires are used, and the accuracy is not great enough if fewer wires are used. Another, more complex method of construction was adopted for a few- models Dut the results obtained v/ore not much more accurate than those obtained with the simpler 'wire construction met-iod. The other construe bion method was to use tain aluminum strips (resulting in '"flat wires") rather than circular copper wires (fig. 6), the reason for tliis type of construction being thst a m.ore nearly continuous distri- bution of vorticity and a corresponding sm.oother induced- velocity field could be obtained. The effect of eddy currents in the aluminum strips proved to be more imiportant than was originally exeected, however, and the induced magnetic field was only slightly smioother than with circular wires and Vvas net so regular; thus difficulties in fairing the m.easured Induced magnetic field proved to be about the sam.e for both tvpes of model. Correction for Finite Thiclcness of Wires Calculations by the lifting-surface theory are usually made for points in the plane of the vortex sheet. Because the v\rires representing the vortices are of finite thickness, the magnetic-field strength must be measured at several vertical heights and extrapolated to zero, that is, to the center of the vi/ires. Except near the v/ing tips and the leading edge of the wing, this extrapolation is usually linear and can thus be made quite accurately. The fact that measurements are made above the wires simplifies the fairing problem, because the va-riation in the magnitude of the vertical component of Induced velocity about the mean value (fig. 5) decreases vi/lth increase in vertical height. 10 NACA ARR No. L5D23 Correction for Finite L-ength of Traillng-Vortex Sheet For the steady-state condition of a finite-span lifting surface, the tr ailing-vortex 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 electromagnstic-analogy models. The v/lres representing the incremental vortices were connected in series? the closing loops were about O.7 span behind the v;ing for one model and about 2 spans behind the v»'lng for another model. (See fig. 5.) Because the correction for the approximation of the infinite length of the trailing- vortex sheet is fairly small - about a 5-P©rcent correction for a 0.7-spsn wake and less than 1-percent correction for a 1.5- "to 2.0-span v.ake - it appears that wakes need not be longer than I.5 spans. More dovv'nwash is contributed by the closing loops than by the m.issing trailing vortices. The correction is slm.ply the difference between the down- wash contribution of the closing loops and the contribution of the m.isslng part of the tralllng-vortex sheet. The corrections are small and can usually be estim.ated by assujning that the span loading is a slm^ple rectangular loading or the sum of tv/o rectangule-r loadings. The corrections may also be determined from measure- ments of the induced field at 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 1 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 m.odel reflected from the closing loops. Such an image would cancel the effect of the closing loops and w^ould double the length of the wake. The rem.aining error is relatively sm.all. SUGGESTED METHOD OF f,TEASURING MAGNETIC -FIELD STRENGTH Several m.ethods of measuring the magnetic-field strength were Investigated. The fundamental principle of the method selected as being the siir.plest and as requiring the least special equipment and the least development work is given herein. This method consists in passing alternating currents tiirough the wires representing the vortex sheet and measuring the m^agne tic-field strength by m.eans of the NAG A ARR No. L5D23 H voltage induced In a small search coil. A discussion of other possible methods of measuring bhe magnetic-field strength is given in the appendix, Basic Principles According to the principles of electromagnetic induction (reference 11), the electromotive force induced in a fixed circuit (the search coil, in this case) by chan<-^ine; the mai^ietic flux throui'h the coil is eauel to the time rate of change of flux linkages with the coil. If an alternating current is passed through the wires representing the ].ifting wing ^aid the v»8ke, the magnetic- field strength will e iso be alternating at the saine frequency and witli the sa.ie wave shape. This fact may be seen from equation (2), because the instantaneous value of dli is proportional to the instantaneous value of 1-. A simple method of determining the magnetic-field strength, therefore, is to measure m^lth an electronic voltmeter the voltage induced in a small search coil (fig. 7) when alternating current is cass&d through the v/ires representing the vortices (fig. 5). The search coil 3s directional; that is, this coil measures only the component of magnetic-field intensity along the axis of the coil. Practical Problems Upper-frequency limit .- Because the voltage induced in the search coil is proportional to the rate of change of flux, it would seem desirable to use a very high- frequency current so that the voltage Induced in the search coll would be very large and thus somewhat easier to measure. The maximum, frequency that may be used, however, is about J)00 cycles, because the capacitance between the closely spaced wires in the model representing the vortex sheet becomes Important above this frequency. If greater frequencies are used, capacitative reactance becomes sufficiently low to have perceptible effect. vVhen the capacitative reactance between wires becomes small, the current is not the same in all wires even tliough they are connected in series. The maximum allov/sble frequency was determined from, measuremients of the capacitative reactance of the electromagnetic-analogy model used for the preliminary tests, to be described in the section "Preliminary Tests with Electromagnet! c-Aiialogy Model." The capacitative reactance, and thus the leakage current of this model, became measurable above about 5*^^"^ cycles. 12 XAGA ARR No. L5D25 Several other models tested have had about the sane or a higher limiting frequency; therefore, a 300-^7016 limit is believed to be conservative. Wave shape .- The fact thst the frequency is limited to about 500 cycles also requires that the wave shape of the current in the vor-tex sheet be very nearly sinusoidal; that is, any departure from a slr.iple sine wave means that higher harmonic frequencies are also present. Usually the most nearly sinusoidal wave shape is obtained by putting enough condensers in series with the m.odel to balance the inductance of the model; in this v;ay a pure resistance load is put on the generator. The easiest ^vay to determine the wave shape is to look at the voltage output of the search coil by means of an oscilloscope. This v/ave 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 s is equal to K — . The .amplitude of the third harmonic as seen on the oscilloscope is, then, three times the am/olitude of the actual third harm^onic of the current. In other words, the 7/ave 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 m,?ke the voltage output of the search coil appear s atisf actor;/, no further test is necessary. Practically all alternators have vvave shapes 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 electromagnetic-analogy model. Probably the easiest way to m.ake these wave forms Identical is to connect the Plelmiholtz coil and the electromagnetic-analogy miOdel in series. The Relmlioltz coil and the electromagnetic- analogy model should be placed as far apart as possible and should be oriented in siich a v/ay that dovmwash measure- ments on the analogy m.odel are lanaffected by the field of the Helmholtz coil and that the HelmJaoltz field is unaltered by the analogy-model field while the search coil is being calibrated. It would also be advantageous to effect ari arrangement such that the search-coll calibration could be checked frequently. A fairly satisfactory arrangement v;as emiployed for the preliminary tests, (See fig. 8.) The leads from the search coil v/ere long enough to go to either the Helmholtz coil or the analogy m.cdel. IT AC A ARR No, L5D25 I5 Sear ch colls . - The o'otlmum size of the search coll depends unon tvvo factors. First, the search coil must be small enough to make ''point'''' measurements possible. If the coil is too lar^e and the magnetic-field strength varies norilinesrly v/ith position, the effective center of the coil may be too far from the geometric center. If the coil dimensions sre kept small relative to the wing, little error due to this cause will result. The size of the search coil therefore depends upon the size of the electromagnetic-analogy model. The second consideration is that, for accurate voltage measurements, the voltage outiout of the search coil should be at least 0.0001 volt and a minimum value of 0.001 volt is preftrsble. Seversl sizes of coil v>rsre triid with the preliminary model, and it is felt that the maximum-size search coil that gives m-easur-ements fairly close to point measurements Is one with a diameter about 1 percent of the v/ing semi- span and a height about O.5 oercent of the vi/ing semispan. In order to get the desired search-coil voltage output, the model semispan must be at least 5 to U feot and the search coil should have about 1000 turns and be m'ound v/ith about No. 14.3 wire (0.0022-in. diam.). If a larger model is used, larger wire m.ay be u:.ed to wind the coil, although wire as small as 0.001 inch was wound satisfactorily for the coil used for the preliminary tests. It is probably desirable to have a number of search coils (fig. 7) °^ various sizes, hov.-ever, to meet any special conditions. Smaller coils are desirable near the leading ease and the "■o tips of the wing and in other places v^'here the flux field varies rapidly. The search coil must be wound rather carefully so that all loops are a.s nearly perpendicular as possible to the axis of the coil in order to maintain the directional properties of the coil. The coil must then be carefully mounted on the survey apparatus. A test for the correct, mo'onting (or alinem.ent) of the coil is to read the voltage output of the small coil when mounted with its axis vertical at a position en the electroiriagnetic-anslogy model v;here the horizontal component of the magnetic field is stronger than the vertical com.ponent. Regardless of how the search coll is turned about its vertical axis, the voltage cut- put should be the same if the search coil is kept at the sarae place on the model. The leads from the search coil to the electronic voltmeter must be tv/isted so that no large loops are present. ^^i- NAG A ARR No. L5D23 Others/Vise, the voltage measured by the electronic volt- meter will not only be the voltage induced in the search coll 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 m.agnetic- field strength. Ervtraneo us fields .- One of the most Important problems in measuring the magnetic-field strength is to filter out all extraneous fields. It is desirable to make the tests in a wooden building fairly far from electrical distur- bances such ss electric motors and computing m.achines. 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. Sue '^ a filter sucpresses 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 som,e distance from the model. s ?!essuring equipment.- No current should be dravvn. in me esurTng~e arch- coir voltage , which is of the order of only 1 millivolt. Cormr.erclal electronic voltmeters combine the high sensitivity and high resistance demanded. Small am.ounts of povi-er-supply ripple usually exist in these voltm.eters. This power-supply riople interferes with measurem.ents made at integral m.ultlples of the line fre- quency, and care- should be taken to avoid use of these frequencies in testing. PRELIMINARY TESTS WITH ELECTROMAGNETIC -A1\[AL0GY MODEL ■ In order to check the accuracy of the electrcmagnetic- anslogy method of solving lifting-surface problems, a vortex pattern for which the induced velocities had already been calculated by the method of reference 2 v/as inves- tigated. Calculations were made of the vertical component of the velocities induced by a plane vortex sheet repre- senting the lift distribution estim.ated from the two- dimensional theories for an elliptic vdng having an aspect ratio of J. MAC A ARH No. L^DSJ I5 Equipment A s-^iall model of th3 plane vortex sheet v;as constructed of 50 wires representing 50 incrementpl vortices. This Riodel (fig. 5) represents about the sinallesT: nodel (wing spaii of 5o ^t) that will yield satisfactory results. A sinall search coil (fig. ?(?)) of 1000 turns was used to measure the msgnetic-f ield strength. A helmholtz coil (fig, 2) vi^as used to calibrate the search ceil. The power supply was a 5^0-cycle alternator driven by a direct- curx'ent laotor, the speed of which was controlled so that the frequency output was held at 27O cycles. The direct- current field of the alternator was f/djusted to give a current oat".:ut of Q sinperes. The wave shape was very nearly sinusoidal (third harmonic, less than I), percent of the first harmonic). Some trial tests were made in the worlrshop of the Langley Atmosohoric "A'lnd Tvinnel Section. The results proved unsatisfactory, however, because of excessive elec- tric and magnetic interference. The apparatus was then moved to a l?rge wooden buildings in which there v/as very little electric and magnetic interference. The setup is shown in figure 8. The small amount of 6G-cycle electric interference that was still present was eliminated by using a 100-cycle high-pass filter in the electronic- voltmeter circuit. Results A complete survey was m.ade of the induced-velocity field (both the horizontal and vertical components) at from 50 to 100 chcrdvvise points at each of I5 spanwise locations on the model and at several vertical heights. Near the leading edge and the tips of the model, the effect of vertical height was very large and, at all locations, the effect was large enough to require surveys at several heights in order that the results could be extrapolated to zero vertical height. The most im.portani: component of induced velocity comiputed by lifting-surface theory is the vertical component, The only reason for measuring the horizontal component is to check the arrangemient of the 'wires representing the vortices. According to the assumptions of thin-airfoil theory, the horizontal com.ponent of induced velocity is proportional to the pressure distribution. Values of the l6 NACA ARR No. L5D25 horizontal component u of tha induced velocity z measured at a relative vertical hei.tJiht of —rr; = O.OOd are snown in figure 9 along with the theoretical velocity distribution that the v/ing was built to represent. The agreement is satisfactory and indicates that the model was constructed vvjth sufficient accuracy. Chordwise surveys of the messured value of the ver- tical com.ponent 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 tralling-vortex sheet (correction, about one-half of 1 per- cent) and are s\ammarized in figur-e 12. Included in fig- ure 12 are the calculated, values. The agreement betvtreen the two sets of results may be seen to be satisfactory. The time and labor involved in obtaining the solution were considerably less (approximately one-third the man-hours) by the analogy method than by calculation, after the proper experim.ental techjiiaue had been determined. CONCLUDING- REMARKS A m.ethod for m,3klng lifting-surface calculations by means of magnetic measurements of an electromagnetic- analogy ffiodel has been developed. The method is based on the perfect analogy between the strength of the m.agnetlc field around a conductor and the strength of the induced- velocity field around a vortex. Electric coxiductors are arr&nged to represent the vortex sheet. The magnetic- field strength is determdned by meas-'oring, with an elec- tronic voltmeter, the voltage Induced in a smj.all search coil by the slternating current in the vifires representing the vortex sheet. A comparison was made of the downxvash determined by means of a preliminary electromagnetic-analogy model with the doxwiwash obtained by calculation for an elliptic wing having an aspect ratio of J. The accuracy of the magnetic m.easurements compared satisfactorily with the accuracy of the do'ATiwash calculations. Other applications of the m.ethod include solutions of nonlinear lifting-surface problems obtained by placing NAG A ARR No. L5D23 1? the conductors representing the trailing vortices along the fluid lines (Helmholtz condition), A potential-flow solution for the distortion and rolling up of the trailin; vortex sheet npy be obtained. By use of the Prandtl- Glauert rule, the lifting-surface theory may be adapted to include first-order compress ibl 11 ty effects. Langley Memorial Aeronautical Laboratory National Advisory Committee for Aeronautics Langley Field, Va. 18 NACA ARR Ko. L5D25 APPENDIX ALTE-^.!'^ATS ^'''ilTRODS OF M3ASUROTG MAG-NSTIC -FIELD STRENGTH By R. A. Ger diner Several methods could be used to rrie.ke the measurements described in this report. The methods that were considered are : Pickup Device 1. With d-c. field on wing Measuring Instrument a. Search coil (flip coil or collapse of field) Ballistic ^salvsnometer 2. b. Rotating search coil With slip rings Wi th c ommu t a t o r c. Saturated-core magnetometer d. Tors ion -type ra agn e t om^e t e r Yifith a-c. field on wing a. Saturated-core maarietometer b. Bismuth bridge c. . Search coll A-c. electronic voltmeter D-c. aj7"."olif ier and voltmeter Suitable electronic equipment Suitable o"Dtical equipment Suitable electronic equipment Wheats tone bridge A-c. electronic voltm.eter The considerations that led to the choice of the method used (method 2c in the foregoing list) were the simiplicity, the sturdiness and availability of the equipment, the development work required, the probable success and accuracy, the magnitude of field strength required, the minlm.um possible size of the pickup device, and the free- dom from interference of stray fislds. The outstanding NAG A ARR No. L5D25 19 advantages or d5_s advantages of the various methods may be summarized as follows: Metnod l a.- The use of a search coil and ballistic galvanometer is the established method of magnetic-field measurement. In order to secure the necessary sensitivity, however, s r.?ther delicate galvsnometer would have to be used and would probably require a, special vibration-free support. The flip coil would require tare readings of the earth's magnetic field. Vi/ith a stationary coil and collapsing field, the inductance of the wing would prevent the desired instantaneous collapse, the galvanometer v/ould not be used in a true ballistic m.anner, and errors v/ould result. Method lb .- A rotating search coil and associated equipmicnt have been used to make magnetic m.e asurem.ents ; hcvifever, the induced voltages to be i^easured are lo'wer than those usually mieasured by this method. The necessary sliding contacts would probably introduce thermal electromotive forces and variable resistance and would be subject to corrosion. Precision machine work would be necessary to minimise difficulties from, the motor and bearlnss , Method s Ic and 2a .- It is known that a coil containing s hi gll^p e rme abi 1 i ty me't a 1 (permalloy or Mioraetal) will have a large Induced voltage across its terminals as the core becom-es magnetically saturated. This induced voltage is due to the great change in indiictance that occurs at the saturation point. This principle has been used in the measurem.ent of magnetic fields; the field to be measured is superimposed upon a field set up in the core and the change in voltage due to saturation is measured. Although this m.ethod can be made very sensitive, a large amount of electronic equipment is necessary and the presence of a ferromagnetic substance might cause distortion of the field. Method Id .- The use of torsion ma5netom.eter is an accurate methoQ of measuring the earth's field. A sm.all m.agnet suspended by a suitable fiber is deflected by the earth's field. This deflection is m.eas^ored arid, from the knovrn magnetic momient of this magnet, the earth's field may be determined. The measuring element 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 NAG A ARR No. L5D25 Method 2b .- Resistance change due to the presence of magnetic lines tskes place in bismuth. Measurement of the resistance change by use of a bridge is a possible method of measuring the magnetic-field strength. At present a powerful magnetic field is necessary in order to m.ake practical measurements. Considerations of the available power supply eliminated, this method. Method 2c .- The method finally selected - that using an a-c. voltmeter, an a-c. field, and a sm.all search coil for the pickup device - appeared to be the means which would be least troublesom.e and v/hlch would use easily available and sim.ple comrponents. Details of this method are siven in the reoort. NACA ARE No. L5D23 21 REFERENCES 1, S?/anson, Robert S., and CtIIII.'^, Clarence L.; Limita- tions of Lifting-Line Theory for Estimation of Aileron Hijige-Monent Characteristics. NAGA CB No, 5LC2, I9IJ-5 • 2» Cohen, .Doris: A I.'Iothod for Determining the Camber and Twist of a Surface to Support a Given Distri- bution of Lift. KAGA TN No. 8p5, I9I1.2. 5. voii Karr.ian, Ih., and Burgers, J. M.: General Aero- dynamic Theory - Perfect Fluids. Vol. II of Aero- dynamic Theory, div. E, .V. T. Durand, ed,, Julius Springer ( .oerlin ) , 1955 • 4. Bollay, ^(/illiam; A Non-Linear V/ins Theory and Its Application to Rectane^ular wings of Small Aspect Ratio. Z.f .a.IJ.I,!., Bd. I9 , Eeft 1, Feb. 195^-', pp. 21-35. 5* Kinner, ' >« « •K. »< Q ^ .cJ >«, 1 "K. ^ J5 .c: V. ^ 5 <0 < ^ 5^ 9) 8 "V. S^ ^ Q •§ Q.*^ I Qi ^ .^ > "^ ^ Q < V. ^ <: "fc* ^ «:: 'v. ^ Q "K, ^ ^ ^ < .O ■•>» •V. ^ ^ V^ "V. 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I I — il-f 1 •x/c ' I ho ."^ ' '^ n ri >-- a5 y M^^^o.zo -~ L 1^ M - — — ^0.7^ N n >, r "- ^ 25 (0^^:0.40. /^e/at/\^e he/ghf ; -^^ .6] .6 .5 .4- .3 X I O I I \, \ I N N 1 7> s. ^ ^ *7 -TC \ H ^ ^Cko ^ ^ k •Iv \ ^.Z5 y (e)-^-.0.90 J A 3 .Z \ 1 1 NATIONAL 1 1 ADVISORY \ c OMMITTE ; Fo 1 AED ON A JTIC 5 \ i 3, \ N • \ \ 1 \ \ \ \, / \ ^ \ N, "i/c \ ■n □G .75 > \ \ \ ^. \ N, \ N 15 E*.* ^oz '04 fieJof/ve he/ght^ -^t^ W^-0.95. (<^)^-0.60. F/gare // •— £xtropo/af/on fo zero \/ert/c?o/ he/pht oF cfoto obta/ned fro/rj an e/ectromognef/c-ana/ogy mode/ of an e///p'f/c vving ha\//ng an aspect raf/o of 3 • F/at- p/afe type of /oad/ng . NACA ARR No. L5D23 Fig. 12 V) w ^/c / 0,75 f r \* — Y I ; - r — -i! ~ ^, X if £ r ' ^ k .^5 A .Z X • Cat 'co/c ifed ' /?o/ nfs NATIONAL AD\ COMMITTEE FOR AE /ISORY RONAUT ICS o ( .i. ■? ^ I .i > .t / Ffgure /Z.-The none// mens zona/ downwash v^b -^-= for an e///pf/c w/ng hov/ng an aspect ^' max raf/o of 3 w/th f/at- p/ate type of /oad/ng . Compor/son of resu/fs obtained by method of reference Z w/fh data, extrapo/ated to z - O f obta/ned from tests of pre//m/nary e/ectromagnet/o- ana/ogy m octet . UNIVERSITY OF FLORIDA 3 1262 081 04 968 5 UNIVERSITY OF FLORIDA DOCUMENTS DEPARTMENT GAINESVILLE,FL 32611-7011 USA